**Notes**

**Problem 3.1: Rep-Tile Quadrilaterals**

**Problem 3.2: Rep-Tile Triangles**

**Problem 3.3: Scale Factors and Similar Shapes**

**Stretching and Shrinking (Understanding Similarity)**

**Investigation 1: Enlarging and Reducing Shapes**

Step 1

Investigation 2: Similar Figures

Mathematical and Problem Solving Goals

Investigation 3: Similar Polygons

Mathematical and Problem Solving Goals

Step 4

Conclusion

**Stretching and Shrinking Intro**

**Lets THINK!!!**

**Share your thinking with your shoulder partner.**

**Throughout this Unit...**

**Think about the following questions to further your understanding of Similar Figures.**

**1.) What is the same and what is different about 2 similar figures?**

**2.) What determines whether two shapes are similar?**

**3.) When figures are similar, how are the lengths, area and scale factor related?**

**4.) How can I use information about similar figures to solve a problem?**

Vocabulary for prior Units

congruent

parallel

parallelogram

polygon

probability

quadrilateral

ratio

tessellation

transformation

Essential Vocabulary for Unit 1: Stretching and Shrinking

Complementary Angles

Corresponding Angles

Corresponding Sides

Equivalent Ratios

Image

Ratio

Rep-Tile

Scale Factor

Similar

Supplementary Angles

Additional Vocabulary to Know

Fractal

Midpoint

Nested Triangles

Square

Square Root

**Problem 1.1**

**Solving a Mystery**

Essential Question

How can I use the KNOWN size of a smaller object to help estimate the UNKNOWN actual size of a larger object?

**Think about 2 things that are similar to each other.**

Ask yourself...

How are they similar?

How could you SHOW that they are similar?

Ask yourself...

How are they similar?

How could you SHOW that they are similar?

Launch - Solving a Mystery

As you read...

Think about what you know about photographs as they relate to similar figures...

How might this knowledge help you identify the mystery teacher at P.I. Middle School?

Read Page 5 with your shoulder partner and discuss your findings.

Think about the word...

Similar

In your Math Notebook...

Copy...

1.) The word and its definition

2.) A picture example

3.) Use the word in a sentence by describing the picture taken of the mystery teacher.

You MAY work with your shoulder partner!

Please look at the board to help you head your Math Notebook for today's lesson!

Explore!

Problem 1.1: Introduction to Similarity

With your shoulder partner, read PAGE 6.

Answer BOTH part A and part B

SHOW ALL WORK in your Math Notebook

Remember: Use any strategy you wish to help you solve the problem. You and your partner DO NOT have to use the same strategy.

I will be looking and listening for MATH discussions!

You should be using MATH vocabulary with your partner!

If you finish early... CHECK work with your table group.

HAPPY THINKING!!!

Summarize

What strategy did you use to solve Part A?

Hmmmm..... Think About....

What information do you have?

What are we trying to find?

How does the real-life teacher differ from the teacher in the photo?

Possible Strategies:

measure the mystery teacher's height using the length of the magazine as a unit. (teacher is 7 magazines tall)

The REAL magazine is 20 times the size of the one in the picture, so the mystery teacher must be also. (explained in a minute)

measure the height of the magazine and the height of the mystery teacher in inches. Then , divide the height of the teacher by the height of the magazine. More precise version of the first example.

A. 72.5 in. or 6ft 1/2 in. tall

In the picture, the height of "P.I Monthly" is 0.5 in. We know that the height of the real magazine in 10 in. So, the real magazine is 10/0.5 = 20 times larger than the picture. The teacher should also be 20 times larger in real life than he/she is in the picture. The teacher's height in the picture is about 3 5/8 in. So the actual height is about

20 x (3 5/8) = 72.5 in.

What was your answer for Part A? Explain your thinking.

What was your thinking for Part B?

Answers will vary, but something like this is expected.

The figures in the picture look the same as the original shapes EXCEPT in size. The objects in the picture have the same shape as the actual objects.

Homework

Math Book Page 12: ACE (1 and 2)

Math Book Page 14-15: Connections (8-12)

Copy and ANSWER the Essential Question:

How can I use the KNOWN size of a smaller object to help estimate the UNKNOWN actual size of a larger object?

Remember:

Perimeter = AROUND the figure (add up all sides)

Circumference = pi X r^2

Area of square/rectangle = l x w

Area of triangle = 1/2 (b x h)

Ticket Out The Door

1.) What would you expect the range of possible heights for the mystery teacher to be? ( smallest size - largest size) ***Think*** are my answers reasonable?

2.) Can you think of some other times when you might want to use a photograph to estimate the size of something?

3.) In the movie theater, the image of a person is taller than the real person. How can you use the same techniques we used today, to estimate someone's height using their image on the movie screen?

Please answer on a separate sheet of paper... NOT in your Math Notebook

I will be collecting these BEFORE you leave class today

1.1 - Solving a Mystery: Homework Answers

Essential Question:

answers will vary - I will check these weekly when I collect your Math Notebooks

ACE Questions:

1. a. 30 ft

b. 27 ft 6 in.

