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Unit 6 – Geometry Length of section

6-1 Angles 4 days

6-2 Polygons 3 days

6-3 Triangles 3 days

6-4 Congruent and Similar Figures 4 days

6.1 - 6.4 Quiz 1 day

6-5 Pythagorean Theorem 6 days

6-6 Volume 2 days

6-7 Transformations 4 days

Test Review 1 day

Test 1 day

Cumulative Review 1 day

Unit Project 2 days

Total days in Unit 6 – Geometry = 32 days

287

Review QuestionHow do you graph the point (-5, 4)? Left 5, Up 4What quadrant is that point located? II

DiscussionWhat do you think about when I say the word Geometry? Angles, Shapes, Area, Perimeter SWBAT define basic geometry termsSWBAT estimate an angle measureSWBAT draw an angle given a measure

DefinitionsNotice how things build in nature: Atoms → Cells → Organs → BodyNotice how things build in Algebra: Variable → Combining → Solving What is the building block that starts Geometry? Points How do things build in Geometry?

Points → Segments → Rays → Lines → Plane

Point - location (no shape or size) •A ex) here

Line segment – collection of points with defined length A B ex) wood beam

Ray – segment that extends infinitely in 1 direction ex) laser A B

Line – segment that extends infinitely in both directions ex) #’s, spaceA B

Angle – two rays (say angle BAC or CAB)

Vertex – meeting point of two rays (point A)

Type of Angles

Straight – 180°

Right – 90°

288

B

A C

Section 6-1: Angles (Day 1)

Obtuse - angles > 90°

Acute - angles < 90°

Perpendicular – two lines that intersect at right angles

Parallel – two lines that do not intersect

If a line is 180°, what would the angle measure of each of the dotted lines be?

Example 1: Estimate the angle measure.It’s about 40°. It is more than 0° but less than 45°. It is real close to 45°.

Example 2: Estimate the angle measure.It’s about 160°. It is more than 135° but less than 180°. It is closer to 180°.

289

0°

135°

180°

90°

45°

You Try!1. Draw an acute angle; label it BAD – estimate measure2. Draw an obtuse angle; label it EMO – estimate measure3. Draw a 20° angle; label it OMG 4. Draw a 100° angle; label it LMK

What did we learn today?

Label each angle as acute, obtuse, right or straight. Then estimate each angle measure.

1. 2. 3.

4. 5. 6.

7. 8. 9.

Draw an angle for each of the following. Label the angle MNO.

10. 50° 11. 120° 12. 15°

13. 145° 14. 82° 15. 65°

Draw an angle for each of the following. Label the angle CAB.16. acute angle 17. obtuse angle 18. right angle

19. Write a complete solution to the problem. Your solution should include calculations, explanations, units, and a summary sentence. When the clock reads 8:50, the hands form an acute angle. What is the next time the clock will form an obtuse angle?

290

Section 6-1 Homework

Review Question1. Estimate the following angle.

130°

2. Draw a 15° angle.

DiscussionYesterday we estimated the measure of different angles. Today we will find the exact measurements using a protractor. As we have seen before sometimes an estimate is good enough. But sometimes we need an exact answer. For example, a carpenter needs to know an exact measurement…

Has anyone ever seen wood going around the top of a dining room? What is it called? Crown moldingIn order to install this crown molding, you must cut pieces of wood at certain angles to make the corner pieces fit. If you do not cut them at the precise angle, the two pieces of wood won’t fit. To do this, you use a special tool. It is called a miter saw.

Today, we will be trying to find precise angles. We will use a special tool. It is called a protractor.

SWBAT measure an angle using a protractorSWBAT draw a specific angle given a protractor

Example 1: Draw an acute angle and label it BAC. Estimate the angle measure. Remember the drawing from yesterday. Then use a protractor to measure the angle. Compare.

Notice that the protractor has two sets of numbers. How do we know which number to use? Depending on if the angle is acute or obtuseHow do you know that the angle is 70° not 110°? Because the angle is acute, and acute angles are < 90°

Example 2: Draw an obtuse angle and label it BAC. Estimate the angle measure. Remember the drawing from yesterday. Then use a protractor to measure the angle. Compare.

How do you know that the angle is 120° not 60°? Because the angle is obtuse, and obtuse angles are > 90°

291


Example 3: Draw a 50° angle using estimation. Now get the exact measure using a protractor. Then construct one with the protractor. Start with a horizontal ray. Then place the protractor above the ray. Then put a mark above the 50° mark on the outside of the protractor. Finally, connect the starting point of the ray with the mark you just made. Notice that the protractor has two set of numbers. How do we know which number to use? How do you know that the angle is 50° not 130°? Because the angle is acute, and acute angles are < 90°

Example 4: Draw a 160° angle using estimation. Now get the exact measure using a protractor. Then construct one with the protractor. Start with a horizontal ray. Then place the protractor above the ray. Then put a mark above the 160° mark on the outside of the protractor. Finally, connect the starting point of the ray with the mark you just made. Notice that the protractor has two set of numbers. How do we know which number to use? Depending on whether the angle is acute or obtuseHow do you know that the angle is 160° not 20°? Because the angle is obtuse, and obtuse angles are > 90°

You Try!1. Draw an acute angle. Get its exact measurements.2. Draw an obtuse angle. Get its exact measurements.3. Draw a 15° angle.4. Draw a 122° angle.


Estimate each angle measure. Circle your estimation. Then find each angle measure using a protractor. Compare the two values.

1. 2.

3. 4.

292


5. 6.

Draw each angle by estimating. Then use a protractor to draw each angle. Label the angle BAC. Then compare the two angles.

7. 50° 8. 115°

9. 145° 10. 68°

11. 42° 12. 90°

13. Explain why there are two scales on the protractor. Explain how to use each scale.

14. What angle has the same measure on both scales?

293


Review Question1. What is 2x + 30 equal to when ‘x’ is 20? 70 2. Solve: 4x + 2 + 6x + 8 = 50 x = 4We did this back in Unit 3. You will need this skill today.We did this back in Unit 3. You will need this skill today. DiscussionWhat is the measure of BAC? 90°

What happens if I draw a ray that creates two separate angles? The two angles still have to add up to 90°.

What do we know about BAD and DAC? They add up to 90°.

What is the measure of BAC? 180°

What happens if I draw a ray that creates two separate angles? The two angles still have to add up to 180°.

What do we know about BAD and DAC? They add up to 180°.

SWBAT identify complementary and supplementary anglesSWBAT calculate complementary and supplementary angles

294

A

B

C

B A C

B

AC

D

B A C

D

DefinitionsComplementary – angles that add up to 90°

We say: BAD is complementary to DAC

Supplementary – angles that add up to 180°

We say: ABC is supplementary to DBC

Example 1: The two angles are supplementary. Find ABC.x + 60 = 180x = 120

Example 2: The two angles are complementary. Find each angle.

x + (x + 20) = 902x + 20 = 902x = 70x = 35 Now substitute the ‘x’ into both angles. The first angle is 35°. The second angle is 55° (35 + 20).

Example 3: The two angles are supplementary. Find each angle.

(x + 20) + (4x + 10) = 1805x + 30 = 1805x = 150x = 30 Now substitute the ‘x’ into both angles. The first angle is 50° (30 + 20) and the second angle is 130° (4(30) + 10).

295

A B D

C

x° x + 20°

x + 20°4x + 10°

60°

B

A

D

C

A B D

C

You Try!1. Draw complementary angles. Estimate each angle. Then label them.2. Draw supplementary angles. Estimate each angle. Then label them3. Draw complementary angles. Label the smaller one x + 10 and the bigger one 2x + 20. Then find each angle. x = 20 (the angles are 30° and 60°)4. Draw supplementary angles. Label the smaller one 2x + 15 and the bigger one 3x + 15. Then find each angle. x = 30 (the angles are 75° and 105°)


Classify each pair of angles as complementary, supplementary, or neither.1. 2. 3.

4. 5. 6.

7. 8. 9.

296

58º

41º 54º

1 212

1 2

1 2 126º 49º

160º 20º 49º

49º

32º


Find the value of x in each figure.10. 11. 12.

13. 14. 15.

16. 17. 18.

Each pair of angles is either complementary or supplementary. Find the measure of each angle.19. 20. 21.

x = 30, 90º and 90º x = 20, 80º and 100º

x = 10, 45º and 45º

297

xº xº 18º xº 49º

43º

xº86º 117º xº

xº66º

xº

22ºxº

xº 57º

3xºx + 60º 4x + 5º

2x + 25º

2x + 40º4x + 20º


Review QuestionWhat does complementary mean? Angles that add up to 90°What does supplementary mean? Angles that add up to 180° DiscussionHow would you describe a new song that you like? Bomb, Hard, Cool, Great Notice that there are many ways to say the same thing. The same is true in math.In algebra, we say that two things are equal. In geometry, we say that two things are congruent.

