Math News! 4 th Grade Math. Focus Area Topic A

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1 Math News Grade 4, Module 5, Topic A 4 th Grade Math Math Parent Letter This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module 5 of Eureka Math (Engage New York) covers fraction equivalence, ordering, and operations. Focus Area Topic A Decompositions and Fraction Equivalence Number Bonds and Tape Diagrams with Fractions Students decompose fractions as unit fractions, drawing tape diagrams to represent them as sums of fractions with the same denominator in different ways. Focus Area Topic A: Decompositions and Fraction Equivalence Words to Know: Numerator - top number in a fraction- the number of parts of the whole Denominator -bottom number in a fraction- indicates the number of parts into which one whole is divided. Mixed number - number made up of a whole number and a fraction Unit fraction - fractions with numerator 1 Equivalent fractions - fractions that name the same size or amount A number bond is a graphic showing a part/part/whole relationship. In this case, the fractions and Fraction greater than 1 - a fraction with a numerator that is greater than the denominator Compose -change a group of unit fractions with the same denominator to a single non-unit fraction or mixed number Decompose - change a non-unit fraction or mixed number to the sum of its parts or unit fractions combine to make OBJECTIVES OF TOPIC A Decompose fractions as a sum of unit fractions using tape diagrams. Decompose non-unit fractions and represent them as a whole number times a unit fraction using tape diagrams. Decompose fractions into sums of smaller unit fractions using tape diagrams. Decompose unit fractions using area models to show equivalence..

2 In Topic A, students begin to see that representing a fraction as repeated addition of a unit fraction is the same as multiplying that unit fraction by a whole number. Fractions and the Area Model The idea of the number of fractional parts increasing, while the size of the pieces decreases is further investigated as students represent the decomposition of unit fractions in area models. This area model shows. The dotted line decomposes the whole into 2 equal rows. There were 4 pieces but now there are 8. Each fourth was cut into 2 pieces. Even though the parts changed, the area covered by the shaded region did not change. Students continue with decomposition. They represent fractions as the sum of smaller unit fractions. The area model is used to explain how certain fractions are equivalent. Students will write addition and multiplication sentences to explain this equivalence. This activity helps students discover that the number of fractional parts in a whole increases, while the size of the pieces decreases. This information was generously shared by LPSS, Lafayette, LA

3 Math News Grade 4, Module 5, Topic B 4 th Grade Math Math Parent Letter This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module 5 of Eureka Math (Engage New York) covers fraction equivalence, ordering, and operations. Focus Area Topic B Fraction Equivalence Using Multiplication and Division The Area Model and Fraction Equivalence using Multiplication Students create equivalent fractions with area models. Focus Area Topic B: Fraction Equivalence Using Multiplication and Division Words to Know: Whole fraction when the numerator and the denominator are the same e.g., 2 halves, 3 thirds, 4 fourths Non-unit fraction - fraction with numerators other than 1 Equivalent fractions - fractions that name the same size or amount Compose -change a group of unit fractions with the same denominator to a single non-unit fraction or mixed number Decompose - change a non-unit fraction or mixed number to the sum of its parts or unit fractions Example Problem and Answer Students will be asked to decompose the shaded fraction into smaller units using an area model. Then they will need to express the equivalent fraction in a number sentence using multiplication. Below is one way to solve. Can you think of another way? OBJECTIVES OF TOPIC A Use the area model and multiplication to show the equivalence of two fractions. Use the area model and division to show the equivalence of two fractions Explain fraction equivalence using a tape diagram and the number line, and relate that to the use of multiplication and division.

4 The Area Model and Fraction Equivalence using Division Students will learn that division can be used to create a fraction comprised of larger units (or a single unit) that is equivalent to a given fraction. Focus Area Topic B: Fraction Equivalence Using Multiplication and Division Fractions and the Number Line Students use tape diagrams to transition their knowledge of fraction equivalence to the number line. Example Problem and Answer Students are asked to compose the shaded fraction into larger fractional units and express the equivalent fraction in a number sentence using division. This information was generously shared by LPSS, Lafayette, LA

5 Math News Grade 4, Module 5, Topic C 4 th Grade Math Math Parent Letter This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module 5 of Eureka Math (Engage New York) covers fraction equivalence, ordering, and operations. Focus Area Topic C: Fraction Comparison Using Benchmarks on a Number Line to Compare Fractions Here is an example of how students can reason about the size of a fraction compared to. If they want to know the relative size of, they can use a number line. Focus Area Topic C: Fraction Comparison Words to Know: Common denominator - when two or more fractions have the same denominator Comparing fractions determining which fraction is greater than the other and using symbols to express the comparison - symbol for greater than > - symbol for less than < - symbol for equal to = Benchmark - standard or reference point by which something is measured example benchmarks 0,, 1. Using Benchmarks to Compare Fractions Students use benchmarks to compare fractions with different numerators and different denominators. The use of benchmarks is modeled using a number line. Students use the relationship between the numerator and denominator of a fraction to compare to a known benchmark and then use that information to compare the given fractions. Example of Benchmark Use Students might need to compare and. They reason that 4 sevenths is more than 1 half, while 2 fifths is less than 1 half. They then conclude that 4 sevenths is greater than 2 fifths. OBJECTIVES OF TOPIC C " Reason using benchmarks to compare two fractions on the number line. " Find common units or number of units to compare two fractions.

