Fractions Common Core State Standards

Supplemental Fraction Unit for Grade Four based on the Common Core Standards

SCLME  South Carolina Leaders of Mathematics Education 2012 SCLME recommends that district mathematics curriculum leaders support teachers with the implementation of this unit by providing the necessary content knowledge so that students gain a strong conceptual foundation of fractions. 5/3/12

Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

The following sources and websites were used in part or whole in the creation of this fractions unit: Cramer, K., Behr, M., Post T., & Lesh, R. (2009). Rational Number Project: Initial Fraction Ideas. Retrieved from http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-09.html Cramer, K., Wyberg, T., & Leavitt, S. (2009). Fraction Operations and Initial Decimal Ideas. Retrieved from http://www.cehd.umn.edu/ci/rationalnumberproject/rnp2.html North Carolina Department of Public Instruction. Project directed by D. Polly. 4th Grade Fractions. Retrieved from http://maccss.ncdpi.wikispaces.net/file/view/4thGradeUnit.pdf K-5 Math Teaching Resources. http://www.k-5mathteachingresources.com/4thgrade-number-activities.html

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Fractions Common Core State Standards

Time: 9-11 days (60 minute class periods) Background In grade 3, students begin their focus on fractions. In the study of fractions, a variety of concrete and pictorial experiences have students represent a fraction as a part of a whole in an area model, part of a set, distance designated by a point on the number line (a type of linear model). Students begin to build understanding of the idea that a fraction is a relationship of two numbers – the denominator, which names the parts on the basis of how many equal parts are in the whole, and the numerator, which tells the number of equal parts being considered. Adapted from NCTM Focus in Grade 4 Teaching with Curriculum Focal Points Unit Overview To continue the development of fractions, this unit will support the transition to and implementation of the Common Core State Standards. The learning activities provided herein should engage students in both hands-on and minds-on experiences. Students should have multiple opportunities to communicate about their thinking and reasoning in order to build understanding. Teachers should listen carefully to students’ ideas and encourage flexibility in their thinking. Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students 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. In order for students to have a deep conceptual understanding of fractions, they will use a variety of concrete materials and pictorial representations. In using best practices, virtual manipulatives should not take the place of concrete materials. An anchor chart is a visual recording of students’ ideas and thinking about a certain concept and is used to connect past and future teaching and learning. For example, on a piece of chart paper, the teacher recorded students’ ideas about what ½ means to them. This included illustrations and labels of different representations for ½. (See Appendix A) Models for Fractions Area or Region Models – Fractions are based on parts of an area or region. Examples include: circular pie pieces, pattern blocks, regular/square tiles, folded paper strips (any shape), drawings on grids and partitioning shapes on geoboards. Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

Linear or Length Models – With length models the whole is partitioned and lengths are compared instead of area. Materials are compared on the basis of length. Examples include: fraction strips, Cuisenaire rods, number lines, rulers, and folded paper strips. Set Models – In set models, the whole is understood to be a set of objects or group of objects, and subsets of the whole make up fractional parts. Examples; in a set of 6 marbles, ½ is 3 marbles. This concept should be taught with concrete materials so that there would be 2 groups of 3 marbles so that each group is ½ of 6. Content Progression for CCSS 3rd grade limited to fractions with denominators 2, 3, 4, 6, and 8; 4th grade add 5, 10, 12, and 100. Big Ideas Equivalence and Ordering of Fractions Addition and Subtraction of Fractions with Like Denominators Multiplication of a Fraction by a Whole Number Decimal Notation and Decimal Fractions Common Core Standards (Grade 4) *Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or 1 as a sum of fractions 1/b. 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. Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

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. 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 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 x (1/4), recording the conclusion by the equation 5/4 = 5 x (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 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x 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? Understand decimal notation for fractions, and compare decimal fractions. 4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. *Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade. 4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or 1 as a sum of fractions 1/b. 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. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. NOTE: SMP indicates the Standards for Mathematical Practice throughout the document. Lesson Learning Goals: Build and compare fractions in a set. Explain why two fractions are equivalent even through they use different numbers. Materials: 1-inch color tiles (red, blue, yellow, green), task cards A-H, 1-inch graph paper, crayons/markers Engage (8-10 minutes) In this lesson students use 1-inch square tiles to create designs that follow certain criteria. “Using the tiles at your desk, create a design that is one half blue.” Allow students a minute or two to create their design. As they do, circulate around the room looking for simple and creative examples to share with the class. After students complete their designs, discuss some of the differences in the class. Did everyone use the same colors? Does everybody’s design look the same? Why not? How can that be since half of the design had to be blue? Did everyone use the same number of tiles? Why or why not? How did you decide what you were going to do to create this pattern? If we created another design, would you do it differently? How?

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You may need to repeat this activity a few times before starting the Explore section of this lesson. Before moving on, students should see that there are many different options for each design. Just because the problem calls for a fraction in fourths doesn’t mean they need to use four tiles. They also need to understand that they may only receive part of the information needed to solve the problems; they will need to fill in the rest. Explore (12-15 minutes) Students work in groups of two or three to build designs with 1-inch tiles based on the description given on a task card. Each student builds a representation for the card. Once all students in the group have finished, they discuss their designs and decide on which one they will use for their representation for the class. Once the students agree upon the design, each student will copy it onto a sheet of 1inch graph paper. Below the picture they will write a description and an equation of all the colors used in their design.

“Our design for card C has 1/8 yellow, 4/8 green and 3/8 red. 1/8 + 4/8 + 3/8 = 8/8 or 1 whole.” Start with Card A and work towards Card H. Most groups will not be able to finish all 8 cards in the time allotted for the lesson. Explain (12-15 minutes) Bring all the students together and have them share the results of task cards A, B, and C. Suggested questions What did you do for your task card? Do you think that this group’s design fits the directions? How can you prove it? Compare two different designs. How are they similar and different?

