Grade 5 Mathematics - KENILWORTHkenilworthst.org/Academics/Comprehensive_Curriculum/Mathematics... · with sample activities and classroom assessments to guide teaching and ... Grade

Grade 5 Mathematics

Grade 5

Mathematics

Table of Contents Unit 1: Whole Number Review: Addition and Subtraction..........................................1 Unit 2: Whole Number Review: Multiplication and Division.....................................13 Unit 3: Data, Probability, and the Counting Principle.................................................27 Unit 4: Number Theory and Equivalent Fractions.......................................................39 Unit 5: Properties in Geometry ......................................................................................52 Unit 6: Measurement .......................................................................................................65 Unit 7: Addition and Subtraction of Fractions .............................................................81 Unit 8: Measurement and Algebra.................................................................................89

Louisiana Comprehensive Curriculum, Revised 2008 Course Introduction

The Louisiana Department of Education issued the Comprehensive Curriculum in 2005. The curriculum has been revised based on teacher feedback, an external review by a team of content experts from outside the state, and input from course writers. As in the first edition, the Louisiana Comprehensive Curriculum, revised 2008 is aligned with state content standards, as defined by Grade-Level Expectations (GLEs), and organized into coherent, time-bound units with sample activities and classroom assessments to guide teaching and learning. The order of the units ensures that all GLEs to be tested are addressed prior to the administration of iLEAP assessments. District Implementation Guidelines Local districts are responsible for implementation and monitoring of the Louisiana Comprehensive Curriculum and have been delegated the responsibility to decide if

• units are to be taught in the order presented • substitutions of equivalent activities are allowed • GLES can be adequately addressed using fewer activities than presented • permitted changes are to be made at the district, school, or teacher level

Districts have been requested to inform teachers of decisions made. Implementation of Activities in the Classroom Incorporation of activities into lesson plans is critical to the successful implementation of the Louisiana Comprehensive Curriculum. Lesson plans should be designed to introduce students to one or more of the activities, to provide background information and follow-up, and to prepare students for success in mastering the Grade-Level Expectations associated with the activities. Lesson plans should address individual needs of students and should include processes for re-teaching concepts or skills for students who need additional instruction. Appropriate accommodations must be made for students with disabilities. New Features Content Area Literacy Strategies are an integral part of approximately one-third of the activities. Strategy names are italicized. The link (view literacy strategy descriptions) opens a document containing detailed descriptions and examples of the literacy strategies. This document can also be accessed directly at http://www.louisianaschools.net/lde/uploads/11056.doc. A Materials List is provided for each activity and Blackline Masters (BLMs) are provided to assist in the delivery of activities or to assess student learning. A separate Blackline Master document is provided for each course. The Access Guide to the Comprehensive Curriculum is an online database of suggested strategies, accommodations, assistive technology, and assessment options that may provide greater access to the curriculum activities. The Access Guide will be piloted during the 2008-2009 school year in Grades 4 and 8, with other grades to be added over time. Click on the Access Guide icon found on the first page of each unit or by going directly to the url http://mconn.doe.state.la.us/accessguide/default.aspx.

http://www.louisianaschools.net/lde/uploads/11056.doc

http://mconn.doe.state.la.us/accessguide/default.aspx

Louisiana Comprehensive Curriculum, Revised 2008

Grade 5 Mathematics Unit 1 Whole Number Review: Addition and Subtraction 1

Grade 5

Mathematics Unit 1: Whole Number Review: Addition and Subtraction

Time Frame: Approximately three weeks Unit Description Units 1 and 2 provide the closure to whole number work and provide the opportunity for students to begin the process of becoming computationally fluent in the whole number system by the end of the school year. Computational fluency is the level of skill reached when a person is able to execute an algorithm or procedure efficiently and correctly without assistance. Unit 1 focuses on addition and subtraction. Work with whole numbers should be integrated in each subsequent unit. Student Understandings Students solidify their total comprehension of whole numbers and the operations of addition and subtraction. They understand numbers, ways of representing numbers, relationships among numbers, patterns in numbers, compute fluently, and make reasonable estimates. Guiding Questions

1. Can students determine the steps and operations to use to solve a problem without assistance?

2. Can students use mental mathematics and estimation strategies in checking the reasonableness of computations?

3. Can students work proficiently with whole numbers, the operations of addition and subtraction, and their representations?

4. Can students solve simple equations and inequalities involving whole numbers? Unit 1 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 1. Differentiate between the terms factor and multiple, and prime and composite

(N-1-M) 7. Select, sequence, and use appropriate operations to solve multi-step word

problems with whole numbers (N-5-M) (N-4-M)



8. Use the whole number system (e.g., computational fluency, place value, etc.) to

solve problems in real-life and other content areas (N-5-M) 9. Use mental math and estimation strategies to predict the results of

computations (i.e., whole numbers, addition and subtraction of fractions) and to test the reasonableness of solutions (N-6-M) (N-2-M)

10. Determine when an estimate is sufficient and when an exact answer is needed in real-life problems using whole numbers (N-6-M) (N-5-M)

Algebra 12. Find unknown quantities in number sentences by using mental math, backward

reasoning, inverse operations (i.e., unwrapping), and manipulatives (e.g., tiles, balance scales) (A-2-M) (A-3-M)

13. Write a number sentence from a given physical model of an equation (e.g., balance scale) (A-2-M) (A-1-M)

14. Find solutions to one-step inequalities and identify positive solutions on a number line (A-2-M) (A-3-M)

Sample Activities

Activity 1: Tell Me about 12 (GLEs: 1, 8) Materials List: math learning logs, pencils Have students maintain a math learning log (view literacy strategy descriptions). This is a notebook that students can use to record ideas, questions, reactions, and new understandings. Have students write 12 things about the number 12. The ideas can involve operations, number theory, place value, real-life, etc. Some examples are the following: 12 is a 2-digit number, is greater than 10, is the sum of 6 6+ , is a doubles fact, has 1 ten, equals 1 ten + 2 ones, is a factor of 24, has 6 factors, is a multiple of 2, is a number on a clock, is a dozen, is the number of inches in a foot. This is a good activity to determine what students know about numbers and the vocabulary of mathematics. Repeat this activity throughout the year using numbers such as 0, 25, 100 (especially on the hundredth day of school), 1000, 1 million, ¾, 0.1, and 25%. Activity 2: Place Value of Whole Numbers (GLE: 8) Materials List: place value manipulatives, Place Value Chart BLM, pencils, Internet access (optional) Have students work in groups of 4. Give each group a set of place value manipulatives, such as beans and cups, digi-blocks, connecting cubes, base-ten blocks, or other manipulatives and a copy of the Place Value Chart BLM. For part of the activity, it is easier if students can take the materials apart. Give students a number, such as 134. Using the materials, have each student in the group model the number in a different way and record their models in the place value chart.



Hundreds Tens Ones

1 3 4 13 4 134 1 34

These are all different ways to model 134. The first way is the standard way to model the number. Students need to understand that there is 1 hundred in 134, 13 tens in 134, and 134 ones in 134. There are 3 tens in the tens place and the 3 tens have a value of 30, but there are 13 tens in the number. Have students model a few more numbers. Bring in the idea of expanded notation: 1 hundred + 3 tens + 4 ones = 1(100) 3(10) 4(1) + + = 100 30 4+ + . Expand these ideas to larger numbers. Draw a place value chart similar to this one on the board and give students a second copy of the Place Value Chart BLM.

Millions Thousands Ones

hund

red

mill

ions

ten

mill

ions

mill

ions

hund

red

thou

sand

s

ten

thou

sand

s

thou

sand

s

hund

reds

tens

ones

Write the number 3,248 in the correct columns in the chart. Use a strip of paper to cover the digits 2, 4, and 8, and the words above the digits. Have students read what is uncovered, 3 thousands. Uncover the digit, 2. Have students read this as 32 hundreds. Uncover the digit, 4. Have students read this as 324 tens. Finally, uncover the digit, 8. This is read as 3248 ones. Continue with other numbers. Use numbers from real-life applications, such as the moon is 233,812 miles from Earth. The Guinness Book of World Records is a great place to find these applications. The website www.nctm.org has a good lesson on understanding a million. Access the page directly at http://illuminations.nctm.org/index_d.aspx?id=367. The teacher can also go to the www.nctm.org website. From the NCTM.org site, click on Lessons, then click on Illuminations. This will open a new page. Click Lessons and Resources, check Grades 3–5, and NUM. In the search box, type Count on Math. When Count on Math comes up, click on it and then select Making Your First Million. It involves real-life applications and estimation.

http://www.nctm.org

http://illuminations.nctm.org/index_d.aspx?id=367

http://www.nctm.org



Activity 3: Comparing Whole Numbers (GLE: 8) Materials List: place value manipulatives, a deck of cards for each group Have students work in groups of 4. Give each group two numbers to model using the place value manipulatives, and ask them to compare the numbers. Use measurement units to reinforce measurement and to put the problems in context, such as “Which is the greater distance—456 miles or 495 miles?” Have students discuss how they decided which number was greater. Make sure to give some pairs of numbers that do not have the same number of digits. After modeling 2-and 3-digit numbers, have students apply their ideas of place value and comparing to larger numbers. For practice on comparing larger numbers, use a deck of cards with the tens and face cards removed. Have each student in a group pick a card. The four students in each group should make a 4-digit number using the cards selected. Ask two groups to stand and compare the numbers that were made. Make sure to sometimes ask for the smaller number. To make larger numbers, have each student choose 2 cards so that the group can make an 8-digit number. Activity 4: What’s Your Number Name? (GLE: 8, 9) Materials List: name tags or index cards, pencils, math learning log Through SPAWN writing (view literacy strategy descriptions) prompts, such as “What if?” the teacher can create a thought-provoking activity related to the use of numbers. Ask students to write in their math learning logs (view literacy strategy descriptions) this prompt: “What if your name was made up of numbers, not letters? What might be your number name, and why would you choose that number? Do you see any problems with using numbers rather than letters?” After inviting students to share what they wrote and discussing some of the ideas from their writing, have students meet in groups of 4. Give students name tags and have them write a number name for themselves. The name should be a number with 7 digits. Tell students that they can use any of the digits from 0 to 9, but they can use each digit no more than two times. They must use at least one zero. Have students “meet” the other students in their group by correctly saying the number names. Students should then order the number names in their group from smallest to largest. Have students compare their group’s largest number name to the next group’s largest number name, and so on, to find the largest number in the room. Ask questions involving place value and expanded notation about this number name. Begin to introduce rounding, by asking whose number is closest to 5 million. Activity 5: Change the Digit (GLE: 8) Materials List: calculators (can be 4-function or scientific), overhead calculator (if available) In 1983, it was estimated that the United States had 37,133,296 trucks and buses on the road. Have students enter this number into their calculators. Give instructions such as these: You are



allowed to change only one digit at a time. Change the digit in the hundreds place to a zero. What did you do? (subtracted 200) Change the 7 to a 2. What did you do? (subtracted 5,000,000). Ask one student to read the number each time it is changed. Continue until all the digits have been changed. A way to change this activity is to start with zero and add numbers to each place, giving instructions such as: Start with 0. Place a 5 in the hundreds place. What did you do? (added 500). An overhead calculator would enhance this activity. Activity 6: Mental Math: Compensation, Compatible Numbers, and Breaking Apart Numbers (GLE: 7, 8, 9) Materials List: math learning logs, pencils Give students a problem such as 49 34+ to compute mentally. Tell them to give an exact answer, not an estimate. Some students may use compensation. They may think: I can add 50 34+ to get 84, but I added one too many, so 84 1 83− = . Or they may think: I can add 50 33+ to get 83. I took 1 from the 34 and gave it to the 49. Some students may break apart the numbers to add 40 30 70 + = and9 4 13+ = , so 70 13 83+ = . Or they may think 49 30 79+ = , so 79 4 83+ = . There are probably countless other way that students can find the answers; just make sure they explain their reasoning. Students need to see and hear how others use mental math strategies. Continue to give other problems involving subtraction, not just addition. Subtraction is often harder to do mentally than addition. Counting up can help students to subtract mentally. For example, for the problem 83 – 65, some students might think: I add 5 to 65 to get to 70. From 70 to 83 is 13, so 5 + 13 = 18. Sometimes, a combination of strategies works best. Write the following player scores on the board. Ask students what math strategies they would use to determine the total scores of the following basketball players:

• Player A scored 16 +11+14 = • Player B scored 12+17 +13 = • Player C scored 7 +15+13 =

Some students may use compatible numbers and properties to find the scores of each player (16 + 14 + 11 = 30 + 11 = 41; 17 + 13 + 12 = 30 + 12 = 42; 7 + 13 + 15 = 20 + 15 = 35), then use compensation to find the total: 40 + 40 + 35 + 3 = 115 + 3 = 118. Ask students to give examples of when they might use a certain strategy. A critical part of becoming proficient in mental math strategies is to be able to explain your reasoning. Give students the problem, 84 – 35. In their math learning logs (view literacy strategy descriptions), have them explain in words, numbers, or pictures how they would use mental math to find the answer. Activity 7: Actual Answers and Estimates (GLEs: 8, 10) Materials List: newspapers or Internet access, paper, pencils Discuss with students what determines whether an exact answer or an estimate is appropriate for a given situation. Use the following as examples that require either an estimated or exact answer:



• An estimate is all that is needed when a friend asks you for the temperature or you want to know about how long a bus trip takes. When you talk about estimates, you often use the words about, close to, or approximately.

• An exact number is needed when you want to determine the number of meteorologists that work for a TV station, or you want to find out how many scheduled stops a bus will make.

Have each student go on the Internet or look in a newspaper for numbers in news stories. There are many websites for newspapers such as www.usatoday.com, www.nytimes, and www.nola.com. Consider assigning each student a different section of a newspaper, such as the front page, classified, or sports. It might be interesting to see which section has more estimated numbers. Ask students to find three numbers that are exact and three numbers that are estimates and copy the full sentences about the numbers. Have students discuss times that they need to find an exact answer that involves addition and subtraction and times that an estimate will do. For example, I may want to know approximately how much money I spent at the store. I could estimate the cost of each item before I added them. Activity 8: Types of Estimation (GLE: 9) Materials List: Types of Estimation BLM, pencils Before beginning the estimation activities, have students complete a vocabulary self awareness chart (view literacy strategy descriptions). Provide students with the Types of Estimation BLM. Do not give students definitions or examples at this point.

Word + √ – Example Definition rounding front-end estimation

compatible numbers

clustering Ask students to rate their understanding of each word with either a “+” (understands well), a “√” (some understanding), or a “–” (don’t know). During, and after completing the estimation activities such as Activities 9, 10, and 11, students should return to the chart and fill in examples and definitions in their own words. The goal is to have all plus signs at the end of the activities. Activity 9: Rounding Numbers (GLEs: 8, 9, 10) Materials List: paper, pencils, Internet access Although there are other estimation strategies, such as front-end estimation, compatible numbers, and clustering, rounding is an important strategy. Begin rounding by discussing why multiples of 10 are used in rounding. Help students to realize that rounding makes it possible to use numbers

http://www.usatoday.com

http://www.nytimes

http://www.nola.com



that are easy to compute. Emphasize that when you say round to the nearest 10 (or 100, or 1000) you are asking students to find the closest 10 (or 100, or 1000) To do this, help them to determine what two tens, (or two hundreds, or two thousands, etc.) a number is between. Drawing a number line can help. Give students a number such as 289. (If your students are proficient in rounding with smaller numbers, use larger numbers, but ask the same types of questions.) Ask questions such as these: If you are rounding to the nearest hundred, between which two hundreds is 289? (200 and 300) Which hundred is it closer to? (300) If you are rounding to the nearest ten, between which two tens is 289? (28 tens and 29 tens) To which ten is it closer? (29 tens or 290) Discuss the fact that numbers such as 5, 50, 500 are exactly in the middle, so if given a number such as 250, the number should be rounded up to 300. Be sure to include some of the rounding problems that confuse students such as the following problems: Round 25 to the nearest hundred (0), round 98 to the nearest 10 (100), and round 245 to the nearest hundred (200). Extend the rounding ideas to larger numbers. Instruct students to explain how they arrived at their answers. Give real-world examples such as these: 200,000 people live in Shreveport, do you think this is an estimate or an exact figure? If I had rounded the population to the nearest hundred-thousand, which of the following could have been the actual population? 262,461; 198,364; 209,999; or 252,125. (198,364; 209,999) Explain your reasoning. Teachers can find the populations of different cities in the United States at www.census.gov/popest/cities. Activity 10: Compatible Numbers (GLEs: 8, 9) Materials List: Compatible Numbers BLM, pencils Compatible numbers are numbers that are easy to work with, or “nice numbers.” For example, if asked to estimate 26 + 23, the students could think 25 + 25 or 30 + 20. Give students a number. Ask them to give a number that would be compatible to that given number. For example, in addition, what is a number that is compatible to 25? (Possible answers: 0, 5, 10, 25, 100, etc.) All of the numbers are easy to add to 25. In multiplication, what is a number that is compatible to 25? (Possible answers: 0, 4, 100, etc.) All of these numbers can easily be multiplied by 25. Very often, compatible numbers involve sums or products of 10, 100, 1000, etc. Give addition and subtraction word problems such as the ones on the Compatible Numbers BLM and ask students to solve the problems using compatible numbers.

http://www.census.gov/popest/cities



Activity 11: Estimation Strategies (GLEs: 7, 8, 9, 10) Materials List: pencils, math learning logs Four estimation strategies used at this grade level include rounding, front-end estimation, using compatible numbers, and clustering. Give students this problem: Tom’s family drove 129 miles on Monday, 351 miles on Tuesday, and 275 miles on Wednesday. Approximately how many miles did they drive on the 3 days? Ask, “If I say ‘approximately,’ am I looking for an exact answer or an estimate?” Write 129 351 275+ + on the board. Ask students to estimate the sum and be ready to explain their reasoning. Some may round to the nearest 100; others to the nearest 10. Their answers would be 800 and 760, respectively. Some may use compatible numbers, thinking 125 + 275 is 400 and 400 350 750+ = . Others may use front-end estimation. They would get 100 + 200 + 300 = 600. Front-end estimation must be adjusted. They may think 29 75+ is about 100, plus 50 more, so the estimate is actually closer to 750. Give an example of a problem that could involve clustering, such as these: There are 4 grades at Washington Elementary. There are 202 students in kindergarten, 198 in 1st grade, 217 in 2nd grade, and 189 in 3rd grade. About how many students are in the 4 grades? Ask, if I say “about,” am I looking for an exact answer or an estimate? Write 202 198 217 189+ + + on the board. The numbers cluster around 200, so 200 4 800× = is a good estimate. The teacher may want to give a problem each day involving estimation. In their math learning log, (view literacy strategy descriptions) have students always compare the results from the different types of estimation they used and justify their estimation strategy. Activity 12: Addition and Subtraction (GLEs: 7, 8, 9) Materials List: paper, pencils Even though the operations of addition and subtraction should have been mastered, you may need to review these concepts. Give problems in context and have students estimate before they actually add or subtract the numbers. Writing the answer in a complete sentence can help students determine if their answers are reasonable. For addition, focus on partial sums, emphasizing the place value of the digits. Give an example such as this: The school play ran for two nights. The first night, 368 people attended; the second night, 459 people attended. How many attended in the two nights? Ask students to estimate the answer. Accept estimates between 700 and 900. The following problems are examples of using partial sums.

368 300 60 8

459 400 50 9 700 110 17

= + ++ = + +

+ +

or



368 +459 17 8 9 17+ = 110 60 50 110+ = 700 300 400 700+ = 827 For subtraction, use a number line and the strategy of counting up. The school is trying to sell 459 tickets to the play. They have sold 368. How many more must they sell to reach their goal? 459 368− = _____ or 368 + _____ = 459 +2 +30 +59 2 30 59 91+ + = 368 370 400 459 or + 100 100 9 91− = 368 459 468

– 9

Make sure that you give subtraction problems involving zeros. For example: 508 –149 Remind students that there are 50 tens in this number, so 508 could be renamed as 49 tens + 18. 49 18 508 –149 This is also a good problem to use the count up strategy. To reinforce the idea that addition and subtraction are inverse operations, show students how to check a subtraction problem by using addition and why it works. Continue giving students addition and subtraction problems, in context, always asking for an estimate first. To reinforce estimation, have students discuss which strategy they used. Have students write the word problems that you will use. Sometimes, give an answer to their written problems and ask students if the answer you stated is reasonable or not. Encourage students to write problems that have more than one step, involving both addition and subtraction. Encourage students to try to do problems mentally before they use paper and pencil. Estimation, mental math, and computation should be fully integrated at this grade level.



