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: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
problem_activity: Activity
The purpose of this task is for students to sketch a graph from a story. In order to make the sketch, students must select two quantities from the story to graph, decide which is the independent variable and which is the dependent variable, and create and label their axes based on their decisions (MP4).
Monitor for displays that are correct but different from each other in important ways for students to focus on during the discussion. For example, they may differ by what variables were graphed such as:
distance from home as a function of time
distance from park as a function of time
total distance traveled as a function of time
Digital
Launch
Arrange students in groups of 2–3. Distribute tools for creating a visual display. Before students begin, it may be necessary to demonstrate how to “create a set of axes” so that a first-quadrant graph can be sketched and is large enough to be seen from a distance. Group work time followed by a whole-class discussion.
If using the digital activity, the Activity Narrative is still valid, and student graphs will be used to drive the class discussion. What varies from the print activity is students will have access to an applet to create their graphs. Students will be able to share their graphs by projecting their screen. They may need to save their graph by either exporting the image or “printscreen” to a word document.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as the applet or graphing software. Some students may benefit from a checklist or list of steps to be able to use the applet or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Conversing, Representing, Writing: MLR2 Collect and Display.
As students work in groups, capture the vocabulary and phrases students use to describe the relationship between the variables they selected over time. Listen for students who refer to the story to justify their decisions as they create the shape of the graph. Record their language on a visual display that can be referenced in future discussions. This will help students to produce and make sense of the language needed to communicate about the relationship between quantities represented by functions graphically and in story contexts.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
You will use the tools in the applet for creating a visual display that shows your response to each question.
Here is a story: “Noah was at home. He got on his bike and rode to his friend’s house and stayed there for awhile. Then he rode home again. Then he rode to the park. Then he rode home again.”
Sketch a graph of this story.
What are the two quantities? Label the axes with their names and units of measure. (For example, if this were a story about pouring water into a pitcher, one of your labels might say “volume (liters).”)
Which quantity is a function of which? Explain your reasoning.
Based on your graph, is his friend’s house or the park closer to Noah's home? Explain how you know.
Read the story and all your responses again. Does everything make sense? If not, make changes to your work.
Note: To change the values on the axes, select the Move Graphics tool, click near the end of the axis you want to change, and drag it to grow or shrink it.
[IMG]
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. Distribute tools for creating a visual display. Before students begin, it may be necessary to demonstrate how to “create a set of axes” so that a first-quadrant graph can be sketched and is large enough to be seen from a distance. Group work time followed by a whole-class discussion.
If using the digital activity, the Activity Narrative is still valid, and student graphs will be used to drive the class discussion. What varies from the print activity is students will have access to an applet to create their graphs. Students will be able to share their graphs by projecting their screen. They may need to save their graph by either exporting the image or “printscreen” to a word document.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as the applet or graphing software. Some students may benefit from a checklist or list of steps to be able to use the applet or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Conversing, Representing, Writing: MLR2 Collect and Display.
As students work in groups, capture the vocabulary and phrases students use to describe the relationship between the variables they selected over time. Listen for students who refer to the story to justify their decisions as they create the shape of the graph. Record their language on a visual display that can be referenced in future discussions. This will help students to produce and make sense of the language needed to communicate about the relationship between quantities represented by functions graphically and in story contexts.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Your teacher will give you tools for creating a visual display. With your group, create a display that shows your response to each question.
Here is a story: “Noah was at home. He got on his bike and rode to his friend’s house and stayed there for awhile. Then he rode home again. Then he rode to the park. Then he rode home again.”
Create a set of axes and sketch a graph of this story.
What are the two quantities? Label the axes with their names and units of measure. (For example, if this were a story about pouring water into a pitcher, one of your labels might say “volume (liters).”)
Which quantity is a function of which? Explain your reasoning.
Based on your graph, is his friend’s house or the park closer to Noah's home? Explain how you know.
Read the story and all your responses again. Does everything make sense? If not, make changes to your work.
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?
It is the year 3000. Noah’s descendants are still racing around the park, but thanks to incredible technological advances, now with much more powerful gadgets at their disposal. How might their newfound access to teleportation and time-travel devices alter the graph of stories of their daily adventures? Could they affect whether or not the distance from home is a function of the time elapsed?
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
Students may try and start graphing before they have clearly articulated and labeled their axes with their chosen variables. Encourage groups to make sure all are in agreement on what variables they are graphing before they create their display.
Activity Synthesis
Select previously identified groups to share their visual displays for all to see throughout the discussion. Give students 2–3 minutes of quiet think time to review the 3 displays and identify differences and similarities. Invite students to share what they identified and record and display the responses for all to see. If not brought up by students, ask:
“What is the same and different about the two quantities that each group chose to use in their visual display?”
“Did each group use the same units of measure? Why does it make sense to use the unit of measure seen in any of these examples?”
“Why does it make sense to have time be the independent variable in this situation?"
“Do all of these examples make sense in relation to the situation?”
Conclude the discussion by refocusing students on the input-output pairs described by the different graphs. For example, on a graph where distance from home is a function of time, there should be three inputs where the output is zero since he starts at home, returns home from his friend’s house, and ends at home.
Alternatively, have the class do a “gallery walk” in which students leave written feedback on sticky-notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:
“What is one thing that group did that would have made your project better if you had done it?”
“What is one thing your group did that would have improved their project if they did it too?”
“Does their answer make sense in the situation?”
“Is their answer about which house is closer to the park clear and correct?”
“If there was a mistake, what could they be more careful about in similar problems?”


aligns: no