standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Narrative
The purpose of this activity is for students to find all the possible pairs of whole-number side lengths given the area of a rectangle. Each group is assigned 2 areas for which they find all the possible rectangles. They draw and cut out the possible rectangles with that area. In the next activity, they will display the rectangles in a gallery walk. To find all possible rectangles with a given area, students may use tiles but they may also start to observe patterns such as if there are an even number of rows then the number of tiles in the rectangle is an even number (MP7).
Areas to assign (in square units):
Group A: 11, 27
Group B: 25, 5
Group C: 16, 8
Group D: 9, 18
Group E: 24, 12
Group F: 14, 28
Group G: 15, 30
Group H: 19, 20
This activity uses
MLR7 Compare and Connect
. Advances: representing, conversing.
Engagement: Provide Access by Recruiting Interest.
Students may benefit from 2–3 minutes of independent work time to make sense of and begin the task before joining a group.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Required Materials
Materials to Gather
Glue or tape
Inch tiles
Scissors
Tools for creating a visual display
Materials to Copy
Centimeter Grid Paper - Standard
Required Preparation
Each of the 8 groups needs tools for creating a visual display.
Launch
8 groups
Give each group access to inch tiles, grid paper, poster paper, scissors, and glue.
Activity
MLR7 Compare and Connect
“You are going to be given 2 numbers. Each number represents the area of a rectangle. With your group, draw all the possible rectangles with that area and create a poster for each area. Inch tiles are available if you find them helpful.”
“Your poster should show the rectangles with each of your assigned areas. Include details such as area and side lengths to help others understand your thinking.”
Assign each group 2 area values.
15 minutes: small-group work time
Student Facing
Your teacher will assign 2 numbers to your group. Each number represents the area of a rectangle.
On grid paper:
Draw all the possible rectangles that have the given area.
Label the area and the side lengths.
Use each pair of side lengths only once.
(For example, if you draw a rectangle with 4 units across and 6 units down, you don’t need to also draw a rectangle with 6 units across and 4 units down because they have the same pair of side lengths.)
When you think you've drawn all the possible rectangles for both areas, cut out your rectangles and put them on a poster for each area you were assigned.
Display your poster for all to see.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
Students may list only some of the factor pairs for a given number. Consider asking: “Are there any other rectangles you can draw?” or “How can you be sure that you have drawn all of the possible rectangles?”
Activity Synthesis
“Before we walk around and look at all the posters, take a minute to reflect on the numbers you worked with. What did you notice and wonder as you worked on this activity?”
1–2 minutes: quiet think time


aligns: no