Unit One Organizer:€¦ · Web viewh. Solve and create word problems involving addition and subtraction to 100 without. regrouping. Use words, pictures and concrete models to interpret

Grade 1 Mathematics Frameworks

Unit 2Understanding Operations

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Georgia Performance Standards FrameworkFirst Grade Mathematics Unit 2 1st Edition

Unit Two: Understanding Operations (6 weeks)

TABLE OF CONTENTS

Overview..............................................................................................................................3 Key Standards & Related Standards..................................................................................4 Enduring Understandings.....................................................................................................6 Essential Questions..............................................................................................................7

Concepts & Skills to Maintain ............................................................................................8

Selected Terms and Symbols...............................................................................................9

Classroom Routines...........................................................................................................10

Strategies for Teaching and Learning ..............................................................................10 Evidence of Learning ........................................................................................................10 Tasks .................................................................................................................................11

Combining Examples ............................................................................................12 Joining/Separating Examples ................................................................................18 Comparing Examples ............................................................................................24 Tiles in a Pile ........................................................................................................29 Domino Fact Family .............................................................................................34 Ten in My Hands ..................................................................................................37 One More, One Less, Ten More, Ten Less ...........................................................41 Calculator Skip Counting ......................................................................................45 Let’s Share ............................................................................................................50 Bean Stick Place Value .........................................................................................55 How Many Ways? .................................................................................................58 Two Digit Flip Book .............................................................................................60 Exploring Coins ....................................................................................................63 Exploring Bills ......................................................................................................67 Culminating Task: My Story Book Problem .....................................................71

OVERVIEW

Georgia Department of EducationKathy Cox, State Superintendent of Schools

MATHEMATICS GRADE 1 UNIT 2: Understanding Operations August 2009 Page 2 of 73

Copyright 2009 © All Rights Reserved


In this unit students will: construct a solid understanding of addition and subtraction situations and operations cultivate an understanding of how addition and subtraction affect quantities and are

related to each other use multiple representations to create, describe, and solve story problems compose and decompose numbers up to 10 continue building addition and subtraction fact fluency count to 100 skip count by 2s, 5s, and 10s ( from any number up to 100) develop base ten understanding using a variety of models, diagrams, and number

sentences as a foundation for place value knowledge develop informal strategies for sharing quantities fairly between two to five people use money (coins and bills) for collecting and exchanging quantities up to $1.00 with

coins and up to $20 with bills

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis through the use of calendars, centers, and games.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.

KEY STANDARDS AND RELATED STANDARDS





KEY STANDARDS

M1N1. Students will estimate, model, compare, order, and represent whole numbers up to 100.

a. Represent numbers up to 100 using a variety of models, diagrams, and number sentences. Represent numbers larger than 10 in terms of tens and ones using manipulatives and pictures.

b. Correctly count and represent the number of objects in a set using numerals.c. Compare small sets using the terms greater than, less than, and equal to.d. Understand the magnitude and order of numbers up to 100 by making ordered

sequences and representing them on a number line.e. Exchange equivalent quantities of coins by making fair trades involving combinations

of pennies, nickels, dimes, and quarters up to one dollar; count out a combination of coins needed to purchase items up to one dollar.

f. Identify bills ($1, $5, $10, $20) by name and value and exchange equivalent quantities by making fair trades involving combinations of bills; count out a combination of bills needed to purchase items that total up to twenty dollars.

M1N2. Students will understand place value notation for the numbers 1 to 99. (Discussions may allude to 3-digit numbers to assist in understanding place value.)a. Determine to which ten a given number is closest using tools such as a sequential

number line or chart. b. Represent collections of less than 30 objects with 2-digit numbers and understand the

meaning of place value. c. Decompose numbers from 10 to 99 as the appropriate number of tens and ones.

M1N3. Students will add and subtract numbers less than 100, as well as understand and use the inverse relationship between addition and subtraction.

a. Identify one more than, one less than, 10 more than, and 10 less than a given number.b. Skip-count by 2s, 5s, and 10s, forward and backwards; to and from numbers up to

100.c. Compose/decompose numbers up to 10 (e. g. 3+5=8, 8=5+2+1).d. Understand a variety of situations to which subtraction may apply: taking away from

a set, comparing two sets, and determining how many more or how many less.e. Understand addition and subtraction number combinations using strategies such as

counting on, counting back, doubles and making tens.f. Know the single-digit addition facts to 18 and corresponding subtraction facts with

understanding and fluency. (Use strategies such as relating to facts already known, applying the commutative property, and grouping facts into families.)

g. Apply addition and subtraction to 2 digit numbers without regrouping (e.g.15 + 4, 80-60, 56 + 10, 100-30, 52 + 5).

h. Solve and create word problems involving addition and subtraction to 100 without regrouping. Use words, pictures and concrete models to interpret story problems and





reflect the combining of sets as addition and taking away or comparing elements of sets as subtraction.

M1N4. Students will count collections of up to 100 objects by dividing them into equal parts and represent the results using words, pictures, or diagrams.

a. Use informal strategies to share objects equally between two to five people.

RELATED STANDARDS

M1N4. Students will count collections of up to 100 objects by dividing them into equalparts and represent the results using words, pictures, or diagrams. c. Identify, label, and relate fractions (halves, fourths) as equal parts of a whole using

pictures and models. d. Understand halves and fourths as representations of equal parts of a whole.

M1P1. Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.

M1P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.

M1P3. Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.b. Communicate their mathematical thinking coherently and clearly to peers, teachers,

and others.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely.

M1P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce

a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.

M1P5. Students will represent mathematics in multiple ways.





a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ENDURING UNDERSTANDINGS

When two quantities are joined, the result is more than either of the original quantities.

When one quantity is removed (taken away/subtracted) from another, what is left is less than the original quantity.

Addition is indicated by joining or counting on. Subtraction is indicated by removing, taking away, counting back, or comparing. Addition and subtraction are inverse (opposite) operations. The sum of the parts is equal to the whole quantity. When one quantity is compared to another, the first quantity is either more than,

less than, or equal to the second quantity. Quantities can be shared fairly between two to five people or sets. Problems can be solved in a variety of ways such as modeling with objects or

pictures, counting strategies, number facts, or words (explanations). Problems and solutions can be represented in a variety of ways, including

concrete objects, pictures, number sentences, and words. When adding, the order of numbers does not change the answer. This is not true

for subtraction. Various combinations of numbers and operations can be used to represent the

same quantity. The characteristics of coins and bills identify their values.

ESSENTIAL QUESTIONS

How can we represent a set of objects using numerals? What happens when we join two quantities or take one from another? How can we find the total when we join two quantities? How can we find what is left when we take one quantity from another? How can we find the difference when we compare one quantity to another? How can we compare one quantity to another? How can we represent problem situations? What happens when we change the order of numbers when we add (or subtract)? Why? How can we show that addition and subtraction are related through fact families? How can we use different combinations of numbers and operations to represent the same

quantity? How can we use a 99 chart to help us add and subtract from a given number?





How can we represent a number using tens and ones? How can we represent a number in variety of ways? How can we equally share among two to five people? How can we identify the values of various coins and bills?

CONCEPTS AND SKILLS TO MAINTAIN

Counting to 30 Patterning Sorting Number words through 10 Writing numbers through 20 to label sets Ordinal numbers (1st – 10th) Comparing sets of 1-10 objects (equal to, more than, or less than) One to one correspondence Equivalence Basic geometric shapes Spatial relationships Estimation using five or ten as a benchmark Modeling addition and subtraction Name and value of coins Calendar time and daily schedule Measurement – comparing and ordering





SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

Add: Combine, join, and count on; +

Compare: How quantities relate to one another using terms like more, less, or equal, lighter/heavier

Counting strategy: A strategy that uses the counting sequence by counting on or back from an initial quantity. Objects (fingers, counters, tally marks) may be used to keep track of the counts.

Doubles plus one: A strategy using a known fact that is close to what is needed to determine the exact fact that is needed. For example, to solve a problem by adding 6 and 7 without remembering the fact for 6 + 7, a child might say, “The answer is 13 because 6 and 6 is 12, and 7 is one more than 6, so I need to add one more to 12.”

Estimate: Determining an approximate number or measure.

Fluency: According to the National Research Council report, Adding it up: Helping children learn mathematics ( 2001) , procedural fluency is defined as the knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Quantity: The amount of objects.

Recall facts: The exact fact needed to solve the problem is remembered. For example, to solve a problem by adding 6 and 7, a child might say, “I know that 6 and 7 is 13.”

