Download Comparing Circle Parts

Rates & Proportions
Packet
Name ______________
Period ______
1
Discovering a Proportion Property
A proportion is an equation made of two equivalent fractions.
DIRECTIONS: Use your knowledge of equivalent fractions and reasoning to find out which eight are
really equal. Scribble out the ? if you are sure the pair of fractions make a proportion.
a
2? 7

6 21
d
b
4? 6

8 10
1? 4

9 36
e
6?5

8 6

g
f
2? 6

5 15
4? 6

10 15
h
24 ? 6

8 2
j
i
9? 3

4 2
10 ? 15

6 9

k
3 ? 2

12 8

c
l
8 ? 12

10 20
12 ? 2

18 3


What relationship is always true about a proportion,

2
a c
 ?
b d
Discovering a Proportion Property cont.
DIRECTIONS: Use the relationship you described on the other side to find the missing number in each
proportion.
j
k
3 2

9
8 2

12


l
m
10

25 10
6


3
6

n
o
1

4 20
6


8
12

Explain how you found the missing numbers in these problems.
3
8 20

12
Choose and use a
variable.

Multiply the corners.
Multiply the two
numbers on one side.
Divide both sides to
solve the equation.




4
2 Ways to Solve Proportions
#1 Multiplying the Corners
Directions: Match the steps below by drawing a line to finish the sentence
The first step to solving proportions is
The second step is
The third step is
dividing both sides
multiplying the corners
turning it to an equation
Show the steps you need to use to solve the example below.
1st step
2nd step
3rd step
2
R
=
6
33
#2 The “Equivalent” Shortcut
For some proportions, you can use a shortcut to find the answer using fewer steps. This
shortcut uses equivalent fractions to solve for the variable. Equivalent fractions are
made by multiplying the numerator and denominator by the same number.
Directions: List 5 equivalent fractions for the fraction below.
1
4
=
=
=
=
=
Directions: Now solve the proportion below by making an equivalent fraction.
2
R
=
5
45
2  ___ 
5  ___ 
R
45
R = _______
How do you know when you can use the shortcut?
When one denominator or one numerator is a multiple of the other, you can use this
shortcut. (45 is a multiple of 5).
5
2 Ways to Solve Proportions cont.
Directions: Circle the proportions that can be solved using the shortcut. Solve these proportions first. Now solve
the remaining proportions using the regular method.
3
5
=
R
100
4
10
=
L
35
1
8
S
5
=
8
2
2
28
=
3
T
2
C
=
7
35
5
12
=
W
96
11
B
=
99
54
7
15
=
R
45
=
R
72
Ryan was absent today. Explain to him in words how you know when you can use the shortcut and
when you can’t.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
6
Proportions Practice
DIRECTIONS: Show that the two fractions are equivalent by multiplying the corners (or the X
pattern).
2 8

Example:
3 12
2 10

5 25
2 3

8 12
2 • 12 = 3 • 8
24 = 24
DIRECTIONS: Write the three steps to solving an equation. Then use the three steps to solve the
proportions below.
1. _______________________________________
2. _______________________________________
3. _______________________________________
2 10

5 d
3
p

12 16
2 3

g 12
4 h

6 21
5 25

g 30
1
h

2 20
7
3 6

8 t
3 9

7 t
Proportions Practice II
A proportion is _______________________________________.
DIRECTIONS:
1st - Make up a proportion of your own and write it in the bold box below, including the steps to
solve it.
2nd - When each person in your team has written a unique proportion, pass your paper to the person
on your left. Write your proportion, but not the steps to solve it, on your teammate’s paper in
one of their empty boxes.
rd
3 - Pass the papers again to the left and repeat until you have your own paper back. Then solve the
other three proportions.
8
Topic ________________
Topic Sentence
Concrete Detail
Concrete Detail
Concrete Detail
Commentary
Concluding Sentence
9
10
Proportions Skills Quiz 1
30

