Unit #1 Scale Factor

Matching sides of two or more polygons are called corresponding sidesMatching angles are called corresponding angles.

Similar FiguresTwo figures are similar if• the measures of the corresponding angles are equal• the ratios of the lengths of the corresponding sides are proportional

Similar figures have the same shape but not necessarily the same size.

Finding missing angles

• Hint: Remember angles are the same in corresponding angles!

What is angle D? <D

Finding Missing lengths

111 y

___ 100 200

____ = Write a proportion using corresponding side lengths.

The cross products are equal.200 • 111 = 100 • y

The two triangles are similar. Find the missing length y

y is multiplied by 100.22,200 = 100y

22,200 100

______ 100y 100

____ = Divide both sides by 100 to undo the multiplication.

222 mm = y

Indirect Measurement

Indirect Measurement uses similar figures and proportions to find height of objects you cannot

measure directly

Word Problems 1. Underline the question2. Set up your answer3. Draw it out 4. Find the similar shapes in your

drawing (triangles) 5. Use similar figures and proportions

to solve your problem!!!!

A tree casts a shadow that is 7 ft. lawn. Ken, who is 6 ft tall, is standing next to the tree.

Kens has a 2-foot long shadow. How tall is the tree?

Step 1: underline the questionStep 2: Set up your answer: The tree is __________tall.

Step 3: Draw it out

Step 4: Find the triangles

2 7

__ 6 h

__ =

h • 2 = 6 • 7 2h = 42 2h 2

___ 42 2

___ =

h = 21

Write a proportion using corresponding sides.

The cross products are equal.

h is multiplied by 2.

Divide both sides by 2 to undo multiplication.

The tree is 21 feet tall.

Step 5: Proportions

Example #2 Rockets

A rocket casts a shadow that is 91.5 feet long. A 4-foot

model rocket casts a shadow that is 3 feet long. How tall is

the rocket?

Step 1: underline the questionStep 2: Set up your answer: The Rocket is __________tall.

Step 3: Draw it out Step 4: Find the triangles

91.5 3

____ h 4

__ =

4 • 91.5 = h • 3 366 = 3h

366 3

___ 3h 3

___ =

122 = h

Write a proportion using corresponding sides.

The cross products are equal.

h is multiplied by 3.Divide both sides by 3 to undo multiplication.

The rocket is 122 feet tall.

Step 5: Proportio

ns

The map shown is a scale drawing. A scale drawing is a drawing of a real object that is proportionally smaller or larger than the real object. In other words, measurements on a scale drawing are in proportion to the measurements of the real object.

SCALE

Is it set up right?Why or why not?

1. The scale on the map is 3 cm: 10 m. On the map the distance between two cities is 40 cm. What is the actual distance?

2. The scale on the map is 10 in: 50. On the map the distance between two schools is 30 in. What is the actual distance?

3 1040 x

10 3050 x

NO

Yes

The scale on a map is 4 in: 1 mi. On the map, the distance between two towns is 20 in. What is the actual

distance? 20 in. x mi

_____ 4 in. 1 mi

____ =

1 • 20 = 4 • x 20 = 4x 20 4

___ 4x 4

___ = 5 = x

Write a proportion using the scale. Let x be the actual number of miles between the two towns.The cross products are equal.

x is multiplied by 4.

Divide both sides by 4 to undo multiplication.

5 miles

Question #1 Find side GF

10 cm5 cm

7cm

6 cm

Answer: 12 cm

Question #2 Determine if the ratios are proportional. Explain.

92 23121 34

No, cross proportions don’t equal (2783 doesn’t equal 3128)

Question #3

You want to leave your server a 20% tip. The total bill

comes to $54.50. How much should you leave for a tip?

$10.90

Question #4 A scale on a map reads 5 in: 50 miles. If two lakes

are 11 inches apart on the map, what is the actual

distance?

110 Miles

Question #5 How do you know if

two figures are SIMILAR?

Angles are the EXACT SAME. Side lengths have to be proportionally similar

Question #6 On a sunny afternoon, a

goalpost casts a 70 ft shadow. A 6 ft football player next to the goal post has a shadow 20 ft long. How tall is the

goalpost?

21 ft

Question #7 If all angles are congruent, are

these two shapes SIMILAR

39

6

18Yes!

The cross products are equal! 54=54

OR 3/9 is equal to 6/18Scale Factor = 2

Question #8

< A ___________

EF ___________

BA __________

<F ___________

Answer

< A __<D_________

EF ____BC_______

BA ____ED______

<F ____<C_______

Question #9 Is this a proportion?

2520

108

Yes!

The cross products are equal! 200=200

Question #10

The following rectangles are similar. What is the length of

side RS?