Separate: Take away, remove, subtract

Subtract: Take away, remove, and count back; –





CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include obvious activities such as taking attendance, doing a lunch count, determining how many items are needed for snack, lining up in a variety of ways (by height, age, type of shoe, hair color, eye color, etc.), daily questions, 99 chart questions, and calendar activities. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, how to open and close a door, how to do just about everything! An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of the routines are important to the development of students’ number sense, flexibility, and fluency, which will support students’ performances on the tasks in this unit. See unit 1 for suggestions concerning specific ideas for classroom routines.

99 Chart Post several 99 charts in the classroom. Have one at your math calendar/bulletin board area. It is important to use a 99 chart rather than a hundred’s chart for several reasons. First, a 99 chart begins with zero where a hundred’s chart begins with 1. We need to include zero because it is one of the ten digits and just as important as 1-9. Second, a 99 chart puts the decade numerals (10, 20, 30, etc.) in the correct row. For instance, the number 20 is the beginning of the 20’s family; therefore it is at the beginning of the 20’s row. Lastly, the 99 chart ends with the last two digit number, 99. The number 100 should begin a whole new chart because it is the first three digit number. Please see unit 1 for additional ideas for using a 99 chart.

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99





STRATEGIES FOR TEACHING AND LEARNING

Students should be actively engaged by developing their own understanding. Mathematics should be represented in as many ways as possible by using graphs, tables,

pictures, symbols, and words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher feedback,

peer feedback, and metacognition which includes self-assessment and reflection. Math journals are an excellent way for students to show what they are learning about a

concept. These could be spiral bound notebooks that students could draw or write in to describe the day’s math lesson. Students can go back and look at things they have done in the past. Journals could also serve as a tool for a nine week review.

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:

Understand and use basic addition and subtraction facts in everyday situations Compose and decompose numbers up to 10 Know the addition and subtraction facts to 18 (learning basic facts should be on-going

throughout the year) Count fluently up to 100 Understand a variety of representations as a foundation for place value Use informal strategies for sharing quantities fairly between two to five people Use multiple representations to create, describe, and solve story problems Use money (coins and bills) for collecting and exchanging on quantities up to $1.00 with

coins and up to $20 with bills Use skip counting as a means to extend number patterns





TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all first grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).

Task Name Task Type/Grouping Strategy Content Addressed

Combining Examples Learning Taskdepends on story problem

Adding quantities,Word problems

Removing/Taking Away Examples

Learning Taskdepends on story problem

Subtracting quantities,Word problems

Comparing Examples Learning Taskdepends on story problem

Adding and Subtracting quantities,Word problems

Tiles in a Pile Learning TaskPartners Comparing sets using terms

Ten in My Hands Learning TaskSmall group Addition and subtraction combinations

Domino Fact Family Learning TaskIndividual Addition and subtraction combinations

One More, One Less, Ten More, Ten Less

Learning TaskIndividual Addition and subtraction combinations

Calculator Skip Counting Learning TaskIndividual Skip counting

Let’s Share Learning TaskIndividual

Sharing objects betweentwo to five people

Bean Stick Place Value Learning TaskPartners Understanding place value

How Many Ways? Learning TaskIndividual Representing numbers in many ways

Two Digit Flip Book Performance TaskIndividual Representing numbers in many ways

Exploring Coins Performance TaskIndividual/Partners Purchasing items with coins

Exploring Bills Performance TaskIndividual/Partners Purchasing items with bills

Culminating Task: My Story Problem Book

Culminating TaskIndividual

Creating story problems, representing numbers to 100

Learning Task: Combining Example





STANDARDS ADDRESSED

M1N3. Students will add and subtract numbers less than 100 as well as understand and use the inverse relationship between addition and subtraction.

e. Understand addition and subtraction number combinations using strategies such as counting on, counting back, doubles and making tens.

f. Know the single-digit addition facts to 18 and corresponding subtraction facts with understanding and fluency. (Use strategies such as relating to facts already known, applying the commutative property, and grouping facts into families.)

h. Solve and create word problems involving addition and subtraction to 100 without regrouping. Use words, pictures and concrete models to interpret story problems and reflect the combining of sets as addition and taking away or comparing elements of sets as subtraction.




M1P5. Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical

ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

What happens when we join two quantities? How can we find the total (or sum) when we put two quantities together? How can we represent problem situations?





MATERIALS

Paper Various Manipulatives Pencils Crayons

GROUPING

Depending on the story problem, this task could be solved with students working as a whole-class, small groups, or independently.

TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Background Knowledge This lesson should be repeated often with varying story problems of different problem

types. The lesson is presented in stages (concrete – pictorial – abstract). Students are familiar

with these stages from kindergarten. All stages are important in developing student understanding.

Students are now working to extend their strategies within each of the stages through different problem types. Students will advance towards the different stages of modeling, using manipulatives, creating pictures, utilizing number sentences (equations), and fact fluency at different speeds throughout the year.

Allow each student to work at a level which is appropriate for his or her ability. Reinforce students at their individual level through questioning.

The focus is on understanding the story situation NOT on key words such as in all, altogether, total, etc. Use a variety of problems including those that do not contain key words.

Stage 1: Student Modeling (Acting Out a story problem)

The teacher will tell a story similar to: The other day I saw 6 dogs chasing a ball in the park. Then 7 more dogs came and joined them in playing ball. How many dogs were playing in the park?

After telling the story, suggested questions include: What is my story about? What happened first? What happened next? What are the dogs doing in the park? What are we trying to find out? How can we show/represent this? Can we act out the story?





How many dogs were there at the beginning of the story? (Feel free to let the students act and bark like dogs.)

How many dogs are there at the end of the story?

Continue questioning as needed until the students can answer questions and understand what is happening in the story.

Stage 2: Using Manipulatives

Use the story from stage 1.

After re-telling the story, suggested questions include: Instead of acting like dogs what could we use to represent the dogs? Are there any objects I could use to help me find out how many dogs I found

altogether?

Allow students to work at their seat with various manipulatives (red and yellow counters, buttons, connecting cubes, square tiles, etc.) to help them find the total number of dogs. As students are combining sets, walk around and continue to question students as they represent the number of dogs with manipulatives.

Suggested questions include: How many red counters do you have? How many yellow counters do you have? What do these counters represent? Could we use something else? Why do you have ____________ counters? What strategies are you using to combine the sets of dogs? How did you find the total number of dogs?

Stage 3: Drawing Pictures


After re-telling the story, suggested questions include: What math tools could I use to help me find out how many dogs I had in all? How can

we show/represent this?

Many students will need to begin working at their seat with various manipulatives to combine the sets. Other students will be ready to solve the problem by using a picture. Allow students to use crayons, pencils, and paper to draw a picture to solve the problem. Encourage students to make picture representations that make sense to them (dots, squares, circles, tally marks, actual dogs, etc.)

Suggested questions include:





Describe your picture for me? How many dogs did you draw? What strategies are you using to combine the sets? How did you find the total number of dogs? If a partner looks at your picture will they be able to understand it?

*After doing this type of activity several times have students swap pictures with a neighbor and have them answer any questions their neighbor may have about their picture and vise-versa. Allow several students to share their representations.

Stage 4: Creating Number Sentences

First grade students should begin creating number sentences to describe their story problems after the use of manipulatives or the creation of a picture. Students should be introduced to the terms addition, addends, sum, total, number sentence, and equation throughout this unit. As students are creating number sentences for their story problems, suggested questions include:

What are your addends? How did you figure out those were the addends? Why did you add _______ + _________? How did you know this was an addition problem? Which sign did you use? Why? What is your sum? How did you figure that out? How can you check your sum to make sure it is correct? Is there another way you could have solved this problem? Can you find a neighbor

who solved it a different way? How is it different from what you did? How can we show/represent this?

Alternative Story Problems This morning I was walking in my backyard and I found five red ladybugs on the ground.

Then I saw six yellow ladybugs on a leaf. How many ladybugs did I find in the back yard? Today at lunch I had six chicken nuggets on my tray. Mrs. Brown didn’t want her

nuggets and gave me four more. How many nuggets did I have in all?

Background KnowledgeTeachers should expose students to a variety of combining problem types including joining and part-whole problems. Students do not need to know the names of the different types of problems but they should have experience in solving all of the different types. Examples of each problem type are provided below.

