Fourth Grade Unit 4: Operations with Fractions2 weeksIn this unit students will: Identify visual and written representations of fractions Understand representations of simple equivalent fractions Understand the concept of mixed numbers with common denominators to 12 Add and subtract fractions with common denominators Add and subtract mixed numbers with common denominators Convert mixed numbers to improper fractions and improper fractions to mixed fractions

Understand a fraction ab as a multiple of

1b . (for example: model the product of

34 as 3 x

14 ).

Understand a multiple of ab as a multiple of

1b , and use this understanding to multiply a fraction by a whole number.

Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. Multiply a whole number by a fraction

Make a line plot to display a data set of measurements in fractions of a unit ( 12 , 14 , 18 )

Solve problems involving addition and subtraction of fractions by using information presented in line plots Solve multi-step problems using the four operations

Unit ResourcesUnit 4 Overview video Parent Letter Number Talks Resources Vocabulary Cards Prerequisite Skills Assessment Sample Post Assessment

Topic 1: Operations with FractionsBig Ideas/Enduring Understandings: Fractions can be represented in multiple ways including visual and written form. Fractions can be decomposed in multiple ways into a sum of fractions with the same denominator. Fractional amounts can be added and/or subtracted. Mixed numbers can be added and/or subtracted. Mixed numbers and improper fractions can be used interchangeably because they are equivalent. Mixed numbers can be ordered by considering the whole number and the fraction. Proper fractions, improper fractions and mixed numbers can be added and/or subtracted. Fractions, like whole numbers can be unit intervals on a number line. Fractional amounts can be added and/or multiplied. If given a whole set, we can determine fractional amounts. If given a fractional amount we can determine the whole set. When multiplying fractions by a whole number, it is helpful to relate it to the repeated addition model of multiplying whole numbers.

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A visual model can help solve problems that involve multiplying a fraction by a whole number. Equations can be written to represent problems involving the multiplication of a fraction by a whole number.

Multiplying a fraction by a whole number can also be thought of as a fractional proportion of a whole number. For example, 14 x 8 can be interpreted as finding

one-fourth of eight. Data can be measured and represented on line plots in units of whole numbers or fractions. Data can be collected and used to solve problems involving addition or subtraction of fractions.Essential Questions (Select a few questions based on the needs of your students): How are fractions used in problem-solving situations? How can equivalent fractions be identified? How can a fraction represent parts of a set? How can I add and subtract fractions of a given set? How can I find equivalent fractions? How can I represent fractions in different ways? How are improper fractions and mixed numbers alike and different? How can you use fractions to solve addition and subtraction problems? How do we add fractions with like denominators? How do we apply our understanding of fractions in everyday life? What do the parts of a fraction tell about its numerator and denominator? What happens when I add fractions with like denominators? What is a mixed number and how can it be represented? What is an improper fraction and how can it be represented? What is the relationship between a mixed number and an improper fraction? Why does the denominator remain the same when I add fractions with like denominators? How can I model the multiplication of a whole number by a fraction? How can I multiply a set by a fraction? How can I multiply a whole number by a fraction? How can I represent a fraction of a set? How can I represent multiplication of a whole number? How can we model answers to fraction problems? How can we write equations to represent our answers when solving word problems? How do we determine a fractional value when given the whole number? How do we determine the whole amount when given a fractional value of the whole? How is multiplication of fractions similar to repeated addition of fraction? What does it mean to take a fractional portion of a whole number? What strategies can be used for finding products when multiplying a whole number by a fraction? How do we make a line plot to display a data set?

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Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

MGSE4.NF.3 Understand a fraction ab with a numerator >1 as a sum of unit fractions 1.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify

decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of

operations and the relationship between addition and subtraction.d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models

and equations to represent the problem.MGSE4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number e.g., by using a visual such as a number line or area model.a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the

equation 5/4 = 5 × (1/4).b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to

express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For

example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Represent and interpret data.

MGSE4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (12 ,14 ,18 ). Solve problems involving addition and subtraction of fractions with

common denominators by using information presented in line plots. For example, from a line plot, find and interpret the difference in length between the longest and shortest specimens in an insect collection.Use the four operations with whole numbers to solve problems.MGSE4.OA.3 Solve multistep word problems with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a symbol or letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Vertical ArticulationThird Grade Fraction, Measurement, & Problem-Solving StandardsDevelop understanding of fractions as numbers

MGSE3.NF.1 Understand a fraction 1b as the quantity formed by 1 part when

Fifth Grade Fraction, Measurement, & Problem-Solving StandardsUse equivalent fractions as a strategy to add and subtract fractions MGSE5.NF.1 Add and subtract fractions and mixed numbers with unlike

denominators by finding a common denominator and equivalent fractions to produce like denominators.

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a whole is partitioned into b equal parts (unit fraction); understand a fraction ab

as the quantity formed by a parts of size1b

. For example, 34

means there are three 14

parts, so 34 = 14 + 14 + 14 .

MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.

a. Represent a fraction 1b on a number line diagram by defining the interval

from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that

each part has size 1b

. Recognize that a unit fraction 1b

is located 1b

whole unit

from 0 on the number line. b. Represent a non-unit fraction ab on a number line diagram by marking off a

lengths of 1b (unit fractions) from 0. Recognize that the

resulting interval has size ab and that its endpoint locates the non-unit

fraction ab on the number line.

MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models. Compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions with denominators of 2,

3, 4, 6, and 8, e.g., 12 = 24 , 46 = 23 . Explain why the fractions are equivalent,

e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are

equivalent to whole numbers. Examples: Express 3 in the form 3 = 62 (3 wholes

is equal to six halves); recognize that 31 = 3; locate

44 and 1 at the same point

of

MGSE5.NF.2 Solve word problems involving addition and subtraction of fractions, including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + ½ = 3/7, by observing that 3/7 < ½.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions. MGSE5.NF.3 Interpret a fraction as division of the numerator by the

denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Example: 35 can be interpreted as “3 divided by 5 and as 3 shared by 5”.

MGSE5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.a. Apply and use understanding of multiplication to multiply a fraction or

whole number by a fraction. Examples: ab×𝑞 as

ab×q1 and

ab×cd =acbd

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths.

MGSE5.NF.5 Interpret multiplication as scaling (resizing), by:a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. Example: 4 x 10 is twice as large as 2 x 10.b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

MGSE5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

MGSE5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

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a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.Represent and interpret data. MGSE3.MD.4 Generate measurement data by measuring lengths using rulers

marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

Solve problems involving the four operations, and identify and explain patterns in arithmetic. MGSE3.OA.8 Solve two-step word problems using the four operations.

Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Represent and interpret data. MGSE5.MD.2 Make a line plot to display a data set of measurements in

fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Instructional StrategiesBUILD FRACTIONS FROM UNIT FRACTIONS BY APPLYING AND EXTENDING PREVIOUS UNDERSTANDINGS OF OPERATIONS ON WHOLE NUMBERSStudents extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: operations, addition/joining, subtraction/separating, fraction, unit fraction, equivalent, multiple, reason, denominator, numerator, decomposing, mixed number, rules about how numbers work (properties), multiply, multiple.

Standard NF.3In Grade 3, students added unit fractions with the same denominator. Now, they begin to represent a fraction by decomposing the fraction as the sum of unit fraction

and justify with a fraction model. For example, 34=14 +14 +14

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Students also represented whole numbers as fractions. They use this knowledge to add and subtract mixed numbers with like denominators using properties of number and appropriate fraction models. It is important to stress that whichever model is used, it should be the same for the same whole. For example, a circular model and a rectangular model should not be used in the same problem.Understanding of multiplication of whole numbers is extended to multiplying a fraction by a whole number. Allow students to use fraction models and drawings to show their understanding.

NF.3a – Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such as 23 , they should be able to join

(compose) or separate (decompose) the fractions of the same whole.

Example: 23= 13+ 13 - Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions.

Students need multiple opportunities to work with mixed numbers and be able to decompose them in more than one way. Students may use visual models to help develop this understanding:

Example: 114 –

34 =? →

44 +

14 = 54 →

54 − 34 = 24 or 12

Example of word problem:

Mary and Lacey decide to share a pizza. Mary ate 36 and Lacey ate

26of the pizza. How much of the pizza did the girls eat together?

Possible solution: The amount of pizza Mary ate can be thought of as 36 or

16 +

16 +

16 . The amount of pizza Lacey ate can be thought of as

16 +

16 . The total amount of

pizza they ate is 16+ 16+ 16+ 16+ 16 or

56 of the pizza.

NF.3b - Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

Students should justify their breaking apart (decomposing) of fractions using visual fraction models. The concept of turning mixed numbers into improper fractions

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needs to be emphasized using visual fraction models and decomposing.

Example:

NF.3c - Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

A separate algorithm for mixed numbers in addition and subtraction is not necessary. Students will tend to add or subtract the whole numbers first and then work with the fractions using the same strategies they have applied to problems that contained only fractions.

Example:

Susan and Avery need 838 feet of ribbon to package gift baskets. Susan has 318 feet of ribbon and Avery has 538 feet of ribbon. How much ribbon do they have

altogether? Will it be enough to complete the project? Explain why or why not.

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The student thinks: I can add the ribbon Susan has to the ribbon Avery has to find out how much ribbon they have altogether.

Susan has 318 feet of ribbon and Avery has 5

38 feet of ribbon. I can write this as 3

18 + 5

38 . I know they have 8 feet of ribbon by adding the 3 and 5. They also have

18

and 38 which makes a total of

48 more. Altogether they have 8

48 feet of ribbon. 8

48 is larger than 8

38 so they will have enough ribbon to complete the project. They will

even have a little extra ribbon left, 18 foot.

