Download Unit 1 Ratios and Proportional Relationships: Fractions

Unit 1 Ratios and Proportional Relationships:
Fractions and Ratios
Introduction
In this unit, students will review patterns, fractions, and ratios. Students will recognize when two
quantities are proportional, and use unit ratios to recognize equivalent ratios. Students will also
use tape diagrams to solve ratio problems, and solve word problems by writing and solving
proportions.
There are three ways in which ratios are more general than fractions:
1) A ratio allows comparisons between two parts, and does not just compare a part to the whole.
2) A ratio can compare more than two numbers.
3) The numbers in a ratio can be fractions or decimals.
In this unit, students will understand the first property of ratios, and will be exposed to the
second property in extensions. The third property of ratios will be addressed later in the year.
Signaling. In these lesson plans, we often suggest that all students signal their answers
simultaneously (e.g., by flashing thumbs up and thumbs down). For a complete description of
signaling see Introduction, p. A-20.
NOTE: Even though fractions often appear in line with the text in our lesson plans (e.g., 1/2),
remember to either always stack fractions when you show them to your students (e.g.,
1
)
2
or to
introduce the non-stacked notation.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-1
RP7-1
Patterns
Pages 1–2
Standards: preparation for 7.RP.2
Goals:
Students will use simple patterns to extend number sequences.
Prior Knowledge Required:
Can count forward and backward from any number between 0 and 100
Vocabulary: decreasing, gap, increasing, pattern, repeating, rule, sequence, term
Materials:
5 strips of paper for each student (see Extension 1)
a pair of scissors for each student (see Extension 1)
tape (see Extension 1)
BLM Patterns (Advanced) (p. B-62, see Extension 2)
BLM Pascal’s Triangle (p. B-63, see Extension 3)
Introduce patterns. Tell students to pretend that you are looking for house number 347 on a
street. You look left and see numbers 1, 3, and 5, and you look right and see numbers 2, 4, and
6. ASK: How can I predict which side of the street house 347 will be on? (the left, because that
is where the odd numbers are) Tell students that anything that helps you make predictions is a
pattern. Being able to make predictions is useful because some things are difficult or tedious to
check by hand. ASK: Do you want to check both sides of the street the whole way? (no) SAY:
From the pattern, we can see what will happen without checking.
Repeating, increasing, and decreasing patterns. Tell students that there are lots of ways to
make something predictable. Write on the board the following three patterns:
2, 3, 5, 2, 3, 5, ______
3 ,
21 ,
5 ,
7 ,
9 ,
19 , 17 ,
_____
15 , _____
Have volunteers predict the next term in each sequence. (2, 11, and 13) SAY: All of these terms
are easy to predict. In the first sequence, you are using the same numbers over and over again.
In the next sequence, you are doing the same thing over and over again—adding 2 each time.
Show this by writing “+2” in the circle between each pair of numbers. ASK: Why is the third
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-2
sequence easy to predict? (you are subtracting 2 each time) Show this by writing “–2” in the
circle between each pair of numbers.
Write each rule beside its pattern:
2, 3, 5, 2, 3, 5, ______
2, 3, 5, then repeat.
+2
+2 +2
+2
3 , 5 , 7 , 9 , _____
Start at 3 and add 2 each time.
–2
–2
21 , 19 , 17
Start at 21 and subtract 2 each time.
–2
–2
, 15 , _____
Exercises: Write the rule for the pattern.
a) 4, 1, 4, 1, 4, 1, …
b) 2, 3, 4, 5, 6, …
c) 20, 18, 16, 14, 12, …
d) even, odd, even, odd, …
e) 19, 18, 17, 16, …
Bonus: What is the rule for the pattern of the sequence of musical notes do, re, mi, …?
Answers: a) 4, 1, then repeat; b) start at 2 and add 1 each time; c) start at 20 and subtract 2
each time; d) even, odd, then repeat; e) start at 19 and subtract 1 each time; Bonus: do, re, mi,
fa, so, la, ti, then repeat
For each sequence above, ask students whether they would call it an increasing pattern, a
decreasing pattern, or a repeating pattern. (parts a) and d) are repeating, b) is increasing, and
c) and e) are decreasing) SAY: Some sequences do not have a rule. You need to check every
term to make sure the rule holds.
Exercises:
1. Does the sequence have a rule?
a) 1, 3, 1, 3, 1, …
b) 1, 3, 1, 3, 2, …
c) 20, 22, 24, 25, …
d), 20, 22, 24, 26, …
Answers: a) yes—1, 3, repeat; b) no; c) no; d) yes, start at 20 and add 2 each time
2. Write the first five terms of the pattern from the rule.
a) 2, 2, 3, then repeat
b) start at 8 and add 2 each time
c) 0, 9, then repeat
d) start at 30 and subtract 3 each time
Answers: a) 2, 2, 3, 2, 2; b) 8, 10, 12, 14, 16; c) 0, 9, 0, 9, 0; d) 30, 27, 24, 21, 18
3. Each sequence below has a rule like the ones above. Find the next term.
a) 2, 6, 10, 14, _____
b) 7, 6, 5, 4, ______
c) 8, 9, 10, 8, 9, ______
d) 3, 7, 11, 15, _____
e) 5, 11, 17, 23, _____
f) 34, 31, 28, _____
Bonus:
g) 99, 101, 103, _____
h) 654, 657, 660, _____
Answers: a) 18; b) 3; c) 10; d) 19; e) 29; f) 25; Bonus: g) 105, h) 663
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-3
4. Make a sequence and extend it to solve these problems.
a) A ski resort sells ski tickets for $30 a day and rents skis for $5 an hour. How much does it
cost to rent skis for 7 hours?
b) Ben is reading a book that is 94 pages long. He reads 6 pages a day. How much does he
have left after 5 days?
Bonus: By the end of August, Kate had saved $84. During the school year, she spends $4 a
month on school supplies. How much does she have left by the end of June?
Answers: a) $65, b) 64 pages, Bonus: $44
Sequences that don’t have patterns. Write on the board:
3, 7, 8, 11, 13, 25, _____
ASK: Can you predict the next term of the sequence? (no) Why not? (there isn’t a pattern) Tell
students that some sequences don’t have patterns, and it is just as important to notice when
something doesn’t have a pattern as when it does. SAY: When terms increase or decrease by
the same amount, it is easy to see the pattern, but when terms increase or decrease by different
amounts, there might not even be a pattern.
Exercises:
1. Describe the sequence as one of the following:
A: increases by the same amount
B: increases by different amounts
a) 9, 14, 19, 24, 29 b) 10, 12, 15, 17, 20 c) 210, 214, 218, 222
Answers: a) A, b) B, c) A
2. Describe the sequence as one of the following:
A: decreases by the same amount
B: decreases by different amounts
C: repeats
a) 31, 29, 26, 24, 21
b) 54, 47, 40, 33, 26
c) 8, 5, 1, 8, 5, 1
d) 70, 60, 40, 30, 10
Answers: a) B, b) A, c) C, d) B
3. Describe the sequence as one of the following:
A: decreases by the same amount
B: increases by different amounts
C: increases and decreases
a) 56, 54, 52, 50
b) 73, 83, 53, 63
c) 57, 62, 67, 72
Answers: a) A, b) C, c) B
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-4
More complicated sequences. Tell students that some sequences increase by different
amounts but still have a pattern. Write on the board:
2 ,
5 ,
9 ,
14 , ______, ______
Have a volunteer fill in the first three gaps. (+3, +4, +5) Ask whether anyone sees a pattern in
the gaps, and have a volunteer continue the sequence of gaps. (+6, +7) SAY: If you can find a
pattern in the gaps, then you can continue the gaps and that means you can continue the
sequence. Have a volunteer use the gaps to continue the sequence. (20, 27)
Exercises: Write the next term.
a) 1, 3, 6, 10, ______
b) 24, 23, 21, 18, ______
c) 3, 5, 9, 15, ______
Bonus: 200, 300, 500, 800, _____
Answers: a) 15, b) 14, c) 23, Bonus: 1,200
SAY: You will need to find the gaps between the gaps to solve this one.
Bonus: 0, 1, 4, 10, 20, 35, ____
Answer: 56
Now show students the following sequence:
3
,
4
, 7
,
11
,
18 ,
29
,
______
Ask a volunteer to fill in the first five gaps (+3, +4, +7, +11)—the first gap is left without a circle
on purpose. ASK: Does anyone see a pattern in the gaps? (they are the same as the sequence)
So what should the next gap be? (+18) Then what is the next term? PROMPT: What is 18 more
than 29? (47)
Exercises: Extend the sequence.
a) 7 , 9 , 16 , 25 , _____, _____.
b) 8 , 11 , 19 , 30 , _____, _____
Bonus: How can you get each term in the sequences above from the previous two terms?
Answers: a) 41, 66; b) 49, 79; Bonus: add them
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-5
Extensions
1. Each student will need three strips of paper 11″ × 1″, two strips of paper 22″ × 1″, a pair of
scissors, and tape. Students make five objects as follows:
i) Tape the ends of an 11″ × 1″ strip of paper so that it looks like a ring.
ii) Do the same thing with another 11″ × 1″ strip of paper, but this time turn one of the ends once
before you tape them together.
iii) Repeat with another 11″ × 1″ strip of paper, but this time turn one end twice before taping.
iv) Repeat with two more strips of paper, now 22″ × 1″ and, this time, turning one of the ends
three times and the other one four times.
Students investigate how many sides the resulting object has by trying to color only one side of
the object. Some objects (those made by turning one or three times) will have only one side!
These objects are called Mobius strips. Students can then predict what will happen with 5 turns
(one side) and 6 turns (two sides). What about 99 turns? (one side) 100 turns? (two sides)
If students enjoy this activity, there are other interesting activities that they can do with Mobius
strips, as an online search can reveal.
2. Have students complete BLM Patterns (Advanced). Students will discover the number of
lines that can join 8 dots (see the picture below) by extending the pattern that consists of the
number of lines that join 1 dot, the number of lines that join 2 dots, the number of lines that join
3 dots, and so on.
Answer:
Number of Dots
Number of Lines
1
0
2
1
3
3
4
6
5
10
6
15
7
21
8
28
3. Have students complete BLM Pascal’s Triangle. Students will use what they know about
adding odd and even numbers to predict what numbers in the next row are even or odd.
Answers: 1. by adding them; 2. odd + even is always odd; 3. the eighth row is: O, E, E, E, E, E,
E, E, O, so the ninth row will be: O, O, E, E, E, E, E, E, O, O; 4. all odd, because odd + even =
odd.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-6
(MP.8) 4. What is the next term in these patterns?
a) 3, 6, 12, 24, _____
b) 2, 6, 18, 54, _____
c) 2, 3, 6, 18, 108, _____
d) 2, 5, 10, 50, 500, 25,000, _____
Answers: a) 48, b) 162, c) 1,944 (multiply 18 × 108), d) 12,500,000
(MP.1) 5. These sequences are made by adding the same number each time. Find the missing
terms.
a) 3, ____, 11
b) 14, _____, ____, 20
c) 59, _____, _____, _____, 71
d) 100, _____, _____, _____, _____, _____, 850
Selected solution: d) Start by guessing to add 100 to each term. Doing so gets to 700, not 850.
