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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_2
standard_description: Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
problem_activity: Activity
In the previous activity, students noticed trends in the data from the scatter plot. In this activity, the association is made more precise by looking at equations and graphs of linear models for the data to determine the slope. The numerical value of the slope is then interpreted in the context of the problem (MP2).
Launch
Remind students that earlier we looked at the price and mileage of some used cars. We saw that for these used cars, the price tends to decrease as the mileage increases. Display the scatter plot and linear model for the data.
[IMG]
Tell students that an equation for the line is
\(y=\text-0.073x+17,\!404.485\)
. From the equation we can identify the slope of the line as -0.073. Ask students to think about what that slope tells us and give quiet think time. Select 1–2 students to share their thinking. (It means that for every increase of one mile, the model predicts that the price of the car will decrease by
$
0.073.) Tell students that in this activity they will determine what the slope of the model means for three different sets of data.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present only one scatter plot at a time.
Supports accessibility for: Organization; Attention
Speaking, Writing: MLR8 Discussion Supports.
Provide a sentence frame to scaffold students’ language when describing the meaning of slope in each representation. For example, “If the _________ increases by 1 _________ , the model predicts that ___________ (increases/decreases) by ___________ .”
Design Principle: Maximize meta-awareness; Optimize output (for explanation)
Student Facing
For each of the situations, a linear model for some data is shown.
What is the slope of the line in the scatter plot for each situation?
What is the meaning of the slope in that situation?
\(y=5,\!520.619x-1,\!091.393\)
[IMG]
\(y=\text-0.011x+40.604\)
[IMG]
\(y=0.59x-21.912\)
[IMG]
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
The scatter plot shows the weight and fuel efficiency data used in an earlier lesson along with a linear model represented by the equation
\(y = \text-0.0114 x +41.3021\)
.
[IMG]
What is the value of the slope and what does it mean in this context?
What does the other number in the equation represent on the graph? What does it mean in context?
Use the equation to predict the fuel efficiency of a car that weighs 100 kilograms.
Use the equation to predict the weight of a car that has a fuel efficiency of 22 mpg.
Which of these two predictions probably fits reality better? Explain.
Student Response
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for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is to develop a quantitative sense of trends based on linear models of the data.
Consider asking some of the following questions:
“What was the easiest way to find the slope for each situation?” (The coefficient of the
\(x\)
-coordinate in the equation.)
“Do the answers for the meaning of the slopes make sense in the contexts of the problem?” (Yes. Diamonds are usually expensive, so a 1 carat increase is likely to cost a lot more. A 1 kilogram difference in the weight of a car is not very much, so it may lower the gas mileage, but not by much. A 1 degree temperature increase is not significant, so it should not need a lot more energy to cool off a building, but some would be needed.)
“What is the difference between a positive slope and negative slope in your interpretations?” (A positive slope means both variables increase together. A negative slope means that when one variable increases, the other decreases.)
“The model for energy consumption and temperature predicted a 0.59 kilowatt hour increase in energy consumption for every 1 degree increase in temperature. Estimate how much the temperature would need to increase to raise energy consumption by 6 kilowatt hours.” (A little more than 10 degrees.)


aligns: no