LENGTHS AND ANGLES

Teacher Edition

List of Activities for this Unit:

ACTIVITY / STRAND / DESCRIPTION
0 –Pre-assessment for Geometry terms / ME / Vocabulary grid for students to see what they know
1 – What’s my size / ME / Measuring items
2 –Triangle Classification / ME / Identifying different types of triangles
3 – My Friend the Protractor / ME / Using a protractor to measure lengths and angles
4 – Yeah, I’ve got an Altitude / ME / Exploring triangle altitudes
5 – 30º 60º 90º triangles / ME / Using Pythagorean theorem to develop rule for 30º 60º 90º triangles
6 – From the 60s / ME / Looking for number patterns from a 60 degree angle
7 – I’m only 45 / ME / Using Pythagorean theorem to develop rule for 45º 45º 90º triangles
8 – Lengths and Angles Practice Problems / ME / Applying skills on lengths and angles
COE Connections / Bead design #1
Kitchen Work Triangle
MATERIALS / Ruler
Protractor
Scissors
Graph Paper
30º-60º-90º triangles for group work (one per group Pbm #28)
Warm-Ups
(in Segmented Extras Folder)

Vocabulary: Mathematics and ELL

accurate / compare / estimate / modifications / related
acute / conjecture / grid / obtuse triangle / right triangle
acute triangle / cross-section / hypotenuse / opposite / scalene triangle
altitude / cylinder / interior / perimeter / sealant
appropriate / determine / isosceles triangle / perpendicular / vertex
characteristics / diagonally / lateral surface area / precision
classification / encountered / linear dimensions / predict
classify / equilateral triangle / measure / protractor

Essential Questions:

  • What are appropriate units of measures?
  • What is meant by “exact value”?
  • How do you use measurement tools appropriately?
  • How do you categorize triangles by characteristics?
  • What properties of triangles such as equilateral, isosceles and right triangles assist in determining solutions to problems?
  • What is meant by classifying triangles?
  • What is a 30°-60°-90° right triangle?
  • What is a 45°-45°-90° right triangle?
  • How does a change in one linear dimension affect the area?
  • How are units of measure converted within the US system?
  • How can a conclusion be supported using mathematical information and calculations?
  • How do you find the volume of a triangular prism?

Lesson Overview:

  • Before allowing the students the opportunity to start the activity: access their prior knowledge with regards to how to use measurement tools appropriately (i.e. rulers and protractors). Access their prior knowledge regarding how to find the volume of a triangular prism, equilateral triangles and how the special properties of an equilateral triangle can be used to assist in solving a problem. Allow for collaborative time for the students, possibly activities around the room, discuss with the students.
  • How to use the Pythagorean Theorem to find missing lengths and to identify right triangles.
  • What mathematical information should be used to support a particular conclusion?
  • 7.48 gallons = 1 cubic foot
  • How will the students make their thinking visible?
  • Use resources from your building: rulers, protractors, scissors, extra supplemental material, graph paper

Performance Expectations:

5.3.BIdentify, sketch, and measure acute, right, and obtuse angles.

5.3.CIdentify, describe, and classify triangles by angle measure and number of congruent sides.

5.3.FDetermine the perimeters and areas of triangles and parallelograms.

8.2.F Demonstrate the Pythagorean Theorem and its converse and apply them to solve problems.

8.4.CEvaluate numerical expressions involving non-negative integer exponents using the laws of exponents and the order of operations.

G.3.CUse the properties of special right triangles (30°–60°–90° and 45°–45°–90°) to solve problems.

Performance Expectations and Aligned Problems

Chapter 13“Lengths and Angles” Subsections: / 1-
What’s My Size / 2-
Triangle Class-ification / 3-
My Friend the Pro-tractor / 4-
Yeah, I’ve Got Altitude / 5-
30°-60°-90° triangles / 6-
From the 60’s / 7-
I’m only 45 / 8-
Practice Problems
Problems Supporting:
PE 5.3.B≈ 5.3.C / 4 - 17 / 20, 21
Problems Supporting:
PE 5.3.F / 19 (partially)
Problems Supporting:
PE 8.2.F / 24 / 27 – 33 / 36 / 42, 43, 45, 47, 49, 51, 52 / 54 - 60
Problems Supporting:
PE 8.4.C / 24 / 27 -29, 32, 34, 35 / 42, 43, 45, 47, 51 / 54 - 60
Problems Supporting:
PE G.3.C / 27 -33 / 34 – 41 / 42 – 53 / 54 – 56, 59, 60

Assessment: Use the multiple choice and short answer items from Measurement and Geometric Sense that are included in the CD. They can be used as formative assessments attached to this lesson or later when the students are being given an overall summative assessment

For the vocabulary table on the next page you may want to have the students revisit and refine their understanding of each of the terms as they work through this lesson.