2. a. approx. 5 ft 7 in.

b. approx. 7 ft 2 1/2 in.

Connections Problems:

8. perimeter = 50 km

area = 131.25 km2

9. perimeter = 42m

area = 75m2

10. circumference = 55.29 m

area = 243.28m2

11. perimeter = 43mm

area = 75mm2

12. perimeter = 67.8cm

area = 125cm2

**Remember**

ALL work must be shown

ALL problems must be attempted

Place a STAR next to any problems you have questions on.

ALL Math Notebooks will be collected on FRIDAYS

**Problem 1.2**

**Stretching a Figure**

Head your Math Notebook for today's lesson

Essential Question

How can I make similar figures and compare the approximate measurements of corresponding parts?

Launch 1.2

Members of the Mystery Club want to make a poster that shows their logo. To do this, they need to ENLARGE the logo found on the flyer they have already designed.

Turn to Page 7 in your Math Book.

Read Page 7 with your shoulder partner.

Talk about Michelle's clever way to enlarge the drawing using rubber bands. How do you think she will do it?

Now, I will demonstrate one method for using rubber bands to enlarge a figure!! (TE page 19)

I have a super machine called a stretcher that will help me draw a copy of this figure. My machine has 2 parts. Watch me carefully while I make the stretcher before your eyes!

Materials Needed:

Labsheet 1.2A (right handed) or 1.2B (left handed)

#16 rubber bands (2 per person)

Blank Paper

Tape

Angle Rulers

Notice that I put one end of my stretcher on a point, called the anchor point, and hold it down securely without covering up any more of the band then necessary.

I put my marker through the other end and stretch the bands until the knot is just above the outside part of my figure.

I move my marker as I trace the figure with the knot, trying to keep the knot over the original figure.

I DO NOT look at the marker, ONLY the knot as I draw.

Look at my drawings, what happened? What can you tell me about them?

Explore 1.2

You are now going to make your own

STRETCHER!

Please do NOT work ahead!

I will come around to make sure you have functional stretcher.

Remember

TRACE the figure you are trying to enlarge with the knot of your stretcher.

Look at the knot while you trace... NOT your pencil.

Once you have traced the whole figure. Ask yourself the following questions...

1.) A stretcher made from 2 rubber bands lets us draw a figure that has been enlarged by a factor of ______?_______

2.) What would happen if I made a three-band stretcher?

3.) Do you get exactly the same drawing if you switch the ends of your 2 band stretcher?

4.) How could we use something like a rubber-band stretcher to make an image smaller than the original?

Discuss you answers with your table group!

Summarize 1.2

Image

Relationship

Corresponding

Corresponding Angles

Hmm... I wonder?

What do you notice about the original figure and the enlarged image?

The word IMAGE (during this lesson) refers to a drawing made using the stretcher.

What is the relationship between the side lengths of the original figure and the side lengths of the image?

Corresponding objects are those that appear in the same place in two similar situations. It often happens with angles as shown below. Angle A on the left is the corresponding angle to K on the right, because they are in the same location in the two similar shapes. We say A corresponds to K.

A

K

What is the relationship between the measures of the corresponding angles? How do you know?

1. How does the area change?

2. What happens if we change the anchor point?

Homework 1.2

Applications: Math Book Page 12 - 13 (#3)

Connections: Math Book Page 15 (#13 - #14)

Remember

the word OF means to multiply

SHOW ALL YOUR WORK

Math Notebooks can be collected at random

Ticket out the Door

Answer the Essential Question:

How can I make similar figures and compare the approximate measurements of corresponding parts?

1.2 Homework Answers: Stretching a Figure

#3:

a.) The new lengths are 2 (scale factor) times the original length

b.) The perimeter of the new figure is 2 (scale factor) times the original perimeter

c.) The angles remain the same

d.) Area of the new figure is 4 times the original area. It takes 4 copies of the original figure to cover its stretched image

#13:

a.) Diameter of the image circle is 2 times as long as the diameter of the original circle.

b.) Area of the image circle is 4 times as big as the area of the original circle.

c.) Circumference of the image circle is 2 times as long as the circumference of the original circle.

#14:

a.) 30 b.) 96 c.) 96 d.) 105 e.) 300 f.) 300

#15:

B

**Problem 1.3**

**Scaling Up and Down**

Essential Question

How can I use percents to describe size change, and then make accurate comparisons of the measurements of the similar figures?

Launch 1.3: Scaling Up and Down

Recap

Talk about the different methods we have used to stretch figures.

Think

If you wanted to make a really good enlargement of something, would you use the rubber-band stretcher?

What other ways do you know of to make a larger copy of something?

We use percents to enlarge or reduce figures, using a photocopier! The percents are put into the photocopier, and the machine enlarges or reduces the figure automatically.

Okay... Lets try!

Partner Reading:

Table Partners

Read and Discuss

Remember to use MATH vocabulary and language

Read Page 10 in your Math Book

Lets look at an example of enlarging and shrinking

TE page 28 (Transparency 1.3)

Estimate the percent Daphne entered into the photocopier in order to get the smaller image on the left. Write your answer in your Math Notebook.