Activity 13: Addition and Subtraction Story Chains (GLEs: 7, 8, 9) Materials List: paper, pencils Use math story chains (view literacy strategy descriptions) to practice addition and subtraction. Math story chains involve a small group of students writing a story problem using the math concepts being learned and then solving the problem. The first student writes the opening sentence of the problem. There are 436 students at Washington Elementary. The student passes the paper to the student sitting to the right. That student writes the next sentence in the story. There are 521 students at Lincoln Elementary. The paper is passed again to the right. This student writes the question for the story. How many more students are at Lincoln than at Washington? The paper is now passed to the fourth student who must solve the problem and write the answer in a complete sentence. Answer: There are 85 more students at Lincoln Elementary. Tell students that their problems can involve more than one operation and can involve estimation. Activity 14: Balances/Scales (GLEs: 12, 13, 14) Materials List: number balances or scales, objects to count, paper, pencils Introduce students to the concepts involved in solving equations and inequalities. Emphasize that adding or subtracting the same amount to both sides of an equation or inequality does not change the relationship. Using a balance or a scale can help them understand this concept. Have the students use the balances or scales and similar objects (marbles, tiles, plastic counters) to create an equation or an inequality. Have them write number sentences to show the reading of the balances or scales: 8 6> , 15 15= , 4 + 1 = 2 + 3, etc. Using the balance, show students that adding or subtracting the same amount to both sides of an equation or inequality does not change the relationship. For 15 15= , if you add 5 to both sides, you still get an equation. It becomes 20 20= . For8 6> , if you subtract 4 from both sides, the left side of the inequality will still be greater than the right side. The inequality will now read 4 2> . Also, give problems such as 4 + 3 = 9 – □. Apply these ideas to larger numbers that cannot be done on the scales. If 595 = 595, does the equation change if I subtract 110 from both sides? (No, 485 = 485) If 1043 < 1141, does the inequality change if I add 56 to both sides? (No, 1099 < 1196)



Activity 15: Bean Math (GLEs: 12, 13, 14) Materials List: beans, small cups, Equation Mats BLM, Inequality Mats BLM, pencils, paper Have students work in pairs to create equations using beans, small cups and the Equation Mats BLM. The cup should represent the variable or unknown value. To model a number sentence such as, 7 15x + = , ask students to place an empty cup and 7 beans next to it on one side of the equal sign and 15 beans on the other side. To find the solution to the number sentence, ask students to use metal math, thinking about how many beans should be put in the cup to have this side of the number sentence equal 15. Or have them remove 7 beans from both sides of the equation to find that 8x = . Emphasize that 7 beans were added, so to get x alone, you need to subtract 7 from both sides. The equation x + 7 = 15 can be written as x + 7 – 7 = 15 – 7. Therefore, x = 8. Subtraction is harder to model using beans and cups. For x – 3 = 7, place a cup on one side of the mat and 7 beans on the other. Ask, how many beans have to be in the cup so that you can subtract 3 and still have 7. There must be 10 beans in the cup. Since 3 is being subtracted from x, to get x alone, add 3 to both sides of the equation. The equation x – 3 = 7 can be written as x – 3 + 3 = 7 + 3. Therefore, x = 10. Remind students that if you do something to one side of the equation, you must do it to the other side. Have one student model a number sentence and another student solve it, explaining what they are doing as they go along. Students should write the equations that are being modeled throughout the lesson. Also introduce the idea of solving inequalities by modeling 3 5x + > . Give students a copy of the Inequality Mats BLM. Ask students what number of beans could be put in the cup to make the sentence true. (3, 4, 5 …) Students can also use a number line to solve the problems, but at this time I would use just whole numbers in the solution.

Assessments

General Assessments

• Portfolio assessment could include the following: o Anecdotal notes made during teacher observation. o Any of the journal entries, or one of the explanations from the specific activities o Corrections to any of the missed items on tests

• On any teacher-made written tests, the teacher could include at least one of the following. o One problem that requires the use of manipulatives or drawings such as this:

Using some type of base-ten manipulative or drawing, the students will show why 184 < 203.

o One question that requires the student to explain his/her reasoning such as this: How many hundreds are in the number 1541?

o One problem involving real-life such as this: Since numbers and mathematics are used all the time during the day, the students will list two times that an exact answer is needed to answer a question, and two times that an estimate is all that is needed.



• Journal entries could include the following: o The students will explain how they would estimate the answer to the following

problem: 409 298 − = _____ o Mr. Mistake worked the following problem, 176 + 489, and got an answer of 565.

The student will explain why his answer is not reasonable, and what mistake he made.

o The students will explain in writing how to mentally find the sum of 57 and 34. Activity-Specific Assessments

• Activity 1: For the number 12, the students will write an addition, a subtraction, a multiplication, and a division expression that equals 12 and write two different expressions equal to 12 that involve two operations. ( possible answers: 4 + 8; 13 – 1; 3 × 4; 24 ÷ 2; 8 + 5 – 1; 4 × 4 – 4)

• Activity 3: Write the digits 4, 3, 1, 9 on the board. Have students use these 4 digits to

write a number greater than 4,319 and another number that is less than 4,319. (possible answers: 4,913 and 4, 139)

• Activity 6: Give students the Are They Equal BLM. Students must explain their

reasoning on each problem.

• Activity 9: 2, 3, 5, 9 The students will use each digit in the rectangle to write a number that …

o rounded to the nearest thousand is 3000. (possible answer 3,295) o rounded to the nearest hundred is 3900. (3,925) o rounded to the nearest ten is 3590. (3,592)

• Activity 13: Give students the numbers 225, 500, and 62. Have them write word

problems that involve … o only addition. o only subtraction. o both addition and subtraction.


Grade 5 Mathematics Unit 2 Whole Number Review: Multiplication and Division 13

Grade 5 Mathematics

Unit 2: Whole Number Review: Multiplication and Division Time Frame: Approximately four weeks Unit Description Units 1 and 2 provide the closure to whole number work and provide the opportunity for students to begin the process of becoming computationally fluent in the whole number system by the end of the school year. Computational fluency is the level of skill reached when a person is able to execute an algorithm or procedure efficiently and correctly without assistance. Unit 2 focuses on multiplication and division. Work with whole numbers should be integrated in each subsequent unit. Student Understandings Students solidify their total comprehension of whole numbers and the operations of multiplication and division. They understand numbers, ways of representing numbers, relationships among numbers, patterns in numbers, can compute fluently, and can make reasonable estimates. Guiding Questions

5. Can students determine the steps and operations to use to solve a problem without assistance?

6. Can students use mental mathematics and estimation strategies in checking the reasonableness of computations?

7. Can students work proficiently with whole numbers, the operations of multiplication and division, and their representations?

8. Can students solve simple equations and inequalities involving whole numbers?

9. Can students identify a simple rule for a sequence pattern problem and find missing elements?



Unit 2 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 1. Differentiate between the terms factor and multiple, and prime and composite

(N-1-M) 7. Select, sequence, and use appropriate operations to solve multi-step word

problems with whole numbers (N-5-M) (N-4-M) 8. Use the whole number system (e.g., computational fluency, place value, etc.) to

solve problems in real-life and other content areas (N-5-M) 9. Use mental math and estimation strategies to predict the results of

computations (i.e., whole numbers, addition and subtraction of fractions) and to test the reasonableness of solutions (N-6-M) (N-2-M)

10. Determine when an estimate is sufficient and when an exact answer is needed in real-life problems using whole numbers (N-6-M) (N-5-M)

Algebra 12. Find unknown quantities in number sentences by using mental math, backward


13. Write a number sentence from a given physical model of an equation (e.g., balance scale) (A-2-M) (A-1-M)

14. Find solutions to one-step inequalities and identify positive solutions on a number line (A-2-M) (A-3-M)

Measurement 23. Convert between units of measurement for length, weight, and time, in U.S.

and metric, within the same system (M-5-M) Patterns, Relations, and Functions 33. Fill in missing elements in sequences of designs, number patterns, positioned

figures, and quantities of objects (P-1-M)

Sample Activities Activity 1: Patterns in Multiples (GLEs: 1, 8, 33) Materials List: Hundred Chart BLM, crayons, pencils, math learning logs Distribute two copies of the Hundred Chart BLM to each student. Have students work in groups of 4. Assign students the following numbers: Student 1—numbers 2 and 8; Student 2—numbers 3 and 7; Student 3—numbers 4 and 6; Student 4—numbers 5 and 9. Students should shade the multiples of their assigned numbers on different hundred charts. They should also list the multiples of each number along the side of the chart in a column. Have students search for patterns in the charts and the lists. As a whole class, have students discuss the patterns they discovered. The focus of this activity is practice of multiplication facts and practice in searching for patterns, not on number theory. Have students choose any one number from 2 to 9. In their math learning logs (view literacy



strategy descriptions), have students write about the patterns they see in the multiple charts and lists for their number. To demonstrate that students have learned the content, use the professor-know-it-all strategy (view literacy strategy descriptions). Form groups of 3-5 students who have all chosen the same number. Tell them that one group will be called on randomly to come to the front of the class to be a team of “professor-know-it-alls” about their number. Invite questions from other groups. The team should be able to answer questions about patterns that they saw on the hundred chart and on the list of multiples. Other students should listen for accuracy and logic in the know-it-alls answers to their questions. After about 5 minutes or so, ask a new group to take its place in front of the class. Activity 2: Multiplication Patterns (GLEs: 8, 9) Materials List: calculators for each student, paper, pencils In this activity, students will look for patterns to help them multiply mentally. Have students work in groups, but provide each student with a calculator. Have students use the calculator to find 4 1; 4 10; 4 100; 4 1000; 40 10; 40 100× × × × × × . Have them discuss the patterns that they see. Repeat the activity using these problems: 4 6; 4 60; × × 4 600; 4 6000; 40 60; 40 600× × × × . Again, have students discuss any patterns they see. Ask questions such as these: How can you tell how many zeros the product will have? What is an easy way to multiply 40 60× ? If I multiply 4 600× , how many hundreds do I get? (24 hundreds) Is there another name for this answer? (2 thousand 4 hundred) Now have students find the answers to the following problems: 20 5; 200 5; 2000 5;÷ ÷ ÷ 200 50; 2000 50÷ ÷ . Ask students to describe the patterns they see. Ask questions similar to the ones asked for multiplication. You may want to do this part of the activity when you get to division. Activity 3: Multiplication Properties (GLE: 8, 9) Materials List: Multiplication Properties BLM, pencils Before beginning the multiplication activities, have students complete a vocabulary self awareness chart (view literacy strategy descriptions). Provide students with the Multiplication Properties BLM. Do not give students definitions or examples at this point.

Word + √ – Example Definition Commutative Property

Associative Property



Distributive Property

Identity Property

Zero Property Ask students to rate their understanding of each word with either a “+” (understands well), a “√” (some understanding), or a “–” (don’t know). During, and after completing any multiplication activities, such as activities 4, 6, 10, 11, and 13, students should return to the chart and fill in examples and definitions in their own words. The goal is to have all plus signs at the end of the activities. Activity 4: Properties and Mental Math (GLEs: 8, 9) Materials List: Grid Paper BLM, pencils, math learning logs The multiplication properties can help students use mental math to multiply. Give students problems such as 4 × 18 × 25. Help them see that using the Commutative Property to change the order of 18 and 25 makes an easier problem. 4 × 25 = 100 and 100 × 18 = 1800 Give students problems that the Associative Property would make easier, such as 34 × 4 × 5. Grouping the 4 × 5, and multiplying it first to get 20, leaves the easier problem of 34 × 20, or 680. Give examples of the identity and zero properties of multiplication, such as 310 × 1 = 310 and 562 × 0 = 0. The Distributive Property is extremely useful when multiplying mentally. Distribute the Grid Paper BLM. Have students draw a rectangle 6 units long and 13 units wide. Since 13 is the same as 10 + 3, have them draw a line to show 2 rectangles, one that is 6 × 10 and one that is 6 × 3.



This is an illustration of the Distributive Property. 10 + 3 13 6 6 × 13 = 6 × (10 + 3) = (6 × 10) + (6 × 3) = 60 + 18 = 78 Continue with other examples using the grid paper and then have students use the distributive property to multiply mentally. In their math learning logs, (view literacy strategy descriptions) have students explain how they could use the distributive property to solve the problem 8 × 49. Activity 5: Is an Estimate Okay? (GLEs: 8, 10) Materials List: paper, pencils, Hit the Target BLM, 1 calculator per group Discuss with students what determines whether an exact answer or an estimate is appropriate for a given situation. Use the following as examples that require either an estimated or exact answer:

• You are at the store buying 6 DVD’s at $22.98 each. You give the clerk $150.00 and want to make sure your change is correct. Do you need an exact amount or an estimate? (exact)

• You are at the store buying 3 DVD’s at $18.98 each and you want to know if a $50 bill will be enough. Do you need an exact amount or an estimate? (estimate)

Have students give examples of multiplication problems that would require an exact answer or an estimate. Sometimes when you estimate, you need to know if your estimate is an underestimate (less than the exact answer) or an overestimate (more than the exact answer.) Give examples, such as the following, and ask students to tell if you need an underestimate or an overestimate, or if it doesn’t matter.

• The weight of your suitcase when flying (underestimate) • The amount of fuel the plane will need to get you to your destination

(overestimate) • The average age of the passengers (doesn’t matter)



Estimation strategies used at this grade level include rounding and compatible numbers. Remind students that compatible numbers are numbers that are easy to compute with mentally, such as 25 × 4. Give an example of a real-world problem, such as this: A school bought 260 books for $39 each. About how much did they spend on the books? If you round both numbers down, you might use 200 × $30, and get $6,000, an underestimate. If you round both numbers up, you might use 300 × $40, and get $12,000, an overestimate. You could use compatible numbers, such as 250 × $40, and get $10,000. Have students create other problems, estimate the answers, and compare the results for the different types of estimation they used. Distribute the Hit the Target BLM to students in groups of 4. Each student should choose 3 numbers from Circle A and create a 3-digit number. Next they choose 1 number from Circle B. The object is to get the product closest to the target number. The student with the closest product scores a point. The student with the most points wins the game. If 2 students are the same amount from the target score, they both score a point. Have a calculator available to check the multiplication. Activity 6: Multiplication (GLEs: 7, 8, 9) Materials List: place value materials, Grid Paper BLM, pencils, paper For more review of the meanings and properties of multiplication, go to www.nctm.org, click on Lessons and Resources, Illuminations, Lessons, Grades 3–5, Num, then type in multiplication facts. The lesson, All About Multiplication, focuses on the meanings of multiplication and the lesson, It’s in The Cards, focuses on the properties of multiplication. For multiplying by one- and two-digit numbers, put students in groups of 4 and provide a set of place value blocks and two copies of the Grid Paper BLM. Give the following problem: There are five 5th grade classes in the school. Each class has 23 students. How many students are in the 5th grade? Write the problem 5 23× on the board and ask students to find the product using any of the materials they want. Ask them to use words or drawings to record how they found the product. Some may show 5 groups of 23 blocks, some may draw a 5 23× rectangle, and some may use the distributive property. Discussion is critical. Give other problems to multiply, discussing the strategies students used. Expand this to include multiplying by two-digit numbers. In order to use the place value blocks, keep the numbers in the teens or twenties. Have students connect the partial products to the blocks or drawings on grid paper. You can also connect the partial products to the standard algorithm. For example:

http://www.nctm.org



26 26 ×14 ×14 24 104 80 260 60 364 200 364 Emphasize that the traditional algorithm is a shortcut for the expanded form. Continue giving students multiplication problems, in context. Have students write the word problems that you will use. At this time, encourage students to write problems that have more than one step, involving addition, subtraction, or multiplication. Activity 7: Meanings of Division (GLE: 8) Materials List: counters, pencils, paper Have students work in pairs or small groups. Each pair will need about 40 counters. This activity focuses on the inverse relationship of multiplication and division, and on the two meanings of division, sharing and repeated subtraction. When you think of multiplication as equal groups of an amount, you can write the following equation:

number of groups × number in each group = whole If you know the whole amount and the number of groups, you use sharing division to find the number in each group.

whole ÷ number of groups = number in each group

If you know the number in each group and the whole amount, you use repeated subtraction division to find the number of groups.

whole ÷ number in each group = number of groups

Understanding both meanings using simple problems can help students later in more difficult division problems and with division of fractions. Give students many examples to model the division such as the following:

There are 5 groups of 4 students in the class. How many students are in the class? (20 students) There are 20 students in the class. If they are separated into 10 groups, how many are in each group? (2) What type of division is this? (sharing) There are 20 students in the class. If I send 4 students to each learning station, how many learning stations do I need? (5) What type of division is this? (repeated subtraction)

Continue with additional problems.

4 × 6 4 × 20 10 × 6 10 × 20



Activity 8: Estimation in Division (GLEs: 8, 9, 23) Materials List: paper, pencils, math learning logs The types of estimation used for division at this grade level are rounding and compatible numbers. Give students a problem and have them estimate using both of these methods. Use measurement units to practice measurement concepts and to put problems in context. Example 1: If there are 8 ounces in one cup, about how many cups are in 62 ounces? Rounding: 60 ÷ 8 = ?

This problem is still not very easy. You know it is more than 7 cups, but less than 8 cups. Since you used 60 rather than 62, you know that 7 is an underestimate.

Compatible numbers: 64 ÷ 8 = 8 This problem was easy to divide. Eight cups is an overestimate because you used 64 instead of 62.

Example 2: If there are 12 inches in one foot, about how many feet are in 34 inches. Rounding: 30 ÷ 10 = 3 feet Compatible numbers: 36 ÷ 12 = 3 feet Both estimation methods give 3 feet. Continue giving division problems using other measurements units. In their math learning logs, (view literacy strategy descriptions) have students explain how they could estimate the number of feet in 43 inches. Activity 9: Division (GLEs: 7, 8, 9) Materials List: base-ten blocks, Grid Paper BLM, pencils, paper Have students work in groups and provide each group with base-ten blocks and grid paper. Tell students that you ran off 500 flyers to advertise the school fair. If each person in the class takes an equal amount, how many flyers will each person get and how many will be left over? Allow students to use any of the materials they want to model the problem. Observe how the students approach the problem. Did the students use estimation techniques such as rounding or compatible numbers? Did the students use repeated subtraction? Did they make a rectangular array with the blocks? Did they draw a rectangle on the grid paper? Suppose there are 23 students in the class. Ask questions such as these: Can you give each person a set of 100 flyers? (No) Could you give each person a set of 10 flyers? (Yes) How many sets of 10 could you give each person? (2 sets) How many sets of flyers did



you give out altogether? (46 sets) How many are left? (40) Can you give each person some individual flyers? (Yes) How many will you give each person? (1) How many did you use? (23) How many are left over? (17) You may want to allow students to write the division problem the second way. 1 21 20 50023 50023 21 R 17 46 460 40 40 23 23 17 17 Continue giving students other division problems. Have students write the word problems that you will use. Encourage students to write some problems that involve multiple steps using any of the four operations. Activity 10: Multiplication and Division Story Chains (GLEs: 7, 8) Materials List: paper, pencils Use math story chains (view literacy strategy descriptions) to practice any of the four operations with an emphasis on multiplication and division. The first student writes the opening sentence of the problem. I have $512 to spend for the class. The student passes the paper to the student sitting to the right. That student writes the next sentence. I want to buy 18 video games at $25 each. The paper is passed again to the right. This student writes the question for the story. Do I have enough money to buy all 18? The paper is now passed to the fourth student who must solve the problem and write the answer in a complete sentence. Answer: Yes, I have enough money because 20 × 25 is 500, and I have more than

$500. Students in the story chain groups should talk about the accuracy of the answer and the logic of the story problem. If necessary, revisions to the story chain should be made.



Activity 11: What’s the Operation? (GLEs: 7, 8) Materials List: paper, pencils, Counting on Frank – optional Have students work in small groups. Give students different whole number examples using prices of children’s tickets and adult tickets or prices of matinee and evening tickets. For example, children’s tickets for the show are $3 and adult tickets are $7. If 20 children’s tickets and 12 adult tickets were sold, how much money was collected? Post the different ticket prices on the board or a chart, and have each group write a multi-step word problem that uses at least two operations for each example. Ask student groups to share their problems with other groups. Have groups compare the way they solved the problems. The book, Counting on Frank, provides wonderful practice for multiplication, division, estimation, and measurement. Activity 12: Which Method Would You Use? (GLEs: 7, 8, 9) Materials List: Which Method? BLM, paper, pencils, math learning logs Give students the Which Method? BLM. Using the numbers in the table below, have students write two problems for each of the following methods: mental math, calculators, paper and pencil. For the first problems for each method, use two numbers. For the second problem, use three numbers. Any operation can be used, but students must use every operation at least once. For example, a student might use mental math to solve the problem 4,381 × 100.

4,381 38 2,000100 99 8,296200 635 62 19 1 4

In their math learning logs, (view literacy strategy descriptions) have students explain their reasoning for one of the mental math problems. Activity 13: Rolling the Number Cube (GLE: 8, 9)

Materials List: two number cubes per group, paper, pencils, 1 calculator per group Have students work in small groups with two number cubes and 1 calculator given to each group. Students in each group should take turns acting as the “checker”, using the calculator. For the first activity, ask students to take turns rolling the number cube five times, while group members keep a running total in their heads. Students try to be the first one in the group to say the correct total at the end of each person’s five rolls.



For the second activity, ask a student to roll two number cubes and write a 2-digit number of his or her choice. Then he or she should roll the number cubes a second time, write a second 2-digit number and all students should mentally add the two numbers. Or, as an adaptation, students could mentally subtract the smaller number from the larger number. For the third activity, have a student roll two number cubes, write a 2-digit number, and then roll a single number cube. All students should mentally multiply the 2-digit number by the single digit number. For the fourth activity, have a student roll two number cubes, write a 2-digit number, and then roll a single number cube. All students should mentally estimate the quotient of the two numbers. In each activity, the “checker” with the calculator should confirm the answers. Activity 14: Does It Balance? (GLEs: 12, 13, 14) Materials List: number balances, objects to count, paper, pencils Introduce students to the concepts involved in solving equations and inequalities. Have the students use the balances or scales and similar objects (marbles, tiles, plastic counters) to create an equation or inequality. Have them write number sentences to show the reading of the balances of scales: 8 > 6, 5 = 5, etc. Using the balance, show students that multiplying or dividing both sides of the equation or inequality by the same amount (as long as you are multiplying or dividing by a positive number) does not change the relationship. For 5 = 5, if you multiply both sides by 2 (or double the amount), you get 10 = 10. For 8 > 6, if you divide both sides by 2 (or take ½ of each side), the left side of the inequality will still be greater than the right side. The inequality now reads 4 > 3. Apply the ideas to larger numbers that cannot be done on the scales. If 482 = 482, does the equality change if I divide both sides by 2? (No, 241 = 241) If 45 < 54, does the inequality change if I multiply by 3? (No, 135 < 162) Activity 15: Multiplication Equations (GLEs: 12, 13) Materials List: beans, small cups, Equation Mats BLM Have students work in pairs to create equations using beans, small cups, and the Equation Mats BLM. The cups should represent the variable, or unknown. To model an equation such as 2x = 8, ask students to place two empty cups on one side of the mat and 8 beans on the other side. When solving a multiplication equation, students need to find the value of one x, or one cup. They could put 4 beans in each cup, so 1x or x = 4. Do many



examples of multiplication equations. Begin an introduction of 2-step equations by using an equation such as 3x + 1 = 10. Students would subtract 1 bean first, and then divide the 9 remaining beans into groups of 3. The solution to the equation is x = 3. Activity 16: Input/Output Tables (GLEs: 8, 33) Materials List: Input/Output Table BLM, pencils Give students a copy of the Input/Output Table BLM. Give students a rule, such as “Multiply by 10.” Call out a number to be put in the input column, and ask students to find the output number using the rule. For example, if the input number is 5, the output number is 50. You can review the facts for all operations using input/output tables. Next, give students 4 or 5 input and output numbers, and have them guess the rule. An example follows:

Input Output150 30 100 20 50 10 10 2

What is my rule? (Divide by 5.) Continue with examples involving all four operations and possibly two operations together. My rule could be double the input number and add 3. Activity 17: What is the Pattern? (GLE: 33) Materials List: paper, pencils, T-Tables BLM Distribute the T-Tables BLM to students. Draw a pattern such as this on the board. Figure 1 Figure 2 Figure 3 Figure 4 Ask students to draw the next two figures in the pattern. Ask students to describe the pattern in different ways. Have them enter the numbers into one of the T-Tables. Lead them to see that the number of dots is related to the number of the figure. (2 times the figure number or n = 2f) Draw another pattern on the board. Figure 1 Figure 2 Figure 3 Ask questions similar to the ones asked above. This time the relationship of the number of dots to the figure is the number times itself or the figure number squared, or n = f × f or n = f2.) Continue drawing other patterns.