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Time permitting, give the students 8 tiles of any color and tell them as a class you need to make a design that is ½ red, ¼ green, 1/8 yellow and 1/8 blue. Ask students to describe how they know how many tiles of the region match a specific fraction. Elaborate (10-12 minutes) Have students create their own task cards. Students should use 24 total tiles and use the denominators 2, 3, 4, 6, 8 and 12. Students need to make sure that the fractions add up to 24/24ths or 1 whole. As students work, check to make sure that they have completed the puzzle and have written fractions in simplest form. Evaluation of Students Formative: As students are building the designs, circulate throughout the room checking for misunderstandings. Are students using only the minimum number of tiles? Can they use more? How did they choose the number of tiles? Why did they choose the number of tiles of each color? Review the students’ description for clarity. Summative: Have students collect their descriptions of each task card they were able to finish, and staple them together to create a book. Plans for Individual Differences Intervention: Students who are struggling with this activity may need help determining the number of tiles that will be found in their design. These students may need to start with very basic designs, using the minimum number of tiles. Extension: Students who are ready will create task cards for a defined number of tiles for class use. For example, the student may use 16 tiles to create a design with ¼ blue, 1/8 green, ½ red and the rest yellow.

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Fractions Common Core State Standards

Fraction Task Cards A-H

Card A

Card E

Build a design that is… Build a design that is… one-fourth red one-half red one-fourth green one-fourth yellow Card B Card F Build a design that is… Build a design that is… two-thirds yellow five-twelfths blue one-sixth red two-sixths green Card C Card G Build a design that is… Build a design that is… one-eighth yellow one-fifth red four-eighths green four-tenths green two-fifths blue Card D Card H Build a design that is… Build a design that is… one-third blue one-third yellow two-thirds red one-sixth red one-half green Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

Lesson 2: Race to One Standards: 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b. 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. SMP 3 Construct viable arguments and critique the reasoning with others. SMP 7 Look for and make use of structure. Lesson Learning Goals: Name equivalent fractions. Apply knowledge of equivalent fractions while playing a game. Materials: Fractions Cards, Race to One game board, Race to One rules, counters for each game Lesson: Engage (8-10 minutes) Give each pair of students a game board. They must fill in the fractions on each fraction bar before playing (halves, thirds, fourths, fifths, sixths, eighths, and tenths). Introduce the game Race to One by playing a practice game with the class. Using just the fraction cards that are equal to or less than one, shuffle the cards and place them face down. Start by placing one counter on each fraction bar at a location that is less than 3/4. (Students will start at the beginning of the fraction bar during their game.) Select a card from the pile and discuss the possible moves available to the students. The player can move one or more than one counter during each play, but he must move the full amount on the chosen card. If a player is not able to move the full amount, he will lose his turn. Once a player moves a counter exactly to the number 1 on any fraction bar, they collect the counter. Place a new counter at the beginning of that fraction bar so every play has 7 counters available. Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

Play a few hands so students become familiar with the rules of the game. Explore (20-30 minutes) Students play the game Race to One in pairs. Move throughout the room observing how the students are playing the game. Suggested questions: • Which counter are you moving? • Are there other counters that you could also move? How do you know? • Which move will help you get more counters closest to one? Explain (12-15 minutes) Bring the class back together after students have played the game for about 20 – 30 minutes. Continue your practice game from the beginning of the class, but have students decide which counters to move and how far. Discuss the possibilities and give reasons for each choice. Elaborate Continue to play this game.

(10-20 minutes)

If students need an extension, tape two Race to One boards together, and make the game Race to Two. In this version, all the cards can be used. Evaluation of Students Formative: As students are playing the game, observe them and pose questions to check for mathematical understanding. Suggested questions are in the Explore section. Summative: If teachers want a summative assessment, pose an additional follow-up task: You have the card ¾. Name 3 possible moves that you can make on the game board. • One move involves 1/2 • One move that includes 1/4 • One move that includes 1/6 Plans for Individual Differences Intervention: For students who are struggling to find equivalent fractions, provide fraction manipulatives (fraction bars, fraction tiles) to help them. Extension: Play Race to Two the entire time if students need an extension. Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

Race to One Game Rules 1. Shuffle the fraction cards that are equal to or less than 1. Place them face down. 2. Place seven counters on the game board, one at the beginning of each fraction bar. 3. Player 1 draws the first card off the top of the deck of fraction cards. Move a counter (or counters) the total amount shown on the card. You can move one or more than one counter on every turn. You must move the full value of the fraction on the fraction card. Example: Player 1 chooses 3/5; they can move one counter 3/5 on the fifths line or 6/10 on the tenths line. They can also move more than one counter the following ways: ½ and 1/10, 1/5 and 4/10, or 1/3, 1/6, and 1/10. 4. Player 2 draws the next card off the top of the deck of fraction cards and moves his counter or counters the total found on their card. Players take turns flipping cards and moving counters. 5. When a counter lands exactly on one, the player has won the counter. Once a player has a won a counter, another counter is placed at the beginning of the fraction bar so that there are always 7 counters being played at one time. 6. If you are unable to move the amount found on the fraction card, your turn is over.

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Fractions Common Core State Standards

Fraction Cards

1 2 3 4 4 5 3 8 3 10 3 2 4 4 6 5

1 3 1 5 1 6 5 8 7 10 3 3 5 4 7 5

2 3 2 5 5 6 7 8 9 10 4 3 6 4 2 6

Developed by SC Leaders of Mathematics Education 2011-2012

1 4 3 5 1 8 1 10 2 2 2 4 5 5 3 6 14

Fractions Common Core State Standards

Fraction Cards

4 6 9 6 8 8 12 8 6 10 12 10 1 1

6 6 2 8 9 8 2 10 8 10 13 10

7 6 4 8 10 8 4 10 10 10 14 10

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8 6 6 8 11 8 5 10 11 10 15 10

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Lesson 3: Fraction Buckets Standards: 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or , , ,, =

< > =

< > =

< > =

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Fractions Common Core State Standards

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Fractions Common Core State Standards

Lesson 4: Fraction Chain Standards: 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or and how do you know? Can you justify your answer without using a mathematical algorithm?