Estimate the percent Daphne entered into the photocopier in order to get the larger image on the right. Write your answer in your Math Notebook.

Work with your table partner to answer these two questions.

When finished discussing page 10, You and your partner can answer the questions on page 11.

Explore 1.3

On page 10 in your Math Book

Answer A-E for Problem 1.3: Corresponding Sides and Angles

You may use both your ruler and angle ruler at your desk

I will be listening to your discussions, make sure they are math related!

I will be watching you measurement skills

Be sure you are comparing features such as;

length

area

angle size

You may compare with fractions or percents... But keep the Essential Question in mind... You will have to answer it!

Summarize 1.3

Lets talk about Problem 1.3

Lets look at

angle measures

side measures

area

Lets talk about strategies

Did you add or multiply?

Did you subtract or divide?

WHY?

Lets Think...

If you want to enlarge/reduce a figure by 25% will the image be larger or smaller than the original figure?

What number will I put into the copier?

multiply by

1.25

divide by

1.25

Homework 1.3: Scaling Up and Down

Make sure you show ALL your work (even on Multiple Choice Problems)

All notebooks will be collected on Fridays (graded)

Notebooks may be collected at random throughout the week.

Math Book page 13 and 14 (5, 6 and 7)

Math Book page 16 (19 and 20)

Ticket

out the

Door

1.) Essential Question:

How can I use percents to describe size change, and then make accurate comparisons of the measurements of the similar figures?

2.) How is the photocopy similar to the rubber-band method of creating similar figures?

3.) Any questions you have about the first 3 lessons? Anything you need help with, or you do not understand? You could also tell me the most interesting thing you have learned thus far.

Homework Answers: 1.3 - Scaling Up and Down

#5

a.) 50%: You could have used the side of a piece of paper to compare the side lengths of the floor plan.

b.) The line segments in the reduced plan are half as long as the corresponding line segments in the original plan

c.) Area of the whole house in the original plan is about 4 times the area of the reduced plan. The relationship between a room in the original plan and in the reduced plan is the same as the relationship between the whole plan.

#6

C, height to width ratio is the same as in the original figure.

#7

Angle measures do NOT change in each case. Side lengths and perimeter are:

a.) 2 times as long (200%)

b.) 1.5 times as long (150%)

c.) 1/2 times as long (50%)

d.) 3/4 times as long (75%)

#19

a.) Circumference = 25.13 cm Area = 50.27 cm2

b.)Radius = 6 cm Diameter = 12 cm Circumference = 37.7 cm Area = 113.1 cm2

c.) Radius = 2 cm Diameter = 4 cm Circumference = 12.57 cm Area = 12.57 cm2

#20

a.) Both statements are accurate

b.) You can use similar statements in comparing sizes of shapes. For example, for question 19b, you could say: "The Diameter of the image circle is 2 inches longer than the diameter of the original circle.

c.) The second method is more appropriate because each size will be enlarged or reduced by the same factor. However, the exact amount of increase or decrease of the lengths will be different.

Mathematical Reflection

Please write your answers in complete sentences.

Re-state the question in your answer, so your reader knows what you are answering.

Pretend you are the TEACHER... use math language and vocabulary to explain your answer.

You may (whisper) talk with your table if you get stuck, or have questions.

Everyone needs to write their own paper.

This will serve as your ticket out the door, and it WILL be graded!

Turn to page 20 in your math book, and begin

Homework - End of Stretching and Shrinking: Investigation 1

Math Book page 15 - 16 (16, 17, 18)

Math Book page 17-18 (21 and 24)

Remember to SHOW YOUR WORK

Answers to Homework - Investigations 1

#16

G

#17

C

#18

H

#21

a.) The width and height would be 2 times as large as the first picture

width = 6 ft

height = 4 ft

area = 24 square ft

b.) The width and height would be 1.5 times as large as the first picture

width = 4.5 ft

height = 3 ft

area = 13.5 square feet

#24

a. ) About 1.57 square in.

b.) About 1.57 square in.

c.) Path (1): along the outer circle

Path (2): along the outsides of the two smaller circles. Both paths are the same length (3.14 in. long each) You can see this by the similarity of the large circle to the smaller one. The scale factor from the smaller to the larger circle is 2. So, the circumference of the large circle is twice as long as the circumference of the small one. Hence, walking along half of the circumference of the large circle is the same distance as walking along the full circumference of the small one, the same length as path (2)

Use algebraic rules to produce similar figures on a coordinate grid

Focus attention on both lengths and angles as criteria for similarity

Contrast similar figures with non-similar figures

Understand the role multiplication plays in similarity relationships

Understand the effect on the image if a number as added to the x- and y- coordinates

Develop more formal ideas of the meaning of similarity, including the vocabulary of scale factor

Understand the relationships of angles, side lengths, perimeters and areas of similar polygons

**Problem 2.1**

Drawing Wumps

2 Days

Drawing Wumps

2 Days

**Problem 2.2**

Hats Off to the Wumps

Hats Off to the Wumps

**Problem 2.3**

Mouthing Off and Nosing Around

Mouthing Off and Nosing Around

**Mathematical Reflections**

Essential Question

How can I use algebraic rules to produce similar figures on a grid?