Activity 18: One Grain of Rice (GLEs: 8, 9, 33) Materials List: One Grain of Rice: A Mathematical Folktale (optional), calculators, paper, pencils, math learning logs Read the book, One Grain of Rice: A Mathematical Folktale, to the students. When you begin reading, have students predict the number of grains of rice on the 30th day. In their math learning logs, (view literacy strategy descriptions) have students explain why they predicted their amount. As you read the book, write the amount of rice for each day as a sequence. 1, 2, 4, 8, 16, … Allow students to change their predictions after day 5, day 10, day 15, and day 20. Each time students change their predictions, they should explain their reasoning in their math learning logs. Ask students to describe any patterns they see. (Each day the amount doubles.) If you do not have the book, just focus on the pattern in the sequence. You can also show this pattern of growth on a calculator. Enter1 2 , , , × = = = . (On some 4-function calculators enter 2 1, , ,× = = = .) Because the numbers get so large so quickly, you can ask questions about place value, estimation, and comparing with large numbers.

Sample Assessments

General Assessments

• Portfolio assessment could include the following: o Anecdotal notes made during teacher observation. o Any of the journal entries, or one of the explanations from the specific

activities o Corrections to any of the missed items on the tests

• On any teacher-made written tests, the teacher could include at least one of the following.

o One problem that requires the use of manipulatives or drawings such as this: Using some type of base-ten manipulative or drawing, show 6 × 24.

o One question that requires the student to explain his/her reasoning such as this: How you solve the problem 56 × 3 mentally?

o One problem involving real-life such as this: Since numbers and mathematics are used all the time during the day, list two times that an exact answer is needed to answer a question, and two times that an estimate is all that is needed.

• Journal entries will include the following: o Explain how you would estimate the answer to the following problem: 135

× 19 = _____ o Mr. Mistake worked the following problem 76 4× and got an answer of

2824. Explain why his answer is not reasonable, and tell the mistake he made.



o Explain in writing how to mentally find the product of 52 and 7. Activity-Specific Assessments

• Activity 1: Solve the following problem. Paulo counted by 5’s, Su counted by

3’s, and Chemika counted by 4’s. They all counted to 200. What same numbers do they all say? (60, 120, 180, …)

• Activity 4: Which property or properties would you use to multiply 6 × 90 × 5

mentally? (Possibly commutative property 6 × 5 × 90) What is your answer? (2700)

• Activity 5: Suppose you wanted to buy 29 packs of crackers at $0.89 each. What

estimate could you use to find the total cost? (Possibly 30 × $0.90 or $27.00)

• Activity 9: Solve the following problem. Suppose 770 students are going on a field trip; 32 students can ride on each bus. Explain how to find how many buses will be needed.

• Activity 12: Which method, mental math, paper and pencil, or a calculator,

would you use to solve the problem 8, 296 × 100? (Hopefully, mental math) Explain your reasoning.

• Activity 16: Look at the following input/output table. What is the rule? (Multiply

by 2 or double the input number.)

Input Output5 10 8 16 10 20 12 24

• Activity 17: The teacher will draw the following pattern on the board.

Figure 1 Figure 2 Figure 3 Figure 4

Have students continue the pattern by drawing the 5th and 6th figures. They should describe the pattern in words and using an equation. (The number of dots is the figure number doubled minus 1; d = 2f – 1.)


Grade 5 Mathematics Unit 3 Data, Probability, Counting Principle 27

Grade 5 Mathematics

Unit 3: Data, Probability, and the Counting Principle Time Frame: Approximately four weeks Unit Description This unit uses graphical settings to review and practice the operations on whole numbers. This unit includes a review of chance (probability) from grade 4, especially the use of counting with lists, tables, and tree diagrams to describe a sample space. It also includes the use of fractions to describe the probability of given events. Student Understandings Students organize, display, and interpret data. They practice the operations on whole numbers in different graphical representations. They also predict the probability of simple experiments and understand that the measure of the likelihood of an event can be represented by a number from 0 to 1. Guiding Questions

1. Can students work proficiently with whole numbers and their operations in graphical settings?

2. Can students organize, display and interpret data? 3. Can students identify and/or create equivalent ratios? 4. Can students enumerate the possible outcomes for events involving number

cubes, spinners, and coins? 5. Can students identify the outcomes of an experiment? 6. Can students represent the probability associated with an event as a common

fraction? Unit 3 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 2. Recognize, explain, and compute equivalent fractions for common fractions

(N-1-M) (N-3-M) 7. Select, sequence, and use appropriate operations to solve multi-step word

problems with whole numbers (N-5-M) (N-4-M)



8. Use the whole number system (e.g., computational fluency, place value, etc.) to

solve problems in real-life and other context areas (N-5-M) 11. Explain concepts of ratios and equivalent ratios using models and pictures in

real-life problems (e.g., understand that 23 means 2 divided by 3) (N-8-M) (N-

5-M) Data Analysis, Probability, and Discrete Math 28. Use various types of charts and graphs, including double bar graphs, to

organize, display, and interpret data and discuss patterns verbally and in writing (D-1-M) (D-2-M) (P-3-M) (A-4-M

29. Compare and contrast different scales and labels for bar and line graphs (D-1-M)

30. Organize and display data using spreadsheets, with technology (D-1-M) 31. Compare and contrast survey data from two groups relative to the same

question (D-2-M) 32. Represent probabilities as common fractions and recognize that probabilities

fall between 0 and 1, inclusive (D-5-M) Patterns, Relations, and Functions 33. Fill in missing elements in sequences of designs, number patterns, positioned


Sample Activities Activity 1: Graphing Vocabulary (GLE: 28) Materials List: Graphing Vocabulary BLM, pencils Before beginning the graphing activities, have students complete a vocabulary self awareness (view literacy strategy descriptions) chart. Provide students with the Graphing Vocabulary BLM. A sample grid is shown below. You may choose to include different words. Do not give students definitions or examples at this point.

Word + √ – Example Definition axis scale mean median mode range cluster gap



Ask students to rate their understanding of each word with either a “+” (understands well), a “√” (some understanding), or a “–” (don’t know). During and after completing the graphing activities 2 through 9, students should return to the chart and fill in examples and definitions in their own words. Some words will have a “+”, a “√”, and a “–” by the end of the activities. The goal is to have plus signs for every word at the end of the activities. Activity 2: Spinning Bar Graph (GLEs: 28, 29, 30) Materials List: Spinners BLM, Bar Graph BLM, colored pens or crayons, pencils, paper, paper clips, computers – optional, spreadsheet program – optional, transparent spinners – optional Have students work in pairs. Provide each pair of students the Spinners BLM, the Bar Graph BLM, colored pens or crayons, and a transparent spinner or a paper clip and a pencil to use as a spinner. Start the activity with this question: If you spin Spinner A 20 times, about how many times do you predict it will land on red? Survey the class and tally the numbers. In each pair, one student should spin Spinner A 20 times. The other student should record the outcome of each spin. On the Bar Graph BLM, each pair of students should make a bar graph of the outcomes. Discuss the labels that should be shown on the horizontal and vertical axes. Allow students to choose the intervals for the scale. Compare the graphs that the different pairs of students make, looking for differences in the results of the spins and for differences in the appearance when various scales are used. Compare the results of spinning red with the initial predictions. If students need more practice, use Spinner B. As a variation of this activity, students could collect data on classmates’ favorite colors, input this information into a spreadsheet, and create a bar graph. They can make another bar graph with a different vertical scale and describe the difference between the two graphs. If computers are not available for each group of students, one computer could be used for the whole class. Begin to introduce the idea of probability. Have students look at Spinner B. Ask questions such as the following. Which color do you think you have the best chance of spinning? (yellow) Why? (There are more yellow sections than blue or red sections.) What are your chances of spinning green? (impossible) What are your chances of spinning red, yellow, or blue? (certain) Which color are you least likely to spin? (blue) Activity 3: Survey Preferences (GLEs: 7, 28, 29, 31) Materials List: paper, pencils, Bar Graph BLM, math learning logs Have students work in groups of 3 or 4. Each group should write a survey question such as, “Do you prefer cats or dogs as pets?” or “Do you own a computer?” Have each group



survey the boys and girls in the class separately. Distribute the Bar Graph BLM. Have students tally the data and make a double bar graph of the data, showing the preference for each gender group. Have students present their graphs to the class, pointing out the title, vertical and horizontal axes, scale interval, and bars. Have students write word problems to go along with their graphs. Some word problems should involve multiple steps. In their math learning logs (view literacy strategy descriptions), have students answer the following question: How does changing the scale on a bar graph change the look of the bar graph? Activity 4: Line Plots (GLEs: 8, 28, 31) Materials List: paper, pencils Through SPAWN writing (view literacy strategy descriptions) prompts, such as “What if?” a thought-provoking activity can be created related to the use of graphs to display information. Ask students to answer this prompt: What if you were a clothing designer and you want to design a really popular shirt. You are unsure as to the number of buttons you should put on the shirt. What could you do to collect information on the number of buttons on the typical shirt that your classmates wear? How could you organize and display the data you collect to convince the owner of the company as to the number of buttons you want to put on the shirt? Use the different responses as a springboard to discuss different types of graphs. If no one thinks of a line plot, do the following activity. As a whole class, collect data on the number of buttons that students have on their clothing that day. (In schools where students wear uniforms, collect data the number of buttons on students’ favorite shirts.) Use a line plot to collect the data. Generally, the number of buttons will fall between 0 and 15. Discuss the information, using terms such as range, clusters, and gaps. Have students find the mode, median, and mean. Students will practice operations on whole numbers when finding the mean. As a whole class, have students decide on another question. Separate students into small groups. Instruct each group to survey a total of 20 people outside the classroom on this question and make a line plot of the data. Have students compare the results of the data from the different groups. Activity 5: M&M Spreadsheet Math (GLEs: 7, 28, 29, 30) Materials List: Bar Graph BLM, M&Ms®, computer, paper, pencils, spreadsheet program – optional Using one-serving size bags of M&Ms® per group, have students sort the candy by color and make a group bar graph by hand on the Bar Graph BLM. Make sure that the students name the graph and label both axes.



To incorporate technology using the data gathered, input the data into a spreadsheet. Have the students construct a data table and make a bar graph from the options menu. Compare the computer-generated bar graph to the one they made by hand. Using the same data, have students create a small group pictograph. Have the students use a symbol that represents a number other than one. Allow different groups to use different amounts for their symbols and compare the graphs. Have students write a math story chain (view literacy strategy descriptions) about the information collected in the graphs. Each student in the group contributes a part to the story chain. An example of a story chain could be: Our group had 8 red M&M’s in the bag. We also had 11 brown M&M’s in the bag. How many more brown M&M’s did we have than red ones? Students in the story chain groups should talk about the accuracy of the answer and the logic of the story problem. If necessary, revisions to the story chain should be made. Activity 6: Stem and Leaf Plots (GLEs: 7, 28) Materials List: paper, pencils, timing device with seconds Have students work in pairs to do one of the following activities: snap their fingers, jump on one foot, or tap the top of their head. Act as the timekeeper. Allow 30 seconds. Have one partner count as the other partner performs the activity. As a class, collect and record the data in a stem and leaf plot. Make a second stem and leaf plot of the same data to organize it into numerical order. Ask questions about the range, clusters, gaps, median, mode, and mean. If you feel that the students need more practice, repeat the activity using the other hand or foot. Have students write word problems or questions from the plot, making sure that some of the problems are multi-step. Activity 7: Line Graphs (GLEs: 28, 29) Materials List: textbooks or newspapers, paper, pencils, Internet access (optional) Using tables found in social studies books, math textbooks, science books, or newspapers such as USA Today, ask students to use the data to make a line graph. Remind students that line graphs show trends or changes over time. Have students discuss any patterns or trends that they observe in the data. Have students compare scales for the graphs and discuss which scale best shows the data. Discuss the advantages and disadvantages of using different scales with the same data.



The website, www.nctm.org, has a lesson that can be used to introduce scatterplots. Go to the website, click on Lessons and Resources, Elementary, Illuminations, Lessons, and in the search box, type Using Math in Everyday Life, Skin Weight. Activity 8: Features of Graphs (GLE: 28) Materials List: Types of Graphs BLM, pencils This activity will help students summarize the important features of different types of graphs. On the board, or overhead projector, draw a word grid (view literacy strategy descriptions) like the one below. To take full advantage of word grids, the grids should be co-constructed with students, so as to maximize participation in the learning process. Provide students with the Types of Graphs BLM. In the first column, have students help list the types of graphs they have been studying. In the other column headings, have them think of features of the graph, such as these: needs horizontal and vertical axes, needs a scale, shows trends. In each box, they should write A for always, S for sometimes, and N for never. Once the word grid is complete, students should be given time to review the differences and similarities among graphs. They can quiz each other over the information in the grid in preparation for tests or other class activities. Sample Word Grid

Types of graphs

needs a horizontal

and vertical

axis

shows trends

bar graph A S

line plot N S line

graph A A

A = always S = sometimes N = never Activity 9: Which Graph is Best? (GLEs: 28, 29, 30, 31) Materials List: paper, pencils, math learning logs, computer – optional, spreadsheet program – optional Have each group of students design a survey question and ask 12 other students in the class (or 12 people outside of class) the survey question. Students should record the answers in a frequency table. Give the same question to two groups. Suggested questions: How tall are you in inches? What is your favorite subject? What kind of transportation do you use to get to school? How many brothers do you have?

http://www.nctm.org



On the following day, have each group of students graph the data, present their findings, and explain how they made their graphs. Allow students to choose the type of graph they want to use. Hopefully, they will find that some graphs will not work for their given information. In their math learning logs (view literacy strategy descriptions), have students write about why they chose a certain graph to display their data. To incorporate technology into the lesson, have students use a spreadsheet to create their graph. Using the results of the data table and graphs, have students compare and contrast their graphs. They should see differences in the overall graphs, the scales, and the labels. Activity 10: Writing Ratios (GLE: 2, 11) Materials List: paper, pencils, paper cups, 2-color counters Have students count different items in the classroom, such as the number of boys, girls, teachers, desks, chairs, doors, windows, clocks. On the board, make a table of their counts. Discuss that ratios are comparisons of two quantities, and that they can be used to compare a part to another part, a part to the whole, or the whole to a part. Ratios can be written in 3 forms. Examples are 15 to 14, 15 : 14, and 15

14 . Have students choose 2 items to compare, such as boys and girls in the classroom, and have them write as many ratios as they can about the two quantities. Each ratio should be written in all 3 forms. Provide pairs of students with 20 two-color (e.g., red/yellow) counters and a paper cup. Have the first student shake the counters in the cup and then empty the counters onto a desk. Ask the other student to determine the ratio of red counters to yellow counters. Once the ratio has been determined, the first student should give another equivalent ratio. For example, if the 20 counters are emptied from the cup and produce 14 red counters and 6 yellow counters, then the ratio is 14

6 . An equivalent ratio would be 7 red counters to 3 yellow counters. This can be shown by creating 2 stacks of red counters with 7 in each stack versus 2 stacks of yellow counters with 3 in each stack. If the ratio is in lowest terms, then repeat the shake. Note: Even though fractions are studied in Unit 4, students should be able to write ratios as fractions and to find equivalent ratios using manipulatives. Activity 11: Vocabulary Ratios (GLE: 11) Materials List: paper, pencils Using math vocabulary or spelling words for the week, have the students choose a word and the ratio of consonants to vowels, the ratio of vowels to consonants, and the ratio of vowels to the total number of letters in the word, etc. When completed, have student volunteers call out the correct ratio for their word. Challenge students to write a word that has a consonant to vowel ratio of 3 to 1, such as the word “that” or the word “students.”



Activity 12: Ratios in Patterns (GLEs: 2, 11, 33) Materials List: pattern blocks, paper, pencils Create a repeating pattern on the board or overhead projector such as this: core Your pattern should show the core and 3 repetitions. Create a table such as this:

Number of blocks In Core After 1st Repetition

After 2nd Repetition

After 3rd Repetition

Number of Triangles 1 2 3 4 Number of Squares 4 8 12 16 Ratio of Triangles

to Squares 14 2

8 312 4

16

Ask students if they see any relationships between the number of triangles and squares. They should say something such as, for every triangle there are 4 squares. Ask questions such as these: If you continued the pattern completing each repetition until you had 5 triangles, how many squares would you have? (20) Write a ratio between the number of triangles and squares. If you continued the pattern until you had 28 squares, how many triangles would you have used? (7) Write this as a ratio of triangles to squares. Explain that all of these ratios show the same comparison, and are called equivalent ratios. Using pattern blocks, have students make their own patterns and a table for their patterns. Lead students to see that they can simply use the original ratio and multiply the numerator and denominator by the same number to get an equivalent ratio. Activity 13: Ratios in Recipes (GLEs: 2, 11) Materials List: Sample Recipes BLM, paper, pencils On the board or on the overhead projector, show the sample recipe for lemonade from the Sample Recipes BLM. Have students write ratios for different ingredients, such as, 6 cups of water to 2 cans of juice is a 6:2 ratio. Ask them what the equivalent ratio would be if they doubled the recipe or tripled it. (12:4, 18:6) Distribute the Sample Recipes BLM to students. Have students make a table showing equivalent ratios for different pairs of ingredients from any of the recipes. If there are two ingredients that are multiples of 2, ask students what would be the ratio if you divided the recipe in half. If you want to use other recipes, just write ratios for the ingredients that are whole numbers.

…



Activity 14: Is It Likely? (GLE: 32) Materials List: Is It Likely? BLM, math learning logs Provide students with the Is It Likely? BLM showing the following categories: impossible, not likely, equally likely, likely, and certain. Have students give examples of each type of event in the real world. Give students an event and ask into which category they would place the event. Examples include: the likelihood of seeing a red car on the way home, of one of your parents winning the lottery, of getting a head on a coin toss, or of getting an even number on the toss of a number cube with the numbers 1, 2, 3, 4, 5, 6 on it. Introduce the term probability as the likelihood of an event happening. In their math learning logs (view literacy strategy descriptions), have students write examples of certain, impossible, and equally likely events. Activity 15: How Many Kinds of Pizza? (GLE: 32) Materials List: paper, pencils, Internet access (optional) Tell students that you have a friend who wants to offer lunch specials at her pizza restaurant. She will offer one-topping pizzas for a special price. The toppings on Monday will be pepperoni, mushrooms, or extra cheese, and the types of crust will be crispy, thick, or whole wheat. What are the combinations she could make for lunch specials? Have the students make an organized list, draw a picture, or make a tree diagram. Have students use a variable to represent each choice, such as C, to represent the crispy crust. Have students think of other lunch specials that a restaurant could offer and list the combinations or outcomes. NCTM, www.nctm.org, has a lesson on combinations of ice cream cones and flavors. Once on the website, go to Lessons and Resources, Elementary, Illuminations, Lessons, and in the search box type Combinations. Ask probability questions, such as this: What is the probability that the first customer will choose a mushroom pizza with whole wheat crust? Activity 16: What are the Outcomes? (GLE: 32) Materials List: coins, paper, pencils Give each pair of students a coin to flip. Ask them to list the possible outcomes, using the variables H for heads and T for tails. Give each pair of students a second coin. Have them flip the two coins at the same time. Encourage them to use a tree diagram as a way of finding the outcomes. Give the students a 3rd coin and have them use a tree diagram to find the possible outcomes for tossing 3 coins at a time. Introduce the term probability as the ratio of favorable outcomes to the possible outcomes. When students tossed 3 coins, how many outcomes were there? (8 outcomes) How many of the outcomes had all heads?

http://www.nctm.org



(1) What is the probability of getting all heads when flipping 3 coins at a time? ( 18 )

Continue asking questions about the probability of certain events. Activity 17: Probability on a Number Line (GLEs: 32) Materials List: paper, pencils, Spinners BLM Draw a number line on the board from 0 to 1. Tell students that this is a likelihood or probability line. At 0, an event is impossible. As you move closer to 1, the event is more likely to happen, until you reach 1, and then the event is certain to happen. Mark the halfway point. Ask students to tell you the fraction for this point, plus tell the likelihood of an event occurring. ( 1

2 , equally likely) Do the same for the points 14 and 3

4 . Your number line should look something like this:

Impossible Not Likely

Equally Likely Likely Certain

| | | | | 0 1

4 12 3

4 1 Give statements such as these and have students tell you where they fall on the line. Josie said their chances of winning were close to 4 out of 5. (likely) Sam said that the chance of getting a head and a tail on the toss of 2 coins was 2

4 . (equally likely) Have students look at Spinner A. Ask different students to write situations where the probability would be 0, ¼, ½, ¾, or 1. Do the same for Spinner B. Activity 18: Predicting Rolls (GLEs: 32) Materials List: number cubes, blank number cubes Hold up a number cube and discuss the possible outcomes when rolling the number cube. Ask students to predict the chance or probability of rolling a 3. (1 out of 6, or 1

6 ) Ask them what the probability is of not rolling a 3. (5 out of 6, or 5

6 ) Discuss concepts of positive and negative outcomes for rolling or not rolling any number 1 through 6. Talk about theoretical or mathematical probability. Provide each group of students with a number cube that you have modified by drawing or gluing symbols to each side of the number cube.). Ask each group of students to perform an experiment of rolling the number cube a specified number of times (e.g., 20) and record the results of their rolls. Place each group’s findings in a class chart. Class discussion should include analysis comparing the theoretical probability to the actual or experimental results. Teacher Note: Octagonal number cubes are available and would provide a variation for this activity.



