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: 
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.

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: 
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, 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_pos_0

HS_2
standard_description: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
problem_activity: Activity
In groups, students rotate through five different stations, where they measure one or more quantities using different units, and answer a series of summary questions afterward. Here are the quantities being measured and the units used at each station:
Station 1:
Volume of a box, in cubic inches and cubic centimeters.
Station 2:
Length, in meters and feet.
Station 3:
Volume of water, in gallons, quarts, and liters.
(If desired, you can have students measure water with actual containers instead of watching the video
https://vimeo.com/illustrativemathematics/water
.)
Station 4:
Weights and masses of 2–3 objects, in ounces, pounds, grams, and kilograms.
(You can have students weigh actual objects, use the digital simulation
http://ggbm.at/eQQVYB7D
, or use the paper simulations from the blackline master. If using one of the simulations instead of a real scale, prepare some real objects labeled with their weight or mass for students to hold and feel the weight of.)
Station 5:
Volume of salt, in milliliters and teaspoons.
You will need the blackline master for this activity. Page 1 is a net for the box needed for station 1. If you are using the paper scale simulation instead of a real scale or the applet, pages 2–13 are the parts needed to assemble Station 4.
Digital
Launch
Tell students they will further investigate the idea of using different units to measure the same set of items. Introduce the five stations, what students are expected to do at each, the protocol for rotating through them, and the questions to answer at the end. Then, demonstrate how to use the straightedge to measure a level teaspoon of salt. If students do not use a level teaspoons of salt, they will not be able to answer the last set of questions about volume.
Arrange students into 5 groups and assign a starting station for each group.
If students have devices, Stations 3 and 4 can be digital.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “I noticed _____ so I think…”
Supports accessibility for: Language; Social-emotional skills
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support small-group discussion. As students rotate through stations, encourage students to solidify their own understanding by pressing for details and questioning their peers' explanations. Provide sentence frames for students to use, such as "I agree/disagree because . . .”, "How do you know . . .”, and “Can you give an example?” This will help students clarify their reasoning about comparing different measurements for the same quantity using different units.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Station 1
Each large cube is 1 cubic inch. Count how many cubic inches completely pack the box without gaps.
Each small cube is 1 cubic centimeter. Each rod is composed of 10 cubic centimeters. Count how many cubic centimeters completely fill the box.
|                   | cubic inches | cubic centimeters |
|-------------------|--------------|-------------------|
| volume of the box |              |                   |
Station 2
Your teacher showed you a length.
Use the meter stick to measure the length to the nearest meter.
Use a ruler to measure the length to the nearest foot.
|                            | meters | feet |
|----------------------------|--------|------|
| length of ________________ |        |      |
Station 3
Watch the video.
Count how many times you can fill the quart bottle from the gallon jug.
Count how many times you can fill the liter bottle from the gallon jug.
Video
Quarts and Liters
available at
https://player.vimeo.com/video/304136597
.
|                   | quarts | liters |
|-------------------|--------|--------|
| 1 gallon of water |        |        |
Station 4
Use the applet to record the weights of different objects in different units.
Record their weights in ounces, pounds, grams, and kilograms.
| object | ounces | pounds | grams | kilograms |
|--------|--------|--------|-------|-----------|
|        |        |        |       |           |
|        |        |        |       |           |
|        |        |        |       |           |
Station 5
Count how many level teaspoons of salt it takes to fill the graduated cylinder to 20 milliliters, 40 milliliters, and 50 milliliters.
Pour the salt back into the original container.
|                                                 | milliliters | teaspoons |
|-------------------------------------------------|-------------|-----------|
| small amount                           of salt  |             |           |
| medium amount                           of salt |             |           |
| large amount                           of salt  |             |           |
After you finish all five stations, answer the following questions with your group.
Which is larger, a cubic inch or a cubic centimeter? Did more cubic inches or cubic centimeters fit in the cardboard box? Why?
Did it take more feet or meters to measure the indicated length? Why?
Which is bigger, a quart or a liter? Explain your reasoning.
Use the data from Station 4 to put the units of weight and mass in order from smallest to largest. Explain your reasoning.
About how many teaspoons of salt would it take to fill the graduated cylinder to 100 milliliters?
If you poured 15 teaspoons of salt into an empty graduated cylinder, about how many milliliters would it fill?
How many milliliters per teaspoon are there?
How many teaspoons per milliliter are there?
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
Tell students they will further investigate the idea of using different units to measure the same set of items. Introduce the five stations, what students are expected to do at each, the protocol for rotating through them, and the questions to answer at the end. Then, demonstrate how to use the straightedge to measure a level teaspoon of salt. If students do not use a level teaspoons of salt, they will not be able to answer the last set of questions about volume.