Example:

Timothy has 418pizzas left over from his soccer party. After giving some pizza to his friend, he has 2

18 of a pizza left. How much pizza did Timothy give to his friend?

Solution:

Timothy had 418 pizzas to start. This is

338 of a pizza. The x’s show the pizza he has left which is 2

48 pizzas or

208 pizzas. The shaded rectangles without the x’s are the

pizza he gave to his friend which is 138 or 1

58 pizzas.

Mixed numbers are formally introduced for the first time in Fourth Grade. Students should have ample experiences of adding and subtracting mixed numbers where they work with mixed numbers or convert mixed numbers into improper fractions. Keep in mind Concrete-Representation-Abstract (CRA) approach to teaching fractions. Students need to be able to “show” their thinking using concrete and/or representations BEFORE they move to abstract thinking.

Example:

While solving the problem, 334 + 2

14 , students could do the following:

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NF.3d – Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

Example:

A cake recipe calls for you to use ¾ cup of milk, 14 cup of oil, and

24 cup of water. How much liquid was needed to make the cake? Use an area model to solve.

About the Math: Addition and subtraction of fractions with like denominators can easily be solved using an algorithm of adding or subtracting the numerators and keeping the same denominator. However, prior to this, students need instruction on the conceptual understanding of adding and subtracting fractions and what a reasonable answer looks like. Questions such as will the answer when you add 7/8 + 3/8 be more or less than a whole and why, should be part of instruction. Concrete materials should be used to introduce addition and subtraction prior to moving to the algorithm.Students need to see that a fraction can be decomposed just like whole numbers. There are many different ways to decompose a fraction. Understanding this helps students see the value of fractions and enhances their fraction sense.7/10 = 1/10 + 1/10 +1/10 + 1/10 +1/10 + 1/10 + 1/10 or 3/10 + 3/10 + 1/10 or 5/10 + 2/10 or 4/10 + 2/10 + 1/10.A mixed number is a whole number and a fraction. Students need to see that 3 ¼ is the same as 3 + ¼. This can be connected to the previous standard of decomposing fractions.

Drawing a picture and writing an equation are two effective strategies when solving word problems with fractions. Writing an equation helps students translate word

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phrases into numbers. Encourage students to use these two strategies instead of looking for key words. Looking for key words should be avoided because key words can indicate different operations depending on the context in the problem.

Standard NF.4NF.4a - Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

This standard builds on students’ work of adding fractions and extending that work into multiplication.

Example: 36= 16+ 16+ 16= 3 ×

16

Number Line:

Area Model:

Students need many opportunities to work with problems in context to understand the connections between models and corresponding equations. Contexts involving a whole number times a fraction lend themselves to modeling and examining patterns. This standard builds on students’ work of adding fractions and extending that work into multiplication.

Examples:

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If each person at a party eats 38 of a pound of roast beef, and there are 5 people at the party, how many pounds of roast beef are needed? Between what two whole

numbers does your answer lie?

A student may build a fraction model to represent this problem:

About the math: A unit fraction is a fraction that describes one part of the whole. Unit fractions always have a numerator of one. Students need to see that fraction parts can be counted, just like we count whole numbers. So if we count 1 orange, 2 oranges, 3 oranges, etc., we can also count 1 fourth, 2 fourths and 3 fourths. So if 3 oranges can be thought of as 3 groups of one orange, then 3/4 can be represented as 3 groups of ¼ or 3 x 1/4.

Prior to teaching the procedure for multiplying a whole number by a fraction, students need to understand conceptually why the answer is reasonable. If I have 5 groups 4th Grade Unit 4 11 2015-2016

of 1/6, how many 1/6s are there? You can add 1/6 + 1/6 + 1/6 + 1/6 + 1/6 to equal 5/6. If you want to multiply 4 x 2/3, students need to think of this as 4 groups of 2/3 or 8 groups of 1/3. When added to show four groups of two-thirds, 2/3 + 2/3 + 2/3 + 2/3 = 8/3 or 2 2/3. Or this can be shown as 4 X 2/3 or 4/1 x 2/3= 8/3 or 2 2/3.

NF.4b - Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

This standard extends the idea of multiplication as repeated addition.For example,

Students are expected to use and create visual fraction models to multiply a whole number by a fraction.

About the Math: Students may need to explore the concept of doubling one factor while halving another factor with whole numbers before being able to understand the concept with fractions.  When exploring equality (e.g., 5 x 14 = 10 x 7), use arrays to focus on the physical properties of doubling and halving to build conceptual understanding, rather than just teaching the rule abstractly. 

NF.4c - Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

This standard calls for students to use visual fraction models to solve word problems related to multiplying a whole number by a fraction.

Example:

In a relay race, each runner runs 12 of a lap. If there are 4 team members how long is the race?