Joining





1. Result Unknown There are six cats in a yard. Four more cats walked into the yard. How many cats are in the yard? 6+4=___

2. Change Unknown Mr. Pup gave haircuts to four dogs. How many more dogs need a haircut if there are eleven dogs in the shop? 4+__=11

3. Start Unknown There was a herd of cows. Three baby cows were born. Now there are eight cows. How many cows were in the herd to start with? __+3=8

Part-Part-Whole 1. Whole Unknown

A school of fish has four males and eight females. How many fish are in the school? 4+8=

2. Part Unknown There are ten mice in a cage. Three of the mice are black and the rest are grey. How many grey mice are in the cage? 3+___=10

More Examples

Problem Type Action or Situation Number Sentence

Some Addition Types

Join, Result Unknown

Pete puts 25 mice in a cage, and then puts in 12 more. How many are in the cage? 25 + 12 =

Join, Start UnknownPete has some mice in a cage. He puts 12 more in. Now he has 37. How many were in the cage

to begin with? + 12 = 37

Join, Change Unknown

Pete has 25 mice in a cage. He puts some more in. Now he has 37. How many did he add to the

cage?25 + = 37

Compare, Compare Quantity Unknown

Pete has 25 mice. Max has 12 more mice than Pete. How many mice does Max have? 25 + 12 =

Part-Part-Whole, Whole Unknown

Pete has 12 white mice and 13 brown mice. How many mice does he have altogether? 12 + 13 =

Part-Part-Whole, Part Unknown

Pete has 25 mice. 12 are white and the rest are brown. How many mice are brown?

12 + = 25+ 12 = 25

Questions for Teacher Reflection





Are students able to represent and solve a variety of addition problem solving situations?

At which stage are students working?

How can I help scaffold students to be successful in the next stage?

What strategies do students use for solving problems?

DIFFERENTIATION

Extension Give the answer to a story problem such as “10 frogs.” Students create their own story

problem that has the answer they were given.

Intervention Allow students to work through the stages at a pace that is appropriate to their

developmental level. This will provide students with the remediation they need to understand the concept of comparing numbers. Continue to allow them to work with manipulatives as much as needed before moving to the next stage. At times, partner them with students who are very articulate about their mathematical thinking so they can hear (through conversations) how these students have made sense of the problems.





Learning Task: Removing/Taking Away Example

STANDARDS ADDRESSED

M1N3. Students will add and subtract numbers less than 100 as well as understand anduse the inverse relationship between addition and subtraction. e. Understand addition and subtraction number combinations using strategies such as



h. Solve and create word problems involving addition and subtraction to 100 withoutregrouping. Use words, pictures and concrete models to interpret story problems and reflect the combining of sets as addition and taking away or comparing elements of sets as subtraction.





phenomena.

ESSENTIAL QUESTIONS

What happens when we take one quantity from another? How can we find what is left (the difference) when we take one quantity from another? How can I represent problem situations? Can I change the order of numbers when I subtract? Why or Why not?

MATERIALS






GROUPING



Background Knowledge This lesson should be repeated often with varying story problems of different problem

types. The lesson is presented in stages (concrete – pictorial – abstract). Students are familiar

with these stages from kindergarten. All stages are important in developing student understanding.



The focus is on understanding the story situation NOT on key words. Be sure students understand what the problem is asking – not just look for the bigger number. Use a variety of problems including those that do not contain key words.


The teacher will tell a story similar to: Ten friends were playing outside together on a summer day. Three of the friends had to go home to eat lunch. How many friends were left outside?

After telling the story, suggested questions include: What is my story about? What happened first? What happened next? Were there more (or fewer) friends at beginning of the story or at the end of the story?

How did you figure that out? How can we show/represent this? Can we act out the story? How many friends were there at the beginning of the story? (Feel free to let the

students act out this situation if necessary) How many friends were taken away? How many friends were left at the end of the story? How do you know?

Continue questioning as needed until the students can answer the questions and understand what is happening in the story.







After re-telling the story, suggested questions include: Instead of acting out the story what could we use to represent the friends? Are there any objects I could use to help me find out how many friends were still

outside?

Allow students to work at their seat with various manipulatives (red and yellow counters, buttons, connecting cubes, square tiles, etc.) to help them find the number of kids left outside. As students are working on the separating problems, walk around and continue to question students.

Suggested questions include: How can we show/represent this? How many counters are you starting with? Why? How many counters are you taking away? Why? Can you explain to me how you are using your counters to help you figure out

how many friends are left? What strategies are you using to help you find out how many friends are left? Is your neighbor finding the number of friends left a different way? Can you do

it a different way? How did you find out how many friends were left? Can you explain that to a

neighbor?



After re-telling the story, suggested questions include: What math tools could I use to help me find out how many friends were left playing

outside? How can we show/represent this story?

Many students will need to begin working at their seat with various manipulatives to combine the sets. Other students will be ready to solve the problem by using a picture. Allow students to use crayons, pencils, and paper to draw a picture to solve the problem. Allow students to use picture representations that make sense to them (dots, squares, circles, tally marks, actual friends, etc.) Allow students to share representations.

Suggested questions include: (In teacher modeling, use dots, etc. to show various ways to represent friends to help students not get bogged down with pictures.)





How can we show/represent this story? Describe your picture for me. Do you think a friend would be able to make sense of your picture? How many friends did you draw? How did you show the friends that left for lunch? What strategies are you using to help you subtract/take away the friends that left? How did you find the number of friends that were still outside? Can you think of a different way to find out how many friends were outside? Did your neighbor solve it the same way you did or differently? How do you know?


First grade students should begin creating number sentences to describe their story problems after the use of manipulatives or the creation of a picture. Students should be introduced to the terms subtraction, difference, and number sentence throughout this unit. As students are creating number sentences for their story problems, suggested questions include:

Which number did you subtract from? Why? How many did you subtract from ______? Why did you subtract _______ - _________ and not the other way around? Which number did you place first when you subtracted? Will I get the same answer if I change the order of numbers when I subtract? How did you know this was a subtraction problem? Which sign did you use? Why? What is the difference? What does that mean? How did you find the difference? What strategies did you use? How can we show/represent this story? How can you check the difference to make sure it is correct? Is there another way you could have solved this problem? Did your neighbor solve it the same way you did or differently? How do you know?

Alternative Story Problems There were eight bunnies at the pet store. Three went home with a new family. How many

bunnies were still at the pet store? Today at lunch I had twelve chicken nuggets on my tray. I ate seven of my chicken

nuggets. How many did I not eat?

CommentTeachers should expose students to a variety of separating problem types. Students do not need to know the names of the different types of problems but they should have experience in solving all of the different types. Examples of each problem type are provided below.





Removing/Taking Away Problems

1. Result Unknown

There were thirteen chattering monkeys. Five fell asleep. How many monkeys are still chattering? 13-5=__

2. Change Unknown

There were ten toads sitting near a pond. Some jumped away. Four toads are still sitting near the pond. How many toads jumped away? 10-___=4

3. Start Unknown

An elephant had some peanuts. She ate six of them. Now she has two peanuts left. How many peanuts did the elephant have to start with? ___- 6=2

Problem Type Action or Situation Number Sentence

Some Subtraction Types

Separate, Result Unknown

Pete has 37 mice in a cage. 12 escape. How many are left? 37 – 12 =

Separate, Start Unknown

Pete has some mice in a cage. 12 escape. 25 are left. How many did he have at the start? – 12 = 25

Separate, Change Unknown

Pete has 37 mice in a cage. Some escape. 25 are left. How many escaped? 37 - = 25

Compare, Difference Unknown

Pete has 37 mice. Max has 25 mice. How many more mice does Max have than Pete? 37 – 25 =

Compare, Referent Unknown

Pete has 37 mice. He has 12 more mice than Max. How many mice does Max have? 37 – 12 =

Questions for Teacher Reflection Are students able to represent and solve a variety of addition problem solving situations?








DIFFERENTIATION

Extension Allow students to work with two digit numbers without regrouping. Ask students to create and solve their own story problems.

Note: Students typically create addition problems. Check students’ problems to make sure they are creating subtraction problems too.


developmental level. This will provide students with the remediation they need to understand the concept of comparing numbers. Continue to allow them to work with manipulatives as much as needed. At times, partner them with students who are very articulate about their mathematical thinking so they can hear (through conversations) how these students have made sense of the problems.





Learning Task: Comparing Example

STANDARDS ADDRESSED




h. Solve and create word problems involving addition and subtraction to 100 withoutregrouping. Use words, pictures and concrete models to interpret story problems and reflect the combining of sets as addition and taking away or comparing elements of sets as subtraction.





phenomena.

ESSENTIAL QUESTIONS

What happens when we compare one quantity to another? How can we find the difference when we compare one quantity to another? How can we represent and solve a variety of comparison problem situations?

MATERIALS






GROUPING



Background Knowledge This lesson should be repeated often with varying story problems of different problem types. The lesson is presented in stages (concrete – pictorial – abstract). Students are familiar with

these stages from kindergarten. All stages are important in developing student understanding.



