Skate Park TaskTask Description Standards

In this task, students are asked to reason conceptually about the relationship between angles and their opposite sides in a triangle, calculate the horizontal and vertical lengths of a skate park ramp using trigonometric ratios and the Pythagorean theorem.

G.SRT.C.6 M2.G.SRT.C.6G.SRT.C.8a M2.G.SRT.C.8a

Mathematical Understandings Extending Understandings

Upon successful completion of these tasks students will demonstrate the ability to:

Recognize that side ratios in right triangles are properties of the angles in the triangle.

Find missing sides and angles of a right triangle, given other sides and angles.

Recognize when it is appropriate to use Pythagorean Theorem and when it is appropriate to use trigonometric ratios to solve triangles.

Use trigonometric ratios to solve real-world problems.

To move students towards deeper understanding, they should be given opportunities to:

Explain when it is appropriate to use Pythagorean Theorem and when it is appropriate to use Trigonometric ratios to solve triangles.

Explain their solution path and reasoning for their calculations, as well as their reasoning for selecting the appropriate strategy.

Determine if the triangles are similar and justify their answer using precise mathematical vocabulary.

Supporting Strategies

If students are struggling to access this grade level task, additional supports and strategies could be employed as students are engaging with the task.

Key Terms Sentence Frames Scaffolded Questions

trigonometric ratios (trig ratios)

sine (sin)

cosine (cos)

tangent (tan)

Pythagorean Theorem

Sine, cosine, and tangent are called ____________ _________.

The ratio for sine is _______.

The ratio for cosine is _______.

The ratio for tangent is ________.

The Pythagorean Theorem is ________.

What type of triangle does this situation represent?

What are two methods that can be used to solve for missing side lengths of a right triangle?

What information do you need to know before you can use the Pythagorean Theorem to solve for a missing side of a right triangle?

What information do you need to know before you can use trigonometric ratios to solve for a missing side of a right triangle?

Additional ResourcesModel word problems using trigonometric ratios and the Pythagorean TheoremSolve word problems using the Pythagorean Theorem and trigonometric ratiosWorking with trig ratiosUnderstand that similar triangles share angle measures and side ratios

Skate Park Task

A group of friends decide to build a skate ramp. The board used to make the ramp is 50 inches, and it will form a 35-degree angle with the ground.

a. Without doing any calculations, Kira says that the horizontal distance (the distance on the ground the ramp covers) will be greater than the vertical distance (the height of the ramp). Sketch the problem situation described and then explain how she might know this without performing any calculations.

b. Determine the horizontal and vertical distances of the ramp.

c. After building the ramp, the group decided they want both the vertical and horizontal distances of the ramp to be half as long. Will this also decrease the ramp length by one half? Explain your reasoning.

Student Work Part aQuestion: Without doing any calculations, Kira says that the horizontal distance (the distance on the ground the ramp covers) will be greater than the vertical distance (the height of the ramp).

a. Sketch the problem situation described and then explain how she might know this without performing any calculations.

StandardsG.SRT.C.6M2.G.SRT.C.6

Assessing Questions

The student sketches the situation correctly, recognizes that a 45-45-90 triangle has congruent legs, and argues that Kira is correct by noting that “the angle across from the vertical distance is less than 45 degrees so it will be shorter and the horizontal distance will be longer”.

Which two angles would have to be 45°?

How do you know the lengths of the legs are the same if the acute angles are both 45°?

What relationship exists between angles and opposite sides that allows you to claim that one leg is shorter or longer than the other?

The student sketches the situation correctly. The student incorrectly states that 35 is the pivotal measure for the change in length “if the vertical distance is more than the horizontal then the degree angle with the ground would be more than 35°.” The student does not justify this thinking.

Can you prove the angle with the ground would need to be more than 35° for the vertical distance to be more than the horizontal distance? How much more?

What triangle property could you use to justify your conclusion?

The student sketches the situation correctly. However, the student incorrectly states “the angle formed by the ramp and the ground would have to be greater than 90° for the vertical distance to be greater than the horizontal distance”.

Sketch a picture of the ramp with the angle formed by the ramp and the ground greater than 90°. What do you notice?