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Example:

Heather bought 12 plums and ate 13 of them. Paul bought 12 plums and ate

14 of them. Which statement is true? Draw a model to explain your reasoning.

a. Heather and Paul ate the same number of plums.b. Heather ate 4 plums and Paul ate 3 plums.c. Heather ate 3 plums and Paul ate 4 plums.d. Heather had 9 plums remaining.Possible student solutions:If both Heather and Paul bought the same amount of plums, all I need to do is compare what fraction of the plums each person ate. I know that one third is larger than one fourth so Heather must have eaten more plums. The correct solution must be that Heather ate 4 plums and Paul ate 3 plums.

I know that Heather ate one third of 12 which is four because 13 x 12 is

123 , or 4. I know that Paul ate one fourth of 12, which is three because

14 x 12 is

124 , or 3. The

correct answer is choice b.

About the Math: Students need to understand what a reasonable answer looks like when multiplying fractions. If students have been led down the incorrect idea that when you multiply whole numbers the product is always greater. Giving students “rules” such as that sets them up for confusion as they continue to learn about rational numbers. This is not true when dealing with fractions. Students should focus on what an answer will look like prior to actually calculating the answer. Questions like: Does this answer make sense? How do you know it is going to be less than a certain number? Using word problems in a context helps students make sense.

Standard MD.4This standard provides a context for students to work with fractions by measuring objects to an eighth of an inch. Students are making a line plot of this data and then adding and subtracting fractions based on data in the line plot.

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Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: data, line plot, length, fractions.

Example:

Students measured objects in their desk to the nearest 12 , 14 , or

18 inch. They displayed their data collected on a line plot. How many objects measured

14 inch?

12

inch? If you put all the objects together end to end what would be the total length of all the objects?

Student Solution:

Since 28 =

14 , there are three objects that measured

14 of an inch. Since

48 is equal to

12 , there are 2 objects that have a length of

12 of an inch. The total length of all

the objects is 88 +

48 + 48 +

38 +

18 which is 2

48 inches. Then, add

28 +

28 +

28 +

68= 128 , which is 1

48 inches.

2 48 inches + 1

48 inches is 4 inches in total length.

Example:Ten students in Room 31 measured their pencils at the end of the day. They recorded their results on the line plot below.

Possible questions:4th Grade Unit 4 14 2015-2016

• What is the difference in length from the longest to the shortest pencil?• If you were to line up all the pencils, what would the total length be?

• If the 518" pencils are placed end to end, what would be their total length?

Data has been measured and represented on line plots in units of whole numbers, halves or quarters. Students have also represented fractions on number lines. Now students are using line plots to display measurement data in fraction units and using the data to solve problems involving addition or subtraction of fractions.

Have students create line plots with fractions of a unit (12 ,14 , or

18 ) and plot data showing multiple data points for each fraction.

Pose questions that students may answer, such as: “How many one-eighths are shown on the line plot?” Expect “two one-eighths” as the answer. Then ask, “What is the total of these two one-eighths?” Encourage students to count the fractional numbers as they would with whole-number counting, but using

the fraction name.

“What is the total number of inches for insects measuring 38 inches?” Students can use skip counting with fraction names to find the total, such as, “three-eighths,

six-eighths, nine-eighths. The last fraction names the total. Students should notice that the denominator did not change when they were saying the fraction name. Have them make a statement about the result of adding fractions with the same denominator.

“What is the total number of insects measuring 18 inch or 58 inches?” Have students write number sentences to represent the problem and solution such as, 18 +

18 +

58= 78 inches.

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Use visual fraction strips and fraction bars to represent problems to solve problems involving addition and subtraction of fractions.

Standard OA.3The focus of standard OA.3 is to have students use and discuss various strategies. It refers to estimation strategies, including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be structured so that all acceptable estimation strategies will arrive at a reasonable answer. Students need many opportunities solving multistep story problems using all four operations.

This standard references interpreting remainders. Remainders should be put into context for interpretation. Ways to address remainders:• Remain as a left over• Partitioned into fractions or decimals• Discarded leaving only the whole number answer• Increase the whole number answer by one• Round to the nearest whole number for an approximate resultExamples: On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day. About how many miles did they travel

total? Some typical estimation strategies for this problem:

Your class is collecting bottled water for a service project. The goal is to collect 300 bottles of water. On the first day, Max brings in 3 packs with 6 bottles in each container. Sarah wheels in 6 packs with 6 bottles in each container. About how many bottles of water still need to be collected?

Chris bought clothes for school. She bought 3 shirts for $12 each and a skirt for $15. How much money did Chris spend on her new school clothes?4th Grade Unit 4 16 2015-2016

3 × $12 + $15 = 𝑎 There are 29 students in one class and 28 students in another class going on a field trip. Each car can hold 5 students. How many cars are needed to get all the

students to the field trip? (12 cars, one possible explanation is 11 cars holding 5 students and the 12th holding the remaining 2 students) 29 + 28 = 11 × 5 + 2

Write different word problems involving 44 ÷ 6 = ? where the answers are best represented as: Problem A: 7 Problem B: 7 r 2 Problem C: 8 Problem D: 7 or 8 Problem E: 7 26

Possible solutions: Problem A: 7 Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches did she fill? 44 ÷ 6 = p; p = 7 r 2. Mary can fill 7 pouches completely.

