7 Flex Day (Instruction Based on Data) Recommended Resources: Decomposing Fractions Pizza Share Give Em Chocolate 8 Use cuisinaire rods and tape diagrams to model equivalent fraction values. Determine whether two fractions are equivalent using models I. Provide an opportunity for students to build fractions with tiles/strips and make observations about the different ways they can compose that value before explicitly teaching them about equivalence II. Suggested activity for modeling (after students had a chance to explore) is to give every student a piece of construction paper and scissors to follow along with your modeling to show all of the different ways to show ½ (tip: you may record all of the equivalent relationships in a chart so that students can also see a numerical pattern between numerators and denominators): First cut the paper into two halves, then cut each half in half to show that ½ = 2/4. Then cut each fourth into half to show that ½ = 2/4 = 4/8 and so on Use the two pictures below to answer the questions. 1.) The rectangle in picture (a) has length 1. What fraction does the shaded part represent? (1 pt) 2.) The rectangle in picture (b) also has length 1. What fraction does the shaded part represent? (1 pt) 3.) Use the two pictures to explain why the two fractions represented above are equivalent. (2 pts) My Math Chapter 8 Lesson 3 Lessons 5.4 Modify resources to align with objective. Provide students with tiles/strips to build equivalent values and then record their findings in a chart to demonstrate all of the equivalent relationships they find III. Something to Think About/Write About/Talk About: Does every fraction have an equivalent? Prove it! 11 Page
9 Decompose fractions using area models to demonstrate equivalence Use visual (area) models to explore and justify the following statements: 1) The shaded part of the rectangle below represents 1/5 Lessons 5.5 & 5.6 Every fraction is equivalent to an infinite number of other fractions. Any fractional quantity can be subdivided into an infinite number of equal pieces while maintaining the value of the original fraction Which of the following rectangles is also shaded to represent 1/5? **Explain what happens to the number and size of each part of a fraction when you create different equivalents (higher quantity = smaller size, while smaller quantity = larger size) A. B. **Teaching Tip: Emphasize the word infinite today this could easily be introduced in a creative HOOK J C. D. 2) Jose said the two models below were equal or equivalent because in both two parts are shaded. So Jose says it s the same fraction of the whole. Do you agree or disagree with Jose? 12 Page
10 Analyze the patterns in numerators and denominators in equivalent fractions to infer the algorithm for creating equivalent fractions Lesson Idea for Interactive Mini-Lesson: 1) Eli cut a pizza into 6 equal slices. He ate ½ of the pizza. (1 pt) Red Rectangles Which fraction best shows the part of the pizza Eli ate? A. 1/6 B. 1/3 C. 3/6 D. 3/2 2) Look at the rectangles below and answer parts a- d (a) What fraction of Rectangle A is shaded? (b) What fraction of Rectangle B is shaded? (c) What fraction of Rectangle C is shaded? (d) What conclusions can you draw about the fractions you found in parts (a)-(c)? Explain. 13 P a g e
11-12 Use area models and multiplication to show the equivalence of two fractions Pacing: 2 days Use the Engage NY resources first to connect the procedure to the conceptual area model; use My Math as needed for re-teach and additional practice Explain that when you multiply the numerator and denominator by the same number, it does not change the value of the fraction but only changes the number and size of its parts. Use visual models to support your thinking. Explore what happens you do not multiply both the numerator and denominator by the same number Suggested activity for lesson closing (think/write/talk about it) 1.) Use the following fraction to answer the three questions below: 1) James and Benito each have a bag of pencils. Some pencils are sharpened and some are not. James's bag has: A total of 5 pencils Exactly 2 sharpened pencils. Benito's bag has a total of 10 sharpened pencils. Benito has the same fraction of sharpened pencils in his bag as James has in his bag. (a) Exactly how many of Benito's pencils are sharpened? ( 1 pt) (b) In the drawing below, draw pictures of the pencils in James's bag and the pencils in Benito's bag. Use numbers to show the fractions of sharpened and unsharpened pencils in each bag. (2 pts) Lessons 5.7 & 5.8 My Math Chapter 8, Lesson 4 4 / 5 a.) How does the value of the fraction change if you double only its numerator? Explain and illustrate: 14 Page b.) How does the value of the fraction change if you double only its denominator? Explain and illustrate: (c) How does the value of the fraction change if you double both the numerator and denominator? Explain and illustrate your thinking with a visual: (c) Benito's bag has a total of 10 pencils inside, and James's bag has a total of 5 pencils inside. How can the fraction of sharpened pencils in James's bag be the same as the fraction of sharpened pencils in Benito's bag, even though they have a different number of pencils? Explain