SWBAT locate congruent angles using the definition of vertical, corresponding, and alternate interior angles

DefinitionsCongruent – equal to; (in algebra we use = ; in geometry we use )

Parallel – two lines that do not intersect each otherThe following diagram is called two parallel lines cut by a transversal. We say that line L is parallel to line P. This is written as L | | P.

CONGRUENT ANGLES:vertical angles - form an X (1 and 4; 2 and 3; 5 and 8; 6 and 7)corresponding angles - form an F (1 and 5; 3 and 7; 2 and 6; 4 and 8)alternate interior angles - form a Z (3 and 6; 4 and 5)

*These relationships only exist if the lines are parallel.

Example 1: Find the other seven angles. Give a reason for your answer. For example, 4 is 100º because it is vertical to the 100 º angle.

(Supp. to 100 º)(Vertical to

(Alt. Int. to (Alt. Int. to (Vertical to

(Vertical to

298

1 23 4

5 67 8

100°

L

P

245 6

7 8

3

Example 2: Find the other angles. Give a reason for your answer.

60º or 120º (Supp., Vertical, Alt. Int.)

You Try!1. Find the other seven angles. Give a reason for your answer. 70º or 110º

2. Find the other seven angles. Give a reason for your answer. 85º or 95º

3. Draw supplementary angles. Label the bigger angle 3x + 20. Label the smaller angle 2x + 10. Find each angle. x = 30, 70º and 110º


1. Classify each pair of angles as complementary, supplementary, or neither. Then estimate each angle measure.a. b. c.

299

1 212

1 2

Section 6-1 Homework (Day

70°

85°

60°120° 120°120°60° 120°

60°60°

2. Find the value of x in each figure.a. b. c.

d. e. f.

3. Each pair of angles is either complementary or supplementary. Find the measure of each angle.a. b. c.

4. Find the measure of each of the remaining angles.a. b. c.

300


38º

xº86º 117º xº

xº66º

xº +10 x º 4x + 5º

4x + 5º

3x + 20º4x + 20º

50° 40°

100°

70°

Section 6-2: Polygons (Day 1)

Review QuestionWhat do corresponding angles look like? ‘F’What do alternate interior angles look like? ‘Z’ What do vertical angles look like? ‘X’

DiscussionWhat is a polygon? The students will say shape.Is a circle a shape? YesIs it a polygon? No

DefinitionsPolygon – closed figure formed by three or more line segments

You can use the classroom as an example of a three dimensional polygon when the door is closed. The classroom is not a polygon when the door is open. Is it a polygon if you open one of the closet doors? Yes, it is still a closed figure.

SWBAT name polygons with sides 3 through 8SWBAT distinguish between regular and irregular polygons

Give a reason as to why the shapes may or may not be polygons.

Naming Polygons

Regular Polygons – all angles and sides are congruentExplain why each of the polygons above are irregular. Triangle: different angles, Quadrilateral: different sides, Pentagon: different angles, Hexagon: different sides, Heptagon: different sides, Octagon: different angles

301

Yes, closed

Yes, closed

No, not line segments

No, open

No, open

Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon

You Try!1. Draw a regular version of each polygon.

Discuss how tick marks are used to show congruency.


On line 1 write the name of the polygon.If it isn’t a polygon, explain why not on line 2.On line 3 write regular or irregular.If it is irregular, explain on line 4.

1. 2. 3.

1. ______________ 1. ______________ 1. _______________2. ______________ 2. ______________ 2. _______________

3. ______________ 3. _____________ 3. _______________4. ______________ 4. _____________ 4. _______________

4. 5. 6.

1. ______________ 1. ______________ 1. _______________2. ______________ 2. ______________ 2. _______________

3. ______________ 3. _____________ 3. _______________4. ______________ 4. _____________ 4. _______________

302


7. 8. 9.

1. ______________ 1. ______________ 1. _______________2. ______________ 2. ______________ 2. _______________

3. ______________ 3. _____________ 3. _______________4. ______________ 4. _____________ 4. _______________


d. e. f.


x = 82, 92º and 88º x = 14, 33º and 57º x = 20, 75º and 105º

303


46º

xº82º

101º xº xº62º

xº + 10 x + 6 º 2x + 5º

3x + 15º

x + 55º4x + 25º

Review QuestionWhat is the difference between a regular and irregular polygon? Regular – all sides/angles congruent

DiscussionToday we will be using a new formula. This formula is going to allow us to calculate the sum of the interior angles of any polygon. I would like for us to come up with the formula together. So here we go:

1 x 180° = 180° 2 x 180° = 360° 3 x 180° = 540°

Notice the relationship between the number of sides and the number of triangles that you can draw inside each polygon. To find the sum of the interior angles of a polygon, subtract two from the number of sides then multiply by 180. The formula is written as follows: (n – 2) x 180°

SWBAT derive the formula for the sum of the interior angles of a polygonSWBAT apply the formula to calculate the sum of the interior angles of any polygonSWBAT calculate each angle of a regular polygon by applying the formula

Example 1: Draw a regular octagon. What do the interior angles add up to?(n – 2) x 180°(8 – 2) x 180°6 x 180° = 1080° What would each angle have to be? 1080°/8 = 135°

Example 2: Draw a 10 sided polygon. What do the interior angles add up to?(n – 2) x 180°(10 – 2) x 180°8 x 180° = 1440° What would each angle have to be? 1440°/10 = 144°

Example 3: What regular polygon has angles that are 108° each?Use guess and check to arrive at the answer.8 sides = 1080°/8 = 135°10 sides = 1440°/10 = 144° (answer is getting bigger; we need to guess smaller)5 sides = 540°/5 = 108°

You Try!1. Find the sum of the interior angles of a hexagon polygon. Then find the measure of each angle.sum = 720º one angle = 120º2. Find the sum of the interior angles of a regular 12 sided polygon. Then find the measure of each angle. sum = 1800º one angle = 150º3. What regular polygon has angles that are 156° each? 15 sided polygon4. What regular polygon has angles that are 162° each? 20 sided polygon

304

3 4

5



Find the sum of the interior angles for each regular polygon. Then find the measure of each angle.

1. Pentagon 2. Hexagon 3. Decagon

4. 15 sided 5. 12 sided 6. 80 sided

7. Heptagon 8. Octagon 9. 25 sided

10. Each angle in a regular polygon is 108°. What polygon is it?




14. Find the measure of each remaining angle.a. b. c.

305


44º

55° 40°

95°

72°


2

4

3

1

5

12

34

60°80°

100°x

120°

120°

120°

Review QuestionHow do you find the sum of the interior angles of a polygon? (n -2)/180

DiscussionYesterday, we talked about interior angles. What are interior angles? Angles on the inside.Today, we are going to talk about exterior angles. What are exterior angles? Angles on the outside.

In any polygon, the sum of the exterior angles is always 360°. You can visualize putting all five of these exterior angles together. They would form a circle (360°). This is the case for any polygon. Let’s make sure: Draw a square with its four exterior angles. Notice each interior and exterior angle would be 90°. They would total 360°.

Let’s do the same thing for a triangle with equal angles. Notice that each interior angle is 60°. This would make each of the three exterior angles equal to 120°. Therefore, they total to 360° as well.

SWBAT find the measure of an exterior angle

Example 1: Find ‘x’.

x + 60 + 80 + 100 = 360x + 240 = 360

x = 120°

306


x

110°

120° 110°

80°

50°

x

Example 2: Find the measure of each exterior angle in a regular hexagon.

The exterior angles add up to 360°. Since there are 6 angles:6x = 360x = 60Each exterior angle would be 60°


1. Find ‘x’.

a. b.

2. Find the measure of each exterior angle in the following regular polygons.

a. Triangle b. Pentagon

c. Octagon d. 15 sided

3. Find the sum of the interior angles for each regular polygon. Then find the measure of each angle.

a. Pentagon b. Hexagon c. Quadrilateral

d. 18 sided e. 22 sided f. 50 sided

4. Each interior angle in a regular polygon is 60°. What polygon is it?



7. Draw a regular and irregular hexagon.

307

Section 6-2 In-Class Assignment

Review QuestionWhat letters do you vertical, corresponding, and alternate interior angles form?‘X’, ‘F’, and ‘Z’ respectively

DiscussionWhat do we need in order to have a triangle? 3 sides, closed, adds up to 180°Can anyone write a sentence putting all of these ideas together?

Triangle – 3 sided polygon whose angles add up to 180°

SWBAT draw a triangle given its nameSWBAT name a triangle by its angles and sides

Discussion Some famous people are known by one name. For example, how many of you know who Lebron is?Can you name some other famous people who are known by one name?Beyonce, Obama, Eminem, Tiger

How many of you know who John in Junior High is?What else do you need to know? Last nameLet’s say we are talking about John Grabowski. What does his first name tell you? Gender/individualWhat does his last name tell you? His family/nationality

Triangles go by two names as well: a first name and last name.Triangles’ first names describe their angles.Triangles’ last names describe their sides.

Types of triangles – by angles

1. Acute – 3 acute angles

2. Obtuse – 1 obtuse angle

3. Right – 1 right angle

Hmmm?!?Why can you only have one obtuse angle in a triangle? It would add up to more than 180°. Why can you only have one right angle in a triangle? It would add up to more than 180°.

