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: 
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.

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: 
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, 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_pos_0

HS_2
standard_description: Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
problem_activity: Activity
Students continue to use diagrams to represent the ratio of ingredients in a recipe as well as mixtures that contain multiple batches. They come to understand that a change in the number of batches changes the quantities of the ingredients, but the end product tastes the same. They then use this observation to come up with a working definition for equivalent ratio.
Launch
Launch the task with a scenario and a question: “Let’s say you are planning to make cookies using your favorite recipe, and you’re going to ‘double the recipe.’ What does it mean to double a recipe?”
There are a few things you want to draw out in this conversation:
If we double a recipe, we need to double the amount of
every ingredient
. If the recipe calls for 3 eggs, doubling it means using 6 eggs. If the recipe calls for
\(\frac13\)
teaspoon of baking soda, we use
\(\frac23\)
teaspoon of baking soda, etc.
We expect to end up with twice as many cookies when we double the recipe as we would when making a single batch.
However, we expect the cookies from 2 batches of a recipe to taste
exactly the same
as those from a single batch.
Tell students they will now think about making different numbers of batches of a cookie recipe.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, "I noticed _____, so I . . .", "In my diagram, _____ represents . . .", and “To find an equivalent ratio, first I _____ because . . . .” This will help students produce mathematical language as they make sense of equivalent ratios using diagrams.
Supports accessibility for: Language; Social-emotional skills
Student Facing
A recipe for one batch of cookies calls for 5 cups of flour and 2 teaspoons of vanilla.
Draw a diagram that shows the amount of flour and vanilla needed for
two
batches of cookies.
How many batches can you make with 15 cups of flour and 6 teaspoons of vanilla? Show the additional batches by adding more ingredients to your diagram.
How much flour and vanilla would you need for 5 batches of cookies?
Whether the ratio of cups of flour to teaspoons of vanilla is
\(5:2\)
,
\(10:4\)
, or
\(15:6\)
, the recipes would make cookies that taste the same. We call these
equivalent ratios
.
Find another ratio of cups of flour to teaspoons of vanilla that is equivalent to these ratios.
How many batches can you make using this new ratio of ingredients?
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
For the fourth question, students may not multiply both the amount of flour and the amount of vanilla by the same number. If this happens, refer students to the previous questions in noting that the amount of each ingredient was changed in the same way.
Activity Synthesis
Invite a few students to share their responses and diagrams with the class. A key point to emphasize during discussion is that when we double (or triple) a recipe, we also have to double (or triple) each ingredient. Record a working (but not final) definition for equivalent ratio that can be displayed for at least the next several lessons. Here is an example: “Cups of flour and teaspoons of vanilla in the ratio
\(5:2\)
,
\(10:4\)
, or
\(15:6\)
are equivalent ratios because they describe different numbers of batches of the same recipe.” Include a diagram in this display.


aligns: no