The focus is on understanding the story situation NOT on key words. Be sure students understand what the problem is asking – not just look for the bigger number. Use a variety of problems including those that do not contain key words.


The teacher will tell a story such as: Twelve girls and five boys stayed after school for the Math Club. How many more girls than boys were in the math club?

After telling the story, suggested questions include: What is this story about? What are we trying to find out? Can we act out the story? How many boys are in the math club? How can we show/represent this? How many girls are in the math club? How can we show/represent this? How many more girls are in the math club than boys? How can we show/represent

this? How do you know this is correct? Is there another way to figure this out?

Continue questioning as needed until the students can answer questions and understand what is happening in the story.


Use the story from stage 1.After re-telling the story, suggested questions include:





What type of math tools could you use to help you solve this problem? Did your neighbor solve this problem differently?

Allow students to work at their seat with various manipulatives (red and yellow counters, buttons, connecting cubes, square tiles, etc.) to help them find the difference between the two candies. As students are working on the comparing problems, walk around and continue to question students.

Suggested questions include: How can we show/represent this? How many counters are you starting with? Why? How are you representing the boys and girls? Can you explain to me how you are using your counters to compare the number of

boys and girls? What strategies are you using to help you find out the difference between the number

of boys and girls? How did you find out the difference? Is a neighbor doing it a different way? Can you do it a different way? Can you

explain that to a friend?



After re-telling the story, students should be asked questions like: How many boys are there? How many girls are there? What should you do with those two numbers? How do you know?

Many students will need to begin working at their seat with various manipulatives to compare the sets. Other students will be ready to solve the problem by drawing a picture. Allow students to use crayons, pencils, and paper to draw a picture to solve the problem. Allow students to use picture representations that make sense to them (dots, squares, actual flowers, etc.)

Suggested questions include: How can we show/represent this problem? Describe your picture for me? How many boys did you draw? How many girls did you draw? What strategies are you using to help you compare the number of boys and girls?

How will this help us understand the problem? How did you find out how many more girls than boys were in the club? How is your picture like/different from your neighbor’s?





Can you explain your neighbor’s picture?


First grade students should begin creating number sentences to describe their story problems after the use of manipulatives or the creation of a picture. Students should be introduced to the terms subtraction, difference, number sentence, and equation throughout this unit. As students are creating number sentences for their story problems, suggested questions include:

How can we use numbers to represent this story? Why did you select these numbers? What number sentence did you create to help you solve this problem? Why? How did you know where to place each number in the sentence? Which sign did you use? Why? What is the difference? How did you find the difference? What strategies did you use? How can you check the difference to make sure it is correct? Is there another way you could have solved this problem? Did your neighbor solve it the same way you did? Can you explain what your

neighbor did, how it is alike and different?

Alternative Story Problems In my bag of candy, I have thirteen yellow candies and eight red candies. How many

more red candies do I have than yellow candies? I picked some flowers for my mother. I gave her twenty daisies and fifteen roses. How

many more daisies did I give her than roses?

Background Knowledge Teachers should expose students to a variety of comparing problem types. Students do not need to know the names of the different types of problems but they should have experience in solving all of the different types. Examples of each problem type are provided below.

Comparison

1. Difference Unknown

One litter has eleven kittens. Another litter has six kittens. How many more kittens does the first litter have? 11-6=____ or 6+___=11

2. Compare Quantity Unknown

One flock of sheep has five lambs. The second flock has eight more than the first flock. How many lambs does the second flock have? 5+8=__





3. Referent Unknown

There are 14 geese in the pond. This is two more than the number of geese on the grass. How many geese are on the grass? 14-2=___ or ___+2=14

Questions for Teacher Reflection Are students able to represent and solve a variety of comparison problem solving

situations?




DIFFERENTIATION

Extension Allow students to work with larger numbers. Ask students to create and solve their own story problems.


developmental level. This will provide students with the remediation they need to understand the concept of comparing numbers. Continue to allow them to work with manipulatives as much as needed. At times, partner them with students who are very articulate about their mathematical thinking so they can hear (through conversations) how these students have made sense of the problems.





Learning Task: Tiles in a Pile

STANDARDS ADDRESSED

M1N1. Students will estimate, model, compare, order, and represent whole numbers up to 100. a. Represent numbers up to 100 using a variety of models, diagrams, and number

sentences. Represent numbers larger than 10 in terms of tens and ones using manipulatives and pictures.

c. Compare small sets using the terms greater than, less than, and equal to.






phenomena.

ESSENTIAL QUESTIONS

How can we compare one quantity to another? How can we represent a set of objects using numerals?

MATERIALS

30 numbered bags with varying amounts of tiles in each bag (bags should be clear)

Tiles in a Pile Recording Sheet





GROUPING

Partners


Background Knowledge Students should be familiar with counting sets of objects and comparing them using the terms for greater than, less than, or equal to. (Symbols are not introduced until 2 nd grade .)

Give each student a bag of tiles. Students should first estimate the number of tiles in their bag. They should use words and numbers to record the number of tiles they estimated.

Students will then empty their bag into a pile. Next, have students count the tiles in their pile. Using words, pictures, and numbers,

they should record the number of tiles they counted. Last, have students compare the number of tiles in their pile to the number of tiles in

their partner’s pile and write a number sentence representing the comparison using the words greater than, less than, or equal to.

Have students place the tiles back in their bag. They can choose another bag and start again.

While students are working, ask questions like: What did you estimate as your total number of tiles? Why did you choose this

number? How does estimating help us? In other words, why should we estimate first before

counting something? How are you counting your tiles? Are you using a counting strategy? (grouping, skip

counting, counting on, etc.) What strategies are you using to keep track of your total? How did you represent the total number of tiles in the bag? How does the number of tiles in your bag compare to the number of tiles in your

partner’s bag? Who has more/less? How do you know?

Allow several students to share their work. Allow students to describe how they determined a reasonable estimate and why that is important to do before counting, then demonstrate the counting strategies they used to determine the total number of tiles. When choosing students to share, try to choose students who can demonstrate/model a variety of strategies, including skip counting, forming groups, using addition strategies, or using base-ten understanding. Finally, have the students explain how their number of tiles compared to their partner’s tiles. Allow time for the students to address questions, receive compliments, and listen to suggestions from the audience. As the students share, ask questions such as:





How did you decide upon that number for your estimate? What strategy did you use to count the tiles in your pile? How did you represent/show your work on the paper? How did you keep track of the tiles you had already counted? How did you avoid mixing

up the tiles you had already counted with the tiles you had not yet counted?

Comments Adjust the number of counters used in this task to the individual needs of your

students. Early in first grade, you may need to use 40-50 1-inch tiles, Unifix cubes, 1-cm cubes or other counters. You may return to this task later in the year using 95-100 tiles.

Observe students’ strategies for completing the task. Attend to their counting (one-to-one correspondence and sequence of number names), keeping track, addition strategies (if used), and errors. Offer support to individual students as needed for both content and language development. Select students to share their strategies based on what you observed. Begin with students who used the most basic strategies (counting by ones) and progress to students who used more advanced/efficient counting strategies (skip counting, forming groups, using addition strategies, or using base-ten understanding).

Using words cards with the vocabulary terms would be very helpful in creating a deeper understanding of the concepts.

Questions for Teacher Reflection Can students count the tiles with one to one correspondence and use the correct sequence

of number names? Are students using strategies to make counting more accurate and time efficient? If so,

which strategies? (skip counting, forming groups, using addition strategies, or using base ten understanding)

Can students explain their counting strategies? Are students making reasonable estimates before counting? Do they understand the

importance of doing this? Can students accurately compare two sets of tiles using the words greater than, less than,

or equal to?

DIFFERENTIATION

Extension The teacher may begin to expose students to the comparison of the sums of number

sentences. For example, 5+2 _____ 6+3 (Again, focusing on the terms, NOT symbols.)





Intervention If a student is having difficulty counting his or her tiles correctly, provide him or her with

a 99 board. The student can place a tile on the board beginning with one. The student can continue to place tiles on his or her 99 board and count up to determine the total number.





Task Name: Domino Fact Families

STANDARDS ADDRESSED M1N3e. Understand addition and subtraction number combinations using strategies such

as counting on, counting back, doubles and making tens. M1N3f. Know the single-digit addition facts to 18 and corresponding subtraction facts

with understanding and fluency (Use strategies such as relating to facts already known, applying the commutative property, and grouping facts into families.)