Arrange students into 5 groups and assign a starting station for each group.
If students have devices, Stations 3 and 4 can be digital.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “I noticed _____ so I think…”
Supports accessibility for: Language; Social-emotional skills
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support small-group discussion. As students rotate through stations, encourage students to solidify their own understanding by pressing for details and questioning their peers' explanations. Provide sentence frames for students to use, such as "I agree/disagree because . . .”, "How do you know . . .”, and “Can you give an example?” This will help students clarify their reasoning about comparing different measurements for the same quantity using different units.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Station 1
Each large cube is 1 cubic inch. Count how many cubic inches completely pack the box without gaps.
Each small cube is 1 cubic centimeter. Each rod is composed of 10 cubic centimeters. Count how many cubic centimeters completely fill the box.
|                                                   | cubic                                 inches | cubic                                 centimeters |
|---------------------------------------------------|----------------------------------------------|---------------------------------------------------|
| volume of                                 the box |                                              |                                                   |
Station 2
Your teacher showed you a length.
Use the meter stick to measure the length to the nearest meter.
Use a ruler to measure the length to the nearest foot.
|                                                              | meters | feet |
|--------------------------------------------------------------|--------|------|
| length of                                 __________________ |        |      |
Station 3
If not using real water, open
https://vimeo.com/illustrativemathematics/water
.
Count how many times you can fill the quart bottle from the gallon jug.
Count how many times you can fill the liter bottle from the gallon jug.
|                                                   | quarts | liters |
|---------------------------------------------------|--------|--------|
| 1 gallon                                 of water |        |        |
Station 4
If not using a real scale, open
http://ggbm.at/eQQVYB7D
.
Select 2–3 different objects to measure on the scale.
Record the weights in ounces, pounds, grams, and kilograms.
| object | ounces | pounds | grams | kilograms |
|--------|--------|--------|-------|-----------|
|        |        |        |       |           |
|        |        |        |       |           |
|        |        |        |       |           |
Station 5
Count how many level teaspoons of salt fill the graduated cylinder to 20 milliliters, 40 milliliters, and 50 milliliters.
Pour the salt back into the original container.
|                                                       | milliliters | teaspoons |
|-------------------------------------------------------|-------------|-----------|
| small amount                                 of salt  |             |           |
| medium amount                                 of salt |             |           |
| large amount                                 of salt  |             |           |
After you finish all five stations, answer these questions with your group.
Which is larger, a cubic inch or a cubic centimeter?
Did more cubic inches or cubic centimeters fit in the cardboard box? Why?
Did it take more feet or meters to measure the indicated length? Why?
Which is larger, a quart or a liter? Explain your reasoning.
Use the data from Station 4 to put the units of weight and mass in order from smallest to largest. Explain your reasoning.
About how many teaspoons of salt would it take to fill the graduated cylinder to 100 milliliters?
If you poured 15 teaspoons of salt into an empty graduated cylinder, about how many milliliters would it fill?
How many milliliters per teaspoon are there?
How many teaspoons per milliliter are there?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
People in the medical field use metric measurements when working with medicine. For example, a doctor might prescribe medication in 10 mg tablets.
Brainstorm a list of reasons why healthcare workers would do this. Organize your thinking so it can be followed by others.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
At Station 1, students may count the number of base-10 centimeter rods rather than the number of centimeter cubes. Remind them that the question prompts for the number of cubes.
At Station 2, students may need reminders about measuring objects at the zero marking on the ruler and about keeping the ruler going straight, both of which will affect the answer. Show them they can measure along the edge of the object to make sure the ruler is not veering off in one direction or another.
At Station 4, students may be unclear about how to change the output unit on the scale for each object. Consider showing the class ahead of time. Students who are able to distinguish between weight and mass might say they cannot accurately compare their measurements. Clarify that we are talking only about the weight of the objects on Earth’s surface.
At Station 5, some students may consistently use under-filled or rounded teaspoons of salt, so their data will not reveal the
\(5 : 1\)
ratio of milliliters to teaspoons. Repeat the demonstration of how to measure a level teaspoon for them.
Students may answer 3 milliliters for the question about 15 teaspoons because they divided by 5 instead of multiplying by 5. Encourage them to pay attention to which unit is bigger and ask what that tells them about which numerical value should be larger.
Activity Synthesis
Though much of the discussion will take place within groups, spend a few minutes ensuring that everyone understands the answers to the five questions. To conclude the activity, invite students to share anything that surprised them from the measuring work.


aligns: no