options: Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
problem_activity: Activity
In this activity, students write and graph equations to represent two constraints in the same situation and use tables and graphs to see possible values that satisfy the constraints. The work prompts them to think about a pair of values that simultaneously meets multiple constraints in the situation, which in turn helps them make sense of the phrase "a solution to both equations."
Monitor for these likely strategies for completing the tables, from less precise to more precise:
Guessing and checking
Using the graph (by identifying the point with a given
\(x\)
- or
\(y\)
-value and estimating the unknown value, or using technology to get an estimate)
Substituting the given value of one variable into the equation and solving for the other variable
Identify students who use each strategy and ask them to share later.
Launch
Arrange students in groups of 2 and provide access to graphing technology. Explain to students that they will now examine the same situation involving two quantities (raisins and walnuts), but there are now two constraints (cost and weight).
Give students 3–4 minutes of quiet time to complete the first set of questions about the cost constraint, and then pause for a whole-class discussion. Select previously identified students to share their strategies, in the order listed in the Activity Narrative. If one of the strategies is not mentioned, bring it up.
Ask a student who used the graph to complete the table to explain how exactly the graph was used. For example: "How did you use the graph of
\(4x+8y=15\)
to find
\(y\)
when
\(x\)
is 2?" Make sure students recognize that this typically involves hovering over different points on the graph or tracing the graph to get the coordinates.
Ask students to proceed with the remainder of the activity. If time permits, pause again after the second set of questions (about the weight constraint) to discuss how the strategies for completing the second table is like or unlike that of completing the first table. Alternatively, give students a minute to confer with their partner before moving on to the last question.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Here is a situation you saw earlier: 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.
Write an equation to represent this constraint. Let
\(x\)
be the pounds of raisins and
\(y\)
be the pounds of walnuts.
Use graphing technology to graph the equation.
Complete the table with the amount of one ingredient Diego could have bought given the other. Be prepared to explain or show your reasoning.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "raisins (pounds)"}, {"_attributes": {"scope": "col"}, "_value": "walnuts (pounds)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {}]}, {"td": [{"_value": "0.25"}, {}]}, {"td": [{}, {"_value": "1.375"}]}, {"td": [{}, {"_value": "1.25"}]}, {"td": [{"_value": "1.75"}, {}]}, {"td": [{"_value": "3"}, {}]}]}]}]}
Here is a new piece of information: Diego bought a total of 2 pounds of raisins and walnuts combined.
Write an equation to represent this new constraint. Let
\(x\)
be the pounds of raisins and
\(y\)
be the pounds of walnuts.
Use graphing technology to graph the equation.
Complete the table with the amount of one ingredient Diego could have bought given the other. Be prepared to explain or show your reasoning.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "raisins (pounds)"}, {"_attributes": {"scope": "col"}, "_value": "walnuts (pounds)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {}]}, {"td": [{"_value": "0.25"}, {}]}, {"td": [{}, {"_value": "1.375"}]}, {"td": [{}, {"_value": "1.25"}]}, {"td": [{"_value": "1.75"}, {}]}, {"td": [{"_value": "3"}, {}]}]}]}]}
Diego spent
$
15 and bought exactly 2 pounds of raisins and walnuts. How many pounds of each did he buy? Explain or show how you know.
Student Response
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Anticipated Misconceptions
Some students might think that the values in the second table need to reflect a total cost of
$
15. Clarify that the table represents only the constraint that Diego bought a total of 2 pounds of raisins and walnuts.
The idea of finding an
\((x,y)\)
pair that satisfies multiple constraints should be familiar from middle school. If students struggle to answer the last question, ask them to study the values in the table. Ask questions such as: "If Diego bought 0.25 pound of raisins, would he meet both the cost and weight requirements?" and "Which combinations of raisins and walnuts would allow him to meet both requirements? How many combinations are there?"
Activity Synthesis
Invite students to share their response and reasoning for the last question. Display the graphs representing the system (either created by a student or as shown in the Student Response). Discuss with students:
“Can you find other combinations of raisins and walnuts, besides 0.25 pound and 1.75 pounds, that meet both cost and weight constraints?” (No)
“How many possible combinations of raisins and walnuts meet both constraints? How do we know?” (One combination, because the graphs intersect only at one point.)
Explain to students that the two equations written to represent the constraints form a
system of equations
. We use a curly bracket to indicate a system, like this:
\(\begin{cases}\begin {align} 4x + 8y &= 15\\ x+ \hspace{2mm}y&=2 \end{align} \end{cases}\)
Highlight that the
solution to the system
is a pair of values (in this case, pounds of raisins and walnuts) that meet both constraints. This means the pair of values is a solution to both equations. Graphing is an effective way to see the solution to both equations, if one exists.


correct_option_index: 8