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
Previously, students matched diagrams and equations to situations with an unknown quantity. Here, they generate such equations, using a letter for the unknown quantity, solve problems, and explain how they know their answers makes sense. Students should be encouraged to use any solving strategy they feel comfortable with. If not yet addressed, mention that any letter can be used for the unknown quantity in their equation.
While this activity is focused on independent practice, encourage students to discuss the problem with a partner if needed. Though the task asks students to write an equation first, students may complete the task in any order that makes sense to them.
Students reason abstractly and quantitatively when they write an equation that represents the situation (MP2). They also practice making sense of a problem and its solution in terms of the context (MP1).
MLR5 Co-Craft Questions:
Keep books or devices closed. Display only the problem stem, without revealing the question, 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
Engagement: Develop Effort and Persistence.
Differentiate the degree of difficulty or complexity. Some students may benefit from starting with a familiar example or one with more accessible values before working independently on the three parts to the activity.
Supports accessibility for: Social-Emotional Functioning, Conceptual Processing
Launch
Groups of 2
“Stones are sometimes used to play games. One of the games that can be played with stones is mancala. In mancala, stones are moved from pit to pit and players try to capture the other player's stones. The picture shows a game of mancala in progress.”
“Now we're going to solve a problem about the stones in a game of mancala.”
“What are some ways you will be able to determine if your answer makes sense?” (I can estimate an answer using rounding. I can think about the size of the numbers in the problem.)
Share and record responses.
Activity
“Take some independent time to work on this problem. You can choose to solve the problem first or write the equation first.”
5–7 minutes: independent work time
Monitor for different ways students:
write an equation
represent the problem, such as by using a tape diagram
decide their answer makes sense, such as thinking about the situation or by rounding
Student Facing
Kiran is setting up a game of mancala. He has a jar of 104 stones.
From the jar, he takes 3 stones for each of the 6 pits on his side of the board.
How many stones are in the jar now?
[IMG]
Write an equation to represent the situation. Use a letter for the unknown quantity.
Solve the problem. Explain or show your reasoning.
Explain how you know your answer makes sense.
Student Response
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Advancing Student Thinking
If students don't find a solution to the problem, consider asking:
“What is this problem about? What can be counted or measured in this situation?”
“How could you represent the problem?”
Activity Synthesis
Invite students to share the equations they wrote.
Discuss differences in equations students wrote. Consider asking: “_____ wrote _____ and _____ wrote _____. How are those equations alike and different?” (I used a different letter for my unknown. I wrote
\(3\times6\)
instead of
\(6 \times 3\)
.)
Have several students share different strategies used to solve the problem. Try to feature a student-drawn tape diagram.
Consider asking:
“Did anyone solve the problem in a different way?”
“Did anyone use a tape diagram to solve?”
“How did you know if your answer made sense?”


aligns: no