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: Activity
In this activity, students use what they know about finding the area of a square from a given side length to think about the converse problem: how do you find the side length of a square with a given area? Students solve this problem in general in the next lesson, but this lesson sets them up for that work.
First, they work with squares whose side lengths lie along grid lines, so both the side lengths and areas can be found simply by counting length or area units.
They also see a “tilted” square whose area can be determined by counting square units or by decomposing and rearranging the square or a related figure. In this case, the side lengths are not apparent, so students have to reason that since the area is the same as one of the earlier squares, the side lengths must also be the same. They should check that this appears to be true by tracing one of the squares on tracing paper and seeing that it does, in fact, line up with the other.
The last question asks them to use the insights from the previous questions, tracing paper, or a ruler to estimate the side lengths of some other tilted squares (MP5). In the next lesson, they will see other techniques for estimating those side lengths as well.
Monitor for students who use these strategies for the fourth question:
reason that the side lengths are between 5 and 6 because the areas are between 25 and 36 (MP3)
measure with tracing paper or a ruler
Launch
Provide access to tracing paper. Students in groups of 2. 3 minutes of quiet work time followed by a partner discussion on the first three problems. Pause students and make sure everyone agrees on the answers for the first three questions. Students continue with the fourth question, then a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “I noticed _____ so I…”, “By comparing triangles I…”, or “Another strategy could be _____ because….”
Supports accessibility for: Language; Social-emotional skills
Conversing, Writing: MLR5 Co-Craft Questions.
Before revealing the questions in this activity, display the diagram of the three squares and invite students to write possible mathematical questions about the diagram. Ask students to compare the questions they generated with a partner before sharing questions with the whole class. Listen for and amplify questions about the side lengths and area of each square. If no student asks about the side lengths or area of each square, ask students to adapt a question to align with the learning goals of this lesson. Then reveal and ask students to work on the actual questions of the task. This routine will help develop students’ meta-awareness of language as they generate questions about the side lengths and areas of squares in the coordinate plane.
Design Principle(s): Maximize meta-awareness
Student Facing
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What is the side length of square A? What is its area?
What is the side length of square C? What is its area?
What is the area of square B? What is its side length? (Use tracing paper to check your answer to this.)
Find the areas of squares D, E, and F. Which of these squares must have a side length that is greater than 5 but less than 6? Explain how you know.
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Student Response
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Activity Synthesis
A key take-away from this activity is that if the area of a square is in between the areas of two other squares, then the side lengths are also in between, which in the case of the squares in this task can be reinforced by estimating with tracing paper.
Select previously identified students to share their solutions. Sequence with a reasoning strategy followed by a measuring strategy using a ruler or tracing paper. Use the measuring strategy to verify the reasoning strategy.


aligns: no