Developed by SC Leaders of Mathematics Education 2011-2012

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Lesson 5: Kellen’s Candy Company Standards: 4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b. 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. SMP 4 Model with mathematics. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Lesson Learning Goals: Decompose a whole unit into an addition equation where all the fractions have the same denominator and the sum is one whole. Materials: connecting/pop cubes, chart paper, crayons/markers, math journals Lesson: Engage (10-12 minutes) In today’s activity students build Special Bars from different colored pop cubes. Each color will represent a different flavor of candy. The bars come in different sizes depending on the number of candies the buyer wants. The teacher will need to make a bar using 8 total pop cubes prior to beginning class. “Today we are going to pretend to visit a special candy store called Kellen’s Candy Company. At the company they have a very unique candy bar called the Special Bar. This bar is special because the buyer of the bar is able to pick out all the flavors that will be in the bar. This way each bar is different and the buyer can get exactly what he wants. As a treat, each person who visits the store receives a free 8 piece candy bar at the end of his visit.” To personalize this task teachers may want to use their name, example: Mr. Smith, Smith Bar. Students could even use their names when designing a bar of their own.

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“Let’s look at the Special Bar that I made on my visit.” Share with students a bar you created that has 8 pieces.

Take a moment to discuss the flavors that are possible. (See possible flavors below.) Suggested Questions: • Which flavor of candy did I choose most often? • Which flavor of candy did I choose least often? • How do you know which candy I chose most often? • How much of my bar is flavored blueberry? cherry? banana? lime? … The students’ answers should be in fraction form. You are not asking how many pieces are certain flavors, but how much of the bar is that flavor. As students tell you the fraction for each flavor, record the fractions on the board. If I add up the all the fractions 3/8 + 4/8 + 1/8, I will get 8/8 which is the whole candy bar. Create an anchor chart with 8/8 in the center to model various equations that represent the decomposition of the whole. Later in the lesson, students can add additional equations to model 8/8 as a decomposed representation. Today you are going to build Special Bars of different sizes and record them in your math journal. First you will build a Special Bar that has 8 pieces of candy. Then you will represent the bar by drawing it in your notebook. After that you will write an equation to show the sizes of your Special Bar. You will repeat the process with Special Bars of different sizes. (2, 3, 4, 5, 6, 8, 10, or 12 pieces) Explore Building and Recording Special Bars

(18-20 minutes)

Students work on building and recording different sized Special Bars. They start with a bar that has 8 pieces of candy. As the students are building and recording the bars, the teacher should be asking questions of the students. • How many (flavor) pieces do you have? • How many more pieces would you need to complete a bar? • Which do you have more of? less of? the same amount of? • What does your equation look like? • How are you determining which fractions to use in your equation? • How does your representation compare to the candy bar on the anchor chart? If it is a different equation, add to the anchor chart. Make sure the representations and equations that are being recorded are correct.

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Explain (12-15 minutes) Students rebuild their favorite Special Bar from the day. Bring the Special Bar and the equations for the bar to a large group meeting. Students share their drawings and discuss the equation that represents it. Have students determine the equation before the presenting student shares it. Elaborate (8-10 minutes) Students write a story problem about their Special Bar. For example, Sandy’s Special Bar was 4/10 Cotton Candy, 5/10 Marshmallow, and 1/10 Orange. Her dog, Ripley, ate all of the cotton candy pieces while she was at school. How much of her Special Bar was remaining? Students are given part of a bar, and need to complete the rest of the bar. For example, I have 7/12 of my bar complete with banana and chocolate. I don’t want any more banana or chocolate, but I want two more flavors. What are some of my options? Evaluation of Students Formative: As you are working with the students are they able to describe each section of the bar in fraction form? Can they create equations that equal a whole? Summative: I have a bar with 3 licorice, 3 cotton candy, 2 apple, and 4 orange pieces. Draw what the bar looks like. Write an equation that represents my Special Bar. Plans for Individual Differences Intervention: Limit the number of types of candy per Special Bar. Start with only two colors, and then continue to add one at a time. Extension: Build the Mega Special Bar which is only sold for Valentine’s Day. The Mega Special Bar has 100 pieces of candy, and can have up to 10 different types of candy. Have students determine the equation to represent their Mega Special Bar. Possible Flavors for the Colored Connecting Cubes Red – Cherry Blue – Blueberry Light Green – Lime White – Marshmallow Brown – Chocolate Black – Licorice Yellow – Banana Pink – Cotton Candy Dark Green – Apple Orange - Orange Developed by SC Leaders of Mathematics Education 2011-2012

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Lesson 6: Who am I? Puzzles Standards: 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or ) one half.” Which of these fractions does this clue help us eliminate? 1/4 and 1/2 Discuss with the class why this clue helps us determine which choices to eliminate. Show the second clue to the puzzle: “My denominator is larger than or equal to my numerator.” How does this help us get closer to the answer? This will eliminate the fraction 5/4, leaving us 3/4 and 4/4. Show the last clue: “I cannot be written as a whole number.” The only fraction left that can be written as a whole number is 4/4, which can be written as 1, so the answer has to be 3/4. After the class has discussed how to use the clues to solve the puzzles, explain that they will be working with a partner to solve more puzzles.