ESSENTIAL QUESTIONS What happens when we join two quantities or remove one quantity from another? How are addition and subtraction related through fact families?

MATERIALS Dominoes Domino Fact Family Recording Sheet




Tiles In a Pile Bag # _____Estimated Number of Tiles Counted Number of Tiles

My Number of Tiles Circle one: My Partner’s Number of Tiles

greater than

less than

equal

Tiles In a Pile Bag # _____Estimated Number of Tiles Counted Number of Tiles

My Number of Tiles Circle one: My Partner’s Number of Tiles

greater than

less than

equal


Learning Task: Ten in My Hand

STANDARDS ADDRESSED







M1P4. Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce


ESSENTIAL QUESTIONS

What happens when we join two quantities?

MATERIALS

Deck of Number Cards (without face cards) Counting Manipulatives

GROUPING

Small Groups (three to four)






Remove the face cards from several decks of playing cards. If you prefer, you may use index cards and make sets of cards with 0 – 10 on them (number, dots, pictures, or words). Separate students into groups of three or four. Have a student shuffle the cards and deal five cards to each of the other players in the group. Place the extra cards face down in the middle of the group as the draw pile. Students should try to make a total of ten with their cards (for example, 6+4 or 5+2+3, etc.). If students are able to combine the cards to make a ten then they can lay the set of cards face up on the floor. Then they draw cards to replace the number of cards laid down on the floor. If the student cannot make a ten, he or she should draw a new card from the card pile to add to his or her hand. The game ends when all the cards have been pulled from the middle deck or the teacher has called time. The winner is the student with the most sets of cards that have a sum of ten.

Questions for Teacher Reflection What strategies are students using to determine cards that equal a sum of ten? Can students explain the strategies they are using to determine the sums of ten? Do students create sets of ten using more than two cards?

DIFFERENTIATION

Extension If this game is easy for students then you can use word cards to play the game or change

the directions to include subtraction. For instance, if a student had 9, 7, 5, 4, 2 in their hand then they could add the 9 and 5 then subtract the 4 to make ten. Or add the 7, 5, and 2 then subtract 4 to make ten!

Intervention Some students may have difficulty solving the part-whole problem type. Teachers may

provide manipulatives to help students determine the missing addend. For example, the teacher may provide a student with a tower of ten cubes. If the student has a four card and wants to know the card he or she needs to make ten, the student can remove four cubes from the tower of ten cubes and count what is left. You would want to use centimeter cubes for this activity.

The cards could also include a ten-frame as shown below to allow students to place the number of cubes on the card in the ten-frame and count the empty boxes. This will help the student determine the card needed and begin to develop strategies to solve part-whole problem types.





4Technology Connection Students can practice adding sums of ten with this game found at the website http://www.mathsyear2000.org/magnet/minus3/sumtox/index.html





Learning Task: Domino Fact Families

STANDARDS ADDRESSED




M1P1. Students will solve problems (using appropriate technology).e. Build new mathematical knowledge through problem solving.f. Solve problems that arise in mathematics and in other contexts.g. Apply and adapt a variety of appropriate strategies to solve problems.h. Monitor and reflect on the process of mathematical problem solving.

M1P3. Students will communicate mathematically.e. Organize and consolidate their mathematical thinking through communication.f. Communicate their mathematical thinking coherently and clearly to peers, teachers,

and others.g. Analyze and evaluate the mathematical thinking and strategies of others.h. Use the language of mathematics to express mathematical ideas precisely.


ideas.d. Select, apply, and translate among mathematical representations to solve problems.e. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

What happens when we join two quantities? What happens when we take one quantity from another? How are addition and subtraction related through fact families?

MATERIALS

Dominoes Domino Fact Family Recording Sheet

GROUPING





Individual


Background KnowledgeSuccess in this task depends on students understanding that once they can recall a particular addition fact, they can use that fact to solve a variety of related addition and subtraction tasks. The set of related facts is called a family. Before this, students will see number facts as unrelated, and hence might feel burdened by how many facts there are to learn. The use of fact families can help students to see relationships between several addition and subtraction facts.

Part IPrepare baggies of dominos (around eight dominos) for each child in the class. Draw several demonstration dominos on the board. You can also make large demonstration dominos by cutting a piece of poster paper in half and drawing dots on either side. Ask students to look at both halves of the domino and tell how many dots are in each square. Suggested questions include:

What is the sum if you add the dots in each square? Will the sum change if I flip the domino (180 degree turn)? Why do you think this?

(Special Note: This is not about students understanding degrees and angles. It is included for teachers to identify the specific way to flip the domino.)

Is there another way we could write the addition sentence and still get the same sum? What would the difference be if you subtracted the number of dots in one of the squares

from the sum? What type of relationship are you noticing? What are the number sentences in the fact family for this domino?

Part IIAllow children to work independently to record the fact families that their dominos show. They will need to draw each domino and label it with their fact family using the recording sheet. Observe the counting strategies that the children utilize. Notice if they are counting on, counting each individual dot, or recalling the answers because they know their addition and subtraction facts. (Note: Students are only being exposed to the concepts of vertical and horizontal. It is not the focus of the task Teachers are encouraged to use the terms in vocabulary. Students are not responsible for this yet.)

Suggested questions include: How many dots are on the left side of your domino (looking at it horizontally)? How many dots are on the top of your domino (looking at it vertically) How many dots are on the right side of your domino (looking at it horizontally)? How many dots are on the bottom of your domino (looking at it vertically)? What is the sum if you add the dots in each of your domino squares? Is there another way you could write the addition sentence and still get the same sum?





What would the difference be if you subtracted the number of dots in one of the squares from the sum?

What are the numbers in this fact family? What type of relationship are you noticing? What are the number sentences in the fact family for this domino? How are you determining the fact families for each of your dominoes? Why is important to know fact families?

Allow several students to share their work. Allow students to discuss how they determined the number sentences in their fact family and the strategies they used to create these number sentences. Allow time for the student who is sharing to answer questions, receive compliments, and listen to suggestions from the audience. Questions for Teacher Reflection

What are the students’ counting strategies when determining the number of dots on their dominoes?

Are students counting on, counting each individual dot, or recalling the answers because they know their addition and subtraction facts?

Can students explain their counting strategies? Can students create addition number sentences to describe their dominoes? Are they looking at the domino both horizontally and vertically? Are they writing number

sentences both horizontally and vertically? Can students create the corresponding subtraction number sentence to describe their

dominoes? Are students able to understand the relationship between number sentences within the

domino fact families?

DIFFERENTIATION

Extension Allow students to create their own fact families with larger numbers.

Intervention Give students note cards that have an addition or subtraction number sentence written on

each card. Have students group the cards into fact families. Once they get the hang of this, give them three of the four number sentences and they have to figure out the one that is missing. As they get better with the “game” start decreasing the number of sentences you give them. Once they can tell you three of the sentences when you have only given them one, change the game. You give them all the number sentences in the family and they have to tell you who the individual family members are.





Name: _______________________ Date: ____________

Domino Fact Family Recording Sheet

____ + ___= ___

____ + ___= ___

____ - ___= ___

____ - ___= ___

____ + ___= ___

____ + ___= ___

____ - ___= ___

____ - ___= ___

____ + ___= ___

____ + ___= ___

____ - ___= ___

____ - ___= ___

____ + ___= ___

____ + ___= ___

____ - ___= ___

____ - ___= ___

_____ _____

+ _____ + ____________ _______

_____ _____

- _____ - ____________ _______

_____ _____

+ _____ + ____________ _______

_____ _____

- _____ - ____________ _______

_____ _____

+ _____ + ____________ _______

_____ _____

- _____ - ____________ _______

_____ _____

+ _____ + ____________ _______

_____ _____

- _____ - ____________ _______





Learning Task: One More, One Less, Ten More, Ten Less

STANDARDS ADDRESSED

M1N3. Students will add and subtract numbers less than 100 as well as understand anduse the inverse relationship between addition and subtraction. a. Identify one more than, one less than, 10 more than, and 10 less than a given number.




ESSENTIAL QUESTIONS

How can we use a 99 board to help us add and subtract from a given number?

MATERIALS

Counting Manipulatives (chips, cubes, etc. – transparent counters are allow students to see the numbers)

99 chart Large demonstration 99 chart

Comment99 ChartIt is important to use a 99 chart rather than a hundred’s chart for several reasons. First, a 99 chart begins with zero where a hundred’s chart begins with 1. We need to include zero because it is one of the ten digits and just as important as 1-9. Second, a 99 chart puts the decade numerals (10, 20, 30, etc.) in the correct row. For instance, the number 20 is the beginning of the 20’s family; therefore it is at the beginning of the 20’s row. Lastly, the 99 chart ends with the last two digit number, 99. The number 100 should begin a whole new chart because it is the first three digit number.





