few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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


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).

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

Example 3:
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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_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: Warm-up
The goal of this opening activity is to activate, through a familiar context, what students know about functions from middle school.
Students first encounter a relationship in which two quantities—the number of bagels bought and price—do not form a function. They see that for some numbers of bagels bought, there are multiple possible prices. The relationship between the number of bagels bought and the
best
price, however, do form a function, because there is only one possible best price for each number of bagels.
Students contrast the two relationships by reasoning about possible prices, completing a table of values, and observing the graphs of the relationships.
Launch
Arrange students in groups of 4. Give students a few minutes of quiet time to think about the first question, and then a couple of minutes to share their thinking with their group. Pause for a class discussion.
Invite students to share an explanation as to how each person in the situation could be right. If possible, record and display students’ reasoning for all to see. After the reasoning behind each price is shared, direct students to the table in the second question. Ask students to write down “best price” in the header of the second column and then complete the table.
Student Facing
[IMG]
[IMG]
A customer at a bagel shop is buying 13 bagels. The shopkeeper says, “That would be $16.25.”
Jada, Priya, and Han, who are in the shop, all think it is a mistake.
Jada says to her friends, “Shouldn’t the total be \$13.25?”
Priya says, “I think it should be \$13.00.”
Han says, “No, I think it should be \$11.25.”
Explain how the shopkeeper, Jada, Priya, and Han could all be right.
Your teacher will give you instructions for completing the table.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "br": [{}], "_values": ["number", "of\u00a0bagels"]}, {"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}]}, {"td": [{"_value": "2"}, {}]}, {"td": [{"_value": "3"}, {}]}, {"td": [{"_value": "4"}, {}]}, {"td": [{"_value": "5"}, {}]}, {"td": [{"_value": "6"}, {}]}, {"td": [{"_value": "7"}, {}]}, {"td": [{"_value": "8"}, {}]}, {"td": [{"_value": "9"}, {}]}, {"td": [{"_value": "10"}, {}]}, {"td": [{"_value": "11"}, {}]}, {"td": [{"_value": "12"}, {}]}, {"td": [{"_value": "13"}, {}]}]}]}]}
Student Response
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Activity Synthesis
Consider displaying a table to summarize the different possibilities for calculating the price of 13 bagels (or the prices for 6 or more bagels), as well as the table showing the best price for each number of bagels bought.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number", "of bagels"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["shopkeeper\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Jada\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Priya\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Han\u2019s", "price"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "1.25"}, {}, {}, {}]}, {"td": [{"_value": "2"}, {"_value": "2.50"}, {}, {}, {}]}, {"td": [{"_value": "3"}, {"_value": "3.75"}, {}, {}, {}]}, {"td": [{"_value": "4"}, {"_value": "5.00"}, {}, {}, {}]}, {"td": [{"_value": "5"}, {"_value": "6.25"}, {}, {}, {}]}, {"td": [{"_value": "6"}, {"_value": "7.50"}, {"_value": "6.00"}, {}, {}]}, {"td": [{"_value": "7"}, {"_value": "8.75"}, {"_value": "7.25"}, {}, {}]}, {"td": [{"_value": "8"}, {"_value": "10.00"}, {"_value": "8.50"}, {}, {}]}, {"td": [{"_value": "9"}, {"_value": "11.25"}, {"_value": "9.75"}, {"_value": "8.00"}, {}]}, {"td": [{"_value": "10"}, {"_value": "12.50"}, {"_value": "11.00"}, {"_value": "9.25"}, {}]}, {"td": [{"_value": "11"}, {"_value": "13.75"}, {"_value": "12.25"}, {"_value": "10.50"}, {}]}, {"td": [{"_value": "12"}, {"_value": "15.00"}, {"_value": "12.00"}, {"_value": "11.75"}, {"_value": "10.00"}]}, {"td": [{"_value": "13"}, {"_value": "16.25"}, {"_value": "13.25"}, {"_value": "13.00"}, {"_value": "11.25"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number", "of bagels"]}, {"_attributes": {"scope": "col"}, "_value": "best price"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "1.25"}]}, {"td": [{"_value": "2"}, {"_value": "2.50"}]}, {"td": [{"_value": "3"}, {"_value": "3.75"}]}, {"td": [{"_value": "4"}, {"_value": "5.00"}]}, {"td": [{"_value": "5"}, {"_value": "6.25"}]}, {"td": [{"_value": "6"}, {"_value": "6.00"}]}, {"td": [{"_value": "7"}, {"_value": "7.25"}]}, {"td": [{"_value": "8"}, {"_value": "8.50"}]}, {"td": [{"_value": "9"}, {"_value": "8.00"}]}, {"td": [{"_value": "10"}, {"_value": "9.25"}]}, {"td": [{"_value": "11"}, {"_value": "10.50"}]}, {"td": [{"_value": "12"}, {"_value": "10.00"}]}, {"td": [{"_value": "13"}, {"_value": "11.25"}]}]}]}]}
Ask students,
“’Number of bagels’ and ’price’ do not form a
function
, but ’number of bagels’ and ’best price’ do form a function. Why is this? What do you recall about functions?”
“If we graph the relationship between ’number of bagels’ and ’price,’ what do you think the graph would look like?”
“If we graph the relationship between ’number of bagels’ and ’best price,’ what would the graph look like?”
After students make some predictions, display the two graphs for all to see. In the first graph, solid blue dots represent the shopkeeper’s price, open green circles Jada’s price, red squares Priya’s price, and yellow triangles Han’s price. In the second graph, each X represents the best price for each number of bagel.
[IMG]
Because there are multiple possible prices when the number of bagels is 6 or greater, price is not a function of the number of bagels bought.​​​​​​
[IMG]
Because there is only one best price for a particular number of bagel, best price is a function of the number of bagels bought.
Emphasize that a function assigns one output to each input. Clarify that the word “function” in mathematics has a very specific meaning that does not necessarily agree with how “function” is used in everyday life (for instance, as in the sentence “The function of a bridge is to connect two sides of a river.”).


aligns: no