Question #11 A postcard is 6 inches wide and 14 inches long. When the postcard is

enlarged, it is 10 inches wide.

What is the length of the enlarged postcard?

Unit #2 Measurement

The customary system is the measurement system used in the

United States. It includes units of measurement for

length, weight, and capacity.

WHAT IS IT?

What is a benchmark?

If you do not have an instrument, such as a ruler, scale, or measuring cup, you can estimate the length, weight, and capacity

of an object by using a benchmark. It helps you visualize actual

measurements!

Customary Units of Length

Unit Abbreviation Benchmark

Inch in. Width of your thumb

Foot ft Distance from your elbow to your wrist

Yard yd Width of a classroom door

mile mi Total length of 18 football fields

Lengths

What unit of measure would provide the best

estimate?A doorway is about 7_____________high

Feet

Customary Units of WeightUnit Abbreviation Benchmark

Ounce oz A slice of breadPound lb A loaf of bread

Ton T A small car

Weight

What unit of measure would provide the best

estimate?A bike could weigh 20 _____?

lb.

CapacityCustomary Units of Capacity

Unit Abbreviation BenchmarkFluid ounce fl oz A spoonful

Cup c A glass of juicePint pt A small bottle of salad dressing

Quart qt A small container of paintGallon gal A large container of milk

What unit of measure would provide the best

estimate?A large water cooler holds about

10 _____ of water.

Gallons

Customary Conversion Factors1 foot = 12 inches1 yard = 3 feet1 yard = 36 inches1 mile = 5,280 feet1 mile = 1,760 yards1 pound (lb) = 16 ounces (oz)1 Ton (T) = 2,000 pounds1 gallon = 4 quarts 1 quart= 2 pints1 pint = 2 cups1 cup = 8 fluid ounces (fl oz)

To convert from one unit of

measurement to another unit of measurement,

use a proportion

Cool, we’ve been using this since November!

Important! Same measurements in the numerator

And Same measurements in the denominator

inches inchesfeet feet

not feet inchesinches feet

How many feet are in 3 miles?

1 35,280 ?mile miles

ft

5,280 3 15,840 115,840 feet

A book weighs 60 ounces. How many pounds is this?

1 ?16 60poundounces ounces

1 60 60 163.75pound

Metric System:

King Henry Doesn’t Usually Drink Chocolate Milk

Memorize      this!

Naming the Metric Units• Meter units measure Length

• Gram units measure mass or weight

• Liter units measure volume or capacity.

LengthUnit Abbreviation Relation to a

meterBenchmark

Millimeter

mm .001 m Thickness of a dime

Centimeter

cm .01 m Width of a fingernail

Decimeter

dm .1m Width of a CD case

Meter m 1 m Width of a single bed

Kilometer Km 1,000 m Distance around a city block

Mass Unit Abbreviatio

nRelation to a

gramBenchmark

Milligram mg .001 g Very small insect

Gram g 1 g Large paper clip

Kilogram kg 1,000 g Textbook

Capacity

Unit Abbreviation

Relation to a liter

Benchmark

Milliliter

mL .001 L A drop of water

Liter L 1 L Blender Container

Example #1:(1)Look at the problem. 56 cm = _____

mm

Look at the unit that has a number. 56 cm

On the device put your pencil on that unit. k h d U d c m

km hm dam m dm cm mm

Example #1:

k h d U d c m

km hm dam m dm cm mm

2. Move to new unit, counting jumps and noticing the direction of the jump!

One jump to the right!

Example #1:3. Move decimal in original number

the same # of spaces and in the same direction.

56 cm = _____ mm

56.0.

Move decimal one jump to the right. Add a zero as a placeholder.

One jump to the right!

Example #1:

56 cm = _____ mm

56cm = 560 mm

Question¼ lb=

________________oz

4

Question8 cups =

________________fl oz

16

Question346 yards=

________________inches

12,456 inches

Question10 quarts =

________________gal

2.5

QuestionAn average cat weighs

15__________(ounces, pounds, tons)

Pounds

Question130 g = ________kg

.13 kg

QuestionWhat would be a reasonable measurement for the

distance from NGMS to NGHSCustomary

Mile

QuestionHow many pints are in a gallon?

8

QuestionHow long is the line?

3.5 inches

Unit #3 Symmetry

A rotation is the movement of a figure around a point. A point of rotation can be on or outside a

figure.

The location and position of a figure can change with a rotation.

A full turn is a 360° rotation. So a turn

is 90°, and a turnis 180°.