Explore Developed by SC Leaders of Mathematics Education 2011-2012

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Students work in pairs to solve the remaining Fraction Puzzles. As the students are working, observe how they are solving the puzzles. What strategies do students use to get started? What clues do they not understand? When students are finished with the remaining puzzles, they attempt to write their own fraction puzzles in their math journals. Choose any five fractions, and write clues that will help eliminate a fraction or two at a time, but keep the others. Remember that fraction denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 100. Have students use symbols and words for greater than, less than, or equal to. Students ask classmates to solve their puzzles. Explain (20 minutes) As a class, discuss how students solved the puzzles. What clues were most helpful, and what clues were least helpful? Which clues did students need help with? Share some of the puzzles that the students made. If time permits, work as a class to solve a few of the puzzles that students created.

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1/4

Puzzle 1

Puzzle 2

Who am I?

Who am I?

1/2

3/4

4/4

5/4

• I am > (greater than)one half. • My denominator is larger than or equal to my numerator. • I cannot be written as a whole number. • I am .

2/8

2/3

3/4 2/5

7/10

• My numerator is an even number. • I am > (greater than) one half. • I am written in simplest form. • I am .

Puzzle 3

Puzzle 4

Who am I?

Who am I?

4/6

9/12

3/5

5/12

• I am > (greater than)1/4. • My denominator is a multiple of three. • I can be simplified. • When I am simplified, my numerator and denominator are less than four. • I am .

6/8

1/2

5/12

1/4

8/10

2/3

• I am < (less than) one half. • I am greater than one third. • My denominator is a multiple of three. • I am simplified. • I am .

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2/4 • • • •

Puzzle 5

Puzzle 6

Who am I?

Who am I?

3/9

1/5

7/12

9/10

I am greater than 1/4. I cannot be simplified. I am closer to 1 than one half. I am .

6/10

5/4

1/5

4/6

3/8

• I am < (less than) one. • My denominator is even. • I can be written in a different way. • I am another way to say 2/3. • I am .

Puzzle 7

Puzzle 8

Who am I?

Who am I?

4/8

5/9

1/3

3/12

• I am greater than one fourth. • I am not another way to write 1/2. • I am written in lowest form. • I am less than one half. • I am .

2/10

7/8

4/9

2/10

9/6

4/12

• I can be simplified to a simpler fraction. • I am less than one. • My denominator is a multiple of three. • I am closer to one half than I am to zero. • I am .

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Lesson 7: Fraction Tangrams Note: This lesson could take more than one day depending on the needs of your students. Standards: 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or 1 as a sum of fractions 1/b. 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. SMP 1 Make sense of problems and persevere in solving them. SMP 8 Look for and express regularity in repeated reasoning. Lesson Learning Goals: Decompose a fraction on a number line. Explain how they decomposed a fraction into smaller fractions. Materials: Race Handout Lesson: Engage (15-18 minutes) Today we are going to get ready for a relay race. Before we run the race, we need to make a plan. Each team will have three people on it. Each person on a team has to run in the race, but they do not need to run the same distance. There are certain places during the race where you can hand off the baton to the next runner. You are going to get a chance to plan the distances of each runner on your team before the race begins. Let’s work together to make a plan for a team. Today’s race will have different places that a team can hand off their baton to the next runner. Draw a line on the board with a start and finish line. Mark 5 additional locations, equal distance apart, where students can hand off the baton. This will break the track into 6 separate sections. Students may have a difficult time with the concept that there are 5 locations to hand off, but 6 sections to the race. This is a good time to discuss the fact that the distance between the marks is what we are considering and not the marks. Have students talk with their teammates to determine some possibilities to setting up the race. Remember that each person doesn’t have to run the same distance. Share a few of the students’ ideas, and ask what fraction of the race each student will need to run. Developed by SC Leaders of Mathematics Education 2011-2012

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S

X

X

F

In this race the first runner runs 3/6 of the race, the second runner runs 2/6, and the final runner runs 1/6. Write an equation for each idea. 3/6 + 2/6 + 1/6 = 6/6 or 1 whole. Look for multiple ways to set up the race. Explore (15-18 minutes) Students work on planning four different races. For each race the student teams need to find multiple ways to set up each race. They record the distance each runner will run, and then write an equation that will equal one whole. Explain (12-15 minutes) Have students share their possibilities for each race and discuss their favorite and the reason why they chose it. Make the connection between the races and a number line from 0 – 1. How are these similar? Elaborate (12-15 minutes) Set up a race outside using cones as hand off positions. Have the students run the race according to their plans. What are some possibilities if we had only 2 people on a team? 4 people? Evaluation of Students Formative: While students are working, observe them and pose questions to check for mathematical understanding. Summative: Students’ work from the Explore phase can be used as a summative assessment. Plans for Individual Differences Intervention: If students are having difficulties, provide them with fractions manipulatives (fraction bars, fraction tiles) to help them visualize the idea of decomposing a whole unit. Extension: If students are in need of an extension, have them design a relay race that is 2 laps long so they have to decompose the number 2. You could also have them design a race that is 2 ½ laps long. Developed by SC Leaders of Mathematics Education 2011-2012

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Race 1

S

F

This race has 6 different sections to run. What are some possibilities that your team can run? Runner 1

Runner 2

Runner 3

Equation

Which one of your options is your favorite one? Explain why.

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Race 2

S

F

This race has 4 different sections to run. What are some possibilities that your team can run? Runner 1

Runner 2

Runner 3

Equation

Which one of your options is your favorite one? Explain why.

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Race 3

S

F

This race has 10 different sections to run. What are some possibilities that your team can run? Runner 1

Runner 2

Runner 3

Equation

Which one of your options is your favorite one? Explain why.

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Race 4

S

F

This race has 8 different sections to run. What are some possibilities that your team can run? Runner 1

Runner 2

Runner 3

Equation

Which one of your options is your favorite one? Explain why.

