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: Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.) Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
problem_activity: Narrative
The purpose of this activity is for students to understand that the same division expression can be used to represent both types of division situations. Students are given two situations and asked to match a division expression to one of the situations, but the expression matches both situations given. It is okay if students do not recognize that the expression matches both situations in the activity, because it will be discussed in the activity synthesis. Students learn that the number we are dividing by is called the
divisor
and understand that the divisor can represent the size of the groups or the number of groups. When students explain that a divisor can be interpreted differently based on the situation it represents, they reason abstractly and quantitatively (MP2).
MLR2 Collect and Display.
Circulate and collect the language students use as they consider the two situations. Listen for and clarify any questions about the context. On a visible display, record words and phrases such as: put in groups, split, divide, number of groups, etc. During the synthesis, add “divisor” to the display and highlight connections to any related language.
Advances: Conversing, Reading
Launch
Groups of 2
“Let’s look a bit closer at division expressions. Take a minute to read these two situations.”
1 minute: quiet think time
Activity
“Work with your partner to decide which situation the expression represents.”
2–3 minutes: partner work time
Student Facing
Consider these two situations.
A. Han has 21 toy cars. He puts the same number of cars in each of 3 boxes. How many cars will be in each box?
B. Han has 21 toy cars. He wants to put 3 cars in each box. How many boxes will he need?
Which situation does the expression
\(21 \div 3\)
represent? Explain your reasoning.
[IMG]
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
Invite students to share their responses and reasoning.
“How can the same expression represent two different situations?” (Both situations involve the same numbers, 21 and 3. Both situations involve putting 21 objects into equal groups. In one case, the 3 is the number of objects in the group, but in the other, it is the number of groups. Both situations are talking about 21 divided by 3, just in different ways.)
“We noticed that the number that we are dividing by, 3, can have two different meanings. It can mean 3 groups or 3 objects in each group.”
“When we divide, the number we divide by is called the
divisor
. In the expression
\(27 \div 3\)
, the divisor is 3.”


aligns: no