Adding 150 each time gets to 1,000, so try adding 125, which works. Or, there are six additions
required to get from 100 to 850, a difference of 750, so each addition must be
750 ÷ 6 = 125.
Answers: a) 7; b) 16, 18; c) 62, 65, 68; d) 225, 350, 475, 600, 725
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-7
RP7-2
T-tables
Pages 3–4
Standards: preparation for 7.RP.2
Goals:
Students will use T-tables to extend patterns.
Students will also use T-tables to compare patterns where the part of the rule that tells you what
to do is the same.
Prior Knowledge Required:
Can extend a sequence made by adding or subtracting the same number
Can find the difference between two numbers
Can multiply by 2
Can divide small 2-digit numbers by 2
Vocabulary: increasing, decreasing, pattern, sequence, term, T-table, rule
Materials:
calculator
(MP.4) Using T-tables to see patterns. Draw on the board:
Figure 1
Figure 2
Figure 3
Tell students to pretend that they are babysitting a child who is making houses out of toothpicks,
and they want to give the child only the exact number of toothpicks they need for the next
house. Tell students that you want to know, without actually making the house, how many
toothpicks will be needed. Tell students you are going to make a T-table. NOTE: The lines in the
central part of the table resemble a “T,” hence the name “T-table.” Then draw on the board:
Figure Number
1
2
3
Number of Toothpicks
Ask volunteers to complete the table. (Number of Toothpicks: 6, 9, 12; gaps: +3, +3)
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-8
Demonstrate an easy way to count the number of toothpicks in each figure by grouping
toothpicks together in easy groups to add:
3
+3
+3
ASK: How does the number of toothpicks change as another floor is added to the house. (add 3
each time) Write “+3” in a circle between each pair of terms to show the gap between terms.
ASK: How many toothpicks are needed for the next house? (15)
In the following exercise, encourage students to group the toothpicks to make it easier to count.
Exercise: Make a T-table to determine how many toothpicks are in Figure 5.
Figure 1
Answer: 20
Figure 2
Figure 3
A T-table with rules for headings. Tell students that a T-table can have the rules as headings.
Write on the board:
Subtract 5
36
Add 4
3
SAY: The first row tells you what number to start at, and the column heading tells you what to
do. Demonstrate completing the first column and have a volunteer complete the second column.
(answers: first column: 36, 31, 26, 21; second column: 3, 7, 11, 15)
Exercise: Extend both sequences in the table to four terms.
Subtract 5
Add 4
30
30
Answers: first column: 30, 25, 20, 15; second column: 30, 34, 38, 42
Comparing sequences with the same “what to do” part of the rule—addition and
subtraction. Tell students that T-tables like the ones above are useful for comparing
sequences made from different rules. Remind students that a rule for a pattern has two parts: a
part that tells you where to start and a part that tells you what to do. Point students’ attention to
the exercise above, and ASK: What part of the rules are the same, the start part or the “what to
do” part? (the start part)
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-9
Write on the board the rules:
Start at 2 and add 4
Start at 5 and add 4
ASK: Which part is the same? (the part that tells you what to do)
Exercise: Complete the T-table.
Add 4
Add 4
2
5
Answers: first column: 2, 6, 10, 14; second column: 5, 9, 13, 17
Have a volunteer write their answers on the board for each sequence. Tell students that when
only the starting point in the rules are different—when the “what to do” part is the same—then
they can compare the sequences. ASK: How can you get the second sequence from the first
sequence? Suggest that students look at the rows. Cover up the rows you are not looking at so
students can concentrate on one row at a time. ASK: How can we get from 2 to 5? (add 3)
How can we get from 6 to 9? (add 3) Check the third and fourth rows.
Exercises: How can you get from the first sequence to the second sequence?
a)
b) Subtract 3 Subtract 3
Add 3
Add 3
0
4
51
53
3
7
48
50
6
10
45
47
9
13
42
44
c)
Add 10
4
14
24
34
Add 10
1
11
21
31
e)
Subtract 5
20
15
10
5
Subtract 5
18
13
8
3
d)
Add 7
3
10
17
24
Add 7
0
7
14
21
Answers: a) add 4, b) add 2, c) subtract 3, d) subtract 3, e) subtract 2
Rules with multiplying and dividing. Tell students that the rule for a pattern can involve
repeating multiplication or division, not just addition or subtraction.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-10
Exercises: Write the first four terms of the pattern.
a) Start at 2 and multiply by 3.
b) Start at 6 and multiply by 2.
c) Start at 32 and divide by 2.
d) Start at 0 and multiply by 4.
Answers: a) 2, 6, 18, 54; b) 6, 12, 24, 48; c) 32, 16, 8, 4; d) 0, 0, 0, 0
Comparing sequences with the same “what to do” part of the rule—multiplication and
division.
Exercises: How can you get from the first sequence to the second sequence?
a) Multiply by 2 Multiply by 2
b)
Divide by 2
Divide by 2
1
3
200
400
2
6
100
200
4
12
50
100
8
24
25
50
c)
d)
Multiply by 2 Multiply by 2
Divide by 2
Divide by 2
5
1
1,200
400
10
2
600
200
20
4
300
100
40
8
150
50
Answers: a) multiply by 3, b) multiply by 2, c) divide by 5, d) divide by 3
(MP.3, MP.4) Explaining why the relationship between the columns holds. Tell students
that when both sequences use adding the same number or subtracting the same number, you
can always get from one sequence to the other by adding or subtracting the same number.
Write on the board:
Add 4
8
12
Add 4
6
10
Tell students that there are two ways to get from 8 to 10 in the chart:
subtract 2
8
12
6
10
add 4
add 4
8
12
6
10
subtract 2
SAY: When you start with the same number and add or subtract the same numbers, it doesn’t
matter which operation you do first: 8 + 4 – 2 gives the same answer as 8 – 2 + 4. The same is
true when you multiply or divide. 8 × 4 ÷ 2 gives the same answer as 8 ÷ 2 × 4.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-11
(MP.3) Exercises:
1. a) Explain how you know that the ? in the table below can be solved by multiplying 6 × 9.
(MP.5) b) Which is easier, multiplying 6 × 9 or multiplying 18 × 3?
Multiply by 3
Multiply by 3
2
18
6
?
Answers: a) 2 × 9 × 3 = ? and 2 × 3 ×_____ = ?, so the _____ is 9; b) 6 × 9 will be easier for
most students because it is a known fact
2. Massimo made this table:
Multiply by 5
Multiply by 5
3
6
15
?
He says that because you add 3 in the first row, you have to add 3 in the second row, so ? = 18.
Explain how you know the answer is incorrect.
(MP.3) Bonus: Explain why the reasoning Massimo used is incorrect. Hint: Use the order of
operations in your explanation.
Answers: 6 × 5 is not 18, so the answer is incorrect, Bonus: When you combine addition and
multiplication, the answer will change depending on which operation you do first, so adding 3,
then multiplying by 5 gets a different answer than multiplying by 5 first, then adding 3.
Predicting terms from one sequence to another. Draw on the board:
Multiply by 2
6
Multiply by 2
2
768
Tell students you have a riddle for them: you extended the first sequence for many terms, and
you want students to figure out what the term is in a certain row but in the second sequence.
Ask students to explain how they find the answer. Students can use a calculator. (The rule to
get from the first sequence to the second is “divide by 3,” so the missing term is 768 ÷ 3 = 256.)
(MP.8) Exercises: The rows between the first term and the last term are hidden. What is the
missing number in the table?
a)
b)
Multiply by 3 Multiply by 3
Add 11
Add 11
5
10
5
2
1,310,720
1,732
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-12
c)
Subtract 5
1,000,000
Subtract 5
2,000,000
d)
Divide by 2
1,000,000
Divide by 2
3,000,000
45
15,625
Answers: a) 2,621,440; b) 1,729; c)1,000,045; d) 46,875
Extensions
1. a) How many 11s are in the sequence 1 3 3 5 5 5 7 7 7 7 …?
b) How many 7s are in the sequence 1 2 2 2 2 3 3 3 3 3 3 3 …?
Answers: a) 6, b) 19
(MP.7, MP.8) 2. a) What is the second sequence when the first sequence is 997?
Add 7
Subtract 7
500
500
997
b) Use a calculator and only one computation to find what the second sequence is when the first
sequence is 64.
Multiply by 2
Divide by 2
2
500,000
64
Solutions: a) The two sequences always add to 1,000, so the second sequence is 3 when the
first sequence is 997, or add 497 in the first column to get 997, so subtract 497 in the second
column; the answer is 3; b) 500,000 ÷ 32 or 1,000,000 ÷ 64. Either way, the answer is 15,625.
(MP.8) 3. A path can only go up or right along each edge. How many paths are there from A to
B in each figure? Use the pattern to determine how many paths there would be in Figures
5 and 6.
Figure 1
a)
B
A
b)
Figure 2
B
A
B
A
Figure 3
B
A
B
A
Figure 4
B
A
B
A
B
A
Answers: a) 1, 2, 3, 4, 5, 6; b) 1, 3, 6, 10, 15, 21
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-13
(MP.7) 4. a) Complete the chart.
1×2=
1
2×3=
1+2=
3×4=
1+2+3=
4×5=
1+2+3+4=
5×6=
1+2+3+4+5=
Answers: first column: 2, 6, 12, 20, 30; second column: 3, 6, 10, 15
b) How can you get the second column from the first?
Answer: b) divide by 2
c) What is 1 + 2 + 3 + 4 + … + 100?
Solution: c) The sum in part c) is the number in the 100th row of the second column. So the
answer will be half of the number in the 100th column. In row 100, the first column would be 100
× 101 = 10,100. So the sum in part c) is 10,100 ÷ 2 = 5,050.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-14
RP7-3
Lowest Common Multiples
Pages 5–6
Standards: preparation for 7.RP.A.2
Goals:
Students will identify multiples of a number, common multiples of two numbers, and the lowest
common multiple (LCM) of two numbers.
Prior Knowledge Required:
Knows the times tables
Vocabulary: common multiple, lowest common multiple (LCM), multiple, times table,
whole number
Materials:
BLM Multiples Charts (p. B-64)
Review whole numbers. Remind students that the whole numbers are the numbers 0, 1, 2, 3,
and so on. Write “whole number” on the board and contrast whole numbers to fractions, which
represent parts of a whole.
Introduce multiples. The multiples of a whole number are the numbers you get by multiplying
that number by another whole number. For example, 6 is a multiple of 2 because 2 × 3 = 6.
Write “multiple” and “multiply” on the board. Point out how similar the words are—only the last
letter of each is different—“e” versus “y.” Emphasize that this makes it easy to remember what a
multiple is: You get the multiples of a number by multiplying that number by whole numbers.
