Geometric Properties and Practical Drawings

GHSGT-Math-Obj. 30-Geometric Properties

INTERVENTION

LESSONS

FOR GHSGT

MATH OBJECTIVE 30

Geometric Properties and Practical Drawings

Geometric Properties and Practical Drawings

(Lesson 1 Circles)

GHSGT Objective:

#30 Applies geometric properties, such as the sum of the angles of a polygon property, percent of area of a circle determined by the central angle measure in a pie chart, or parallel sides and angle relations for parallelograms, to practical drawings.

Warm-up / Activator

It is very important that all students be able to use geometric properties and relationships to solve problems in mathematics and in everyday life. Write the colors blue, red, pink, and purple on the board and allow each student to place a tally mark next to his or her favorite color. Determine the number of votes for each color and what percent of the class voted for each color. Discuss whether a circle graph could be used to display the results of the data collection?

Mini Lesson

The central angle has its vertex located at the center of the circle.

Angle A is a central angle of circle A.

When a circle graph is created, the sum of the measure of all of the central angles must be 360°. Each central angle also represents a percentage of the circle and these percentages must add up to 100%.

Example: Suppose the following data represents the results of a survey of 2000 participants asked to identify their favorite vacation location:

Survey of Most Popular Vacation Locations
Miami, Florida 600
Las Vegas, Nevada 450
New York, New York 750
O’ahu, Hawaii 200

In order to represent the results of the survey using a circle graph we need to determine what percent of the total each location represents. In order to do this we divide the number for each location by the total number of participants and write the fraction as a percent. The measure of the central angle associated with each of the four locations is determined by multiplying the fraction represented by each location by 360°.

Miami, Florida and

This means that the 600 people who reported Miami, Florida as their favorite vacation location account for 30% of the 2000 participants surveyed. It also means that the central angle associated with this location in a circle graph of the data has a measure of 108°.

Work Session (Allow students to work in pairs)

Use the provided data to find the percent of participants who selected the other locations as their favorite vacation spot as well as the measure of the central angle associated with each. Then use the information to draw a circle graph to represent the data.

Vacation Location / Number of People / % of Total / Central Angle
Miami, Florida / 600 / /
Las Vegas, Nevada
New York, New York
O’ahu, Hawaii

A dartboard has four wedges with the following color percentages: 35%, 10%, 40%, and 15%.

Find the measure of the central angle for each color.

Choose four colors, draw and label the dartboard

with the given color percentages. Share your dartboard

with the class.

The hands on a clock create many different central angles as time passes. How would you determine the measure of the central angle formed at 1:00? What percent of the clock face is represented by the hands at 1:00?

Determine the percent of the clock face represented by each of the following times and the measure of the central angle formed.

a. 3:00b. 5:05c. 6:00

Draw a circle graph to represent how Maria spent the last 24 hours.

Maria’s activities / Time allotted (in hours)
Sleeping / 9
Eating / 2
Physical activity / 6
Shopping / 3
Relaxation & Pampering / 4

Closure

Allow students to share their circle graphs with the class. Have students discuss what they learned today and how circle graphs are used to display data. Ask them to search and find examples of circle graphs in print media bring them to class to share.

Geometric Properties and Practical Drawings

(Lesson 2 Polygons)

GHSGT Objective:

Warm-up / Activator

Take out a sheet of paper and list the numbers 1-10 vertically. Now write the name of each number using as many different languages and forms as possible. Share your list with the class. Who was able to complete the most lists for all numbers 1-10? What number was translated into the most languages and forms?

Mini Lesson

A polygon is a closed figure with sides that are line segments. Polygons are named according to the number of sides and angles they have. The intersection of two sides of a polygon is known as a vertex and the lines drawn by connecting nonadjacent vertices are known as diagonals.

A polygon is a regular if all of the angles are congruent and all of the sides are the same length. The polygon drawn above is a regular hexagon.

