Aliquippa School District



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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 |

Review Question

How do you graph the point (-5, 4)? Left 5, Up 4

What quadrant is that point located? II

Discussion

What do you think about when I say the word Geometry? Angles, Shapes, Area, Perimeter

SWBAT define basic geometry terms

SWBAT estimate an angle measure

SWBAT draw an angle given a measure

Definitions

Notice how things build in nature: Atoms → Cells → Organs → Body

Notice 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, space

A B

Angle – two rays (say angle BAC or CAB)

Vertex – meeting point of two rays (point A)

Type of Angles

Straight – 180°

Right – 90°

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°.

You Try!

1. Draw an acute angle; label it BAD – estimate measure

2. Draw an obtuse angle; label it EMO – estimate measure

3. 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?

Review Question

1. Estimate the following angle.

130°

2. Draw a 15° angle.

Discussion

Yesterday 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 molding

In 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 protractor

SWBAT 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 obtuse

How 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°

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 obtuse

How 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.

What did we learn today?

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

1. 2.

3. 4.

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?

[pic]

Review Question

1. What is 2x + 30 equal to when ‘x’ is 20? 70

2. Solve: 4x + 2 + 6x + 8 = 50 x = 4

We 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.

Discussion

What 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 angles

SWBAT calculate complementary and supplementary angles

Definitions

Complementary – 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 = 180

x = 120

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

x + (x + 20) = 90

2x + 20 = 90

2x = 70

x = 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) = 180

5x + 30 = 180

5x = 150

x = 30

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

You Try!

1. Draw complementary angles. Estimate each angle. Then label them.

2. Draw supplementary angles. Estimate each angle. Then label them

3. 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°)

What did we learn today?

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

1. 2. 3.

4. 5. 6.

7. 8. 9.

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

[pic]

Review Question

What does complementary mean? Angles that add up to 90°

What does supplementary mean? Angles that add up to 180°

Discussion

How 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

Definitions

Congruent – equal to; [pic] (in algebra we use = ; in geometry we use [pic])

Parallel – two lines that do not intersect each other

The 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.

[pic](Supp. to 100 º)

[pic](Vertical to [pic]

[pic](Alt. Int. to [pic]

[pic](Alt. Int. to [pic]

[pic](Vertical to [pic]

[pic](Vertical to [pic]

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

What did we learn today?

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

a. b. c.

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.

[pic]

Review Question

What do corresponding angles look like? ‘F’

What do alternate interior angles look like? ‘Z’

What do vertical angles look like? ‘X’

Discussion

What is a polygon? The students will say shape.

Is a circle a shape? Yes

Is it a polygon? No

Definitions

Polygon – 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 8

SWBAT 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 congruent

Explain 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

You Try!

1. Draw a regular version of each polygon.

Discuss how tick marks are used to show congruency.

What did we learn today?

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. _______________

7. 8. 9.

1. ______________ 1. ______________ 1. _______________

2. ______________ 2. ______________ 2. _______________

3. ______________ 3. _____________ 3. _______________

4. ______________ 4. _____________ 4. _______________

10. Find the value of x in each figure.

a. b. c.

d. e. f.

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

a. b. c.

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

Review Question

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

Discussion

Today 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 polygon

SWBAT apply the formula to calculate the sum of the interior angles of any polygon

SWBAT 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 = 120º

2. Find the sum of the interior angles of a regular 12 sided polygon. Then find the measure of each angle. sum = 1800º one angle = 150º

3. What regular polygon has angles that are 156° each? 15 sided polygon

4. What regular polygon has angles that are 162° each? 20 sided polygon

What did we learn today?

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?

11. Each angle in a regular polygon is 90°. What polygon is it?

12. Each angle in a regular polygon is 162°. What polygon is it?

13. Find the value of x in each figure.

a. b. c.

14. Find the measure of each remaining angle.

a. b. c.

Review Question

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

Discussion

Yesterday, 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 = 360

x + 240 = 360

x = 120°

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 = 360

x = 60

Each exterior angle would be 60°

What did we learn today?

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?

5. Each interior angle in a regular polygon is 90°. What polygon is it?

6. Each interior angle in a regular polygon is 135°. What polygon is it?

7. Draw a regular and irregular hexagon.

Review Question

What letters do you vertical, corresponding, and alternate interior angles form?

