Fractions

Lesson 7

Objective

Identify fractions in a whole that is not partitioned into equal parts.

Common Core Standards

Core Standards

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  • 3.NF.A.1 — Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Foundational Standards

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  • 2.G.A.3

Criteria for Success

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  1. Partition a shape into fractional units in more than one way.
  2. Understand that equal shares of identical wholes need not have the same shape.
  3. Determine the fraction that parts of a whole represent when those pieces are not equal in shape or size by partitioning them further to identify their equal parts.

Tips for Teachers

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Too often, when students are asked questions about what fraction is shaded, they are shown regions that are portioned into pieces of the same size and shape. The result is that students think that equal shares need to be the same shape, which is not the case. On the other hand, sometimes visuals do not show all of the partitions” (Van de Walle, Teaching Student-Centered Mathematics, Grades 3–5, vol.2, p. 211). Thus, this lesson tries to address both of these potential misconceptions and deepen students’ conceptual understanding of fractions. Having students explain what it meant by “equal parts” also provides opportunities for students to attend to precision (MP.6) (NF Progression, p. 7).

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Anchor Tasks

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Problem 1

Start with a square sheet of paper and make folds to construct a new shape. Explain how you know the shape you constructed has the specified area.

  1. Construct a square with exactly $$\frac{1}{4}$$ the area of the original square. Convince yourself and then your partner that it is a square and has $$\frac{1}{4}$$ of the area.
  2. Construct a triangle with exactly $$\frac{1}{4}$$ the area of the original square. Convince yourself and then your partner that it has $$\frac{1}{4}$$ of the area.
  3. Construct another triangle, also with $$\frac{1}{4}$$ the area, that is not the same shape as the first one you constructed. Convince yourself and then your partner that it has $$\frac{1}{4}$$ of the area.
  4. Construct a square with exactly $$\frac{1}{2}$$ the area of the original square. Convince yourself and then your partner that it is a square and has $$\frac{1}{2}$$ of the area.
  5. CHALLENGE: Construct another square, also with $$\frac{1}{2}$$ the area, that is oriented differently from the one you constructed in #4. Convince yourself and then your partner that it has $$\frac{1}{2}$$ of the area.

Guiding Questions

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References

youcubed Paper Folding

Paper Folding is made available by youcubed under the CC BY 4.0 license. Accessed March 29, 2019, 12:31 p.m..

Modified by The Match Foundation, Inc.

Problem 2

Below is an image of a tangram set. Determine what fraction of the overall set each shape is.

Guiding Questions

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Problem Set & Homework

Discussion of Problem Set

  • Which circles have $$\frac{1}{2}$$ shaded? How do you know?
  • How did you determine the fractional part of each square in #2? In Part (f), did you need to make every single piece the same size or were you able to determine the fractional part of some pieces without breaking the whole shape into the smallest possible shape that could tile the whole square?
  • What shape did you draw for #3? How do you know the shaded part represents $$\frac{3}{8}$$ of the whole shape?
  • What made finding the fraction of each shape that each color represents in #4 different from the fraction of each shape that each region represents in #2? Was it easier or more difficult to do that?
  • How did you solve #5?

Target Task

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Problem 1

Are each of the fractional pieces below $$\frac{1}{2}$$? Explain why or why not.

Problem 2

Determine the fraction of the whole each piece represents.

Mastery Response

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