308

Section 6-3: Triangles (Day 1)

Types of triangles – by sidesHave the students form the particular triangle on their desk with rulers. Then draw them next to their definition.

1. Equilateral – 3 congruent sides

2. Isosceles – 2 congruent sides

3. Scalene – 0 congruent sides

Example 1: Name the triangle.

Right Scalene


Acute Equilateral


Obtuse Isosceles

You Try!Draw each triangle in your notes (with tick marks). Make each of the triangles on your desk using rulers.1. Right Isosceles2. Acute Scalene3. Obtuse Equilateral Can’t do it; demonstrate using meter sticks4. Acute Isosceles5. Obtuse Scalene6. Right Equilateral Can’t do it; demonstrate using meter sticks

What did we learn today? 309

50°

1. Find the missing measure in each triangle. Then classify the triangle as acute, right, or obtuse.

a. b. c.

2. Classify each triangle by its angles and its sides .

a. b. c.

d. e. f.

3. Find each angle.

a. b.

90º, 60º, 30º 80º, 55º, 45º

310


x°

x°

40°

70°

x° 35°

15°

x°

3x°

2x°

80°

x°

x + 10°

A

B1

C

Review QuestionWhat do the interior angles of a triangle add up to? 180°

DiscussionYesterday we talk about the interior angles of triangles. Today we are going to talk about how the interior angles are related to the exterior angles.

A + B = 1

Let’s try to figure out why? We can put some numbers in for the angles and see what happens.

Notice that the exterior angle has to be 150° based on it being a straight line. Notice that this is equal to the sum of the other two interior angles. This is true for any exterior angle. Any exterior angle is equal to the sum of the opposite two interior angles.

Let’s draw in the other two exterior angles so we can see how they are equal to the sum of the other two interior angles.

SWBAT find missing interior and exterior angles of a triangle

Example 1: Find the missing angle.

The ‘?’ is 130° because of our new rule. The exterior angle is equal to the sum of the two opposite interior angles of a triangle.

You can also do it the old school way (using knowledge from 6-1 and 6-2). You can find the ‘?’ by finding the missing angle in the triangle (50°). Then use your knowledge of straight lines and subtract that 50° from 180° to get 130°.

311

60°

30° ?90°

70°

60° ?

Section 6-3: Triangles (Day 2)

90°

60°

30° ?

120°

90°

Example 2: Find the missing angle.

The ‘?’ is 70° because of our new rule. The exterior angle is equal to the sum of the two opposite interior angles of a triangle.

You can also do it the old school way (using knowledge from 6-1 and 6-2). You can find the ‘?’ by finding the other missing angle in the triangle (30°) because of our knowledge of straight lines. Then calculate the ‘?’ by using the fact that the interior angles of a triangle add up to 180°.

Example 3: Find all of the angles in the triangle.

We know that 2x + x = 150° because of our new rule. So let’s solve that equation.2x + x = 1503x = 150x = 50

Therefore, the angles in the triangle are 50°, 100°, and 30°.

You Try!1. Draw a triangle with two angles that are 45° and 90°. What is the angle measure of the opposite exterior angle? 135°

2. Find ‘x’. 60°


312

80°

150°?

x

150°2x

95°

155° x

SHAPE \* MERGEFORMAT

1. Find the missing angle.

a. b.

c. d.

2. Find all of the angles in the triangles.a. b.

3. Draw a triangle with two angles that are 90° and 50°. What is the angle measure of the opposite exterior angle?

4. Draw a triangle with two angles that are 30° and 70°. What is the angle measure of the opposite exterior angle?

5. Draw a triangle. Label the angle measure of each interior angle. Now draw in the three exterior angles. Write in their angle measure. Show how each exterior angle is equal to the sum of the two opposite interior angles.

6. Find each missing angle in the triangle.a. b.

313

Section 6-3 Homework (Day

80°?

50° 20°40° ?

x85°

150° x 140°90°

2x + 5x

x125°3x 160°

x°

80°

x° 60°

x + 30° 2x + 30°

85°

x

x


Review QuestionWhat are the three possible first names of a triangle? Acute, Obtuse, RightWhat are the three possible last names of a triangle? Isosceles, Scalene, Equilateral

DiscussionHow do you get better at something? Practice

Today will be a day of practice. First, we will go to the boards to practice some problems. Then you will do a worksheet that has some problems from today as well as some review problems.

SWBAT to answer questions from the last two sections as well as review problems

You Try!1. What do the interior angles of an octagon add up to? 1080°2. What does each exterior angle of a pentagon equal? 72°3. Draw “the picture” (Two parallel lines cut by a transversal). Label the angles 1-8. Estimate angle one. Then find the other seven angles. Answers may vary.



1. Estimate every angle in each diagram.a. b. c.

2. Each pair of angles is either complementary or supplementary. Find the measure of each angle.

a. b. c.

314

Section 6-3: Triangles (Day 3) (CCSS: 8.G.5)


95º x º 5x + 20º

5x + 20º

x + 15º4x + 5º

3 21

4 1 2 2

1

3. Find the measure of each remaining angle.a. b.

4. Find each angle in the triangle.a. b.

5. Find the sum of the interior angles for each regular polygon. Then find the measure of each angle.

a. Triangle b. Pentagon c. 18 Sided

6. Find the measure of each exterior angle in the following regular polygons.

a. Quadrilateral b. Hexagon


315

125°?

? ?

?

??

?

45°

70°

60°

x°

x + 10° 2x + 10°

x°

125° ?

?? ?

? ? ?

8. Find the missing angle.

a. b.

c. d.

9. Solve.

a. -15 – (-9) = b. c.

d. -20.64 ÷ 8.6 = e. -15.2 + (-3.9) = f. (2.4)(-3.1) =

10. Solve.

a. 4z – 5 = 11 b. -3x + 4 = -8 c.

d. 6x + 2 = 2x + 14 e. 3(2x + 4) = 24 f. 4x – 8 = 4x – 8

11. Graph each line.a. y = 3x + 1 b. y = -4x – 3 c. y = 2

316

80°?

40° 20°50° ?

x85°85°

140° x 130°90°x

x60°

Review QuestionWhat two things are used to classify triangles? Angles, sidesName the two types of triangles that are not possible. Obtuse Equilateral, Right Equilateral

DiscussionWhat does identical mean? The same.What do you know about identical twins? Same features.

Today we are going to talk about identical figures. However, in math we use the term congruent instead of identical. Can someone give me an example of identical twins in our school?

Draw two triangles that would be comparable to the identical twins.

How do you think their angles are related? They are the same. How do you think their sides are related? They are the same.

SWBAT find angles and sides of congruent triangles

DefinitionCongruent Figures – equal to each other

- same shape, same size- same angles, same sides

Congruent triangles are like identical twins.

ABC DEF

The order in which the triangles are named matters!

317

A C DF

Section 6-4: Congruent and Similar Figures (Day 1) (CCSS: 8.G.5,

B E

A C

B

FD

E

1088

Example 1: ABC DEF. (Draw a picture)Given: A = 70º; B = 50º; AC = 5; AB = 8; EF = 10Label the values of all missing angles and sides in both triangles.

C = 60º, BC = 10, D = 70º, E = 50º, F = 60º, DE = 8, DF = 5Notice that the longest side is opposite the largest angle.

Example 2: Given , find all of the missing angles and sides.

FD = 8 ft DE = 3 ft D = 75° F = 35° E & B = 70°

You Try!1. Draw two congruent acute isosceles triangles. Label the triangles ABC and DEF. Given: B = 50º; C = 80º; AB = 6; BC = 4; DF = 4 Label the values of all missing angles and sides in both triangles.A = 50º, AC = 4, D = 50º, E = 50º, F = 80º, DE = 6, EF = 4

Question: What is causing two of the sides of the triangles to be congruent? Opposite Angles

2. Draw an obtuse scalene triangle. Make up your own values for the angles and sides of the triangle. Make sure the biggest side is across from the biggest angle and the smallest side is across from the smallest angle. Draw another obtuse scalene triangle that is congruent to the first triangle.

3. Draw a right isosceles triangle. Make up your own values for the angles and sides of the triangle. Make sure the biggest side is across from the biggest angle and the smallest side is across from the smallest angle. Draw another right isosceles triangle that is congruent to the first triangle.


318

10 ft

E

DF

8 ft

B

A C

3 ft

75º 35º

50°

70°5

1. Given , B = 40°, C = 60°, AB = 6, BC = 8, DF = 4. Draw the two triangles. Then find all of the missing sides and angles.

2. Given , find all of the missing angles and sides.

a.

b.

3. Draw a pair of right scalene triangles that are congruent to each other. Make up your own values for the angles and sides of the triangles.

4. Draw a pair of acute isosceles triangles that are congruent to each other. Make up your own values for the angles and sides of the triangles.