Pre-Assessment  Geometry Vocabulary

List everything you know (write a definition, draw a picture, etc.) about each geometry vocabulary term.

WHAT’S MY SIZE?

Activity Goals:

  • Determine a unit of measurement for an item and accurately measure the item.
  • Explain the reasons for differences in estimating and actual measurements.

1. Look around the classroom for the following items. Estimate (educated guess) the length of the following items, without using any measuring tools, and enter those estimates in the table. Include the unit of measure you were thinking of when you estimated.

Object / Estimated / Actual / Difference
Whiteboard
Height of the door
Length of your table or desk
Length of an unsharpened pencil
Diameter of a penny

2. Why did you choose the units of measure that you chose? How accurate do you think your measurements are? Support your answer using words.

Answers will vary: for example: smaller measurement units are used to measure small items; a penny could be measured in centimeters or inches. Bigger objects can be measured with bigger units: for example: the height of the door and the whiteboard were measured in feet, yards, or meters since those measure larger units.

3. Measure the items that you previously estimated. Compare your estimates to the actual measurements. Place your answers in the table. Calculate the difference between the actual and estimated measures. List some possible reasons for the difference between the actual and estimated measures.

Answers will vary______

TRIANGLE CLASSIFICATION

4. These are acute triangles These are not acute triangles

Write a definition for an acute triangle.

An acute triangle is a triangle

with three acute angles -OR-

A triangle with three angles measuring less than 90.

5. These are right triangles These are not right triangles

Write a definition for a right triangle.

A right triangle is a triangle with one right angle -OR- a triangle with one angle measuring 90º.

6. These are obtuse triangles These are not obtuse triangles

Write a definition for an obtuse triangle.

An obtuse triangle is a triangle with one obtuse angle -OR- a triangle with one angle measuring greater than 90º and less than 180º.

7. These are scalene triangles These are not scalene triangles

Write a definition for a scalene triangle.

A scalene triangle is a triangle that has three unique length sides…no two sides have the same length.

8. These are isosceles triangles These are not isosceles triangles

Write a definition for an isosceles triangle.

An isosceles triangle is a triangle that has at least two sides congruent. (An isosceles triangle has at least two angles of the same measure.) Equilateral triangles are a special case of isosceles triangles.

9. These are equilateral triangles These are not equilateraltriangles

Write a definition for an equilateral triangle.

An equilateral triangle is a triangle that has all three sides congruent.

10. Review the triangles given below this table. Place the letter of each triangle in all the appropriate

boxes in thetable.

Acute / Obtuse / Right
Equilateral / E
Isosceles / D & F / B / A
Scalene / E / C / G

ABC

D

E F G

11. Can an acute triangle also be scalene? ______YES____ Why? ______

See triangle “E” above.

12. Can a right triangle also be scalene? ______YES_____ Why? ______

See triangle “G” above.

13. Can an obtuse triangle also be right? ______NO_____ Why? ______

90º (a right angle measure) + an angle > 90º (an obtuse angle measure) + a third angle measure is > 180º. The maximum amount of angle measures (degrees) in a triangle in a plane is 180º.

14. Can an obtuse triangle also be acute? ______NO______Why? ______

An acute triangle has all three angles less than 90º; therefore, no obtuse angles are possible.

15. Can a scalene triangle also be isosceles? _____NO_____ Why? ______

“At least two sides congruent” in the definition of isosceles triangle means a scalene triangle cannot be isosceles. Recall a scalene triangle has three unique side lengths.

16. Can a right triangle also be isosceles? ______YES____ Why? ______

See triangle “A” above.

17. Can an equilateral triangle also be isosceles? __YES_____ Why? ______

All equilateral triangles have at least two congruent sides. All equilateral triangles are isosceles, but not all isosceles triangles are equilateral triangles. An equilateral triangle is a more restrictive classification than an isosceles triangle.

MY FRIEND THE PROTRACTOR

Activity Goals:

  • Use a protractor to accurately measure and draw angles, with a precision of ±3 degrees.
  • Classify triangles.

18. As a class, practice drawing angles with a precision of ±3 degrees. Use a protractor to draw the following angles: 25º, 72º, 147º, and 90º.

Many students need practice using a protractor these problems are to assess their proficiency measuring angles.

19. Measure and label, to the nearest tenth of a centimeter, each side of the triangles below; put the side length measures on the triangles.

Measure, to the nearest degree, each angle of the triangles; label the angles with their appropriate angle measurements.

A. B.

C.D.

  1. Classify each triangle from problem 19 according to the size of the angles of the triangles.

Triangle / Acute, Obtuse, or Right / Equilateral, Isosceles, Scalene
A / Obtuse / Scalene
B / Acute / Isosceles
C / Right / Scalene
D / Obtuse / Scalene

YEAH, I’VE GOT AN ALTITUDE!