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Lesson 9: Math Situations Standards: 4.NF.3 Understand a fraction a/b with a >1 as a sum of fractions 1/b. 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. Lesson Learning Goals: Solve word problems using addition of fractions. Solve word problems using subtraction of fractions. Write equation to represent solutions involving addition and subtraction of fractions. Materials: blue cubes, green cubes, yellow cubes (6 of each); paper bag to hold 18 blocks; large piece of paper for drawing the model (1 piece per group); chart markers Lesson: Engage 10-15 minutes Present opportunities for students to informally share and represent strategies involving addition and subtraction with fractions. Students will create, add, and subtract fractions in the context of visiting animal habitats. Introduce the lesson by saying, “A third grade teacher named Ms. Lawson has a problem I think we can help her solve. Her 12 students have been studying animal habitats, and she wants them to visit some habitats and then share what they learn with the class. She would like to divide her students into 3 groups so they can gather information from 3 different locations. Now, here’s the problem. Ms. Lawson wants to be fair in choosing which group each student will join. So this is her idea. She wants us to try out her plan for deciding how the groups are set up. She will use our opinions to help her make the final plans.” Explore 20-25 minutes Say, “So let’s try her idea and see what we think.” Ask the following questions and write the answers on the board: 1. What do we know about the teacher’s problem? (12 students, 3 destinations, 3 groups) 2. What does she want us to do? (Try her plan to assign students to groups and give her our opinion.) Developed by SC Leaders of Mathematics Education 2011-2012

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Read Ms. Lawson’s idea to the class: a. Use green cubes to represent the pond group, yellow cubes to represent the cave group, and blue cubes to represent the forest group. b. Put the cubes into a bag, and have each student draw out a cube without looking. c. The students will divide into 3 groups based on the color cubes they draw. Students who draw green cubes will visit a pond, students who draw yellow cubes will visit a cave, and students who draw blue cubes will visit a forest. Say, “Okay, we need 12 of you to pretend you are Ms. Lawson’s students.” Randomly select 12 students (for example, draw popsicle sticks that have students’ names on them). As the teacher draws each name, the students draw a cube from the bag and form 3 groups. When the 12 students are standing in groups, the class discusses the outcome of the drawing, as the teacher writes the results on the board. The class divides into small groups to complete the following: Directions:  Decide as a group the answers to these questions.  Have a recorder write your answers.  Decide who will share the group’s answers during class Answers discussion time. 1. What fractions can be used to represent the students who pond = ______ will visit each location? cave = ______ forest = ______ 2. On the piece of paper given to your group, draw a (on separate fraction model to represent the results of the group drawing. paper) 3. What fraction represents the part of Ms. Lawson’s class who did not go to the pond? 4. Write an equation to represent your thinking for question #3. 5. What fraction represents the difference between the part of the class that went to the forest and the part that went to the cave? 6. Write an equation to represent your thinking in question #5. 7. What fraction represents the total of all 3 groups of students? 8. Does your group think this is a good way for Ms. Lawson to choose which group each student will join? Explain your opinion. Developed by SC Leaders of Mathematics Education 2011-2012

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Explain The groups share their answers and opinions to the questions.

10-15 minutes

Elaborate 10-15 minutes Give students the following problems to solve independently. Discuss student solutions and strategies. (adapted from www.k-5mathteachingreasources.com)

Write an equation to show your thinking for each problem situation. In each equation, underline the answer to the question. 1. Tom and Ben ordered a pizza for lunch. They each ate 1/3 of the pizza. How much pizza was eaten? Equation: __________________ How much pizza was left? Equation: __________________ 2. On Monday I spent 3/12 of my homework time reading and 9/12 working on a math project. How much more time did I spend on my math project than on reading? Equation: _______________ 3. Liam and Sam shared a chocolate bar. Liam ate 7/12 and Sam ate 5/12. Who ate more? Answer: _________________ How much more? Equation: ___________________ 4. I ate 4/12 of a box of donuts. My friend ate 1/12 more than I did. What fraction of the box of donuts did we eat in all? Equation: _______________ Evaluation of Students Students complete the following performance task: Read the following story:

Shanreese, Jackson, and Tony shared a giant cookie. Shanreese ate 1/8 of the cookie, Jackson ate 5/8, and Tony ate the rest. 1. Using the information in the story, create and write a question that could be solved by adding fractions with like denominators. Write an equation to represent your thinking. 2. Using the information in the story, create and write a question that could be Developed by SC Leaders of Mathematics Education 2011-2012

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solved by subtracting fractions with like denominators. Write an equation to represent your thinking. Plans for Individual Differences Intervention: Using fraction models, work with students needing extra practice to discuss and solve the following problems. Write an equation to show their thinking for each question.

1. 6/8 of a set of pencils need to be sharpened. What fraction of the pencils does not need to be sharpened? 2. Three friends ate 4/6 of a birthday cake. After dinner dad ate 1/6 of the remaining cake. How much of the cake was left? 3. Gracie’s mom cut a pan of brownies into 12 equal squares, and gave Gracie and each of her party guests one square. If there were 7 guests at the party, what fraction of the brownies was eaten? How much was left?

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Lesson 10: Multiply a Whole Number by a Unit Fraction Standard: 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. c. Understand a fraction a/b as a multiple of 1/b. SMP 1 Make sense of problems and persevere in solving them. SMP 3 Construct viable arguments and critique the reasoning of others SMP 4 Model with mathematics SMP 8 Look for and express regularity in repeated reasoning. Lesson Learning Goals: Represent a fraction a/b as the numerator times the unit fraction. Materials: fraction circles or squares, handouts, math journals Lesson: Engage (8-10 minutes) On an anchor chart or board, ask students to answer the following question: “What does 3 x 7 mean to you?” Student responses may include the following: 21, 3 groups of 7 objects, 3 sets of 7, 7 + 7 + 7, or draw some type of picture (see below).