5. What is wrong with the following picture?

6. Explain why the longest side of a triangle must be across from the largest angle. (include pictures)

7. Name the two types of triangles that can’t be formed. Then explain why they are impossible. (include pictures)

319

6

10

8


B

A C60º

10 m

4 m

E

D F

7 m

E

DF

8 ft

B

A C

6 ft11 ft

45º

65º

50º

Review QuestionWhat does congruent mean? SameWhat do congruent figures have in common? Angles, Sides

DiscussionYesterday, we talked about congruent figures. We said they were like identical twins. Today, we are going to talk about similar figures. Similar figures are like brothers. Can someone give me an example of a big brother and little brother in our school that look the same except one is bigger? (Think Dr. Evil and Mini Me from Austin Powers.)

Draw two right triangles that would be comparable to a big brother and little brother.

How do you think their angles are related? SameHow do you think their sides are related? ProportionalWhat does proportional mean? Two things are related by the same ratio. (twice as big, ¼ as small)

What is the missing side?

Why can’t we just increase the 6 (height) by 4 to get the answer to the other height? That’s not proportional

How do you solve a proportion? Cross multiply, then divide

SWBAT find angles and sides of similar triangles

DefinitionSimilar – alike- same shape, different size- same angles, proportional sides

320


12

6

8

?

Similar triangles are like brothers.

ABC ~ DEFThe order in which the triangles are named matters!

Example 1: ABC ~ DEF.

Find all of the missing angles and sides.

The angles are easy to find as they are the same is each triangle. (A and D = 90º, E = 50º, C and F = 40º)

Find DE. Find EF.

Are there other ways to setup the proportions? Yes. You can switch the order of any of the proportions as long as you go in the same order in the second fraction as you do in the first fraction.

321

B

E

D F12

A C D

E

F

B

A C

6

8

1050º

Example 2: ABC ~ DEF

Find all of the missing angles and sides.

The angles are easy to find as they are the same is each triangle. (A and D = 60º, B = 80º, C = 40º)

Find DE. Find BC.

You Try!1. Draw two similar right scalene triangles. Label them triangles ABC and DEF.The longest side of the larger triangle is 10. The longest side of the smaller triangle is 4. The base of the shorter triangle is 2. Find the length of the base of the larger triangle. 5

2. The two triangles are similar. Find the two missing sides and all of the missing angles.

15/2

10

3. Draw two congruent acute isosceles triangles. Label the triangles ABC and DEF. Givens: A = 70º; B = 55º; AB = 12; BC = 14; DF = 12Label the values of all missing angles and sides in both triangles.AC = 12, DE = 12, EF = 14, D = 70°, E = 55°, F & C = 55°


322

A

B C E

D

F

12 8

9

4

80º 40º

4

8

5 6 ? 15151

?

80º

60º

80º

40º 40º 60ºA

B

C D

E

F

1. The two figures in each example are similar to each other. Find the missing side.

a. b.

c. d.

e. f.

323

3

4

6

x 7 3

x 9

14

12

x

18 20

x

24

30

x

9

4

6

x

5

6

3


2. Given , B = 60° , C = 80°, AB = 8, BC = 4, DF = 6. Draw the two triangles. Then find all of the missing sides and angles.

3. Given ABC ~ DEF, find the missing sides and angles.

4. Given ABC ~ DEF, B = 50° , C = 10°, AB = 5, BC = 15, DE = 10, DF = 25. Draw the two triangles. Then find all of the missing sides and angles.

5. Determine whether each pair of figures is similar by using proportions.

a.

b.

324

9

7

18

14

8

15 24

40

75º 50º A

B

6 12

E

F

4

6 C D

Review QuestionWhat does congruent mean? SameWhat do congruent figures have in common? Angles, SidesWhat does similar mean? AlikeWhat do similar figures have in common? Angles, Proportional Sides

DiscussionWe know that if three angles in one triangle are congruent to three angles of another triangle then the triangles are similar. What if only two of the angles are congruent to each other? They would be similar as well because if two of the angles are the same then the 3rd would be the same as well.

For example: Triangle 1: 90°, 60°, ? Triangle 2: 90°, 60°, ? What is the 3rd angle of each triangle? 30°

This is called the Angle-Angle (AA) Similarity. It states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

SWBAT solve application problems using similar triangles

Example 1: A six foot man casts a 10 foot shadow. A building casts a 120 shadow. Find the height of the building.

Demonstrate the problem by acting it out. Assume the clock in the room is the sun. Then have one student be the man and another taller student the building. Finally, draw a picture of the problem on the board, labeling all of the known parts. Notice that two angles in one triangle are congruent to two angles in the other triangle. Therefore, the triangles are similar.

325

6 ft.

10 ft.

? ft.

120 ft.


6 = 10x 120

10x = 720x = 72 feet

Example 2: Joe rides his bike 30 miles in 4 hours. How far can he ride in 7 hours?Why would calculating an 8 hour ride be easier? It would be multiplied by 2.

30 miles = x miles4 hours 7 hours

4x = 210x = 52.5 miles

Example 3: Which is the better deal on ice cream? 2 gallons for $2.80 or 5 gallons for $7.50

$2.80 = $7.50 2 5$1.40 or $1.50

Hmmm?!?When is it appropriate to use a proportion to solve a problem? When you are comparing two things and you are missing one of the items. It does not matter what we are comparing. It could be height, distance or money.

Notice we can cross multiply to see which fraction is smaller. ($14 and $15)


1. Given , B = 65° , C = 85° , AB = 10, BC = 4, DF = 7. Draw the two triangles. Then find all of the missing sides and angles.


326


6

2.5

30º 12 13

A

B

C

D

E

F

For problems 3-6, write a proportion then solve.

3. On a small scale map, the length from Boston to New York City is 15 centimeters, and from New York City to Washington, D.C. the length is 18 centimeters. On a larger map, the length from Boston to New York City is 25 centimeters. Find the length from New York City to Washington, D.C. on the larger map. (Draw a picture.)30 cm

4. If a tree 6 feet tall casts a shadow 4 feet long, how high is a flagpole that casts a shadow 18 feet long at the same time of day? (Draw a picture.)27 feet

5. Jimmy earns $152 in 4 days. At that rate, how many days will it take him to earn $532?14 days

6. A research study shows that three out of every twenty pet owners got their pet from a breeder. Of the 100 animals cared for by a veterinarian, how many would you expect to have been brought from a breeder? 15 pets

7. Are the two triangles similar? Why or why not? a.

b.

8. Determine if the following example would represent a similar/proportional relationship. Comparing the height of an ant to a dinosaur and the height of a human to a skyscraper. (Draw a picture of each example. Make up values for each height to prove or disprove proportionality.)

327

40º 40º

90º A

B

C

D

E

F

60º

50º

70º

60º


Review QuestionWhat does congruent mean? The sameWhat do congruent figures have in common? Angles, SidesWhat does similar mean? AlikeWhat do similar figures have in common? Angles, Proportional Sides

SWBAT solve application problems using similar trianglesSWBAT solve application problems using proportionsSWBAT determine when it is appropriate to use a proportion to solve a problem

DiscussionWhen is it appropriate to use a proportion to solve a problem? When you are comparing two things and you are missing one of the items. It does not matter what we are comparing. It could be height, distance, or money.

How do you get better at something? PracticeTherefore, we are going to practice solving problems that involve angles, triangles, and congruent/similar figures today. We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.



1. Set up a proportion. Then solve.

a. If 5 quarts of iced tea costs $6.25, how much would 7 quarts cost? $8.75

b. In a recipe, 2 cups of sugar makes 25 cookies. How many cups of sugar will you need to make 85 cookies? 6.8 cups

c. An 8 x 10 picture is enlarged so that the width of the new photo is 12 inches. What is the height of the new photo? 1 ft 3 in

d. Johnny has to do 500 practice problems before the PSSA test. He did 125 in 5 days. How many total days will it take him to get done with all 500? 20 days

e. A flagpole casts a 42 foot shadow. A 6 foot man casts an 8 foot shadow. How tall is the flagpole? (Include a picture.) 4 ft 8in

f. When trying to find the distance between two cities on a map, Jimmy noticed that Pittsburgh was at the 1in. mark and Erie was at the 3.5 in. mark. If one inch equal 50 miles, how far is it from Pittsburgh to Erie? 125 miles

328



4. Classify each triangle by its angles and its sides. Then make up sides and angles that would be appropriate for each triangle.

a. b. c.

5. Two of the angles in a triangle are 30° and 75°. What type of triangle is it?

329

115°

110º xº 2x + 10º

2x + 20º

2x + 5º3x + 15º

60°

6. Find each angle.

a. b.

7. Given , B = 40°, C = 70°, AB = 8, BC = 8, DF = 5. Draw a picture of each triangle correctly labeled. Then find all of the missing angles and sides.


9. What do the interior angles of a regular octagon add up to? What is the measure of each angle?

330

30º

x x

70°

2x° 2x + 10°

8

6

40º 12

15

A B

C D E

F

*Calculators will be useful in this section.