Sketch a picture of the ramp where the vertical distance is greater than the horizontal? What would have to be true of the angle formed by the ground for the vertical distance to be longer?

What is the measure of the angle where the

horizontal line and the ground meet? What would be the measure of the angle where the horizontal line and the ramp meet?

What do we know about three angles of a triangle?

It is not clear in the student’s drawing that he/she understands the board is the entire hypotenuse of the triangle in this situation (The vertical line with an arrow causes confusion). The student also asserts that the distance on the ground “looks longer”, which is not mathematically precise nor backed by mathematics.

Tell me what your arrow represents.

What geometric shape is formed by the board, the ground, the 35° angle, and the vertical distance from the ground?

What angle measures do you know in that figure?

Which side of the triangle would represent the 50 inch board? What is that side called?

Is there any additional information you could add to your drawing based on geometric properties?

How can you use the given angle to help you decide which leg of the right triangle would be longer?

Student Work Part bQuestion:

b. Determine the horizontal and vertical distances of the ramp.

StandardsG.SRT.C.8aM2.G.SRT.C.8a

Assessing Questions

The student determines the correct vertical distance by setting up and solving the trigonometric ratio for sin(35). The student determines the correct horizontal distance by using the Pythagorean Theorem to find the missing side length of the right triangle.

Explain why you chose to solve for A using trig ratios and to solve for B using Pythagorean Theorem.

Could another trigonometric ratio be used to solve this problem?

The student determines the correct horizontal distance by setting up and solving the equation x=sin(35) • 50 The student determines the correct vertical distance by setting up and solving the equation y=cos(35) • 50

Explain why you chose to solve the problem using trig ratios. Could you have used a different method to solve for either side?

Can you prove your side length values hold true using the Pythagorean Theorem?

The student set up incorrect proportions relating side lengths and angles. The student may have been trying to use the law of sines to find the side lengths, but forgot to use the sine of each angle within each ratio which resulted in incorrect answers.

Tell me which method you chose to solve for the side lengths and why.

What is the relationship between the side lengths and angles in a right triangle?

Is there another method you could use since this is a right triangle?

The student uses the Pythagorean Theorem to calculate the legs as though they are the same length, and then gives a general statement that one leg is less than that measurement and the other leg is greater than that measurement.

Why did the Pythagorean Theorem not give you actual lengths for the legs?

What is another method to find missing pieces in a right triangle when Pythagorean Theorem does not work?

How can you use trigonometric ratios to find the actual length and height of the ramp?

Student Work Part cQuestion: After building the ramp, the group decided they want both the vertical and horizontal distances of the ramp to be half as long.

c. Will this also decrease the ramp length by one half? Explain your reasoning.

StandardsG.SRT.C.6G.SRT.C.8a

M2.G.SRT.C.6M2.G.SRT.C.8a

Assessing Questions

Note: This question has multiple solution paths that could align to either standard.

The student halves the side values given in Part B and uses the Pythagorean Theorem to prove that the hypotenuse would be 25, half of the original ramp length.

What does this tell you about the two triangles? (similar)

Will this work every time? For example, if the horizontal and vertical lengths are doubled in size, will the length of the ramp be doubled?

The student incorrectly concludes, based on incomplete math, that the ramp length will decrease by more than half. The student reasons through using the Pythagorean Theorem to cut A and B

in half, but stops at c2

4

without consideringc2

4= c2

.

What do A, B, and C in your equation represent in the context of the problem?

Explain your conclusion that the ramp length will decrease by more than half.

While the student does not provide explicit calculations, the student reasons that the ramp length will also decrease by one half, noting that “each measurement has to be proportional to the original measurements.” The student also provides a model illustrating a smaller triangle within the original triangle and states “if you were to make the shape smaller.” This is an understood reference to similar triangles.

How do you know the measurements have to be proportional?

Which measurements are proportional?

Explain what you mean by making “the shape smaller”.

The student provides the correct response “Yes”. However, the student’s justification of asserting that the ramp length will also decrease by one half “because horizontal distance and ramp length are directly proportional,” does not refer to similar triangles or show

Which measurements are proportional and why?

How does your claim of “longer” support vertical and horizontal lengths that have been halved?

appropriate calculations to support the claim.