Explore (20-25 minutes) The class will study the fraction 3/6 and determine the unit fraction (1/6) that is repeated to create the fraction. A unit fraction is written as a fraction where the numerator is one and the denominator is a positive integer. Have students draw a picture in their math journal to represent 3/6. Lead a discussion concluding with 3/6 = 1/6 + 1/6 + 1/6 which is the same as 3 groups (sets) of 1/6 or 3 x 1/6. Have students repeat process with 7/5. The unit fraction will be 1/5 so it is 7 groups (sets) of 1/5 or 7 x 1/5. Students could use fraction circles, squares, etc. to show what 7/5 looks like. Have students complete the handout Multiplying Whole Numbers and Fractions. Developed by SC Leaders of Mathematics Education 2011-2012

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Explain (10-15 minutes) After students finish Multiplying Whole Numbers and Fractions, bring class together. Have students share answers and the strategies they used to find them. Discuss any difficulties that students had or that you saw during the explore stage. Elaborate (20-25 minutes) Assign Multiplying Whole Numbers and Fractions 2. Choose 1-2 problems and have students place answers on an anchor chart or Promethean board. Discuss any difficulties that students had or that you saw. Evaluation of Students Formative: While students are working, observe them and pose questions to check for mathematical understanding. Summative: Collect Multiplying Whole Numbers and Fractions 2. Allow students to edit and rewrite how they solved the problem. Plans for Individual Differences Intervention: If students are having difficulties, provide them with whole number multiplication practice. Extension: Have students create word problems that contain whole numbers and units fractions. Then have them show how to find a solution by drawing a picture, writing out in words, and writing a multiplication sentence.

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Multiplying Whole Numbers and Fractions For each picture shown below, write an addition and multiplication sentence. 1.

Unit is

Words: 3 groups of ____ Addition Sentence: ____ + ____ + ____ = ____ Multiplication Sentence: ____ x ____ = ____ ____________________________________________________________________________ 2.

Unit is

Words: ___ groups of ____ Addition Sentence: ____ + ____ + ____ = ____ Multiplication Sentence: ____ x ____ = ____ ___________________________________________________________________________ 3. Unit is

Addition Sentence: ____ + ____ + ____ + ____ + ____ = _____ Words: ___ groups of ___ Multiplication Sentence: ____ x ____ = _____ Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

Multiplying Whole Numbers and Fractions 2 For each picture shown below, write an addition and multiplication sentence. 1.

Unit is

Words: 5 groups of ____ Addition Sentence: ____ + ____ + ____ + ____ + ____ = ____ Multiplication Sentence: ____ x ____ = ____ ____________________________________________________________________________ 2.

Unit is

Words: ___ groups of ____ Addition Sentence: ____ + ____ + ____ + ____ = ____ Multiplication Sentence: ____ x ____ = ____ ____________________________________________________________________________ 3. Unit is

Words: ___ groups of ____ Addition Sentence: ____ + ____ + ____ = ____ Multiplication Sentence: ____ x ____ = ____

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Lesson 11: Multiply a Whole Number by a Unit Fraction Using a Number Line Standard: 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a/b as a multiple of 1/b. SMP 1 Make sense of problems and persevere in solving them. SMP 3 Construct viable arguments and critique the reasoning of others SMP 4 Model with mathematics SMP 8 Look for and express regularity in repeated reasoning. Lesson Learning Goals: Use a number line as a visual fraction model to represent the multiplication of two whole numbers. Use a number line as a visual fraction model to represent a fraction a/b as a multiple of 1/b. Materials: handouts Lesson: Engage (8-10 minutes) Using the anchor chart created in Lesson 10, ask students how they would use a number line to represent 3 x 7.

0 7 14 21 Ask students how they would use a number line to represent 1/6 + 1/6 + 1/6 which equals 3/6. Use an anchor chart or board to record ideas. Explore__________________________________________ ________(20-25 minutes) As a class, study the fraction 3/6 and review the unit fraction (1/6). A unit fraction is written as a fraction where the numerator is one and the denominator is a positive integer. Go over several different examples and then have students complete Practice with Number Lines.

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Explain (10-15 minutes) After students complete Practice with Number Lines, bring class together. Have students share answers and the strategies they used to solve them. Discuss any difficulties that students had or that you saw during the explore stage. Elaborate (20-25 minutes) Assign Practice with Number Lines 2. Choose 1-2 problems and have students place answers on an anchor chart or Promethean board. Discuss any difficulties that students had or that you saw. Evaluation of Students Formative: While students are working, observe them and pose questions to check for mathematical understanding. Summative: Collect Practice with Number Lines 2. Allow students to edit and rewrite how they solved the problem. Plans for Individual Differences Intervention: If students are having difficulties, provide them with whole number multiplication practice. Extension: Have students create word problems that contain whole numbers and units fractions. Then have them show how to find a solution by drawing a picture, writing out in words, using a number line and writing a multiplication sentence.

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Practice with Number Lines 1. Use the number line to represent 2 x 6.

0

2. Use the number line to represent 4/7. What is the unit fraction?______

0

3. Use the number line to represent 2/9. What is the unit fraction?______

0

4. Use the number line to represent 7/5. What is the unit fraction?______

0

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Practice with Number Lines 2 1. How do you represent 2/3 on a number line?

