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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
problem_activity: Activity
Following the review from the previous activity, students are asked to use these calculations to compare two groups more formally. Students are shown one quantifiable method of determining whether the two groups are relatively close or relatively very different in the discussion following the activity involving describing the difference of the measures of center as a multiple of the variability (MP2). The important idea for students to grasp from this activity is that the measures of center and measures of variability of the groups work together to give an idea of how similar or different the groups are.
As students work to compare heights of the two families, monitor for students who:
Create dot plots of the data and look at the overlap visually.
Compute measures of center to compare the groups numerically.
Also compute measures of variability to compare the measures of center for each group accounting for the spread of values.
Launch
Keep students in groups of 2.
Introduce students to the idea that both the measures of center as well as the measures of variability are important when comparing data sets by asking students about these contexts.
“Two groups of adults have mean weights that are different by 10 pounds. Are the two groups very different in weight?”
“Two groups of 8 year olds have weights that are different by 10 pounds. Are the two groups very different in weight?”
“Two groups of birds have weights that are different by 3.2 ounces (0.2 pounds). Are the two groups very different in weight?”
Without knowing more about the exact numbers, it seems that the adults might not be that different. A typical adult might weigh 180 pounds and, depending on diet and activity, even one person’s weight might go up or down by 10 pounds in a few weeks. It might be hard to tell these two groups apart even with the 10 pound difference in means.
On the other hand, 8 year olds typically weigh about 50 pounds, so a 10 pound difference can mean a lot more. A group with an average weight of 50 pounds is 25% heavier than a group with an average weight of 40 pounds.
For the birds, it might be hard to say how different the groups are unless we have more information. If the two groups contain the same type of bird (e.g., parrots), we might expect the weights to be fairly consistent, so a 3.2 ounce difference could be a lot. If the two groups each contain a variety of types (from hummingbirds up to emus), the 3.2 ounce difference might not be very much.
Based on these examples, it helps to know not only the means (or medians), but also the variability of the groups. We saw this in the previous activity with the overlap in the dot plots.
For the work in this activity, tell students to try to give some values to help back up their comparison of the two groups.
Allow students 1 minute of quiet think time to examine ways of approaching the problem, followed by partner work time and a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge of measures of centers. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Conversing: MLR2 Collect and Display.
Use this routine to capture the language students use as they compare the heights of these two families. Circulate and listen to student talk during small-group and whole-class discussion. Record the words and phrases students use on a display for all to see. Invite students to borrow from—or add more language to—the display throughout the remainder of the lesson. This will help students read and use mathematical language during their paired and whole-group discussions.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness
Student Facing
Compare the heights of these two families. Explain or show your reasoning.
The heights (in inches) of Noah’s family members: 28, 39, 41, 52, 63, 66, 71
The heights (in inches) of Jada’s family members: 49, 60, 68, 70, 71, 73, 77
Student Response
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Student Facing
Are you ready for more?
If Jada's family adopts newborn twins who are each 18 inches tall, does this change your thinking? Explain your reasoning.
Student Response
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for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is for students to begin to be more formal in their comparison of data from different groups. In particular, a general rule is established that will be used in this unit.
Select students to share their approaches in the sequence outlined in the Activity Narrative.
The heights of Noah’s and Jada’s families overlap more than the heights of the gymnastics and volleyball teams. The difference in means for the two families is about the same as between the two teams.
The difference between the mean heights of the volleyball and gymnastics teams is
\(78.17 - 62.75\)
, or 15.42 inches.
The difference between the mean heights of Jada’s and Noah’s families is
\(66.86 - 51.43\)
, or 15.43 inches. The variability in heights for the families is greater than the variability in heights for the teams.
Explain: The difference between the means is not enough information to know whether or not the data sets are very different. One way to express the amount of overlap is to divide the difference in means by the (larger) mean absolute deviation.
Demonstrate for students how to do this calculation for the volleyball and gymnastics teams:
The difference in means is more than 6 times the measure of variability, because
\(15.42 \div 2.46 \approx 6.3\)
.
Leave the calculation for the two teams displayed. Ask students to do the calculation for Jada’s and Noah’s families.
The difference in means is a little more than 1 time the measure of variability, because
\(15.43 \div 13.22 \approx 1.2\)
.
As a general rule, we will consider it a large difference between the data sets if the difference in means is
more than twice
the mean absolute deviation. If the mean absolute deviation is different for each group, use the larger one for this calculation.
For students who ask why twice the MAD is used rather than some other value, defer the question for later in the unit. A later lesson titled Do They Carry More? will address this question in more detail.


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