Activity 19: What Colors Are in the Bag? (GLEs: 28, 32) Materials List: paper bags, cubes or slips of paper Give each group of students a bag filled with 10 cubes: 2 blue, 4 red, 1 green, and 3 yellow. Do not let the students look inside the bag. Have students pull one block, write the color, and replace the block. Have students do this 20 times. After the 20th pull, have them predict the probabilities that they will pull a certain color. Have them count the blocks and see how their experimental probabilities compared to the theoretical probabilities. You may want students to use a line plot or tally chart to collect the information.

Sample Assessments General Assessments

• Portfolio assessment could include the following: o Anecdotal notes made during teacher observation o Any of the journal entries, or one of the explanations from the specific


• On any teacher-made written tests, include at least one of the following. o One problem that requires the use of manipulatives or drawings such as

this: Using 2-color counters, display a set and write all of the ratios possible comparing the quantities of each color.

o One problem that requires the student to explain his/her reasoning such as this: Explain how you could find three ratios equivalent to 2

3 . o One question involving real-life such as this: Name an event in real life

that is impossible to happen, one that is not likely, one that is likely, and one that is certain to happen.

• Journal entries could include the following: o In Ms. Thomas’ class, there are 9 boys and 11 girls. Using ratios, three

classmates described the situation as 9 to 11, 11 to 9, and 9 to 20. Explain how all three classmates could be correct.

o Using drawings, explain why the ratio 36 is equivalent to 1

2 . Activity-Specific Assessments

• Activity 4: The student should make a second type of graph with data collected in the activity and discuss in writing which graph better displays the data and why.



• Activity 9: The students should write three questions that could be answered by a graph that he/she constructs.

• Activity 10: The students should look around the room and find two quantities

whose ratio is 2 to 1; two quantities whose ratio is 1 to 5, and two quantities whose ratio is 2: 3.

• Activity 11: The students should write his/her last name on a piece of paper, then

write the ratio of vowels to consonants in the three different forms. • Activity 13: Tell the student that a good recipe for cookies calls for 2 cups of

brown sugar and 4 cups of flour. The student should write a ratio of brown sugar to flour and find equivalent ratios if the recipe is doubled, tripled, or cut in half.

• Activity 15: The students should make a tree diagram to find the combinations or

outcomes of the following problem: Suppose you have to wear a uniform to school each day. You have a choice of red pants or blue pants, and a choice of a red shirt, a blue shirt, or a white shirt. How many combinations can you make and what are they?

• Activity 17: The student should draw four spinners. Spinner 1 will show the

probability of spinning blue is 12 . Spinner 2 will show the probability of spinning

yellow is 0. Spinner 3 will show the probability of spinning yellow is 1. Spinner 4 will show the probability of spinning blue is 1

2 , but it should look different from Spinner 1.


Grade 5 Mathematics Unit 4 Number Theory/Equivalent Fractions 39

Grade 5 Mathematics

Unit 4: Number Theory and Equivalent Fractions Time Frame: Approximately four weeks Unit Description The focus of this unit is equivalent fractions, comparison of fractions, and the number theory properties that provide the basis for such equivalencies and comparisons. The work with factors and multiples is critical to work with fractions, but also provides review of multiplication and division facts. Student Understandings Students develop an understanding of different representations of fractions such as parts of wholes, parts of collections, locations on number lines, and as divisions of whole numbers. Students recognize and generate equivalent forms of commonly used fractions, mixed numbers, and decimals. Students use models, benchmarks, and equivalent forms to judge the size of fractions. Guiding Questions

1. Can students identify fractions using region models, set models, and linear models?

2. Can students use factorization methods to talk about prime and composite numbers and multiples and factors in number contexts?

3. Can students identify or develop equivalent fractions related to a given fraction?

4. Can students compare fractions? 5. Can students describe mixed numbers and improper fractions and convert

between these forms? 6. Can students convert between decimals and fractions or mixed numbers?



Unit 4 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 1. Differentiate between the terms factor and multiple, and prime and composite

(N-1-M) 2. Recognize, explain, and compute equivalent fractions for common fractions

(N-1-M) (N-3-M) 3. Add and subtract fractions with common denominators and use mental math to

determine whether the answer is reasonable (N-2-M) 4. Compare positive fractions using number sense, symbols (i.e., <, =, >), and

number lines (N-2-M) 5. Read, explain, and write a numerical representation for positive improper

fractions, mixed numbers, and decimals from a pictorial representation and vice versa (N-3-M)

6. Select and discuss the correct operation for a given problem involving positive fractions using appropriate language, such as sum, difference, numerator, and denominator (N-4-M) (N-5-M)

8. Use the whole number system (e.g., computational fluency, place value, etc.) to solve problems in real-life and other content areas (N-5-M)

Sample Activities Activity 1: All About One-Half (GLEs: 2, 4, 5, 6) Materials List: paper, pencils Have students use brainstorming (view literacy strategy descriptions) to determine what they know about the fraction 2

1 . Some examples are the following: 21 = 4

2 , 1 is the numerator, 2 is the denominator, 2

1 < 1, 21 > 4

1 , 41 + 4

1 = 21 , 2

1 is in simplest terms, 21 =

0.50, 21 of $10 is less than 4

1 of $1000, 21 is 1 ÷ 2, etc. This activity provides insight

about students’ knowledge of fractions. Have each student draw a picture of 21 . Using

their drawings, discuss the different models for fractions; area or regions, sets, a linear model, or a division problem. Consider repeating this activity later on in the unit using a mixed number, improper fraction, or a decimal.



Activity 2: Vocabulary Cards (GLE: 1) Materials List: index cards, pencils To develop students’ knowledge of key vocabulary, have them create vocabulary cards (view literacy strategy descriptions) for terms related to number theory. Distribute 3 x 5 or 5 x 7 inch index cards to each student, and ask them to follow your directions in creating the sample card. On the board, place a targeted word in the middle of the card, as in the example. Ask the students to provide a definition. It is best if a term can be defined in students’ own words. Write the definition in the appropriate space. Next, have students list the characteristics or write a description, give one or two examples, and illustrate the term. For activities 2 – 6 involving number theory, as students come across key terms, have them create vocabulary cards for each term. Allow time for students to review their cards, and quiz a partner on the terms to hold them accountable for accurate information on the cards. Activity 3: Make a Rectangle (GLE: 1, 8) Materials List: Grid Paper BLM, blank construction paper, scissors, pencils Have students work in groups. Provide them with the Grid Paper BLM, construction paper, and scissors. On the grid paper, have students draw and cut out rectangles for each of the numbers 1–30. For example, for the number 4, students could draw the following: Consider assigning each group certain numbers so that students do not have to make all of the numbers. Have students place the cut-out rectangles for each number on the sheets of construction paper, with the corresponding number as the heading. Students should consider a 4 2× rectangle the same as a 2 4× rectangle. The dimensions of the rectangles are the factors of the number. The number, 4, has factors of 1, 2, and 4. The number, 4, is a multiple of 1, 2, and 4. The number, 4, is divisible by 1, 2, and 4. Introduce the

Definition: A multiple is the product of a whole number and any other number.

Characteristics: Multiples are whole numbers. Multiples can be even or odd.

Examples: Multiples of 3 are 3, 6, 9, 12 …

Illustration: ○ ○ ○ ○ ○ ○ 6 is a multiple of 3. I can make groups of 3 with no leftovers.

Multiple



concepts of prime and composite numbers. Numbers with only one possible rectangle are prime; numbers with more than one rectangle are composite. Although a 1 1× rectangle can be drawn for the number 1, one is considered neither prime nor composite. The definition of a prime number requires that the two factors be different. Have students describe their rectangles for a specific number. For the number 8, students would show 1 8× and 2 4× rectangles. Introduce the idea of factor as the whole number lengths and widths of each rectangle. 1, 2, 4, and 8 are factors of 8. Ask students to see if they found any rectangles that had a length equal to a width. The corresponding numbers are called square numbers. Hold up sheets for the numbers 6 and 8. Ask, what are the factors of each number? (for 6: 1, 2, 3, 6; for 8: 1, 2, 4, 8) What factors do the numbers have in common? (1 and 2) What is the greatest factor they have in common? (2) This is a way to informally introduce the concepts of common factors and greatest common factor. Activity 4: The Sieve of Eratosthenes (GLE: 1, 8) Materials List: Hundred Chart BLM, paper, pencils, fraction calculators - optional Provide each student with the Hundred Chart BLM. Begin by saying a prime number has exactly two different factors, itself and 1. Have students discuss why 1 is neither prime nor composite. Next, have students circle the number 2 and then cross out all the numbers that are multiples of 2. Circle the first number after 2 that is not crossed out – the number 3. Cross out each number that is a multiple of 3. Note that some numbers get crossed out more than one time. When finished, find the next number not crossed out (5), circle it, and repeat the process. Continue until all numbers have been crossed out or circled. Questions for student discussion are as follows: What types of numbers get crossed out? (composite) How do the crossed out numbers relate to the numbers that remain? (They are multiples of at least one of the remaining numbers.) What types of numbers remain? (prime) Explain that this process filters out the composites. A sieve is a filtering device, like a strainer. The method of eliminating multiples of primes used in this exercise was named after Eratosthenes, the mathematician who first used the technique. Another way to determine if a number is prime is to use a fraction calculator. For example, to determine if 43 is prime, enter 4 3 / 4 3 SIMP = . If the display reads 1/1, the number is prime. The only factors of 43 are 43 and 1. If the display reads something like 2/2, the number has other factors and is not prime.



Activity 5: What’s My Rule? (GLE: 1, 8) Materials List: Venn Diagrams BLM, pencils Distribute the Venn Diagrams BLM to students, or use on the overhead projector. Graphic organizers (view literacy strategy descriptions), such as Venn diagrams, can help students organize their thoughts about number theory. Begin by using Venn diagram I. Think of a rule, such as multiples of 11, but do not tell students the rule. Ask each student to call out a number less than 100. If the number fits the rule, place it inside the circle. If it doesn’t fit the rule, place it outside the circle. The object is for students to guess the rule. In Venn Diagram II, use a rule for A, such as multiples of 2, and a rule for B, such as multiples of 3. Do not tell students the rule. Again, have students call out numbers. Place the numbers in the corresponding circles, in the overlap, or outside the circles. Students should guess the rules. The numbers in the overlap will be common multiples of 2 and 3, and will also be multiples of 6. The numbers outside the circles will be numbers that are not multiples of 2 or 3. Activity 6: Finding Factors (GLE: 1, 8) Materials List: Finding Factors BLM, paper, pencils, math learning logs Give each pair of students the Finding Factors BLM. The student with the fewest letters in his/her last name will go first. Ask this student to choose one of the numbers for which he/she receives that many points. This number should be crossed out. The second student receives all of the factors of the chosen number as points, except the number itself. These numbers are also crossed out. For example, if Student 1 chooses 30 and gets 30 points; Student 2 gets the factors (except 30) 1, 2, 3, 5, 6, 10, and 15, for a total of 42 points. Student 2 then chooses a number and Student 1 gets the factors as points. Both students must get points. Student 2 could not choose 4 at this point because the only factors that Student 1 could get would be 1 and 2, and both of those numbers have been crossed out. The winner is the student with the most points. You may want to play the first game against the class. After students have played the game a few times, have them write in their math learning logs (view literacy strategy descriptions), what they think is a good choice for the first number chosen and what is a bad choice for the first number chosen. They must explain their reasoning. For example, a good choice is the largest prime number, 29. Their opponent gets only 1 point. Another good choice is the largest odd square number, or 25. A bad first choice would be 24 because the opponent gets 1, 2, 3, 4, 6, 8, and 12. As an extension, enlarge the mat to include numbers to 50. This activity is also a good review of multiplication and division facts.



Activity 7: What about Fractions? (GLEs: 2, 6) Materials List: What about Fractions? BLM, pencils Before beginning the fraction activities, have students complete a vocabulary self awareness (view literacy strategy descriptions) chart. Provide students with the What about Fractions? BLM. Do not give students definitions or examples at this point.

Word/Phrase + √ – Example Definition numerator

denominator

mixed number

improper fraction

equivalent fraction

simplest form

Ask students to rate their understanding of each word with either a “+” (understands well), a “√” (some understanding), or a “–” (don’t know). During, and after completing fraction activities, students should return to the chart and fill in examples and definitions in their own words. Some words may have a “–”, a “√”, and a “+” by the end of the activities. The goal is to have plus signs for all words at the end of the activities. Activity 8: Fractions of Shapes (GLEs: 2, 3, 4, 6) Materials List: transparency of Circle BLM, Square BLM, pencils, fraction tiles or circles – optional, Internet access – optional Using the overhead and a transparency of the Circle BLM, demonstrate dividing the circle in half. Label each half with 1

2 using colored overlays or water-based overhead markers. To give the activity a more real-world feel, call the circle a pizza. Discuss the terms numerator and denominator. The denominator shows the number of pieces that the pizza has been cut into and the numerator shows the number of pieces that you get to eat. Draw a line perpendicular to the first line through the center of the circle and ask students what the pizza is divided into now. (fourths) Continue drawing lines to show eighths and sixteenths. Shade parts of the circle and ask what fraction would describe it. An example would be 1

4 , which could also be called 28 or 4

16 . Give the students the Square BLM (maybe call it a pan of brownies) and have them divide it the same way (halves, fourths, eighths, and sixteenths). Have students make up various problems and use them in number sentences. Having colored fraction tiles and circles for the students will make this activity easier. Next, have students write



comparison statements using their fractional pieces. For example, they could write 1 12 4> .

Informally introduce addition and subtraction, by saying 1 1 24 4 4+ = . You want students to

see that 1 14 4+ equals 2

4 , not 28 . Continue informally with addition and subtraction

questions. Each time, write the equation on the board. The website www.nctm.org has a lesson on the region model of fractions. Go to the site, select, Lessons and Resources, Elementary, Illuminations, Lessons, Grades 3-5. In the search box, type Fun with Fractions: Developing the Region Model. Activity 9: Creating Equivalent Fractions (GLEs: 2) Materials List: Equivalent Fractions BLM, paper, pencils, Internet access – optional, fraction circles – optional Use student sets of fraction circles, if available. If not, provide each student with the Equivalent Fractions BLM. Students should cut out the three circles. Have students create a model of 1

2 by having them fold one circle along a diameter to make two congruent parts. Using another circle, have students fold it into four congruent parts. Now, have students repeat folding with a third circle to create eighths. Once the folding is complete, have students cut the fractional parts and label them. Using the fractional parts, have students create number sentences by showing that 1 2

2 4= , 1 24 8= . Having students working

in pairs, challenge them to create as many equivalencies as they can with their models. The activity can be repeated with thirds, sixths, and twelfths. Students can think of a clock to estimate 3

1 and 32 . They can draw lines from the center to 12:00, to 4:00, and to

8:00 to make thirds. Another way to fold the circles to get sixths is to fold the circle in half and then make two folds to make equal sections. Make sure that the edges meet to insure equal sections. This divides the circle into six equal parts. As an extension, students can visit http://www.learningplanet.com/sam/ff/index.asp to practice matching equivalent fractions in a game format. The game is called Fraction Frenzy. Activity 10: Sets of Fractions (GLEs: 2, 6) Materials List: paper, pencils, Internet access – optional Have students draw 3 small circles on their papers and shade 2 of the circles. Ask what fraction represents the number of shaded circles. ( 2

3 ) Ask what the numerator, 2, represents and what the denominator, 3, represents. Underneath the first set, have students draw a second set exactly like the first set. Ask questions such as: How many circles in all? How many circles are shaded? What fraction represents the number of shaded circles? ( 4

6 ) Discuss the fact that these fractions are equivalent.

http://www.nctm.org

http://www.learningplanet.com/sam/ff/index.asp



Show that 2 sets of 23 means 2 sets of 2 4

62 sets of 3 or . Continue modeling with other fractions, emphasizing equivalent fractions. The website www.nctm.org has a lesson on the set model of fractions. Go to the site, select Lessons and Resources, Elementary, Illuminations, Lessons, Grades 3-5. In the search box, type Fun with Fractions: Developing the Set Model. Activity 11: Fractions on a Number Line (GLEs: 2, 3, 4, 6) Materials List: paper, pencils, Internet access – optional Provide students with a blank sheet of 1

28 11× inch paper. Tell them that they are going to look at the parts of a ruler. They are going to enlarge the section between 0 and 1. Have students turn their paper sideways and draw a line across the top of the paper from end to end. Draw a line on each side of the paper that starts at the top line and runs about halfway down the page. Label the left line as 0 and the right line as 1. Have students fold the paper in half. Show the students which way to fold. On the fold line, draw a line from the top line about half as long as the 0 and 1 lines. Ask questions, such as these: When you folded the paper, how many sections were formed? (2) How far is it from the 0 line to the fold line? ( 1

2 unit) What should we label the fold line as? ( 12 ) How much farther is

it to the 1 line? ( 12 unit more) How many halves are in 1? (2)

0 1 2

1 22

Continue having students fold the paper in halves. Label fourths and eighths. Consider folding all the way to sixteenths depending on the ability of the class. As each fold line is labeled, emphasize equivalent fractions. Discuss the fact that the first fraction that you wrote for each fold line is the simplest form of that fraction. Have students also compare fractions such as 1

4 and 24 , and 1

4 and 12 . Informally ask questions involving addition and

subtraction. What is 1 18 8+ ? What is 3 1

8 8+ ? What is 3 14 4− ? The website www.nctm.org has a

lesson on the linear model of fractions. Go to the site, select Lessons and Resources, Elementary, Illuminations, Lessons, Grades 3-5. In the search box, type Fun with Fractions: Developing the Length Model.

http://www.nctm.org

http://www.nctm.org



Activity 12: Decimals, Fractions, and Mixed Numbers (GLEs: 2, 5) Materials List: Decimal Squares BLM, transparency of Decimal Squares BLM, paper, pencils Provide each student with a copy of the Decimal Squares BLM, and make a transparency of it. Begin by discussing the top large square. Have students tell what they know about it. (It is made up of 100 smaller squares. There are 10 rows and 10 columns. There are 10 squares in each row, and 10 squares in each column.) On your transparency, shade the small square at the top left, and have students do the same on their Decimal Squares BLM. Ask questions such as these: How many small squares are shaded? (1) How many squares in all? (100) What part of the large square is shaded? ( 100

1 ) Show students that this can also be written as the decimal, 0.01. Consider also discussing that percent means parts per hundred. The amount could also be written as 1%. On a sheet of paper, have students write 100

1 = 0.01 = 1%. Shade the next square in that column. Have students do the same. The amount is now 100

2 . On their paper, have students write 100

2 = 0.02 = 2%. Consider looking at other equivalencies such as 1002 = 50

1 . Continue shading and recording different amounts. Pay particular attention to 100

10 , 10020 ,

10025 , 100

50 , 10075 , and 100

100 . Once they have shaded one entire large square, have students shade one small square in the second large square. Students should see that the amount shaded is 100

101 , 1 1001 , 1.01, or

101% Continue shading the other amounts of squares. Activity 13: Fractions and Money (GLEs: 2, 4, 5) Materials List: paper, pencils, play money Ask students to write fractions as parts of a dollar. For example, ask them to show a dime as a fractional part of a dollar. Students may reason that there are 10 dimes in a dollar, so 1 dime would be 1

10 . Others may reason that a dime is worth 10 cents, so the fractional part would be 10

100 . Stress that these are equivalent fractions. 10110 100= . Remind students to

write 110 as the decimal 0.1 and 10

100 as the decimal 0.10. Have students represent 2 dimes, 5 dimes, 7 dimes, etc. Ask questions about comparisons of two fractions, such as, which is greater 5

10 or 710 and the comparison of two decimals, such as, which is greater 0.09 or

0.1? Continue with questions about nickels, pennies, quarters, and 50¢ pieces. To incorporate numbers greater than 1, give amounts such as 11 dimes, 6 quarters, etc.



Activity 14: Different Names for Fractions Greater than One (GLE: 5) Materials List: paper, pencils, 3 sets of fraction circles for each group Provide each group of students with 3 sets of fraction circles. Have them place the 3 whole units in the center of the table. Using the 1

2 pieces, place one piece on one of the whole units. Ask what fraction has been displayed. ( 1

2 ) Place a second piece on the same circle and ask what amount has been displayed. Some students will say 1 whole, and some students will say 2 halves. Stress that 1 = 2

2 . Continue adding the 12 pieces until

they have covered all 3 circles. Ask students to explain why 122 names the same amount

as 52 . Do the same activity with thirds.

Continue having students display fractional amounts greater than 1 and naming each amount in two different ways. Fractions greater than or equal to 1 are called improper fractions in many textbooks. Activity 15: Fractions Greater than One on a Number Line (GLEs: 5) Materials List: paper, pencils This activity is an extension of Activity 11. Tell students that they are going to enlarge another part of the ruler, this time the part between 1 and 2. Use a new sheet of paper, not the old sheet used in Activity 11. Have students turn their paper sideways and draw a line across the top of the paper from end to end. Draw a line on each side of the paper that starts at the top line and runs about halfway down the page. Get as close to the edges as possible. Label the left line as 1 and the right line as 2. Fold the paper in half. On the fold line, draw a line from the top line about half as long as the 1 and 2 lines. Be sure that the students label the fold line as 1

21 , not just 12 .

