few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Standard options: 
A. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
B. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
C. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)


Your response:
C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_lesson_009431

1
options: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
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.
| raisins (pounds) | walnuts (pounds) |
|------------------|------------------|
|        0         |                  |
|       0.25       |                  |
|                  |      1.375       |
|                  |       1.25       |
|       1.75       |                  |
|        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.
| raisins (pounds) | walnuts (pounds) |
|------------------|------------------|
|        0         |                  |
|       0.25       |                  |
|                  |      1.375       |
|                  |       1.25       |
|       1.75       |                  |
|        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
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
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: 3