0 What is the unit fraction of 2/3? _________ Words: ___ groups of ___ Multiplication Sentence: ____ x ____ = _____

2. How do you represent 3/10 on a number line?

0 What is the unit fraction of 3/10? ________ Words: ___ groups of ___ Multiplication Sentence: ____ x ____ = _____

3. How do you represent 5/3 on a number line?

0 What is the unit fraction of 5/3? ________ Words: ___ groups of ___ Multiplication Sentence: ____ x ____ = _____ Developed by SC Leaders of Mathematics Education 2011-2012

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Lesson 12: Multiply a Whole Number by a Fraction Using Pictures and Fraction Circles Note: This lesson could take more than one day depending on the needs of your students. Standard: 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. d. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. e. 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. SMP 1 Make sense of problems and persevere in solving them. SMP 3 Construct viable arguments and critique the reasoning of others SMP 4 Model with mathematics SMP 8 Look for and express regularity in repeated reasoning. Lesson Learning Goals: Use pictures and fraction circles to find the product of a whole number and a fraction. Explain that the expression a x b can be read as “a groups of b”. Materials: fraction circles, 2 handouts (Multiplying Fractions and 4 Word Problems) Lesson: Engage (8 – 10 minutes) Say, “Today we are going to be using pictures and fraction circles to multiply whole numbers and fractions. I want you to draw a picture that you can use to solve the following word problem. Write a number sentence for the problem.” 1) Riley wants to give five cookies to each of his three friends. How many cookies will he need? Ask students to share their pictures and number sentences. A possible picture might look like this.

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Students’ pictures should show three groups of five cookies. (Teachers – it is important for students to notice that this problem can be modeled using the multiplication sentences: 3 x 5 = 15. Also help students make connections between the words 3 groups of 5 and the mathematical expression 3 x 5.) 2) Write 2 x 4 = on the board. Ask students to explain what the 2 and the 4 stand for in this problem. The 2 should stand for the number of groups and the 4 should stand for the number of objects in each group. Ask students to draw a picture to find the answer 2 x 4 = . (Teachers – walk around as students work and ask 2-3 students to share their pictures. Have the students show and explain their work. Be sure to emphasize that they draw 2 groups of 4 objects.) Explore (20 - 25 minutes) Let students work in groups to solve the following problems with their fraction circles. Have them also write a multiplication sentence that would answer this problem. 1) Seth has several pizzas he wants to share with his friends. Seth wants to give each of his 4 friends 2/5 of a pizza. How much pizza will he give away? Teachers – The mathematical sentence will be 4 x 2/5 = 8/5 = 1 3/5 where the 4 represents the number of groups, the 2/5 represents the amount in each group, and 8/5 or 1 3/5 represents the total amount of pizza given away. The students should put 4 groups of 2/5 as shown below and the sum should 8/5 or 1 3/5. Ask: How is this problem similar to the cookie problem we did previously? 2) Mariah uses 1/3 cup of brown sugar for each batch of chocolate chip cookies she makes. How much brown sugar will she need if she makes 5 batches? Teachers – Ask the students to solve the problem with their fraction circles. They will need two sets of fraction circles to be able to find the answer. Ask what the 5 means in the sentence and what the 1/3 means (5 groups of 1/3). The final answer is 5/3 or 1 2/3. Have students record the answer both as an improper fraction and a mixed number. Have students complete Multiplying Fractions. Explain (10 – 15 minutes) After students complete Multiplying Fractions, bring the class together. Have students share answers and strategies they used to find them. Discuss any difficulties that students had or that you saw during the explore stage.

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Elaborate (20-25 minutes) Assign Multiplying Fractions 2. Choose 1-2 problems and have students place answers on an anchor chart or Promethean board. Discuss any difficulties that students had or that you saw. Evaluation of Students Formative: While students are working, observe them and pose questions to check for mathematical understanding. Summative: Collect Multiplying Fractions 2. Allow students to edit and rewrite how they solved the problem. Plans for Individual Differences Intervention: If students are having difficulties, provide them with whole number multiplication practice. Extension: Have students create word problems that contain whole numbers and fractions. Then have them show how to find a solution by drawing a picture, writing out in words, and writing a multiplication sentence. They may use fraction circles if needed.

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Fractions Common Core State Standards

Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

Multiplying Fractions 2

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Lesson 13: Multiply a Whole Number by a Fraction Using Fraction Circles, Pictures, and Mental Images Standard: 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 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. 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. SMP 1 Make sense of problems and persevere in solving them. SMP 3 Construct viable arguments and critique the reasoning of others SMP 4 Model with mathematics SMP 8 Look for and express regularity in repeated reasoning. Lesson Learning Goals: Multiply a whole number and a fraction using fraction circles, pictures, and mental images. Materials: fraction circles, handouts (3 pages) Lesson: Engage (8-10 minutes) Write 2/3 on the board. Ask the students to picture this fraction in their minds. Ask a few students to describe what they are picturing. Write a 5 and a multiplication sign before the 2/3. Ask the students to picture what the following statement would represent - 5 x 2/3. Ask the students to draw a picture of this problem to find the product in their journal. Walk around as students work and ask two or three students to share their pictures. Have the students show and explain their work. Be sure to emphasize that they draw 5 groups of 2/3. Sample student work for 5 groups of two-thirds below.

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Explore Assign Multiplying Fractions.

(20-25 minutes)

Explain (10-15 minutes) After students finish Multiplying Fractions, bring class together. Have students share answers and strategies they used to find them. Discuss any difficulties that students had or that you saw during the explore stage. Elaborate (20-25 minutes) Assign Multiplying Fractions 2. Teachers – use your discretion about which problems to assign. After students finish, choose 1-2 problems and have students place answers on an anchor chart or Promethean board. Discuss any difficulties that students had or that you saw. Evaluation of Students Formative: While students are working, observe them and pose questions to check for mathematical understanding. Summative: Collect the 4 word problems Multiplying Fractions 2 handout. Allow students to edit and rewrite how they solved the problem. Plans for Individual Differences Intervention: If students are having difficulties, provide them with whole number multiplication practice. Extension: Have students create word problems that contain whole numbers and fractions. Then have them show how to find a solution by drawing a picture, writing out in words, and writing a multiplication sentence. They may use fraction circles or squares if needed.