Ask students what is another name for the 1 on their rulers when they fold it in halves. ( 2

2 ) Ask what would be other names for 121 and 2 using halves. ( 3

2 , 42 ) Fold the ruler

again, and ask the same questions about fourths. Don’t bother with eighths and sixteenths. Just make sure that students get the idea that 1

21 is the same as 2 12 2+ or 3

2 . Activity 16: Mixed Numbers, Decimals, Improper Fractions (GLEs: 4, 5) Materials List: paper, pencils, play money, overhead projector Using the overhead, paper bills and quarters, model 1

41 dollars, 54 dollars, and $1.25, so

that students see how mixed numbers, decimals, and improper fractions are related. Draw pictures of 2

32 bricks, 3.5 cookies, and 94 pizzas on the overhead or board. Have students



write the correct numerical representation. Check for accuracy. Next, put a mixed number, a decimal, and an improper fraction on the board and have the students draw pictures to match. Assign each student a different set of three numbers (mixed, decimal, and improper) and have them illustrate each number. Ask students to locate their three numbers on a number line. Activity 17: Fractions Near 0, 1

2 , and 1 (GLE: 4) Materials List: paper, pencils, fraction circles or strips Provide each group of students with a set of fraction circles or strips. Have them place a 12 piece in the center of the table. On the board, or on the overhead projector, make a table like this.

Fractions Near 0 Fractions Near 12 Fractions Near 1

Call out a fraction. Have students model the fraction and decide where to place the fraction in the table above. Once about 10 fractions have been called out, ask students to look for patterns in each column. What is alike about the fractions in each column? How can you decide if a fraction is near 0, 1

2 , or 1? These ideas will help students when comparing fractions and estimating fractional amounts. Activity 18: Comparing Fractions (GLE: 4) Materials List: paper, pencils, fraction circles, math learning logs Give each group of students two sets of fraction circles. Have them place 2 whole units in the center of the table. Have one student display 1

4 on one of the circles, and a second student display 3

4 on the other circle. Write 14 __?__ 3

4 on the board. Ask students to discuss which fraction is larger and why. Continue having students model fractions with the same denominator. You want them to see that if the denominators are the same, the larger numerator will show the larger fraction. Tell the students to think of eating a pizza. If two pizzas are cut into the same size pieces (the denominator), they get more pizza if they get more pieces (the numerator.) Continue the activity with fractions in which the numerators are the same, but the denominators are different. You want them to see that if the numerators are the same, the fraction with the smaller denominator will be larger. If a pizza is cut into 2 pieces, the pieces will be larger than a pizza that is cut into 6 pieces. Both of these ideas are critical to working with fractions. Next, give fractions that do not have the same numerator or denominator. To help them compare fractions, encourage students to model the fractions



using the fraction circles or a number line. Encourage students to use number sense when comparing fractions. 2

5 is less than 12 and 5

7 is greater than 12 , so 52

5 7< . In their math learning logs, (view literacy strategy descriptions), have students use pictures, numbers, and/or words to find ways to show that 3

2 of a pizza is larger than 52

of that same pizza. Activity 19: How Big Is the Fraction? (GLEs: 2, 4, 5) Materials List: How Big is the Fraction? BLM, paper, pencils Distribute the How Big Is the Fraction? BLM to students.

= 0 Between 0 and 1

2 = 12 Between

12 and 1 = 1 Between

1 and 2

Call out fractions, decimals, and mixed numbers and have the students place them in the proper columns. Examples include 2

4 , 06 , 4

4 , 53 , 1

21 , 0.5, 1.2.


• Portfolio assessment could include the following: o Anecdotal notes made during teacher observation o Either one of the journal entries, or one of the explanations from the

specific activities o Corrections to any of the missed items on the tests


this: Show how to compare 12 and 3

4 using manipulatives or drawings. o One problem that requires students to explain their reasoning such as this:

Explain how to find three fractions equivalent to 13 .

o One problem involving real-life such as: Name 3 places where you see fractions, mixed numbers, or decimals in real life.

• Journal entries could include the following: o Answer the following question and explain your reasoning. Can 1

4 ever be larger than 1

2 ? o Explain the difference between prime and composite numbers, and give an

example of each.



Activity-Specific Assessments

• Activity 3: On grid paper the student should draw all of the rectangles for the number 24, then list all of the factors of 24.

• Activities 8 and 9: The student should draw 4 congruent rectangles and use these

rectangles to show that 81 2 42 4 8 16= = = .

• Activity 11: The student should discuss how he/she can easily tell that 1

8 is to the left of 1

2 on the number line.

• Activities 14 and 15: Using fraction circles, a number line, or drawings, the student should explain why 51 1

4 4 41 1= + = .

• Activities 17 and 19: The student should name a fraction, mixed number, or decimal equal to 0, between 0 and 1

2 , between 12 and 1, equal to 1, and between 1

and 2.


Grade 5 Mathematics Unit 5 Properties in Geometry 52

Grade 5 Mathematics

Unit 5: Properties in Geometry Time Frame: Approximately four weeks Unit Description This unit focuses on geometric concepts involving plane figures, and includes the concepts of transformations, rotational symmetry, angle measurement, and coordinate graphing. Student Understandings Students can analyze characteristics and properties of polygons and circles. They can apply transformations, symmetry, and spatial reasoning to solve problems. Guiding Questions

1. Can students classify and describe the properties of circles and polygons? 2. Can students recognize motions in a plane (reflections, rotations, translations)

and use the appropriate language to discuss these motions? 3. Can students recognize and discuss line and rotational symmetry in figures? 4. Can students measure and identify types of angles? 5. Can students identify and plot points on a coordinate grid?

Unit 5 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Measurement 15. Model, measure, and use the names of all common units in the US and metric

systems (M-1-M) 21. Measure angles to the nearest degree (M-3-M) Geometry 24. Use mathematical terms to classify and describe the properties of 2-

dimensional shapes, including circles, triangles, and polygons (G-2-M) 25. Identify and use appropriate terminology for transformations (e.g., translation

as slide, reflection as flip, and rotation as turn) (G-3-M) 26. Identify shapes that have rotational symmetry (G-3-M) 27. Identify and plot points on a coordinate grid in the first quadrant (G-6-M)



Sample Activities Activity 1: Geometry Vocabulary (GLE: 21, 24, 25) Materials List: index cards, pencils Have students work in groups to think of geometry vocabulary words. They should write each word in large print on an index card. Have each group write at least 20 words. The words could be ones such as square, pyramid, intersecting, symmetry, flip, angle, congruent. After writing their 20 words, have the groups sort them into categories that make sense to them. The categories could be ones such as types of lines, 2-dimensional shapes, or 4-sided shapes. To demonstrate that students have learned the content, use the professor-know-it-all (view literacy strategy descriptions) strategy. Tell students that one group will be called on randomly to come to the front of the class to be a team of “professor-know-it-alls.” Select one group to come up and place their index cards on the floor, or tape them to the wall. The team should be able to explain their categories and answer questions about their sorting choices. Other students should listen for accuracy and logic in the “professors” answers to their questions. After about 5 minutes or so, ask a new group to take its place in front of the class. The new group should add their words to these categories or make new categories. Sometimes a word will be placed into more than one category, such as radius could be placed with circles or with line segments. Discussion is critical in this activity. This is a way to informally see what the students know about geometry and provides a chance to review terms from 4th grade. Do not worry if the students do not think of all of the vocabulary words. This is just an introduction. Activity 2: More Geometry Vocabulary (GLE: 21, 24, 25) Materials List: index cards, pencils To develop students’ knowledge of key vocabulary, have them create vocabulary cards (view literacy strategy descriptions) for terms related to geometry. Distribute 3 x 5 or 5 x 7 inch index cards to each student, and ask them to follow directions in creating the sample card. On the board, place a targeted word in the middle of the card, as in the example. Ask the students to provide a definition. It is best if a word can be defined in students’ own words. Write the definition in the appropriate space. Next, have students list the characteristics or a description and illustrate the term. Throughout the unit, as students come across key terms, have them create vocabulary cards for each term. Allow time for students to review their cards and quiz a partner on the terms to hold them accountable for accurate information on the cards.



You may want to review the terms point, line, line segment, ray, and plane from 4th grade. Activity 3: Arm Angles (GLE: 21) Materials List: paper, pencils, students themselves, transparency of Measuring Angles BLM Begin the activity by reviewing the terms, angle and vertex. Tell students that angles are measured in degrees. Make an overhead transparency of the Measuring Angles BLM. Discuss the different types of angles on the page (acute, right, obtuse, and straight). Ask students to estimate the measures of the acute and obtuse angles, but do not have them actually measure them in this activity. Ask students to model different types of angles (acute, obtuse, right, and straight) with their arms. After students have modeled these four types of angles, choose a degree measure between 0°and 180° , such as 45° . Have students model it and explain how they decided how wide to open their arms. They certainly won’t be exact, but if there are large differences in the width of their spread arms, discussion should follow. Have them model measures such as 170 , 5 , 30 , and 92° ° ° ° . For each measure, ask students to name the type of angle. Activity 4: Angle Measurement (GLE: 21) Materials List: Measuring Angles BLM, protractor, paper, pencils, index cards Provide pairs of students with a protractor and the Measuring Angles BLM. Demonstrate how to use a protractor to measure angles A – C. Ask students to measure angles 1-6 with their protractors and record the angle measure to the nearest degree. It may help students to extend the rays to measure the angles. Have both students in each pair measure the

Definition: A quadrilateral with only 1 set of parallel lines.

Characteristics: 4 sides, 1 set of parallel sides, can have right angles, can have symmetry

Illustration

Trapezoid



angles. Make sure that their measurements are close. Consider accepting an error range of 2 degrees either way in the measurements. Students should classify each angle. Call out different angle measures in whole degrees. Using their protractors, have student pairs draw the appropriate angle on index cards and exchange with other student pairs who will verify the measurements of the drawings. On one of the angles, such as a 30° angle, ask students how they know that the angle measures 30° and not 150° . Since many protractors have a double scale, this will help the students avoid errors when using a protractor. Activity 5: Circles (GLEs: 21, 24) Materials List: paper plates, protractors, pencils, colored pencils Have students discuss some of the properties of circles. Circles are round, have no straight sides, are not polygons, are the base of a cone or cylinder, and the distance around them is called the circumference. Give each student a paper plate and a protractor. These are special plates that have 36 crimps or fingers around them. They can be found at most grocery stores and the styrofoam ones work better than the paper ones. The edges look similar to ∩∩∩ with crimps all around the plate. Students may have trouble finding the center, so consider labeling the center of each plate. The crimps have lines that extend toward the center. Each line should be labeled with multiples of 10, starting with 0° . ∩∩∩ 0°10° 20° 30° When students have labeled the full plate, also label the crimp at 0° as360° . Discuss the fact that a circle has 360° in it. Have students draw a line from one side of the plate, starting at 0° , go through the origin, and continue to the other side of the plate. Ask where the line ends? (180°) Talk about the terms diameter and radius. Draw a radius from the center to the 90°mark. Ask, “What kind of angle is formed by the intersection of this radius and the drawn diagonal?” (right) Draw another radius from 0° to 270° . “What kind of angle is formed by the intersection of this radius and the drawn diameter?” (right) Introduce the term central angle. A central angle in a circle is any angle that has a vertex at the center of the circle. This activity can help students better understand turns or rotations of 90 , 180 , 270 ,° ° ° and 360° and can be used to introduce circle graphs. Have the students use different colored markers to draw the different radii. Additionally, have students use their protractors to measure some of the different angles that they have drawn on the plates to reinforce using a protractor.



Activity 6: Common Shapes (GLE: 21, 24) Materials List: Common Shapes BLM, paper, pencils, math learning logs Give students a copy of the Common Shapes BLM. Have students trace or draw the shapes in their math learning logs (view literacy strategy descriptions). They should write descriptions of each shape, using appropriate mathematical terms. Give attention to number of sides, relationships between sides (parallel, perpendicular, same length), number of angles, relationships between angles (angles all same measure, acute, right, obtuse), circumference, diameter, and radius. On the board or overhead, post some of the terms you expect students to include in the descriptions. Once completed, students can find a partner to compare and clarify shapes and terms. Activity 7: Polygons (GLE: 24) Materials List: transparency of Common Shapes BLM, markers, Internet access – optional Make an overhead transparency of the Common Shapes BLM. Ask questions such as these: All of these shapes are polygons except one. Which one is not a polygon? (C) Why? (It has no straight sides.) Look at the other figures. Write a definition for polygon. What is a definition for quadrilateral? (a polygon with 4 sides) Which of the figures are quadrilaterals? (B, D, F, H, I, K) What is a definition for trapezoid? (a quadrilateral with only one set of parallel sides) Which of the figures are trapezoids? (B, K) Why is the triangle not a trapezoid? (It does not have 4 sides and it does not have a set of parallel sides.) Continue asking questions about parallelograms, rectangles, squares, rhombuses, etc. Help students see that although the most specific name for a square is square, it has many other names. Discuss regular and non-regular polygons. Play the game, “What’s My Shape?” Think of a shape on the Common Shapes BLM, but do not tell students the shape. Allow students to ask questions to determine the shape that you are thinking of. Some types of questions allowed are the following: Does your shape have 4 sides? Does your shape have any sets of parallel lines? Does your shape have any right angles? As you answer, yes or no, to the questions, cover the shapes that could not be your shape. These shapes are now eliminated. For the question, does your shape have 4 sides? Figures A, C, E, G, J, and L would all be covered because they do not have 4 sides. The website www.nctm.org has a good lesson on what distinguishes a rectangle from a more general parallelogram. Once on the site, go to Lessons and Resources, Elementary, Illuminations, Lessons, Grades 3-5. In the search box, type Geometry, Rectangles and Parallelograms. The lessons are in alphabetical order. This lesson can be accessed directly at http://illuminations.nctm.org/index_d.aspx?id=350.

http://www.nctm.org




Activity 8: Classes of Triangles (GLEs: 15, 21, 24) Materials List: What Kind of Triangle? BLM, paper, pencils, rulers, protractors Give each group of 4 students the What Kind of Triangle? BLM. Assign 2 students in each group triangles A, B, and C. Assign the other 2 students, triangles D, E, and F. Using rulers, have students measure the sides of the triangles to the nearest centimeter. Using protractors, ask students to measure the angles to the nearest degree. Students should label each triangle by the types of sides and by the types of angles. Consider allowing a 2 degree difference in the angle measures, but tell them that the measures of the 3 angles of a triangle must equal 180°. The angles will be easier to measure if students extend the length of the sides. Have students then draw, using rulers and protractors, six different types of triangles. Ask questions such as these: Did anyone draw a right scalene triangle? Is it possible? If not, why not? Did anyone draw an equilateral obtuse triangle? Is it possible? If not, why not? Activity 9: Geoboard Polygons (GLE: 21, 24, 27) Materials List: Dot Paper Geoboards BLM, geoboards – optional, paper, pencils, Internet access – optional Place students into groups of four. Give students geoboards or use the Dot Paper Geoboards BLM. Have each student make, or draw, a quadrilateral. Students in each group should have different figures, either ones that are different sizes or different shapes. Have two students in each group discuss how their figures are alike and how they are different. Try to get them to focus on parallel lines, types of angles, symmetry, etc. Have students make, or draw, other figures such as a shape with four sides and no right angles, a shape with three sides and one obtuse angle, or a shape with six sides. Do this activity one day with 3-sided shapes, the next with 4-sided, and then the third day focus on figures with more than four sides. The National Library of Virtual Manipulatives has geometry lessons using virtual geoboard manipulatives. Go to this site at www.nlvm.usc or directly access this site at http://matti.usu.edu/nlvm/nav/category_g_2_t_3.html. This site contains lessons that use geoboards, attribute blocks, and tangrams. Consider introducing coordinate geometry by having students label each vertex of one of the figures drawn.

http://www.nlvm.usc

http://matti.usu.edu/nlvm/nav/category_g_2_t_3.html



Activity 10: Real-world Geometry (GLE: 24) Materials List: paper, pencils Have students find examples of different plane figures in the classroom, at home, outside, or in a grocery store. Give the students a list of the figures you want them to find. Some figures may be difficult to find. Students may find things such as polygons used as faces of traffic signs, or a rectangle as the face of a windowpane. Have students make sketches of the figures and note the location of each one. If a student says that a cereal box is a type of rectangle, stress that the faces of the box are rectangles, but the box itself is a rectangular prism. Have students, in a group of 4, select a plane geometric figure (other than a rectangle) of their choice. Ask them to demonstrate their understanding of the properties of their chosen figure by completing a RAFT writing (view literacy strategy descriptions) assignment. This form of writing gives students the freedom to project themselves into unique roles and to look at content from unique perspectives. From these roles and perspectives, RAFT writing can be used to describe a point of view, envision a potential job or assignment, or solve a problem. It is the kind of writing that when crafted appropriately, should be creative and informative. R – Role (role of the writer – salesperson) A – Audience (to whom the RAFT is being written – billboard sign makers) F – Form (the form the writing will take – a letter) T – Topic ( the subject of the focused writing – to get billboard sign makers to switch from the standard rectangular billboard sign to the shape chosen by the group. The groups should share their writing with the class. Students should listen for accuracy and logic in each RAFT. Activity 11: Faces and Bases of 3-D Figures (GLE: 24) Materials List: solid figures, paper, pencils, math learning logs Display solid figures such as cubes and other rectangular prisms, cylinders, cones, spheres, and different types of pyramids. Discuss the terms faces, bases, edges, and vertices. Have students choose two of the figures and in their math learning logs (view literacy strategy descriptions), and write a paragraph about how the figures are alike and how they are different. Make sure that students include the types of plane figures that are parts of the 3-D figures. For example, a square pyramid is composed of a square base and 4 triangular faces. Students should share with the class their comments about the different figures.



Activity 12: Geometric Shapes: Transformations (GLE: 25) Materials List: Move the Figures BLM, pencils, paper, Internet access - optional Provide each student with the Move the Figures BLM. Develop the definition of translation by having students trace Figure A, slide it to a new location, and then trace the figure again. Be sure to reinforce the use of correct terminology. Remind students that they may have referred to the movement in earlier grades by calling it a slide, but the correct mathematical term is translation. Repeat the activity with reflection and rotation. Introduce the term transformation indicating that translations, reflections, and rotations are different types of transformations. To reinforce the skill and the terminology, have students trace or draw Figure B on a sheet of paper. Give students a transformation to perform on this shape. Students should draw and label the transformation. Continue the process until students can perform the given transformation upon demand. NCTM, www.nctm.org, has a lesson on transformations. Once on the site, go to Lessons and Resources, Elementary, Illuminations, Lessons, Grades 3-5, Geometry, Paper Quilts. Or access the page directly at http://illuminations.nctm.org/index_o.aspx?id=104. In this lesson, students investigate fractional parts of a whole and use transformational geometry to make 4-part quilt squares. Activity 13: Rotations (GLE: 24, 25) Materials List: Geoboard Rotations BLM, geoboards, pencils, Internet access – optional Give students a geoboard and the Geoboard Rotations BLM. Ask students to make a figure of their choice on a geoboard. This activity is better if the figures are not too complex, but make sure that the figures do not have symmetry. Have students draw their figure on paper geoboard 1 and label it as “original.” Ask students to visualize how the figure would look if it were rotated 90° . Have them rotate the figure 90° clockwise and draw it on one of the three remaining paper geoboards, but do not label it. Ask students to place the geoboard back in the original position, then rotate it 180° and draw this figure in one of the two remaining paper geoboards. Have students put the figure back in its original position and then rotate it either 270° clockwise or 90° degrees counterclockwise. Ask them to draw that figure on the remaining paper geoboard, exchange papers, and determine which rotation is shown by each picture. The National Library of Virtual Manipulatives has geometry lessons using virtual tangram manipulatives. Go to this site at www.nlvm.usc or directly access this site at http://matti.usu.edu/nlvm/nav/vlibrary.html. Using tangrams is one way to help students with transformations. Students must rotate and reflect the pieces to make figures.

http://www.nctm.org

http://illuminations.nctm.org/index_o.aspx?id=104

http://www.nlvm.usc

http://matti.usu.edu/nlvm/nav/vlibrary.html



Activity 14: Rotational Symmetry (GLE: 26) Materials List: Rotational Symmetry BLM, paper, pencils Provide groups of students with 2 copies of the Rotational Symmetry BLM. They should cut out the figures on the second copy. Have students begin with the equilateral triangle. They should place the cut-out triangle on top of the other equilateral triangle. Ask them to tell how many times they could rotate the triangle and have it “match itself,” or fit back into the original triangle. Once the students understand that the triangle can be matched three different times, introduce the term rotational symmetry. If a shape, when rotated less than 360° , still looks the same, the shape has rotational symmetry. Have students continue with the rest of the figures. Figures A, C, E, F, H, and I have rotational symmetry. Extend the activity by having students determine the number of degrees for the rotational symmetries. For example, a square has rotational symmetry at90 , 180 , 270 , and 360° ° ° ° . Activity 15: Symmetry: Line and Rotational (GLE: 26) Materials List: pattern blocks, paper, pencils Use the SQPL (view literacy strategy descriptions) strategy to challenge students to further explore the concept of symmetry. Put the following statement on the board for students. “If a figure does not have line symmetry, it cannot have rotational symmetry.” In groups of 4, have students brainstorm (view literacy strategy descriptions) different questions they might need to answer to determine whether this statement is true or false. Each group should present one question to the class. Give the class time to read each question. Have students discuss which questions would help to prove or disprove this statement. If a shape can be folded on a line so that the two halves match, it has line symmetry. Some shapes can be folded in more than one way, so they have more than one line of symmetry. Give each group of students a set of pattern blocks. Have them determine the number of lines of symmetry for each block. To help students understand rotational symmetry, have them take one of the pattern block shapes and trace around it to form an outline. As they rotate the shape from 0° to 360°, if it can fit into the outline without being flipped over, it is said to have rotational symmetry. The order of symmetry is the number of times it will fit into the outline. This parallelogram ABCD has an order of two.