6 Focus Area Topic C: Fraction Comparison Comparing Fractions using Related Numerators Students learn to use like numerators to compare fractions. They compare using the size of the fractional units as in this example: 3 fifths is less than 3 fourths because fifths are smaller than fourths. This reasoning is extended as students learn to find an equivalent fraction when they see a relationship between the numerators. Let s say we want to compare 2 eighths and 4 tenths. Compare fractions with unrelated denominators using area models Students compare fractions by finding like units or common denominators. In this method, the unit fractions are the same in each model or equation. Since the units are equal, students can easily compare. Throughout this topic, it is important for students to continue drawing area models, tape diagrams, and number lines. Students need to able to compare fractions and justify their comparisons by using a visual fraction model. Notice how the area model is used in the following example. Let's say we want to compare 3 fourths and 4 fifths. First, we draw 2 almost square rectangles that are the same size. These squares are our models. Each model represents 1 whole. First, notice the relationship between the numerators. We can make them the same if we multiply the numerator two by 2. If we multiply the numerator by 2, then we have to multiple the denominator by 2. That will give us an equivalent fraction. We will partition one rectangle with vertical lines into 4 parts and shade to show. The other rectangle will be partitioned into 5 parts using horizontal lines and shaded to show. This tape diagram shows shaded. Since our fractions do not have like denominators, we will find equivalent fractions that do. The dotted lines show how each eighth gets partitioned into 2 parts when we multiply. Now we have " shaded. We can compare using the numerators. We think about the size of the fractional units. is less than " tenths. " because sixteenths are smaller than Now, both models have the same number of units, 20. That means we can compare. Our " model is now showing and " our " model is now showing. " This information was generously shared by LPSS, Lafayette, LA

7 Math News Grade 4, Module 5, Topic D 4 th Grade Math Math Parent Letter This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module 5 of Eureka Math (Engage New York) covers fraction equivalence, ordering, and operations. Focus Area Topic D: Fraction Addition and Subtraction Subtraction of Fractions Students need to see the addition and subtraction of fractions as joining and separating parts referring to the same whole. In the following example, we will use a number line to model our thinking. Let s find the value of 1 -. Focus Area Topic D: Fraction Addition and Subtraction Words to Know: Mixed number - number made up of a whole number and a fraction less than one Unit fraction - fractions with numerator 1 Non-unit fraction - fractions with numerators other than 1 Compose and Decompose Whole Numbers In both the addition and subtraction of fractions, students will need to compose and decompose whole numbers into fractions. In this example, we use a number bond to rename 1 whole as. OBJECTIVES OF TOPIC D Use visual models to add and subtract two fractions with the same units. Use visual models to add and subtract two fractions with the same units, including subtracting from one whole. Add and subtract more than two fractions. Solve word problems involving addition and subtraction of fractions. Use visual models to add two fractions with related units using the denominators 2, 3, 4, 5, 6, 8, 10, and 12.

8 Focus Area Topic D: Fraction Addition and Subtraction Addition of Fractions In this topic, students create tape diagrams and/or number lines to represent and solve fraction addition and subtraction word problems. Consider this example: Mark mixed cup of apple juice, cup of orange Example Problems and Answers Problem Students are asked to use three fractions to write two subtraction and two addition number sentences. Answer juice, and cup of grape juice for his fruit punch. How many cups of juice did he put in his fruit punch in all? We will begin by writing a number sentence. When counting, we consider the items being counted units as in one tree, two trees, three trees, and so on. Fractional units work the same way. In our example, the fractional units are fourths. Imagine if they were cars. Our number sentence would read like this: 3 cars + 2 cars + 1 car =? or Problem Students are asked to find the difference in two ways and use a number bond to show the decomposition. If we wanted to solve and find the total number of cars, we would add 3 plus 2 plus 1. We would not add the word car, would we? That's just silly. It works the same with our fractions. The denominator (fourths) is telling us how many parts we need to make 1 whole. It's telling the unit being counted. Let's model this problem using a tape diagram. Problem Show one way to solve each problem. Express sums and differences as a mixed number when possible. This information was generously shared by LPSS, Lafayette, LA

9 Math News Grade 4, Module 5, Topic E 4 th Grade Math Math Parent Letter This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module 5 of Eureka Math (Engage New York) covers fraction equivalence, ordering, and operations. Focus Area Topic E: Extending Fraction Equivalence to Fractions Greater than 1 Decomposition of Fractions Greater than 1 Understanding what a fraction or a mixed number is equal to will help students successfully add and subtract with fractions. In this topic, students will learn to use decomposition and visual models to add and subtract fractions less than 1 to and from whole numbers. Focus Area Topic E: Extending Fraction Equivalence to Fractions Greater than 1 Words to Know: Line Plot- shows data on a number line with an X or other mark to show frequency. Fraction greater than 1 - a fraction with a numerator that is greater than the denominator -- sometimes called an improper fraction Mixed number - number made up of a whole number and a fraction less than one Unit fraction - fractions with numerator 1 Non-unit fraction - fractions with numerators other than 1 Equivalent fractions - fractions that name the same size or amount OBJECTIVES OF TOPIC E Add a fraction less than 1 to, or subtract a fraction less than 1 from, a whole number using decomposition and visual models. Add and multiply unit fractions to build fractions greater than 1 using visual models. Decompose and compose fractions greater than 1 to express them in various forms. Compare fractions greater than 1 by reasoning using benchmark fractions and by creating common numerators or denominators. Solve word problems with line plots.