How can I compare these similar figures?

How can I show that 2 figures are NOT similar?

Launch 2.1: Drawing Wumps

How many of you know how to play Tic-Tac-Toe?

How many X's or O's do you need in a row to win?

Today, we will be playing a different game of Tic-Tac-Toe. You will need 5 in a row to win.

We will play EVEN number partners, against odd number partners. Even number partner will go first.

Please Draw the following shape on your desk (dry-erase marker please)

Explore 2.1

With a partner (at your table) Read pages 21 - 22.

I will come around with your materials while your read.

Answer Problem 2.1 (A-C). Parts A and B will be done on Coordinate paper, Part C will be answered in your Math Notebook

I will complete Mug Wump on the Board

Questions to think about (2.1)

1.) The points for Zug are found from the points for Mug. The rule is (2x,2y). What do you think this rule tells us to do to a Mug point to get a Zug point?

2.) What do the other rules tell you to do for Lug, Blug and Glug?

3.) Go through your table and compute the new value of the x and y for each point. Remember that you are always starting with Mug's x- and y- coordinates. Then locate the point and connect them in sets as you did for Mug.

A.) Multiply each coordinate by 2 or double the numbers for the coordinates

A.) For Lug, we multiply Mug's x-coordinate by 3 and keep Mug's y-coordinate the same. For Bug, we multiply each coordinate by 3. For Glug, we keep the x-coordinate the same and multiply the y-coordinate by 3.

A.) Answers will be placed on the board

Summarize 2.1: Drawing Wumps

1.) How would you describe to a friend the growth of the figures that you drew?

2.) Which figures seem to belong to the Wump family and which do not?

3.) Are Lug and Glug related? Did they grow into the same shape?

4.) In earlier units in CMP, we learned that both angles and he lengths of the edges help determine the shape of the figure. How do the corresponding angles of the 5 figures compare?

5.) How do the lengths in similar Wump's compare?

A.) They all increase in size. Some grew taller and wider, while one just grew taller and one just grew wider.

A.) Mug, Zug and Bug have the same shape, but Lug and Glug are distorted

A.) No: Lug is wide and short while Glug is narrow and tall.

A.) Watch as I compare the figures on the board. Remember when you measure angles , you measure the amount of turn between the edges, not the lengths of the edges. In Mug, Zug and Bug, the corresponding angles are equal. In Mug and Lug (or Glug), the corresponding angles (except the right angles) are NOT equal.

A.) In Mug, Bug and Zug, the corresponding legths grow the same way. They are multiplied by the same number. Notice, that if the coeffiecients of x- and y- are the same, the shapes are similar. If the coefficients are different, the figure will change more in ONE direction than the other and will be distorted. *** If you multiply x and y by 1 in order to get a new figure, the figures are similar... Even more.. they are CONGRUENT!. You will get a figure of exactly the same size and shape. You will see this in Problem 2.2

Homework: 2.1: Drawing Wumps

Math Book pages 28-29 (1 and 2)

Math Book page 33 (14 and 15)

Math Book pages 34-35 (20 - 28)

You will have 2 nights to complete these problems.

Please show ALL your work

Notebooks will be collected and Graded every FRIDAY!

2-DAY

Ticket Out the Door

How can I use algebraic rules to produce similar figures on a grid?

How can I compare these similar figures?

How can I show that 2 figures are NOT similar?

Homework Answers 2.1

#1

a.) Sum and Crum are imposters

b.) I will show these on the board

c.) Glum and Trum are members. Sum and Crum are imposters

d.) For Glum: Mouth lengths, nose lengths and perimeters are 1.5 times as long as the corresponding lengths Mug. The angles are the same. The areas are 2.25 times as large (1.5 x 1.5 = 2.25) The mouth height is 1.5 units and the width is 6 units. The nose width is 3 units and the height is 1.5 units

For Turn: Mouth lengths, nose lengths and perimeters are 4 times as long as the corresponding lengths and perimeter of Mug. The angles are the same. The areas are 16 times as large. The dimensions of the mouth are 16 units by 4 units and the nose has a width of 8 units and a height of 4 units.

e.) For Sum: The height of the mouth and the height of the nose are 2 times as long while the width of the mouth and the width of the nose are 3 times as long as the corresponding lengths of Mug. The mouth is 12 units wide and 2 units high and the nose is 6 units wide and 2 units high.

For Crum: The heights of the mouth and the nose are the same as the corresponding heights of Mug. The width of the mouth and nose is 2 times as long as the corresponding widths of Mugl

f.) Yes, the findings support the prediction that the imposters wil be Sum and Crum. Imposters are those who have different scale factors applied to both the x- and y-coordinates, while family members have the same scale factor applied.