Skip counting to find multiples. List on the board the first five multiples of 4, including 0. Have
students continue the list by writing the next five multiples of 4. ASK: Is 26 a multiple of 4? (no)
How can you tell? (it is not on the list; it is between 24 and 28, which are right next to each
other) Point out that we can list all the multiples of 4 by skip counting: 0, 4, 8, …
Exercises: Skip count to decide whether each number is a multiple of 3.
a) 8
b) 15
c) 18
d) 22
(MP.6) Why we need multiples to be whole number multiples. Ask students to use a
calculator to calculate 2 × 1.5. ASK: What is 2 × 1.5? (3) Is 3 a multiple of 2? (no) Why not? (we
pass it when counting by 2s) But don’t we get 3 by multiplying a number by 2? (yes, but the
number we multiply 2 by is a decimal number, not a whole number)
Explain that if 2 times anything—even a decimal—was called a multiple of 2, then any number
would be a multiple of 2, and the definition wouldn’t be very useful.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-15
The special cases of 0 and 1.
(MP.3) Exercises:
a) Write a multiplication equation that proves that 0 is a multiple of 3.
b) Which numbers is 0 a multiple of? How do you know?
c) What are the multiples of 0? How do you know?
d) What are the multiples of 1? How do you know?
Answers: a) 0 × 3 = 0; b) all whole numbers, because any whole number can be multiplied by 0
to make 0; c) only 0; because 0 times anything is 0; d) all whole numbers, because counting by
1s gets all whole numbers
Common multiples. Draw two number lines to 12 on the board—one for the multiples of 2 and
the other for the multiples of 3. Mark the multiples of 2 on the first number line with Xs. Point out
that you start at 0 and mark every second number. To mark the multiples of 3, you start at 0 and
mark every third number. Have a volunteer do so. The number lines on the board should now
look like this:
2:
0
1
2
3
4
5
6
7
8
9
10
11 12
0
1
2
3
4
5
6
7
8
9
10
11 12
3:
Referring to the number lines, ASK: Which numbers are multiples of both 2 and 3?
(0, 6, and 12)
Exercises: Draw four number lines to 12 on grid paper.
a) What numbers on the number lines are multiples of both 2 and 5?
b) What numbers on the number lines are multiples of both 3 and 6?
Bonus: Draw number lines to 16 to find multiples of 3 and 5.
Answers: a) 0 and 10; b) 0, 6, and 12; Bonus: 0 and 15
Tell students that a number that is a multiple of two numbers is called a common multiple of the
two numbers. SAY: The numbers 2 and 3 both have 12 as a multiple, so that is something the
numbers 2 and 3 have in common. That is why 12 is called a common multiple of 2 and 3.
Introduce lowest common multiples. Remind students that 0 is a multiple of every number.
Because of that, it’s not interesting as a multiple. When we want to list multiples, we are
interested only in the multiples that are not 0. Then tell students that the smallest common
multiple of two numbers, not including 0, is called the lowest common multiple of the numbers.
Exercises: Find the lowest common multiple. Write your answer in sentence form. Example:
The lowest common multiple of 6 and 10 is 30.
a) 2 and 3
b) 2 and 5
c) 3 and 6
Answers: The lowest common multiple of … a) 2 and 3 is 6; b) 2 and 5 is 10; c) 3 and 6 is 6
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-16
Tell students that “lowest common multiple” is often written as LCM. Write on the board:
Lowest Common Multiple.
A shortcut for finding lowest common multiples. Have students list the multiples of 3 up to
3 × 10 (not including 0): 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. Point out that you can use the list to
find the lowest common multiple of 3 and 4 by finding the first multiple of 4 (not including 0) on
the list. (12)
Exercises: Use the list of multiples of 3 to find the LCM of …
a) 2 and 3
b) 3 and 5
c) 3 and 8
d) 3 and 6
e) 3 and 7
Answers: a) 6, b) 15, c) 24, d) 6, e) 21
(MP.3, MP.5) Listing multiples of the larger number is easier than listing multiples of the
smaller number. Tell students that you want to find the common multiples of 3 and 8. Write on
the board the multiples of 3 and have students stop you when they see a multiple of 8. (24)
Then list the multiples of 8 and have students stop you when they see a multiple of 3. (again,
24) ASK: Did you get the same answer both ways? (yes) SAY: It doesn’t matter which list you
make—the multiples of 3 or the multiples of 8—because you will get the same answer either
way. So you might as well do the one that requires less work. ASK: Which requires less work,
listing the multiples of 3 until you find a multiple of 8, or listing the multiples of 8 until you find a
multiple of 3? (listing the multiples of 8) Discuss why that is: 8 is more than 3, so it takes fewer
multiples of 8 than multiples of 3 to get to the same number.
Exercises:
1. List multiples of the larger number (not including 0) until you find a multiple of the smaller
number. This is the LCM of the two numbers.
a) 3 and 7
b) 4 and 8
c) 4 and 10
d) 5 and 9
e) 3 and 10
Answers: a) 7, 14, 21; b) 8; c) 10, 20; d) 9, 18, 27, 36, 45; e) 10, 20, 30
2. Use your knowledge of the times tables to answer these questions.
a) Is 70 a multiple of 9?
b) Is 36 a multiple of 4?
c) Is 28 a multiple of 3?
d) Is 32 a multiple of 8?
Answers: a) no, 70 is between 63 = 9 × 7 and 72 = 9 × 8; b) yes, 36 = 4 × 9;
c) no, 28 is between 27 = 3 × 9 and 30 = 3 × 10; d) yes, 32 = 4 × 8
Patterns in common multiples. Have students complete BLM Multiples Charts. The page
shows a list of all the multiples of 2, 3, 4, and 5, up to 10 times the number and asks students to
list the common multiples from the charts for the pairs of numbers. When all students complete
at least parts a) and b), take up the answers in a chart on the board. (answers follow in the chart
on the next page)
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-17
Numbers
First Few Common Multiples (not including 0)
a)
3 and 4
12, 24
b)
2 and 3
6, 12, 18
c)
3 and 5
15, 30
d)
2 and 5
10, 20
e)
2 and 4
4, 8, 12, 16, 20
f)
4 and 5
20, 40
(MP.8) Challenge students to determine from the table a way to get the second lowest common
multiple, not including 0, from the lowest common multiple. (double it) Point out that all the
common multiples of two numbers are the multiples of the lowest common multiple. For
example, the common multiples of 3 and 4 are all the multiples of 12. Change the title in the
table to “First Five Common Multiples (not including 0)” and have students continue their lists on
the BLM. (answers: a) 12, 24, 36, 48, 60, b) 6, 12, 18, 24, 30, c) 15, 30, 45, 60, 75, d) 10, 20,
30, 40, 50, e) 4, 8, 12, 16, 20, f) 20, 40, 60, 80, 100.)
Extensions
(MP.1, MP.3, MP.7) 1. Explore the patterns in the ones digits of the multiples of …
a) 2 and 8
b) 3 and 7
c) 4 and 6
What do you notice? In particular, if you know the pattern in the ones digits for multiples of 2,
how can you get the pattern in the ones digits for multiples of 8? Why is this the case?
Answers: a) The pattern for 2 is 0, 2, 4, 6, 8, repeat. The pattern for 8 is 0, 8, 6, 4, 2, repeat;
b) The pattern for 3 is 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, repeat. The pattern for 7 is 0, 7, 4, 1, 8, 5, 2, 9,
6, 3, repeat; c) The pattern for 4 is 0, 4, 8, 2, 6, repeat. The pattern for 6 is 0, 6, 2, 8, 4, repeat.
In each case, after 0, the patterns are the reverse of each other. For example, if you read 2, 4,
6, 8 (the pattern for 2) in reverse order, you get 8, 6, 4, 2 (the pattern for 8).
To understand the reason for the overall pattern, first note that each pair of numbers adds to 10.
Then look at the case of 2 and 8. We get the pattern for 2 by adding 2, so to reverse it we can
subtract 2. But adding 8 to a number gives the same ones digit as subtracting 2 from the same
number, because the results are 10 apart. For example, 15 + 8 = 23 and 15 − 2 = 13 both have
ones digit 3.
(MP.1, MP.8) 2. Calculate the LCM of 2 and various numbers. What is the LCM of 2 and any
even number? What is the LCM of 2 and any odd number?
Answer: The LCM of 2 and any even number is the even number itself. The LCM of 2 and any
odd number is double the odd number.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-18
3. (MP.4) a) There are two tasks that Nurse Rani has to perform in a hospital. One has to be
performed every 3 hours; the other has to be performed every 4 hours. Both tasks are first
performed at midnight. Rani cannot do both things at the same time, so she needs another
nurse to help her. When will she next need another nurse to help her?
b) Cam visits the library every 4th day of the month, starting on the 4th, and swims every 6th day
of the month, starting on the 6th. When will he swim and go to the library on the same day? How
many times in a month will Cam visit the library and the swimming pool on the same day?
Answers: a) at noon; b) on the 12th and 24th—twice in a month
(MP.1) 5. Find the LCM of 3, 4, and 5.
Answer: Any multiple of 3, 4, and 5 is also a multiple of 3 and 4, so it must be a multiple of 12.
Any multiple of both 12 and 5 must be a multiple of 60, so the LCM of 3, 4, and 5 is 60.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-19
RP7-4
Models of Fractions
Pages 7–8
Standards: preparation for 7.RP.A.2
Goals:
Students will understand fractions as equal parts of a whole, including when the whole is a set.
Vocabulary: denominator, fraction, numerator, ordinal numbers, part, whole
Materials:
A banana or other piece of food
Grid paper
Two-color counters (or coins with heads and tails as the two “colors”) or a geoboard and an
elastic for each student
A meter stick
Fractions refer to equal parts. Bring a banana (or some easily broken piece of food) to class.
NOTE: Check student records for possible food allergies before deciding what to bring.
Break it in two very unequal pieces. SAY: This is one of two pieces. Is this half the banana?
Why not? Emphasize that the parts have to be equal for either of the two pieces to be a half.
Introduce fraction notation. Tell students that we use fractions to name a part of a whole.
Fractions have two numbers: a top number, or the numerator, and a bottom number, or the
denominator. The denominator tells you how many equal parts are in the whole. The numerator
tells you how many of the equal parts are in the part you are naming. Write several fractions on
the board (e.g., 1/3, 2/5, 5/8), and have students signal the numerator or denominator.
Naming fractions. Do Exercise 1, part a) together as a class, then have students do the
remaining parts.
Exercises:
1. Name the fraction of the diagram that is shaded.
a)
b)
c)
Answers: a) 3/4, b) 2/6, c) 7/9
2. Which diagram shows 1/4? For each of the other diagrams, give a reason why you didn’t
choose it?
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-20
Answer: A has 1/4 shaded; B doesn’t have equal parts; C has five parts instead of four; D has
two shaded parts, not one.
3. Extend the lines to make equal parts. What fraction is shaded?
Bonus: Add lines to make equal parts. What fraction is shaded?
Answers: a) 3/8, b) 5/8, c) 1/4, Bonus: 5/16
The equal parts of a fraction don’t need to have the same shape, only equal area. Draw on
the board:
ASK: How many equal parts are in the left shape? (four) How many equal parts are in the right
shape? (four) What fraction of each shape is shaded (one fourth) SAY: The same fraction is
shaded, even though different shapes are shaded.
