Introduce Circle Duets Game. For this lesson, the red circle is the whole. To play the Circle Duets game, organize students into groups of four—two teams with two players on each team. Use a display of the Circle Duets pages in the Student Activity Book to discuss the directions.
Before play begins, direct students to the Compose and Decompose Mixed Numbers pages in the Student Guide. Discuss Questions 1–3, which highlight two important aspects of the game: representing addition problems using circle pieces and writing number sentences from the resulting sums.
As you discuss Question 1, use the following prompts:
- What fraction circle pieces did Jackie trade?
(Possible response: First she traded the pink [12] piece for an aqua [16] and an orange [13] piece.)
- Why do you think she did that? (She needed a 16-piece to complete the first circle. If she took 16 away from 12, she would have 13 left over.)
- How do you know she made a correct trade? (I can put an aqua and orange piece on top of the pink. Together they are the same size as one pink.)
- Did she make another trade? (Yes, after she filled the first circle, she traded six 16-pieces for one whole circle piece [red].)
- Could she have made any different trades? (She could have traded the 12-piece for three 16-pieces.)
- What would have been her answer? (126)
- Is that the same as 113? (Yes, because 26 takes up the same space as 13. I can put two 16-pieces on top of the 13-piece to show they are the same.)
Ask similar questions to those above when you discuss Question 2.
- Is 2112 more or less than 216? How do you know? (2112 is less than 216. Possible response: The black pieces are 112 of the circle. They are smaller than the 16 [aqua] pieces.)
To clarify the rules, play a sample round of the game with three students using display circle pieces and the display game pages.
Play Circle Duets. As students play the game, they need not find common denominators to find the sums or reduce their answers to simplest form. They can simply use the circle pieces to add and make trades until they can represent the sum as a proper fraction or a mixed number. Instead of concentrating on procedures at this point, encourage students to focus on the reasonableness of the results and modeling the equivalent fractions.
As students add fractions in the game, circulate and ask:
- Should the sums be less than one or more than one? Less than two or more than two?
Discuss Game Strategies. After students have had the opportunity to play Circle Duets, reconvene as a class to discuss some of the strategies students used during the game. Encourage students to share their invented strategies and use their fraction circle pieces to model the following problems. Have a set of display fraction circle pieces and the display of the Circle Duets pages ready if students need them to explain their solutions.
- Linda spun 34 and 13 and Jackie spun 16 and 23 playing Circle Duets. Estimate to see if you think their team total will be more than 1 or less than 1.
(Possible response: I think their team total will be a lot more than 1.)
- Will their team total be more than 2 or less than 2? (Possible response: I think their team total will be less than 2.)
- Who would like to explain how they estimated the total? (Possible response: I know 13 + 23 is one whole. 34 + 14 would make another whole, but Jackie only spun 16, so that would make less than one whole. I think the team total is close to 2.)
- Work with a partner to find each girl's total. One partner should find Linda's total and the other should find Jackie's total. What is each girl's total? (Linda's total is 34 + 13 = 1312 . Jackie's total is 16 + 23 = 56.)
- Who would like to show how to add the improper fraction 1312 to 56 to find their exact team total? (Possible response: To find 1312 + 56, I found a fraction equivalent to 56 with a denominator of 12, 1012 . 1312 + 1012 = 2312 , or 11112 .)
- Is that sum reasonable? How do you know? (Possible response: Yes; I estimated the sum to be close to 2, and 11112 is very close to 2.)
- Did anyone add the mixed number 1112 to 56? Show us how. (Possible response: I used fraction circle pieces. I know 1 red circle is the same as 1212 or 12 black pieces, and then I had one more black piece. I knew I had to trade the 5 aqua pieces for 10 black pieces so I would have pieces of all one color. 12 + 1 + 10 black pieces is 2312 .)
- Did anyone solve 1112 + 56 differently? (Possible response: At first, I only thought about 112 + 56. I rewrote 56 as 1012 . 112 + 1012 = 1112 . Then I added 1 + 1112 = 11112 .)
Have students save the game materials, or collect the pages to save for Lesson 6. They will have another opportunity to play Circle Duets during the Workshop.
