few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Example 2:
Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Concept/skill: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
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
The purpose of this activity is for students to compare two different proportional relationships represented in different ways using the skills they have worked on over the past three lessons. Working in groups, students compare the relationships, responding to questions about their rate of change, which rate of change is higher, and one other situation-based question. Groups make a visual display for their problem set to explain each of their responses and convince others of their accuracy.
Identify groups using a variety of representations to share during the Activity Synthesis.
Launch
Remind students that in previous lessons they identified representations of and created representations for a single proportional relationship. In this activity, they will consider representations of two
different
proportional relationships and make comparisons between them.
Arrange students in groups of 2–3. Assign to each group (or ask groups to choose) one of the three question sets. Tell groups that they will make a visual display for their responses to the questions. The display should clearly demonstrate their reasoning and use multiple representations in order to be convincing. Let them know that there will be a gallery walk when they finish for the rest of the class to inspect their solutions’ accuracy.
If time allows, ask groups to complete all three problems and make a visual display for just one.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, present one question at a time.
Supports accessibility for: Memory; Organization
Student Facing
Elena babysits her neighbor’s children. Her earnings are given by the equation
\(y=8.40x\)
, where
\(x\)
represents the number of hours she worked and
\(y\)
represents the amount of money she earned.
Jada earns
$
7 per hour mowing her neighbors’ lawns.
Who makes more money after working 12 hours? How much more do they make? Explain your reasoning by creating a graph or a table.
What is the rate of change for each situation and what does it mean?
Using your graph or table, determine how long it would take each person to earn
$
150.
Clare and Han have summer jobs stuffing envelopes for two different companies.
Han earns
$
15 for every 300 ​​​​​envelopes he finishes.
Clare’s earnings can be seen in the table.
| number of                             envelopes | money                             in dollars |
|-------------------------------------------------|----------------------------------------------|
|                       400                       |                      40                      |
|                       900                       |                      90                      |
By creating a graph, show how much money each person makes after stuffing 1,500 envelopes.
What is the rate of change for each situation and what does it mean?
Using your graph, determine how much more money one person makes relative to the other after stuffing 1,500 envelopes.  Explain or show your reasoning.
Tyler plans to start a lemonade stand and is trying to perfect his recipe for lemonade. He wants to make sure the recipe doesn’t use too much lemonade mix (lemon juice and sugar) but still tastes good.
Lemonade Recipe 1 is given by the equation
\(y=4x\)
where
\(x\)
represents the amount of lemonade mix in cups and
\(y\)
represents the amount of water in cups.
Lemonade Recipe 2 is given in the table.
| lemonade mix (cups) | water (cups) |
|---------------------|--------------|
|         10          |      50      |
|         13          |      65      |
|         21          |     105      |
If Tyler had 16 cups of lemonade mix, how many cups of water would he need for each recipe? Explain your reasoning by creating a graph or a table.
What is the rate of change for each situation and what does it mean?
Tyler has a 5-gallon jug (which holds 80 cups) to use for his lemonade stand and 16 cups of lemonade mix. Which lemonade recipe should he use? Explain or show your reasoning.
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?
Han and Clare are still stuffing envelopes. Han can stuff 20 envelopes in a minute, and Clare can stuff 10 envelopes in a minute. They start working together on a pile of 1,000 envelopes.
How long does it take them to finish the pile?
Who earns more money?
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
Some students may confuse the values for the rate of change of a situation. For example, Lemonade Recipe 1's equation,
\(y=4x\)
, shows that the rate of change is 4 cups of water per cup of lemonade mix. Students may switch these values and think that the rate of change is 4 cups lemonade mix per cup of water. Ask students who do this to explain where in the original representations they see the rate of change. Students may need to list a few values or sketch a graph in order to see their mix-up between the two quantities.
Activity Synthesis
Begin with a gallery walk for students to see how other groups answered the same set of questions they did and how students answered questions about the other two contexts.
Invite groups to share the strategies they used with the various representations. Consider asking groups the following questions:
“What representations did you choose to answer the questions? Why did you pick them?”
“What representation did you not use? Why?”
“How did you decide what scale to use when you made your graph?”
“Now that you have seen the work of other groups, is there anything about your display you would change if you could?”
Representing and Conversing: MLR7 Compare and Connect.
During the gallery walk, invite students to discuss “what is the same and what is different?” about the representations on the posters and then share with the whole class. Look for opportunities to highlight representations that helped students answer the questions and decide which scales to use for the graph. This will help students make connections and describe the usefulness of each type of representation.
Design Principle(s): Optimize output


aligns: yes