few_shots: 
Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Concept/skill: 
Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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.

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.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_pos_0

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fl_problem_001097_neg_0

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fl_problem_001894_neg_0

HS_0
standard_description: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
problem_activity: Activity
This info gap activity gives students an opportunity to determine and request the information needed to find and graph the transformation of a given function. In each case, the given function is a trigonometric function which has been translated horizontally or vertically and also reflected or scaled. The problem card gives the graph and equation of the trigonometric function. The student with the problem card figures out which transformations to apply to the graph and equation. The data cards give multiple ways for students to figure out the transformation. Monitor for students who
ask for the explicit transformations and the order in which they are applied
ask for the amplitude, midline, and period of the new function
The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Here is the text of the cards for reference and planning:
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Launch
Tell students they will continue to work on identifying transformations between two functions. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.
Provide students with access to small sheets of graph paper.
Engagement: Develop Effort and Persistence.
Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing:
This activity uses
MLR4 Information Gap
to give students a purpose for discussing information necessary to find and graph the transformation of a given function. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
Silently read the information on your card.
Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
Read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need.
Explain to your partner how you are using the information to solve the problem.
When you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Suppose we considered the function
\(T\)
which is the sum of
\(Q\)
and
\(S\)
. Is
\(T\)
also periodic? If yes, what is its period? If no, explain why not.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Focus on these points:
The order of the transformations matters. For problem card 1, if the vertical stretch is done before the vertical translation it gives a different result, namely
\(Q(x) = 4\sin(2x) + 14\)
. For problem card 2, if the horizontal stretch is done first and then the horizontal translation the result would be
\(S(x) = \text-\cos(\pi (x + \pi))+3\)
For the first card, the vertical translation will take
\(\sin(2x) + 3\)
to
\(\sin(2x )+ 5\)
. For the vertical stretch, make sure students understand that both
\(\sin(2x)\)
and
\(5\)
need to be scaled by 4. The vertical stretch by 4 multiplies the output of the function by 4 and in this case the output, after the translation, is
\(\sin(2x) + 5\)
.
For the second card, the horizontal translation changes
\(\text-\cos(x) + 3\)
to
\(\text-\cos(x+\pi) + 3\)
and then the horizontal scaling by
\(\frac{1}{\pi}\)
gives
\(\text-\cos(\pi x + \pi) + 3\)
. Unlike problem card 1, the scaling does not influence the constant 3 because the scale is horizontal while the 3 translated the graph vertically.
Ask students if they considered asking for the amplitude, period, and midline. In a sense, this information may be more straightforward to process because the order of the transformations is already encoded in this information. But the student with the problem card may not realize that this information is available.
Finally, ask students if they considered sketching the graph first and using this to find the transformed function. The sketch can help show major features of the graph (amplitude, midline, period) and serve as a guide for finding a valid equation.


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