few_shots: 
Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
problem_activity: Activity
In this activity, students apply what they have learned previously about surface area and volume to different situations (MP4). Students have to consider whether they are finding the surface area or volume before answering each question. In addition, students apply proportional reasoning to find the cost of the vinyl that is needed. This is an opportunity for students to revisit this prior understanding in a geometry context.
As students work on the task, monitor for students who are using different methods to decompose or compose the base of the object to calculate the area.
Launch
Arrange students in groups of 2. Make sure students are familiar with the terms “foam” and “vinyl.” For example, it may help to explain that many binders are made out of cardboard covered with vinyl. In the diagram, all measurements have been rounded to the nearest inch.
Give students 3–5 minutes of quiet work time followed by time to share their answers with a partner. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge by reviewing an image or video of a foam play structure. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
At a daycare, Kiran sees children climbing on this foam play structure.
[IMG]
Kiran is thinking about building a structure like this for his younger cousins to play on.
The entire structure is made out of soft foam so the children don’t hurt themselves. How much foam would Kiran need to build this play structure?
The entire structure is covered with vinyl so it is easy to wipe clean. How much vinyl would Kiran need to build this play structure?
The foam costs 0.8¢ per in
3
. Here is a table that lists the costs for different amounts of vinyl. What is the total cost for all the foam and vinyl needed to build this play structure?
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "sup": [{"_value": "2"}], "_values": ["vinyl (in", ")"]}, {"_attributes": {"scope": "col"}, "_value": "cost ($)"}]}, {"td": [{"_value": "75"}, {"_value": "0.45"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "125"}, {"_value": "0.75"}]}]}]}]}
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
When he examines the play structure more closely, Kiran realizes it is really two separate pieces that are next to each other.
[IMG]
How does this affect the amount of foam in the play structure?
How does this affect the amount of vinyl covering the play structure?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
For the first question, students may try to figure out the total mass of foam needed instead of the total volume. Point out that they are not given any information about how much the foam weighs and prompt them to look for a different way of answering the question “how much foam?”
Some students might see the 0.8¢ for the unit price and confuse that with
$
0.80. Remind students of their work with fractional percentages in a previous unit and that 0.8¢ must be less than 1¢.
Activity Synthesis
Select previously identified students to share how they calculated the area of the base.
Consider asking some of the following questions:
“Are there other ways to calculate the area of the base?”
“How did you know if you had to calculate surface area or volume for this problem?”
“If Kiran buys a big block of foam that is 36 inches wide, 20 inches deep, and 10 inches tall and cuts it into this shape, what shapes would he be cutting off?” (A rectangular prism for the step on the left side and a triangular prism for the slide part on the right.)
“How much more would the big block of foam cost than your calculations?” (This method wastes a volume of
\(36 \boldcdot 20 \boldcdot 10 - 4,\!960 = 2,\!240\)
cubic inches of foam. This would be an extra
$
17.92 since
\(2,\!240 \boldcdot 0.008 = 17.92\)
.)
If Kiran decides not to cover the bottom of the structure with vinyl, how much would he save?” (This reduces the area of vinyl needed by another
\(36\boldcdot 20 = 720\)
square inches. This would save
$
4.32 since
\(720 \boldcdot 0.006 = 4.32\)
.)
Conversing, Listening: MLR7 Compare and Connect.
Use this routine when students share their strategies for calculating the volume of the play structure. Ask students to consider what is the same and what is different about each approach. In this discussion, listen for and amplify comments that refer to the way the figure was decomposed. Draw students’ attention to the various ways area and perimeter of the base were found and how these were represented in each strategy. These exchanges can strengthen students’ mathematical language use and reasoning based on volume and surface area.
Design Principle(s): Maximize meta-awareness


aligns: yes