15 50
3 20

12


6
10

12 18


11

8
24
Rates on a Double-sided Numberline
A rate is a comparison of two amounts using division. One of the most
common rates is speed (distance compared to time).
Rates are usually expressed as a ratio.
RATES EXPERIMENT #1:
________ walks ___________ in ___________________
DIRECTIONS: Fill in this information on the double-sided number line below and follow directions to
use it to answer the question your teacher will ask.
0
0
Question from the teacher:
Proportion:
Equation:
Answer to the question in a sentence:
Use a double-sided number line, a proportion and an equation to solve
the question below.
1.
Marcos can swim 4 laps in 5 minutes. At this rate, how long will it take to swim 12 laps?
12
Rates on a Double-Sided Numberline II
A rate is a comparison of two amounts using division. Speeds may not
always involve distance but always have time as the second amount.
Rates are usually expressed as a ratio.
RATES EXPERIMENT #2:
________ claps ___________ times in ___________________
DIRECTIONS: Fill in this information on the double-sided number line below and follow directions to
use it to answer the question your teacher will ask.
0
0
Question from the teacher:
Proportion:
Equation:
Answer to the question in a sentence:
Use a double-sided number line, a proportion and an equation to solve
the question below.
1. Marisol was driving at 60 mph. At this rate, how long will it take to drive 330 miles?
13
Create Your Own Rates Problem
DIRECTIONS: In the indicated spot below, create a rate problem of your own. Use a double-sided
numberline, a proportion, and a sentence to explain your answer. When you are finished, draw a
picture to illustrate your problem.
My Rate Problem:______________________________________________
______________________________________________________________
______________________________________________________________
My Work:
14
Rates Check
For each problem, show:
 A double-sided number line
 A proportion
 A multiplication equation
 A solution
1. The factory can make 60 bicycles in 8 hours.
How many can it make 20 hours?
3. Maribel earns $34 in 4 hours of work. How
long will it take her to earn $85?
2. A car gets 24 miles per gallon of gasoline
(mi/gal). How many gallons of gasoline would
the car need to travel 144 miles?
4. Nicolas can run 3 miles in 20 minutes. How far
can he run in 2 hours?
15
Introduction to Inequalities
Compare these two mathematical sentences. One is an equation and one is an inequality.
x=3
x<3
We read: _______________________
We read: _______________________
On a number line the equation looks like:
On a number line the inequality looks like:
0
0
Here are two more complicated mathematical sentences that can be solved.
2x = -8
2x > -8
We read: _______________________
We read: _______________________
We solve using division:
We solve using division:
2x = -8
2x > -8
On a number line the solution looks like:
On a number line the solution looks like:
0
0
Big Ideas so far:
16
Here are two more mathematical sentences that need two steps to solve:
5x – 8 = 12
5x – 8 < 12
We read: _______________________
We read: _______________________
We solve using addition then division:
We solve using addition then division:
5x – 8 = 12
5x – 8 < 12
On a number line the solution looks like:
On a number line the solution looks like:
0
0
Inequalities are solved using the same inverse operations that are
used to solve equati ons. Instead of having one solution, the answer
to an inequality is a range of numbers.
Solve these inequalities as you would solve equations. Show the solution on the line.
A
2x – 9 < 17
B
4x + 6 < –2
0
C
𝑥
2
− 5 > −6
0
0
D
17
𝑥
2
+4>5
0
Conversions on a Double-Sided Numberline
A conversion ratio is another kind of rate. Instead of using time,
conversion ratios compare two of the same kinds of amounts
with different units.
Here are some common unit conversions you should know.
1 foot = 12 inches
1 pound = 16 ounces
1 meter = 100 centimeters
1 yard = 3 feet
Use the conversions above, a double-sided number line, a proportion and an
equation to solve each of the questions below.
1. How many ounces are in a 5 pound bag of potatoes?
2. A poster has a length of 30 inches. How many feet is that?
3. Yolanda is 152 centimeters tall. How many meters is this?
4. A football field is 120 yards long. How many feet is that?
**5. One papaya weighed 56 ounces and another papaya weighed 40 ounces. Together, how many
pounds do the two papayas weigh?
18
Multi-Step Rate Problems
Read and compare the two problems below.
Elizabeth rode 20 miles on Monday
and 25 miles on Tuesday. Her
average speed was 10 mph. How
much time did she spend riding?
Elizabeth rode her bike 45 miles. Her
average speed was 10 mph. How
much time did she spend riding?
Write what you notice: ____________________________________________________________
Solve the problems here:
Some multi-step problems can be done with two separate number lines.
On his first day driving across the
country, Miguel drove 380 miles in 6
hours. On his second day he went
470 miles in 8 hours. Which day had
the faster average speed? By about
how much?
Two of the three problems below are multi-step problems.
