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: 
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_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.
problem_activity: Narrative
The purpose of this activity is for students to find combinations of coins with a value of 100 cents and understand that a dollar is the same as 100 cents. They add within 100 by grouping like coins, using a ten, or using counting on strategies. They find coin combinations that add up to 100 cents or 1 dollar and write the leftover amount as cents. This work builds on students’ understanding of a hundred.
Engagement: Develop Effort and Persistence.
Chunk the tasks by question and check in with students to provide feedback and encouragement after each question. Look for the students to be using a strategy and ask them to verbalize their solution to ensure they are on the right track.
Supports accessibility for: Language; Organization
Required Preparation
Add a dollar bill to the money poster showing the front and back to display in the launch.
Launch
Groups of 2
Display the poster with the dollar added.
[IMG]
“We just saw that there are many different ways to use coins to make 100 cents. One dollar is also the same as 100 cents.”
“If I have a dollar and I found a quarter, how much money would I have?” (125 cents or \$1 and 25¢)
30 seconds: quiet think time
Display a dollar and a quarter and write $1 and 25¢.
[IMG]
“When you write dollar amounts you use the \$ symbol. Use the ¢ symbol for the amount that cannot be grouped to make a dollar.”
Activity
“Now, you are going to work with a partner to make dollars and cents.”
10 minutes: partner work time
Monitor for students who show Priya's money in different ways to share in the Lesson Synthesis.
Student Facing
[IMG]
Andre emptied his pockets and found these coins.
[IMG]
How much money does he have? Show your thinking.
Han emptied his pockets and found these coins.
[IMG]
How much money does he have? Show your thinking.
Priya has $1 and 18¢ in her pocket.
If Priya only had coins in her pocket, what coins could she have?
Represent Priya’s coins:
If Priya had 1 dollar bill and some coins, what coins could she have?
Represent Priya's money:
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students show the value of Andre and Han's coins only in cents, consider asking:
“How did you find the total value of the coins?”
“How many dollars does _______ have? How many leftover cents would he have?”
“How could you say how much money _______has using dollars and cents?”
Activity Synthesis
“I noticed that when you were creating a coin collection for Priya, there were different solutions.”
“How is it possible that everyone represented the same amount of money?” (Different coins can show the same amount, but you need a different number of coins.)
Invite students to share different combinations of coins they used to make $1.
Share and record responses.


aligns: no