Problem B: 7 r 2 Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches could she fill and how many pencils would she have left? 44 ÷ 6 = p; p = 7 r 2; Mary can fill 7 pouches and have 2 pencils left over.

Problem C: 8 Mary had 44 pencils. Six pencils fit into each of her pencil pouches. What is the fewest number of pouches she would need in order to hold all of her pencils? 44 ÷ 6 = p; p = 7 r 2; Mary needs 8 pouches to hold all of the pencils. Problem D: 7 or 8 Mary had 44 pencils. She divided them equally among her friends before giving one of the leftovers to each of her friends. How many pencils could her friends have received? 44 ÷ 6 = p; p = 7 r 2; Some of her friends received 7 pencils. Two friends received 8 pencils.

Problem E: 7 26

Mary had 44 pencils and put six pencils in each pouch. What fraction represents the number of pouches that Mary filled?

44 ÷ 6 = p; p = 7 26 ; Mary filled 7

26 pencil pouches.

There are 128 students going on a field trip. If each bus held 30 students, how many buses are needed? (128 ÷ 30 = b; b = 4 R 8; They will need 5 buses because 4 buses would not hold all of the students).

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Students need to realize in problems, such as the example above, that an extra bus is needed for the 8 students that are left over.

Estimation skills include identifying when estimation is appropriate, determining the level of accuracy needed, selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of situations using various estimation strategies. Estimation strategies include, but are not limited to: front-end estimation with adjusting (using the highest place value and estimating from the front end, making adjustments to the estimate by taking into account

the remaining amounts), clustering around an average (when the values are close together an average value is selected and multiplied by the number of values to determine an estimate), rounding and adjusting (students round down or round up and then adjust their estimate depending on how much the rounding affected the original values), using friendly or compatible numbers such as factors (students seek to fit numbers together - e.g., rounding to factors and grouping numbers together that have

round sums like 100 or 1000), using benchmark numbers that are easy to compute (students select close whole numbers for fractions or decimals to determine an estimate).Common MisconceptionsNF.3 & NF.4 - Students think that it does not matter which model to use when finding the sum or difference of fractions. They may represent one fraction with a rectangle and the other fraction with a circle. They need to know that the models need to represent the same whole.

MD.4 - Students use whole-number names when counting fractional parts on a number line. The fraction name should be used instead. For example, if two-fourths is represented on the line plot three times, then there would be six-fourths.Differentiation-Increase the RigorStandard NF.3a

Why is ¼ + ¼ not equal to 28 ?

Will 6/10 + 6/10 be greater than one whole? Explain how you know. How can you figure out what ¼ + ½ is without finding a common denominator? Explain how you found the answer. Decompose 11/12 in three different ways. Ester ate ¾ of a small pizza on Monday, and she ate ¼ of a large pizza on Tuesday. She says that she has eaten a whole pizza. Is she correct? Why or why not?

Explain your answer. Think of a time in your life where you might need to add or subtract fractions.

Standard NF.3b How many ways can you decompose 7/8? Show your representations. What set of numerators can you find to make this equation true? Is there a different answer? 5/10 + 1/10 + 4/10 = __/10 + __/10 + __/10 Lily is having a sleepover with 2 friends. They order one party size submarine sandwich, and it is cut into 12 equal parts. They eat the entire sandwich, but each

person has a different number of parts. What is one way the sandwich was shared? Write an equation to represent your answer equal to 12/12. Is there a different way the friends could have shared the sandwich?

Play “Can You Get There In _____ Jumps.” Using a number line, give students a fraction such as 10/12. Ask students to get there in 3 jumps on the number line. One answer might be 2/12, to 5/12, to 10/12 and can be represented by 2/12 + 3/12 + 5/12. Then ask if they can get there in 4 jumps?

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Standard NF.3c If the sum of two mixed numbers is 6, what could the two addends be? Is the sum of 4 2/5 + 2 4/5 over or under 7? Explain why or why not. Is the difference of 5 2/3 – 3 1/3 over or under 2? Explain why or why not. How can you demonstrate how 4 ½ is equal to 9/2? Use models, drawings, and/or equations to explain your thinking. Write two problems that have a difference of 3 4/10.

Standard NF.3d At noon, the bakery had 1 whole pumpkin pie and 5/12 of a pumpkin pie available to sell. At the end of the day, 3/12 of a pie was left. How much pumpkin pie did

the bakery sell during the afternoon? Shelly needs 1 3/8 cups of oats for a cookie recipe. How many cups of oats does Shelly need if she is tripling the recipe? (This question is not exclusive to

multiplication of fractions; repeated addition can be used). The answer is 5 1/6. Write a story problem involving addition and/or subtraction to result in this answer. Joan has 3 ¾ foot of yarn. She uses 1 ¼ foot of the yarn to make a bracelet. Then she gave her sister 1 ¼ foot of yard for her bracelet. How much yarn does she

have left? Represent this problem visually. Don came home and found a fraction of a large pizza on the counter. He eats 3/8 of the pizza and now there is 2/8 of a pizza left. What fraction of the pizza was on

the counter when he got home?

