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: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
problem_activity: Activity
In this activity, students identify what equivalent ratios have in common (a ratio equivalent to
\(a:b\)
can be generated by multiplying both
\(a\)
and
\(b\)
by the same number) and generate equivalent ratios (MP8). It is at this point in the unit where students will explicitly define the term equivalent ratios (MP6).
Launch
Arrange students in groups of 3–4. Provide each group with tools for creating a visual display.
Summarize what we know so far about equivalent ratios. When we double or triple a color recipe, the ratios of the amount of ingredients in the mixtures are equivalent to those in the original recipe. For example,
\(24:9\)
and
\(8:3\)
are equivalent ratios, because we can think of
\(24:9\)
as a mixture that contains three batches of purple water where a single batch is
\(8:3\)
.
When we make multiple batches of a food recipe, we say the ratios of the amounts of the ingredients are equivalent to the ratios in a single batch. For example,
\(3:6\)
,
\(1:2\)
, and
\(9:18\)
are equivalent ratios because they correspond to the amount of the ingredients in different numbers of batches of tuna noodle casserole, and they all taste the same.
In this activity, we’ll write a definition for
equivalent ratios
.
When students pause after question 5, have a whole class discussion about the first five questions. Then assign each group a different ratio to use as their example for their visual display. Some possibilities:
\(4:5\)
\(3:2\)
\(5:6\)
\(3:4\)
\(2:5\)
Writing and speaking: Math Language Routine 1 Stronger and Clearer Each Time.
This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response. Students meet together in 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as: “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response that reflects ideas from their partners, and improvement on their writing. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for generalization)
How It Happens:
Use this routine to provide students a structured opportunity to revise and refine their response to “How do you know when ratios are equivalent and when they are not equivalent?” Allow students 2–3 minutes to individually create first draft responses.
Invite students to meet with 2–3 other partners for feedback.
Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help teams strengthen their ideas and clarify their language: “Can you explain how...?”, “You should expand on….”, “Can you give an example of equivalent ratios?”, and “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.
Signal for students to move on to their next partner and repeat this structured meeting.
Close the partner conversations and invite students to revise and refine their writing in a second draft. Students can borrow ideas and language from each partner to strengthen the final product.
Provide these sentence frames to help students organize their thoughts in a clear, precise way: “I know when ratios are equivalent/not equivalent when….” and “An example of this is…because….”
Here is an example of a second draft:
“I know when ratios are equivalent when I multiply both parts of one ratio by the same number and I get the other ratio. For example, I know that
\(5:3\)
is equivalent to
\(30:18\)
because when I multiply 5 by 6, I get 30, and when I multiply 3 by 6, I get 18, so 6 is the same number used to multiply both parts. But
\(5:3\)
is not equivalent to
\(15:12\)
because when I multiply 5 by 3, I get 15, and when I multiply 3 by 4, I get 12. So since a different number is used to multiply to get the second ratio, they’re not equivalent.”
If time allows, instruct students to compare their first and second drafts. If not, the students can continue on with the lesson by returning to their first partner and creating the visual.
Student Facing
The ratios
\(5:3\)
and
\(10:6\)
are
equivalent ratios
.
Is the ratio
\(15:12\)
equivalent to these? Explain your reasoning.
Is the ratio
\(30:18\)
equivalent to these? Explain your reasoning.
Give two more examples of ratios that are equivalent to
\(5:3\)
.
How do you know when ratios are equivalent and when they are
not
equivalent?
Write a definition of
equivalent ratios
.
Pause here so your teacher can review your work and assign you a ratio to use for your visual display.
Create a visual display that includes:
the title “Equivalent Ratios”
your best definition of
equivalent ratios
the ratio your teacher assigned to you
at least two examples of ratios that are equivalent to your assigned ratio
an explanation of how you know these examples are equivalent
at least one example of a ratio that is
not
equivalent to your assigned ratio
an explanation of how you know this example is
not
equivalent
Be prepared to share your display with the class.
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 incorporate recipes, specific examples, or batch thinking into their definitions. These are important ways of thinking about equivalent ratios, but challenge them to come up with a definition that only talks about the numbers involved and not what the numbers represent.
If groups struggle to get started thinking generally about a definition, give them a head start with: “A ratio is equivalent to
\(a:b\)
when . . .”
If students include “or divide” in their definition, remind them that, for example, dividing by 5 gives the same result as multiplying by one-fifth. Therefore, we can just use “multiply” in our definition.
Activity Synthesis
Each group will share their visual display as they explain their definitions. Highlight phrases or explanations that are similar in each display. Make one class display that incorporates all valid definitions. This display should be kept posted in the classroom for the remaining lessons within this unit. It should look something like:
Equivalent Ratio
A ratio is equivalent to
\(a:b\)
when both
\(a\)
and
\(b\)
are multiplied by the same number.


correct_option_index: 1