0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

GROUPING

Individual


Background KnowledgeStudents should have experience working with a 99 board. Students should also be familiar with adding and subtracting from a number. Please see Classroom Routines in Unit 1 for more ideas.

Part IGive each student a 99 board and a counter. Give students a specific number to place their chip on. Tell students that you will give them a series of directions about where to move their chip. These directions should be written on the board or on an overhead with the directions that you have yet to say covered. As you give students directions, uncover the directions, so that at the end all the directions are shown in order. A sample game is provided. As each step is read, students should move their chip to the new number. Directions should be adjusted based upon student experiences.

Begin with your chip on the number 67. Add ten. Subtract one. Subtract ten. Add one. Add one. Add ten. Subtract ten. Where am I?





After the game, students should have their chip on the number 68. Discuss with students where their chip landed. Have a student demonstrate their movements on a large 99’s board. Encourage students to share their thinking and strategies during the game. Suggested questions include:

On what number did you begin? How did you know what number to move to when I said, “add ten?” Did you have to count out ten spaces on the board? If so, is there another way to add ten

without counting out all ten spaces? What strategies did you use to help you find the correct number quickly? What do you think is difficult about this game? How did you get past these difficulties?

Part IIStudents create number sentences that correspond to the set of directions.

Comments The teacher may begin with fewer clues the first few times the game is played. However, as the students become more efficient with adding and subtracting one and ten, the teacher should add more clues. The teacher could also speed up the wait time between clues.

Questions for Teacher Reflection What strategies are students using to add and subtract one and ten to a given number? Are students using the 99 chart to help them with counting on or counting back? Are students gaining fluency in the addition and subtraction of one and ten to a given

number? Can students explain the strategies they are using to add and subtract one and ten to a

given number?

DIFFERENTIATION

Extension Students create their own game directions and play the game with a partner.

Intervention Have students record each move with number sentences so they can see where they end

in relation to where they start.

Technology Connection Teachers may choose to place this on their interactive whiteboard for use in this activity. http://www.pixelpump.co.nz/flashmedia/hundreds3.swf but you will have to change the board from a hundred’s chart to a 99 chart.




http://www.pixelpump.co.nz/flashmedia/hundreds3.swf


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99





Learning Task: Calculator Skip Counting

This lesson is closely adapted from the NCTM E-example which can be found at the website http://standards.nctm.org/document/eexamples/chap4/4.5/index.htm.

STANDARDS ADDRESSED

M1N3. Students will add and subtract numbers less than 100 as well as understand anduse the inverse relationship between addition and subtraction. b. Skip-count by 2’s, 5’s, and 10’s forward and backwards – to and from numbers up to

100.






phenomena.

ESSENTIAL QUESTIONS

How can we use 99 chart to help us skip count?

MATERIALS

99 chart – 1 per student and demonstration chart Calculators Counting Manipulatives (chips, cubes, etc.) NCTM website http://standards.nctm.org/document/eexamples/chap4/4.5/index.htm




http://standards.nctm.org/document/eexamples/chap4/4.5/index.htm


GROUPING

Individual


Background Knowledge Counting by multiples begins at zero, whereas skip-counting can begin at any number. This task will explore both multiples and skip-counting. However, the definitions of multiples and skip-counting is not the focus of this task. Extending number patterns is one form of skip-counting. When students are “adding four” to complete a number pattern, they are really skip-counting by four.

Students should have previous experiences exploring 99 charts and calculators.

Part IGive each student a 99 board, a calculator, and counters. Throughout this activity, chart the results of the calculator skip counting on a demonstration 99 chart as the students use their own 99 chart.

Using the website: http://standards.nctm.org/document/eexamples/chap4/4.5/index.htm lead students in creating several number patterns using the calculator and 99’s chart simultaneously. Begin with counting by 5’s. Have students press five on the calculator and place a marker on the five on the 99 chart. Now, have students add five more on the calculator. Their calculator should show ten. Students should then place a marker on the ten on their board. Continue the process to 99.

Now you will need to have some discussion about where 100 is since you are using a 99 chart. So be sure to ask “Where is 100? Why is it not on this board? (because it is a three digit number, this board only contains one and two digit numbers) Where should it be? (at the beginning a the next row, on a new board). Why? (because it is the first three digit number, it starts a whole new family/place for numbers)

Encourage students to notice patterns with skip counting by asking questions such as: What patterns are you noticing? Can you predict which number will come next? Which number are we adding each time? How are the 99 board and calculator working together?

Variation: Students could use a bingo stamper or marker instead of counters to avoid movement of cubes.




http://standards.nctm.org/document/eexamples/chap4/4.5/index.htm


Part II Repeat Part I skip-count by 2’s and 10’s. Begin at 100 and skip-count backwards by 2’s, 5’s, and 10’s. Select random numbers and skip-count forward by 2’s, 5’s, and 10’s. Select random numbers and skip-count backwards by 2’s, 5’s, and 10’s.

Part IIIDraw an empty “number pattern” and write a number in one of the pattern spaces. Have students fill-in the remaining numbers based on directions given, such as:

Count forward by 2’s Count backward by 5’s

Examples:15

24

87

CommentsCreating patterns gives students opportunities to describe what is being repeated or how the pattern grows and to explain what should come next. Teachers can use colors and shapes to help highlight numerical patterns, such as the ones created by a first-grade class that marked every fifth and tenth day during the school year on a number line and a 99 chart using a highlighted square or triangle.

As students work with multiple representations of numbers and with counting, their knowledge of number concepts grows more sophisticated. When students place markers on 99 charts to highlight different counting sequences, they can see the relationships among the numbers and make connections between words and symbols. Linking numerical patterns to a visual display, as demonstrated in the interactive calculator and board is a way in which technology facilitates additional experiences, since students can view many different patterns in a brief time.





0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

Questions for Teacher Reflection What patterns are students noticing? What role does counting sequences play in helping students understand number

concepts? Are students able to predict the next number? What naturally occurring classroom activities could serve as a context for teaching

counting sequences? How can teachers use these various representations and the resulting conversations to

assess students' understanding and plan worthwhile instructional tasks?

DIFFERENTIATION

Extension Have students create their own number pattern with skip-counting. Students exchange

number patterns and complete the pattern following the “skip-counting rule.”

Intervention Some students may have difficulty managing the calculator and 99 board at the same

time. These students may benefit from beginning only with the 99 board during this lesson and adding the calculator at a later date.

Students can work with a partner. One student uses the calculator and the other student uses the 99 board to record the results.

0 1 2 3 4 5 6 7 8 9





10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99





Learning Task: Let’s Share

STANDARDS ADDRESSED

M1N4. Students will count collections of up to 100 objects by dividing them into equalparts and represent the results using words, pictures, or diagrams. a. Use informal strategies to share objects equally between two to five people.




phenomena.

ESSENTIAL QUESTIONS

How can a quantity be equally shared among two to five people?

MATERIALS

The Doorbell Rang by Pat Hutchins or A Fair Bear Share by Stuart J. Murphy

Paper Bags Paper Candies Manipulatives to serve as cookies

GROUPING

Individual


Part I Read aloud The Doorbell Rang by Pat Hutchins to introduce the concept of sharing objects equally (this begins the foundational development for division). As you read, model the scenario with students and cookies (manipulatives). Suggested questions include:





How many cookies did the children begin with? How can they share the cookies with __________ children? How many will each child get? Is it important to share things equally? Why? What do you think would have happened if they didn’t share the cookies equally? Have you ever had to share something with someone else? How did you do it? Were the

portions equal? What happened?

Part IIGive each student ten bag cutouts and twenty candy cutouts. Tell students a variety of scenarios. Have them place the correct number of bags in their work area according to your scenario. Then, have students sort the correct number of candies into equal groups.

Sample Scenario 1 Philip had 12 pieces of candy. He gave all his candy to his two best friends. How many pieces of candy did each of his friends get?

Sample Scenario 2Julie had 20 pieces of candy. She gave away her candy to five of her friends. How much candy did each friend receive?

As students are sharing, ask questions like: How many candy pieces do we have altogether? How many equal groups do you need? How many candy pieces are in each group? What strategies did you use to help sort the candy? Does each bag have the same amount of candy? If not, why not?

Continue with several different scenarios similar to the ones above. After each scenario, have students explain their processes and strategies for sharing the candy into equal groups and why it is important to do so.