90° turn 180° turn 270° turn 360° turn does not fit fits does not fit fits again Have students draw outlines for the pattern blocks to help them decide if the shape has rotational symmetry. Note: All of the pattern blocks have rotational symmetry except the trapezoid. At the conclusion of the lesson, draw students’ attention to their SQPL questions and allow them to reflect on which ones were answered. Activity 16: Properties of Geometric Figures (GLEs: 21, 24, 25, 26) Materials List: Properties of Geometric Figures BLM, pencils This activity will help students summarize the important attributes of different types of geometric figures. On the board, or overhead projector, draw a word grid (view literacy strategy descriptions) like the one below. To take full advantage of word grids, they should be co-constructed with students, so as to maximize participation in the learning process. Provide students with a blank word grid, the Properties of Geometric Figures BLM. In the first column, have students help you list the types of geometric figures they have been studying. In the other column headings, have them think of properties of the geometric figure, such as these: has only 3 sides, must have 4 right angles. Students should write A for always, S for sometimes, and N for never. Once the word grid is complete, students should be given time to review the differences and similarities among geometric figures. They can quiz each other over the information in the grid in preparation for tests or other class activities. Sample Word Grid

Properties

Figure has only 3 sides

has at least 1

right angle

has rotational symmetry

all sides congruent

equilateraltriangle A N A A

trapezoid N S N N square N A A A

A = always S = sometimes N = never



Activity 17: Coordinate Grids (GLE: 27) Materials List: masking tape, paper, pencils, The Fly on the Ceiling – optional Using masking tape, make a coordinate grid on the floor or on the playground. Emphasize the first quadrant only. Introduce the terms x-axis and y-axis. Have one student walk four spaces to the right on the x-axis and three space up on the y-axis. Explain that the point where the student is standing can be labeled ( )4,3 . When labeling a point, the x-coordinate should always be listed first. Have other students walk to other points and write the coordinates. Place an object on the grid and ask students to explain how they would walk to get to that point. The book, The Fly on the Ceiling, is a good way to introduce the purpose of the coordinate plane. Activity 18: Plot That Figure! (GLEs: 24, 27) Materials List: Plot that Figure BLM, pencils, Internet access – optional, rulers Provide students with sheets 2 copies of the Plot that Figure BLM. Call out the coordinates of vertices (using whole number coordinates) for various two-dimensional shapes. The students should plot the points, label the points, and connect them in the order presented to create a geometric shape. Ask students to identify the shape they created. Here are some points you may want to use. Grid 1 Point A (1, 2); Point B (6, 2); Point C (5, 4); Point D (2, 4). Connect A to B, B to

C, C to D, and D to A. (trapezoid) Grid 2

Point A (1, 5); Point B (4, 5); Point C (4, 1). Connect A to B, B to C, and C to A. (right scalene triangle) Grid 3

Point A (2, 5); Point B (4, 5); Point C (4, 1); Point D (2, 1). Connect A to B, B to C, C to D, and D to A. (rectangle) Grid 4

Point A (1, 2); Point B (4, 2); Point C (2, 4); Point D (2, 6); Point E (1, 6). Connect A to B, B to C, C to D, D to E, and E to A. (pentagon)

On the 2nd copy of the BLM, have students draw simple figures, name the figures, and state the coordinates of each vertex. The website http://mathforum.org/cgraph has a good lesson called Chameleon Graphing. The lesson provides an introduction to graphing in a coordinate plane.

http://mathforum.org/cgraph







this: Choose 3 types of triangles and use a ruler and protractor to accurately draw an example of each kind of triangle chosen.

o One problem that requires the student to explain his/her reasoning such as this: Can a triangle be both right and equilateral?

o One question involving real-life such as this: Describe places at home where a rectangle, a square, a circle, and a triangle could be found.

• Journal entries could include the following: o Explain how rectangles and trapezoids are alike and how they are

different. o Explain how to plot the point ( )2, 4 on a coordinate grid. o Write an autobiography/self-description from a shape’s point of view.

Read the description to a partner and see if the partner can guess the shape.


• Activity 4: Provide students with protractors and the Angle Measures BLM. Have students measure the angles to the nearest degree and classify the angles.

• Activity 6: Provide students with the Which Shape Does not Belong? BLM. Have

students decide which shape does not belong and explain why. A. B. C. D.

• Activity 7: Draw a rectangle on the board. Have students use geometric terms to describe 3 properties of the figure and also write all of the names of the figure.



• Activity 12: Give students the Grid Paper BLM. Have students draw the figure below and label it as “original figure.”

Have students draw two translations, two rotations, and two reflections of the original figure. Each drawing should be labeled.


Grade 5 Mathematics Unit 6 Measurement 65

Grade 5 Mathematics

Unit 6: Measurement Time Frame: Approximately five weeks Unit Description The focus of this unit is on measurement in both the U.S. and metric systems. Student Understandings Students understand the measurable attributes of objects and can apply the appropriate techniques and tools to determine and estimate measurement. They can make conversions of units within the same system for U.S. and metric measurements and make comparisons between the two systems. Guiding Questions

1. Can students recognize, select appropriate tools and units, and make and interpret measures for contexts involving length, weight/mass, capacity, temperature, and time?

2. Can students convert between units of length, weight/mass, capacity, and time measurements within the same systems for U.S. and metric system measurements?

3. Can students compare measurements between U.S. and metric systems? 4. Can students estimate measurements?

Unit 6 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 8. Use the whole number system (e.g., computational fluency, place value, etc.) to

solve problems in real-life and other content areas (N-5-M) 11. Explain concepts of ratios and equivalent ratios using models and pictures in

life problems (e.g., understand that 2/3 means 2 divided by 3)(N-8-M) (N-5-M) Measurement 15. Model, measure, and use the names of all common units in the U.S. and metric

systems (M-1-M)



GLE # GLE Text and Benchmarks 16. Apply the concepts of elapsed time in real-life situations and calculate

equivalent times across time zones in real-life problems (M-1-M) (M-6-M) 17. Distinguish among the processes of counting, calculating, and measuring and

determine which is the most appropriate strategy for a given situation (M-2-M) 18. Estimate time, temperature, weight/mass, and length in familiar situations and

explain the reasonableness of answers (M-2-M) 19. Compare the relative sizes of common units for time, temperature, weight,

mass, and length in real-life situations (M-2-M) (M-4-M) 20. Identify appropriate tools and units with which to measure time, mass, weight,

temperature, and length (M-3-M) 22. Compare and estimate measurements between the U.S. and metric systems in

terms of common reference points (e.g., l vs. qt., m vs. yd.) (M-4-M) 23. Convert between units of measurement for length, weight, and time, in U.S.

and metric, within the same system (M-5-M)

Sample Activities

Activity 1: Units of Measure (GLEs: 15, 17, 20) Materials List: Objects in the classroom Before the class begins, have a list of objects to be measured, counted, or computed written on the board or on an overhead transparency. Include examples such as classroom temperature, outdoor temperature, number of math books in the room, number of books in the library, teachers in the school, length of the school building, distance to the buses, number of students in each class, square yards of carpet in a classroom, number of water fountains, weight of a box of books or a student desk, number of soft drinks needed for a class party, and the size of the playground. Have students discuss each example and decide whether they should measure, count, or compute to find the information. For example, to find the number of math books in the room, they should count. To find the distance to the buses, they would measure. To find the square yards of carpet in a classroom, they could measure and then calculate. Have students name other objects that could be measured, counted, or computed. The activity could be extended by asking what tools you would use for the objects to be measured, and by asking for the correct units of measurement.



Activity 2: Measurement Vocabulary (GLEs: 15, 17, 20) Materials List: Measurement Vocabulary BLM, pencils Have students work in groups to answer the following questions. Why is it necessary to measure? What questions are answered by measuring? When is measuring done rather than counting? Help students to see that measuring determines how tall, how wide, how heavy an object is. Questions are answered such as these: How cold is the temperature outside today? How fast did Michael Phelps swim in the 200-meter Individual Medley at the 2004 Olympics? Help students to understand that attributes such as time, temperature, length, and capacity are measured rather than counted. This activity will help students think about the tools and units that are used to measure certain attributes. Ask students to think of other attributes that can be measured. The attributes should include length, capacity, weight/mass, time, temperature, area, and volume (this is a minimal list; they may have more). Length, or linear measurement, includes height, width, distance, circumference, perimeter, etc. On the board, or overhead projector, draw word grids (view literacy strategy descriptions) like the ones below. Provide students with a copy of the Measurement Vocabulary BLM. The top grid will involve tools, and the bottom grid will involve units.

Tools for Measuring Attribute ruler scale thermometer length temperature Y = yes N = no

Units Used for Measurement Attribute inch centimeter pound hour length temperature Y = yes N = no In the 1st column on the top grid, have students list the attributes they want to consider. In the other column headings, have them list different tools used to measure. If the tool is used to measure an attribute, students should write Y for yes. If not, they should write N for no. In the 1st column on the bottom grid, have students list the attributes again. In the other column headings, have them list some of the measurement units that they know. If the unit is used for measuring an attribute, students should write Y for yes. If not, they should write N for no. Do not worry if they do not list all of the tools and units. Other activities will address both.



Once the word grid is complete, students should be given time to review which tools and units are used to measure certain attributes. They can quiz each other over the information in the grid in preparation for tests or other class activities. Activity 3: Linear Units (GLEs: 15, 18, 19, 20, 22) Materials List: Linear Units BLM, scissors, index cards, pencils, card stock Give students a copy of the Linear Units BLM run on cardstock. There are blank cards in case other units are studied. Have them cut out and shuffle the cards. Students should sort the cards into U.S. and metric units. For each unit, have students discuss benchmarks that can help them remember approximately how large the units are. For example, a millimeter is about the width of an eyelash. Have students place the U.S. unit cards in order from smallest to largest. To the side of these cards, have them order the metric unit cards. Ask the students to see if there are units in each system that are about the same size or that would be used to measure the same object. For example, a meter is a little longer than a yard. An inch is about 1

22 times the length of a centimeter, but both could be used to measure the length of a desk. Ask students what they could use to measure the length of very small items in the U.S. system. Students should think about 2

1 , 41 , and 1

8 of an inch. Have students match the unit cards with the abbreviation cards. To develop students’ knowledge of key vocabulary, have them create vocabulary cards (view literacy strategy descriptions) for terms related to linear measurement. Distribute 3 x 5 or 5 x 7 inch index cards to each student, and ask them to follow directions in creating the sample card. On the board, place a targeted unit in the middle of the card, as in the example below. Ask students to list what attributes the unit is used to measure, list a benchmark that would help them remember the size of the unit, write its abbreviation, and list what tools could be used to measure using that unit. Allow time for students to review their cards and quiz a partner on the terms to hold them accountable for accurate information on the cards.

Attribute Measured: length

Benchmark: The length of a paper clip is about 1 inch.

Abbreviation: in.

Tools: rulers, measuring tapes

Inch



Activity 4: U.S. and Metric System Lengths (GLEs: 15, 18, 19, 20, 22) Materials List: Measuring Length BLM, math learning logs, pencils, linear measuring devices such as rulers, measuring tapes, yardsticks, or meter sticks Provide students with the Measuring Length BLM. Working in groups, ask students to estimate the length of common objects in the classroom. Examples are doors (height and width), desktops, math books, folders, paper, and unsharpened pencils. The selection of materials to be measured should include some objects that provide common reference points between U.S. and metric measures (i.e., the width of a door is about one yard or one meter). Spend time discussing the reasonableness of estimates before having students actually measure the objects. Students must decide which unit and which tool they should use to measure each object. After measuring, have students discuss their findings. Ask questions such as these: What unit did you use to measure the height of the door? Would an answer of 25 feet be a reasonable estimate of the height of the door? In their math learning logs (view literacy strategy descriptions), have students choose one object that they measured and explain why they chose a particular tool and unit to use to measure the object. Activity 5: Measuring the Distance Around a Shape (GLEs: 15, 18) Materials List: Distance Around a Shape BLM, pencils, linear measuring devices, Internet access – optional Since Activity 4 involves linear measurements, this is a good time to introduce perimeter. Provide students in groups with the Distance Around a Shape BLM, rulers, and measuring tapes. Have two students use metric tools and two students use U.S. tools. Ask students the purpose of finding the distance around an object. Have each group decide on four objects to measure the distance around. Explain that they are to find the perimeter of a face of the object. If they choose an object with a circular face, they are finding the circumference. Have students estimate the perimeter before measuring to the nearest inch or centimeter, depending on the tool used. Spend time discussing the reasonableness of students’ estimates. To emphasize the relationship between US and metric units, ask questions such as this: Did it take more centimeters or inches in finding the perimeter of your object? The NCTM website www.nctm.org has a lesson on perimeter and area called Junior Architects. Go to the website, click Lessons and Resources, Elementary, Illuminations, Lessons, and in the search box, type Junior Architects or by going directly to http://illuminations.nctm.org/LessonDetail.aspx?id=U172. Students design a clubhouse using linear measurement, perimeter, and area. Another good lesson on measurement and perimeter may be accessed directly by typing: http://www2.edc.org/mathpartners/pdfs/3-5%20Geometry%20and%20Measurement.pdf. Go to lesson 4, page 20 to view the

http://www.nctm.org

http://illuminations.nctm.org/LessonDetail.aspx?id=U172

http://www2.edc.org/mathpartners/pdfs/3-5%20Geometry%20and%20Measurement.pdf

http://www2.edc.org/mathpartners/pdfs/3-5%20Geometry%20and%20Measurement.pdf



lesson. In this lesson, students estimate, measure, and compare perimeters of different objects. Activity 6: Units of Time (GLEs: 15, 18, 19, 20) Materials List: Units of Time BLM, scissors, index cards, pencils, card stock, clocks with movable hands Give students a copy of the Units of Time BLM run on cardstock. There are blank cards in case other units are studied. Have students cut out and shuffle the cards. For each unit, have students brainstorm (view literacy strategy descriptions) activities they estimate would take approximately that long. For example, their favorite TV show might last 1 hour. Have students place the unit cards in order, from least amount of time to greatest amount of time. Have students match the unit cards to the abbreviation cards. Discuss the tools that are used to measure time. Give students different times to model on an analog clock. Relate times shown on analog clocks to digital displays. Activity 7: Elapsed Time (GLEs: 15, 16, 18) Materials List: paper, pencils, clocks with movable hands Using an actual class schedule, model problems of elapsed time. If you do not have clocks with moveable hands for each pair of students, model with one or two clocks. Examples: If it’s 9:30 A.M. and lunch is at 11:20 A.M., how much time will pass between the two times? If it is 12:30 and school ends at 3:15, how much time is left in the school day? Using the clock, show the hands moving while counting out loud. “It is 1 hour from 12:30 to 1:30, two hours to 2:30, 30 minutes to 3:00, and 15 more minutes to 3:15. This is one hour, two hours, thirty, forty-five minutes. It is two hours and forty-five minutes until the school day ends.” (Have students estimate times, such as it’s about three hours until the school day ends.) Ask questions such as this: Is 1 hour a reasonable amount of time remaining in the school day? Using the board or overhead, give various times so students can find the amounts of time elapsed. Consider showing elapsed time on an open number line. To show elapsed time from 12:30 to 3:15, use a number line similar to this and say, 30 min 2 hours 15 min 12:30 1:00 3:00 3:15 “from 12:30 to 1:00 is 30 minutes, from 1:00 to 3:00 is 2 hours, and from 3:00 to 3:15 is 15 minutes. The school day ends in 2 hours and 45 minutes.” Students will not have access to clocks with movable hands for testing, and they need to be shown other strategies.



Activity 8: How Long Does It Take? (GLEs: 15, 16, 18) Materials List: paper, pencils, clocks with movable hands Have students work in pairs. Each student will need a clock face with movable hands. (If these are not available, have at least 2 clocks for demonstration.) The night before have students time an activity, such as watching a movie or playing a computer game, to the nearest minute. Ask students to record the type of activity, the starting time, and the ending time. Have one student share his/her information with the rest of the class. Instruct the first student in each pair to show the starting time and the second student to show the ending time. Have the students estimate the amount of time it took to do the activity. Ask the students to discuss how they could move the hands on the first clock to find the amount of time that passed, or elapsed, to get the second time. Continue using different activities and times. Once students are comfortable with finding the elapsed time, give students a starting time, and the amount of time it would take to do the activity. Ask them to find the ending time and to discuss how they found it. Re-introduce the number line model from Activity 7. Use math story chains (view literacy strategy descriptions) to practice elapsed time. The first student writes the opening sentence of the problem. The party started at 4:45 P.M. The student passes the paper to the student sitting to the right. That student writes the next sentence. The party was over at 6:15 P.M. The paper is passed again to the right. This student writes the question for the story. How long was the party? The paper is now passed to the fourth student who must solve the problem, write the answer in a complete sentence, and discuss how the answer was found. Answer: The party was 1 hour 30 minutes long. Students in the story chain group should talk about the accuracy of the answer and the logic of the story problem. If necessary, revisions to the story chain should be made. Activity 9: Planning a Trip (GLEs: 8, 16) Materials List: paper, pencils, Internet access Making schedules are a good way to help students practice elapsed time. Go to www.nctm.org, click Lessons and Resources, Elementary, Illuminations, Lessons, and in the search box, type Planning a Trip. In this activity, have teams of students plan trips making a travel schedule and a budget for their trip. The activity involves elapsed time. Making the budget will provide a review of operations on whole numbers.

http://www.nctm.org



Activity 10: Time Zone Math (GLE: 8, 16) Materials List: paper, pencils, Internet access – optional Using a time zone map, review with students the need for the different time zones across the United States. Most 5th grade textbooks have time zone maps in them. If not, you can get maps online at sites such as, www.time.gov, www.WorldAtlas.com, www.worldtimezone.com. Ask questions such as this: If the sun is rising at 6 A.M. on the East coast, what time is it on the West coast? Have students brainstorm (view literacy strategy descriptions) to find examples of events that will have different starting times because of the location of the event. For example, if an awards ceremony is scheduled in New York for 8 P.M., what time will it be in California? A football game that is broadcast at noon in the Central time zone actually begins at 1 P.M. in the Eastern time zone. When students grasp this concept, have them work simple problems involving duration of and arrival time for flights. For example, if it takes three hours to fly from New York to New Orleans, what time will the plane arrive if it left New York at 2 P.M.? Have students create simple word problems across time zones. Ask volunteers to share their problems with the class and have them checked for accuracy. An extension of this activity would be to use airlines’ flight schedules. Have students determine the number of flying hours/minutes from city to city using times within the same time zone and between different time zones. Web sites, such as www.orbitz.com, www.expedia.com, or any of the major airlines, have online flight schedules and information. Activity 11: Temperature (GLEs: 15, 18, 22) Materials List: Temperature BLM, pencils, thermometers, Internet access – optional To demonstrate different temperatures on both Fahrenheit and Celsius thermometers, measure the temperature of a cup of ice water, a cup of water at room temperature, and a cup of hot water. Record the temperatures in a chart such as this one drawn on the board or on an overhead transparency.

Cold Warm Hot Ice Water Water at Room Temperature. Hot Water

______°F ______°C ______°F ______°C ______°F ______°C Have students either measure other temperatures or look for temperature measurements in books, newspapers, on the Internet at sites such as, www.accuweather.com, or a local television site, such as, www.wwltv.com. Provide students with the Temperature BLM. In the proper column, cold, warm, or hot, have students record what was measured and the temperatures in both °F and °C. For warm temperatures, you might want to say “medium” or “comfortable.” Ask questions to compare the temperatures in the two systems.

http://www.time.gov

http://www.WorldAtlas.com

http://www.worldtimezone.com

http://www.orbitz.com

http://www.expedia.com

http://www.accuweather.com

http://www.wwltv.com



Activity 12: Weight/Mass Units (GLEs: 15, 18, 19, 20, 22) Materials List: Weight/Mass Units BLM, scissors, index cards, pencils, card stock Give students a copy of the Weight/Mass Units BLM run on cardstock. There are blank cards in case other units are studied. Have them cut out and shuffle the cards. Students should sort the cards into metric and U.S. units. For each unit, have students discuss benchmarks that can help them remember approximately how large the units are. For example, a slice of bread weighs about an ounce, while a loaf of bread weighs about a pound. Have students order the U.S. unit cards from smallest to largest. On the side of these cards, have students order the metric cards. Ask the students to see if there are units in each system that are about the same size or that would be used to measure the same object. For example, a kilogram is about 2 pounds. Both units could be used to weigh a puppy. Have students match the unit cards with the abbreviation cards. To develop students’ knowledge of key vocabulary, have them create vocabulary cards (view literacy strategy descriptions) for terms related to weight/mass measurement. Distribute 3 x 5 or 5 x 7 inch index cards to each student, and ask them to follow your directions in creating the sample card. On the board, place a targeted unit in the middle of the card, as in the example below. Ask students to list what attributes the unit is used to measure, list a benchmark that would help them remember the size of the unit, write its abbreviation, and list what tools could be used to measure using the unit. Allow time for students to review their cards and quiz a partner on the terms to hold them accountable for accurate information on the cards.

Attribute Measured: weight

Benchmark: A slice of bread weighs about 1 ounce.