Exercises: What fraction of the whole shape is covered by …
a) the shaded triangle?
b) the shaded square?
Answers: a) 1/8, b) 1/8
Finding the whole from a part. Draw three rectangles on the board as shown: the first one
30 cm long, the second one 45 cm long, and the third one 60 cm long.
Tell students that the rectangle shown is 3/5 of a full rectangle and have a volunteer finish the
full rectangle, using a ruler or meter stick to measure each part. Emphasize that the given
rectangle is 3 out of 5 equal parts, so they need two more parts. For the third rectangle, the
volunteer will first need to break the rectangle into three equal parts using a meter stick, and
then create two more equal parts.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-21
Exercises: Use grid paper to draw three rectangles that are 6 grid squares long. Then draw the
whole if that rectangle is …
a) 2/3 of the whole
b) 2/5 of the whole
c) 3/5 of the whole
Answers: a) 9 squares long, b) 15 squares long, c) 10 squares long
Reading fraction names using ordinal numbers. Point out the connection between how we
say fractions and ordinal, or position, numbers: The denominator is said as an ordinal number
(e.g., one third, one fourth, one fifth). The only exception is that we don’t say “one second” for
1/2—we say “one half.” Write some fractions on the board and have volunteers read the fraction
names:
3
4
5
8
7
100
(three eighths, four sevenths, five hundredths)
Showing fractions on a number line. Tell students that fractions can be shown on a number
line. Draw on the board:
0
1
Remind students that the numbers on the number line show how far the location is from 0. So,
0 is 0 units away from 0 and 1 is 1 unit away from 0. Now divide the number line into two equal
parts. Point to the middle mark and SAY: This is one half as far from 0 as 1 is, because it is 1 of
2 equal parts away from 0. Label the point as 1/2:
1
2
0
1
Now draw on the board:
0
1
Ask volunteers to tell you what fraction of the distance from 0 to 1 each arrow covers. (1/3 and
2/3) Then label the distances on the number line:
0
1
3
2
3
1
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-22
Exercise: Draw a number line divided into 5 equal parts. Label each mark.
Answer:
0
1
5
2
5
3
5
4
5
1
You can take a fraction of a set. Tell students that a whole can be a group of people, such as
the students in your class. ASK: What fraction of students in our class are girls? Tell students
that every student is an equal part of the set, so they can find the fraction of students who are
girls by counting the number of girls (this will be the numerator) and the total number of students
(this will be the denominator). Find this fraction together.
Fractions of sets of shapes. Draw on the board:
Exercises:
a) What fraction of the shapes are …
i) shaded?
ii) squares?
iii) triangles?
b) What fraction of the triangles are shaded?
c) What fraction of the squares are shaded?
Bonus: What fraction of the big squares are unshaded?
Answers: a) i) 4/5, ii) 3/5, iii) 2/5; b) 1/2; c) 3/3 = 1, or all of them; Bonus: 2/3
Events as parts of a set. Point out that events can be parts of a set, too. ASK: A basketball
team played 5 games and won 2 of them. What fraction of the games did the team win? (2/5)
Exercises: Team A won 3 games and lost 1 game. Team B won 4 games, lost 1 game, and tied
2 games. For each team, answer these questions:
a) How many games did the team play altogether?
b) What fraction of the games did the team win?
c) What fraction did the team not lose?
Answers: Team A: a) 4, b) 3/4 c) 3/4; Team B: a) 7, b) 4/7, c) 6/7
Ensure that students understand that they must first determine the total number of members of
a set before they can find the fraction made by any part.
Complementary parts of a set. Draw on the board:
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-23
ASK: What fraction of the square is shaded? (5/9) What fraction is unshaded? (4/9) What do the
numerators add to? (9) SAY: The numerators add up to the same number as the denominator
because, together, the shaded and unshaded parts make the whole shape.
Exercises:
1. What fraction is unshaded if the fraction shaded is ...
3
4
1
b)
c)
a)
8
7
3
Answers: a) 5/8, b) 3/7, c) 2/3
2. A team won 7/12 of its games. If there are no ties, what fraction of the games did the team
lose?
Answer: 5/12
Extensions
1. Draw a 4 × 4 square on grid paper. Color half the square in as many ways as you can. One
way is shown below.
2. What word do you get when you combine …
a) the first 2/3 of “sun” and the first 1/2 of “person”?
b) the first 1/2 of “grease” and the first 1/2 of “ends”?
c) the first 1/2 of “wood” and the last 2/3 of “arm”?
Try making up your own questions.
Answers: a) super, b) green, c) worm
(MP.8) 3. What fraction of the multiples of 2, up to 30, are also multiples of 5? up to 50? up to
100? up to 3,000?
Answer: always 1/5
4. Draw the set: There are 7 triangles and squares altogether; 2/7 of the shapes are triangles,
3/7 of the shapes are shaded, and 2 triangles are shaded.
Answer: Students’ drawings should show five squares and two triangles. Both triangles and
one square should be shaded.
(MP.2, MP.3) 5. There are 9 circles and triangles.
a) Can you draw a set so that 7/9 are circles and 4/9 are striped?
b) Can you draw a set so that 7/9 are circles and 4/9 are triangles?
c) What is the same and what is different in parts a) and b).
Answers: a) yes; b) no, because the numerators add to more than the denominator; c) same: in
both parts, the fractions are the same; different: in a), it is possible for a shape to be both a
circle and striped; in b), it is not possible for a shape to be both a circle and a triangle.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-24
RP7-5
Comparing Fractions Using Equivalent Fractions
Pages 9–11
Standards: preparation for 7.RP.A.2
Goals:
Students will generate equivalent fractions by multiplying the numerator and denominator by the
same number.
Students will compare fractions with unlike denominators.
Prior Knowledge Required:
Can name fractions from a picture
Understands that fractions show same-sized parts
Vocabulary: equivalent fraction
Materials:
BLM Fraction Strips (p. B-65)
Comparing fractions using a picture. Show several pairs of fractions on fraction strips from
BLM Fraction Strips (photocopy the BLM onto a transparency, cut out the strips, and display
them in pairs on an overhead projector). Have students name the fractions represented and say
which fraction is greater. Do not include any examples of equivalent fractions yet. Sample pairs:
1
3
and
2
5
5
8
and
10
12
6
6
and
10
12
Comparing fractions with the same denominator. Now include examples of fractions that
have the same denominator. Sample pairs:
5
6
and
10
10
8
6
and
12
12
Point out that eight twelfths of something is always greater than six twelfths of the same thing,
because 8 is more than 6—eight pieces of size one twelfth is more than six pieces of size one
twelfth. ASK: Which sign, < or >, goes between the two fractions? (>) Remind students that the
bigger (wider) side of the sign faces toward the bigger (greater) number.
Write on the board:
3/5
1/5
SAY: Fifths are like any other object—three of them is more than one of them. ASK: Which is
more: two eighths or three eighths? (three eighths) four sevenths or three sevenths?
(four sevenths)
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-25
Exercises: Write the correct sign between each pair.
Answers: a) >, b) <, c) >
Introduce equivalent fractions. Have volunteers compare several pairs of fractions on fraction
strips from BLM Fraction Strips again, but this time include examples in which the two
fractions are equivalent and examples in which they are not. Sample pairs:
1
5
and
2
10
6
3
and
9
5
6
8
and
9
12
3
6
and
5
10
2
6
and
3
10
2
8
and
3
12
1
6
and
2
12
(sample answers: 1/2 = 5/10, 6/9 > 3/5, 6/9 = 8/12, 3/5 = 6/10, 2/3 > 6/10, 2/3 = 8/12, 1/2 =
6/12)
Tell students that when two fractions look different, but actually show the same amount, they
are called equivalent fractions.
Creating equivalent fractions by breaking each part into two parts. Draw on the board:
Have students name the fraction that is shaded. (3/4) Write the fraction under the picture. Then
draw an identical picture and tell students you are going to break the shaded pieces into two
parts:
SAY: Now 6 of 7 parts are shaded. ASK: Does this show 6/7? (no) Why not? (not all the parts
are the same size) Point out that if we want the picture to show a fraction, we need all the parts
to be the same size. So if we break the shaded parts into two, we need to break all the other
parts into two as well. Extend the line to show:
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-26
SAY: Now there are twice as many shaded parts and twice as many parts altogether. ASK:
What fraction does this picture show? (6/8) Write the fraction under the picture.
Point out that by breaking all the equal parts into halves, we keep the parts equal, but now we
have a different name for the fraction.
Exercises: Break each part of the circle in half to create a pair of equivalent fractions.
Answers:
Creating equivalent fractions by breaking each part into any number of parts.
Draw on the board:
Have volunteers name the equivalent fractions shown by the pictures.
(2/3 = 4/6 = 6/9 = 8/12 = 10/15) Then write on the board:
For each picture, ASK: How many times more shaded pieces are there? How many times more
pieces are there altogether? SAY: When all the pieces—shaded and unshaded—were divided
into 4, there are 4 times as many shaded pieces and 4 times as many pieces altogether. That
means if you multiply the bottom of a fraction by 4, you need to multiply the top of the fraction by
4 to make the fractions equivalent.
Exercise: For the diagrams below, divide each piece into equal parts so that there is a total of
12 pieces. Then write the equivalent fractions with the multiplication statements for the
numerators and denominators.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-27
Finding equivalent fractions by multiplying the numerator and denominator by the same
number. Refer to the exercises above, pointing out that now that the denominators are all the
same, the fractions are easy to compare. Write on the board:
3
8 10
<
<
12 12 12
So
<
<
Have students write the fractions in order from least to greatest and have a volunteer write the
answer on the board.
1 2 5
< <
4 3 6
Now write on the board:
3
=
4
5
=
7
SAY: The denominator we use has to be a multiple of both 4 and 7 and it’s easier to use the
smallest number we can. ASK: What is the lowest common multiple of 4 and 7? (28) Write this
denominator in both fractions:
3
=
4 28
5
=
7 28
Have volunteers fill in the missing numerators.
Exercises: Compare the fractions by using the LCM of the denominators.
SAY: In these exercises, only one denominator needs to change.
SAY: In these exercises, both denominators need to change.
SAY: For these exercises, it will be a bit trickier to find the smallest common multiple of the
denominators. You can’t just multiply the denominators.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-28
Extensions
1. What numbers can go in the box:
4
?
>
9 45
Answer: 0, 1, 2, …, 19
2. Explain how you can use your fingers and your hands to show that 1/2 and 5/10 are
equivalent fractions.
3. (MP.1) a) Find a fraction equivalent to 2/3 so that:
i) its denominator is 3 more than its numerator
ii) its denominator is a multiple of 5
iii) its numerator and denominator add to 20
(MP.3) b) Can you find a fraction equivalent to 3/5 whose denominator is 17 more than its
numerator? Explain how you know.
Answers: a) i) 6/9, ii) sample answer: 10/15, iii) 8/12; b) No, because for each fraction
equivalent to 3/5, the difference between its denominator and numerator is an even number.
