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: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
problem_activity: Warm-up
By now, students understand that lengths in a scaled copy are related to the original lengths by the scale factor. Here they see that the area of a scaled copy is related to the original area by the
square
of the scale factor.
Students build scaled copies of a single pattern block, using blocks of the same shape to do so. They determine how many blocks are needed to create a copy at each specified scale factor. Each pattern block serves as an informal unit of area. Because each original shape has an area of 1 block, the
\(\text{(scale factor)}^2\)
pattern for the area of a scaled copy is easier to recognize.
Students use the same set of scale factors to build copies of three different shapes (a rhombus, a triangle, and a hexagon). They notice regularity in their repeated reasoning and use their observations to predict the number of blocks needed to build other scaled copies (MP8).
Digital
Launch
Arrange students in groups of 3–4. Distribute pattern blocks and ask students to use them to build scaled copies of each shape as described in the task. Each group would need at most 16 blocks each of the green triangle, the blue rhombus, and the red trapezoid. If there are not enough for each group to have a full set with 16 each of the green, blue, and red blocks, consider rotating the blocks of each color through the groups, or having students start with 10 blocks of each and ask for more as needed.
Give students 6–7 minutes to collaborate on the task and follow with a whole-class discussion. Make sure all students understand that “twice as long” means “2 times as long.”
Using real pattern blocks is preferred, but the Digital Activity can replace the manipulatives if they are unavailable.
Student Facing
Use the applets to explore the pattern blocks. Work with your group to build the scaled copies described in each question.
[IMG]
How many blue rhombus blocks does it take to build a scaled copy of Figure A:
Where each side is twice as long?
Where each side is 3 times as long?
Where each side is 4 times as long?
How many green triangle blocks does it take to build a scaled copy of Figure B:
Where each side is twice as long?
Where each side is 3 times as long?
Using a scale factor of 4?
How many red trapezoid blocks does it take to build a scaled copy of Figure C:
Using a scale factor of 2?
Using a scale factor of 3?
Using a scale factor of 4?
Make a prediction: How many blocks would it take to build scaled copies of these shapes using a scale factor of 5? Using a scale factor of 6? Be prepared to explain 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.
Print
Launch
Arrange students in groups of 3–4. Distribute pattern blocks and ask students to use them to build scaled copies of each shape as described in the task. Each group would need at most 16 blocks each of the green triangle, the blue rhombus, and the red trapezoid. If there are not enough for each group to have a full set with 16 each of the green, blue, and red blocks, consider rotating the blocks of each color through the groups, or having students start with 10 blocks of each and ask for more as needed.
Give students 6–7 minutes to collaborate on the task and follow with a whole-class discussion. Make sure all students understand that “twice as long” means “2 times as long.”
Using real pattern blocks is preferred, but the Digital Activity can replace the manipulatives if they are unavailable.
Student Facing
Your teacher will give you some pattern blocks. Work with your group to build the scaled copies described in each question.
[IMG]
How many blue rhombus blocks does it take to build a scaled copy of Figure A:
Where each side is twice as long?
Where each side is 3 times as long?
Where each side is 4 times as long?
How many green triangle blocks does it take to build a scaled copy of Figure B:
Where each side is twice as long?
Where each side is 3 times as long?
Using a scale factor of 4?
How many red trapezoid blocks does it take to build a scaled copy of Figure C:
Using a scale factor of 2?
Using a scale factor of 3?
Using a scale factor of 4?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may come up with one of these arrangements for the first question, because they assume the answer will take 2 blocks to build:
[IMG]
You could use one pattern block to demonstrate measuring the lengths of the sides of their shape, to show them which side they have not doubled.
Students may also come up with:
[IMG]
for tripling the trapezoid, because they triple the height of the scaled copy but they do not triple the length. You could use the process described above to show that not all side lengths have tripled.
Activity Synthesis
Display a table with only the column headings filled in. For the first four rows, ask different students to share how many blocks it took them to build each shape and record their answers in the table.
|  scale factor   | number of blocks to build Figure A | number of blocks to build Figure B | number of blocks to build Figure C |
|-----------------|------------------------------------|------------------------------------|------------------------------------|
|        1        |                                    |                                    |                                    |
|        2        |                                    |                                    |                                    |
|        3        |                                    |                                    |                                    |
|        4        |                                    |                                    |                                    |
|        5        |                                    |                                    |                                    |
|       10        |                                    |                                    |                                    |
|      \(s\)      |                                    |                                    |                                    |
| \(\frac{1}{2}\) |                                    |                                    |                                    |
To help student notice, extend, and generalize the pattern in the table, guide a discussion using questions such as these:
In the table, how is the number of blocks related to the scale factor? Is there a pattern?
How many blocks are needed to build scaled copies using scale factors of 5 or 10? How do you know?
How many blocks are needed to build a scaled copy using any scale factor
\(s\)
?
If we want a scaled copy where each side is half as long, how much of a block would it take? How do you know? Does the same rule still apply?
If not brought up by students, highlight the fact that the number of blocks it took to build each scaled shape equals the scale factor times itself, regardless of the shape (look at the table row for
\(s\)
). This rule applies to any factor, including those that are less than 1.


aligns: yes