Geometry – Right Triangles Grades 9-10 Time length: give or take 8 days

Day 1: Pythagorean Theorem

Materials: Ziplocs with straws for each student; square and triangle image handouts; real-

life examples handouts; calculators; powerpoint presentation

Standards:

HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Objectives:

The learners will be able to apply the Pythagorean Theorem to solve for missing sides in

right triangles.

Learning Activities:

1. The teacher will begin the lesson by explaining to the students that they will be

discussing the Pythagorean theorem

a. Their goal by the end of class is to be able to accurately use the Pythagorean

Theorem to solve for missing sides in right triangles.

2. The teacher will use a powerpoint to guide the lesson.

3. The students will each receive a bag of straws varying in lengths. They will be asked

to form two right triangles with the 6 straw pieces

a. They will be asked if they notice any similarities between the two triangles

i. For example – between the right angles and the opposite side lengths

4. First, the teacher will review a few key ideas and definitions:

a. Right triangles

b. Hypotenuse and legs of triangles

c. Squares and square roots

5. Next they will be given a handout

a. The first line of the table will be completed as a class

b. They will be asked to complete the handout with their small groups (the

groups are simply the students they are sitting at the tables with)

c. The teacher will walk among the groups to help out where needed

6. Then, the class will discuss the handout, noting the patterns

a. This will lead to the teacher describing the Pythagorean Theorem:

or

b. The teacher will ask the students if they think this pattern would happen for

all right triangles or for only a select few

c. The answer is all – the teacher will present a few examples on the board

7. Next the teacher will talk about real-life examples.

a. One example will be completed as class, and then the students will do

examples in their groups

b. A student will complete the examples on the board

8. Next, the teacher will introduce the converse of the Pythagorean theorem

a. this can be done on the board

i. if , then the triangle is a right triangle

ii. if , then the triangle is acute

iii. if , then the triangle is obtuse

b. To solve these problems, one must square the side lengths of a triangle and

plug them into the equation to figure out which type of triangle it is.

c. Next, the teacher will do 2 or 3 examples on the board

9. Quick review of what Pythagorean theorem is and how it can be used

10. Homework: practice problems

a. Some would be taken from the book, but I would write a few real-life

problems if the book did not include any

i. The geometry books I have seen do a good job of incorporating real-

life examples, so the homework for day 1 would probably come from

problems listed in the textbook.

Assessment:

An informal assessment will continuously be taken through observation as the teacher

walks around and the students participate in the lesson. A formal assessment will be taken

when the students turn in their handouts and example problems.

Reflection:

To be completed after the lesson is taught.

Day 2: Special Right Triangles

Materials: exploration handouts for each student, calculators if needed

Standards:

HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Objectives: The learners will be able to find side lengths in 45-45-90 triangles and 30-60-90 triangles using the relationships between the legs and the hypotenuse.

Learning Activities:

1. The lesson will begin with a quick review of Pythagorean Theorem:

a. Have a student come to the board and teach this to the class

b.

2. Then, quickly review what an Isosceles triangle is:

a. Two congruent angles, two congruent sides

3. Draw right triangle on the board

a. “If this is both a right triangle and an isosceles triangle, what do the measures of the

other two angles have to be?”

i. Measure of two acute angles must equal 90⁰ (180-90{measure of right

angle}=90⁰ left between the two remaining angles.

ii. 90⁰÷2=45⁰ (both angles must be equal)

iii. Therefore, this is called a 45-45-90 triangle, named after the angle

measures

iv. Also, label the legs and the hypotenuse on the board

1. Remind students: hypotenuse is ALWAYS opposite of 90⁰ angle

4. Activity with partners (teacher: pair students up with someone sitting next to them; do not

let the students choose their own partners)

a. Give handout (included at end of unit)

b. Only do 45-45-90 part of handout for now

c. Walk through handout on the board when every group seems to be finished

i. Proof of the rule:

1. Example: both legs are 1 unit long, then we can use the Pythagorean

theorem to solve for the hypotenuse length:

d. Put the 45-45-90 RULE on the board somewhere students will be able to see it for

remainder of lesson:

45-45-90 Triangle:

5. Complete 2 examples on the board using the 45-45-90 triangle rule. Encourage students to

put examples in notebooks for examples

a. If a=2, find the hypotenuse (draw the triangle)

b. If hypotenuse =4 , find the lengths of the legs (draw the triangle)