4. Compare 151/300 and 201/400 by comparing how much more than 1/2 each fraction is.
Answer: 151/300 > 201/400
(MP.3, MP.4) 5. You can use red and white modeling clay to compare the fractions 2/3 and 3/5:
Make a ball that is 2/3 red (2 parts red and 1 part white) and another ball that is 3/5 red (3 parts
red and 2 parts white). The ball that is darker red has the greater fraction of red clay.
How could you use a drink mix powder, water, and your taste buds to decide which fraction is
greater: 3/10 or 1/4?
Answers: Make a drink that is 3/10 drink mix powder (3 parts powder and 7 parts water) and
another drink that is 1/4 drink mix powder (1 part powder and 3 parts water). The drink that
tastes stronger has a greater fraction of drink mix.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-29
RP7-6
Fractions and Ratios
Pages 12–13
Standards: 7.RP.A.2
Goals:
Students will understand ratios as a way to compare one part of a whole to a different part of a
whole.
Prior Knowledge Required:
Can interpret fractions
Vocabulary: part-to-part, part-to-whole, ratio
Ratios compare parts to parts. Draw on the board:
ASK: What fraction of the shapes are squares? (3/5) What fraction are circles? (2/5) SAY: If I
want to know how much of the whole set are squares, I can use fractions. But if I want to
compare the squares to the circles, I can’t use fractions because the squares are not a part of
the set of circles. If I want to compare two parts, I need to use ratios. Write on the board:
The ratio of squares to circles is 3 to 2 or 3 : 2.
Exercises: Write the ratios for the set below.
a) circles to squares
b) circles to triangles
c) triangles to squares
Bonus: circles to polygons
Answers: a) 2 : 3 or 2 to 3, b) 2 : 1 or 2 to 1, c) 1 : 3 or 1 to 3, Bonus: 2 : 4 or 2 to 4
Ratios can compare parts to wholes. SAY: You can also use ratios to compare parts to the
whole.
Exercises: A team won 2 games, lost 3, and tied 1 game. What is the ratio of …
a) games won to games played?
b) games tied to games played?
c) games lost to games played?
Answers: a) 2 : 6, b) 1 : 6, c) 3 : 6
SAY: A ratio is called a part-to-part ratio when it compares one part to another part. A ratio is
called a part-to-whole ratio when it compares a part to the whole.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-30
Exercises:
1. Is the ratio part-to-part or part-to-whole?
a) the ratio of girls to boys in a class
b) the ratio of girls to students in a class
c) the ratio of vowels to letters in the word “ratio”
d) the ratio of vowels to consonants in the word “ratio”
e) the ratio of triangles to squares in a group of shapes
f) the ratio of triangles to polygons in a group of polygons
g) the ratio of circles to polygons in a group of shapes
Answers: a) part-to-part, b) part-to-whole, c) part-to-whole, d) part-to-part, e) part-to-part,
f) part-to-whole, g) part-to-part
2. Find the part-to-whole ratio.
a) circles to all shapes
b) shaded circles to all circles
c) small circles to all circles
d) squares to polygons
Answers: a) 2 : 5, b) 6 : 7, c) 3 : 7, d) 3 : 7
Part-to-whole ratios as fractions.
Exercises: For each set above, find the fraction of the set that is …
a) circles.
b) shaded circles.
c) small circles.
d) squares.
Answers: a) 2/5, b) 6/7, c) 3/7, d) 3/7
(MP.7) Compare the fractions to the part-to-whole ratios. They are the same! Emphasize that a
part-to-whole ratio is just like a fraction, but with a colon between the part and the whole instead
of a dividing line. A fraction is a special kind of ratio, so ratios are more general than fractions
and extend the concept. We’ll see later that you can compare and order ratios just as you can
fractions.
Exercises: Write a ratio and a fraction.
a) small circles to circles = ____ : _____
_____ of the circles are small
b) shaded circles to circles = _____ : _____
_____ of the circles are shaded
c) striped circles to circles = ____ : _____
_____ of the circles are striped
Answers: a) 7 : 10 and 7/10, b) 3 : 10 and 3/10, c) 4 : 10 and 4/10
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-31
Changing part-to-part ratios to part-to-whole ratios and fractions. Tell students that if a set
only has two types of things, such as circles and triangles or girls and boys, then you can find a
part-to-whole ratio from knowing only the part-to-part ratio. Write on the board:
There are 3 girls for every 5 boys.
g g g b b b b b
So there are 3 girls for every _____ students.
ASK: If there are 3 girls and 5 boys, how many students are there altogether? (8) SAY: So if
there are 3 girls for every 5 boys, then there are 3 girls for every 8 students. Write “8” in the
blank. Tell students that you just made a part-to-whole ratio from a part-to-part ratio: the ratio of
girls to students is 3 : 8. ASK: What is the fraction of students who are girls? (3/8) What is the
fraction of students who are boys? (5/8)
Exercises: Find two part-to-whole fractions from each part-to-part ratio:
a) The ratio of girls to boys is 3 : 4.
b) The ratio of circles to triangles is 2 : 5.
c) The ratio of a team’s wins to losses is 7 : 5. Ties are not allowed.
d) The ratio of left-handed to right-handed people is 2 : 7.
Answers: a) 3/7 of the students are girls and 4/7 are boys, b) 2/7 of the shapes are circles and
5/7 are triangles, c) 7/12 of the games were won and 5/12 were lost, d) 2/9 of the people are
left-handed and 7/9 are right-handed
Deciding which is more.
Exercises: Complete the chart.
Ratio of circles Fraction that
to squares
are circles
2
a)
2:1
3
b)
2:3
c)
3:5
d)
1:4
e)
4:1
Picture
Are there more
circles or squares?
circles
(MP.3) ASK: How can you tell from the ratio whether there are more circles or squares? (if the
first number is bigger than the second number) How can you tell from the fraction whether there
are more circles or squares? (if the fraction that is circles is more than half, there are more
circles than squares)
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-32
Exercises:
a) If the ratio of girls to boys is 74 : 71, are there more girls or boys? b) The ratio of parks to
schools in City A is 19 : 21 and in City B it is 12 : 11. Which city has more schools than parks?
Answers: a) girls, b) City A
Extensions
1. Describe the set in three different ways using the ratio 2 : 5.
Answer: 2 : 5 = circles to squares = big to small = shaded to unshaded
2. Introduce ratios of 3 or more quantities, and then have students find the ratio of circles to
squares to triangles in the set below.
Point out that fractions cannot compare three numbers at a time, so ratios are more general
than fractions.
Answer: 3 : 2 : 4
(MP.1) 3. Draw a set of shapes with the following ratios:
a) shapes to circles = 5 : 3
circles to squares = 3 : 1
b) circles to polygons = 3 : 5
squares to circles = 2 : 3
Sample answers:
a)
triangles to shapes = 1 : 5
triangles to squares = 1 : 2
b)
(MP.3) 4. Can there be a set of shapes with the ratio “squares to polygons = 3 : 2”? Explain.
Answer: No, because the squares are a part of the set of polygons, so there cannot be more
squares than polygons.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-33
RP7-7
Equivalent Ratios
Pages 14–16
Standards: 7.RP.2
Goals:
Students will understand ratios as a comparison through multiplication and will create equivalent
ratios by skip counting.
Students will use equivalent ratios to solve one-step word problems.
Prior Knowledge Required:
Can name fractions equivalent to a given fraction
Vocabulary: equivalent ratios
Tell students that a recipe for orange-banana juice calls for 3 oranges and a banana, and draw
on the board:
Tell students that you want to make lots of juice. ASK: If I use two bananas, how many oranges
would I need? (6) Show this on the board, too:
SAY: The recipe looks like it calls for just 3 oranges and 1 banana, but what it actually says is
that for every banana you use, you need 3 oranges. SAY: The ratio of oranges to bananas is 3
to 1, but it is also 6 to 2. ASK: If I use 1 more banana, how many more oranges would I need?
(3) Have a volunteer draw it on the board. ASK: Now what ratio is showing? (9 oranges to
3 bananas) Write on the board:
oranges to bananas = 3 : 1 = 6 : 2 = 9 : 3
Point out that the ratio 3 to 1 doesn’t just describe a single situation, but a whole sequence of
possible situations. Ask a volunteer to continue the picture to find the next ratio it’s also
showing. (12 : 4) SAY: All these ratios are equivalent.
In the exercise below, some students might benefit from using red blocks and blue blocks
instead of drawing pictures.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-34
Exercise: Finish the picture so that there are 2 shaded circles for every unshaded circle. Then
write a sequence of equivalent ratios.
Answer:
So 2 : 1 = 4 : 2 = 6 : 3 = 8 : 4
Then challenge a volunteer to find the next ratio without using the picture. (10 : 5) ASK: How did
the volunteer know that? (add 2 more shaded circles and 1 more unshaded circle) Point out that
students can skip count by 2s to continue the first sequence and by 1s to continue the second
sequence.
Exercises: Skip count to write two more equivalent ratios.
Bonus: 1 : 5,000
a) 4 : 1
b) 1 : 3
c) 10 : 1
Answers: a) 8 : 2 and 12 : 3, b) 2 : 6 and 3 : 9, c) 20 : 2 and 30 : 3, Bonus: 2 : 10,000 and 3 :
15,000
Tell students that a recipe for making glue calls for 2 cups of flour and 3 cups of water. ASK: If I
use 4 cups of flour, how much water would I need? (6 cups) Explain that you find cups of flour
and water hard to draw, so you will just use circles for cups of flour and squares for cups of
water. Draw on the board:
Remind students that the recipe actually says that you need 2 cups of flour for every 3 cups of
water. So if you add 2 more cups of flour, you need 3 more cups of water. Add 2 more circles to
the drawing. ASK: How many squares do I need to draw? (3) Do so, then write on the board:
2 : 3 = _____ : ______
Ask a volunteer to write the equivalent ratio from the picture. (4 : 6)
Exercises:
1. Draw 3 circles and 5 squares, twice. Then write two equivalent ratios.
Answer:
3 : 5 = 6 : 10
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-35
2. There are three apples for every four bananas. Draw a picture to write a sequence of three
equivalent ratios.
Answer:
So 3 : 4 = 6 : 8 = 9 : 12.
SAY: Every time you add three apples, you add four bananas, so you can skip count by 3s to
get the apple part of the ratio, and skip count by 4s to get the banana part of the ratio.
In the exercises below, some students might find keeping track of the skip counting easier if
they write the equivalent ratios in a vertical format instead of a horizontal format. For example,
in part a) …
4:5
8 : 10
12 : 15
Exercises:
1. Skip count to write two more equivalent ratios.
a) 4 : 5
b) 3 : 10
c) 5 : 2
Bonus: 21 : 13
Answers: a) 4 : 5 = 8 : 10 = 12 : 15, b) 3 : 10 = 6 : 20 = 9 : 30, c) 5 : 2 = 10 : 4 = 15 : 6,
Bonus: 21 : 13 = 42 : 26 = 63 : 39
2. Skip count to write a sequence of three equivalent ratios for the situation.
a) There are 3 cups of oatmeal for every 5 cups of raisins.
b) There are 3 cups of flour for every 10 mL of vanilla.
c) There are 5 boys for every 4 girls in a class.
