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: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
problem_activity: Activity
The purpose of this activity is to let students investigate what happens to the standard deviation using different data sets. The goal is for students to make conjectures about what standard deviation measures and how relative size of the standard deviation can be estimated from the shape of the distribution. In particular, students should recognize that adding or subtracting the same value from each value in the data set will change the mean by the same amount, but the standard deviation remains unchanged. Multiplying or dividing each value in the data set by the same value scales both the mean and standard deviation by the same value.
Monitor for students who
Check values without much thought until the mean or standard deviation is correct.
Use a starting set of values, then modify a few of them up or down to match the statistics given.
Use symmetry to get the mean correct.
Adjust the data by multiplying all of the values by a number to change the spread of the distribution to get close to the standard deviation.
Identify students who create data sets with different types of distributions for “10 different numbers that have a mean of 1” and “10 different numbers with a standard deviation as close to 2.5 as you can get in one minute.”
Launch
Demonstrate how to find standard deviation and mean using the technology available. In Algebra 1, students are dealing with the entire population, not sampling, so the population standard deviation is used.
Open the Spreadsheet & Statistics tool from Math Tools, or navigate to
https://www.geogebra.org/classic/spreadsheet
. Enter the values in column A. Select all of column A, and then choose the button that looks like a histogram and “One Variable Analysis.” Click the button that says
\(\Sigma x\)
. The population standard deviation is labeled
\(\sigma\)
. Additionally, the spreadsheet command =SD(A1:A10) will compute the population standard deviation for data in cells A1 through A10.
Arrange students in groups of 2. Give students time to work through the first two questions, followed by a whole-class discussion.
Conversing: MLR 5 Co-Craft Questions.
Display only the first line of this task (“Use technology to find the mean and the standard deviation for the data in the dot plots”) and the dot plots, and ask pairs of students to write possible mathematical questions about the distributions. Then, invite pairs to share their questions with the class. This helps students produce the language of mathematical questions and talk about the relationships between the two quantities in this task (mean and standard deviation).
Design Principle(s): Cultivate conversation
Student Facing
Use technology to find the mean and the standard deviation for the data in the dot plots.
What do you notice about the mean and standard deviation you and your partner found for the three dot plots?
Invent some data that fits the conditions. Be prepared to share your data set and reasoning for choice of values.
Partner 1
​Partner 2
Dot plots:
[IMG]
[IMG]
[IMG]
Dot plots:
[IMG]
[IMG]
[IMG]
​​​​​​
Conditions:
10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 6.
10 numbers with a standard deviation three times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
Conditions:
10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 12.
10 numbers with a standard deviation four times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of this discussion is to understand that the standard deviation is a measure of variability related to the mean of the data set. The discussion also provides an opportunity for students to discuss what they notice and wonder about the mean and standard deviation.
Select previously identified students in the order listed in the narrative to share their distributions and methods for finding values that worked.
For at least one student’s data set that had a standard deviation close to 2.5, ask them how they might adapt the data so that it has a mean of 1. (Shift the data up or down so that it has a mean of 1.)
Here are some questions for discussion.
“What do you think that standard deviation measures? Why do you think that?” (I think that it measures variability because it behaved like MAD. When all the values were the same it was zero.)
“Why is the standard deviation the same for {1,2,3,4,5} and {-2,-1,0,1,2}?” (For each data set: 1) the values to the left of the mean are a distance of 2 and 1 from the mean and the values to the right of the mean are a distance 2 and 1 from the mean, and 2) the middle value is 0 away from the mean. This results in the standard deviation being the same.)
“Why is the standard deviation different for {-4, -2, 0, 2, 4} and {-4, -3, -2, -1, 0}?” (The values for the first data set are twice the distance from the mean as the values in the second data set. That makes the standard deviation for the first set greater than the standard deviation of the second data set.)
“When is the standard deviation equal to zero?” (It is zero when all the values are the same as each other or when there is no variability.)
“Was your mean the same as your partner’s mean in the fifth match?” (Answers vary. Sample response: My mean was much different from my partners. Mine was 5 and theirs was 10.)
“How did using technology help or hinder your mathematical thinking about standard deviation and mean?” (It really helped me because I did not get bogged down with the calculations and I could look for patterns in the data, the data displays, and the statistics. In particular, having to come up with my own data and then see what happened to the statistics was really helpful.)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer with standard deviation at the center, connecting to it all the related concepts mentioned during the discussion.
Supports accessibility for: Language; Organization


aligns: no