21.

The perpendicular distance from the base of a triangle to the vertex opposite the base is called the altitude of the triangle. The length BD for each triangle above is the altitude of the triangle.

Measure and label all the sides (in centimeters) and angles (in degrees) in each triangle.

  1. What is the measure of angle ADB in all 3 triangles? ______90º______
  1. What type of triangles are the 3 above? And why do you know this? ______

From left to right the triangles are:

Equilateral because all sides are equal (2.8 cm) and therefore congruent.

Right Scalene because it has a right angle and all sides are different lengths.

Obtuse Scalene because the angle 133º is obtuse and the lengths of the sides are all different.

  1. Are there any triangles that do not have an altitude? If true draw them, if not true tell why.

NO,

The perpendicular distance from the base of a triangle to the vertex opposite the base is called the altitude of the triangle. Two points determine the base, the third point is not on the base line segment otherwise no triangle is formed, and the distance from the third point to the base line segment is the altitude of the triangle determined by the three points. So if you have a triangle you are guaranteed an altitude.

NOTE: any side of a triangle can be used as the base of the triangle. This means orientation of the triangle does not matter for our altitude discussion.

______

22. Triangle ABC is an equilateral triangle with altitude labeled a.

a. What is the length of m?___2 units because all sides are equal in length and m = 2___

b. What is the length of n? ___1 unit because it is one half of m = one half of 2=1

23. Explain how you find the value of the altitude a?

Use the Pythagorean Theorem.

______

24.Find the length of a.

Show your work using words, numbers and/or diagrams. The conventional form of the Pythagorean Theorem is a2 + b2 = c2 -OR- in words (one leg)2 + (other leg)2 = hypotenuse2

12 + a2 = 22

a2 = 22 - 12

a = (Square root both sides.)

a =

25. a. Without measuring, what is the measure of angle DBC?

30º

b. Without measuring, what is the measure of angle DCB?

60º

c. Without measuring, what is the measure of angle CDB?

90º

26. From 25 a, b, c, above what are the angle measures of triangle BCD? This type of triangle is often encountered in real world applications and is referred to by the angle measuresof the triangle BCD. (The triangle’s name is the answers and order of a, b, c from problem 25 above. )

30º-60º-90º

______

30°-60°-90°Triangles

27. Shown below is a 30°-60°-90° triangle that has a legof length 1 and the hypotenuse of length 2 units. Call this triangle T1. (Note: many calculator answers are approximations. Do not use your calculator for these problems! Find the exact answer algebraically.)

For any right triangle:

A leg is the name foreither of the sides of a right triangle that form the right angle.

A hypotenuse is the name of the side opposite the right angle.

a. What is the name of the process that allows you to
calculate the length of the other leg of this triangle?

The Pythagorean Theorem______

b. What is the formula? ____

(one, short, leg)2 + (other, long, leg)2 = hypotenuse2

c. Calculate the exact length of the leg labeled x.

Show your work using words, numbers and/or diagrams.

12 + x2 = 22

x2 = 22 - 12

x = (Square root both sides.)

x =

With a partner or a small group (no more than 4 students), examine the characteristics of the 30°-60°-90° triangle you are given. Compare your triangle to T1find all the numeric values of your triangle and share your results with the class.

28. Now consider the 30°-60°-90° triangle, T2, which has one leg of length 2 units and the hypotenuse of length 4 units.

  1. How is thisT2 triangle related to the T1triangle above?

_Corresponding sides are twice as long.______

______

b. Calculate the exact length of the leg labeled g. ______

Show your work using words, numbers and/or diagrams.

22 + g2 = 42

g2 = 42 - 22

g = (Square root both sides.)

g = = = 2

c.How are the hypotenuses for triangles T1 and T2 related?

T2 is twice the length of T1

______

29. Now consider the 30°-60°-90° triangle, T3, which has a hypotenuse of six and a short leg of three.

a. How is this triangle related to T1?

All lengths of T3 are three times those of T1.

______

b. Calculate the exact length of the second leg. ______

Show your work using words, numbers and/or diagrams.

32 + (m)2 = 62

(m)2 = 62 - 32

(m) = (Square root both sides.)

(m) = = = 3

c. How are the hypotenuses for triangles T1 and T3 related?

The hypotenuse of T3 is three times the length of the hypotenuse of T1.

______

30. Now consider the 30°-60°-90° triangle, T4, which has a hypotenuse of eight and a short leg of

four.

a. How is this triangle related to T1?

All lengths of T4 are four times those of T1.

______

b. Calculate the exact length of the second leg. ___4_____

Show your work using words, numbers and/or diagrams.