Standard NF.4a Josh noticed that 1/3 + 1/3 + 1/3 + 1/3 was the same as 4 x 1/3. Do you agree or disagree with Josh’s observation? Explain your thinking. What pattern do you notice when multiplying a whole number by a fraction? Why do you think this pattern occurs? When multiplying a whole number by a fraction, what happens to your product? Why is your product less than your original whole number? What two factors can be multiplied to equal a product of 6/8?

Standard NF.4b What is the relationship between decomposing fractions and multiplying fractions by a whole number? (halving and doubling) Why is 3 x 2/6 the same as 6 x 1/6? What equation could equal the same product as 2 x 8/10?

Ab Standard NF.4c Kim runs 2/3 mile every day. How far does she run in one week? Between what 2 whole numbers does then answer lie? Ms. Howard is making punch. The punch uses ¾ pint of orange juice for one serving. If she makes 8 servings, will she use more than 8 pints or less than 8 pints?

Draw a model to justify your reasoning.

Ab Standard MD.4 What is the relationship between line plots and number lines? What questions could be answered by analyzing the data on this line plot? Write down as many questions as you can think of. Questions can include what is the

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difference between the longest and shortest amount of time studying? How many students study 45 minutes or less?

Ten people collect data on how far a frog jumps. (Lengths ¼ foot = 2 jumps; ½ foot = 5 jumps; 1 foot = 2 jumps; 1 ¼ = 1 jump). No frog jumped ¾ of a foot. When making a line plot for this data, can you skip ¾ foot on the graph? Why or why not? (even if no data is collected for a particular value, it must be included on the line plot. Line plots are essentially number lines, and we cannot eliminate a section on the number line. Data does not have to start at 0, but whatever section of data is shown, it must show all of the intervals).

Data was collected to reflect the amount of pie eaten by people in a pie eating contest at the fair. The table to the right shows the data. Construct a line plot to reflect the data. How many people participated in the pie eating contest? How many pies were consumed during the contest?

Standard OA.3 There are 583 students in Suzy’s school. 99 third grade students left the school on a field trip. There are about 20 students in each class. How many classrooms are

being used today? Explain your answer. The school bought apples to give to students. They have 30 boxes with 8 apples in each box and they have 20 boxes with 10 apples in each box. Each student

needs 3 apples for the week. How many students can the school feed? Why is it important to consider the remainder when answering a problem? Give a real-life example of when it is important to drop the remainder? Give a real-life

example of when you need to round the remainder. Zoe is having a wedding. She has 178 guests attending. The party location can set up tables with 10 at each table OR tables with 8 at each table. How many tables

will Zoe need under each situation? Write a division problem that has 15 R2 as the quotient. Barry’s family donated 11 cases of tomato soup to the local food kitchen. Each case has 12 cans of soup. The shelter already has 16 cans of tomato soup. How

many cans of tomato soup does the food kitchen have now? The food kitchen uses 20 cans of tomato soup each week. How many weeks will go by before the food kitchen needs more tomato soup?

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

Decompose fractions Add and subtract mixed numbers with like denominators Solve word problems with addition/subtraction of fractions with like denominators Multiply a fraction by a whole number Solve word problems of multiplying a fraction by a whole number

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Make a line plot and solve problems by adding and subtracting fractions with like denominators Solve multi-step problems with whole numbers using the four operations

Additional AssessmentsAdopted ResourcesMy MathChapter 8: Fractions8.9 Mixed Numbers8.10 Mixed Numbers and Improper Fractions

Chapter 9: Operations with Fractions9.1 Hands-On: Use Models to Add Like Fractions9.2 Add Like Fractions9.3 Hands-On: Use Models to Subtract Like Fractions9.4 Subtract Like Fractions9.5 Problem-Solving Investigation: Work Backward9.6: Add Mixed Numbers9.7: Subtract Mixed Numbers9.8: Hands-On: Model Fractions and Multiplication9.9 Multiply Fractions by Whole Numbers

Chapter 11: Customary Measurement11.8: Display Measurement Data in a Line Plot

*These lessons are not to be completed in consecutive days as it is too much material. They are designed to help support you as you teach your standards.

