Compare Fractions. To prepare to read the story, ask students to compare some fractions. Distribute fraction circle pieces and Centimeter Grid Paper to students. Display the fraction sets listed below.
1/2 3/8
1/2 2/3
3/4 9/10
3/4 2/3
Ask students to decide which fraction is smaller.
- What tools might you use? (Possible responses: fraction circle pieces, fraction strips, a drawing, a drawing on the Centimeter Grid Paper, counters)
Assign one set of fractions to each pair of students. Ask students to show how they decided which fraction was smaller. As students are working, identify students who will prepare a display and share their solutions with the rest of the class. When most students have completed their work, discuss each fraction pair.
Use questions similar to the following to support this discussion:
- Linda thinks 1/2 is smaller than 3/8 because the denominator is larger. Do you agree with her? Why or why not? (Possible response: I do not agree with Linda. The denominator tells how many pieces are in the whole. If I divide something into 8 equal pieces the pieces are pretty small. If I divide the whole into 2 equal pieces each piece is pretty big.)
- John thinks 9/10 is smaller than 3/4 because the pieces are smaller. Do you agree with him? Why or why not? (Possible response: I do not agree with John. He is right that the whole is divided into smaller pieces but 9/10 is almost all of those pieces or one whole. 3/4 is also just one piece shy of a whole, but the piece is larger and therefore less of the whole.)
Display the fraction sets below.
1/2 of 50 and 1/2 of 1000
1/2 of a brownie and 1/2 of a pan of brownies
1/2 of 100 and 9/10 of 10
3/4 of 6 and 2/3 of 12
Again, ask students to decide which fraction is smaller and again, assign one set of fractions to each pair of students. As students are working, identify students that will prepare a display and share their solutions with the rest of the class. When most students have completed their work, discuss each fraction pair.
Use questions similar to the following to support this discussion:
- Sam thinks 1/2 is 1/2 so the fractions are equal. Do you agree with Sam? (Possible response: No, the wholes are not the same. 1/2 of 50 is not the same as 1/2 of 1000).
- Compare these sets of fractions to the first sets of fractions. What is different? (Possible response: In the second set the whole is defined. In the first set, the whole stays the same but in the second the fractions compared have different wholes.)
Read the Clever Tailor Story. Tell students that you will read the story of “The Clever Tailor” together and that the tailor uses fraction comparisons to find her way in the world. Use the following discussion prompts to highlight the mathematics in the story. The first time you read the story, read it in its entirety without stopping for discussion. Then, reread it more slowly, drawing out the mathematics.
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- What is three-fourths of four flies? (three flies)
- What is three-fourths of 8 flies? (six flies)
Repeat the question, substituting other numbers that can be manipulated evenly, such as 12 and 20. Students can use drawings or counters to solve the problems.
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- What might the giant think “Three-fourths at One Blow!” means? (He might think it means that the tailor kicked three-fourths of a group of people at one blow.)
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- Is half a loaf of bread a lot to take in one bite? (Yes, it is more than most people can bite, though it does depend on the size of the loaf.)
- Do you think the tailor can eat more than the giant? (Answers will vary.)
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- Why was the giant afraid? (Because he compared 1/2 to 2/3 and knows 2/3 of the same whole is bigger.)
- Which is bigger, one-half of a whole or two-thirds of the same whole? (Two-thirds is bigger than one-half of the same whole.)
Ask students to use tools to compare 1/2 to 2/3 of the same whole. Students may reference the display made at the beginning of the lesson.
- Is the tailor comparing fractions of the same whole? (No)
Now ask students to draw a picture to help them compare 1/2 of a loaf to 2/3 of a slice of bread.
- Which is bigger, 1/2 of a loaf or 2/3 of a slice of bread? ( 1/2 of a loaf)
- Give another example where half of something is bigger than two-thirds of something else. (For example, half of a jumbo pizza is bigger than 2/3 of a personal pizza.)
- Is “three-fourths the height of a tree” a big jump? (Yes, it's a very big jump.)
- How high is three-fourths of our [classroom wall]?
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- Why is the giant even more afraid? (Because he compared 9/10 to 3/4 and knows 9/10 of the same whole is bigger.)
- Which is bigger, nine-tenths of a whole or three-fourths of the same whole? ( 9/10 of the same whole.)
Ask students to use tools to compare 3/4 to 9/10 . Students may reference the display made at the beginning of the lesson.
- Is the tailor comparing fractions of the same whole? (No)
Now ask students to draw a picture to help them compare 3/4 of a tree to 9/10 of a meterstick.
- Give another example where 3/4 of a something is bigger than 9/10 of something else. (For example, 3/4 of an Olympic-size pool is bigger than 9/10 of a kiddie pool.)
The tailor is able to call all of her sisters and the giant is able to call 9/10 of his brothers. Draw a picture to compare these fractions.
- Does the giant have reason to be afraid? (No, because she can only call 2 sisters and he can call 9 brothers. He keeps comparing the fractions as if they were describing part of the same whole.)
- Give another example where 9/10 of something is larger than 1 whole of something else.
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- How much is half of $100? ($50)
- How much is 9/10 of $100? ($90)
- Do you think the tailor was a good friend to the giant? Why or why not?
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- If the younger brothers each get one-fourth and the oldest brother gets one-half, each brother gets one piece. Is this a fair way to share the land? (No, this plan is not fair because the pieces are not the same size.)
- Draw a picture showing how the oldest brother wants to divide up the land. (See picture on the next page.)
- How do you think the land should be divided among the brothers? (The land should be divided into thirds.)
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- How much money does the tailor have now? (The tailor got $90 from the giant and $10 from the brothers, totaling $100.)
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- How many is three-fourths of eight robbers? (six robbers)
- How much money does each robber have? ($1000)
- How much money do the eight robbers have altogether? ($1000 times eight robbers is $8000.)
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- Is the tailor's plan for sharing the money fair? (No, she had less money than the robbers at this point.)
- How much is half of the robbers' money? ($4000)
- How much is half of the tailor's money? ($50)
- Explain how the tailor ended up with $4050. (The robbers gave her half of their $8000 or $4000. She gives them half of her $100 or $50. The tailor now has $4050 and the robbers each have $500 plus the $50 to share among the eight of them.)
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- Who is the fancy lady? (The tailor!)
- Explain how the tailor used mathematics to trick the giant and the robbers.
Refer students to Questions 1–4 in The Clever Tailor Problems section in the Student Guide. Ask students to work on answering these questions with a partner. For Question 1, suggest that students draw a picture or use their tools to decide how much money each robber would receive if they shared the $50. In Question 4 students are challenged to make their own scene for the story of “The Clever Tailor.” Students should be encouraged to draw a picture that tells their story. Have students share their stories with a small group of their peers.
Use Check-In: Questions 3–4 in “The Clever Tailor” section of the Student Guide to assess students' abilities to recognize that the same fractional parts of different-size unit wholes are not equal [E4], and compare fractions using models [E10].