6. Explain how 30-60-90 triangles are formed:

a. Draw an equilateral triangle

b. Bisect an angle, creating a line perpendicular to opposite side

c. How long is y in comparison to x?

i. Y is twice as long as X; y=2x

ii. Label short leg, long leg, and hypotenuse

7. Activity with partners (keep same partners as previous activity)

a. Have the students complete 30-60-90 aspect of handout

b. Complete table on board and ask for the conclusions that the students reached

through completion of the handout

c. Put the 30-60-90 RULE on the board with the other rule so students can refer to it as

needed:

(This triangle is half of the triangle on

the left)

30⁰

X

Y

60⁰

8. Give 2 examples on the board using 30-60-90 triangles (More examples can be use

completed if students need them.)

a. Find the long leg and the hypotenuse if the short leg=3 (draw a triangle on the

board)

b. Find the short leg and the hypotenuse if the long leg= equals 6 (draw a triangle)

9. Remainder of class is work time – students will be given homework problems from the book

again

a. Problems from the previous days’ assignments can also be reviewed during work

time, and any issues can be addressed.

b. Again, the books I have seen seem to have a good variety of problems for the

students to work with. I would select the problems I think are appropriate and

supplement additional problems where necessary.

Assessment: An assessment will be taken through observation of student participation, especially

when the students are working in small groups. Another assessment will be taken when the

students turn their homework worksheet in.

Reflection: To be completed after lesson is taught

Days 3 and 4: Trigonometric Functions - (this lesson will more than likely need to be spread over

2 days)

Materials: ribbon, handouts, calculators

Standards:

HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Objectives:

The learners will be able to describe and list the relationship between sides in right triangles using

the trigonometric functions.

The learners will use the trigonometric functions to solve for missing side lengths in right triangles.

Learning Activities:

Introduction: Locating opposite sides, adjacent sides, and hypotenuse of a right triangle with

respect to a given angle.

1. Teacher will explain to the students that we will be creating ratios between side lengths

within right triangles

a. The teacher will explain that a ratio is a relationship between two side lengths.

i. In this case, the ration will be written in fraction form

2. The teacher will begin by putting the following image on the board:

a. The teacher will explain that the angle we are focusing on is labeled with an “A.”

b. The teacher will then point out that the opposite side is directly across from the

angle, and the adjacent side is one of the legs creating the angle, but is not the

hypotenuse

c. The teacher will also stress the idea that if we focused on the other acute angle in

the triangle, the opposite and adjacent sides would change. (drawing another

triangle on the board to illustrate this would be beneficial)

3. Activity:

a. Put the students in groups of 3 (again, teacher should pick the groups, not the students)

i. The students will need to create 6 right triangles on their desk or on the floor using ribbon

1. They can use a book corner as a template for the right angle

ii. The students will then be asked to draw a copy of each triangle on a sheet of paper (each student should do a separate sheet)

iii. The students will label one acute angle with an “A”

1. They will also label each of the sides with a, b, and c

2. They will then be asked to label the sides opposite, adjacent, or hypotenuse in respect to their angle “A”

4. As a full class, the teacher will explain that the different ratios are important when solving for missing side or angle lengths

a. Introduce sine, cosine, tangent

b. What do they look like on a calculator? (sin, cos, tan)

c. What do their relationships look like?

i. For right triangles they look like this:

1.

2.

3.

d. Draw a right triangle on the board and label the sides a, b, c.

1. Point out one of the acute angles and label it “A”

a. Ask the students to write in their notebooks the relationships for sine, cosine, and tangent in relationship to angle “A”

b. Have them compare with a neighbor

c. Write the relationships on the board

5. Give the students a handout in which they are simply finding the three trigonometric ratios in respect to a given acute angle

a. This handout should be corrected in class so students can easily see their mistakes

***This would be a great stopping point if there is minimal time left in the class, if not, continue with the lesson

6. Once the students have successfully found trigonometric ratios in respect to given acute angles, the lesson will move into the idea of how to actually solve for side lengths with the trig functions:

1.

a. Read, “sine of the given angle measure is equal to the opposite side length divided by the hypotenuse length”

2.

a. Read, “cosine of a given angle measure is equal to the adjacent side length divided by the hypotenuse length”

3.

a. Read, “tangent of a given angle measure is equal to the opposite side length divided by the adjacent side length”

7. Then, the teacher will explain how important it is to choose the correct trig function when solving for missing side lengths

a. Explain: first, figure out which acute angle measure is given. If there are two acute angle measures given, pick one that you want to use.

b. Next: find the two side lengths will be used (one that is given and one that is labelled with a variable, meaning we need to find the length )

c. Third, figure out the relationship between the two side lengths and the angle

d. Decide if it is necessary to use sine, cosine, or tangent to solve for the missing length

i. Refer to SOH CAH TOA to help make the decision

e. Then, plug in the known aspects to your chosen equation and solve

i. One of the best ways to solve these equations is to cross multiply

8. Do at least 2 examples on the board that the students can add to their notebooks

a. Check for understanding – do more examples if needed

X ft.