Polygons may be convex (the line drawn from any two vertices lies within the polygon). If the polygon is not convex, it is concave.

Convex Concave

a.b. c.d. e.

Work Session (Allow students to work in pairs)

Explain why each of the following is not a polygon.

Complete the following table for polygons with 3, 4, 5, …10 sides.

Name of polygon / Picture of Polygon / Number of sides / Number of vertices / Number of diagonals

Discuss any patterns you see in the table. From what language do the prefixes used to name the polygons originate?

Determine whether each polygon is regular or not, convex or concave, and classify the polygon according to the number of sides it has.

a.b.c.d.

Finally, mark each question either true or false. If false, explain why.

TrueFalse

e. All quadrilaterals have four vertices.□ □

f. All polygons have at least three sides.□ □

g. A heptagon has 6 sides.□ □

h. A nonagon has nine diagonals.□ □

i. The lengths of all the sides of a regular □ □

polygon are congruent.

j. All polygons have twice as many diagonals□ □

as number of sides.

Closing

Allow students to share their answers. Ask leading questions to be sure that students understand the concepts taught today. Have each pair of students deliver a 1-minute summation of what they learned today.

Geometric Properties and Practical Drawings

(Lesson 3 Sum of Angles of a Polygon)

GHSGT Objective:

Warm-up / Activator

Very often company logos, safety signs, and other architectural designs are made using polygons. Discuss where you have seen polygons used in this way. Imagine you are in a contest to create a logo for the math club at your school. The rules state that at least three polygons must be used in the logo. Draw the logo that you think best represents a math club. Share your results with the class and explain why you chose the polygons that you used.

Mini Lesson

The sum of the measure of the interior angles of a polygon can be found using the formula 180° (n - 2) where n is the number of sides of the polygon. Do you recall what 180° and (n – 2) represents? Share your answers with the class.

Consider the pentagon. By drawing the diagonals, we observe three non overlapping triangles. Therefore, the sum of the measure on the interior angles of the pentagon is. If the pentagon is regular (all sides are congruent and all angles are congruent), the measure of each of the five congruent angles is

Consider a polygon whose interior angles have a sum of 1260°. It is possible to use our formula to find the number of sides for the polygon.

Therefore, the polygon has 9 sides and can be classified as a nonagon.

Work Session

Use the formula to find the sum of the interior angles of the following polygons

a. b.c.

Determine the number of sides for a polygon with the gives interior angle sum. Classify the polygon according to the number of sides it has.

d. 1980°e. 2520°f. 1620°

Now find the measure of each angle of the following regular pentagons.

g. regular hexagonh. regular decagoni. regular 15-gon

The polygon is regular. What is the measure of ?

j.k.l.

Closure

Have students discuss where they have observed polygons used in real life situations. Ask leading questions to assess student understanding of the concepts covered in this lesson. Have students share what they learned today.

Geometric Properties and Practical Drawings

(Lesson 4 Parallelograms)

GHSGT Objective:

Warm-up / Activator

What is a quadrilateral? There are many different kinds of quadrilaterals. The five most common are the square, the rectangle, the rhombus, the parallelogram and the trapezoid. Make a list of their similarities and differences and share your list with the class. Use a Venn diagram to show how these quadrilaterals are related. Which sets of the common quadrilaterals are mutually exclusive?

Mini Lesson

Use the graph paper to draw a parallelogram and label the vertices M, A, T, and H (counter clockwise from the top right corner). Use appropriate math tools to find the measure of opposite angles and opposite sides. Use your results to complete each of the following conjectures:

Conjecture: The opposite angles of a parallelogram are ______.

Consecutive angles are angles that share a common side. Examine the sum of the measures of each pair of consecutive angles. Did you find that the sum is the same for both pairs? What is the sum?

Conjecture: The consecutive angles of a parallelogram are ______.

Conjecture: The opposite sides of a parallelogram are ______.