‘X’, ‘F’, and ‘Z’ respectively

Discussion

What 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 name

SWBAT 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 name

Let’s say we are talking about John Grabowski.

What does his first name tell you? Gender/individual

What 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°.

Types of triangles – by sides

Have 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

Example 2: Name the triangle.

Acute Equilateral

Example 3: Name the triangle.

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 Isosceles

2. Acute Scalene

3. Obtuse Equilateral Can’t do it; demonstrate using meter sticks

4. Acute Isosceles

5. Obtuse Scalene

6. Right Equilateral Can’t do it; demonstrate using meter sticks

What did we learn today?

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

Review Question

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

Discussion

Yesterday 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°.

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 = 150

3x = 150

x = 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°

What did we learn today?

[pic]

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.

[pic]

Review Question

What are the three possible first names of a triangle? Acute, Obtuse, Right

What are the three possible last names of a triangle? Isosceles, Scalene, Equilateral

Discussion

How 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.

What did we learn today?

[pic]

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.

3. Find the measure of each remaining angle.

a. b.

[pic]

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

7. Each interior angle in a regular polygon is 135°. What polygon is it?

8. Find the missing angle.

a. b.

c. d.

9. Solve.

a. -15 – (-9) = b. [pic] c. [pic]

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. [pic]

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

Review Question

What two things are used to classify triangles? Angles, sides

Name the two types of triangles that are not possible. Obtuse Equilateral, Right Equilateral

Discussion

What 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

Definition

Congruent Figures – equal to each other

- same shape, same size

- same angles, same sides

Congruent triangles are like identical twins.

[pic]

(ABC ( (DEF

The order in which the triangles are named matters!

[pic]

Example 1: (ABC ( (DEF. (Draw a picture)

Given: (A = 70º; (B = 50º; AC = 5; AB = 8; EF = 10

Label the values of all missing angles and sides in both triangles.

(C = 60º, BC = 10, (D = 70º, (E = 50º, ( F = 60º, DE = 8, DF = 5

Notice that the longest side is opposite the largest angle.

Example 2: Given [pic], 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.

What did we learn today?

1. Given [pic], (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 [pic], 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)

Review Question

What does congruent mean? Same

What do congruent figures have in common? Angles, Sides

Discussion

Yesterday, 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? Same

How do you think their sides are related? Proportional

What 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

[pic]

SWBAT find angles and sides of similar triangles

Definition

Similar – alike

- same shape, different size

- same angles, proportional sides

Similar triangles are like brothers.

(ABC ~ (DEF

The 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.

[pic]

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.

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.

[pic]

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 = 12

Label the values of all missing angles and sides in both triangles.

AC = 12, DE = 12, EF = 14, D = 70°, E = 55°, F & C = 55°

What did we learn today?

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

a. b.

c. d.

e. f.

2. Given [pic], (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.

Review Question

What does congruent mean? Same

What do congruent figures have in common? Angles, Sides

What does similar mean? Alike

What do similar figures have in common? Angles, Proportional Sides

Discussion

We 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.

6 = 10

x 120

10x = 720

x = 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 miles

4 hours 7 hours

4x = 210

x = 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)

What did we learn today?

1. Given [pic], (B = 65° , (C = 85° , AB = 10, BC = 4, DF = 7.

Draw the two triangles. Then find all of the missing sides and angles.

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

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.)

Review Question

What does congruent mean? The same

What do congruent figures have in common? Angles, Sides

What does similar mean? Alike

What do similar figures have in common? Angles, Proportional Sides

SWBAT solve application problems using similar triangles

SWBAT solve application problems using proportions

SWBAT determine when it is appropriate to use a proportion to solve a problem

Discussion

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.

How do you get better at something? Practice

Therefore, 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.

What did we learn today?

[pic]

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

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

a. b. c.

3. Find the measure of each remaining angle.

a. b.

[pic]

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?

6. Find each angle.

a. b.

7. Given [pic], (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.

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

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

*Calculators will be useful in this section.

Review Question

What is 42? 16

What is ? 4

What is the opposite of adding? Subtracting

What is the opposite of squaring? Square Rooting

Discussion

There 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

Definitions

Pythagorean 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

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 → 100

20 → 400

30 → 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. 17

2. Draw a right triangle whose leg is 9 and hypotenuse is 15. Find the other leg. 12

3. Draw a right triangle whose legs are 12 and 16. Find the hypotenuse. 20

4. Draw a right triangle whose leg is 6 and hypotenuse is 9. Find the other leg. (Estimate) 7

What did we learn today?