Review QuestionWhat is 42? 16What is ? 4

What is the opposite of adding? SubtractingWhat is the opposite of squaring? Square Rooting

DiscussionThere are right angles all around us. Today’s lesson is based solely on right triangles. When you build a wall it has to form a right angle in order to be sturdy. Look at where the walls in the classroom meet the floor and ceiling. They are all right angles. How can we ensure that they are right angles?

We can start to build the floor so it is 3 feet long. Then we can start to build the wall so it is 4 foot tall. Then we can check the diagonal distance between the top of the wall and the edge of the floor. If the distance is greater than 5 feet, then we have an obtuse angle. If the distance is less than 5 feet, then we have an acute angle. All you have to do is adjust the sides until the diagonal distance is exactly 5 feet. This concept is also used to set up and line football fields.

SWBAT find the third side of a right triangle given the other two sides

DefinitionsPythagorean Theorem

a2 + b2 = c2

Legs – two shorter sides of a right triangle (a, b)Hypotenuse – longest side of a right triangle (c)

How do we know what the shorter sides are? They are across from the shorter angles.How do we know which side is the hypotenuse? It is across from the right angle.

Example 1: 62 + 82 = c2

36 + 64 = c2

100 = c2

10 = c

331

ac

b

6

8

?

Section 6-5: Pythagorean Theorem (Day 1) (CCSS: 8.EE.2,

Example 2: 102 + 242 = c2

100 + 576 = c2

676 = c2

26 = c (spiral back to square roots lesson)*Remember we can find the square root of numbers by using our number sense

10 → 10020 → 40030 → 900

676 is “between 20 and 30.” Also, it is closer to 900 so the answer has to be closer to 30. Finally, it has to be 26 because 26 times 26 ends in a 6.

Example 3: 52 + b2 = 132

25 + b2 = 169-25 -25 b2 = 144

b =12

You Try!1. Draw a right triangle whose legs are 8 and 15. Find the hypotenuse. 172. Draw a right triangle whose leg is 9 and hypotenuse is 15. Find the other leg. 123. Draw a right triangle whose legs are 12 and 16. Find the hypotenuse. 204. Draw a right triangle whose leg is 6 and hypotenuse is 9. Find the other leg. (Estimate) 7


1. Find the missing side of each right triangle.

a. b. c.

25 cm 26 ft 15 mm

332

10

24

?

5

?

13


c15 cm

20 cm

c24 ft

10 ft

17 mm8 mm

a

d. e. f.

10 m 16 in 36 ft

2. Draw and label a triangle. Then estimate the missing side of each triangle.

a. a = 5 in, b = 5 in, c ≈ ? 7 b. b = 13 cm, c = 16 cm, a ≈ ? 9

c. a = 2 m, b = 12 m, c ≈ ? 12 d. a = 20 ft, c = 50 ft, b ≈ ? 46

3. Plot the points A (-6, 8) B (-6, 0) C (0, 0) on the coordinate system. Connect the three lines to form a triangle. Then find the length of AC. 10

4. Using sentences, explain one use of the Pythagorean Theorem. Draw a picture to demonstrate its use.

333

6 m

8 m

c 34 in

30 in

b 15 ft 39 ft

a

Review QuestionIn a right triangle there are legs and a hypotenuse.How do we know what sides are the legs? Shorter sidesHow do we know which side is the hypotenuse? Across from the right angle

DiscussionIf you had a triangle with sides 1, 2, and 3 feet respectively, would it form a right triangle? The students will say it is. Suckers!!! No, 12 + 22 ≠ 32

SWBAT determine if three sides of a triangle would form a right triangle using the Pythagorean Theorem

Example 1: Is it a right triangle? Why or Why not?

62 + 102 = 122

36 + 100 = 1442

136 ≠ 144These three sides do not form a right triangle.

You can construct a triangle using rulers with these dimensions on the floor to prove that they do not form a right triangle.

Example 2: Three sides of a triangle are 12, 20, and 16. Do they form a right triangle?How do you know where to put each of the values? a, b are the legs, c is the hypotenuse122 + 162 = 202 144 + 256 = 4002

400 = 400These three sides form a right triangle.

If the hypotenuse was 21, what would that tell us about the angle? It is greater than 90°. This could help us if we were building a wall to see if it was “square.” If our wall was off a few degrees, would that matter? How would that mistake be magnified on a 50 story building? Greatly.

You Try!For problems 1-2, reinforce that square roots can be found using mental math1. Draw a right triangle whose legs are 8 and 15. Find the hypotenuse. 172. Draw a right triangle whose leg is 5 and hypotenuse is 13. Find the other leg. 123. Draw a right triangle whose legs are 3 and 6. Find the hypotenuse. (ESTIMATE) 74. Prove that 12, 14, 28 do not form a right triangle. 340 ≠ 7845. Prove that 7, 25, 24 form a right triangle. 625 = 625


334

10

6

12


1. Find the missing measure of each right triangle.

a. b. c.

24 cm 20 ft 5 mm

d. e. f.

20 m 400 in 17 ft

2. Estimate the missing measure of each triangle. a. a = 6 in, b = 12 in, c ≈ ? 13 b. b = 7 cm, c = 9 cm, a ≈ ? 6

c. a = 8 m, b = 9 m, c ≈ ? 12 d. a = 9 ft, c = 16 ft, b ≈ ? 13

3. The measures of the three sides of a triangle are given. Determine if each triangle is a right triangle by using the Pythagorean Theorem.

a. 16 km, 30 km, 34 km b. 8 mm, 15 mm, 9 mm

b. 10 mi, 20 mi, 30 mi d. 14 in, 50 in, 48 in

4. Tommy drives 5 miles west to Oakdale. Then he drives 12 miles north to Wexford. Finally, he drives directly home from Wexford. How far did he drive in all? (Draw a picture.) 30 miles

5. Find each angle.

a. b.

x = 20, 20º, 80º, 80º x = 75, 85º and 95º

335

26 cm 10 cm

b

29 ft 21 ft

a

c3 mm

4 mm

b

15 m

25 m 500 in

300 in

b 8 ft

c

15 ft

x +10º x + 20 º x

x+60 x+60


Review QuestionHow can you prove if a triangle is a right triangle? Put the 3 sides of the triangle into the Pythagorean Theorem and see if it “works”

DiscussionA Pythagorean triple is a group of three integers that satisfy the equation: a2 + b2 = c2. For example, 3, 4, 5 is a Pythagorean triple because 32 + 42 = 52. Notice that if we multiply each of the triples by an integer a new triple is formed. We know that 3, 4, 5 “works.”

6, 8, 10 (times 2) 62 + 82 = 102

9, 12, 15 (times 3) 92 + 122 = 152

Other “famous” Pythagorean triples: (5, 12, 13) (8, 15, 17) (7, 24, 25). We can multiply any of these by an integer to create a new triple.

SWBAT use Pythagorean triples to solve right triangles mentallySWBAT apply the Pythagorean Theorem to word problems Example 1: In a Walmart catalog, an HD TV is listed as being 50 inches. This distance is the diagonal distance across the screen. If the screen measures 30 inches in height, what is the actual width of the screen?

a2 + 302 = 502

a2 + 900 = 2500 - 900 - 900a2 = 1600a = 40 in Notice that the answer of 30, 40, 50 is the Pythagorean Triple 3, 4, 5 just multiplied by 10.

Example 2: Plot the points (0, 5) and (12, 0) on the coordinate plane. What is the diagonal distance between these two points?

52 + 122 = c2

25 + 144 = c2

169 = c2

13 = c Notice that we can find the distance between two points on the coordinate plane using the Pythagorean Theorem.

336

30 in 50 in

?


5

12

?

(12, 0)

(0, 5)



a. b. c.

60 cm 40 ft 34 mm



a. 50 km, 130 km, 120 km b. 8 mm, 15 mm, 34 mm

For problems 4-7, draw a picture then solve using the Pythagorean Theorem.

4. Each side of a square is 3 ft. Find the length of the diagonal. (Round decimal answer to the nearest tenths.) 4.2 ft

5. Scott wants to swim across a river that is 300 meters wide. He begins swimming from one side of the shore. When he gets across the river, he realizes that he ended up 160 meters down river from where he started because of the current. How far did he actually swim from his starting point? 340 m

6. Suppose the height of a hill on a roller coaster is 150 feet off of the ground. The distance on the ground from the start of the hill to the end of the hill is 80 feet. How far do you actually travel on the roller coaster’s track? 170 ft

7. In baseball, the bases are 90 feet apart. If you want to throw from 1st base to 3rd base, how far is it? (Round decimal answer to the nearest tenths.) 127.3 ft

337


65 cm 25 cm

b

58 ft 42 ft

a

c16 mm

30 mm

8. Ancient Egyptians used the Pythagorean Theorem to check if their pyramids were “level.” They would use blocks that were 3 feet by 4 feet.

a. How could they check if the blocks were stacked “level?”

b. Why were 3’ by 4’ blocks a good choice of dimensions for each block?