2.)

a.) Answers will vary

b.) Answers will vary

c.) The rule is = multiply x- and y-coordinate by the same number k: (kx,ky)

d.) Choose different numbers multipplying the x- and y-coordinates: (kx,ky), where k is not equal to r.

14.) No; because 1 is NOY equal to 3/4. The image will look shorter because it will shrink vertically.

15.)

a.) (6,6)

b.) (9,6)

c.) (3/2, 1)

20.) 2 21.) 1/2 22.) 3/4 23.) 4/3 or 1 1/3

24.) 5/2 or 2 1/2 25.) 4

26.)

a.) .72 / .04 = 18 servings

b.) One possible answer = 4 x 4 1/2 = 18 servings

27.)

a.) .8 / .3 = 2 pizzas, and .66 of another. Or a remainder of .2 of the block of cheese.

b.) Will show on the board

28.)

a.) .5

Essential Question:

What effect does a number ADDED to the x- and y-coordinates have to the resulting image?

What is the role of multiplication in similarity relationships?

Cause

Effect

Launch 2.2

This problem is related to drawing the Wump family and imposters.

In this case, we will be looking at hats for the Wump family!

Remember... The MAIN POINT of the problem is to look back over your drawings and make sense of what adding OR multiplying (the rule) does to the image.

After you have drawn all your images (hats), your

CHALLENGE

is to find a way to

PREDICT

what will happen to the image

ONLY

by analyzing the

RULE

...

NOT

drawing the figure

!

Launch 2.2: Hats Off to the Wumps

Work with a Partner... You can choose (Remember to Choose someone who you will WORK with!!)

I will be listening to your discussions... You will have to work BY YOURSELF... if you and your chosen partner cannot work together.

Read and complete Problem 2.2 on pages 24-25 in your Math Book.

Remember... SHOW ALL YOUR WORK!

Hint... You can use different colors for each hat... This will help you differentiate (tell them apart)

Happy Thinking... You may begin.

Challenge!!!!!!!!!!!

1.) What rule would give the LARGEST possible image on the grids provided?

2.) Make up a rule that would place the image in another quadrant.

Summarize 2.2: Hats Off to the Wumps

1.) Are the images similar to the original? Why or why not?

A.) For (x + 2, y + 3), (x - 1, y + 4) and (.5x, .5y) the images are similar to the original because they are the same shape. For (x + 2, 3y) and (2x, 3y), the images DO NOT keep the same shape, so they are NOT similar.

2.) What rule would make a hat with line segments 1/3 the length of Hat 1's line segments?

A.) 1/3x + 2, 1/3y +3)

3.) What happens to a figure on a coordinate grid when you add to or subtract from its coordinates?

A.) It relocates the figure on the grid

4.) What rule would make a hat the same size as Hat 1, but moved up 2 units on the grid?

A.) (x + 2, y + 5)

5.) What rule would make a hat with line segments twice as long as Hat 1's line segments and moved 8 units to the right?

A.) (2x + 10, 2y + 3)

6.) Describe a rule that moves Hat 1 and does not produce a similar figure.

A.) (x + 4, 3y + 3) *** notice the coefficients are different ***

7.) What are the effects of multiplying each coordinate by a number

A.) If the numbers (coefficients) are the same, then the figures are similar. If the numbers (coefficients) are different, then the figures are not similar.

8.) What effect does the rule (5x - 5, 5y + 5) have on the original hat?

A.) The figure would be similar. Its sides will be 5 times as large and the image will be moved to the left five units and up 5 units.

9.) What about the rule (1/4x, 4y - 5/6)?

A.) This rule would NOT give a similar figure. The figure is shrunk horizontally and stretched vertically. It is also moved down 5/6 of a unit.

10.) Make up a rule that will shrink the figure, keep it similar and move it to the right and up.

A.) Many possibilities... One is...

(2/3x + 2, 2/3y + 1)

Homework: 2.2 - Hats Off to the Wumps

Math Book page 29 (3 and 4)

Math Book page 33 (16 and 17)

Math Book page 34 (18)

Answers...

3.)

a.) I will check your drawings

b.) The side lengths and perimeter of triangle PQR are 1.5 times the side length and perimeter of triangle ABC. The angle measures of triangle ABC and PQR are the same and the area of triangle PQR is 2.25 times (the scale factor squared) the area of triangle ABC

c.) In comparing triangle ABC to triangle FGH, the side lengths of triangle FGH grew by a different size scale factors. Therefore, the perimeter of triangle FGH did not grow by the same scale factor as the side lengths, and the angle measures are not the same. Finally, the area of triangle FGH is the same as the area of triangle ABC. (NOTE: doubling the base, and halving the height makes the areas equal.

d.) Triangle PQR is similar to triangle ABC since the corresponding lengths are enlarged by the same factor.

4.)

a.) I will check your drawings

b.) Choose any number k greater than 1. The rule is (kx, ky)

c.) Choose any positive number s smaller than 1. The rule is (sx, sy)

16.) A 17.) J

18.)

a.) (17/3, 9/4)

b.) (5/6, 1/6)

c.) (17/12, 1/20)

1.) What happens to a figure on a coordinate grid when you add to or subtract from it's coordinates?