Now draw the diagonals of your parallelogram and label the point of intersection S. Compare MS and TS as well as AS and HS. What does this tell you about the relationship between the diagonals?

Conjecture: The diagonals of a parallelogram ______each other.

Work Session

Decide whether you are given enough information to decide that WORK is a parallelogram. Explain your reasoning.

a. WSO ≡KSR, WSK ≡ OSRb. ∆WKS ≡ ∆RSO

c. WO=OK, KR=OKd. WO=KR, WK=OR

Now complete each of the following problems related to parallelograms.

e. What is the measure of OKR? f. Three vertices of a parallelogram have coordinates (0, 1), (3, 7), and

(4, 4). What are the coordinates of the fourth quarter vertex?

Determine the value of m and n so that the quadrilateral is a parallelogram.

g. h.

Find the length of side OR if the perimeter of parallelogram WORK if its perimeter = 116m and WO = 20m

Finally, Show that (2, -1), (1, 3), (6, 5) and (7, 1) are the vertices of a parallelogram by verifying that opposite sides have the same slope, or by verifying that opposite sides have the same length.

Closing

Have students share their solutions. As the final problem is discussed have students consider other strategies that can be used to show that a quadrilateral is a parallelogram. Have students share with the class what they learned today.

GHSGT Assessment (Standard #30)

Name______

1. What is the measure of ?

a. 53°b. 57°c. 110°d. 70°

2. What is the measure of ?

a. 53°b. 57°c. 110°d. 70°

3. In parallelogram ABCD, is parallel to _____.

a. b. c. d.

4. In parallelogram ABCD, is congruent to _____.

a. b. c. d.

5. Find the value of x in the parallelogram below.

a. 40°b. 83°c. 33°d. 120°

6. What is the value of y in the parallelogram below?

a. 75°b. 105°c. 15°d. 165°

7. Identify the polygon by the number of sides.

a. nonagonb. hexagonc. octagond. heptagon

8. Determine whether the polygon below is regular or not regular,

and classify it as convex or concave.

a. regular, concaveb. not regular, concave

c. not regular, concaved. regular, convex

9. Describe the polygon.

a. equilateral and equiangularb. not equilateral or equiangular

c. equiangulard. equilateral

10. Identify a diagonal in the octagon .

a. b. c. d.

11. Describe the polygon.

a. octagonb. decagonc. pentagond. heptagon

12. Find the sum of the measure of the interior angles for the hexagon.

a. 720°b. 1080°c. 900°d. 540°

13. What is the measure of the missing angle?

a. 110°b. 150°c. 120°d. 140°

14. Find the sum of the interior angles of a polygon with 14 sides.

a. 2160°b. 166°c. 2520°d. 360°

15. Find the measure of an interior angle of a regular pentagon.

a. 72°b. 540°c. 18°d. 108°

16. Refer to the data in the table below.

a. A line graph could be used to display the data.

b. A circle graph could not be used to display the data because there

must be more than three parts.

c. A circle graph could be used to display the data.

d. A circle graph could not be used to display the data because the

percents do not add up to 100%.

17. The circle graph below shows what kind of media kids aged 8 to 18

would bring to a desert island. What are the two most popular choices?

a. CDs/radio and TVb. computer/internet and video games

c. video games and TVd. CDs/radio and computer/internet

18. For the circle graph below, what did most students choose as their favorite candy?

a. chocolateb. sour candyc. hard candyd. other

19. According to the circle graph below, which of the following is false?

a. Ten percent of the students said volleyball was their favorite sport.

b. More people chose football than baseball as their favorite sport.

c. The sport that was chosen most often as the favorite sport was basketball.

d. Soccer was chosen by most students as their favorite sport.

20. The circle graph shows how satisfied Americans are with the current steps that are being taken to protect the environment. This graph shows that most Americans ______.

a. have no opinionb. are very satisfied

c. are not too satisfiedd. are somewhat satisfied