Questions for Teacher Reflection Are students able to determine how many bags are necessary from listening to the

scenarios? Are students able to determine the total number of candies to sort from listening to the

scenarios? What strategies are students using to sort the candies into equal groups? Can students sort objects equally between two to five people? Can they explain why this

is an important thing to do?





DIFFERENTIATION

Extension Teachers may choose to expand the numbers of equal groups or the number of objects

that students begin with.

Intervention Students who are having difficulty may benefit from working in a small group

setting. During this time, the teacher can share actual objects with the students. Students can talk through strategies with the teacher and other students in the small group for sharing the objects equally between the students.













Learning Task: Bean Stick Place Value

STANDARDS ADDRESSED

M1N2. Students will understand place value notation for the numbers 1 to 99. (Discussions may allude to 3-digit numbers to assist in understanding place value.)b. Represent collections of less than 30 objects with 2-digit numbers and understand the

meaning of place value. c. Decompose numbers from 10 to 99 as the appropriate number of tens and ones.





phenomena.

ESSENTIAL QUESTIONS

How can we represent a number using tens and ones?

MATERIALS

beans glue craft sticks place value mat card set with various numbers from 1-99

GROUPING

Partners






Background Knowledge Students should have prior experiences counting numbers up to thirty and representing these numbers using tens and ones. Teachers may want to have a tub of beans and craft sticks for students to make the bean sticks prior to the task. The focus of the task is using the bean sticks rather than creating them.

Part IPlace 4 beans on the overhead projector. Keep the beans available for students to see for approximately 5 seconds. Cover them up and have students tell you how many beans were visible. Repeat the process with 9. Next, show students twenty-two beans. Ask students if they were able to determine how many beans were displayed. If so, ask students what strategies they used to determine the correct number of beans. Discuss with students how grouping the beans into groups of ten could have helped them count faster. Show students the number 34, but have the beans already grouped by ten. Discuss the benefit of grouping objects when counting.

Show students bean sticks with ten beans glued onto a craft stick. Ask students if this reminds them of any math tools (base-ten blocks) we use in our classroom. Draw connections between bean sticks and ten rods.

Have students create at least ten bean sticks for an activity tomorrow. Students should count out ten beans and then group them together by gluing them on a craft stick to create a bean stick.

Part IIReview the concept of grouping and place value by showing students numbers with beans on the overhead. Have students discuss the number of beans shown and what strategies they used to determine the number of beans shown.

Tell students that they will now work with a partner to play a place value game using their bean sticks. One student will pull a card and read the number to their partner. For example, 76. The partner will then have to place 7 ten bean sticks and 6 individual beans on his or her place value mat. The student will double check his or her partner’s work. If the partner is correct, he or she will get a point. Students can play until the teacher calls time. The student with the most correct numbers wins.

While students are playing, the teacher should walk around and observe the students as they create the numbers using the bean sticks. Suggested questions include:

Which number card did your friend choose? How many groups of ten are in your number? How many ones are in your number? What strategies are you using as you represent your number using tens and ones?





Questions for teacher reflection Are students able to group ten individual beans onto a bean stick? Are students able to represent the number shown on the number card using groups of ten

and ones? Can students explain their number representations? What strategies are students using to correctly represent the number? Are students mindfully checking their partner and engaging in conversations about how

to change examples that are not correct?

DIFFERENTIATION

Extension Have students who are comfortable with the ones and tens place value positions create a

bean representation for 100. Allow them some freedom here, do not force them to make a square (“flat”), see if they can come up with that configuration on their own.

Intervention Use smaller numbers, stay between 11-30 until students demonstrate a clearer

understanding of the task. Make sure students who are having difficulty are paired with a student who is very articulate with their mathematical thinking. Encourage more conversation about how they are choosing to represent the number with the beans and bean sticks.

Technology Connection Students may play this place value game for additional practice in working with tens and ones. http://www.dositey.com/addsub/tenoneex.htm





Learning Task: How Many Ways?

STANDARDS ADDRESSED

M1N1. Students will estimate, model, compare, order, and represent whole numbers up to 100. a. Represent numbers less than 100 using a variety of models, diagrams, and number

sentences. Represent numbers larger than 10 in terms of tens and ones using countersand pictures.

M1N2. Understand place value notation for the numbers between 1 and 100. (Discussionsmay allude to 3-digit numbers to assist in understanding place value.) c. Decompose numbers between 11 and 19 as one ten and the appropriate number of

ones.





phenomena.

ESSENTIAL QUESTIONS

How can we use different combinations of numbers and operations to represent the same quantity?

How can we represent a number in different ways?

MATERIALS

Chart paper Student task sheet

GROUPING

Individual




46



Background KnowledgeStudents should have experience representing numbers in a variety of ways including through place value models, tens/ones, money (coins/bills), number line, addition number sentences, subtraction number sentences, words, and pictures.

Part IGather students on the carpet. Select a number up to 100. Using chart paper and post-its have students represent the specified number in as many ways as possible.

Part IIGive students a number up to 100. Tell students to represent the number in as many ways possible. Let them know that you will be looking to see a variety of number sentences, pictures, models, and words to show the number. As students are working on their representations ask questions such as:

How does this represent the number ______ ? What other ways could you represent the number ________? What strategies did you use in creating these number sentences? Can you use both addition and subtraction in an example? (For instance if the number

they were making is 25 can they say 30 plus 5 minus 10)

Allow time for each student to share one way that they represented the number that is different from what others have presented. Have students explain their thinking about why they chose this representation and describe any strategies they may have used in developing their representation.

CommentsTeachers should begin with smaller numbers in the beginning of the year and increase to larger numbers throughout the year. The teacher should encourage students to use a variety of representations, including place value models, addition number sentences, subtraction number sentences, combination of addition and subtraction number sentences, words, and pictures.

Questions to consider for teacher reflection Are students able to use different combinations of numbers and operations to represent

the same quantity? Do students use a variety of representations for the number, including place value

models, addition number sentences, subtraction number sentences words, and pictures? What strategies are students using to correctly represent the number?





Performance Task: Two Digit Flip Book

STANDARDS ADDRESSED



M1N2. Understand place value notation for the numbers between 1 and 100. (Discussionsmay allude to 3-digit numbers to assist in understanding place value.)

b. Represent collections of less than 30 objects with 2-digit numbers and understand the meaning of place value.

c. Decompose numbers from 10 to 99 as the appropriate number of tens and ones.







phenomena.

ESSENTIAL QUESTIONS

How can we represent a number in variety of ways?





MATERIALS

pre-made flip books- You can use this website to help you make flip books http://www.readwritethink.org/materials/flipbook/

Various counting manipulatives White boards, paper, etc.

GROUPING

Individual


Background InformationStudents should be familiar with the representation of numbers through pictures, base ten models, words, numerals, addition number sentences, and subtraction number sentences. Teachers should also have a math word wall with the spellings of various number words available for students.

Revisit the task How Many Ways (can I make the number___) with students. Give students a number such as 46 and have them represent it in as many ways as possible using a piece of paper or white board.

After students are warmed up, tell students that today they will make a flip book to show how they can represent a number in several different ways. Tell students that they will write the number using numerals on the cover page of the flip book. Then, they will represent the number in several different ways for the remainder of the flip book. Each page will be labeled with the sections picture, base ten models, numeral, words, addition number sentences, and subtraction number sentences. Give each student a number based on their personal developmental level. As students are working on their flipbooks, ask questions such as:

Which number were you given? What picture do you see in your head when you hear/see this number? How will you represent this number in words? Where can you look in the room to help you with the spelling of your number word? How many groups of ten are in your number? How many ones are in your number? What strategies are you using as you represent your number using tens and ones? How many addition number sentences can you create for your number? How many subtraction number sentences can you create for your number? How will you self-check these number sentences?

Questions to consider for teacher reflection





Are students able to represent their number using pictures, words, number sentences, and base ten models?

Can students explain their number representations? What strategies are students using to correctly represent the number?

Technology Connection Teachers may use the website: http://www.readwritethink.org/materials/flipbook/ to assist with the creation of the two-digit flipbooks.





Performance Task: Exploring Coins

STANDARDS ADDRESSED

M1N1. Students will estimate, model, compare, order, and represent whole numbers up to 100. e. Exchange equivalent quantities of coins by making fair trades

involving combinationsof pennies, nickels, dimes, and quarters, and count out a

combination needed to purchase items up to a dollar.






phenomena.

ESSENTIAL QUESTIONS

How can we identify the values of various coins?

MATERIALS

Various coins Bill Observation Sheet Class Bill Observation Chart

GROUPING





Individual, then Partners


Background Knowledge

Students should be familiar with the values of coins. Students should also be able to count small sets of coins.