Multi or Single
Multi or Single
A plane flew 200 mph for
3 hours then 250 mph for 2
hours. How far did it fly?
A plane carrying 250
passengers can fly 300
mph. How long will it take
to fly 750 miles?
19
Multi or Single
A plane uses 55 gallons of
fuel per hour. If it flies for
4 hours in the morning and
2 hours in the afternoon,
how much fuel will it use?
Understanding a Strange Property of Inequalities
0
1
1) On the number line above, show the points A and B. A = 2 and B = 4.
Write an inequality using the letters A and B. ____________________
2) Describe what would happen to the location of A and B if both values were multiplied by 2.
_______________________________________________________________________________________________________________________
0
1
Show the location and write an inequality for the expressions 2A and 2B. ____________________
3) Forget about multiplying by 2 and describe what would happen to the location of A and B if 7 is subtracted
from both values.
_______________________________________________________________________________________________________________________
0
1
Show the location and write an inequality using the expressions A - 7 and B - 7. ______________
4) Forget about subtracting 7 and describe what would happen to the location of A and B if both values were
multiplied by -2.
_______________________________________________________________________________
0
1
Show the location and write an inequality for the expressions -2A and -2B. __________________
20
5) Forget about multiplying by -2 and describe what would happen to the location of A and B if both values
are divided by 2.
_______________________________________________________________________________________________________________________
0
1
𝐴
𝐵
Show the location and write an inequality for the expressions 2 and 2 . ______________________
6) Forget about dividing by 2 and describe what would happen to the location of A and B if both values are
divided by -2.
______________________________________________________________________________________________________________________
0
𝐴
1
𝐵
Show and write an inequality for the expressions −2 and −2. ______________________
Look back at the five operations that were done to A and B. List them here, along with noting if the
inequality remained the same or if it reversed directions.
Five operations done to A and B
Did the inequality stay or switch?
_________________________
___________________________
_________________________
___________________________
_________________________
___________________________
_________________________
___________________________
_________________________
___________________________
Explain clearly with a sentence the answer to this question. Under what conditions does an inequality
switch directions when an operation is done to both sides?
__________________________________________________________________________________
Explain why this happens.
__________________________________________________________________________________
21
Rates Practice Problems: Decide first Multi or Single?
1. Victor can finish 3 spelling words in a minute. He has 5 minutes before school and 10
minutes at nutrition to work on his spelling. How many words can he finish?
2. If Jocelyn’s sunflower grows 3 inches per day, how much does it grow in 2 weeks?
3. Jose’s Tacos sells 20 fish tacos for every 5 plates of nachos. If they sell 300 fish tacos
on Tuesday, how many plates of nachos did they sell?
4. To mix the perfect color of purple paint for her bedroom, Veronica needs a 2:4 ratio
of red to blue. If she uses 3 gallons of red paint, how many cans of blue does she use?
5. A snowboarder slides down a steep slope at a speed of 1400 feet per minute for 8 minutes and then
continues down an easier slope at 1000 feet per minute for 12 more minutes. How far did she come
down?
22
Product Units
Use this place to write information from the board about the class experiment.
Solve these problems.
1. A painter estimates that it will take 80 man-hours to prepare and paint a large
house. Give at least three combinations of numbers of workers and number of
hours that could do this job.
2. The factory can usually use all five of its machines to fill a large order in 6 days. If
two machines are down for repairs, how long will it take to fill the same size order?
23
Solving Equations and Inequalities
Solving ________________ and ________________ are both done using
inverse operations on both sides. The solutions look different, however, because
________________ have only a single answer while ________________ have a whole
range of answers. These solutions can be shown on a __________ __________.
We must be careful when solving inequalities because the direction will
________________ whenever we multiply or ________________ by a
________________ number.
Solve these inequalities and equations. Show the solution on the line.
A
𝑥
2
+5>7
B
0
C
𝑥
4
− 2 < −3
0
4𝑥 + 3 = 23
0
D
24
−3𝑥 − 4 < 5
0
Exponents Review
Name _________________________________
Write each in standard form.
1.
33 =
5.
5 2 =
2.
54 =
6.
 24 =
3.
4 3 =
7.
20=
2
2
4.  3  =
 


