Rectangle Madness

20 min

Teacher Prep
Setup
5 minutes of quiet think time followed by whole-class discussion.

Narrative

This first activity helps students understand the geometric process that they use in later activities to connect the greatest common factor with related fractions. The first question helps students focus on the effect on side lengths of decomposing a rectangle into smaller rectangles. The second question has students analyze a rectangle that has been decomposed into squares. The third question has students themselves decompose a rectangle into squares. As students work with each rectangle, they make use of the structure to approach the problems (MP7). In the next activity, students relate this process to greatest common factors and fractions.

This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
 

Launch

Tell students to close their books or devices (or to keep them closed). Display rectangle ABCD for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display responses, without editing or commentary, for all to see. If possible, record the relevant reasoning on or near the image.

If naming shapes by vertices does not come up during the conversation, ask students to discuss this idea.

Tell students to open their books or devices, and arrange students in groups of 2. Give 7–10 minutes for students to complete the problems, and follow that with a whole-class discussion.

As students work, use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to decompose rectangles. Display words and phrases such as “rectangle,” “square,” “split,” ”divide,” “segment,” and “pieces.”

Student Task

  1. Rectangle ABCDABCD is not a square. Rectangle ABEFABEF is a square. Use the possible segment lengths to find the missing segment length.

    A rectangle is labeled ABCD with horizontal side BC below horizontal side AD. Vertical line segment FE is drawn creating square ABEF.

    1. If segment AFAF is 5 units long and segment FDFD is 2 units long, how long would segment ADAD be?

    2. If segment BCBC is 10 units long and segment BEBE is 6 units long, how long would segment ECEC be?

    3. If segment AFAF is 12 units long and segment FDFD is 5 units long, how long would segment FEFE be?

    4. If segment ADAD is 9 units long and segment ABAB is 5 units long, how long would segment FDFD be?

  2. Rectangle JKXWJKXW has been decomposed into 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.  

    Segment JKJK is 33 units long and segment JWJW is 75 units long. Find the areas of all of the squares in the diagram.

  3. Rectangle ABCDABCD is 16 units by 5 units.

    A rectangle labeled ABCD is indicated with horizontal side AB below horizontal side DC.

    1. In the diagram, draw a line segment that decomposes ABCDABCD into two regions: a square that is the largest possible and a new rectangle.

    2. Draw another line segment that decomposes the new rectangle into two regions: a square that is the largest possible and another new rectangle.

    3. Keep going until rectangle ABCDABCD is entirely decomposed into squares.

    4. List the side lengths of all the squares in your diagram.

Sample Response

  1. In rectangle ABCDABCD:
    1. 7 units
    2. 4 units
    3. 12 units
    4. 4 units
  2. 1,089 square units, 81 square units, 36 square units, and 9 square units.

  3. Sample response:

    Rectangle decomposed into squares.

Activity Synthesis (Teacher Notes)
Direct students’ attention to the reference created using Collect and Display. Ask students to share how they decomposed rectangle ABCDABCD in the last problem. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.
The goal of this activity is to familiarize students with decomposing rectangles into squares before using this tool with fractions. Here are some questions for discussion:
  • “Do you think every rectangle can be decomposed into squares?” (Yes.)
  • “What challenges did you run into while decomposing rectangles? How did you resolve them?” 
  • “If a line segment of length xx units is decomposed into two segments of lengths yy and zz it can be represented with the equation x=y+zx=y+z. How else can this relationship be represented?” (x−z=yx-z=y)
Extension

Rectangle VWYZ is decomposed into 3 squares.
Rectangle VWYZ is decomposed into 3 squares. YZUX is a large square. UVTS and XTSW are two equal sized squares that are stacked vertically. Both are on the right of YZUX.

  1. The diagram shows rectangle VWYZVWYZ which has been decomposed into 3 squares. What could the side lengths of this rectangle be?

  2. How many different side lengths can you find for rectangle VWYZVWYZ?

  3. What are some rules for possible side lengths of rectangle VWYZVWYZ?

Extension Response

Sample responses:

  1. Height is 2 units, width is 3 units.
  2. There are an infinite number of solutions.
  3. The length of line segment XWXW can be any number. Then the width of VWYZVWYZ is 3 times the length of line segment XWXW, and the height of VWYZVWYZ is 2 times the length of line segment XWXW.
Standards
Building On
  • 4.MD.3·Apply the area and perimeter formulas for rectangles in real world and mathematical problems. <em>For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.</em>
  • 4.MD.A.3·Apply the area and perimeter formulas for rectangles in real world and mathematical problems. <span>For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.</span>

30 min