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: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
problem_activity: Activity
In this activity, students graph three time-distance relationships along with the one from the previous lesson, “Tyler’s Walk.” One of these is not a proportional relationship, so students must pay close attention to the quantities represented. The purpose of this activity is to give students many opportunities to connect the different features of a graph with parts of the situation it represents. In particular, they attach meaning to any point that is on a graph, and they interpret the meaning of the distance when the time is 1 second as both the constant of proportionality of the relationship and the person’s speed in the context in meters per second. Comparing different but related situations and their graphs supports students as they make sense of the situation.
Digital
Launch
Arrange students in groups of 2–3. Provide access to colored pencils and rulers.
Tell students that this activity is tied to the activity titled "Tyler's Walk" from the previous lesson. All references to Tyler going to the bumper cars come from the statements in that activity.
The digital version has an applet with options to change line colors and hide points. You may want to demonstrate the applet before students use it, perhaps graphing Tyler's data from the previous activity together. Note: the applet can graph lines, rays, or segments. Your class can decide how to represent the data.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use a different color for each person to highlight the connection between the table, graph, and constant of proportionality.
Supports accessibility for: Visual-spatial processing
Student Facing
Diego, Lin, and Mai went from the ticket booth to the bumper cars.
1. Use each description to complete the table representing that person’s journey.
Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "Diego's time (seconds)"}, {"_attributes": {"scope": "col"}, "_value": "Diego's distance (meters)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {}]}, {"td": [{"_value": "15"}, {}]}, {"td": [{"_value": "30"}, {"_value": "50"}]}, {"td": [{"_value": "1"}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "Lin's time (seconds)"}, {"_attributes": {"scope": "col"}, "_value": "Lin's distance (meters)"}]}]}], "tbody": [{"tr": [{"td": [{}, {"_value": "0"}]}, {"td": [{}, {"_value": "25"}]}, {"td": [{"_value": "20"}, {"_value": "50"}]}, {"td": [{"_value": "1"}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "Mai's time (seconds)"}, {"_attributes": {"scope": "col"}, "_value": "Mai's distance (meters)"}]}]}], "tbody": [{"tr": [{"td": [{}, {"_value": "0"}]}, {"td": [{}, {"_value": "25"}]}, {"td": [{"_value": "40"}, {"_value": "50"}]}, {"td": [{"_value": "1"}, {}]}]}]}]}
2. Using a different color for each person, draw a graph of all four people’s journeys (including Tyler's from the other day).
Drag the names to the correct lines to label them.
If you choose to, you can use the Paint Brush tool to change the color of each line. Select the tool, click on a color in the palette below the graph, and then click on a line. Click on the Move tool (the arrow) before changing to a new paint brush color.
You can hide any points you create with the checkbox below the graph.
3. Which person is moving the most quickly? How is that reflected in the graph?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2–3. Provide access to colored pencils and rulers.
Tell students that this activity is tied to the activity titled "Tyler's Walk" from the previous lesson. All references to Tyler going to the bumper cars come from the statements in that activity.
The digital version has an applet with options to change line colors and hide points. You may want to demonstrate the applet before students use it, perhaps graphing Tyler's data from the previous activity together. Note: the applet can graph lines, rays, or segments. Your class can decide how to represent the data.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use a different color for each person to highlight the connection between the table, graph, and constant of proportionality.
Supports accessibility for: Visual-spatial processing
Student Facing
Diego, Lin, and Mai went from the ticket booth to the bumper cars.
Use each description to complete the table representing that person’s journey.
Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["Diego\u2019s", "time", "(seconds)"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["Diego\u2019s", "distance", "(meters)"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {}]}, {"td": [{"_value": "15"}, {}]}, {"td": [{"_value": "30"}, {"_value": "50"}]}, {"td": [{"_value": "1"}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["Lin\u2019s", "time", "(seconds)"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["Lin\u2019s", "distance", "(meters)"]}]}]}], "tbody": [{"tr": [{"td": [{}, {"_value": "0"}]}, {"td": [{}, {"_value": "25"}]}, {"td": [{"_value": "20"}, {"_value": "50"}]}, {"td": [{"_value": "1"}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["Mai\u2019s", "time", "(seconds)"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["Mai\u2019s", "distance", "(meters)"]}]}]}], "tbody": [{"tr": [{"td": [{}, {"_value": "0"}]}, {"td": [{}, {"_value": "25"}]}, {"td": [{"_value": "40"}, {"_value": "50"}]}, {"td": [{"_value": "1"}, {}]}]}]}]}
Using a different color for each person, draw a graph of all four people’s journeys (including Tyler's from the other day).
[IMG]
Which person is moving the most quickly? How is that reflected in the graph?
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?
Write equations to represent each person’s relationship between time and distance.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Students may expect the graphs to intersect because everyone arrives at the same location. However, they did not arrive there at the same time (with the exception of Tyler and Mai). Because all characters traveled the same distance from the ticket booth and no further, the endpoints of their graphs lie on the same horizontal line
\(y = 50\)
, that is, they have the same
\(y\)
-coordinate. The points will vary in position from right to left depending on the number of seconds after Tyler left the ticket booth it took each person to arrive at the bumper cars. Note this features in a whole-class discussion.
The most important goals of the discussion are to attach meaning to any point that is on a graph, and to interpret the meaning of the distance when the time is 1 second as both the constant of proportionality of the relationship and the person’s speed in the context in meters per second.
These questions may be used to facilitate the class discussion:
"For each graph that shows a proportional relationship, what is the constant of proportionality?" (Tyler's was 1.25, Diego’s was
\(1\frac23\)
, Lin’s was 2.5.)
"How did you find them?" (Answers will vary, but students could have divided a
\(y\)
-coordinate by its associated
\(x\)
-coordinate.)
"Where do constants of proportionality occur in the tables, and where do they occur on the graphs?" (They occur in the tables as the values in the second column that correspond to the value of 1 in the first column; on the graphs as the
\(y\)
-coordinate of points where the
\(x\)
-coordinate is 1.)
"Which is the only graph that does not represent a proportional relationship?"
"A classmate argues that Mai's graph must represent a proportional relationship, because she jogged at a steady rate. How do you answer?" (Mai's graph does not pass through the origin, so it does not represent a proportional relationship. That is, the distance she traveled is not proportional to the time elapsed. What does “time elapsed” mean? It is time elapsed since Tyler left the ticket booth. However, distance vs. time elapsed since
Mai
left the ticket booth until she arrived at the bumper cars is a proportional relationship.)


aligns: no