Unit 9, Lesson 5
Rectangle Madness (Optional)
Accelerated 6
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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.
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.”
Rectangle
If segment
If segment
If segment
If segment
Rectangle
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
Rectangle
In the diagram, draw a line segment that decomposes
Draw another line segment that decomposes the new rectangle into two regions: a square that is the largest possible and another new rectangle.
Keep going until rectangle
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Draw a rectangle that is 21 units by 6 units.
In your rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until the diagram shows that your original rectangle has been entirely decomposed into squares.
How many squares of each size are in your diagram?
What is the side length of the smallest square?
Draw a rectangle that is 28 units by 12 units.
In your rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until the diagram shows that your original rectangle has been decomposed into squares.
How many squares of each size are in your diagram?
What is the side length of the smallest square?
Write each of these fractions as a mixed number with the smallest possible numerator and denominator:
What do the fraction problems have to do with the earlier rectangle decomposition problems?
If students do not see connections between the decomposition of the rectangles and the fraction problems, consider asking:
Building Toward