1. Find the missing side of each right triangle.

a. b. c.

25 cm 26 ft 15 mm

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.

Review Question

In a right triangle there are legs and a hypotenuse.

How do we know what sides are the legs? Shorter sides

How do we know which side is the hypotenuse? Across from the right angle

Discussion

If 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 ≠ 144

These 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 hypotenuse

122 + 162 = 202

144 + 256 = 4002

400 = 400

These 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 math

1. Draw a right triangle whose legs are 8 and 15. Find the hypotenuse. 17

2. Draw a right triangle whose leg is 5 and hypotenuse is 13. Find the other leg. 12

3. Draw a right triangle whose legs are 3 and 6. Find the hypotenuse. (ESTIMATE) 7

4. Prove that 12, 14, 28 do not form a right triangle. 340 ≠ 784

5. Prove that 7, 25, 24 form a right triangle. 625 = 625

What did we learn today?

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

Review Question

How 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”

Discussion

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. 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 mentally

SWBAT 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 - 900

a2 = 1600

a = 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.

What did we learn today?

1. Find the missing measure of each right triangle.

a. b. c.

60 cm 40 ft 34 mm

2. Estimate the missing measure of each triangle.

a. a = 4 in, b = 8 in, c ≈ ? 9 b. b = 10 cm, c = 12 cm, a ≈ ? 7

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. 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

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)

Review Question

How 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”

Discussion

Let’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 plane

SWBAT calculate the slope between two points

SWBAT apply the Pythagorean Theorem to word problems

Activity

1. 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?

[pic]

Review Question

How can you find another Pythagorean Triple that is related to 5, 12, and 13?

Multiply each number by an integer

Discussion

Has 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, Cylinder

Why 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

Definition

Volume – space inside a 3D object

3- Dimensional Figures

Volume of Cone: [pic] Volume of Cylinder: [pic] Volume of Sphere: [pic]

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[pic].

Why is that? Each one of the shapes contains circles. The area of a circle is [pic]

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

V = [pic]

V = [pic]

V = [pic]

Example 2: Find the volume.

V = [pic]

V = [pic]

V = [pic]

Example 3: Find the volume.

V = [pic]

V = [pic]

V = [pic]

You Try!

1. Draw a cone with r = 10 and h = 3. Find the volume. [pic] 3

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

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

What did we learn today?

[pic]

Find the volume.

1. [pic]

2. [pic]

3. [pic]

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

5. Draw a cylinder with r = 2 and h = 3. Find the volume. [pic]

6. Draw a sphere with r = 4. Find the volume. [pic]

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

8. Draw a cylinder with r = 1 and h = 5. Find the volume. [pic]

9. Draw a sphere with r = 10. Find the volume. [pic]

[pic]

Review Question

Why are they called 3-D shapes? They have three dimensions: B, H, W

What does volume mean? Stuff inside a 3D shape

How is this different from area? Stuff inside a 2D shape

Discussion

Let’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 inches

Notice 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 cylinder

SWBAT complete review problems

Example 1: Find the volume.

V = [pic]

V = [pic]

V = [pic] in3

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

V = [pic]

V = [pic]

V = [pic]

What did we learn today?

[pic]

1. Find the volume.

a. [pic]

b. [pic]

c. [pic]

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

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

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

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.)

3. Find the missing measure of each right triangle.

a. b. c.

48 cm 42 ft 40 mm

4. Estimate the missing measure of each triangle.

a. a = 5 in, b = 9 in, c ≈ ? 10 b. b = 11 cm, c = 16 cm, a ≈ ? 12

5. 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. 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.

7. Find the measure of each remaining angle.

a. b.

[pic]

8. Find each angle.

a. b.

x = 30, 30º, 60º

9. Given [pic], (B = 40° , (C = 80°, AB = 8, BC = 6, DF = 4.

Draw the two triangles. Then find all of the missing sides and angles.

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

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

[pic]

Review Question

Why are they called 3-D shapes? They have three dimensions: B, H, W

What does volume mean? Stuff inside a 3D shape

Discussion

Has 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? Change

In 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

Definition

Transformation – 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 right

How 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.

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)

What did we learn today?

[pic]

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. ([pic],[pic]) (0,-[pic]) ([pic],-[pic]) Translated left 2, down 3

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.