Bonus: There are about 5 centimeters for every 2 inches.
Answers: a) 3 : 5 = 6 : 10 = 9 : 15, b) 3 : 10 = 6 : 20 = 9 : 30, c) 5 : 4 = 10 : 8 = 15 : 12,
Bonus: 5 : 2 = 10 : 4 = 15 : 6
Solving word problems by making sequences of ratios. Write the following problem on the
board:
There are 3 boys for every 2 girls in a class.
There are 12 girls in the class.
How many boys are in the class?
Boys Girls
3 : 2
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-36
ASK: What is the next term of the ratio sequence? (6 : 4) SAY: If there are 3 more boys, then
there are 2 more girls, so that’s 6 boys and 4 girls. Record this on the board:
Boys Girls
3 : 2
6 : 4
SAY: We need to continue the sequence until the number of girls is 12. ASK: Is that the first
number or the second number? (the second) Point out that, because we titled the columns
clearly, it makes it clear which number has to be 12. Have a volunteer continue the sequence
until the second number is 12:
Boys
3
6
9
12
15
18
Girls
: 2
: 4
: 6
: 8
: 10
: 12
(MP.6) Point out that if you hadn’t put the title in each column, you might have instead just
looked here (point to the 12 boys and 8 girls) and you would have gotten the wrong answer,
because the ratio you are actually looking for is here (point to the 18 boys and 12 girls).
In the exercises below, encourage students to include the units when appropriate, and to
include titles for each column.
(MP.6) Exercises:
a) There are 3 boys for every 4 girls in a class. There are 12 boys. How many girls are there?
b) There are 5 cups of oatmeal for every 4 cups of raisins. How much oatmeal is needed for 20
cups of raisins?
Bonus: A mixture of green paint calls for 2 spoonfuls of blue paint for every 3 spoonfuls of
yellow paint. How much blue paint is needed for 24 spoonfuls of yellow paint?
Answers: a) 16 girls, b) 25 cups, Bonus: 16 spoonfuls
Finding a part from the total. Write on the board:
There are 4 boys for every girl in an after-school club.
There are 25 students in the club.
How many boys are in the club?
ASK: What is the ratio of boys to girls in the club? (4 : 1) Point out that the 1 is not explicitly
stated, as in “4 boys for every 1 girl,” but is understood. Write on the board:
Boys Girls
4 : 1
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-37
SAY: I want to continue the sequence until I see a 25, but where should I look for the 25?
PROMPT: Is 25 the number of girls? (no) Is it the number of boys? (no) What is 25 referring to?
(the total number of students) SAY: I need to keep track of both the ratio of boys to girls and the
total number of students in the class. Write on the board:
Boys
Girls
4 :
1
Total
ASK: If there are 4 boys and 1 girl, how many students are in the class? (5) Ask a volunteer to
add 4 more boys and 1 more girl to the chart (now there are 8 boys and 2 girls). ASK: How
many is that in total? (10) Tell students that you have to keep going until you get 25 in the total
column, and then have more volunteers continue adding rows to the chart until that is the case:
Boys
4
8
12
16
20
:
:
:
:
:
Girls
1
2
3
4
5
Total
5
10
15
20
25
Exercises:
a) There are 4 boys for every 7 girls in a class of 33 students. How many girls are in the class?
b) There are 6 boys for every 5 girls in a class of 22 students. How many boys are in the class?
c) There are 3 red marbles for every 4 blue marbles in a jar. If there are 28 marbles, how many
marbles are red?
Bonus: The ratio of girls to boys in a school is 12 : 13. If the school has 200 students, how
many girls are there?
Answers: a) 21, b) 12, c) 12, Bonus: 96
Extensions
(MP.3) 1. How can you tell immediately, without doing any calculations, that 3 : 5 and
4,575 : 2,745 are not equivalent ratios?
Answer: In equivalent ratios, whichever term is bigger in the first ratio has to be bigger in the
second.
(MP.1) 2. Sun made 20 cups of green paint by using 2 cups of blue paint for every 3 cups of
yellow paint.
a) How much blue paint and how much yellow paint did she use?
b) Sun meant to use 2 cups of yellow paint for every 3 cups of blue paint. She added some blue
paint to correct her mistake. How much blue paint did she add?
Answers: a) Sun used 8 cups of blue paint and 12 cups of yellow paint, b) Sun needs
12 : ? = 2 : 3, so she needs to have 18 cups of blue paint altogether; she used 8 cups already,
so she needed to add 10 more cups to make 18 cups.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-38
(MP.1) 3. The ratio between two numbers is 1 : 7. The sum is more than 40 but less than 50.
What are the numbers?
Answer: 6 and 42
(MP.1) 4. Find two numbers that add to 12 and are in the ratio.
a) 1 : 2
b) 1 : 3
c) 1 : 5
Answers: a) 4 and 8, b) 3 and 9, c) 2 and 10
(MP.3) 5. Two whole numbers are in the ratio 1 : 3. Rob says they cannot add to an odd
number. Is he right? Explain.
Answer: Yes, the second number is three times the first, so their sum is four times the first.
So the sum is always even.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-39
RP7-8
Ratio Tables
Pages 17–19
Standards: 7.RP.A.2
Goals:
Students will create equivalent ratios using multiplication.
Students will create and identify ratio tables.
Students will determine whether two quantities are proportional from the table comparing their
values.
Prior Knowledge Required:
Understands a ratio in terms of “for every”
Can create a list of equivalent ratios by drawing a picture
Vocabulary: equivalent ratio, proportional, ratio
Using multiplication to find equivalent ratios. Write on the board:
There are 3 oranges for every 2 bananas.
SAY: Every time you add 2 bananas, you have to add 3 oranges. Show this on the board as
follows:
SAY: If you add 2 bananas four times, then you have to add 3 oranges four times. Write on the
board:
bananas: 2 + 2 + 2 + 2
apples: 3 + 3 + 3 + 3
ASK: How can you say the same thing in terms of multiplication? (If you multiply 2 times 4, then
you have to multiply 3 times 4, too.) Write on the board:
2:3
4×2:4×3
Exercises:
1. Multiply both terms by 4 to make an equivalent ratio.
a) 3 : 5
b) 1 : 2
c) 6 : 7
d) 10 : 9
Bonus: 1,200 : 13
Answers: a) 12 : 20, b) 4 : 8, c) 24 : 28, d) 40 : 36, Bonus: 4,800 : 52
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-40
2. What number is each term being multiplied by to make the second ratio?
a)
3:5
b)
2:3
× ___
× _____
× ___
× _____
30 : 50
10 : 15
c)
7:8
× ___
Bonus:
× ___
× _____
14 : 16
Answers: a) 10, b) 5, c) 2, Bonus: 3
1,232 : 412
× _____
3,696 : 1,236
3. Multiply both terms by the same number to make an equivalent ratio.
a)
3:5
b)
2:5
×2
× _____
×3
× ____
____ : ____
____ : ____
Answers: a) 6 :10, b) 6 : 15
4. What number is the first term multiplied by? Multiply the second term by the same number to
make an equivalent ratio.
a) 3 : 4 = 9 : ______
b) 2 : 7 = 8 : ______
c) 3 : 8 = 30 : ______
Bonus: 13 : 7 = 65 : _____
Answers: a) × 3, 12; b) × 4, 28; c) × 10, 80; Bonus: × 5, 35
5. Multiply the first term by the same number the second term was multiplied by.
a) 5 : 8 = _____ : 24
b) 3 : 10 = _____ : 60
Bonus: 7 : 100 = __: 10,000,000
c) 2 : 9 = _____ : 36
Answers: a) 15, b) 18, c) 8, Bonus: 700,000
Introduce ratio tables. Explain that a ratio table is a table in which, if you used the terms in
each row to make a ratio, all the rows would make equivalent ratios. One way to make a ratio
table is to skip count by the numbers in the first row:
First Term
3
6
9
Second Term
5
10
15
SAY: Another way to make a ratio table is to multiply both terms in the first row by the same
number. Do part a) of each exercise below with the class.
Exercises:
1. What number is the first row being multiplied by to make the second row?
a)
3
4
30
40
b)
2
5
8
20
c)
4
7
12
21
Bonus:
4
7
200
350
Answers: a) 10, b) 4, c) 3, Bonus: 50
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-41
2. Find the missing number in each ratio table.
a)
2
7
b)
10
4
5
c)
4
9
8
d)
5
8
27
48
Bonus:
e)
8
9
f)
6
3,600
60
g)
3
110
h)
15
60
42
2,804
84
Answers: a) 35, b) 10, c) 12, d) 30, Bonus: e) 3,200, f) 11, g) 12, h) 1,402
3. What number is the first row being multiplied by to make the second and third rows?
a) 2
b) 2
c)
Bonus:
3
5
4
7
4
7
6
9
8
20
12
21
200
350
10
15
20
50
20
35
4,000
7,000
Answers: a) 3, 5; b) 4, 10; c) 3, 5; Bonus: 50, 1,000
Identifying ratio tables. Write on the board:
3
7
3
7
12
28
12
35
Point to each table in turn and ASK: Is the ratio made by the second row equivalent to the ratio
made by the first row? (yes for the first table, and no for the second table) SAY: So the first table
is a ratio table and the second one isn’t.
Exercises:
1. Is the table a ratio table? (Are the two rows equivalent?)
a) 8
b) 4
c) 4
d)
3
5
90
4
24
12
32
35
8
180
12
17
51
Answers: a) no, b) no, c) yes, d) yes
2. Each table below is not a ratio table. Which row is not equivalent to the first row?
a)
2
5
6
8
b)
4
9
15
12
30
16
c)
5
6
3
20
24
36
25
23
Bonus:
5
9
35,000 63,000
400
7,200
Answers: a) 8 : 30, b) 12 : 3, c) 25 : 23, Bonus: 400 : 7,200
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-42
Identifying proportional quantities. Write on the board:
Amount of juice
(fl oz)
10
20
Price
($)
2
3.50
SAY: The bigger the size of the juice, the more you pay, but that doesn’t mean that twice as
much juice costs twice as much money. ASK: Does the price of 20 fluid ounces cost twice as
much as 10 fluid ounces? (no) Is it more or less than twice the cost? (less) Why might a store
charge less than twice as much for twice as much juice? (sample answers: they want to
encourage you to buy more from them; packaging costs more on smaller amounts)
Tell students that when two quantities are always in the same ratio, such as flour and milk in the
same recipe, then they are said to be proportional. The amount of juice and the price are not
proportional because they are not always in the same ratio. SAY: You can check whether two
quantities are proportional if the table comparing their values is a ratio table.
Exercises: Are the two quantities proportional?
a)
b) Number of
Number of
Price ($)
Sheets
T-Shirts
c)
Price ($)
100
2
1
5
200
3
2
10
500
5
3
15
Distance
Run (m)
Time (min)
500
d)
Distance
Driven (miles)
Time
(hours)
2
100
2
1,000
5
200
4
5,000
35
500
10
Answers: a) no, b) yes, c) no, d) yes
(MP.1, MP.2) Reasons why some quantities are proportional and others are not. While
taking up the answers to the exercises above, ask students to reflect on why their answers
make sense. ASK: Does buying twice as much paper cost more or less than twice as much?
