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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
problem_activity: Narrative
Previously, students multiplied two factors where one factor was a whole number and the other a teen number. The purpose of this activity is for students to make sense of multiplication of a one-digit number and a two-digit number greater than 20. Students analyze representations used to find
\(4 \times 23\)
and articulate reasons for using strategies that are based on the distributive property and place value to find products (MP7). Along the way, they see that decomposing the two-digit factors into tens and ones is particularly helpful for multiplying. They also reinforce what they know about the connections between area diagrams and multiplication expressions.
MLR8 Discussion Supports.
Before partner work, remind students to use words such as decompose, tens, and ones.
Advances: Speaking, Representing
Required Materials
Materials to Gather
Base-ten blocks
Materials to Copy
Centimeter Grid Paper - Standard
Launch
Groups of 2
“Take a minute to make sense of how Clare and Andre represented
\(4 \times 23\)
.”
1 minute: quiet think time
Give students access to grid paper and base-ten blocks.
Activity
“Now, work with your partner to complete the first two problems about
\(4 \times 23\)
.”
8–10 minutes: partner work time
Monitor for the diagrams that students choose in Diego’s problem and for their explanations.
Pause for a discussion. Invite students to share their responses.
If not mentioned by students, clarify that all diagrams can be used to multiply 4 and 23, but not all are equally practical.
Select a student to explain why diagram D might be more productive than the other diagrams.
“Now, work independently to complete the last problem.”
2–3 minutes: independent work time
Student Facing
Here is how Clare and Andre represented
\(4 \times 23\)
.
Clare
[IMG]
Andre
[IMG]
How does each diagram show
\(4 \times 23\)
?
How could we use Clare’s diagram to find the value of
\(4 \times 23\)
?
How could we use Andre’s diagram to find the value of
\(4 \times 23\)
?
Diego tried different ways to partition or split a diagram to help him find the value of
\(4 \times 23\)
.
A
[IMG]
B
[IMG]
C
[IMG]
D
[IMG]
What do you notice about the numbers in his diagrams?
Which diagram would you use to find the value of
\(4 \times 23\)
? Explain your reasoning.
Find the value of
\(3 \times 28\)
.
Show your thinking using diagrams, symbols, or other representations.
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 choose a diagram in the second problem that doesn't help them find the product of
\(4\times23\)
, consider asking:
“Tell me about how you chose which diagram you would use to find the value of
\(4\times23\)
.”
“Is there another diagram that would make finding the value of
\(4\times23\)
easier? How would it make it easier?”
Activity Synthesis
“What representations did you use in the last question to show
\(3 \times 28\)
? How did they help you find the product?” (Base-ten blocks helped me break the 28 into tens and ones. A diagram with a grid helped me break 28 into smaller numbers. A diagram with no grid helped me think about the numbers as I labeled it without worrying about all the little squares.)


aligns: no