[pic]

Review Question

What does moving a polygon left/right do to its coordinates? Changes the x’s

What does moving a polygon up/down do to its coordinates? Changes the y’s

Discussion

When you look in the mirror how is your reflection related to you? It is flipped/reversed.

What does the word reflection mean? Appearance is flipped

Reflection – 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

Definitions

Reflection – flipping an object over the x or y axis

Over y-axis – same y’s, opposite x’s

Over 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.

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 reflected over the x-axis? y coordinates

How 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.

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 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)

What did we learn today?

[pic]

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

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

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

7. a. List the coordinates of ΔRST.

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.

8. a. List the coordinates of ΔKLM.

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.

[pic]

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

Discussion

Has 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 object

How do you make something twice as big, mathematically? Multiply

How do you make something twice as small, mathematically? Divide

Enlarge – multiply

Reduce – 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 values

How are they going to change? Multiply by 3

What 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.

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 [pic]

What is going to change? The x and y values

How are they going to change? Divide by two

What are the new coordinates? ([pic], 1) (-[pic],[pic]) (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)

What did we learn today?

[pic]

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 [pic]

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

7. a. List the coordinates of ΔRST.

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.

8. a. List the coordinates of ΔKLM.

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.

[pic]

Review Question

What is a transformation? Change

What are the three types of transformations that we talked about? Translation, Reflection, Dilation

How are the coordinates affected by being translated up/down or left/right? y’s and x’s increase/decrease

How are the coordinates affected by being reflected over the x-axis? Opposite y’s

How are the coordinates affected by being reflected over the y-axis? Opposite x’s

How are the coordinates affected by being reduced/enlarged? x’s and y’s divided/ multiplied

Discussion

Have 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? Practice

Therefore, we are going to practice our transformations today. Plus we are going to review other topics from this unit.

What unit is this? Geometry

What 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 polygon

SWBAT review the previous topics in Unit 6

What did we learn today?

[pic]

1. Find each angle.

a. b. c.

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

2. Given [pic], (B = 30° , (C = 70° , AB = 9, BC = 13, DF = 5.

Draw the two triangles. Then find all of the missing sides and angles.

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

4. Find the missing side.

a. b.

39 cm 15 ft

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

6. Find the volume.

a. [pic]

b. [pic]

c. [pic]

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

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

f. Draw a sphere with r = 2 in. Find the volume. [pic]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 [pic]

e. (-4, -2) (0, -4) (2, -1)

Reflected over the x-axis

f. (-1, 2) (3, 2) (2, -3)

Translated right 3, down 2

Review Question

What is a transformation? Change

What are the three types of transformations that we talked about? Translation, Reflection, Dilation

SWBAT study for the Unit 6 test

Discussion

How 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

Activity

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

What did we learn today?

[pic]

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.)

[pic]

SWBAT do a cumulative review

Discussion

What 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 Solving

2. Numbers/Operations

3. Pre-Algebra

4. Algebra

5. Systems

6. 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.

[pic]

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%, [pic]?

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

8. -7 + 10 =

a. 3 b. -3 c. 17 d. -17

9. [pic]

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 [pic]?

a. 1/9 b. -1/9 c. -9 d. 9

12. Which of the following is equal to [pic]?

a. 30 b. 31 c. 39 d. 320

13. Which of the following is equal to[pic]?

a. 21 b. 22 c. 29 d. 220.5

14. Which of the following is equal to[pic]?

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. [pic]

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

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. [pic]

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

a. [pic] b. y = 4x + 10 c. [pic] d. [pic]

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. [pic] b. y = 4x + 3 c. [pic] d. [pic]

34. A scatter plot of hours worked and money made would be what type of relationship?

a. Positive b. Negative c. Scattered d. Weathered

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

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[pic]?

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

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

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.

