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 by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Explain and use the relationship between the sine and cosine of complementary angles.
problem_activity: In Class Launch
Use after Unit 4, Lesson 9
Display an image of the flag of Nepal, such as this one, for all to see.
[IMG]
Tell students that this is the flag of Nepal, which is a country in South Asia that borders India and China. Ask students “What do you notice? What do you wonder?”
Students may notice:
It looks like it’s made of triangles.
I could break it up into a rectangle and three triangles.
The bottom left corner looks like a right angle.
Students may wonder:
What shapes is the flag made of?
Why does it look like this?
How could we break up the white symbols into simpler shapes?
If it does not come up during the discussion, ask students how they might break up the shape of the flag into simpler shapes so that they could, for example, find its area.
Then tell students that in this task they will see how to construct this flag, and investigate its shape. They will also investigate the shapes involved in a flag of their choosing.
Teacher Instructions
Students will not have seen how to trisect a line segment, which they will need to do in order to construct
\(AC\)
. They can measure first to see where the points that trisect the line should go. Instructions for trisecting a line segment can be found online. There are many ways of doing it, and some are easier than others, so if students find the instructions themselves they should be encouraged to compare several methods before deciding which one they prefer. If students try different methods, it could be beneficial to invite selected students to show the class what they did. The other steps for constructing the flag should be familiar. Having access to geometry software can streamline the construction process.
If students have trouble getting started with finding the angles, they can try giving
\(AB\)
a specific value and calculating the other side lengths.
Once students have completed the task, invite them to share the flags they chose and show how they adapted them to fit on the triangular banner.
Student-Facing Statement
The constitution of Nepal includes instructions for constructing the flag. Here are the instructions:
Step 1: Draw a line
\(AB\)
of the required length.
\(AB\)
will be the bottom edge of the flag.
Step 2: Draw a line
\(AC\)
perpendicular to
\(AB\)
so that the length of
\(AC\)
is
\(\frac43\)
the length of
\(AB\)
.
Step 3: Mark the point
\(D\)
on the line
\(AC\)
so that
\(AD\)
is the same length as
\(AB\)
. Then join the points
\(D\)
and
\(B\)
with a straight line.
Step 4: Mark the point
\(E\)
on the line
\(BD\)
so that
\(BE\)
is the same length as
\(AB\)
.
Step 5: Draw a line
\(FG\)
that goes through
\(E\)
, is parallel to
\(AB\)
, and is the same length as
\(AB\)
.
\(F\)
should be on the line
\(AC\)
.
Step 6: Join the points
\(C\)
and
\(G\)
with a straight line.
Construct the flag according to the instructions. You can make it any size you want.
Find the measure of each of the angles in the flag.
Imagine that there will be a parade featuring the flag of Nepal. For the parade there will be small flags and large flags. Decide what the sizes of each type of flag should be.
How much material would you need to make each type of flag?
If you made the border of the flags by sewing a ribbon along the edge, how long would the ribbon have to be for each type of flag?
Imagine that there will be a parade featuring a flag that is meaningful to you. (Choose a flag or design your own.) At the parade there will be small flags, large flags, and triangular banners like this:
[IMG]
How will you adapt your flag to fit on the triangular shape?
How much material of each color will you need for each kind of flag?
Teacher Instructions
Students who want an extra challenge can find the areas and perimeters of both the small and large flag in the second question, and then compare the areas to see the effect of the scale factor. The relationship between scaling lengths and scaling areas will be a topic in the next unit, so students do not need to have a full understanding of it at this time, but this is a good opportunity to preview the concept.
Another opportunity for an extra challenge is to try describing the exact ratios between the side lengths using radicals. An especially interesting thing students may notice is that the ratio between
\(AF\)
and
\(AD\)
is the same as the ratio between
\(AD\)
and
\(BD\)
,
\(\tfrac{1}{\sqrt2}\)
.
Having access to geometry software can help students visualize the shapes they are working with.
Once students have completed the task, invite them to share the flags they chose and show how they adapted them to fit on the triangular banner.
Student-Facing Statement
The constitution of Nepal includes instructions for constructing the flag. Here are the instructions:
Step 1: Draw a line
\(AB\)
of the required length.
\(AB\)
will be the bottom edge of the flag.
Step 2: Draw a line
\(AC\)
perpendicular to
\(AB\)
so that the length of
\(AC\)
is
\(\frac43\)
the length of
\(AB\)
.
Step 3: Mark the point
\(D\)
on the line
\(AC\)
so that
\(AD\)
is the same length as
\(AB\)
. Then join the points
\(D\)
and
\(B\)
with a straight line.
Step 4: Mark the point
\(E\)
on the line
\(BD\)
so that
\(BE\)
is the same length as
\(AB\)
.
Step 5: Draw a line
\(FG\)
that goes through
\(E\)
, is parallel to
\(AB\)
, and is the same length as
\(AB\)
.
\(F\)
should be on the line
\(AC\)
.
Step 6: Join the points
\(C\)
and
\(G\)
with a straight line.
Clare is constructing the flag and she starts with
\(AB\)
of length 6 inches. Here is her finished flag shape. Use the instructions to fill in as many lengths and angles as you can.
[IMG]
Imagine that there will be a parade featuring the flag of Nepal. For the parade there will be small flags and large flags. For one of those types, decide how big it should be.
How much material would you need to make the flag?
If you made the border of the flag by sewing a ribbon along the edge, how long would the ribbon have to be?
Imagine that there will be a parade featuring a flag that is meaningful to you. (Choose a flag or design your own.) At the parade there will be rectangular flags and triangular banners like this:
[IMG]
How will you adapt your flag to fit on the triangular shape?
How much material of each color will you need for each kind of flag?


correct_option_index: 1