2.) What are the effects of multiplying each coordinate by a number?

Ticket Out The Door

Essential Question

How can you decide if two figures are similar?

Problem 2.3: Mouthing Off and Nosing Around

In your Math Notebook.... Define

1.) similar

2.) scale factor

You will also need the following information

Please use a Frayer Diagram to help

Word

Define

Example

In your own words, write a sentence describing the word.

Non-Example

Use Page 25 in your Math Book to help you!!

Explore 2.3

Today, we will begin to form a more precise definition of the meaning of similar in mathematics.

You will use the idea of "same shape" to discover that similar figures have corresponding angles that have the same measure.

AND... corresponding sides grow by a common factor. (Scale Factor)

Turn to Page 26 in your Math Book

Work with a partner to answer 2.3 (A-F)

Launch 2.3 - Mouthing Off and Nosing Around

1.) How does Zug's hat compare with Mug's hat?

Review... using your HAT drawing from Problem 2.2

A.) Its side lengths are double that of Mug's hat.

2.) How many Mug hats can you put in Zug's hat?

A.) 4

3.) How do the perimeters compare between Mug and Zug?

A.) Zug's perimeter is double Mug's

4.) The hats of the set of figures are all made from a triangle and a rectangle, but they are not all similar. How can you tell if two rectangles (or triangles) are similar? What information should you collect?

A.) The measures of the corresponding angles are the same (equal or congruent). The lengths of the corresponding sides are related (these answers will differ). You should gather information on perimeter and area.

I will put the answers on the board. :-) (TE pg 48)

Summarize 2.3

1.) I want to grow a new Wump from Wump 1 (Mug). The scale factor is 9. What are the dimensions and perimeter of the new Wump's mouth?

A.) 36 x 9; p=90

2.) If the scale factor is 75, what are the measurements of the new mouth?

A.) 300 x 75

3.) Why are the dimensions 300 x 75? What rule would produce this figure?

A.) The scale factor tells what to multiply the old sides by to get the new sides. Since Mug's mouth is 4 by 1, the new mouth is 4 x 75 by 1 x 75 or 300 by 75, The rule is (75x, 75y)

4.) Why does the perimeter grow the same way as the lengths of the sides of rectangle?

A.) Perimeter is a length, so it behaves like the width and length. Perimeter = 2(l x w), if the scale factor is 2, then the new perimeter equals 2(2l x 2w)

5.) Lets go in the reverse direction. How can you find the scale factor from the original to the image if all you have are the dimensions of the two similar figures?

A.) Divide the length of a side of the image by the length of the corresponding side of the original figure.

Homework 2.3

Math Book pages 30 - 33 (5-13)

Math Book page 34 (19)

5.) D

6.) Z and T are similar. The comparison of small sides with each other and the larger sides with each other gives the scale factor, 2 or 1/2.

7.) a. answers will vary

b. answers will vary

c. The comparison of small sides with each other and the larger sides with each other gives the same scale factor.

8.) a. (1.5x, 1.5y)

b. (1/1.5x, 1/1.5y) or (2/3x, 2/3y)

c. (i) 1.5

(ii) The perimeter of B is 1.5 times as large as the perimeter of A and the area of B is 2.25 times as large as the area of A. The perimeter relationship is given by the same factor as the constant number multiplying the x- and y- coordinates, i.e., the scale factor. The area relationship is given by the square of this number.

d. (i) 1/1.5 or 2/3

(ii) The perimeter of A is 2/3 times as small ass the perimeter of B while the area of A is 4/9 times as small as the area of B. The perimeter is given by the same factor os the constant number multiplying the x- and y- coordinates. The area relationship is given by the square of this number.

9.) a. Corner points of C: (0,0) (o,4) (4,8) (12,4) (8,2) and (10,0)

b.) (i) scale factor = 2

(ii) perimeter of C is 2 times as large. The area of C is "square of 2" or 4 times as large.

c.) (i) 1/2

(ii) Perimeter of A is 1/2 times as small as the perimeter of B. The area of A is 1/4 times as small as the area of B.

(iii) (1/2x,1/2y)

10.) a. 2 b. 1.5 c. 2.5 d. .75

11.) a. Rectangles ABCD and IJKL seem to be similar. Triangles DFE and XYZ seem to be similar. You need to know angle measures to be sure they are similar.

b.) rectangle angles =

A and J (or L), B and I (or K), C and L (or J), D and K (or I)

rectangle sides =

AB and JI (or LK), BC and IL (or KJ), CD and LK (or JI), DA and KJ (or IL)

triangle angles =

F and Z, E and Y, D and X

triangle sides =

FE and ZY, ED and YX, DF and XZ

12.) a. (3x, 3y)

b.) The perimeter of EFGH varies because the perimeter of recatngle ABCD varies. It is 3 times as long as the perimeter of the rectangle ABCD.

c.) Area of rectangle EFGH is 9 times as large as the area of the rectangle ABCD.

d.) The answer to part (b) is the same as the scale factor and the answer to part (c) is the square of the scale factor.

