Rectangle Fractions

20 min

Teacher Prep
Setup
Groups of 2. Provide access to graph paper. Quiet work time followed by partner and whole-class discussion.

Narrative

In this activity, students make the connection between the fraction determined by the original rectangle and the resulting more precise fraction. The two rectangles taken together are designed to help students notice that decomposing rectangles is a geometric way to determine the greatest common factor of two numbers. (This is a geometric version of Euclid’s algorithm for finding the greatest common factor.) Students reason abstractly and quantitatively when switching between numeric and geometric representations (MP2).

Launch

Arrange students in groups of 2. Provide access to graph paper. Students work on problems alone and compare work with a partner.

Student Task

  1. Use this rectangle to answer the questions. Suppose this rectangle is 9 units by 4 units.

    A rectangle with a longer horizontal length than vertical length.

    1. In the rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until your original rectangle has been entirely decomposed into squares.

    2. How many squares of each size are there?

    3. What are the side lengths of the last square you drew?

    4. Write 94\frac94 as a mixed number.

  2. Use this rectangle to answer the questions. Suppose this rectangle is 27 units by 12 units.

    A rectangle with a longer horizontal length than vertical length.

    1. In the rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until your original rectangle has been entirely decomposed into squares.

    2. How many squares of each size are there?

    3. What are the side lengths of the last square you drew?

    4. Write 2712\frac{27}{12} as a mixed number.

    5. Compare the diagram you drew for this problem and the one for the earlier problem. How are they the same? How are they different?

  3. What is the greatest common factor of 9 and 4? What is the greatest common factor of 27 and 12? What does this have to do with your diagrams of decomposed rectangles?

Sample Response

  1. 9-by-4 rectangle.

    1. Sample response:

      A rectangle deconstructed into two large squares and four small squares.

    2. two 4-by-4 squares and four 1-by-1 squares
    3. 1 by 1
    4. 94=214\frac94=2\frac14
  2. 27-by-12 rectangle:

    1. Sample response:

      A rectangle deconstructed into two large squares and four small squares.

    2. two 12-by-12 squares and four 3-by-3 squares
    3. 3 by 3
    4. 2712=2312=214\frac{27}{12}=2\frac{3}{12}=2\frac14
    5. It is similar because there are 2 large squares and 4 small squares. It is different because the squares are not the same size. The larger rectangle split into larger squares is a scaled up version of the smaller rectangle split into smaller squares.
  3. The greatest common factor of 9 and 4 is 1. The greatest common factor of 27 and 12 is 3. The greatest common factor is the same as the side length of the smallest square.
Activity Synthesis (Teacher Notes)

The purpose of this discussion is to make sure students understand that using a decomposition model can help turn fractions into mixed numbers and find the greatest common factor of the numerator and denominator. It is not necessary for students to understand a general argument for why chopping rectangles can help you know the greatest common factor of two numbers. Here are some questions for discussion:

  • “We call fractions like 94\frac{9}{4} and 2712\frac{27}{12} equivalent fractions. What would be true about the rectangles we could draw about other equivalent fractions?” (Equivalent fractions would have rectangles that are decomposed in the same way, but each square would be larger or smaller by the same amount.)
  • “Suppose there is a pair of equivalent fractions where the numbers in one of the fractions are 5 times the numbers on the other fractions. If both fractions were represented with rectangles decomposed into squares, how would the side lengths of the squares compare?” (The rectangle representing the fraction with the larger numbers would have squares whose side lengths are 5 times those of the smaller rectangle.)
Extension

We have seen some examples of rectangle tilings. A tiling means a way to completely cover a shape with other shapes, without any gaps or overlaps. For example, here is a tiling of rectangle KXWJKXWJ with 2 large squares, 3 medium squares, 1 small square, and 2 tiny squares.

Tiling of rectangle KXWJ with 2 large squares, 3 medium squares, 1 small square, and 2 tiny squares.
Tiling of rectangle KXWJ with 2 large squares, KJST and TSUV; 3 medium squares, GUWH, KGHL, and MILN; 1 small square, VMOP; and 2 tiny squares, QONR and PQRX. TSUV and KJST are stacked horizontally. TSUV is on the right of KJST. GUWH, KGHL, and MILN are stack vertically. All 3 are on the right of TSUV. VMOP is below MILN. QONR and PQRX are stacked vertically. Both are on the right of VMOP.  

Some of the squares used to tile this rectangle have the same size.

Is it possible to tile a rectangle with squares where the squares are all different sizes?

If you think it is possible, find an example that works. If you think it is not possible, explain why it is not possible.

Extension Response

Sample response: It is possible to tile a rectangle with only differently-sized squares. One example is a 32-by-33 rectangle that can be tiled with squares of side length 18, 15, 14, 10, 9, 8, 7, 4, and 1. (For more examples of solutions and more history on the matter, research Martin Gardner’s November 1958 column in Scientific American.)

Standards
Building On
  • 5.NF.3·Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. <em>For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?</em>
  • 5.NF.B.3·Interpret a fraction as division of the numerator by the denominator <span class="math">\((a/b = a \div b)\)</span>. Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. <span>For example, interpret <span class="math">\(3/4\)</span> as the result of dividing <span class="math">\(3\)</span> by <span class="math">\(4\)</span>, noting that <span class="math">\(3/4\)</span> multiplied by <span class="math">\(4\)</span> equals <span class="math">\(3\)</span>, and that when <span class="math">\(3\)</span> wholes are shared equally among <span class="math">\(4\)</span> people each person has a share of size <span class="math">\(3/4\)</span>. If <span class="math">\(9\)</span> people want to share a <span class="math">\(50\)</span>-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?</span>
  • 6.NS.4·Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1—100 with a common factor as a multiple of a sum of two whole numbers with no common factor. <em>For example, express 36 + 8 as 4 (9 + 2).</em>
  • 6.NS.B.4·Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.<span> For example, express <span class="math">\(36 + 8\)</span> as <span class="math">\(4 (9 + 2)\)</span>.</span>

25 min