9. A Pythagorean triple is a group of three integers that satisfy the equation a2 + b2 = c2. For example, 3, 4, 5 is a Pythagorean triple because 32 + 42 = 52.

a. Show that 5, 12, 13 is a Pythagorean triple. b. Name two other Pythagorean triples.

c. Show that multiplying all of the integers in a Pythagorean triple by the same number produces another Pythagorean triple.

d. Suppose the integers of a Pythagorean triple are the lengths of the sides of a right triangle. Draw a right triangle using a Pythagorean triple. Multiply the sides of your original triangle by three. Draw this new triangle. How are the two triangles related?

10. Find the diagonal distance between the two points.a. (0, 8) (15, 0) b. (-2, 5) (1, 9)

338

34

?

Review QuestionHow can you prove if a triangle is a right triangle? Put the 3 sides of the triangle into the Pythagorean Theorem and see if it “works”

DiscussionLet’s see if we can put all of this together. We are going to do an activity over the next couple of days that will use the Pythagorean Theorem plus other Algebra skills that we learned in Unit 4. SWBAT plot points on the coordinate planeSWBAT calculate the slope between two pointsSWBAT apply the Pythagorean Theorem to word problems

Activity1. You will design an amusement park. You have to include specific attractions and necessities in your design. First off, it has to be done on a large sheet of graph paper. Next identify the center of your paper, and draw in the x and y axes. The following items must be included in the design: help center, water ride, three different roller coasters, kiddy land, merry-go-round, concessions, gift shop, restrooms, and security desk. Next mark an entrance point of each attraction on the graph paper and draw in the remaining part of the attraction around it. Try to spread the attractions out as much as possible.

2. After planning out the layout and design of each attraction, you must identify its location by using ordered pairs. Use your “entrance points” as the attractions identifiable location.

3. After identifying each attraction’s location with ordered pairs, you will calculate the slope between attractions. To do this mark the direct path to/from the following locations: gift shop to restrooms, security desk to roller coaster #2, kiddie land to merry-go-round, help center to gift shop, restroom to roller coaster #3, concessions to water ride, roller coaster #1 to roller coaster #2, and roller coaster #2 to roller coaster #3. Then use the slope formula: (y – y)/(x – x).

4. Now you are going to determine how large a space you will need to design the park. Using the Pythagorean Theorem, you will calculate how far away certain attractions are from one another. This will provide you with the information needed to expand the park from a scaled blueprint to actual dimensions. Calculate the distance by making right triangles using the selected attraction points from step 3 and the Pythagorean Theorem.

What did we learn day?

339

Section 6-5: Pythagorean Theorem (Day 4-6) (CCSS: 8.EE.2,

Section 6-6: Volume (Day 1)

Review Question How can you find another Pythagorean Triple that is related to 5, 12, and 13?Multiply each number by an integer

DiscussionHas anyone ever seen a 3D movie? What makes it 3D? It comes out at you.In this section, we are going to find volume of 3D shapes. Let’s make sure we know what volume and 3D shapes are.

What are some examples of 3D shapes? Cube, Sphere, CylinderWhy are they called 3-D shapes? They have three dimensions: base, height, width.

What does volume mean? Stuff inside a 3D shape

SWBAT calculate the volume of a sphere, cone and cylinder

DefinitionVolume – space inside a 3D object

3- Dimensional Figures

Volume of Cone: Volume of Cylinder: Volume of Sphere:

If you take a cross section of each shape, what shape would you get? Circle Notice that each one of these equations contains the expression .Why is that? Each one of the shapes contains circles. The area of a circle is

Example 1: Find the volume. (Don’t worry about units. We will discuss them tomorrow.)

340

CylinderCone Sphere

3

4

V =

V =

V =

Example 2: Find the volume.

V = V = V =


V =

V =

V =

You Try!1. Draw a cone with r = 10 and h = 3. Find the volume. 3

2. Draw a cylinder with r = 4 and h = 6. Find the volume.

3. Draw a sphere with r = 1. Find the volume.


341

2

5

3


Find the volume.

1.

2.

3.

4. Draw a cone with r = 5 and h = 5. Find the volume.



7. Draw a cone with r = 4 and h = 6. Find the volume.



342

2

5

4

6

6

Section 6-6: Volume (Day 2)

Review Question Why are they called 3-D shapes? They have three dimensions: B, H, WWhat does volume mean? Stuff inside a 3D shapeHow is this different from area? Stuff inside a 2D shape

DiscussionLet’s talk about what the units should be when we are calculating volume.

What is the perimeter of a square with sides equal to 4 inches? 16 inchesNotice that the units are “normal.” This is because perimeter is a one dimensional unit.

What is the area of a square with sides equal to 4 inches? 16 inches2

Notice that the units are squared. This is because area is a two dimensional unit.

What is the volume of a cube with sides equal to 4 inches? 64 inches3

Notice that the units are cubed. This is because volume is a three dimensional unit.

SWBAT calculate the volume of a sphere, cone and cylinderSWBAT complete review problems


V =

V =

V = in3

343

2 in

6 in

Example 2: What is the volume of a salt shaker that has a radius of 2 inches and a height of 4 inches?

V = V = V =



1. Find the volume.a.

b.

c.

d. Draw a cone with r = 3 ft and h = 7 ft. Find the volume. ft3

e. Draw a cylinder with r = 5 cm and h = 8 cm. Find the volume. cm3

f. Draw a sphere with r = 20 in. Find the volume. in3

344

2 in

4 in

5 in

6 in

1 m

2m

9 ft

2. What is the volume of a waffle ice cream cone that has a radius of 2 inches and a height of 5 inches? (Draw a picture.)


a. b. c.

48 cm 42 ft 40 mm



a. 32 km, 60 km, 68 km b. 7 mm, 9 mm, 8 mm

6. Each pair of angles is either complementary or supplementary. Find the measure of each angle.a. b.


345

110° 1 2 3

7

4 5 6

85º xº 3x + 10º

4x + 10º

48° 1 2

4 3 6

5 1

7

52 cm 20 cm

b

58 ft 40 ft

a

c24 mm

32 mm

8. Find each angle.

a. b.

x = 30, 30º, 60º

9. Given , B = 40° , C = 80°, AB = 8, BC = 6, DF = 4. Draw the two triangles. Then find all of the missing sides and angles.


11. What is the angle measure of each angle in a regular 12 sided polygon?

346

10

80°

30°

15

3 6

90°

2x° x°

30°x°

50°

A

B C

D

E F

Section 6-7: Transformations (Day 1) (CCSS: 8.G.1.a, b, c,

Review Question Why are they called 3-D shapes? They have three dimensions: B, H, WWhat does volume mean? Stuff inside a 3D shape

DiscussionHas anyone seen the movie/cartoon Transformers? Or for you literature people, has anyone noticed a character in a novel going through a transformation?What does the word transform/transformation mean? ChangeIn this section, we are going to talk about three different types of changes to polygons.

SWBAT slide a polygon left/right or up/down on the coordinate plane

DefinitionTransformation – change/movement of a polygon

Three Types of Transformations:

1. Translation – slide left/right or up/down

2. Reflection – flipping over the x or y axis

3. Dilation – enlarge or reduce

Example 1: Plot the following three points: (1, 5) (3, 1) (6, 3). Connect the points to make a triangle. Translation - three units right

What is going to change? x values because we are moving left and rightHow is moving right going to change the x values? It will add three to each x value.What are the new coordinates? (4, 5) (6, 1) (9, 3)Graph the new triangle.

How are the two triangles related? Congruent. Notice the segments and angles are still the same. If they are not the same shape and size then you did something wrong.What would change if we translate a polygon up or down? y values How would the y’s change? They would increase if we move it up and decrease if we move it down.

347

Example 2: Plot the following three points: (1, 2) (5, 2) (1, 5). Connect the points to make a triangle Translation – two units left and four units down

What is going to change? The x values and the y values.What are the new coordinates? (-1, -2) (3, -2) (-1, 1)Graph the new triangle.

How are the two triangles related? Congruent. Notice the segments and angles are still the same. If they are not the same shape and size then you did something wrong.

You Try!Draw the original polygon. List the new coordinates. Graph the new polygon.1. (2, 1) (5, 6) (8, 3) Translate right 1, up 3 New coordinates (3, 4) (6, 9) (9, 6)2. (-2, 3) (-4, 7) (-6, 5) Translate left 2, down 2 New coordinates (-3, 0) (0, 4) (5, 2)3. (-6, 3) (-4, 2) (-1, 5) (-3, 7) Translate right 2, down 3 New coordinates (-4, 0) (-2, -1) (1, 2) (-1, 4)



For each of the problems, do the following:1. Draw the original triangle. Then name it.2. Find the new coordinates after the translation.3. Draw the new triangle.

1. (6, 1) (6, 3) (3, 5) Translated right 2, up 1

2. (1, 6) (-4, 2) (2, 0) Translated left 2, up 4 3. (4.2, -1.5) (1.5, -1.8) (4.1, -5.9) Translated right 1, up 1 4. (-3, 2) (0, -4) (4, 2) Translated left 3, down 4 5. (-1, -2) (-4, 1) (2, -3) Translated right 3, down 1

6. ( , ) (0,- ) ( ,- ) Translated left 2, down 3

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7. a. List the coordinates of ΔRST.

b. List the coordinates of the vertices of ΔRST after a translation 2 units left and 5 units up. c. Draw the image of ΔRST after the translation.