SOH CAH TOA (Copy this onto the board so the students

will be able to refer to it when needed)

15 ft.

58⁰ The acute angle we will use is 58⁰. The two side lengths we will use are

labeled as 15 ft. and X ft. The side length labeled 15 ft. is the Opposite side,

and the side labeled X ft. is the hypotenuse. Therefore, we will use the Sine

function.

=

; If we cross multiply, we will come up with

If we type that into our calculator, we will come up with

9. Give the students 2 problems to do on their own (similar to the one above)

a. Choose students to work the problems out on the board

10. Distribute the homework (worksheet included at end of unit)

a. If time allows, this homework can be worked on during class time so the teacher is available to help.

Assessment: An informal assessment will take place during observation throughout both days of

the lesson. A more formal assessment will take place as the students complete and turn in their

homework.

Reflection: To be completed after lesson is taught.

Day 5: Writing Assignment

Materials: writing prompt and rubric, worksheets, calculators

Standards:

HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Objectives: The learners will be able to explain the special right triangle rules, in writing, to

someone who does not have a strong mathematical background.

Learning Activities:

1. The students will each be given a writing prompt as well as a rubric to how their writing

will be assessed. (included at end of unit plan)

2. The teacher will explain that district-wide, all classes must complete a writing assignment

each semester

3. The writing prompt will ask the students to explain how to solve for the side lengths in the

two special right triangles they learned about.

4. The writing will hopefully be finished in class, but they will be asked to write at least ¾ of a

sheet.

a. If the students are not done, they may complete writing at home

5. After the writing is completed, the students will receive an assignment combining all

previous lessons.

a. This assignment will be homework for those who do not finish writing.

b. This assignment will be a combination of problems from the text that were not

previously assigned.

Assessment:

An assessment will be taken through the use of a rubric as the students turn in their writing. An

informal assessment for the students will be taken when their homework is graded in class. This

assignment will not be collected, but the teacher will not inform the students of that until after it is

completed and graded. The students will be encouraged to use their scores as a guide for studying

for the test at the end of the unit.

Reflection: To be completed after lesson is taught.

Day 6: Solving Right Triangles (putting it all together!)

Materials: assignment from the book, calculators

Standards:

HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Objectives:

The learners will be able to completely solve right triangles. They will be able to find all side and

angle measurements.

Learning Activities:

1. The teacher will explain that solving a right triangle means that all angle measures and all

side lengths must be found.

a. The teacher will also explain that there are a number of strategies that can be used

to make this possible.

b. As the lesson goes on, the strategies will be listed on a board so students are able to

refer to it as they work.

2. The teacher will tell the students that a triangle is able to be completely solved if you know

either of the following:

a. Two side lengths, or

b. One side length and the measure of one ACUTE angle (besides the right angle)

3. There is one other key concept that needs to be taught before allowing the students to

work:

a. Inverse trigonometric functions

i. These inverse functions can be used to find measures of angles

ii. If , then

iii. If , then

iv. If , then

b. The teacher would also show the students how to punch these into the calculator

c. Next, the teacher would provide a few examples of these problems, such as:

A

C

B 20 ft.

15ft.

Approximate the measure of <A to the nearest tenth of a

degree. Since we know the opposite and the adjacent legs to

<A, we would use the tangent function:

4. After a few examples, the teacher should ask the students to name all of the strategies they

can think of that may be useful to solve a triangle.

a. The teacher should, again, list these somewhere the students will be able to see and

use them

b. Here are some examples of what the list can include:

Strategy Reason

Pythagorean Theorem Find side lengths

Special right triangle rules Find side lengths

Trigonometric functions Find side lengths

Inverse trigonometric functions Find missing angle measures

All angles add up to 180⁰ If 2 angles known, subtract to find 3rd

c. Finally, the students should be given a handout or an assignment from the book

asking the students to completely solve triangles. The problems should require the

students to use a variety of strategies. This assignment can also be given from the

book. Problems will be supplement where necessary.

