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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.”

Activity

None

Rectangle is not a square. Rectangle is a square. Use the possible segment lengths to find the missing segment length.
1. If segment is 5 units long and segment is 2 units long, how long would segment be?
2. If segment is 10 units long and segment is 6 units long, how long would segment be?
3. If segment is 12 units long and segment is 5 units long, how long would segment be?
4. If segment is 9 units long and segment is 5 units long, how long would segment be?
Rectangle has been decomposed into 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 is 33 units long and segment is 75 units long. Find the areas of all of the squares in the diagram.
Rectangle is 16 units by 5 units.
1. In the diagram, draw a line segment that decomposes 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 is entirely decomposed into squares.
4. List the side lengths of all the squares in your diagram.

Student Response

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Building on Student Thinking

5.2

Activity

Standards Alignment

Building On

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Activity Narrative

In this activity, students apply the geometric process that they saw in the last activity and are asked to make connections between the result and the greatest common factor of the side lengths of the original rectangle. They work on a sequence of similar problems, allowing them to see and begin to articulate a pattern (MP8). Then they make connections between this pattern and fractions that are equivalent to the fraction made up of the side lengths of the original rectangle.

Launch

Arrange students in groups of 2. Students work on problems alone and check work with a partner.

If needed, briefly review rewriting fractions before students begin. Consider drawing a diagram to support students’ thinking about the different forms representing the same value.

Write a fraction that is equal to the mixed number . (We can think of as , so it is equal to .)
Write a mixed number that is equal to this fraction: . (This is equal to , so it is equal to .)

It is quicker to sketch the rectangles on blank paper, but some students may benefit from using graph paper to support entry into these problems.

If students did not do the previous activity the same day as this activity, remind them of the earlier work:

Draw a rectangle that is 16 units by 5 units.
1. 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.
2. How many squares of each size are there? (Three 5-by-5 squares and five 1-by-1 squares.)
3. What is the side length of the smallest square? (1)

Representation: Internalize Comprehension. Begin with a physical demonstration of drawing line segments to decompose rectangles to support connections between new situations and prior understandings. Consider using these prompts: “Should we start with a square on the left or the right?” “Where should the next square be placed?”
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Activity

None

Draw a rectangle that is 21 units by 6 units.
1. 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.
2. How many squares of each size are in your diagram?
3. What is the side length of the smallest square?
Draw a rectangle that is 28 units by 12 units.
1. 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.
2. How many squares of each size are in your diagram?
3. 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?

Student Response

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Building on Student Thinking

If students do not see connections between the decomposition of the rectangles and the fraction problems, consider asking:

“Tell me more about the squares you decomposed the rectangle into.”
“Where do you see the parts of the mixed number in the squares?”

Activity Synthesis

Invite students to share some of their observations. Ask, “What connections do you see between the rectangle drawings and the fractions?”

At this point, it is sufficient for students to notice some connections between a partitioned rectangle and its associated fraction. They will have more opportunities to explore, so there’s no need to make sure they notice all of the connections right now. Here are examples of things that it is possible to notice using the rectangles in this activity but might not be noticed until students see more examples:

The given fractions had the same numbers as the side lengths of the given rectangles.
The mixed numbers included the numbers of squares. For example, the 28-by-12 rectangle was partitioned into 2 large squares and 3 small squares, and the associated mixed number was .
The size of the smallest square is related to the fraction used to make the mixed number. For example, in the 28-by-12 rectangle, the smallest square had sides of length 4. To make the mixed number, is rewritten as .

MLR8 Discussion Supports. For each observation that is shared, invite students to turn to a partner and restate what they heard, using precise mathematical language.
Advances: Listening, Speaking