Writing and Graphing Systems of Linear Equations
5 min
Narrative
This Math Talk focuses on values meeting a constraint. It encourages students to think about using values in an equation and to rely on what they know about cost in context to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students write and solve systems of linear equations in two variables.
Many students are likely to approach the task by multiplying pounds of raisins by 4 and pounds of walnuts by 8, and then see if the sum of the two products is 15. Some students may arrive at their conclusions by reasoning and estimating. For example, seeing that a pound of walnuts costs $8, students may simply reason that the first combination is not possible. Or they may reason that the last combination is impossible because at $4 a pound, 3.5 pounds of raisins would cost more than $12, so it is not possible to also get 1 pound of walnuts and pay only $15 total.
In describing their strategies, students need to be precise in their word choice and use of language (MP6).
Launch
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.
Supports accessibility for: Memory, Organization
Student Task
Diego bought some raisins and walnuts to make trail mix.
Raisins cost $4 a pound and walnuts cost $8 a pound. Diego spent $15 on both ingredients.
Decide if each pair of values could be a combination of raisins and walnuts that Diego bought.
- 4 pounds of raisins and 2 pounds of walnuts
- 1 pound of raisins and 1.5 pounds of walnuts
- 2.25 pounds of raisins and 0.75 pounds of walnuts
- 3.5 pounds of raisins and 1 pound of walnuts
Sample Response
- No. Sample reasoning: 4 pounds of raisins costs $16 (4⋅4=16) which is already too much.
- No. Sample reasoning: 1 pound of raisins costs $4 and 1.5 pounds of walnuts costs $12 (8 plus half of 8 is the same as 8 plus 4, which is 12). So the total is $16, not $15.
- Yes. Sample reasoning: 2.25 pounds of raisins costs $9 (two 4s plus one fourth of 4 is the same as 8 plus 1, which is 9) and 0.75 pounds of walnuts costs $6 (one fourth of 8 is 2, so three fourths of 8 is 6), so the total is $15.
- No. Sample reasoning: 3.5 pounds of raisins costs $14 (three 4s plus half of 4 is the same as 12 plus 2, which is 14) and 1 pound of walnuts costs $8, so the total is $22, not $15.
Activity Synthesis (Teacher Notes)
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
Conclude by reminding students that the given situation involves a cost constraint. Emphasize that only the third option, 2.25 pounds of raisins and 0.75 pounds of walnuts, satisfy the constraint. One way to check if certain values meet the constraint is by writing an equation and checking if it is true. For example, the equation 4(2.25)+8(0.75)=15 is true. If we replace the weights of raisins and walnuts with other pairs of values, the equation would be false.
Advances: Speaking, Representing
Standards
- A-REI.A·Understand solving equations as a process of reasoning and explain the reasoning
- HSA-REI.A·Understand solving equations as a process of reasoning and explain the reasoning.
20 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Writing and Graphing Systems of Linear Equations
5 min
Narrative
This Math Talk focuses on values meeting a constraint. It encourages students to think about using values in an equation and to rely on what they know about cost in context to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students write and solve systems of linear equations in two variables.
Many students are likely to approach the task by multiplying pounds of raisins by 4 and pounds of walnuts by 8, and then see if the sum of the two products is 15. Some students may arrive at their conclusions by reasoning and estimating. For example, seeing that a pound of walnuts costs $8, students may simply reason that the first combination is not possible. Or they may reason that the last combination is impossible because at $4 a pound, 3.5 pounds of raisins would cost more than $12, so it is not possible to also get 1 pound of walnuts and pay only $15 total.
In describing their strategies, students need to be precise in their word choice and use of language (MP6).
Launch
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.
Supports accessibility for: Memory, Organization
Student Task
Diego bought some raisins and walnuts to make trail mix.
Raisins cost $4 a pound and walnuts cost $8 a pound. Diego spent $15 on both ingredients.
Decide if each pair of values could be a combination of raisins and walnuts that Diego bought.
- 4 pounds of raisins and 2 pounds of walnuts
- 1 pound of raisins and 1.5 pounds of walnuts
- 2.25 pounds of raisins and 0.75 pounds of walnuts
- 3.5 pounds of raisins and 1 pound of walnuts
Sample Response
- No. Sample reasoning: 4 pounds of raisins costs $16 (4⋅4=16) which is already too much.
- No. Sample reasoning: 1 pound of raisins costs $4 and 1.5 pounds of walnuts costs $12 (8 plus half of 8 is the same as 8 plus 4, which is 12). So the total is $16, not $15.
- Yes. Sample reasoning: 2.25 pounds of raisins costs $9 (two 4s plus one fourth of 4 is the same as 8 plus 1, which is 9) and 0.75 pounds of walnuts costs $6 (one fourth of 8 is 2, so three fourths of 8 is 6), so the total is $15.
- No. Sample reasoning: 3.5 pounds of raisins costs $14 (three 4s plus half of 4 is the same as 12 plus 2, which is 14) and 1 pound of walnuts costs $8, so the total is $22, not $15.
Activity Synthesis (Teacher Notes)
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
Conclude by reminding students that the given situation involves a cost constraint. Emphasize that only the third option, 2.25 pounds of raisins and 0.75 pounds of walnuts, satisfy the constraint. One way to check if certain values meet the constraint is by writing an equation and checking if it is true. For example, the equation 4(2.25)+8(0.75)=15 is true. If we replace the weights of raisins and walnuts with other pairs of values, the equation would be false.
Advances: Speaking, Representing
Standards
- A-REI.A·Understand solving equations as a process of reasoning and explain the reasoning
- HSA-REI.A·Understand solving equations as a process of reasoning and explain the reasoning.