Adopted Online Resourceshttp://connected.mcgrawhill.com/connected/

login.do

Teacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

http://www.exemplarslibrary.com/

User: Cobb EmailPassword: First Name

Deluxe Birthday Cake (NF.3a)Harvest Dinner (NF.3a)Portfolio Pizza Party (NF.3a)Pizza Time (NF.3d)Lugging Water 1 (NF.4a)Hanging Airplanes (OA.3)Hot Dogs for a Picnic (OA.3)

Think MathChapter 7: Fractions7.2 Exploring Fractions Greater Than 17.3 Exploring Fractions with Cuisenaire Rods7.4 Reasoning About Fraction with Cuisenaire Rods7.11 Modeling Addition of Fractions7.12 Problem Solving Strategy and Test Prep

Additional ResourcesIllustrative Mathematics (Tasks listed below) https://www.illustrativemathematics.org/content-standards/4/NBT/A*Press CTRL + Click the link to access each of the Illustrative Mathematics tasks:

Everyday Math E-Tool Kit http://media.emgames.com/em-v2/eToolkit/eTools_v1.swf

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Tasks for Standard NF.3aComparing two different pizzasComparing Sums of Unit Fractions

Tasks for Standard NF.3cWriting a Mixed Number as an Equivalent FractionPeachesPlastic Building BlocksCynthia's Perfect Punch

Tasks for Standard OA.3Karl's GardenCarnival Tickets

National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html

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

K-5 Math Teaching Resources (Press CTRL + Click the link to view the resources):NF.3aAdding & Subtracting Fractions (like denominators ) Adding Fractions Using Pattern BlocksThe Chocolate Bar ProblemSense or Nonsense (1) Sense or Nonsense (2)

NF.3bDecomposing FractionsPizza Share

NF.3cMixed Number Word Problems (like denominators) Adding Mixed NumbersSubtracting Mixed Numbers

NF.3dFraction Word Problems (like denominator)Addition Word Problems with FractionsSubtraction Word Problems with Fractions

NF.4aModels for Fraction Multiplication

NF.4bMultiply a Whole Number by a Fraction

NF.4cWhole Number X Fraction Word Problems

MD.4

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Length of Ants Line PlotObjects in My Desk Line Plot

OA.3Interpret the Remainder Word ProblemsMultistep Word Problems

Learn Zillion (Press CTRL + Click the link to view the tutorials):NF.3aDecompose a fraction into a sum of fractions using an area modelAdd fractions with like denominators using a number lineSubtract fractions with like denominators using a number lineSubtract fractions with like denominators using an area modelAdd fractions with like denominators using an area modelAdd fractions by joining partsSubtract fractions by separating partsDecompose fractions

NF.3bAdd fractions with like denominators from two different wholesAdd fractions with like denominators in number sentencesSubtract fractions with like denominators: labeling shapesSubtract fractions with like denominators: labeling setsSubtract fractions with like denominators starting from less than 1 w...

NF.3cAdd mixed numbers using an area model (Lesson 1 of 2)Add mixed numbers using an area model (Lesson 2 of 2)Add mixed numbers using a number lineAdd mixed numbers by finding equivalent fractionsSubtract mixed numbers using an area modelSubtract mixed numbers using a number lineSubtract mixed numbers by finding equivalent fractionsAdding mixed numbers by creating equivalent fractionsAdding mixed numbers using properties of operationsSubtracting mixed numbers by creating equivalent fractionsSubtracting mixed numbers by using properties of operations

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NF.3dAdd fractions with like denominators by decomposing into unit fractionsAdd fractions with like denominators using a number lineAdd fractions with like denominators using visual modelsSubtract fractions with like denominators by decomposingSubtract fractions with like denominators using a number lineSubtract fractions with like denominators using visual modelsSuggested ManipulativesFraction circlesFraction squaresFraction BarsFraction Number LinesGrid PaperGeoboardsCuisenaire RodsFraction TowersColor TilesFraction Number Lines

VocabularyCommon fractionDenominatorEquivalent SetsImproper FractionIncrementNumeratorProper fractionUnit fractionWhole numberMixed Number

Suggested LiteratureFraction FunFraction ActionA Fraction’s Goal – Parts of a WholeGive Me HalfApple FractionsWhole-y Cow! Fractions are FunThe Hershey’s Milk Chocolate Fractions Book

Scaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solvingPractice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation.Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

State Tasks

Task Name Task Type/Grouping Strategy

Content Addressed Standard(s) Description of Task

Pizza Party Scaffolding TaskPartner/Small Group Task

Draw fraction representations, add and

subtract fractions

MGSE.4.NF.3a,b Students design pizzas with correct toppings that are reported as fractions on an order form.

Eggsactly Scaffolding Task Write number sentences to MGSE.4.NF.3a Students use “eggs” to find fractions of a set.

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Individual/Partner Task show addition of fractionsTile Task Practice Task

Partner/Small Group TaskSubtract and add fractions MGSE.4.NF.3a, b, d Students will create designs for a tabletop

using color tiles and decompose the whole into fractional parts.

Sweet Fraction Bar Constructing TaskIndividual/Partner Task

Solve story problems with fractions

MGSE.4.NF.3a, b, d Students solve problems about candy bars that are divided into fractional pieces using addition and subtraction.