13.) Answers will vary. You should mention the fact that the angles in the two figures are different from each other. In the figure on the left , the angles are all the same measure and obtuse. In the figure on the right, there are some obtuse angles and some acute angles

19.)

a.) About 662 km or 410 miles

b.) About 760 km or 470 miles

c.) The scale on the map gives the lengths of the two corresponding sides - one from the map and one from the real world. The ratio of those lengths gives the scale factor between the map and a fictitious map, which is similar to the first, but the size is the same as the distances in the real world.

You will have 2 nights to complete these problems!!

Ticket Out The Door

1.) On grid paper, draw a quadrilateral (or a parallelogram) that is NOT a rectangle

2.) Make a similar quadrilateral using a scale factor of 2.

3.) Compare the corresponding lengths of the two figures.

4.) Compare the measures of the corresponding angles.

5.) How can you decide if two figures are similar?

QUIZ!

OPEN BOOK

Read page 37 in your Math Book

Answer questions 1-4

You may work with a partner

1.) Construct similar quadrilaterals from smaller, congruent shapes.

2.) Connect the ratio of the areas of 2 similar figures to the scale factor.

3.) Construct similar triangles from smaller congruent figures.

4.) Generalize the relationship between scale factor and area.

5.) Generalize the relationship between scale factor and area to scale factors LESS than 1.

6.) Subdivide a figure into smaller, similar figures.

7.) Use scale factors to make similar shapes.

8.) Find missing measures in similar figures using scale factor.

**Essential Question**

**How can I connect the idea of the ratio of the areas of two similar figures to the scale factor? What is the relationship?**

Launch: 3.1 - Rep-Tile Quadrilaterals

Today, we are going to investigate several kinds of shapes to see which shapes are Rep-Tiles.

I am going to show you 2 different patterns with shapes.

The 1st one is a Rep-Tile, the second one is NOT a Rep-Tile!

Look carefully...

Discuss... What is a Rep-Tile?

What is different about the resulting figures that might make one a Rep-Tile and the other NOT a Rep-Tile?

In this problem, you are going to work with quadrilaterals. For each quadrilateral, your

CHALLENGE

is to determine whether it is possible to put together several identical small quadrilaterals to form a larger version of the same quadrilateral.

When it is possible, you will sketch how the shapes fit together and then answer some questions about the

measurements

of the figures.

Explore 3.1 - Rep-Tile Quadrilaterals

With a partner: Read and discuss page 38 - 39 in your Math Books

Complete Problem 3.1 in your Math Notebook. You may work with your table group to answer the questions on page 39.

You will be making a POSTER to display for the class!

I will have a sample poster up for you to use to help you format your table's poster

Summarize: 3.1 - Rep-Tile Quadrilaterals

Copy, then answer the following questions onto your poster. Neatness COUNTS!

Think about the following questions as you are working...

How does the side of the original rectangle compare to the rep-tile figure's side?

How do the angles of the original rectangle compare to the angles of the rep-tile figure?

Remember, the common factor is called the scale factor and it is used to scale up or down to make similar figures.

1.) How many more smaller rectangles do I need to make the next larger rep-tile figure? How many are there all together? (DRAW IT)

2.) How many more do I need to add to this Rep-Tile figure to make to make the next larger Rep-Tile figure? How many all together? (DRAW IT)

3.) Predict what will happen to the next Rep-Tile figure. (the 10th Rep-Tile figure?)

Homework - 3.1

Applications

Pages 44 - 45 (1-3)

Connections

Pages 50 - 51 (22 - 25)

Things to Remember:

If you add the measure of the angles in a triangle, you will always get 180. angle A + angle B + angle C = 180

A straight angle (straight line) has an angle measure of 180 degrees.

Use ratios to find the unknown side lengths of similar figures.

Multiply the numerator and denominator by the same number to find equivalent fractions

CHALLENGE

please work these problems out on a separate sheet of paper.

They are due when we take the Investigation 3 Quiz

1 bonus point for each CORRECT answer (MAX 10 points)

Math Book pages 53 - 56 (33-42) ALL

Ticket Out the Door

Groups should turn in their completed posters!

Homework Answers: 3.1

1.)

a.

No, they are not similar. One of the small figures is a square, so it does not have the same shape as the original rectangle, which is NOT a square.

b.

Yes, they are similar because their coorresponding interior angles are congruent. The side lengths of the larger shape are double that of the smaller shape. The scale factor is 2.

c.

Yes, they are similar because their corresponding interior angles are congruent. The side lengths of the larger shape are triple that of the smaller shape. The scale factor is 3.

d.

Yes, they are similar because their corresponding interior angles are congruent. The side lengths of the larger shape are double that of the smaller one. The scale factor is 2.

2.)

a.

3

b.

The area of the large rectangle is 9 times the area of the small rectangle. (you should have drawn a picture)

3.) a. 1/5 b. The area of the small rectangle is 1/25 the area of the large rectangle. (you should have drawn a picture)

22.)

a.

a=120, b= 60, c=60, d=120. e= 60, f=120, g=60

b.

supplementary angles MUST add up to 180 degrees

23.)

a

. 20

b.

