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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

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_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
problem_activity: Activity
This activity is intended to further students’ understanding of the graphs of proportional relationships in the following respects:
points on the graph of a proportional relationship can be interpreted in the context represented (MP2)
for these points, the quotient of the coordinates is—excepting
\((0,0)\)
—the constant of proportionality
if the first coordinate is 1, then the corresponding coordinate is
\(k\)
, the constant of proportionality.
Students explain correspondences between parts of the table and parts of the graph. The graph is simple so that students can focus on what a point means in the situation represented. Students need to realize, however, that the axes are marked in 10-unit intervals. The discussion questions are opportunities for students to construct viable arguments and critique the reasoning of others (MP3).
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Reading: MLR5 Co-Craft Questions.
To help students make sense of graphs of proportional relationships, start by displaying only the first line of this task (“Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.”) and the graph. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the questions. Listen for and amplify any questions involving correspondences between parts of the table and parts of the graph. This helps students produce the language of mathematical questions and talk about the relationship between distance and time.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.
The point on the graph shows his arrival at the bumper cars. What do the coordinates of the point tell us about the situation?
The table representing Tyler's walk shows other values of time and distance. Complete the table. Next, plot the pairs of values on the grid.
What does the point
\((0, 0)\)
mean in this situation?
How far away from the ticket booth was Tyler after 1 second? Label the point on the graph that shows this information with its coordinates.
What is the constant of proportionality for the relationship between time and distance? What does it tell you about Tyler's walk? Where do you see it in the graph?
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["time", "(seconds)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["distance", "(meters)"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {"_value": "0"}]}, {"td": [{"_value": "20"}, {"_value": "25"}]}, {"td": [{"_value": "30"}, {"_value": "37.5"}]}, {"td": [{"_value": "40"}, {"_value": "50"}]}, {"td": [{"_value": "1"}, {}]}]}]}]}
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?
If Tyler wanted to get to the bumper cars in half the time, how would the graph representing his walk change? How would the table change? What about the constant of proportionality?
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
These questions can be used for discussion or for students who need scaffolding.
"What quantities are shown in the graph?" (Distance in meters that Tyler is from the ticket booth and time elapsed in seconds since he started walking.)
"How far is the ticket booth from the bumper cars?" (50 meters, assuming that Tyler walked in a straight line.) This is an opportunity for attention to precision (MP6) and making explicit assumptions about a situation (MP4).
"Do the values in your table show a proportional relationship? How do you know?" (Based on prior lessons in this unit, students should identify the relationship as proportional because for every point the unit rate is the same.)
"What do the coordinates of the points on the graph show?" (The first coordinate gives amount of time in seconds that elapsed since Tyler started walking. Its corresponding second coordinate shows how many meters away from the ticket booth Tyler was at the corresponding time, assuming that Tyler walked in a straight line.)
Activity Synthesis
After students work on the task, it is important to discuss how the axis labels and the description in the task statement help us interpret points on the graph.
Consider asking these questions:
"What quantities are shown in the graph?" (Distance in meters that Tyler is from the ticket booth and time elapsed in seconds since he started walking.)
"How far is the ticket booth from the bumper cars?" (50 meters, assuming that Tyler walked in a straight line.) This is an opportunity for attention to precision (MP6) and making explicit assumptions about a situation (MP4).
"Do the values in your table show a proportional relationship? How do you know?" (Based on prior lessons in this unit, students should identify the relationship as proportional because for every point the unit rate is the same.)
"What do the coordinates of the points on the graph show?" (The first coordinate gives amount of time in seconds that elapsed since Tyler started walking. Its corresponding second coordinate shows how many meters away from the ticket booth Tyler was at the corresponding time, assuming that Tyler walked in a straight line.)
Ask students for the equation of this proportional relationship. Finally ask where students see
\(k\)
, the constant of proportionality, in each representation, the equation, the graph, the table, and the verbal description of the situation.


aligns: no