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 © 2015

All Rights Reserved

-----------------------

Section 6-1: Angles (Day 1) (CCSS: 8.G.5)

B

A

C

90°

135°

45°

180°

0°

Section 6-1 Homework (Day 1)

Section 6-1: Angles (Day 2) (CCSS: 8.G.5)

Section 6-1 Homework (Day 2)

Section 6-1: Angles (Day 3) (CCSS: 8.G.5)

B

C

A

D

B

C

A

C

A

B

D

C

A

B

D

B

C

A

A

B

D

C

A

B

D

C

60°

x°

x + 20°

x + 20°

4x + 10°

Section 6-1 Homework (Day 3)

2

1

2

1

1

2

49º

126º

2

54º

41º

1

58º

32º

49º

49º

20º

160º

xº

18º

xº

49º

xº

43º

66º

xº

xº

117º

86º

xº

57º

xº

xº

22º

xº

4x + 20º

x + 60º

3xº

2x + 40º

2x + 25º

4x + 5º

Section 6-1: Angles (Day 4) (CCSS: 8.G.5)

2

1

4

L

3

6

5

P

8

7

2

100°

3

4

5

6

7

8

120°

60°

120°

60°

60°

120°

60°

120°

70°

85°

Section 6-1 Homework (Day 4)

2

1

2

1

1

2

xº

38º

xº

22º

55º

xº

66º

xº

xº

117º

86º

xº

3x + 20º

4x + 5º

4x + 20º

x º

xº +10

4x + 5º

100°

40°

50°

70°

Section 6-2: Polygons (Day 1) (CCSS: 8.G.5)

No, not line segments

No, open

No, open

Yes, closed

Yes, closed

Heptagon

Hexagon

Pentagon

Quadrilateral

Triangle

Octagon

Section 6-2 Homework (Day 1)

xº

45º

xº

xº

35º

46º

xº

xº

62º

82º

xº

101º

4x + 25º

x + 55º

2x + 5º

x + 6 º

xº + 10

3x + 15º

Section 6-2: Polygons (Day 2) (CCSS: 8.G.5)

5

4

3

Section 6-2 Homework (Day 2)

xº

46º

xº

15º

xº

44º

95°

72°

55°

40°

Section 6-2: Polygons (Day 3) (CCSS: 8.G.5)

2

1

3

4

5

1

2

4

3

120°

120°

120°

60°

80°

x

100°

Section 6-2 In-Class Assignment (Day 3)

50°

80°

x

x

110°

120°

110°

Section 6-3: Triangles (Day 1) (CCSS: 8.G.5)

Section 6-3 Homework (Day 1)

50°

15°

x°

70°

40°

x°

x°

35°

x°

80°

x + 10°

3x°

2x°

x°

Section 6-3: Triangles (Day 2) (CCSS: 8.G.5)

A

B

1

C

60°

30°

?

90°

120°

60°

90°

30°

?

90°

70°

?

60°

80°

150°

?

x

150°

2x

95°

155°

x

Section 6-3 Homework (Day 2)

[pic]

80°

?

50°

20°

40°

?

x

85°

x

85°

x

150°

x

140°

90°

2x + 5

x

x

125°

3x

160°

x°

80°

x°

60°

x + 30°

2x + 30°

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

Section 6-3 In-Class Assignment (Day 3)

1

4

2

2

1

2

3

1

5x + 20º

x + 15º

4x + 5º

95º

x º

5x + 20º

125°

?

?

?

?

?

?

?

45°

125°

?

?

?

?

?

?

?

x°

70°

x°

60°

x + 10°

2x + 10°

80°

?

40°

20°

50°

?

60°

x

85°

x

85°

x

140°

x

130°

90°

Section 6-4: Congruent and Similar Figures (Day 1) (CCSS: 8.G.5, Prepares for 8.G.2, 8.G.4)

E

B

A

C

D

F

[pic]

E

B

8

50°

8

10

70°

5

D

F

A

C

E

D

F

8 ft

B

A

C

3 ft

10 ft

75º

35º

Section 6-4 Homework (Day 1)

E

D

F

B

A

C

60º

50º

7 m

10 m

4 m

E

D

F

8 ft

B

A

C

6 ft

65º

11 ft

45º

8

6

10

Section 6-4: Congruent and Similar Figures (Day 2) (CCSS: 8.G.5, Prepares for 8.G.2, 8.G.4)

?

6

12

8

B

E

C

F

D

A

[pic]

[pic]

E

B

50º

10

6

8

C

A

F

D

12

[pic]

D

A

12

8

4

B

80º

40º

F

E

C

9

[pic]

E

B

80º

? 151511515/2

5

6

4

80º

60º

A

8

60º

40º

40º

F

D

?

C

Section 6-4 Homework (Day 2)

3

7

6

3

9

x

x

4

24

20

18

12

x

30

14

x

3

6

5

x

6

4

9

x

B

E

4

12

6

75º

50º

F

A

D

C

6

9

18

7

14

8

24

15

40

Section 6-4: Congruent and Similar Figures (Day 3) (CCSS: 8.G.5, Prepares for 8.G.2, 8.G.4)

6 ft.

