few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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: 
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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
problem_activity: Optional activity
The purpose of this activity is for students to use equations to represent situations of percent increase and decrease. Additionally, students identify the original and new amount in the double number lines to reinforce what they learned in earlier lessons (that the original amount pertains to 100%).
As students work on the task, look for students who created various equations for the last question.
Launch
Show students the double number line from the activity in the previous lesson.
[IMG]
Help them make connections to the first problem. Tell them that they can go back to the previous lesson to see the double number lines for the others if that helps.
Arrange students in groups of 2. Give 5–8 minutes of quiet work time. After 5 minutes allow students to work with a partner or to continue to work alone.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using double number line diagrams. If students are unsure where to begin, suggest that they draw a double number line diagram to help organize the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
The gas tank in dad’s car holds 12 gallons. The gas tank in mom’s truck holds 50% more than that. How much gas does the truck’s tank hold? Explain why this situation can be represented by the equation
\((1.5) \boldcdot 12 = t\)
. Make sure that you explain what
\(t\)
represents.
Write an equation to represent each of the following situations.
A movie theater decreased the size of its popcorn bags by 20%. If the old bags held 15 cups of popcorn, how much do the new bags hold?
After a 25% discount, the price of a T-shirt was
$
12. What was the price before the discount?
Compared to last year, the population of Boom Town has increased by 25%.The population is now 6,600. What was the population last year?
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
Students may continue to struggle to recognize the original amount and new amount with the proper percentages on the double number line. Remind them that the original amount always corresponds to 100%.
Activity Synthesis
Select students to share the values they identified as original amount and the new amount for a few problems. Discuss how 100% always corresponds to the original value and when there is an increase in the value the new value corresponds to a percentage greater than the original 100%.
Select students to share the different equations they came up with. Discuss how the distributive property is useful for finding the percentage that corresponds with the new value instead of the percentage of the change.
Discuss how solving problems about percent change may require either multiplying or dividing numbers. It can be confusing, but it helps to first express the relationship as an equation and then think about how you can find the unknown number. Looking at the examples below, the first two require multiplication, but the others require division.
Using the structure A% of B is C:
\((1.5) \boldcdot 12 = c\)
\((0.80) \boldcdot 15 = c\)
\(a \boldcdot (1,200) = 1,080\)
\(a \boldcdot (1.50) = 1.75\)
\((0.75) \boldcdot b = 12\)
\((1.25) \boldcdot b = 61, 600\)
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
To begin the whole-class discussion, use this routine to give students a structured opportunity to revise and refine their response to the first question, "Explain why this situation can be represented by the equation
\((1.5) \boldcdot 12 = t\)
." Give students time to meet with 2–3 partners, to share and get feedback on their initial explanations. Provide listeners with prompts for feedback such as, “What does
t
represent?” “Where is 50% in the equation?” “Can you say that another way?” etc. Invite students to go back and refine their written explanation based on the peer feedback they receive. This will help students understand how situations of percent increase or decrease can be represented by an equation.
Design Principle(s): Optimize output (for generalization); Cultivate conversation


aligns: no