13-14 Use are models and division to find equivalent fractions. Write fractions in simplest form Pacing: 2 days Use the Engage NY resources first to connect the procedure to the conceptual area model; use My Math as needed for re-teach and additional practice Key Point: when you divide the numerator and denominator by the same number, it does not change the value of the fraction but only changes the number and size of its parts. Explain that when you can no longer divide the numerator and denominator by any other factors besides 1, the fraction is now in simplest form. Teaching Tip: Students should have time to explore dividing by different numbers instead of just using the Greatest Common Factor so that they have flexible strategies to use in the future when computing, and also so that they can draw their own conclusion about how using the GCF is a more efficient process. Possible application or hook: Fill in the missing fractional parts with any two numbers that would make each pair equivalent: 1) Alicia opened her piggy bank and counted the coins inside. Here is what she found: 22 pennies 5 nickels 5 dimes 8 quarters What fraction of the coins in the piggy bank are dimes? Write your answer in simplest form 2) Complete the fractional comparisons by filling in each missing numerator/denominator. Show your work:? / 7 = 6 / 14 5 / 8 = 15 /? 8 / 12 =? / 3 3) Laura looks at the following rectangle and says that 1/4 of the rectangle is shaded. Lessons 5.9 & 5.10 My Math Chapter 8, Lesson 5 Modify resources to align with objective.? / 9 =? / 3 4 /? = 12 /? Challenge: 7 /? =? / 15 Do you think she is correct? Explain why or why not by using the words simplify and equivalent. 15 Page
15 Explain equivalent fractions as the same point on a number line and relate equivalence to multiplication and division Students should define equivalent fractions as the same point on a number line, therefore they should see the number line as a valuable tool in determining equivalency In order for the number line to be a valuable tool, it is absolutely essential to attend to precision when locating these points on a number line To help students attend to precision, model how to draw one rectangle above the number line and one below the number line to represent the two different fractions on the same number line i.e. to show 2/3 and 3/5: 1) Molly, Shaina and Rochelle each baked an equal size cake for their class party. Molly cut her cake into 18 equal size slices. At the party, the class ate 12/18 of Molly s cake. (MP.1, MP a.) Draw a number line. Represent the fraction of Molly s cake that was eaten below: (1 pt) b.) The class ate an equal amount of Shaina s cake, but she cut her cake into larger slices. Draw a model that could represent Shaina s cake and the fraction that was eaten: (2 pts) Lesson 5.11 Students with disabilities would benefit from using fraction strips and a larger number line to aid in their precision c.) Explain your reasoning to the fraction that you created to represent the amount of Shaina s cake the class ate. 2 pts (MP. 6 & MP. 7) 16 Flex Day (Instruction Based on Data) Recommended Resources: Eggsactly: 12 and 18 Equivalent Fractions Picking Fractions Pattern Block Puzzles 16 Page
17 Use benchmark fractions on a number line to visually compare the value of the fractions using the symbols >,< and =. 18 Pacing: 2 Days SW understand that you can only compare fractions when referring to the same whole: http://learnzillion.com/lessons/102- compare-fractions-to-the-same-whole The next lesson asks students to begin reasoning without a number line, so provide plenty of opportunities in these two days for students to make observations about all of the fractions they placed less than ½ and greater than ½ (i.e. the numerator is more than half the denominator, etc.) Provide multiple opportunities for students to observe patterns in the relationship between the numerators and denominators so they can begin to reason about fractional values without a number line (i.e. I notice that all of the fractions closest to 1 have numerators that are close to the denominator, etc.). 1. Alex said that 2/5 was closer to 0 than half on the number line. Do you agree with him? Why or why not? 2) There are two cakes on the counter that are the same size. The first cake has 1/2 of it left. The second cake has 5/8 left. Use a number line to determine which cake has more left over: 3) Using the denominator 7, create a fraction that is less than 4/8 and one that is greater than 4/8: < 4/8 > 4) Byron says that 3 / 5 is greater than ½ because the denominator 5 is greater than the denominator 2. Use a number line to determine if he is correct than 3/5 > ½: b. Is his thinking correct? Explain: Lessons 5.12 & 5.13 My Math Chapter 8 Lesson 7 http://learnzillion.com/l essons/98-comparefractions-to-thebenchmark-of-12 http://learnzillion.com/l essons/99-comparefractions-to-thebenchmark-of-14 http://learnzillion.com/l essons/100-comparefractions-to-thebenchmark-of-34 17 Page
19 Reason about fractional values by exploring different strategies for comparing fractions I. Deduce the characteristics of a pair of fractions that help you determine which strategy to use: II. These strategies should be on an anchor chart for students to refer back to 1) Use the more than/less than one half or one whole strategy (i.e. if I am comparing 4/5 and 1/4 I know that 4/5 is greater than ½ and ¼ is less than ½, therefore 4/5 > ¼) 2) Use the closeness to one half/one whole strategy (If I am comparing the fractions 4/5 and 9/10, I know they are both over ½ so I can t use strategy #1. Instead I will think about which one is closest to one whole. Since both of them have numerators that are only 1 part away from completing the whole, I have to reason about the size of each part. Since tenths are smaller than fifths, that means 9/10 is one smaller part away from completing the whole than the 1 larger part missing in 