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Launch

This is the first time students do the Math Talk instructional routine in this course, so it is important to explain how it works before starting.

Explain that a Math Talk has four problems, revealed one at a time. For each problem, students have a minute to quietly think and are to give a signal when they have an answer and a strategy. The teacher then selects students to share different strategies (likely 2–3, given limited time), and might ask questions such as “Who thought about it in a different way?” The teacher then records the responses for all to see and might ask clarifying questions about the strategies before revealing the next problem.

Consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see if the students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with access to sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Activity

None

Find the value of each expression mentally.

Student Response

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Building on Student Thinking

8.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR2: Collect and Display

Materials

To Gather

Blank paper Long straightedge Scissors

Activity Narrative

The purpose of this activity is to investigate scaled copies of rectangles. Students see that if one rectangle is a scaled copy of another, then they can be arranged so that the diagonal of the larger rectangle contains the diagonal of the smaller rectangle.

Encourage students to use what they know about how the rectangles were created rather than measuring each new rectangle (which is likely to introduce errors). Making use of structure helps students build their understanding of what makes something a scaled copy (MP7).

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

Arrange students in groups of 2. Provide each group with 2 whole sheets of paper, scissors, and a long straightedge. Direct students to create rectangles by folding and cutting an -inch by 11-inch piece of paper in half, in half again, and so on, as illustrated in the task statement. (Alternatively, these rectangles could be pre-cut and a set of 5 pre-cut rectangles provided to each group.)

Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language students use to determine which rectangles are scaled copies of the full sheet of paper (Rectangle A). Display words and phrases such as “Rectangle E is the same as Rectangle C but smaller” and “The side lengths of Rectangle C are half of the side lengths of Rectangle A.”

Action and Expression: Provide Access for Physical Action. Provide access to pre-cut materials to reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.
Supports accessibility for: Fine Motor Skills, Organization, Visual-Spatial Processing

Activity

None

Activity Synthesis

The purpose of this discussion is for students to describe features of scaled copies.

Direct students' attention to the reference created using Collect and Display. Ask students to share how they decided that the -inch by -inch rectangle (C) was a scaled copy of the -inch by 11-inch rectangle (A). Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. For example, the display may say “Rectangle E is the same as rectangle C but smaller,” and be revised to a more precise phrase, such as “Both dimensions of Rectangle E are multiplied by 2 to get the corresponding dimensions of Rectangle C.”

Also discuss:

“How did you decide that the -inch by -inch rectangle (B) is not a scaled copy of the -inch by 11-inch rectangle (A)?” (There is no single number that you can multiply by to get and also multiply by to get 11.)
“What happened when we stacked all of the rectangles from largest to smallest so that they all lined up at a corner?” (The diagonals of the rectangles that are scaled copies of one another match up, creating 2 sets of scaled copies.)

8.3

Activity

Optional

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR5: Co-Craft Questions

Materials

To Gather

Activity Narrative

This optional activity revisits previous grade level standards by examining scaled copies that are created by evenly dividing a rectangle into smaller rectangles. Use this activity if students need additional work with scale factor. There are no given dimensions for any of the rectangles, so students must use their understanding of scale factors and the fact that the rectangles are divided evenly. This offers students additional opportunities to see and make use of structure (MP7).

Launch

Arrange students in groups of 2. Tell them they will examine a different set of rectangles and determine scale factors for pairs that are scaled copies of one another.

First, ask students to interpret what is meant by “evenly divided.” (Rectangle is cut exactly in half vertically and horizontally, and one of its quadrants is cut exactly in thirds vertically and horizontally.) Students may want to use a ruler to validate their understanding of what “evenly divided” means.

Give students 3–4 minutes of quiet work time followed by a partner then whole-class discussion.

MLR5 Co-Craft Questions. Keep books or devices closed. Display only the image of Rectangle R without revealing the questions, and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the task. Ask, “What do these questions have in common? How are they different?” Reveal the intended questions for this task and invite additional connections.
Advances: Reading, Writing

Activity

None

Here is a picture of Rectangle R, which has been evenly divided into smaller rectangles. Two of the smaller rectangles are labeled B and C.

Is B a scaled copy of R? If so, what is the scale factor?
Is C a scaled copy of B? If so, what is the scale factor?
Is C a scaled copy of R? If so, what is the scale factor?

Rectangle R subdivided into four equal rectangles. The upper left rectangle is labeled B. The lower right rectangle is subdivided into nine equal rectangles and the lower left one is labeled C.

Student Response

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Building on Student Thinking

Activity Synthesis

The goal of this discussion is to recall the connection between scale factor and a scaled copy. Discuss with students:

“Why is Rectangle B a scaled copy of Rectangle R?” (The length and width of Rectangle R have been multiplied by the same number because the rectangles were evenly divided.)
“How are the scale factors from R to B and B to C related to the scale factor from R to C?” (The scale factor from R to C is the product of the scale factors from R to B and B to C.)
“If we were to stack Rectangle R, Rectangle B, and Rectangle C so that they all shared one vertex, would the diagonal of Rectangle R contain the diagonals of Rectangles B and C?” (Yes.)

Rectangles are made by cutting an -inch by 11-inch piece of paper in half, in half again, and so on, as illustrated in the diagram. Find the lengths of each rectangle and enter them in the appropriate table.

An image of an 8 point 5 by 11 inch rectangle.

Some of the rectangles are scaled copies of Rectangle A (the full sheet of paper). Record the measurements of those rectangles in this table.

rectangle	length of short side (inches)	length of long side (inches)
A		11

Some of the rectangles are not scaled copies of Rectangle A (the full sheet of paper). Record the measurements of those rectangles in this table.

rectangle length of short side (inches) length of long side (inches)
Look at the measurements for the rectangles that are scaled copies of the full sheet of paper. What do you notice?
Look at the measurements for the rectangles that are not scaled copies of the full sheet. What do you notice?
Stack the rectangles that are scaled copies of the full sheet so that they all line up at a corner, as shown in the diagram. Do the same with the other set of rectangles. On each stack, draw a line from the bottom left corner to the top right corner of the biggest rectangle.

What do you notice?
Stack all of the rectangles from largest to smallest so that they all line up at a corner. Compare the lines that you drew. Can you tell, from the drawn lines, which set each rectangle came from? What do you notice?