8. a. List the coordinates of ΔKLM.

b. List the coordinates of the vertices of ΔKLM after a translation 3 units right and 2 units down.

c. Draw the image of ΔKLM after the translation.

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Review Question What does moving a polygon left/right do to its coordinates? Changes the x’sWhat does moving a polygon up/down do to its coordinates? Changes the y’s

DiscussionWhen you look in the mirror how is your reflection related to you? It is flipped/reversed.What does the word reflection mean? Appearance is flippedReflection – flipping an object over the x or y-axis

What is happening to the coordinates when you flip over the y-axis?

* the y’s stay the same, the x’s are the opposite

What is happening to the coordinates when you flip over the x-axis?

* the x’s stay the same, the y’s are the opposite

SWBAT flip a polygon over the x or y-axis on the coordinate plane

DefinitionsReflection – flipping an object over the x or y axisOver y-axis – same y’s, opposite x’sOver x-axis – same x’s, opposite y’s

Example 1: Plot the following three points: (-6, 5) (-6, 2) (-2, 2). Connect the points to make a triangle. Reflection - over the y-axis

What is going to change? The x values.What are the new coordinates? (6, 5) (6, 2) (2, 2)Graph the new triangle.

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What would change if we reflected over the x-axis? y coordinatesHow would they change? They would be the opposite.

Example 2: Plot the following three points: (1, 4) (1, 1) (6, 1). Connect the points to make a triangle. Reflection - over the x-axis

What is going to change? The y values.What are the new coordinates? (1, -4) (1, -1) (6, -1)Graph the new triangle.


You Try!Draw the original polygon first. List the new coordinates. Graph the new polygon.1. (-4, -1) (-1, -1) (-1, -5) Reflected over the y-axis New coordinates (4, -1) (1, -1) (1, -5)2. (2, -3) (2, -6) (6, -3) Reflected over the x-axis New coordinates (2, 3) (2, 6) (6, 3)3. (1, 2) (4, -3) (-2, -1) Translate left 3, up 4 New coordinates (-2, 6) (1, 1) (-5, 3)



For each of the problems, do the following:1. Draw the original triangle. Then name it.2. Find the new coordinates after the transformation.3. Draw the new triangle.

1. (3, 1) (6, 1) (3, 5) Reflected over the y-axis

2. (1, 3) (4, 5) (5, 2) Translated left 2, up 5

3. (4, -1) (1, -1) (4, -5) Reflected over the x-axis 4. (-5, -5) (0, -2) (3, -4) Reflected over the y-axis

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5. (-1, 3) (-4, -3) (2, -2) Translated right 4, down 2

6. (-4, 1) (0, 5) (3, 3) Reflected over the x-axis


b. List the coordinates of the vertices of ΔRST after a translation: 3 units left and 4 units up.

c. Draw the image of ΔRST after the translation.


b. List the coordinates of the vertices of ΔKLM after a reflection over the y-axis.

c. Draw the image of ΔKLM after the reflection over the y-axis.

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Review Question What happens to the coordinates when you flip over the y-axis? The y’s stay the same, the x’s are the opposite

What happens to the coordinates when you flip over the x-axis?The x’s stay the same, the y’s are the opposite

DiscussionHas anyone ever had their pupils dilated? What does that do to your pupils? Enlarges them.What does the word dilation mean? Changing an object’s size

Dilation – reduce or enlarge an objectHow do you make something twice as big, mathematically? MultiplyHow do you make something twice as small, mathematically? Divide

Enlarge – multiplyReduce – divide

SWBAT enlarge or reduce a polygon

Example 1: Plot the following three points: (1, 2) (-1, 3) (0, 4). Connect the points to make a triangle. Dilate - by a factor of 3 What is going to change? The x and y valuesHow are they going to change? Multiply by 3What are the new coordinates? (3, 6) (-3, 9) (0, 12)Graph the new triangle.

How are the two triangles related? Similar. Notice the segments are proportional and the angles are the same.

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Example 2: Plot the following three points: (1, 2) (-1, 3) (0, 4). Connect the points to make a triangle.

Dilate - by a factor of

What is going to change? The x and y valuesHow are they going to change? Divide by two

What are the new coordinates? ( , 1) (- , ) (0, 2)

Graph the new triangle.

How are the two triangles related? Similar. Notice the segments are proportional and the angles are the same.

You Try!Draw the original polygon first. List the new coordinates. Graph the new polygon.1. (-2, -3) (-4, 4) (6, 2) Dilated by a factor of 2 New coordinates (-4, -6) (-8, 8) (12, 4)2. (6, 3) (-6, 6) (0, 3) Dilated by a factor of 1/3 New coordinates (2, 1) (-2, 2) (0, 1)3. (-5, 4) (-3, 1) (-1, 5) Reflected over the y-axis New coordinates (5, 4) (3, 1) (1, 5)4. (-3, -4) (0, 3) (2, 6) Reflected over the x-axis New coordinates (-3, 4) (0, -3) (2, -6)5. (-2, 2) (4, -3) (0, 5) Translated left 2, up 3 New coordinates (-4, 5) (2, 0) (-2, 8)



For each of the problems, do the following:1. Draw the original triangle. Then name it.2. Find the new coordinates after the transformation.3. Draw the new triangle.

1. (2, 1) (3, 4) (-2, 5) Dilated by a factor of 2

2. (1, 4) (0, 1) (5, 2) Translated left 2, up 5

3. (-4, 1) (0, 4) (2, 3) Reflected over the x-axis 4. (-5, 4) (-4, -2) (-1, 1) Reflected over the y-axis

5. (2, 4) (2, 2) (6, 4) Dilated by a factor of

6. (-4, 2) (0, 2) (2, 3) Translated right 3, down 2

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b. List the coordinates of the vertices of ΔRST after a dilation by a factor of 1/3.

c. Draw the image of ΔRST after the dilation.


b. List the coordinates of the vertices of ΔKLM after a reflection over the x-axis.

c. Draw the image of ΔKLM after the reflection over the x-axis.

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Section 6-7: Transformations (Day 4) (CCSS: 8.G.1.a, b, c, 8.G.2,

Review Question What is a transformation? ChangeWhat are the three types of transformations that we talked about? Translation, Reflection, DilationHow are the coordinates affected by being translated up/down or left/right? y’s and x’s increase/decreaseHow are the coordinates affected by being reflected over the x-axis? Opposite y’sHow are the coordinates affected by being reflected over the y-axis? Opposite x’sHow are the coordinates affected by being reduced/enlarged? x’s and y’s divided/ multiplied

DiscussionHave the students put the solutions to problems 1, 2, 3, and 4 on the board. Have the students explain the solutions to problems 1, 2, 3, and 4.

How do you get better at something? PracticeTherefore, we are going to practice our transformations today. Plus we are going to review other topics from this unit.

What unit is this? GeometryWhat other topics have we discussed in this unit? Angles, Triangles, Similar/Congruent, Pythagorean Theorem, Volume

We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.

SWBAT perform a transformation on a polygonSWBAT review the previous topics in Unit 6



1. Find each angle.a. b. c.

x = 15, 30º and 60º x = 36, 36º, 36º and 108º

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xº 85º 2x + 30º2xº xº

3xº xº

2. Given , B = 30° , C = 70° , AB = 9, BC = 13, DF = 5. Draw the two triangles. Then find all of the missing sides and angles.


4. Find the missing side.a. b.

39 cm 15 ft


a. 30 km, 72 km, 78 km b. 5 mi, 20 mi, 15 mi

6. Find the volume.a.

b.

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c 15 cm

36 cm

25 ft20 ft

a

2 m

5 m

12

75°

40°

18

3 6

A

B

C D

E

F

2 in

5 in

c.

d. Draw a cone with r = 2 ft and h = 7 ft. Find the volume. ft3

e. Draw a cylinder with r = 1 cm and h = 3 cm. Find the volume. cm3

f. Draw a sphere with r = 2 in. Find the volume. in3

7. For each of the problems, do the following:a. Draw the original triangle. Then name it.b. Find the new coordinates after the transformation.c. Draw the new triangle.

a. (3, 1) (-3, 4) (0, 5) Dilated by a factor of 3

b. (5, 1) (4, -2) (1, 4) Reflected over the y-axis

c. (1, 5) (0, 3) (-5, 2) Translated left 3, up 4

d. (0, 3) (-3, 3) (4, 1)

Dilated by a factor of

e. (-4, -2) (0, -4) (2, -1) Reflected over the x-axis

f. (-1, 2) (3, 2) (2, -3) Translated right 3, down 2

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5 ft

Unit 6 Review

Review QuestionWhat is a transformation? ChangeWhat are the three types of transformations that we talked about? Translation, Reflection, Dilation

SWBAT study for the Unit 6 test

DiscussionHow do you study for a test? The students either flip through their notebooks at home or do not study at all. So today we are going to study in class.