Abbreviation: oz

Tools: scales, balances

Ounce



Activity 13: How Heavy Is It? (GLEs: 15, 18) Materials List: Measuring Weight/Mass BLM, U.S. and metric scales, objects to weigh, paper, pencils, Internet access – optional Have a variety of objects available for students to weigh on both U.S. and metric scales. Make sure that you have one object that weighs about 1 pound and one that has a mass of about 1 kilogram. Distribute the Measuring Weight/Mass BLM. Have students try to find the objects that are approximately 1 pound or 1 kilogram first. Then have students compare each of the other objects to the 1-pound (or 1 kilogram) object to decide if it weighs more or less than 1 pound (or 1 kilogram). Have them estimate the weight of the object and then actually weigh it. The NCTM web site, www.nctm.org, has a lesson on exploring a balance. Students make and use a 2-pan balance and compare the masses of two objects. Once on the website, click Lessons and Resources, Elementary, Illuminations, Lessons, and in the search box, type Exploration of a Balance, or access the page directly by typing http://illuminations.nctm.org/index_o.aspx?id=76. Activity 14: Capacity Units (GLEs: 15, 18, 19, 20, 22) Materials List: Capacity Units BLM, scissors, index cards, pencils, card stock Give students a copy of the Capacity Units BLM run on cardstock. There are blank cards in case other units are studied. Have them cut out and shuffle the cards. Have students sort them into metric and U.S. units. For each unit, have students discuss benchmarks that can help them remember approximately how large the units are. For example, sodas come in 2 liter bottles. Have students order the U.S. unit cards from smallest to largest. On the side of these cards, have students order the metric cards. Ask the students to see if there are units in each system that are about the same size or that would be used to measure the same object. For example, a quart is about the same as a liter. Both units could be used to measure the amount of water in a bathtub. Have students match the unit cards with the abbreviation cards. To develop students’ knowledge of key vocabulary, have them create vocabulary cards (view literacy strategy descriptions) for terms related to capacity measurement. Distribute 3 x 5 or 5 x 7 inch index cards to each student, and ask them to follow your directions in creating the sample card. On the board, place a targeted unit in the middle of the card, as in the example. Ask students to list what attributes the unit is used to measure, list a benchmark that would help them remember the size of the unit, write its abbreviation, and list what tools could be used to measure using the unit. Allow time for students to review their cards and quiz a partner on the terms to hold them accountable for accurate information on the cards.

http://www.nctm.org

http://illuminations.nctm.org/index_o.aspx?id=76



Activity 15: Container Math (GLEs: 15, 19, 22, 23) Materials List: paper, pencils, measuring items, cereal or rice, Internet access – optional Have students work in groups. Use cereal or rice, measuring cups, milk cartons, or zip lock bags to help students discover 2 cups = 1 pint, 2 pints = 1 quart, 4 quarts = 1 gallon, 1000 milliliters = 1 liter. The zip lock bags can be purchased in pint, quart, and gallon sizes. Have students compare different containers and decide which would hold more—that is, which has the greatest capacity. Ask students to investigate which U.S. measure is closest, in terms of capacity, to a liter. The NCTM web site, www.nctm.org, has a lesson on capacity measurement. This lesson compares the amount of water that each student uses in daily life to the amount of water allowed on a space shuttle. Once on the site, click Lessons and Resources, Elementary, Illuminations, Lessons, and in the search box, type Water, Water. This site can also be found by entering http://illuminations.nctm.org/index_d.aspx?id=289. Activity 16: Estimate! (GLEs: 15, 18, 22) Materials List: paper, pencils, measuring tools Provide students with estimation challenges each day. Have students estimate the temperature (indoor and outdoor) in both Fahrenheit and Celsius and estimate the time certain activities will take. Provide opportunities for students to estimate the weight/mass, capacity, and length of common classroom objects in both systems. In each instance, ask students to explain the soundness of their estimates by making reference to a known benchmark. Measure some items in the classroom, but do not let students see what objects you measured. Give students a measurement which involves decimals, such as 18.64 meters. Have them round this measurement to the nearest whole meter and see if they can guess what object you measured.

Attribute Measured: capacity

Benchmark: A quart holds about the same amount as a liter.

Abbreviation: L or l

Tools: measuring cups or containers

liter

http://www.nctm.org




Activity 17: Measurements in Our Homes (GLEs: 15, 18, 20, 22) Materials List: math learning logs, pencils, real-world objects Each week, bring in real-world objects from home, such as a can of soup, a bottle of lotion, a can of beans, or a bottle of vinegar. For one of the objects discussed that day, in their math learning logs (view literacy strategy descriptions), have students write what attribute they think was measured, what tool they think was used to get the measurement, what unit was used, and to give an estimate of the measurement. This is a good time to discuss some of the metric/U.S. comparisons. Some of the items that you bring will not have both metric and U.S. measurements. Activity 18: Getting Ready for Conversions (GLE: 8, 23) Materials List: paper, pencils, calculator for each pair of students, overhead calculator – optional When students make conversions in the metric system, they will need to multiply and divide by 10, 100, and 1000. Have students work in groups and provide at least one calculator per two students. If available, model the problems on an overhead calculator. Have students multiply any three numbers by 10. Have students call out one of their problems, such as 23 × 10 = 230. After about 5 examples, see if students can find a pattern to multiplying by 10. The place value of each digit in the number shifts one place to the left. If 245 is multiplied by 10, the digit 2 shifts from the hundreds place to the thousands place, the digit 4 shifts from the tens place to the hundreds place, and the digit 5 shifts from the ones place to the tens place. The new digit in the one’s place is zero. There is one zero in 10, so multiplying by 10 shifts all of the digits one place to the left. The new number becomes 2,450. Do the same activity multiplying by 100 and then by 1000 and dividing by 10, then 100, and then 1,000. Activity 19: Conversions (GLEs: 8, 15, 22, 23) Materials List: Conversions BLM, pencils, linear measuring tools, calculator, 2 strips of paper – one 1 in. long and the other 1 ft long One of the problems that students have with conversions is not knowing when to multiply or when to divide. Hold up a strip of paper that is 1 in. long and a strip that is 1 ft. long. Ask: If you were to measure the length of the room, would it take more “inch” strips or more “foot” strips? (inch strips) Why? (Inches are smaller units, so we need more.) Students need to understand that the smaller the unit, the more of that unit they will need. Since inches are smaller than feet, more inches than feet will be needed when measuring an object. To change from a smaller unit to a larger unit, students should divide so that they get fewer of the larger unit.



Have each group of students choose a linear object to measure. Distribute the Conversions BLM. Make sure that the objects selected are longer than 1 yard. Have them measure the object to the nearest inch, foot, and yard. Make a chart such as this one on the board, and have students record their measurements on the Conversions BLM.

Object Measured Inches Feet Yards

As you list the objects measured and the measurements, ask students questions such as these: Why do we have larger numbers in the Inches column than in the Feet column? When you measured your object, did you get more feet or more yards? Why do you think this happened? Remind students that the actual measurements are approximate. See if they can notice any patterns in the tables. Have students use a calculator to divide the number of inches by 12. This number should be close to the number of feet. Or have them multiply the number of feet by 12 to get the number of inches. Have them divide the number of feet by 3. This number should be close to the number of yards. You want students to begin to see that 1 foot = 12 inches, and that 1 yard = 3 feet. Repeat the activity using millimeters, centimeters, and meters. You want students to see that 10 mm = 1 cm, 100 cm = 1 m and 1000 mm = 1 m. Ask the same types of questions as above. Compare the number of yards and meters for each object, and the number of inches and centimeters for each object. Activity 20: Tables of Conversions (GLEs: 8, 11, 15, 22, 23) Materials List: Tables of Conversions BLM, pencils Provide students with the Tables of Conversions BLM. Focus on the conversions for one attribute each day. For example, for the attribute time, begin by asking if we measure the school year in days or months, would we have a larger number of days or a larger number of months? (days) Why? (Days are smaller units than months.) Give real-world problems such as this one. If Steve practiced football 6 days this week for 50 minutes a day, or 300 minutes, how many hours did he practice this week? Have students think equivalent ratios or make a table. 60 min = 1 hr 120 min = 2 hrs 240 min = 4 hrs 300 min = 5 hrs Students may go from thinking 60 min = 1 hr to 120 min = 2 hrs or they may think that there are five 60’s in 300, so that would be 5 hours. Using equivalent ratios will work even if the numbers do not divide evenly. If the time were 250 minutes, how many hours would that be?



60 min = 1 hr. 120 min = 2 hrs 240 min = 4 hrs, but we have 250 minutes. So the numbers of hours is 4 hrs and 10 minutes left over. Students could also set up equivalent ratios like this. For the question, how many inches are in 6 feet, students could write the following: 1 ft = 12 in. 6 ft ? in. What number would replace the box? (72) Why? (12/72 is equivalent to 1/6) Continue asking students to convert from one unit to another. Using ratios or making a table helps with the decision to multiply or divide. Activity 21: If You Made a Million (GLEs: 8, 15, 18, 23) Materials List: paper, pencils, If You Made a Million, pennies, rulers Read the book If You Made a Million to students. There are wonderful opportunities for measurement in it. After reading the book, provide students with an SQPL (view literacy strategy descriptions) activity by posting this statement from the book on the board or overhead projector: “The book states that ‘You can have a five-foot stack of pennies (that’s one thousand of them…).’ Therefore, a five-foot stack of pennies would equal 1000 pennies.” This strategy is used to encourage students to generate questions they would need to verify the statement. Working in groups of four, have students brainstorm (view literacy strategy descriptions) 2 or 3 different questions. Students might ask questions such as: How many pennies are in one inch? (Students can measure this.) How many inches are in one foot? Have groups present one of their questions to the class. Reread the statement. Have students answer the questions and decide if the statement is true. The book is also a great way to review whole numbers.

Sample Assessments

General Assessments




the following: Draw a picture of a house without using a ruler. The house



should be a rectangle 15 cm by 18 cm. The roof should be in the shape of an isosceles triangle with a height of 5 cm. The door should be a rectangle 2 cm by 4 cm. There should be two windows, each one a square with a side length of 30 mm.

o One problem that requires the student to explain his/her reasoning when answering the following question: When changing from one unit of measurement to another unit, explain whether you should multiply or divide. Give an example to support your statement.

o One problem involving real-life such as: Choose a food product or some other object from home and in your math learning log discuss what attributes could be measured, what tools would be used to measure the attribute, and what unit, in both the U.S. and metric systems, would be used.

• Journal entries could include the following: o Explain the process used to change 4 yards into inches. o Explain your reasoning: Two containers can hold the same amount. Do

they have to be the same shape? o Explain your reasoning: Joanie talked on the phone for 1 hour. Ted talked

for 65 minutes and Byron talked for 355 seconds. Who talked the longest? How do you know?


• Activity 1: Have students give an example of a situation where counting is used to find an amount and an example of a situation where measuring is used to find an amount.

• Activity 2: Call out the name of a tool or unit and have students identify the

attribute that would be measured. If pictures of tools can be found, give students copies and have them decide which attribute would be measured by each tool.

• Activity 5: Measure the distance around a sheet of loose-leaf paper to the nearest

inch and the nearest centimeter. • Activity 7: Give students a starting time and an ending time for an activity, and

have them find the elapsed time. Then give students a starting time, the elapsed time and have them find the ending time. Use real-world events to keep students interested.

• Activity 11: Have students give you an estimated temperature, in both Fahrenheit

and Celsius, for the following: a cold winter day, a comfortable room temperature, and a hot day at the beach.



• Activity 12: Give students a list of the weight/mass units and have them explain how they remembered about how big the units were, whether a benchmark was used, and if so, what the benchmark was.

• Activity 21: The last statement in the book, If You Made a Million, says, “So

what would you do if you made a million?” Have students write a short paper on what they would do if they made a million. They should estimate the cost of each of the things they would buy.


Grade 5 Mathematics Unit 7 Addition and Subtraction of Fractions 81

Grade 5 Mathematics

Unit 7: Addition and Subtraction of Fractions Time Frame: Approximately three weeks Unit Description This unit focuses on adding and subtracting fractions with common denominators and on writing equations to model problems involving fractions. Student Understandings Students will use addition and subtraction of fractions to solve real life problems and will determine if their answers make sense and are reasonable. Guiding Questions

1. Can students add or subtract two fractions with a common denominator? 2. Can students check to see if two different answers for a fraction operation

problem are equivalent? 3. Can students work with equivalent fractions to see if they represent the same

amount? 4. Can students write fraction equations and inequalities then represent the

answer to them on a number line? Unit 7 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 2. Recognize, explain, and compute equivalent fractions for common fractions

(N-1-M) (N-3-M) 3. Add and subtract fractions with common denominators and use mental math to

determine whether the answer is reasonable (N-2-M) 4. Compare positive fractions using number sense, symbols (i.e., <, =, >), and

number lines (N-2-M) 5. Read, explain, and write a numerical representation for positive improper

fractions, mixed numbers, and decimals from a pictorial representation and vice versa (N-3-M)



6. Select and discuss the correct operation for a given problem involving positive

fractions using appropriate language such as sum, difference, numerator, and denominator (N-4-M) (N-5-M)

9. Use mental math and estimation strategies to predict the results of computations (i.e., whole numbers, addition and subtraction of fractions) and to test the reasonableness of solutions (N-6-M) (N-2-M)

Algebra 13. Write a number sentence from a given physical model of an equation (e.g.

balance scale) (A-2-M) (A-3-M) 14. Find solutions to one-step inequalities and identify positive solutions on a

number line (A-2-M) (A-3-M) Patterns, Relations, and Functions 33. Fill in missing elements in sequences of designs, number patterns, positioned


Sample Activities Activity 1: All About Fractions (GLEs: 2, 3, 4, 6) Materials List: paper, pencils Assign each group of students a different fraction. The groups should list some things they know about their fraction. To review fraction concepts from Unit 4, use the “professor-know-it-all” strategy (view literacy strategy descriptions). A group will be called on randomly to come to the front of the class to be a team of “professor-know-it-alls” about their fraction. Invite questions from other groups. Some questions might be the following for 6

3 : What is the numerator of your fraction? (3) Is your fraction greater or less than 5

3 ? (less than) What is a fraction equivalent to yours? ( 21 , 12

6 , etc.) Would you draw a picture of your fraction? (Answers will vary.) What is your fraction written as a decimal? (0.5) The team should be able to answer questions involving equivalency, simplest terms, comparison, and different representations. Other students should listen for accuracy and logic in the professor-know-it-alls’ answers to their questions. After about 5 minutes or so, ask a new group to take its place in front of the class. Consider doing this activity again with mixed numbers, improper fractions, and decimals. Activity 2: Make That Fraction (GLEs 2, 4, 5) new Materials List: Make That Fraction BLM, two number cubes labeled 1-6 for each pair of students, pencils, Opinions About Fractions and Decimals BLM Distribute the Make That Fraction BLM to students. Students should work in pairs, but each student needs a copy of the BLM. The first player rolls the two number cubes and uses the number to create one fraction to fit one of the descriptions. For example, if a



student rolls 3 and 5, the student could make 53 for a fraction > 2

1 , or 35 for a fraction > 1.

Players take turns. If a player cannot make a fraction to match a description, the player loses a turn. The first player to complete the entire sheet is the winner. After playing the game, present students with an opinionnaire (view literacy strategy descriptions) to challenge their understanding of fractions and decimals. Opinionnaires force students to take a stand and to defend their reasoning. The emphasis is on the students’ points of view, not the correctness of their opinions. Give each student a copy of the Opinions on Fractions and Decimals BLM. Invite students to share their opinions and discuss the different viewpoints. Have them return to the opinionnaire and adjust their answers and reasoning as needed. By taking a stand on the questions and engaging in discussions about their stands, students make connections from their opinions and ideas to those of their classmates. Activity 3: Fraction Strips (GLEs: 3, 5) Materials List: Fraction Strips A-F BLMs, colored stock paper, scissors, paper, pencils, Internet access-optional Run copies of the Fraction Strips A-F BLMs. Run each on a different color paper. Provide each student with one of the strips in each color. One strip should be left as the whole unit, while the other strips should be cut into halves, fourths, eighths, thirds, and sixths. Give students problems to model such as these: Cheri ate 4

1 of the Power Bar at breakfast and 4

2 of the Power Bar at lunch. How much of the Power Bar did she eat? ( 1 2

4 4+ = 43 ). Charley had 8

3 of a Power Bar in his pocket. He ate 81 of it. How much of the

Power Bar is left? ( 3 18 8− = 2

8 ) When an answer, such as 28 , can be written in a simpler

form, have students use the other fraction pieces to find the simplest form. Emphasize that both answers are correct because they are equivalent fractions. Give problems that involve the whole unit, such as 3 5

8 8+ or 161− . To show fractions greater than 1, have two

students work together. For fractions greater than 1, emphasize writing both mixed numbers and improper fractions. The National Library of Virtual Manipulatives, www.nlvm.usu.edu, has lessons using fraction manipulatives. From the home page, click on Virtual Library, then click in the box for Numbers and Operations, and Grades 3-5. Scroll down to the different fraction activities. As an example, in one of the fraction pieces activities, students make “1” using different sized pieces. These lessons may also be directly accessed by typing the following and selecting a fraction activity: http://nlvm.usu.edu/en/nav/frames_asid_274_g_2_t_1.html?open=activities. Although the GLE calls for adding and subtracting fractions with like denominators only, because students have worked with equivalent fractions, unlike denominators could be introduced.

http://www.nlvm.usu.edu

http://nlvm.usu.edu/en/nav/frames_asid_274_g_2_t_1.html?open=activities



Activity 4: Cover Up/Uncover (GLEs: 3, 13) Materials List: fraction strips from Activity 3 or fraction circles, Fraction Spinners BLM, paper clip, paper, pencils Using the paper strips from Activity 3, have students play the game Cover Up. (This could be done with fraction circles, rather than the strips.) Provide each pair of students with the Fraction Spinners BLM. A paper clip and pencil can be used as a spinner. Have students choose one of the spinners and the corresponding fraction parts, either halves, fourths, and eighths or halves, thirds, and sixths. Have each student lay out the “whole” strip. The first student spins and covers that amount on his/her “whole” strip. Students take turns spinning until they cover the entire strip. Players lose a turn if they cannot cover the strip exactly. To play Uncover, students cover the entire “whole” strip with the fraction pieces, spin, and uncover until the entire strip is uncovered. For each game, have the students write the number sentence, or equation, for each spin. For example, if they have 1

8 covered and spin 38 , the number sentence would be 31 4

8 8 8+ = . Students could use other equivalent pieces to cover their strip. For example, if students spin 8

2 , they could use the 4

1 piece to cover. The focus is more on equivalent fractions than operations with fractions with unlike denominators. Activity 5: Concentrate! Sums and Differences (GLEs: 3, 9) Materials List: Concentration Cards BLM, scissors, fraction strips from Activity 3 or fraction circles Give students sets of the cards from the Concentration Cards BLM. Working in pairs or groups of fours, have the students play Concentration with the cards. Have students take turns flipping one card over from each set, with the goal of matching the problem with the correct answer. Have students practice using mental math to determine whether the answer is reasonable as the game is played. Sometimes students will have to find an answer that is in simplest form. The student or team with the most correct matches wins. If there is a disagreement, have students model the problem with the strips or circles. Activity 6: Operations on a Number Line (GLEs: 2, 3, 5, 9) Materials List: Number Lines BLM, pencils Give students the Number Lines BLM. Have students label each mark on the number lines with all of the names for it. For example, on line A, for 1

2 , also label 42 . For 1, also

label 2 42 4, and for 2, also label 4

2 and 84 . For 1

41 , also label 54 ; for 3

41 , also label 74 ; and

for 121 , also label 2

41 , 32 and 6

4 . Give students addition and subtraction problems such as 3 31 2

4 4 4 4,+ + and 3 12 2− . Bring in estimation questions such as this: If I subtract 1

2 from 42 ,



will my answer be greater or less than 1? If you have done Activities 11 and 15 from Unit 4, the labeling should be easy. The website http://www.visualfractions.com has a lesson that uses number lines or fraction circles to add and subtract fractions. The game, Find Grampy, focuses on estimating fractions on a number line. The game, Cookies for Grampy, focuses on using different pieces to make a whole cookie. To access these games, type the following into your web browser: http://www.visualfractions.com/Games.htm. Use the student questions for purposeful learning (SQPL) strategy (view literacy strategy descriptions) to challenge students about their ideas of adding fractions and equivalency. Put the following statement on the board. “Jorge says 8

483

81 =+ ; Lilitia says 2

183

81 =+ ;

and Mia says 164

83

81 =+ . Who is correct? Have students work with a partner and

brainstorm 2-3 questions that would have to be answered to determine who is correct. Some questions might be ones such as these: Could all of the students be correct? Are all of the answers equivalent? Could we use the number line to check each answer? Are some answers in lowest terms? What mistakes might have been made when adding fractions? Have student pairs present one of their questions to the class. Write the questions on the board. Reread the opening statement. As a class, have students answer each question, and then decide which student had the correct answer. (Both Jorge and Lititia are correct.) Activity 7: Writing about Computation (GLEs: 3, 6) Materials List: paper, pencils Using a scenario of a trip to a pizza parlor, have students write a math story chain (view literacy strategy descriptions) about fractional parts of pizza. Each student in the group contributes a part to the story chain. For example, Student 1 writes: Jeanne ate one-fourth of a large pizza. Student 2 writes: Derrick ate two-fourths of a pizza. Student 3 asks the question: Did the two students eat the entire pizza? Student 4 answers the question and explains the answer. Students in the story chain groups should talk about the accuracy of the answer and the logic of the story problem. If necessary, revisions to the story chain should be made. Have group present their problems to the class. Ask questions of them such as: What operation should you use to solve the problem? Would 8

3 be a reasonable answer? Why, or why not? Although the GLE only calls for addition and subtraction of fractions with like denominators, some students may write story chains that involve unlike denominators. Students should use models to show equivalent fractions so that an answer can be determined.

http://www.visualfractions.com

http://www.visualfractions.com/Games.htm



Activity 8: How Far Is It? (GLEs: 3, 6, 9) Materials List: How Far Is It? BLM, paper, pencils, math learning logs Make a transparency of the How Far Is It? BLM or give students copies of it. Give directions about the maps such as this: Take the shortest route, but stay on the path. Then ask, To find out how far the library is from Mark’s house, should you add or subtract 7

10 and 3

10 ? To find out how much farther Kathy’s house is from Mark’s house than the library, what operation (s) should you use? To emphasize estimation, ask questions such as this: Do you think the distance from the school to Kathy’s house is more or less than 1 mile? Using math learning logs (view literacy strategy descriptions), have students write one addition, one subtraction, and one estimation question about Map B. Have students share their questions with the class. Activity 9: Who Are We? (GLEs: 2, 3, 4, 5) new Materials List: Who Are We? BLM, pencils Have students work in pairs. Provide one copy of the Who Are We? BLM to each pair of students. Students should find two different answers to fit each statement. Discuss each problem in class, listing all of the possible answers. Students do not have to use unlike denominators at any time. However, if students do use unlike denominators, discussion can follow about substituting an equivalent fraction so that they can add the fractions. Activity 10: Fraction Patterns (GLEs: 3, 5, 33) Materials List: paper, pencils, overhead fraction calculator-optional Write the following sequence of fractions on the board. 31 2 4

6 6 6 6, , , , … Have students describe the patterns they see. They may see things such as the denominator stays the same and the numerator increases by 1. Ask them to explain how they could find the next number in the sequence. Ask them what fraction was added to a term to get the next fraction. ( 1

6 ) Give sequences that involve adding 28 or 3

4 . Be sure to give decreasing sequences, such as 6 5 34

8 8 8 8, , , , … If you have an overhead fraction calculator, enter 0 + 6

1 =. As you press the equal key, the calculator will continue to add

61 each time. As the number appears on the display, write it on the board. Have students predict the next number in the pattern.



