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.


Standard options: 
A. 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.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Your response, in json format:
{
  "explanation": "This task involves representing addition with objects such as number cards and dice.",
  "answer": "A"
}

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response, in json format:
{
  "explanation": "This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).",
  "answer": "B"
}

Example 3:
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.


Your response, in json format:
{
  "explanation": "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"
}

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
problem_activity: Activity
The purpose of this task is for students to use the coordinates of a unit circle to determine information about coordinates on circles of different sizes. Returning to the context of clock hands, this activity focuses on the location of the end of a minute hand relative to the center of the clock.
While there are 3 different clock sizes in this activity, students taking advantage of the similarity between the circles (MP7) can use their solutions for the first clock to identify the solutions for the second clock that is 3 times the size of the first.
Monitor for students who express their answers using:
approximate values based on the unit circle display
exact values using
\(\cos(\theta)\)
and
\(\sin(\theta)\)
Launch
If needed, provide students with copies of separate clock faces.
Student Facing
Here is a clock face.
[IMG]
The length of the minute hand on a clock is 5 inches and the center of the clock is at
\((0,0)\)
on a coordinate plane. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
The minute hand on another clock, also centered at
\((0,0)\)
, has a length of 15 inches. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
At a certain time, the end of the minute hand of a third clock centered at
\((0,0)\)
has coordinates approximately
\((7.5,7.5)\)
. How long is the minute hand of the clock if each grid square is one inch by one inch? Explain or show 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?
The center of a clock is
\((0,0)\)
on a coordinate grid. Its hour hand is half the length of its minute hand. The coordinates of the end of the hour hand are about
\((3,0.5)\)
. What are the approximate coordinates of the end of the minute hand? Explain how you know.
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
Select students to share their strategies for identifying the coordinates on the clocks with a minute hand of length 5 inches and 15 inches in the order listed in the Activity Narrative. If no students wrote their answers using exact values, such as
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
for the 5 inch minute hand 10 minutes after the hour, ask students to consider how they could write the exact coordinates. After a brief quiet think time, invite students to share their thinking, highlighting any callbacks to their previous work identifying exact values by using cosine and sine without approximating. If needed, here are some questions for discussion:
“What angle does the minute hand make, with the positive
\(x\)
-axis, at 10 minutes after the hour?” (
\(\frac \pi 6\)
radians.)
“If the minute hand was 1 foot long, could you use sine and cosine to express the coordinates of its tip?” (The coordinates, in feet, are
\(\left(\cos\left(\frac \pi 6\right), \sin\left(\frac \pi 6\right)\right)\)
.)
“What about when the minute hand is 5 inches long? 15 inches long?” (The coordinates, in inches, are
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
and
\(\left(15\cos\left(\frac \pi 6\right),1 5\sin\left(\frac \pi 6\right)\right)\)
.)
Conclude the discussion by selecting students to share how they calculated the length of the minute hand when the coordinates of the end of the minute hand are known. Make sure to highlight these strategies by inviting students to consider each and share a reason why they prefer one over the other:
use the Pythagorean Theorem
recognize the angle as
\(\frac{\pi}{4}\)
radians due to the
\(x\)
- and
\(y\)
-coordinates matching
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their strategies for identifying the coordinates on the clocks, scribe their thinking on a visual display. Highlight the exact coordinates (expressions using cosine and sine) and the approximate coordinates in the same color. Use annotation to draw attention to the scale factors of 5 and 15 in the expressions.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 0

1

2