10 Focus Area Topic E: Extending Fraction Equivalence to Fractions Greater than 1 Comparing Fractions by Finding Common Denominators Sometimes students will need to compare fractions and mixed numbers with unrelated denominators. Data and Line Plots In topic E, students will create line plots to display a given dataset that includes fraction and mixed number values. To do this, they apply their skill in comparing mixed numbers. Example Problem and Answer Problem Example Problem and Answer Answer The student s work below shows data on a number line with an X representing each child s shoe measure. So, for each shoe length, there is an X above the measure on number line that matches it. Seven children measured their shoes so there are seven X marks on the line plot. This information was generously shared by LPSS, Lafayette, LA

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12 Math News Grade 4, Module 5, Topic G 4 th Grade Math Math Parent Letter This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module 5 of Eureka Math (Engage New York) covers fraction equivalence, ordering, and operations. Focus Area Topic G: Repeated Addition of Fractions as Multiplication Words to Know: Compose -change a group of unit fractions with the same denominator to a single non-unit fraction or mixed number Decompose - change a non-unit fraction or mixed number to the sum of its parts or unit fractions Mixed number - number made up of a whole number and a fraction Line plot - display of data on a number line, using an x or another mark to show frequency Focus Area Topic G: Repeated Addition of Fractions as Multiplication Using the Associative Property Multiplying a whole number times a fraction was introduced in Topic A. Here s an example of how a fraction can be decomposed and rewritten as a multiplication sentence. Now students will use the associative property to multiply a whole number times a mixed number. Consider this example. (3 x ) Each plate has of a cake on it. There are 3 plates on each table. There are 5 tables in the room. To find out how much cake there is, first we can think about the plates on each table and write 3 x which would give us the amount of cake on each table,. Now, we can multiply the amount of cake on each table, by 5, the number of tables in the room. Example Problem and Answer OBJECTIVES OF TOPIC G Represent the multiplication of n times a/b as (n a)/b using the associative property and visual models. Find the product of a whole number and a mixed number using the distributive property. Solve multiplicative comparison word problems involving fractions. Solve word problems involving the multiplication of a whole number and a fraction including those involving line plots.

13 Focus Area Topic G: Repeated Addition of Fractions as Multiplication Example Problems and Answers A grocery store had a sale on rice. Eight families each bought 2 pounds of rice. How many pounds of rice did the store sell to these families? Using the Distributive Property Students will explore the use of the distributive property to multiply a whole number by a mixed number. We can use the distributive property to show 4 32 as (4 3 tens) + (4 2 ones). The following area model uses the distributive property. The store sold 20 pounds of rice to these families. Students begin to see the multiplication of each part of a mixed number by the whole number and use the appropriate strategies to do so. In the following example, we use a tape diagram. Understanding Line Plots A line plot is a graph that shows frequency of data along a number line. Example Problems and Answers What is the difference in growth of the tallest and shortest plant? This information was generously shared by LPSS, Lafayette, LA

14 Math News Grade 4, Module 5, Topic H 4 th Grade Math Math Parent Letter This document is created to give parents and students a better understanding of the math concepts found in Eureka Math ( 2013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module 5 of Eureka Math (Engage New York) covers fraction equivalence, ordering, and operations. Focus Area Topic H: Explore a Fraction Pattern Finding Patterns in Fraction Sums Students begin by lining up fraction cards, through or 1 whole, in order and then adding them together. In the following example, they add to find a total of 21 sixths. They explore further and notice they can create sums of 6 sixths or 1 whole by grouping cards together as shown below. Focus Area Topic H: Explore a Fraction Pattern Words to Know: Numerator - top number in a fraction - tells how many equal parts are being describes by the fraction Denominator -bottom number in a fraction - indicates the number of equal parts into which the whole is divided Unit fraction - fractions with 1 as the numerator When students try this activity with an odd number denominator, they begin to see a pattern. Can you see the pattern? Understanding Patterns In mathematics, the elements of a pattern repeat in a predictable manner. Observing and analyzing these patterns is an essential skill students need to sharpen as they move through school. In topic H, students find patterns in the sums of like denominators. OBJECTIVE OF TOPIC H Find and use a pattern to calculate the sum of all fractional parts between 0 and 1. Share and critique peer strategies. When adding the sums of fractions with even denominators, the answer is not just a whole number. It includes a half. When there are odd denominators, the answer is a whole number. This information was generously shared by LPSS, Lafayette, LA