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


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Thought: This task shows one quadratic equation with one variable, x.
Answer: B

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 $P$.


Standard options: 
A. 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.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: 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.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Recognize and represent proportional relationships between quantities.

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

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: Activity
This activity revisits a context seen previously. Students solved problems like this as early as grade 3 without formulating them in terms of ratios or rates (“If 1 cup of rice serves 3 people, how many people can you serve with 12 cups of rice?”). In this activity, they ultimately find an equation for the proportional relationship.
As students find missing values in the table, they should see that they can always multiply the number of food items by the constant of proportionality. When students see this pattern (MP7) and represent the number of people served by
\(x\)
cups of rice (or
\(s\)
spring rolls) as
\(3x\)
(or
\(\frac{1}{2}s\)
), they are expressing regularity in repeated reasoning (MP8).
Only one row in each table is complete. Based on their experience in the previous lesson, students are more likely to multiply the entries in the left-hand column by 3 (or
\(\frac{1}{2}\)
) than to use scale factors, at least for the first and third rows. If they do, they are more likely to see how to complete the last row in each table.
Some students might use unit rates: If 1 cup of rice can serve 3 people, then
\(x\)
cups of rice can serve
\(3x\)
people. So, students have different ways to generate
\(3x\)
as the expression that represents the number of people served by
\(x\)
cups of rice: completing the table for numerical values and continuing the pattern to the last row; or finding the unit rate and using it in the case of
\(x\)
cups.
Monitor students for different approaches as they are working.
Launch
Tell students that this activity revisits a context they worked on in an earlier lesson.
Student Facing
A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 cup of rice serve?
How many people will 3 cups of rice serve? 12 cups? 43 cups?
How many people will
\(x\)
cups of rice serve?
| cups of dry rice | number of people |
|------------------|------------------|
|        1         |                  |
|        2         |        6         |
|        3         |                  |
|        12        |                  |
|        43        |                  |
|      \(x\)       |                  |
A recipe says that 6 spring rolls will serve 3 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 spring roll serve?
How many people will 10 spring rolls serve? 16 spring rolls? 25 spring rolls?
How many people will
\(n\)
spring rolls serve?
| number of spring rolls | number of people |
|------------------------|------------------|
|           1            |                  |
|           6            |        3         |
|           10           |                  |
|           16           |                  |
|           25           |                  |
|         \(n\)          |                  |
How was completing this table different from the previous table? How was it the same?
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
If students have trouble encapsulating the relationship with an expression, encourage them to draw diagrams or to verbalize the relationship in words.
Activity Synthesis
Select students who have used the following approaches in the given order:
Recognize that to move from the first column to the second, you multiply by 3.
Say that 3 is the constant of proportionality.
Recognize that the equation
\(2 \boldcdot  {?} = 6\)
can be used to find the constant of proportionality algebraically.
Say that the 3 can be interpreted as the number of people per 1 cup of rice.
If no student sees that these insights are connected to prior work, explicitly connect them with the lessons of the previous two days. At the end of the discussion, suggest to students that we let
\(y\)
represent the number of people who can be served by
\(x\)
cups of rice. Ask students to write an equation that gives the relationship of
\(x\)
and
\(y\)
. (This builds on work from grade 6, but it may be rusty.) Be sure to write the equation
\(y = 3x\)
where all students can see it and help students interpret its meaning in the context: “To find
\(y\)
, the number of people served, we can multiply the number of cups of rice,
\(x\)
, by 3.”
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students describe their approaches, use color and annotations to scribe their thinking on a display.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing: MLR7 Compare and Connect.
As students explain their different approaches to completing the tables, ask “What is similar, what is different?” about their methods. Draw students’ attention to the different ways the constant of proportionality was represented across the different strategies. For instance, when discussing first question, ask “Where do you see the 3 in each approach?” These exchanges strengthen students’ mathematical language use and reasoning based on constant of proportionality.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 0