Fraction Cookies Bakery Constructing TaskIndividual/Partner Task

Addition with improper and proper fractions

MGSE.4.NF.3a, b, c, d Similar to Pizza Party, students design cookies with toppings by reading fractional amounts on an order form.

Rolling Fractions Practice TaskIndividual/Partner Task

Add and subtract fractions, use mixed numbers and

improper fractions

MGSE.4.NF.3a, b, c, d Students play a game where they create mixed numbers to add and subtract.

Running Trails 3-Act TaskPartner/Small Group Task

Adding fractions, comparing fractions

MGSE.4.NF.3aMGSE.4.NF.2MGSE.4.MD.2

Students develop and solve problems involving maps and fractions.

The Fraction Story Game Performance TaskIndividual/Partner Task

Create story problems with fractions

MGSE.4.NF.3a, b, c, d Students will develop a game that includes story problems that involve addition and subtraction of fractions.

Fraction Field Events Performance TaskIndividual/Partner Task

Solve story problems with mixed numbers

MGSE.4.NF.3a, b, c, d Students solve a problem about a field event using mixed numbers.

Pizza Parlor Performance TaskIndividual/Partner Task

Add and subtract fractions, improper fractions and

mixed numbers

MGSE.4.NF.3a, b, c Similar to Fraction Cookie Bakery, students will fill orders by designing pizzas that have been ordered.

A Bowl of Beans Scaffolding TaskPartner/Small Group Task

Multiply a whole number by a fraction, represent

fractional products

MGSE.4.NF.4a, b, c Students model fractions of a set using multiplication of whole numbers and fractions to find how many are in the set.

Birthday Cake! Scaffolding TaskIndividual/Partner Task

Determine the fraction of a set, determine the set

when given a fraction, find equivalent fractions

MGSE.4.NF.4a, b, c Students use multiplication of fractions by whole numbers to determine the number of candles used on birthday cakes.

Fraction Clues Constructing TaskIndividual/Partner Task

Determine the fraction of a set, determine the set

when given a fraction, find equivalent fractions

MGSE.4.NF.4a, b, c Students create a fraction riddle using color tiles and write clues for the tiles so that other students may try to build it through clues.

Area Models Scaffolding Task Represent multiplication of MGSE.4.NF.4a, b, c Students develop riddles using equivalent 4th Grade Unit 4 25 2015-2016

Individual/Partner Task fractions fractions and solve riddles by multiplying a whole number by a fraction.

How Many CCs CTE TasksIndividual/Partner Task

Multiply a whole number by a fraction in a real-world

context

MGSE.4.NF.4a, b, cMGSE.4.MD.2

Students apply knowledge of multiplying a whole number by a fraction using measurements given about medication being administered.

Who put the Tang in Tangram?

Constructing TaskIndividual/Partner Task

Multiply a fraction by a whole number utilizing

repeated addition

MGSE.4.NF.3a, b, c, dMGSE.4.NF.4a, b, c

In this task, students use tangrams to learn about multiplication of a fraction times a whole number as repeated addition when finding the area of tangrams.

Birthday Cookout Constructing TaskIndividual/Partner Task

Solve story problems that involve the multiplication

of fractions

MGSE.4.NF.4a, b, c Students determine the number of people ordering food at a cookout using multiplication of a fraction by a whole number.

A Chance Surgery CTE TasksIndividual Task

Multiply a whole number by a fraction in a real world

context

MGSE.4.NF.4a, cMGSE.4.MD.2

Students solve a problem involving a field of medicine to apply their knowledge of multiplication of fractions.

Fraction Pie Game Practice TaskIndividual/Partner Task

Write number sentences to show multiplication of

fractions

MGSE.4.NF.4a, b Students will cover 15 wholes on a recording sheet by multiplying a fraction by a whole number to determine how much to shade in on each turn.

How Much Sugar 3-Act TaskIndividual/Partner Task

Multiply a whole number by a fraction

MGSE.4.NF.4a, c Students discover how much sugar is in a 12-pack of Mountain Dew after developing questions and researching what information is needed to find a solution.

Fraction Farm Performance TaskIndividual/Partner Task

Solve story problem that involves the multiplication

of a whole number by a fraction

MGSE.4.NF.4a, c Students find the amount of money a fraction of a field is worth using multiplication of whole numbers and fractions.

Land Grant Performance TaskIndividual/Partner Task

Multiply whole numbers by fractions and represent

products

MGSE.4.NF.4a, c Students will use multiplication of fractions and whole numbers to develop a proposal to the city for the best way to utilize land being given to the city.

What’s the Story? Performance TaskIndividual/Partner Task

Make a line plot to display data to 1/8 inch

MGSE.4.MD.1MGSE.4.MD.2MGSE.4.MD.4

Students will create a line plot using a given set of data. Students will create and answer questions about the data. Students will create

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the context for the data given.

Td

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