How should you study for a test? The students should start by listing the topics.

What topics are on the test? List them on the board- Angles- Polygons- Triangles- Congruent/Similar Figures- Pythagorean Theorem- Volume- Transformations

How could you study these topics? Do practice problems; study the topics that you are weak on

ActivityYou will make up two problems with the correct solution for each one of the seven topics.


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1. A triangle’s three angles are 10°, 50°, and 120° respectively. What type of triangle is it?a. Obtuse b. Acute c. Right d. Isosceles

2. Johnny travels 6 miles west then 8 miles north. What would his diagonal distance be if he wanted to travel back to his original spot?a. 10 b. 11 c. 12 d. 13

3. What is the volume of a sphere with a radius of 4 inches?a. 260.5 in3 b. 262.9 in3 c. 267.9 in3 d. 2679 in3

4. What is the volume of a cylinder with a radius of 4 inches and height of 6 inches?a. 260.5 in3 b. 282.9 in3 c. 301.44 in3 d. 305.35 in3

5. What type of transformation causes the second shape to be similar?a. translation b. reflection c. dilation d. deportation

6. The following problem requires a detailed explanation of the solution. This should include all calculations and explanations.

a. Use proportions to find which two triangles are similar triangles.

b. Explain how you solved the previous problem.

c. If a tree 6 feet tall casts a shadow 4 feet long, how high is a flagpole that casts a shadow 18 feet long at the same time of day? (Draw a labeled picture.)

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6 cm6 cm5 cm5 cm

2 cm

4 cm4 cm

Unit 6 Standardized Test

2 cm 3 cm

UNIT 6 CUMULATIVE

SWBAT do a cumulative review

DiscussionWhat does cumulative mean?All of the material up to this point.

Our goal is to remember as much mathematics as we can by the end of the year. The best way to do this is to take time and review after each unit. So today we will take time and look back on the first five units.

Does anyone remember what the first six units were about? Let’s figure it out together.1. Problem Solving2. Numbers/Operations3. Pre-Algebra4. Algebra5. Systems6. Geometry

Things to Remember:1. Reinforce test taking strategies: guess/check, eliminate possibilities, work backwards, and estimating.2. Reinforce the importance of retaining information from previous units.3. Reinforce connections being made among units.

In-Class

1. What two consecutive even numbers add up to 30? a. 13, 17 b. 14, 16 c. 10, 20 d. 11, 19

2. Who is the best male rapper? a. Nikki Minaj b. Drake c. Lebron James d. Michael Jordan

3. What is the next term: 1, 5, 9, 13, ___? a. 13 b. 14 c. 15 d. 17

4. What is the next term: 1, 4, 16, 64, ___? a. 132 b. 142 c. 152 d. 256

5. Tommy had $75. He spent $5.75 at Wendy’s and $14.99 on a t-shirt. About how much money does he have left? a. $50 b. $55 c. $65 d. $75

6. What set(s) of numbers does ‘-3’ belong? a. C, W, I, R b. W, I, R c. I, R d. R

7. What number is the smallest 15%, .12, 18%, ?

a. 15% b. .12 c. 18% d. 1/10

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8. -7 + 10 = a. 3 b. -3 c. 17 d. -17

9.

a. -1/120 b. -6/5 c. 10/40 d. -10/40

10. Which of the following is equal to (3)3? a. -27 b. 27 c. 9 d. 10

11. Which of the following is equal to ? a. 1/9 b. -1/9 c. -9 d. 9

12. Which of the following is equal to ? a. 30 b. 31 c. 39 d. 320

13. Which of the following is equal to ? a. 21 b. 22 c. 29 d. 220.5

14. Which of the following is equal to ? a. 1 b. 4 c. 5 d. 6

15. Which of the following is equal to 4.2 x 103? a. .0042 b. 42 c. 420 d. 4200

16. Which of the following is equal to 3.1 x 10-4? a. .00031 b. 31000 c. .031 d. 31

17. Which of the following is 345 in scientific notation? a. 3.45 b. 3.45 x 102 c. 345 x 103 d. 3.45 x 103

18. Which of the following is .0246 in scientific notation? a. 2.46 x 102 b. 2.46 x 10-5 c. 2.46 x 10-2 d. 2.46 x 103

19. 38 – (10 + 3) + 22 a. 10 b. 3 c. 47 d. -17

20. 3(2x + 5) a. 6x + 6 b. 6x + 15 c. 5x + 5 d. 6x

21.

a. All Reals b. 36 c. 52 d. 24

22. 6x + 5 = 4x + 2x + 5 a. Empty Set b. All Reals c. 4 d. 5

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23. 3(2x + 4) = x + 22 a. Empty Set b. All Reals c. -2 d. 2

24. Johnny has $200. He makes $7.50/hour. How many hours will it take for him to save $260? a. 10 b. 8 c. 6 d. 2

25. Solve y = 3x – 4; given a domain of {-2, 0, 4}. a. (-2, -10) (0, -4) (4, 8) b. (-2, 10) (0, 4) (4, 8) c. (-2, -10) (0, -4) (4, 0) d. (-10, -2) (4, 1) (1, 3)

26. Which point is a solution to the following equation: y = 4x + 2? a. (2, -1) b. (0, 5) c. (9, 2) d. (1, 6)

27. Which equation is not a linear equation?

a. 2x2 + 3y = 5 b. 3x + y = 1 c. y = 2 d.

28. Write an equation for the following relation: (2, 10) (4, 7) (6, 4)

a. b. y = 4x + 10 c. d.

29. y = 2x – 4 a. b. c. d.

30. y = 3 a. b. c. d.

31. Which of the following is a function? a. (1,4) (2,5) (3,6) (1,-5) b. (1,4) (2,5) (3,6) (2,3) c. (1,4) (2,5) (3,6) d. (1,2) (2,3) (1,4)

32. Which of the following is a function?

a. b. c. d.

33. Write an equation of a line that contains the points (6, 5) and (0, 3).

a. b. y = 4x + 3 c. d.

34. A scatter plot of hours worked and money made would be what type of relationship? a. Positive b. Negative c. Scattered d. Weathered

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35. Solve the following system of equations. y = x + 2

2x + 3y = 11 a. (0, 2) b. (1, 3) c. (3/2, 1/2) d. (-3, 1)

36. Two angles are complementary. One of the angles is 42°. What is the measure of the other angle?a. 40° b. 45° c. 48° d. 50°

37. What is the measure of each angle in a regular hexagon?a. 120° b. 108 c. 90° d. 66°

38. A triangle’s three angles are 40°, 80°, and 60° respectively. What type of triangle is it?a. Obtuse b. Acute c. Right d. Isosceles

39. Two legs of a triangle are 10 and 24. What is the hypotenuse?a. 34 b. 14 c. 40 d. 26

40. What is the volume of a sphere with a radius of 5 inches? a. 533.3 in3 b. 562.9 in3 c. 523.3 in3 d. 5233 in3

41. What type of transformation causes the second shape to be congruent?a. deportation b. reflection c. dilation d. purification

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Unit 6 Hand-In This problem set is intended to challenge the students and encourage students to apply a deep

understanding of problem–solving skills.

1. Two angles are complementary. One angle is 5 more than four times the other. Find the measure of both angles.

2. Two angles are supplementary. One angle is 20 less than three times the other. Find the measure of both angles.

3. Show that 3, 4, and 5 are a Pythagorean triple. Then show that 300, 400, and 500 are a Pythagorean triple. Then explain why this works.

4. Draw two right triangles that are similar to each. Make sure to include the measures of each side of each triangle. (The sides should be proportional. The sides should also work in the Pythagorean Theorem)

5. Draw two acute scalene triangles that are congruent to each other. Make sure to include the sides and angles of each triangle.

6. What is the radius of a cone whose height is 4 meters and whose volume is ?

7. What is the height of a cylinder whose radius is 2 meters and whose volume is ?

8. What is the radius of a sphere whose volume is ?

9. A cube has a volume of 27 cm3. If the length of a side must be a whole number, what is the volume of the next larger cube?

10. Draw a right triangle in the first quadrant. List its coordinates. Then translate it 3 units left and 4 units up. From there reflect it over the x-axis. List its final coordinates.

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Pythagorean Theorem Project

In this unit, we discussed the Pythagorean Theorem. This theorem allows us to check to see if a triangle is a right triangle. We also discussed how this can be used in real life situations. For example, we discussed how to make sure that a wall is “square” by using the Pythagorean triple 3, 4, 5.

Assume that you have a large piece of land that has no markings on it. You are given the challenge of creating a football field on this piece of land. Write a paragraph describing how the Pythagorean Theorem can be used to line a football field.

Find another real life situation where the Pythagorean Theorem can be used. Write a paragraph explaining how the Pythagorean Theorem would be used in this situation.

Jacob Louis Trombetta © 2015All Rights Reserved

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Unit 6 Project

manual63.doc · web viewthe following diagram is called two parallel lines cut by a transversal. we...

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