I see a pattern!

c.How are the hypotenuses for triangles T1 and T4 related?

The hypotenuse of T4 is four times the length of the hypotenuse of T1.

______

31. Fill in the table using your results from 27 - 30.

Table I

Triangle / Length of One (short) Leg / Length of the Other (long)Leg / Length of theHypotenuse
T1 / 1 / 1 / 2
T2 / 2 / 2 / 4
T3 / 3 / 3 / 6
T4 / 4 / 4 / 8
Tn / n / n / 2n


Practice:

32. You are given a30°-60°-90° trianglewith the shortest leg having a length 38 units. Determine the length of the longest leg and of the hypotenuse. ______38 units______

n = 38 from the table in problem 31 above.

33. The value of X is 30º. What is the length of XY and YZ?

a. XY = _10units__b. YZ = _10 units_ c. measure of angle Z = 60º__

Show your work using words, numbers and/or diagrams.

FROM THE 60’S

34. Shown here is a 30°-60°-90° triangle that has one leg of length units. The measure of angle A is 60°.

Calculate the length of the other two sides.

Show your work using words, numbers and/or diagrams.

y = 1 unit

z = 2 units

From the table of problem #31.

35. Shown here is a 30°-60°-90° triangle that has one leg of length 8 units. The measure of angle A is 60°.

Calculate the length of the other two sides.

Show your work using words, numbers and/or diagrams.

y = 8 unit

z = 16 units

From table Iof problem #31…n = 8

36. Calculate the areaof each of the triangles from Table I and record your results in Table II.

The areaof a triangle is the number of unit squares that can be contained within the triangle.

Table II

Triangle / Length of the Short Leg / Length of the Long Leg 2 / Process / Area of triangle
T1 / 1 / 1 / (1• units2)÷ 2 = units / units2
T2 / 2 / 2 / (4• units2) ÷ 2 = 2 units2 / 2 units2
T3 / 3 / 3 / (9• units2) ÷ 2 = units2 / units2
T4 / 4 / 4 / (16• units2) ÷ 2 = 8 units2 / 8 units2
Tn / n / n / (n2 units2) ÷ 2 = units2 / units2

37. Use your table to determine the exact area for a 30°-60°-90° triangle with the shortest leg measuring 6 units.

(36• units2) ÷ 2 = 18 units2

38. a. Compare the linear dimensions(lengths of the sides) of triangles T1 and T2.

Triangle: short leg, long leg, hypotenuse AREA

T1: 1 2 units2

T2: 2 2 4 2 units2

(Each length is twice as long and the area is 4 times larger.)

b. Compare the areas of triangles T1 and T2.

See part a above under area.

39. a. Compare the linear dimensions of triangles T1 and T3.

Triangle: short leg, long leg, hypotenuse AREA

T1: 1 2 units2

T2: 3 3 6 units2

(Each length is three times longer and the area is 9 times larger.)

b. Compare the areas of triangles T1 and T3.

See part a above under area.

40. a. Compare the linear dimensions of triangles T1 and T4.

Triangle: short leg, long leg, hypotenuse AREA

T1: 1 2 units2

T2: 4 4 8 units2

(Each length is four times longer and the area is 16 times larger.)

b. Compare the areas of triangles T1 and T4.

See part a above under area.

41. Make a conjecture(educated guess)of how the area changes when the linear dimensions are changed. Be specific by giving the exact details of your conjecture.

Test your conjecture by using a 30º-60º-90º triangle with a short side of length 14 units and predict the area compared to the area of T1.

If each of the dimensions of a 30º-60º-90º triangle are multiplied by the value n, then the area of the 30º-60º-90º triangle will be n2times the area of the original triangle.

Area of T1 is units2 = (1•) ÷ 2

Area of T14: (14•14 ) ÷ 2 = units2

NOTE: units2 ,where n = 14 units, isn2 times the area of the original,T1, triangle.

Was your prediction correct?

YES

Do you think your conjecture will work with any type of triangle?

YES

How could you “test” your answer to the last question above?

I would generalize my calculations.

E.G. (length of base)•(length of altitude) = Area of any triangle

(n•length of base)•(n•length of altitude) =

n2 •(length of base)•(length of altitude) = n2 •Area of any triangle

True for all triangles, including 30º-60º-90º triangles.

I’M ONLY 45

42. On the grid below is a 45°-45°-90° triangle with legs of length 1 unit. Call this triangle T1.

a. What is the name of the process that allows you to calculate the length of the hypotenuse

of this triangle? The Pythagorean Theorem

b. Calculate the exact length of the hypotenuse of the triangle.

Show your work using words, numbers and/or diagrams.

Let h = hypotenuse

12 + 12 = h2

2 = h2 (Square root both sides)

= h NOTE: the actual answer is ±, but there is no length of -.