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Fractions Common Core State Standards

Multiplying Fractions 2 (page 1)

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Fractions Common Core State Standards

Multiplying Fractions 2 (page 2)

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Lesson 14: Multiply a Whole Number by a Fraction Using a Number Line Note: This lesson could take more than one day depending on the needs of your students. Standard: 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 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. 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. SMP 1 Make sense of problems and persevere in solving them. SMP 3 Construct viable arguments and critique the reasoning of others SMP 4 Model with mathematics SMP 8 Look for and express regularity in repeated reasoning. Lesson Learning Goals: Use number lines to multiply a whole number by a fraction. Use partitioning to find a fractional part of a whole number. Materials: handouts Lesson: Engage (8-10 minutes) Pose the following problem to students: Tasha makes $12 an hour babysitting several children. How much does she make if she babysits these children for 3 hours? Write a multiplication sentence that describes how to find the answer. 3 x 12 = 36 number of groups (hours)

number in each group ($/hour)

amount earned in 3 hours ($)

Teacher – The previous two lessons developed the meaning for the factors in a multiplication sentence. The first factor is typically the number of groups. The second factor is the number of objects in each group. Many real world situations make it difficult to have a fractional number of groups. Show the students how they can solve the word problem using a number line.

0

12

24

36

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Explore (20-25 minutes) Ask students how the number line can be used to show how much Tasha will earn if she babysits 2 hours or 7 hours. Have the students use their pencils or fingers to trace along a number line. Have them start tracing at 0 and make jumps for each hour worked. The number line also demonstrates how multiplication can be demonstrated by repeated addition.

Ask students to use number line 1 from Multiplying Fractions on Number Lines to find the amount of money Tasha will earn if she babysits 5 hours. Encourage them to draw in the jumps on the number line and then write a multiplication sentence that solves the problem. The multiplication sentence should be 5 x 12 = 60. Use the other number lines for more practice. Teachers – Present the following problem: Suppose Tasha works ¾ of an hour. Estimate in your head how much money Tasha will earn. (Ask the students as a group whether this amount would be more than $12 or less than $12. Ask a few students to explain their estimate.)Use the number line to find the exact amount of money that Tasha will earn if she babysits ¾ of an hour and write a multiplication sentence. Sample student work is shown below.

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Fractions Common Core State Standards

Ask several students to come to the board and show how they used the number line to solve this problem. Be sure that they show how they divided their number line and explain how they found their answer. Also make sure they explain the meaning of each number in the multiplication sentence. Sample student work is shown below:

Assign Part 2: Multiplying Fractions on Number Lines (teachers use your discretion). Explain (10-15 minutes) After students finish Part 2: Multiplying Fractions on Number Lines, bring the class together. Have students share answers and the strategies they use to find them. Discuss any difficulties that students had or that you saw during the explore stage. Elaborate (20-25 minutes) Assign Part 3: Multiplying Fractions on Number Lines. Teachers – use your discretion about which problems to assign. After students finish, choose 1-2 problems and have students place answers on the board. Discuss any difficulties that students had or that you saw. Evaluation of Students Formative: While students are working, observe them and pose questions to check for mathematical understanding. Summative: Collect the 4 word problems Multiplying Fractions handout. Allow students to edit and rewrite how they solved the problem. Plans for Individual Differences Intervention: Teacher may need to use smaller number lines. Extension: Ask the students to use the number line II to find the exact amount of money that Tasha will earn is she babysits 6 ½ hours. Developed by SC Leaders of Mathematics Education 2011-2012

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Fractions Common Core State Standards

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Fractions Common Core State Standards

Part 2:

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Fractions Common Core State Standards

Part 3: Multiplying Fractions on Number Lines

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Fractions Common Core State Standards

The following additional word problems are provided as needed. 4.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. SMP 5 Use appropriate tools strategically. Teacher may use this next handout from the website http://www.k-5mathteachingresources.com/ for extra practice. This website has great resources.

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Fractions Common Core State Standards

15 friends want to order pizza for dinner. They predict that each person will eat 1/3 of a pizza. How many pizzas should they order?

Tom picked 10 plums from a tree in the garden and ate 3/5 of them before lunch. How many plums did he eat before lunch?

Mike rode his bicycle 6 kilometers to school. He stopped at his friend’s house after 2/3 of the total journey. After how many kilometers did Mike stop?

On Monday Maria spent 3 hours reading. Of the time she spent reading, ½ was spent reading magazines. For how many hours did Maria read magazines? Developed by SC Leaders of Mathematics Education 2011-2012

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A restaurant uses 1/3 cup of mayonnaise in each batch of salad dressing. How many cups of mayonnaise will be used in 7 batches?

Peter went to the store and bought 8 pounds of apples. If ¼ of the apples were cooking apples, how many pounds of cooking apples did Peter buy?

A farmer owns 4 acres of farmland. He grows potatoes on 3/8 of the land. On how many acres of land does the farmer grow potatoes?

A cookie factory puts 3/6 of a barrel of flour into each batch of cookies. How much flour will the factory use in 7 batches?

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Sue is baking cherry pies for a family dinner. She expects that each of the 15 guests will eat 1/5 of a cherry pie. How many cherry pies should she bake?

Chris had 25 marbles in his collection. He gave 2/5 of his marbles away to his friends. How many marbles did Chris give away?

Two runners ran for 9 kilometers. They stopped for water after 2/3 of the run. After how many kilometers did the runners stop for water?

Mrs. Smith spent 8 hours in the kitchen. Of the time she spent in the kitchen, ¾ was spent making bread. For how many hours did Mrs. Smith make bread?

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Appendix A Sample Anchor Chart

One-half

1 2

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