4/5. Therefore, 9/10 is the larger fraction.) manipulatives are an essential part of building the understanding for this strategy 3) Determine when the fractions have the same amount of different sized parts (i.e. same numerators) and therefore can determine which parts are larger (for instance, if I am comparing 2/3 and 2/5 I see I have the same number of parts, but the 1) Which of the fractions below are less than ½? Circle them: Explain how you chose the fraction(s) less than ½: 2) Explain each strategy using symbols, models or reasoning in words next to each. Then choose which of the comparisons below would best be solved using that strategy and complete them in the example column: Strategy Explain Example Closeness to one whole Same size parts More than/ less than 1/2 Same amount of different size parts ¾ 2 / 8 4 / 8 4 / 5 Closest to 1 More or Less In this lesson, it is important for students to have plenty of hands on opportunities to reason with fractions to develop their fraction sense. Resist the urge to teach four different strategies or rules to follow and allow students the opportunities to play with fractional values and reach their own conclusions about using these strategies BEFORE you explicitly model how to apply the strategies. Between today and tomorrow s lesson it is important for students to build their own understanding of when each strategy works best. Students should have 18 Page
20-21 Given two fractions with unlike numerators and denominators, create equivalent fractions in order to compare using like numerators or denominators. denominators tell me that the parts are different sizes. If a whole is cut into thirds, those are bigger pieces than if the same-size whole is cut into fifths. Therefore, I know 2/3 is larger.) 4) Determine when the fractions have the same sized parts (i.e. same denominators) and therefore you can determine which one has more of the same sized parts by just looking at the numerator (for instance, if I am comparing 3/5 and 2/5 I see that I have the same number of equal sized parts (5), therefore I can look at the numerator to see how many of each equal size part I have. In this case, since the pieces are the same size and 3>2, I know that 3/5 > 2/5) know that 6/8 is the largest fraction. Pacing: 2 days Remember: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12 and 100.) Some different denominators are used to continue to allow students to work with multiples when finding common denominators, but you can modify the problems for struggling students who need to use manipulatives to work within the numbers in the manipulatives. 7 /8 5 / 6 3 / 10 4 / 10 3. Leah and Jamal were swimming laps in an Olympic size pool. They timed each other to see who could swim the farthest in just thirty seconds. Leah swam 11/12 of a lap and Jamal swam 7/8 of a lap. Who swam the farthest in thirty seconds? Use reasoning to explain how you know which one is bigger. 4.) Robbie said that Jamal swam further in thirty seconds because breaking the lap into 8ths means that those parts are bigger and he swam 7 out of 8 of those parts of the lap. Do you agree or disagree with Robbie? 1. Compare the fractions below by creating equivalent fractions with common numerators OR denominators. Make each inequality true by filling in the blank with >, < or =. Show your work for each one. (3 pts) Explain how you came to your solution using strategies from yesterday. 5 / 8 5 / 10 4/ 10 27/100 3/10 2/5 access to manipulatives, fraction strips, number lines and visual models to explore and justify these strategies. Lessons 5.14 & 5.15 My Math Chapter 8, Lesson 6 19 Page
22 Choose the best strategy to efficiently compare fractions Students should be encouraged to determine what strategy makes most sense based on the fractions they are comparing Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12 and 100.) Assess readiness for the lesson with this quick check: 1)Which of the following strategies accurately describes how to compare the following fractions: 6 / 8 4 / 6 a. I see that both fractions are greater than 1 half, but less than one whole. Therefore 6 / 8 = 4 / 6 b. Since the 6 in the fraction 6/8 is 2 parts away from the complete whole, and the 4 in the fraction 4/6 is 2 parts away from the complete whole, I know that the fractions are equivalent. Therefore, 6 / 8 = 4 / 6 are equal. c. Since both the numerator and denominator in the fraction 6/8 are larger than the numerator and denominator in the fraction 4/6, I 3. This is what Andrew says about the fractions from the problem above: Since both fractions are greater than one half and less than one whole, I will think about the size of their parts to figure out which one is closest to one half. 8 parts of the same whole are smaller than 6 parts of the same whole. That means in the fraction 6/8, there are two smaller parts away from the whole, compared to the two larger parts missing from the whole in 4/6. Therefore, 6/8 is the largest fraction: 6 / 8 > 4 / 6 Do you agree with his reasoning? Explain Fraction Game Cards Comparing Fractions Quiz *Students can play Fraction War with a partner by each flipping over a game card and determining the best strategy to compare the pair. The player with the greatest fraction wins. Students should record each pair and illustrate the strategy used on a recording sheet *Continuation of yesterday s lesson differentiate and extend as necessary. Students should have more hands on experiences with comparing fractions using the various strategies and most importantly, deciphering WHEN each strategy is most useful. 20 Page