Activity 11: Number Lines (GLE: 4, 14) Materials List: Number Lines BLM, paper, pencils Give students the Number Lines BLM. Ask students to find a fraction such as 1

4 on number line A. After locating 1

4 , ask them to name 3 fractions that would be greater than 14 . Write 1

4x > on the board. Show them how to draw 14x > on the first number line.

Continue with different fractions and the other number lines, asking questions about greater than, greater than or equal to, less than, less than or equal to, and equal to. Each time, have students write the equation or inequality, and then graph it on the number lines.


• Portfolio assessment could include the following: o Anecdotal notes made during teacher observation o Any of the learning log entries, or one of the explanations from the

specific activities o Corrections to any of the missed items on the tests


this: Using fractions, circles, or drawings, show why 314 4 1+ = .

o One problem that requires the student to explain their reasoning such as this: Why is the sum of 1 1 2

4 4 4+ = and not 28 ? Explain your reasoning.

o One problem involving real-life such as this: Describe a time in your life when it would be necessary to add and subtract fractions.

• Learning log entries could include the following:

o Suppose two fractions are both less than 1. Can their sum be greater than 1? Can their sum be greater than 2? Explain your thoughts using an example.

o Answer the following question. Look at the following sequence. 9 6 312

10 10 10 10, , , … What patterns do you see in the sequence? Explain how you could find the next fraction.


• Activity 6: Have students draw a number line and mark the following points on the line: 0, 3

4 , 121 , 2

2 .



• Activity 8: Using the How Far Is It? BLM, have students make up one addition

and one subtraction word problem involving fractions, using Map A. Students should write the equation and solve the problem.

• Activity 10: Have students work as partners. One student should write a sequence

involving fractions. The second student should continue the pattern and explain the sequence.

• Activity 11: Have students explain the difference in the graphs of an inequality

that says “greater than,” and one that says “greater than or equal to.”


Grade 5 Mathematics Unit 8 Measurement and Algebra 89

Grade 5 Mathematics

Unit 8: Measurement and Algebra Time Frame: Approximately five weeks Unit Description The focus of the unit is measurement — finding the area and perimeter of rectangular objects and converting units within a system. There is also an emphasis on writing and solving equations and looking for patterns. Student Understandings Students develop, understand, and use formulas for perimeter and area and perform measurement conversions within a system. Guiding Questions

1. Can students find the area and perimeter of rectangles? 2. Can students find the unknown’s value in simple measurement formula

situations for area and perimeter? 3. Can students use formulas or simple ratios to convert between measures in the

same system? 4. Can students write number sentence stories and rules associated with patterns

or measurement contexts? Unit 8 Grade-Level Expectations (GLEs) GLE # GLE Text and Benchmarks Number and Number Relations 7. Select, sequence, and use appropriate operations to solve multi-step word

problems with whole numbers (N-5-M) (N-4-M) 8. Use the whole number system (e.g., computational fluency, place value, etc.) to

solve problems in real life and other content areas (N-5-M) 11. Explain concepts of ratios and equivalent ratios using models and pictures in

real-life problems (e.g., understand that 23 means 2 divided by 3) (N-8-M) (N-

5-M) Algebra 12. Find unknown quantities in number sentences by using mental math, backward




13. Write a number sentence from a given physical model of an equation (e.g.,

balance scale) (A-2-M) (A-1-M) 14. Find solutions to one-step inequalities and identify positive solutions on a

number line (A-2-M) (A-3-M) Measurement 15. Model, measure, and use the names of all common units in the U.S. and metric

systems (M-1-M) 23. Convert between units of measurement for length, weight, and time, in U.S.

and metric, within the same system (M-5-M) Patterns, Relations, and Functions 33. Fill in missing elements in sequences of designs, number patterns, positioned


Sample Activities Activity 1: How Many Squares? (GLEs: 8, 12, 13, 15) Materials List: Centimeter Grid Paper BLM, transparency of the Find the Perimeter and Area BLM, paper, pencils Give students at least four copies of Centimeter Grid Paper BLM. Have students draw a rectangle that is 6 cm by 4 cm and count the units around all four sides. Tell them that they have found the perimeter of a rectangle. Define perimeter as the distance around a figure. Make a transparency of the Find the Perimeter and Area BLM. Give students different dimensions for rectangles (include some squares), and have them draw the rectangles. Students should count to find the perimeter. Record the dimensions and the perimeter on the top chart of the BLM transparency. After four or five sets of dimensions, check to see if students see any patterns, and ask them to how to find the perimeter of a rectangle without counting. Have students generate an equation or formula for the perimeter of a rectangle. Students should see that the formula can be written as P l w l w= + + + ; P l l w w= + + + ;

2 2P l w= + ; and 2( )P l w= + . Give some problems in which students need to find the perimeter using one of the formulas. Ask questions about the square figures. Students should see that the perimeter of a square can be found using 4P s= or using any of the formulas for rectangles. Give a few problems where students are given the perimeter and either the length or width, and ask them to find the missing dimension. Repeat the activity, but this time, have students count the squares to find the area of rectangles. Use the same dimensions for the rectangles used in the perimeter part of the activity. Record the dimensions and the area in the bottom chart of the BLM transparency. Remind students that area is shown in square units. Have students look for patterns. Discuss how to find the area without counting squares, and how to generate the



formula for the area of a rectangle. Give additional problems in which students have to find the area or a missing dimension using the formula. Activity 2: Measuring Common Rectangular Objects (GLEs: 8, 15, 23) Materials List: Opinions About Perimeter or Area BLM, linear measuring tools, paper, pencils Present students with an opinionnaire (view literacy strategy descriptions) to challenge their understanding of area and perimeter. The emphasis is on students’ points of view not on the correctness of their opinions. Give each student a copy of the Opinions about Perimeter or Area BLM then have them complete the BLM. Have students share their thoughts with the rest of the class. Encourage discussion of all of the opinions. By taking a stand on the questions and engaging in discussion about their stands, students make connections from their opinions and ideas to those of their classmates. Have students, in groups of four, actually measure to find both the area and perimeter of the floor, chalkboard, and desktop. Have each group use a different unit, such as millimeters, centimeters, meters, inches, feet, and yards. Informally compare the numbers using each unit. Ask a question such as this: When finding the perimeter of the floor, did it take more centimeters or meters? Allow students to return to the opinionnaire and revise any initial responses based on new learning about perimeter and area. Use this opportunity for further discussion and to provide clarification. Activity 3: Area or Perimeter? (GLEs: 7, 8) Materials List: paper, pencils In groups of four, have students write a math story chain (view literacy strategy descriptions) about area or perimeter. The first student should give a fact for the problem such as this: The length of the pen for my dog is 12 feet. The second student should give a second fact, such as this: The width is 8 feet. The third student should write a question, such as this: If I put a fence around the dog pen, how many feet of fencing do I need to buy? Do not allow students to use the word, area or perimeter, in their questions. The fourth student must decide if the story chain problem is asking for perimeter or area, must solve the problem, and explain how it was solved. Have each group present one problem to the class.



Activity 4: Does It Double? (GLE: 8,) Materials List: Centimeter Grid Paper BLM, Does It Double? BLM, paper, pencils, math learning logs Through SPAWN writing (view literacy strategy descriptions) prompts, such as “What if?” a thought-provoking activity can be created related to area and perimeter. Ask students to answer this prompt: If you double the lengths of the sides of a rectangle, both the area and perimeter of the rectangle double. In groups of four, have students brainstorm (view literacy strategy descriptions) how they could find if this statement is true and write their ides in their learning logs (view literacy strategy descriptions). Students should share their response with the class. Distribute one copy of the Centimeter Grid Paper BLM and the Does It Double? BLM to each student. Have each student in the group draw a different small rectangle on the Centimeter Grid Paper BLM. Have students record the area and perimeter of their rectangles on the Does It Double? BLM. Have them draw the same shape, but make each side twice as long. They should find the area and perimeter and record it. Have them draw 3 more rectangles and record the areas and perimeters of the original shapes and the sides-doubled shapes. As a group, students should look at their tables and decide if the prompt statement was correct. Allow students to return to their SPAWN prompt responses and compare their initial brainstormed ideas with what they have learned in the activity about the perimeter and area of rectangles. Activity 5: Area and Perimeter (GLEs: 8, 12, 13) Materials List: Centimeter Grid Paper BLM, paper, pencils Distribute the Centimeter Grid Paper BLM to students and have each student draw a rectangular shape on the paper. Encourage a few students to draw squares. On a separate sheet of paper, have students create a word problem and a corresponding number sentence with a missing element for their figure. An example: My rectangle has one side of 4 and one side of 6. What is the perimeter? ( P = 2(4 + 6) ) What is the area? ( A= 4×6 ) My rectangle has an area of 30 square units and a width of 3 units. What is the length? ( 30 = 3×l ) Working in groups of four, have the groups work each problem without using the drawings. They can check the answers by looking at the drawings.



Activity 6: What is the Area? (GLEs: 8, 15) Materials List: What is the Area? BLM, linear measuring tools, pencils Give students a copy of the What is the Area? BLM. Have them measure the length of each side of Figure A to the nearest centimeter. In groups, have students discuss how they could find the area of the entire figure using their measurements. (They could break the figure into smaller rectangles and add the areas of the smaller rectangles.) Have them find the area of Figure A. (82 sq. cm) Repeat the activity, but this time, have students measure Figure B to the nearest inch, and then find the area. (9 sq. in.) Activity 7: Perimeter Patterns (GLEs: 8, 33) Materials List: Perimeter Patterns BLM, pattern blocks (squares only) or color tiles, paper, pencils Have students work in groups using the squares from pattern blocks or color tiles. Give students the following problem. The length of a side of the pattern block square is 1 inch. What is the perimeter of the square? (4 inches) Put two squares together in a horizontal line. What is the perimeter of the figure? (6 inches) Continue to add one square at a time, stating the perimeter for each new figure. Stop at 5 squares. Give students a copy of the Perimeter Patterns BLM. Have them fill in the table for the first 5 figures. Ask students to predict the perimeters for the 6th and 7th figures. See if they can explain in words how they could find the perimeter for the 10th figure. (double the figure number and add 2) Have students find the perimeter of the 100th figure. (202 inches) Ask students to explain why this rule works. Repeat the activity letting the side of the square equal two units. Activity 8: Area Patterns (GLEs: 8, 33) Materials List: Area Patterns BLM, pattern blocks (squares only) or color tiles, paper, pencils Distribute the Area Patterns BLM. Have students work in groups using the squares from pattern blocks or color tiles. Give students the following problem. The side of the square equals 1 inch. What is the area of the square? (1 sq. inch) Put 1 square above the original square, and 1 square to the right of it. The figure should look like this: Ask students to find the area of the new figure. (3 sq. inches) Continue adding 1 square on the top and 1 to the right for three more figures. Each time, ask the area. Have students record the areas on the Area Patterns BLM. Ask students to predict the areas for the 6th and 7th figures. See if they can explain in words how they could find the area for the 10th figure. (double the figure number minus 1) Have students find the area of the 100th figure.



(199 square inches) Consider repeating the activity making different patterns with the squares. Activity 9: Which is More Money? (GLEs: 7, 8, 11, 15) Materials List: math learning logs, rulers, yardsticks, quarters, dimes, pennies, paper, pencils Present students with this problem: Would you rather have a 10-inch line of dimes or a yard-long line of pennies? Have students work in groups to determine how to solve the problem. Some students will measure for the entire problem. Some may think ratios. Encourage them to place pennies on the ruler until they get to a whole number of inches and then use ratios rather than measure the entire amount. For example, there are about 4 pennies in 3 inches, so this is a ratio of 4 pennies to 3 inches. They could make a chart similar to this one.

pennies 4 8 16 32 48inches 3 6 12 24 36

Since there are 36 inches in 1 yard, what is an equivalent ratio? (48 pennies to 36 inches) So a yard of pennies is worth 48¢. There are about 7 dimes in 5 inches. So in 10 inches, there would be 14 dimes, or it would be worth $1.40. In their math learning logs (view literacy strategy descriptions), have students respond to this problem: Would you rather have a 10-inch line of quarters or a yard-long line of dimes? Explain your reasoning. Activity 10: How Many Minutes? (GLEs: 11, 15, 23) Materials List: paper, pencils, calculators Use the SQPL strategy (view literacy strategy descriptions) to challenge students to further explore the concept of conversions in measurement. Put the following statement on the board for students. “You will sleep 1 million minutes over your lifetime.” In groups of four, have students brainstorm (view literacy strategy descriptions) different questions they might need to answer to determine whether this statement is true or false. Some questions might include the following: How many hours of sleep do I get each night on an average? How many years do I expect to live? How many minutes are in 1 hour? Each group should present one question to the class. Give the class time to read each question. Have students discuss which questions would help prove or disprove this statement. Allow students to use calculators. Encourage students to use equivalent ratios or tables to find the answer. For example, students might start thinking this way. I sleep 8 hours a night, so I sleep 8 × 7 hours in a week. 8 hours 56 hours 1 night 1 week =



Others may change the 8 hours to minutes. 8 hours 240 minutes 1 night 1 night At the conclusion of the lesson, draw students’ attention to their SQPL questions and allow them to reflect on which ones were answered. Activity 11: How Far Can You Jump? (GLEs: 8, 15, 23) Materials List: Celebrated Jumping Frog of Calaveras County, yardsticks, paper, pencils Have students work in groups of 4, providing each group with a yardstick. Read Celebrated Jumping Frog of Calaveras County by Mark Twain before beginning the activity. Have students jump as far as they can. Have partners measure and record each other’s jumps. Have students measure in inches and then convert to feet or yards. Before converting, ask students if they will have more or fewer feet than inches. Have students order the lengths of the jumps in their groups. Do the same type of activity for metric units. Have students measure the distances in centimeters and then convert to millimeters and meters. With millimeters, review ideas about place value, comparison, and all four operations. Have each group write two story problems about the distances. As a follow-up, have the students toss a cotton ball, flick a paper clip, or some other similar activity to measure distances. Activity 12: Missing Measurements (GLEs: 7, 12, 15, 23) Materials List: paper, pencils Have students find the missing quantities in a variety of measurement problems. An example follows: If I have 20 ounces of orange juice left in a half-gallon container, how many ounces have been used? If my desk is 8 feet away from the classroom door and I walk 36 inches in the direction of the door, how much farther do I have to walk in order to reach the door? Ask each group of students to write four similar problems that involve two different units of measure, such as pints and quarts. Make sure that any conversions are within a system. Activity 13: Balancing an Equation (GLEs: 12, 13) Materials List: transparency of Balance Scale BLM, paper, pencils Make a transparency or draw a picture of the Balance Scale BLM on the board. Tell students that an equation is made up of two expressions that are equal. If the two sides are equal, then the equation is balanced. Draw 6 small circles and a box on one side of the

=



balance and 8 small circles on the other side. Ask students to explain how to find the number of circles in the box. Some will use mental math and see that there should be 2 circles in the box. Or they could remove 6 circles from both sides of the balance. Either way, the equation remains balanced. Give other examples that are harder to do mentally. Stress that whatever is added or removed from one side must be added or removed from the other. Give a multiplication example next. Draw 3 boxes on one side, and 6 circles on the other side of the balance. Tell students that each box contains the same number of circles. Ask, “If the number in each box equals n, what equation could you write to represent the balance? ( 3n = 6 ) Ask, How would you find the number of circles in each box?” Stress that if you divide one side by 3, you must divide the other side by 3 to maintain the balance. Give other examples that are harder to do mentally. Equations involving subtraction, such as 4 10x − = and division, such as 2 10x = , are difficult to model using the balance. Tell students that the idea of doing the same thing to both sides and using inverse operations also holds true for these types of equations. Activity 14: What’s the Equation? (GLEs: 12, 13) Materials List: play money (bills only), paper, pencils Have students work in pairs using play money (bills only). Read a problem involving money, such as this: John left the house with $12. When he returned home, he had $8. What equation could you write to show the amount of money that he spent? (12 - 8 = m or12 - m = 8 ) Have students act out each problem to see if they have written the proper equation and then solve the equation. Continue with other problems, making sure some are multi-step problems. Activity 15: Missing Numbers (GLEs: 8, 12) Materials List: Missing Numbers BLM, math learning logs, pencils Distribute the Missing Numbers BLM to students. Have students find the value of each shape in problems 1-3. The same shape must have the same value in the problem. For problem 4, have students explain, in their math learning log (view literacy strategy descriptions), how they solved the problem.



Activity 16: Reasoning About Quantities (GLEs: 12, 13) Materials List: paper, pencils, Internet access The website, http://www.figurethis.org, provides an excellent opportunity to challenge students. Math Challenge #58 involves reasoning about quantities and has problems similar to the ones in Activity 15. Go to the website and click on printing the challenges. An index will appear. The lesson provides a challenge problem, ideas to get started, a complete solution to the challenge, and additional challenge problems. Activity 17: Number Lines (GLEs: 12, 14) Materials List: paper, pencils Review the terms equality and inequality. On the board, write the symbols <, >, ≥, ≤, ≠, and =. Have students read each symbol. Have students look at the equation, x = 4 and draw a number line. Ask students how they could show the solution on their number line. (Put a dot above the number 4.) Have students draw a number line from 0 to 10 on their paper and mark a dot on any whole number. Write n < 2 on the board. Ask students if the point on their number line is a solution. Write the solutions next to it. Continue giving other equalities and inequalities, such as n ≥ 3, n = 3, n < 0. Show students how to graph the solutions on a number line. Provide students with a variety of simple one-step equations and inequalities to solve (e.g., x – 5 = 48 or x + 4 > 10) and graph on student drawn number lines. Activity 18: Calculator Patterns (GLEs: 8, 33) Materials List: overhead calculator, paper, pencils Using an overhead calculator, enter 0 4+ into the calculator. Tell the students that the calculator will start at 0 and add 4 each time you press equal key. Press the equal key 3 times and write 4, 8, 12, ___, ___, ___, on the board. Have students predict the next three numbers in the sequence. Next enter1 5+ , but this time, do not let the students see what you have entered. As you press the equal key (=), write the numbers on the board. (6, 11, 16) Ask students to predict the next three numbers in the sequence. Ask them what the starting number was or what number in the sequence comes before 6. Write this as ___, 6, 11, 16, ___. For subtraction problems, the first number entered is the starting number, while the second number is the constant. The way to enter multiplication problems differs on some calculators. On most scientific calculators, a starting number, such as 1 is entered, then times a constant is entered. An example is1 2 , ,× = = = . On some 4-function calculators, the constant is entered first, then the starting number, so thus 2 × 1 =, =, = is entered. Either way, the three numbers should be 2, 4, and 8. For division, a starting number is

http://www.figurethis.org



entered divided by a constant. One million divided by 10 is a good way to see patterns in dividing by 10. When using the constant feature numbers divided after 1 will show decimals such as 0.1, 0.01, and 0.001. Activity 19: What Number Comes Next? (GLEs: 8, 33) Materials List: paper, pencils Write a pattern on the board that involves addition, such as 113, 123, 133, 143, ___, ___, ___. Ask students to describe what is happening in the pattern. (increasing by 10) Call this a “ 10+ ” pattern. Have students predict the next three numbers. Show a “ 8− ” pattern, such as 80, 72, 64, 56, ___, ___, ___. Have students predict the next three numbers. Do the same with multiplication and division patterns. Also include patterns with fractions, such as 31 2

4 4 4, , ,... This would be a “ 14+ ” pattern. Have each group of four students write

four different patterns, one for each operation. Students in the group should work each pattern, and then choose two patterns to present to the class.


• Portfolio assessment could include the following: o Anecdotal notes made during teacher observation o Any of the learning logs, or one of the explanations from the specific


• On any teacher-made written tests, include at least one of the following: o One problem that requires the use of manipulatives or drawings such as

this: Using a ruler, draw a rectangle, show the dimensions, and find the area and perimeter of the rectangle. One problem that requires the student to explain his/her reasoning such as this: Answer the following question and explain his/her reasoning: Sometimes salads are sold by weight. If you were really hungry, would you choose a salad that weighs 10 ounces or 1 pound?

o One problem involving real-life such as this: Give an example of when you would need to find the area of a figure in real-life.

• Learning log entries could include the following: o Explain your answer to the question: Can you find the area of a square

using the formula A l w= × ? o Explain how to find the answer to this question: If the area of a rectangle

is 20 sq. cm and the length of one side is 5 cm, what is the perimeter of the rectangle? (18 cm)




• Activity 1: On grid paper, the students should draw all of the rectangles that have an area of 16 square units, predicting if the perimeter of each rectangle will be the same and finding the perimeters.

• Activity 2: The students should write a real-life problem that would involve area

and one that would involve perimeter.

• Activity 14: The students should write a story problem involving money. He/she should write the number sentence that represents the problem and then solve the problem.

• Activity 19: The students should create a “ 1

3+ ” pattern and a “ 4÷ ” pattern, then create a pattern of his/her own using one of the 4 operations and see if another student can find the pattern.

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Grade 5 Mathematics - KENILWORTHkenilworthst.org/Academics/Comprehensive_Curriculum/Mathematics... · with sample activities and classroom assessments to guide teaching and ... Grade