few_shots: 
Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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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: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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.

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

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fl_problem_001097_neg_0

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fl_problem_001894_pos_0

HS_0
standard_description: Explain and use the relationship between the sine and cosine of complementary angles.
problem_activity: Activity
In this activity students will apply their knowledge of trigonometry to a real-world context. They will use the safe ladder ratio to determine the safe ladder angle. Then they can use this angle to decide if a ladder is long enough for the given scenario.
Choose if students will use the ratio
\(4:1\)
or measure it themselves. Note that answers will vary if students measure. The image in the task is closer to
\(2:1\)
than
\(4:1\)
, so be prepared to discuss if that much variation is safe.
Launch
Tell students a ladder could tip over backwards if the angle is too large or slide down the wall if the angle is too small.
Provide students the safe ladder ratio by telling them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. On most people that ratio is approximately
\(4:1\)
.”
Or if students will estimate the safe ladder ratio themselves, tell them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. Form groups and measure the triangle one person’s body forms. That is, measure the distance from the floor to their shoulder. Then measure the distance from the front of their chest to their wrist.”
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to help students interpret the language of ratios and angles in context. Display the image and the first sentence of this problem, without revealing the questions that follow. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions with a partner before selecting 1–2 students to share their questions with the class. Listen for and amplify any questions that connect the ratios and angles to the context.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
A good rule of thumb for a safe angle to use when leaning a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground.
What are the angles in the triangle formed by your body and the ladder?
What are the angles in the triangle formed by the ladder, the ground, and the railing? Explain or show your reasoning.
You have a 13 foot long ladder and need to climb to a 12 foot tall roof.
If you put the top of the ladder at the top of the wall, what angle is formed between the ladder and the ground?
Is it possible to adjust the ladder to a safe angle? If so, give someone instructions to do so. If not, explain why not.
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Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
People have various proportions to their body. Suppose that someone’s height to arm ratio is
\(5:1\)
.
What are the angles in the triangle formed by their body and the ladder?
How far off is this from the
\(4:1\)
safe angle?
What could this person do to make the ladder closer to the safe ladder angle?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students might struggle to determine if the ladder can reach the roof. Inform them that the roof height has to be 12 feet but we can use as much of the ladder length as we want up to 13 feet. Invite them to draw a diagram where the ladder extends beyond the top of the building.
Activity Synthesis
Display the image for all to see and invite a student to demonstrate their explanation that the triangles are similar by annotating the image. Emphasize that all of trigonometry is based on similar right triangles. These are just two of the infinite set of right triangles with a
\(4:1\)
ratio and an angle measure of 76 degrees.


aligns: no