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Halves, thirds, and sixths

Alignments to Content Standards: 3.MD.C.6 3.G.A.2 3.NF.A.1 3.NF.A.3.b

Task

  1. A small square is a square unit. What is the area of this rectangle? Explain.

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  2. What fraction of the area of each rectangle is shaded blue? Name the fraction in as many ways as you can. Explain your answers.

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  3. Shade \frac12 of the area of rectangle in a way that is different from the rectangles above.

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  4. Shade \frac23 of the area of the rectangle in a way that is different from the rectangles above.

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IM Commentary

The purpose of this task is for students to use their understanding of area as the number of square units that covers a region (3.MD.6), to recognize different ways of representing fractions with area (3.G.2), and to understand why fractions are equivalent in special cases (3.NF.3.b). Determining the fraction of the area that is shaded for rectangles A-D in part (b) is increasingly complex. Rectangles E, F, and G show that there are many ways for \frac12 of the area to be shaded blue, which implies that there are many ways to represent the fraction \frac12 with area. Rectangle H requires students to see the equivalence of two fractions, neither of which is a unit fraction. Students get a chance to demonstrate what they have learned in part (b) by generating their own representations of fractions in parts (c) and (d).

Note that in third grade, students are limited to working with halves, thirds, fourths, sixths, and eighths. While it would be acceptable in instructional situations to work with other fractions, students should have an opportunity to work extensively with the fractions mentioned in order to develop a deep and flexible understanding of them. In particular, summative assessment should be strictly limited to fractions with denominators 2, 3, 4, 6, and 8.

This is an instructional task. Students would benefit from having tracing paper, colored pencils, and multiple blank copies of the rectangle as they work. Students would also benefit from working in pairs or small groups so they can compare their answers and explain the fraction equivalences to each other.

Attached Resources

  • Lesson Plan - Unit Fractions 1
  • Solution

    1. The area of the the rectangle is 6 square units.
      • \frac16 of rectangle A is shaded blue because the area is 6 square units and one square unit is shaded.
      • \frac36 of rectangle B is shaded blue because the area is 6 square units and 3 square units are shaded. We can also say that \frac12 of rectangle B is shaded blue because half of the squares are shaded. This shows that \frac36 and \frac12 are equivalent fractions.
      • \frac26 of rectangle C is shaded blue because the area is 6 square units and two square units are shaded. We can also say that \frac13 of rectangle C is shaded blue. This is easier to see if we add some more shading:
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        We can see that the part that is shaded blue is the same size and shape as the part that is shaded yellow and the part that is white, so each of these parts is \frac13 of the rectangle. This shows that \frac26 and \frac13 are equivalent fractions.
      • \frac26 of rectangle D is shaded blue because the area is 6 square units and two square units are shaded. We can also say that \frac13 of rectangle D is shaded blue. We can cut figures apart and rearrange them without changing the area (as long as the pieces don't overlap when we are done). We can see that \frac13 of the area is shaded if we rearrange the squares so it looks like rectangle C.
      • \frac36 of rectangle E is shaded blue because the area is 6 square units and 3 square units are shaded. We can also say that \frac12 of rectangle E is shaded blue because half of the squares are shaded. We can also rearrange them to look like rectangle B. This shows that \frac36 and \frac12 are equivalent fractions.
      • \frac36 of rectangle F is shaded blue because the area is 6 square units and 3 square units are shaded. We can also say that \frac12 of rectangle F is shaded blue because half of the squares are shaded. We can also see that the three blue squares form a "piece" of the rectangle that is the same size and shape as the piece formed by the three white squares. This shows that \frac36 and \frac12 are equivalent fractions.
      • \frac36 of rectangle G is shaded there are 6 equal squares and three are shaded. We can also say that \frac12 of rectangle G is shaded blue because half of the squares are shaded. We can also rearrange them to look like rectangle B. This shows that \frac36 and \frac12 are equivalent fractions.
      • \frac46 of rectangle H is shaded there are 6 equal squares and three are shaded. We can also say that \frac23 of rectangle H is shaded blue because as we saw earlier, two blue squares represent \frac13 of the rectangle and two "2-square rectangles" are shaded. This shows that \frac46 and \frac23 are equivalent fractions.
    2. There are many ways to do this.
    3. There are many ways to do this.