Assessment: An assessment will be taken as the students turn in their assignments. This

assessment will be used to directly influence what the teacher reteaches during the review day.

Reflection: To be completed after lesson is taught.

Day 7: Review/Reteach if necessary

**This day is optional, but I feel it would be beneficial. The teacher should use his/her judgment

based on previous assignments to figure out if any reteaching is necessary. In most textbooks, there

are review problems in the back of the book. Those problems are an option for this day. Also,

students could correct any problems they have had in previous assignments. If this day is needed,

the teacher would assign the students to create real-life examples of the Pythagorean Theorem

problems, special right triangle problems, and trigonometric function problems. For the students

who use the review time, this assignment can be homework. For those who do not need the

additional review time, these problems could be completed during class time.

Day 8: Assessment day

Materials: tests for each student, calculators

Standards:

HS.GSRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Objectives: The learners will demonstrate their abilities to use the Pythagorean Theorem, special

right triangle rules, and trigonometric functions to solve right triangles.

Learning activities:

1. The teacher will ask if there are any ideas or questions the students need cleared up.

a. Any problems or reteaching will be done before the test.

2. The students will be complete a test about right triangles.

3. The test should include both full completion triangles as well as problems that require

students to use only one or a few strategies that were taught. The test should also contain

multiple (5 or so) real-life example problems so students understand that these skills are

necessary. (I did not actually create a test because I want to create a test that corresponds

somewhat to the text and the different real-life examples that are used when I am actually

teaching this unit)

Assessment:

The assessment will be taken when the tests are turned in and graded.

Reflection: To be completed after lesson is taught.

Day 2 – Special Right Triangles – in class exploration handout

45-45-90 Triangles

Do you see any patterns among your answers? State your idea below:

30-60-90 Triangles

Do you see any patterns among your answers? State your ideas below:

Length of each leg 1 2 3 4 5 10

Length of hypotenuse

Length of short side, a

1 2 3 4 10

Length of long side, l

Length of hypotenuse, h

45⁰

45⁰

X

Y

X

Find the length of each hypotenuse in the triangle at the

left. The table below will give you a variety of values for

the legs, X. Please simply each square root you find as a

value for Y.

Feel free to draw any additional triangles if needed.

In a 45-45-90 triangle, if the legs have a length of X, then the hypotenuse

has a length of

_______.

l

h

a

60⁰

30⁰

Find the lengths of the long leg and the hypotenuse in the 30-

60-90 triangle shown to the left. Use the ideas we have just

talked about in class regarding these special triangles. The

table below will give you values for the short leg, a. Please fill

in the rest of the table. Keep in mind that you may need to use

the Pythagorean Theorem to find the third side length.

You may draw additional triangles if necessary.

In a 30-60-90 triangle, if the short leg has a length of a, then the long leg has a length of

______, and the hypotenuse has a length of _______.

Day 5 Writing Assignment Prompt and Rubric:

Special Right Triangles:

Imagine you are teaching someone about the rules of finding the measure of sides in the two types

of special right triangles we discussed. You need to explain, in writing, what relationships exist and

how we apply those relationships to find the side lengths. You may use diagrams to help you with

the explanation, but you have to use words to explain your diagrams. Be sure to use correct

terminology, but please explain the terminology when needed.

Student Name: ____________________________________

Category 5 3 1 Score Main idea: the body of your message

The paper is clear and focused. It holds the reader’s attention.

The writer is beginning to define the topic.

The paper has no clear sense of purpose Important details and mathematical ideas are missing.

Organization The paper has thoughtful transitions that connect ideas. The sequencing is logical and effective.

The transitions in the paper sometimes work, and the sequencing follows some logic.

The connections between ideas are difficult to follow, and the sequencing needs work.

Word Choice The technical terms are used correctly and explained or clarified. The language contributes to understanding.

The technical terms are undefined, and the reader may wish for more clarification.

The language is too basic or too technical. Reader may feel very confused and need more explanations.

Grammar The sentences were well constructed. There were no grammatical errors.

There were a few grammatical errors, but they did not take away from the flow or meaning of the piece.

The grammatical errors made the piece difficult to follow and understand.

Presentation The paper was neat and easy to read. The diagrams, if any, aided in the explanations.

The paper was fairly neat, but sometimes difficult to read. The diagrams mostly aided the explanations.

The paper was difficult to read due poor writing or diagrams.

Total: ____/25 points