(less) Why does this make sense? (stores want you to buy more; packaging costs more on
smaller amounts) Does buying two T-shirts cost twice as much as one T-shirt? (yes) Why does
that make sense? (sometimes there is a discount for buying more than one, but sometimes
there isn’t) Does it take longer or shorter than twice the time to run twice the distance? (longer)
Why does that make sense? (because you have to save your energy; people get more tired
when running long distances than running short distances) Does it take twice as much time to
drive twice as far? (yes) Why does that make sense? (cars don’t get tired from driving like
people do from running)
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-43
SAY: To make the same color of green paint, the blue and yellow paint need to be proportional.
But if you want to make different shades of green, then the quantities won’t be proportional.
(MP.2, MP.4) Exercises: Do these mixtures make the same color of paint?
a)
c)
Blue Paint
(cups)
Yellow Paint
(cups)
3
b)
Blue Paint
(tsp)
Yellow Paint
(tsp)
2
1
4
6
5
2
8
9
8
3
12
Blue Paint
(cups)
Yellow Paint
(liters)
Blue Paint
(tbsp)
Yellow Paint
(cups)
600
3
600
3
1,200
6
1,200
6
30,000
15
30,000
150
b)
Answers: a) no, b) yes, c) no, d) yes
Units in ratio tables. Point out that it doesn’t matter what units you use to compare the
quantities, as long as you use one unit for one color for the whole chart. You can compare cups
to cups or cups to liters, but if you compare blue paint to yellow paint, you have to use the same
unit for blue paint throughout the chart and the same unit for yellow paint throughout the chart.
Extensions
(MP.1) 1. The ratio table was made by skip counting. Find the missing terms.
a) 3
b)
30
14
12
Answers: a)
20
3
15
6
b)
7
5
30
14
10
9
45
21
15
12
60
28
20
2. Tessa and Sam are sister and brother. Tessa is 3 years old and Sam is 6 years old. Sam
says that he is twice as old as Tessa, so when Tessa is 5, he will be 10. Is that correct? Explain.
Answer: No, Sam is always 3 years older than Tessa. So when Tessa is 5, Sam will be 8.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-44
(MP.3) 3. The table below was made by skip counting. Is it a ratio table? Explain why or why
not.
2
3
3
5
4
7
5
9
Answer: No. The table was made by skip counting by different numbers than are in the first
row. This means you can’t multiply the numbers in the first row to get the other rows, so the
rows are not equivalent ratios. (Students can also check this directly by looking at the first and
third rows—4 is two times 2, but 7 is not two times 3.)
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-45
RP7-9
Unit Ratios
Pages 20–21
Standards: 7.RP.A.2
Goals:
Students will use unit ratios to compare ratios and to identify equivalent ratios.
Prior Knowledge Required:
Can create equivalent ratios using multiplication
Understands the relationship between multiplication and division
Can identify when two quantities are proportional
Vocabulary: constant of proportionality, equivalent ratios, ratio, unit ratio
Materials:
calculators
NOTE: There are two unit ratios for each ratio. For example, 5 : 20 is equivalent to both the unit
ratios 1 : 4 and 1/4 : 1. Students will learn about ratios with fractions later in the year. For now,
restrict to ratios with whole number terms.
Introduce unit ratios. Tell students that a ratio is called a unit ratio if one of the quantities is
equal to 1. Then SAY: Unit ratios are easy to work with because 1 is easy to multiply and divide
by. Write on the board:
1 pen costs $2
4 pens cost _____
Ask a volunteer to fill in the blank. (8) Then SAY: Because I know that I have to multiply 1 by 4
to get 4, I know that I have to multiply $2 by 4 to get how much 4 pens cost.
Exercises: Multiply to find the missing information.
a) 1 book costs $4
b) 40 miles in 1 hour
3 books cost _____
_____ miles in 2 hours
c) 1 watermelon costs $3
d) 70 heartbeats in 1 min
5 watermelons cost _____
______ heartbeats in 3 mins
Answers: a) $12, b) 80 miles, c) $15, d) 210
Dividing to find equivalent ratios. SAY: Two ratios are equivalent if I can multiply both terms
of one of the ratios by the same number to get the other ratio. That means I can also divide both
terms of the other ratio by the same number to get the first ratio.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-46
Exercises: Divide the first term by the same number the second term was divided by.
a) 10 : 25 = _____ : 5
b) 18 : 45 = _____ : 5
Bonus: 628 : 46 = _____ : 23
c) 30 : 42 = _____ : 7
Answers: a) 2, b) 2, c) 5, Bonus: 314
Dividing to find a unit ratio. Write on the board:
7 notebooks cost $35
1 notebook costs _____
Ask a volunteer to fill in the blank. (5) Then SAY: Because I know that I have to divide 7 by 7 to
get 1, I know that I have to divide $35 by 7 to find out how much 1 notebook costs.
Exercises: Divide to find the missing information.
a) 5 notebooks cost $30
b) 4 sweaters cost $100
1 notebook costs ______
1 sweater costs ________
c) 8 mangoes cost $16
d) 7 avocados cost $7
1 mango costs ______
1 avocado costs _______
Answers: a) $6, b) $25, c) $2, d) $1
(MP.1) Using unit ratios to determine the missing number in a pair of equivalent ratios.
Write on the board:
5 : 15 = 2 : _____
SAY: If one of these was a unit ratio, we could solve the problem easily. Write on the board:
5 : 15 = 1 : ______ = 2 : ______
Ask a volunteer to fill in the blanks (3, 6), and then point out that this strategy of changing the
problem into one we already know how to do is a very useful problem-solving strategy in
general.
Exercises: Complete the ratio tables.
a)
4
20 ÷ 4
b)
30
6 ÷
1
1
×7
×
7
7
Bonus:
3
24
1
32
Answers: a) 1 : 5 = 7 : 35; b) 30 : 6 = 5 : 1 = 35 : 7; Bonus: 3 : 24 = 1 : 8 = 4 : 32
Using unit ratios to compare ratios that are not equivalent. Write on the board:
Store A
3 T-shirts cost $33
Store B
4 T-shirts cost $36
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-47
ASK: How can you tell which store has a better price? (find the cost for 1 T-shirt) Write on the
board, underneath Store A and Store B:
1 T-shirt costs ______
1 T-shirt costs _______
Have volunteers fill in the blanks. ($11 and $9) ASK: So which store offers a better price?
(Store B) Tell students that unit ratios are convenient because when two ratios are different, you
can compare them by comparing their unit ratios.
(MP.4) Exercises:
1. What is the better deal? Use a unit ratio.
a) 8 pencils for 40¢ or 5 pencils for 30¢
b) 5 CDs for $40 or 7 CDs for $49
Before giving the Bonus below, SAY: A unit ratio can have either the first term or the second
term as 1. Sometimes it is easier to put the second term as 1.
Bonus: 600 sheets for $3 or 1,000 sheets for $4.
Answers: a) 8 pencils for 40¢, b) 7 CDs for $49, Bonus: 1,000 sheets for $4 because you get
250 sheets per dollar rather than only 200 sheets per dollar
2. Sally took three bicycle trips.
A. She biked 36 km in 3 hours.
B. She biked 44 km in 4 hours.
C. She biked 50 km in 5 hours.
a) What is the unit ratio for each trip?
b) Does Sally bike faster for long distances or short distances? Why does this make sense?
Bonus: Tim types 78 words in 3 minutes. Yu types 96 words in 4 minutes. Who types faster?
(Students can use a calculator.)
Answers: a) A: 12 km in 1 hour, B: 11 km in 1 hour, C: 10 km in 1 hour; b) Sally bikes faster for
shorter distances. This makes sense because it is harder to keep up the same pace for longer
distances; Bonus: Tim types faster because he types 26 words per minute, and Yu types only
24 words per minute.
Identifying equivalent ratios using unit ratios. Write on the board:
3 : 33
9 : 90
Have volunteers write the unit ratios for each one. (1 : 11 and 1: 10) SAY: The unit ratios are
different, so they are not equivalent. Repeat for 4 : 24 and 7 : 42 (the unit ratios are the same—
1 : 6—so the ratios are equivalent).
Exercises: Find the unit ratios. Are the ratios equivalent?
a) 2 : 12 and 9 : 45
b) 3 to 21 and 7 to 49
c) 6 to 60 and 42 to 420
Bonus: 7 : 56 and 12 : 96
Answers: a) 1 : 6 and 1 : 5, no; b) 1 : 7 and 1 : 7, yes; c) 1 : 10 and 1 : 10, yes; Bonus: 1 : 8 and
1 : 8, yes
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-48
Remind students that in a ratio table, all the rows make equivalent ratios. So each row is
equivalent to the same unit ratio.
Exercises: Find the unit ratio in each row. Is the table a ratio table?
a)
2
12
b)
4
20
c)
3
30
d)
2
24
3
18
9
45
70
7,000
5
60
7
42
40 2,000
800 80,000
8
96
Answers: a) 1: 6, 1 : 6, 1: 6, yes; b) 1 : 5, 1 : 5, 1 : 50, no; c) 1 : 10, 1 : 100, 1 : 100, no; d) 1 :
12, 1 : 12, 1 : 12, yes
Introduce the constant of proportionality. Write on the board:
3
6
12
24
ASK: Is this a ratio table? (yes) What is the unit ratio in each row? (1 : 4) SAY: Because the unit
ratio is always 1 : 4, you can multiply the first column by 4 to get the second column. What you
multiply the first column by to get the second column is called the constant of proportionality.
Exercises: In each ratio table, use the complete row to find the constant of proportionality. Then
find the missing term in the other row.
a)
3 12
b)
20
c)
800,000
Bonus: 7 420
4
34 68
60
6,000
180
Answers: a) 4, 16; b) 2, 10; c) 100, 8,000; Bonus: 60, 3
(MP.7) Another way to see that the unit ratio is the same in a ratio table. Refer students’
attention to the ratio table you drew above. SAY: The ratios tell us two ways to get from 3 to 24.
Write on the board:
×4
3
6
12
24
×2
×2
3
6
12
24
× _____
SAY: 3 × 4 × 2 and 3 × 2 ×_______; since we get the same answer, and two of the numbers are
the same, the third one has to be the same, too. That means we multiply the first numbers in
each row by the same number to get the second numbers in each row.
Reverse ratios and unit ratios. Write on the board:
7 : 35 = 1 : ______
35 : 7 = _____ : _____
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-49
Have a volunteer fill in the first blank. (5) Tell students that you want to find a unit ratio
equivalent to 35 : 7. SAY: The 1 can be either first or second, so to get a unit ratio for the
reverse ratio, you can just reverse the unit ratio you started with. Write 5 : 1 in the blanks.