90

c.

180-x

24.)

a.

6m since the scale factor from the smaller to the larger is 2, side RS is 6m

b.

10m; 10m = 5m x 2

c.

50

d.

50; since the sum of the angles s in triangle STR is 180 and 2 angles are known, 80 and angle y = 50, we know that angle R must be 180 - (80 + 50) = 50. Since the triangles are similar angle C is also 50 since it corresponds to angle R.

e.

R and Q, C and B, R and B, Q and C (complementary angles)

25.)

a.

6

b.

20

c.

8

d.

3

e.

60

f.

15

Essential Question

How can I generalize the relationship between scale factor and area, when the scale factor is GREATER than 1 or LESS than 1?

Launch 3.2 - Rep-Tile Triangles

In this Problem, you will be repeating yesterday's lesson, but with triangles.

Please cut out numerous copies of congruent triangles using the template I gave you.

You will be gluing these on construction paper in order to make Rep-Tiles.

You will also be comparing side length, area, perimeter and angle measurements. (Rulers ready!)

You will present your work to the class.

Explore and Summarize 3.2: Rep-Tile Triangles

Use Math Book pages 40-41

Answer the following questions to summarize...

1.) We seem to get square numbers for the totals each time. Why do these odd numbers add together to give square numbers?

2.) Suppose the area of one triangle is 15 square units and the scale factor between this triangle and a similar triangle is 2.5. What is the area of the similar triangle?

3.) Suppose the area of one triangle is 15 square units and the scale factor between this triangle and a similar triangle is 0.5. What is the area of the similar triangle?

Extend 3.2

Can you subdivide this triangle into smaller congruent triangles? What is the scale factor? How do the perimeters and areas compare?

Homework 3.2

Applications

Pages 45-46 (4-6)

Connections

Pages 51 - 53 (26-31)

Remember...

CHALLENGE

please work these problems out on a separate sheet of paper.

They are due when we take the Investigation 3 Quiz

1 bonus point for each CORRECT answer (MAX 10 points)

Math Book pages 53 - 56 (33-42) ALL

Ticket Out the Door

Turn in your Project!!!

Homework Answers 3.2

4.)

a.

Yes, Scale Factor = 2

b.

Small triangles on the left and right corners are similar, Scale Factor = 1/2, but the other two triangles are NOT similar

c.

NO

d.

Yes, Scale Factor = 2

5.) Shown on board

6.) Answers will vary... I will check them

26.)

a.

2

b.

0.5

c.

1.5

d.

1.25

e.

0.75

f.

0.25

27.)

a.

2/5 = 40/100 = 40% = 0.4

b.

3/4 = 75/100 = 75% = 0.75

c.

3/10 = 30/100 = 30% = 0.3

d.

1/4 = 25/100 = 25% = 0.25

e.

7/10 = 70/100 = 70% = 0.7

f.

7/20 = 35/100 = 35% = 0.35

g.

4/5 = 80/100 = 80% = 0.8

h.

7/8 = 87.5/100 = 87.5% = 0.875

i.

3/5 = 60/100 = 60% = 0.6

j.

15/20 = 75/100 = 75% = 0.75

28.)

a.

No, the birds are not similar (explain why)

b.

Yes, the figures are similar (Explain how you know)

c.

No, The figures are NOT similar (explain why)

d.

No, the lighthouses are NOT similar. (explain why)

29.) True; the corresponding angles will always be equal to each other since they are all 90 degrees and the ratio of any two sides of a square is 1.

30.) False; While the angles of any two rectangles will be the same (90 degrees), it is not the case that the ratios of sides will be equal.

31.) True; the fact that there is a consistent scale factor implies that the shapes are similar, and so the corresponding angle measures are equal. The fact that the scale factor is 1 means that the side lengths are unchanged. Equal angle measures and equal side lengths yield congruent figures.

Essential Question

How can I use a scale factor to find missing measures in similar figures?

Launch 3.3 - Scale Factor and Similar Shapes

Today, you will be challenged to make similar figures when you are given the scale factor, area or perimeter of a new figure!

You may work in pairs to complete today's problem.

Turn to pages 42 and 43 in your Math Book

Explore 3.3

I will be looking for the strategies that are being used during you exploration.

Ask yourself....

If the scale factor is 2.5, what will the new side length look like? How will it compare to the side length of the original figure?

Hint:

IF you notice the the new area is 9 times the original area...REMEMBER... area grows by the SQUARE of the scale factor. SO, in this case you would find the square root of 9 (which is 3), so the scale factors is 3!

Summarize 3.3

Pair up with another group (max. 4 to a group). Review your strategies and compare your answers.

Answer this question as a group... (everyone should record their answer!)

If this rectangle is enlarged by a scale factor of 3, what is the perimeter of the new rectangle?

What is the area of the new rectangle?

If you finish early... I have another CHALLENGE for you ... (Labsheet 3.3B)

7.5

3