8.  

3
3

4 =
Write each in expanded form and then simplified exponential form.
9.
34  34 =
5
10. 5  5
3
4
11. 2  2
6
12. 10

2
=
=
 10 4 =
510
13.
52 =
612
14.
64 =


85
15. 8 5 =
43
16.
49 =
25

Match the equations with the graph. Show how you know they match. Make a graph for the leftover
equation and write an equation for the leftover graph.
Equations to choose from:
y = 3x - 1
y
y = 3x – 3
6
Equation:
3
Show how:
x
-6
-3
3
6
-3
-6
6
Equation:
3
Show how:
y
x
-6
-3
3
6
-3
-6
6
Equation:
3
Show how:
y
x
-6
-3
3
6
-3
-6
6
Equation:
3
Show how:
y
x
-6
-3
3
6
-3
-6
26
y = 2x – 2
Solving Equations and Inequalities Again
________________ equations and inequalities are both done using
____________ ____________ on both sides. The solutions look different, however,
because equations have only a single ________________ while inequalities have a
whole range of ________________. These solutions can be shown on a __________
__________.
We must be careful when solving ________________ because the direction
will ________________ whenever we ________________ or divide by a
________________ number.
Solve these inequalities and equations. Show the solution on the line.
A
𝑥
−3
+5>3
B
0
C
𝑥
5
− 6 = −7
0
5𝑥 − 3 > 17
0
D
27
−2𝑥 + 4 < 10
0
Scientific Notation
Standard Form
Conversion Work
28
Scientific Notation
Slope Practice Page
Follow your flow map to find the slope of each line below.
y
7
Line b
6
Line f
5
4
3
Line c
Cc
2
Line d
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-1
Line h
-2
Line e
-3
Line a
Line g
-4
-5
-6
-7
Slope of line a
Slope of line b
Slope of line c
Slope of line d
Slope of line e
Slope of line f
Slope of line g
Slope of line h
29
Pythagoras Individual Quiz
Name__________________________
32 =
42 =
72 =
82 =
122 =
162 =
172 =
252 =
The Pythagorean Theorem states,
“If you have a ______________ triangle,
then __________ + ____________ = hypotenuse2 .
Find the missing side in each right triangle below. Be sure to show your squares.
Explain clearly using complete
sentences how you did one of
the problems to the left.
13
___________________________________
12
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
9
___________________________________
12
___________________________________
___________________________________
___________________________________
___________________________________
30
Show your calculations to find out if any of these sets of numbers are Pythagorean
Triples.
4,
8,
5,
6
15,
17
3,
9,
4,
5
16,
25
Explain using a complete sentence what it means to be a Pythagorean Triple.
____________________________________________________________________
____________________________________________________________________
Bonus: Show how the Pythagorean Theorem can help you estimate the length of the
hypotenuse.
Explain your estimate:
______________________
?
______________________
5
10
______________________
______________________
_
31