Part I Review the coins and their values with students using large coins and various coin poems and rhymes. Shel Silverstein’s poem SMART is excellent. Teacher may want to record poems/songs on chart paper.

My dad gave me one dollar bill'Cause I'm his smartest son, And I swapped it for two shiny quarters'Cause two is more than one!

And then I took the quartersAnd traded them to LouFor three dimes-I guess he don't knowthat three is more than two!

Just them, along came old blind BatesAnd just 'cause he can't seeHe gave me four nickels for my three dimes, And four is more than three!

And I took the nickels to Hiram CoombsDown at the seed-feed store, and the fool gave me five pennies for them, And five is more than four!

And then I went and showed my dad, and he got red in the cheeksAnd closed his eyes and shook his head-Too proud of me to speak!

Here are other examples of coin value rhymes. The authors are unknown.

Penny, PennyPenny, penny, easily spent -Copper brown and worth one cent. Nickel, Nickel, thick and fat.It’s worth 5 cents – I know that.





Dime, dime, little and thin.I remember – you’re worth 10.

Quarter, quarter, big and bold.It’s worth 25 – I am told!

Dollar Holler Rap

My sweetie knows a song. It’s really kind of funny.It’s all about coins – And learning about money.

Pennies, Nickels, Dimes, and QuartersPennies, Nickels, Dimes, and QuartersNow a Penny means 1. (point 1)And a Nickel means 5. (flash 5)Dimes are worth 10. (show 10)And quarters twenty-five. (hands make 25)

Pennies, Nickels, Dimes, and QuartersPennies, Nickels, Dimes, and Quarters

Five pennies in a nickel. Two nickels in a dime.Five nickels in a quarter.You’ll know it every time.

Pennies, Nickels, Dimes, and QuartersPennies, Nickels, Dimes, and Quarters

Part IIAllow students to play this banking game with a partner. Give each student an envelope with a value on the envelope, for example forty cents. Have the students choose coins to place into the envelope that add up to forty cents. Once students have filled their envelopes with the correct amount, have students trade envelopes with their partner. The partner should check to see that the student has filled the envelope with the correct amount of money.

While students are working the teacher should circulate around the room asking questions such as:

What value is written on your envelope? What collection of coins did you use to represent that amount? Is there another combination of coins you could use to represent that value? What strategies are you using to count your coins?





How are you checking your partner’s work?

Part IIIStudents create their own envelope sheet.

Questions to consider for teacher reflection Are students able to describe the characteristics of the various coins? Can students purchase an item using the correct collection of coins? What strategies are students using to count out a collection of coins? Are students able to make more than one representation for a given amount?

DIFFERENTIATION

Extension Give more advanced students larger amounts on their envelopes. Have students represent money amounts in two different ways.

Intervention Some students may have difficulty counting sets of mixed coins. Start these students off

with coins of the same denomination. For example, using only nickels to make $0.35. Use a hundreds board to connect money to skip-counting.

Technology ConnectionExplore the U.S. Mint website for kids at http://www.usmint.gov/kids/. Discuss the U.S. Mint and take time to learn more about the making of money. There are also games for children to play.




http://www.usmint.gov/kids/


Learning Task: Exploring Bills

STANDARDS ADDRESSED

M1N1. Students will estimate, model, compare, order, and represent whole numbers up to

100. f. Identify bills ($1, $5, $10, $20) by name and value and exchange equivalent

quantities by making fair trades involving combinations of bills; count out a combination of bills needed to purchase items that total up to twenty dollars.






ESSENTIAL QUESTIONS

How can we identify the values of various bills?

MATERIALS

Various fake bills Bill Observation Sheet Class Bill Observation Chart

GROUPING





IndividualTASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Background InformationStudents should have previous experience working with money in the sense of using coins. However, this will be many students’ first experiences in adding dollar amounts.

Part I Give students the Bill Observation Sheet. Begin by giving students a dollar bill. Allow students to share various characteristics observed about their dollar bill and record them on their Bill Observation Sheet. Have students share some of these characteristics to add to the class Bill Observation Chart. Repeat the process with the five dollar bill, ten dollar bill, and twenty dollar bill. While students are making their observations, ask questions such as:

What do you notice about the one dollar bill? What numbers do you see on this bill? What type of pictures do you see on this bill? How much is this bill worth? How can you tell? What are similarities between the bills? Differences?

Part IIAllow students to play a shopping game with a partner. Have one partner play the cashier and the other partner play the shopper. The shopper chooses a picture card which they would like to purchase. He or she must pay the cashier that amount of money using the exact number of dollar bills.

CommentDo this as a class activity first using only one dollar bills. After several examples ask “Is there another way we could make 7 dollars rather than using all one dollar bills?” If students have difficulty thinking about a different representation then asking a few more probing questions such as: Could we make it using a ten dollar bill? Why not? What if we start with the 5 dollar bill, is that enough? How do you know? What else would we need to make 7 dollars? Once students are able to represent an amount in two ways, put them with a partner to play the shopping game.

Variation Have students cut out pictures from magazines of items they like. Then let them place a dollar amount on the pictures and add these items (pictures) to the “store.”

While students are shopping, ask questions such as: How much does that item cost? Which bills can you use to purchase that item? Is there another combination of bills you could use to purchase that item? How are you counting your bills?





Questions for Teacher Reflection Are students able to describe the characteristics of the various bills? Can students purchase an item using the correct collection of bills? What strategies are students using to count out a collection of bills?

DIFFERENTIATION

Extension Allow students to purchase more than one object at a time. Have students represent money amounts in two different ways. Give students different object cards to purchase with prices that mix dollars and cents.

For example: $7.25. The student will have to purchase the objects using both bills and coins.

Intervention Some students may have difficulty counting sets of mixed bills. Start these students off

with bills of the same denomination. For example, continue with the class example where we were only using one dollar bills to make $7.00. Next they could make fair trades.

Technology Connection Explore the U.S. Mint website for kids at http://www.usmint.gov/kids/. Discuss the U.S. Mint and take time to learn more about the making of money. There are also games for children to play.




http://www.usmint.gov/kids/


Name: ____________________________ Date: _______________

Bill Observation Sheet

$6.00 $20.00 $4.00 $12.00

$9.00 $15.00 $2.00 $13.00




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CULMINATING TASK : My Story Problem Book

This culminating task represents the level of depth, rigor, and complexity expected of all first grade students to demonstrate evidence of learning.

STANDARDS ADDRESSED



M1N3. Students will add and subtract numbers less than 100 as well as understand anduse the inverse relationship between addition and subtraction. h. Solve and create word problems involving addition and subtraction to 100 without

regrouping. Use words, pictures and concrete models to interpret story problems and reflect the combining of sets as addition and taking away or comparing elements of sets as subtraction.


M1P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.











phenomena.

ESSENTIAL QUESTIONS

How can we represent problem situations?

MATERIALS

Story Problem Book for each child Various Manipulatives Pencils and crayons

GROUPING

Individual


Background Knowledge In order to complete this culminating activity, students should have prior experiences solving various problem types.

Students will create a story problem book that represents at least three different types of story problems from the lessons used during this unit. Types of lessons that students may create include combining, separating, and comparing problems. Students may also choose to incorporate money and the comparing terms for greater than, less than, or equal to within their story problems. After the creation of their story problems, the students should solve these problems using numbers, pictures, and words.

In supporting the writing of story problems, encourage the inclusion of at least one of each of the various types of problems that have been explored and solved during this unit. If necessary, add additional problems to the book to insure that this variety is represented. Create a Grade 1 Problem-Solving Book for each student. You may want to extend this task by asking students to illustrate the problem. If so, when printing pages for the book, provide enough space for students to illustrate the problem as well as record their solution strategies.

While students are solving problems, you may wish to note what students are able to do on their own (without your support) before providing support needed for successful completion of the task (making personal notes about what support was needed by individual students).





After the peer review, ask the peers to share one of their partner’s solutions, focusing on the type of problem selected, the solution strategy used, and the ways the solutions were represented. Ask students to comment on how problems are alike and different and how solutions are alike and different.

During the revision/improvement process, provide support to individual students as needed.

Questions for Teacher Reflection Are students able to create and solve various types of problems? Are students able to distinguish between different problem types? Are students able to solve their problems using pictures, numbers, and words? Are students able to explain another students’ work? Are they able to understand

someone else’s pictures/representations? Can they explain how it was different or similar to what someone else did?




Documents

Unit One Organizer:€¦ · Web viewh. Solve and create word problems involving addition and subtraction to 100 without. regrouping. Use words, pictures and concrete models to interpret