12__

14__

90°

180°

360°

Clockwise CounterClockwise

QuestionDraw a 180° counterclockwise rotation

AnswerDraw a 180° counterclockwise rotation

QuestionDraw a 90° counterclockwise rotation

Answer

Unit #3 2-D Figures

• The area of a figure is the amount of surface

it covers. • We measure area in

square units.• Example: in , cm , etc.² ²

6 cm

4 cm

A lw 224 6 4cm

Rectangles

#1 Find the area of the figure

Write the formula.

Substitute 15 for l.

Substitute 9 for w.

A = lw

A = 15 • 9A = 135

The area is about 135 in2.

15 in.

9 in.

A parallelogram is a quadrilateral with opposite sides that are parallel.

Base

Height

A=Bh

Why does this work?

9 cm

8 cm

4 cm

A bh 232 8 4cm

#1

AREA OF A TRIANGLE

b

A = 12bh

The area A of a triangleis half the product of itsbase b and its height h.

h

Or A=bh÷2

Find the area of the triangle.A = 12 bh Write the formula.

A = 12 (20 · 12) Substitute 20 for b and 12 for h.

A = 120The area is 120 ft2.

A = 12 (240) Multiply.

AreaOf

Circles

tttttttttttt

Estimate the area of the circle. Use 3 to approximate pi.

A ≈ 3 • 202

A ≈ 1200 m2

19.7 m

A = r2 Write the formula for area.Replace with 3 and r with 20.

A ≈ 3 • 400 Use the order of operations.

Multiply.

Estimate the area of the circle. Use 3 to approximate pi.

r = 28 ÷ 2

A ≈ 3 • 142

28 m

A = r2 Write the formula for area.

Replace with 3 and r with 14.

r = 14

Use the order of operations.

Divide.

r = d ÷ 2 The length of the radius is half the length of the diameter.

A ≈ 3 • 196A ≈ 588 m2 Multiply.

5 ft

Question 1 Find the Area

17 ftAnswer:

85 ft²

r = 20 ÷ 2

A ≈ 3 • 102

20 m

A = r2 Write the formula for area.

Replace with 3 and r with 10.

r = 10

Use the order of operations.

Divide.

r = d ÷ 2 The length of the radius is half the length of the diameter.

A ≈ 3 • 100A ≈ 300 m2 Multiply.

Question 2Find the Area

Find the area of the triangle.A = 12 bh Write the formula.

A = 54The area is 54 in2.

A = 12 (108) Multiply.

24 ft

4 ft12

A = 12 (4 • 24)12

Substitute 4 for b and 24 for h.

12

Question 3Find the Area

Unit #4 3-D Figures

-A closed plane figure formed by three or more line segments that intersect only at their endpoints

-A three dimensional figure in which all the surfaces are polygons

-A flat surface (polygon) on a solid figure

-The segment where two faces meet

-The point where three or more edges meet

BaseA side of a polygon; a face of a

three dimensional figure by which the figure is measured

or classified.

PrismA polyhedron that has two congruent, polygon shaped bases and other faces that are all rectangles.

Prism named after what kind of BASE it has

PyramidA polyhedron with a polygon base and

triangular sides that all meet at a common vertex.

Pyramid named after what kind of BASE it has

CylinderA three dimensional figure with two parallel, congruent circular bases connected by a curved

lateral surface.

ConeA three dimensional figure with one vertex and one circular base.

CubeA rectangular prism with six congruent square

faces.

Question

What is a 3-D shape that has 5

FACES

Pyramid

Which of the nets below could be used to form a pyramid like the one below?

Question

12 in²

12 in²

30 in²

WHAT IS THE

SURFACE AREA

Question

54 in²

FORMULA SHEET

a=Bh Rectangular Prism: l x w

Cylinder Prism: π 𝒓 ²

What is It?!?!• Volume- The number of cubic units needed to ____________ a

space

• Volume is measured in:

FILL

CUBIC UNITS

The figure below is a net for a cube. Suppose the square labeled 3 is the bottom face of this cube. Which number names the square that will be the front face of the cube?

a. 1b. 2c. 5d. 6

How many vertices does the rectangular prism have?

a. 4b. 6c. 8d. 10

Find the volume of the cylinder.

a. 122.46 in3

b. 244.92 in3

c. 367.38 in3

d. 734.76 in3

Find the volume of the cone.

a. 37.68 m3

b. 452.16 m3

c. 113.04 m3

d. 169.56 m3

QuestionHow many 1 inch

cubes can fit in a box that is 5 inches wide, 2 inches tall and 10 inches long?

100 in³

QuestionMatch each situation with the appropriate units of measure. a. 25.3 cm2 _______ volume of a rectangular prismb. 25 m __________ perimeter of a large squarec. 5 mm __________ area of a parallelogramd. 25.64 ft3 __________ length of a mosquito

da

Cb

Find the Surface Area

72 in²

PRACTICE a. Top view b. Front view c. Right view