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: Recognize and represent proportional relationships between quantities.
problem_activity: Activity
This activity is comprised of two major parts: finding the total wall area to be painted and estimating needed supplies and associated costs. Students use a floor plan of a bedroom and a list of its features and measurements to calculate the room’s total wall area. They then use the square footage to make purchasing decisions and estimate costs.
Because of the variations in the bedroom’s walls and features, keeping track of the shapes to be included or excluded from the area calculation may be challenging to students. Look out for likely omissions or missed steps. For examples, students may neglect to:
Include the 3 square feet of wall space above the door.
Account for the south face of the small wall at one end of the closet. (At 0.5 feet wide and 8 feet tall, it accounts for 4 square feet of paintable surface area.)
Account for the area above or below the window.
Convert lengths to a common unit before determining area or before adding areas of different faces of the room. As students work, notice those who organize their work methodically and those who may need organizational support.
Launch
Since students may not be familiar with floor plans, consider spending a few minutes making sense of one as a class before working on the task. Display a floor plan of the classroom and identify where and how different features of the classroom (doors, windows, closets, etc.) are represented on the the plan.
Give students 2–3 minutes to read the task statement individually and to note any questions that come to mind. Afterwards, ask a couple of students to restate the task in their own words, and spend a few minutes answering any clarifying questions students may have.
Arrange students in groups of 2. Give students approximately 15 minutes of quiet think time to complete the first question (determining the total wall area to be painted). Ask students to discuss their work with their partner afterwards, and to pause for a whole-class debrief before moving on to the rest of the task.
Select a few students to share their strategies and solutions for the total wall area. Discuss any disagreements or questions in the shared calculation or approach. Make sure that the class agrees on the total square feet before instructing students to complete the rest of the task individually.
Engagement: Internalize Self Regulation.
Provide a project checklist that chunks the various steps of the project into a set of manageable tasks.
Supports accessibility for: Organization; Attention
Reading: MLR6 Three Reads.
Use this routine to help students make sense of the diagram and the accompanying text before unveiling the task questions. In the first read, students read the floor plan as well as the problem and with the goal of comprehending the situation (e.g., We need to repaint all the walls in a bedroom, except for a door and window.). In the second read, ask students to identify important quantities (e.g., The east wall is 3 yards long; the window is 5 feet long by 3 feet wide; the ceiling is 8 feet high). Use a visual display to help students make connections to the text by adding details to the floor plan. In the third read, reveal the questions and ask students to brainstorm possible solution strategies that connect to the important floor plan details and quantities they have listed. This routine supports students’ reading comprehension, without solving the problem for them.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here is the floor plan for a bedroom:
[IMG]
Imagine you are planning to repaint all the walls in this room, including inside the closet.
The east wall is 3 yards long.
The south wall is 10 feet long but has a window, 5 feet by 3 feet, that does not need to be painted.
The west wall is 3 yards long but has a door, 7 feet tall and 3 feet wide, that does not need to be painted.
The north wall includes a closet, 6.5 feet wide, with floor-to-ceiling mirrored doors that do not need to be painted. There is, however, a smaller wall between the west wall and the closet that does need to be painted on all sides. The wall is 0.5 feet wide and extends 2 feet into the room.
The ceiling in this room is 8 feet high.
All of the corners are right angles.
If you paint all the walls in the room, how many square feet do you need to cover?
An advertisement about the paint that you want to use reads: “Just 2 quarts covers 175 square feet!” If you need to apply two coats of paint on all the walls, how much paint do you need to buy?
Paint can only be purchased in 1-quart, 1-gallon, and 5-gallon containers. How much will all supplies for the project cost if the cans of paint cost
$
10.90 for a quart,
$
34.90 for a gallon, and
$
165.00 for 5 gallons?
You have a coupon for 20% off all quart-sized paint cans. How does that affect the cost of the project?
Student Response
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Activity Synthesis
Begin by asking 1–2 students to explain their calculations on how much paint is needed. Students should see that using the total wall area alone to find the volume of paint to purchase is not going to be adequate, since two full coats are needed, and paint can only be purchased in specific quantities. How students account for these factors in their decisions and calculations may vary. For example, some students may prefer buying fewer containers of a larger quantity while others may prefer buying more containers of a smaller quantity. Some may choose to have as little extra paint as possible, while other may choose otherwise. (To have enough paint, students may choose to purchase 7 quart-sized containers, 2 gallon-sized containers, or 3 quart-sized and 1 gallon-sized containers.)
Once students have a chance to discuss the various ways students chose to purchase paint, discuss their cost estimates. Consider setting up a few ranges of costs (e.g.,
$
80–
$
84.99,
$
85–
$
89.99,
$
90–
$
94.99,
$
95–
$
99.99,
$
100 or more, etc.) and polling students on where their cost estimates land. Their estimates vary depending on the containers chosen, supplies included, and other assumptions or decisions they made.
Ask, “Can all of these values be correct?” Give students a moment of quiet think time before selecting 2–3 to respond and to share how they calculated their estimates. Record and display students’ responses to help everyone make sense of how the different costs are possible. For example, students who decided that a minimum of 7 total quarts of paint are needed may have reasoned as follows: Cost of painting supplies + cost for 1 gallon container + cost of 3 quart containers = total cost.
$
For the final question of the activity, select a student with a successful strategy to share how they applied the coupon. Consider wrapping up the activity by asking how to purchase the paint and use the coupon to get the best deal.


aligns: no