10 ft.

? ft.

120 ft.

Section 6-4 In-Class Assignment (Day 3)

E

B

30º

6

12

13

2.5

C

A

F

D

E

B

40º

40º

C

A

F

D

90º

50º

70º

60º

60º

Section 6-4: Congruent and Similar Figures (Day 4) (CCSS: 8.G.5, Prepares for 8.G.2, 8.G.4)

Section 6-4 In-Class Assignment (Day 4)

2x + 10º

3x + 15º

2x + 5º

xº

110º

2x + 20º

60°

115°

30º

70°

2x + 10°

2x°

x

x

E

D

C

8

40º

12

15

B

A

6

F

Section 6-5: Pythagorean Theorem (Day 1) (CCSS: 8.EE.2, 8.G.6, 8.G.7, 8.G.8)

[pic]

c

a

b

?

6

8

24

10

?

?

5

13

Section 6-5 Homework (Day 1)

a

8 mm

24 ft

10 ft

c

c

20 cm

15 cm

17 mm

15 ft

8 m

39 ft

a

6 m

34 in

b

c

30 in

Section 6-5: Pythagorean Theorem (Day 2) (CCSS: 8.EE.2, 8.G.6, 8.G.7, 8.G.8)

6

10

12

Section 6-5 Homework (Day 2)

4 mm

a

26 cm

b

10 cm

c

3 mm

21 ft

29 ft

c

15 ft

8 ft

b

300 in

500 in

25 m

15 m

b

x

x + 20 º

x +10º

x+60

x+60

Section 6-5: Pythagorean Theorem (Day 3) (CCSS: 8.EE.2, 8.G.6, 8.G.7, 8.G.8)

30 in

50 in

?

(0, 5)

?

5

(12, 0)

12

Section 6-5 In-Class Assignment (Day 3)

30 mm

c

42 ft

a

65 cm

b

25 cm

16 mm

58 ft

?

3

4

Section 6-5: Pythagorean Theorem (Day 4-6) (CCSS: 8.EE.2, 8.G.6, 8.G.7, 8.G.8)

Section 6-6: Volume (Day 1) (CCSS: 8.G.9)

Sphere

Cylinder

Cone

3

4

2

5

3

Section 6-6 Homework (Day 1)

2

5

4

6

6

Section 6-6: Volume (Day 2) (CCSS: 8.G.9)

2 in

6 in

2 in

4 in

Section 6-6 In-Class Assignment (Day 2)

5 in

6 in

1 m

2m

9 ft

32 mm

40 ft

a

52 cm

b

20 cm

24 mm

c

58 ft

3x + 10º

xº

85º

4x + 10º

2

6

7

5

4

3

1

1

48°

110°

1

2

3

7

4

5

6

90°

50°

x°

2x°

30°

x°

A

D

30°

6

10

3

80°

B

F

E

15

C

Section 6-7: Transformations (Day 1) (CCSS: 8.G.1.a, b, c, 8.G.2, 8.G.3, 8.G.4)

Section 6-7 Homework (Day 1)

Section 6-7: Transformations (Day 2) (CCSS: 8.G.1.a, b, c, 8.G.2, 8.G.3, 8.G.4)

Section 6-7 Homework (Day 2)

Section 6-7: Transformations (Day 3) (CCSS: 8.G.1.a, b, c, 8.G.2, 8.G.3, 8.G.4)

Section 6-7 Homework (Day 3)

Section 6-7: Transformations (Day 4) (CCSS: 8.G.1.a, b, c, 8.G.2, 8.G.3, 8.G.4)

Section 6-7 In-Class Assignment (Day 4)

xº

2xº

xº

85º

2x + 30º

3xº

xº

B

E

40°

12

6

3

75°

A

F

C

18

D

25 ft

20 ft

c

a

15 cm

36 cm

2 in

5 in

2 m

5 m

5 ft

Unit 6 Review

Unit 6 Standardized Test Review

6 cm

6 cm

5 cm

5 cm

2 cm

4 cm

4 cm

3 cm

2 cm

UNIT 6 CUMULATIVE REVIEW

In-Class Assignment

Unit 6 Hand-In Problems

Unit 6 Project

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