Exercises: Find the missing unit ratio.
a) 56 : 8 = 7 : 1 so 8 : 56 = ____ : _____
b) 19 : 57 = 1 : 3 so 57 : 19 = ____ : _____
c) 38 : 76 = 1 : 2 so 76 : 38 = _____ : _____
Bonus: 15,732 : 1,748 = 9 : 1, so 1,748 : 15,732 = _____ : _____
Answers: a) 1 : 7, b) 3 : 1, c) 2 : 1, Bonus: 1 : 9
Extensions
1. (MP.2) a) Find the constant of proportionality for the ratio of width (shortest side) to length
(longest side) in each rectangle.
i)
ii)
iii)
iv)
Width
2
1
5
3
Length
6
4
10
18
b) Draw the rectangles from part a). As the constant of proportionality gets bigger, does the
rectangle look more or less like a square?
Answers: a) i) 3, ii) 4, iii) 2, iv) 6; b) less
2. (MP.3) a) Investigate: If you switch the rows and columns of a ratio table, will you get another
ratio table? Check with these ratio tables.
5
10
3
9
2
10
4
28
30 60
12 36
6
30
8
56
Explain why this is the case.
b) Write two pairs of equivalent ratios from the ratio table.
i)
2
5
ii)
2
3
iii)
3
4
iv)
2
7
4
10
6
9
12
16
6
21
Answers: a) The answer is always yes. The reason is because, in each row, you always
multiply the first number by the same number to get the second number in that row. That is what
the unit ratio is. For example, 5 × 2 is 10 and 30 × 2 is 60. That means the ratio made by the
numbers in the first column is equivalent to the ratio made by the numbers in the second
column. b) i) 2 : 5 = 4 : 10 and 2 : 4 = 5 : 10; ii) 2 : 3 = 6 : 9 and 2 : 6 = 3 : 9; iii) 3 : 4 = 12 : 16
and 3 : 12 = 4 : 16; iv) 2 : 7 = 6 : 21 and 2 : 6 = 7 : 21
(MP.4) 3. In her first 300 games of Solitaire, Hanna had 50 wins. After another 50 games (350 in
total), her computer recorded 70 wins. Is she improving?
Answer: Yes; In 300 games, the ratio of wins to games was 1 : 6. In 350 games, the ratio of
wins to games was 1 : 5, which is a better ratio.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-50
RP7-10 Tape Diagrams and Ratio Problems
Pages 22–24
Standards: 7.RP.A.3
Goals:
Students will use tape diagrams to solve word problems with ratios.
Prior Knowledge Required:
Understands the expression “times as many”
Understands a ratio as describing a “for every” situation
Vocabulary: tape diagram
Using pictures for “times as many” problems. Write on the board:
Josh has four times as many stickers as May.
Tell students that you can draw a picture to show the situation.
May’s stickers:
Josh’s stickers:
SAY: No matter how many May has, Josh has 4 times that amount. Write on the board:
May
1 Josh
1 2
1 1 1 2
17
2
2
2
17
17
17 17 SAY: Each box can represent any number of stickers, as long as all the boxes represent the
same number. No matter how many May has, Josh has four times that number.
In the exercises below, students can use “J” and “M” to represent Josh and May.
Exercises: Draw a picture to show the following.
a) Josh has 3 times as many stickers as May
b) Josh has twice as many stickers as May.
c) Josh is twice as old as May.
Answers:
a) J:
b) J:
c) J:
M:
M:
M:
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-51
(MP.2, MP.4) Point out that the pictures for b) and c) are the same because only the contexts
are different, and the picture doesn’t show the context. The picture is just something you use to
help you solve the problem, so you need to include the basic information in the picture only.
Introduce tape diagrams. Tell students that the pictures they used above are tape diagrams. A
tape diagram has two or more strips, one above the other. The strips are made of units of the
same size. Draw on the board:
red marbles:
green marbles:
SAY: The picture shows that we have seven boxes with the same number of marbles in each
box. Suppose we also know that there are 12 more red marbles than green marbles.
Show that on the board:
red marbles:
green marbles:
12
ASK: How many marbles does each box represent? (4) How do you know? (because 3 boxes
represent 12 marbles and 12 ÷ 3 = 4)
Exercises:
1. How many marbles does each box represent?
a) red
green
12
b) red
green
Bonus:
red
green
green
c) red
28
40
3,500
Answers: a) 6, b) 4, c) 10, Bonus: 500
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-52
2. How much does each box represent if …
red apples
green apples
a) there are 30 more red apples than green apples?
b) there are 30 apples in total?
c) there are 30 red apples?
d) there are 30 green apples?
Answers: a) 30, b) 6, c) 10, d) 15
Before doing the next exercise, show students that they can write the number that each box
represents inside the box. This will help them answer the questions.
3. Use the tape diagram to answer the question.
a) There are 10 more girls than boys. How many children are there in total?
girls:
boys:
b) There are 10 children altogether. How many girls and how many boys are there?
girls:
boys:
c) There are 30 boys. How many girls are there?
girls:
boys:
Bonus: There are 2,400 girls. How many children are there altogether?
girls:
boys:
Answers: a) 18, b) 6 girls and 4 boys, c) 18, Bonus: 4,000
Connect tape diagrams to ratios. Write on the board:
girls:
boys:
Tell students that the picture gives some information about how many girls and how many boys
are in a class, but it doesn’t tell us how many students each box represents.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-53
Draw on the board the following table:
If each box
represents …
then there are
_____ girls
and
____ boys
1 student
2 students
3 students
4 students
2
5
Ask volunteers to dictate what to put in each cell. (girls: 2, 4, 6, 8; boys: 5, 10, 15, 20) When the
chart is complete, ASK: What kind of table did you just make? (a ratio table) Point out that each
row is made by multiplying the first row by the number of students, so all the rows are equivalent
ratios. SAY: The tape diagram doesn’t show only 2 girls and 5 boys. It shows all the possible
equivalent ratios: 4 girls and 10 boys, or 6 girls and 15 boys, and so on. The given information
will tell you what each box represents. Write on the board:
There are 3 girls for every 2 boys.
There are 20 students altogether.
(MP.6) ASK: What is missing from the picture? (labeling which bar represents the girls and
which bar represents the boys) Have a volunteer do so:
girls
boys
ASK: How many students does each box represent? (4) How do you know? (because 5 boxes
represent 20, so 1 box represents 4) Write 4 in each box in the picture:
girls
4
4
boys
4
4
4
ASK: How many girls are there? (12) How many boys? (8) In the exercises below, if some
students label the bars incorrectly, encourage them to write the ratios one under the other:
girls : boys
3 : 2
This makes it clear that the girls correspond to the 3 and the boys to the 2.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-54
Exercises: Draw a tape diagram to answer the question.
a) girls : boys = 5 : 3
There are 10 more girls than boys. How many students altogether?
b) girls : boys = 2 : 7
There are 28 boys. How many girls?
c) girls : boys = 4 : 5
There are 36 students altogether. How many girls and how many boys?
Answers: a) 40, b) 8, c) 16 girls and 20 boys
Connecting times as many to ratios.
Exercises: Draw a tape diagram to show the situation. Use “r” for red and “g” for green.
a) The ratio of red apples to green apples is 3 : 1.
b) There are 3 red apples for every green apple.
c) There are 3 times as many red apples as green apples.
Answers: They are all the same tape diagram:
r:
g:
Point out that the answers are all the same because the sentences are just three different ways
of saying the same thing. SAY: If the ratio of red to green apples is 3 to 1, then you have 3 red
apples for every green apple. That means no matter how many green apples you have, you
have 3 times as many red apples.
Extensions
1. a) There are three apples and two oranges for each plum at a fruit stand. There are 420 fruits
altogether. How many of each fruit are there?
b) Check your work by making sure that your answers add to 420.
Answers: a) Use the tape diagram with three bars shown below. 420 ÷ 6 = 70, so there are 70
plums, 210 apples, and 140 oranges; b) 70 + 210 + 140 = 420
plums
apples
oranges
420 (MP.1) 2. Anwar reads the same number of pages every school day. He reads twice as many
pages every weekend day. He finished a book of 108 pages in a week. How many pages does
he read on Monday? How many pages does he read on Sunday?
Solution: Make a tape diagram:
Monday:
Tuesday:
Wednesday:
Thursday:
Friday:
Saturday:
Sunday:
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-55
There are 9 boxes that represent in total 108 pages, so each box represents 12 pages.
So Abdul reads 12 pages on Monday and 24 pages on Sunday.
3. Create a tape diagram and a word problem that would fit the tape diagram. Have a partner
solve the problem.
(MP.1) 4. The ratio of two numbers is 4 : 3 and their product is 300. What are the numbers?
Solution: Make a ratio table and multiply each row until you get a product of 300.
Answer: 20 and 15
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-56
RP7-11 Solving Proportions
Pages 25–26
Standards: 7.RP.A.3
Goals:
Students will solve proportions, and will decide when to use the unit ratio and when to use the
ratio table.
Prior Knowledge Required:
Can create a ratio table from a given ratio
Can determine the unit ratio from a given ratio in which one term is a multiple of the other
Vocabulary: proportion, solve
Solving word problems using equivalent ratios. Write on the board:
5 bus tickets cost $8. How much would 40 tickets cost?
SAY: Start by writing a ratio table for bus tickets to dollars. Write on the board:
Tickets
5
Dollars
8
ASK: Where do I write the 40? (in the Tickets column) Do so and ASK: How can I find the
missing number? (5 × 8 = 40, so use 8 × 8) Ask a volunteer to fill in the missing number. (64)
Show the multiplication on the board:
×8
Tickets
5
40
Dollars
8
64
×8
Explain to students that a proportion is an equation that shows two equivalent ratios, such as
5 : 8 = 40 : 64. SAY: When you know two ratios are equivalent, but you are missing a term,
finding the missing term is called solving the proportion.
Exercises: Write and solve a proportion to answer these questions.
a) Five bus tickets cost $8. How many bus tickets can you buy for $40?
Tickets
Dollars
5
8
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-57
b) Bo can swim 3 laps in 4 minutes. At that rate, how many laps can he swim in 12 minutes?
Laps
Minutes
c) Bo can swim 3 laps in 4 minutes. At that rate, how long would it take to swim 12 laps?
d) When Tina played baseball, she got a hit 2 out of every 3 times at bat. She was at bat 12
times. How many hits did Tina have?
Answers: a) 25 tickets, b) 9 laps, c) 16 minutes, d) 8 hits
Two ways to solve proportions.
Exercises: Find the missing number in the ratio table.
a)
b)
c)
4
12
2
10
3
9
8
6
12
Bonus: Find the answer another way. Make sure you get the same answer both ways.
Answers: a) 24, b) 30, c) 36
Remind students that a ratio table has rows that are equivalent ratios. Then SAY: So by finding
the missing number in the ratio table, you were solving the proportion. Ask volunteers to
suggest two ways to find the answer to part a) above. To summarize, SAY: There are two ways
to solve the proportion. You could use the fact that the second row is 2 times the first row:
4
×2
8
12
×2
ASK: What is 12 × 2? (24) Write that in the empty cell. Then SAY: Or you could use the fact that
the second column is 3 times the first column. ASK: How do I know that the second column is 3
times the first? (because the unit ratio is 1 : 3) Remind students that if the unit ratio in the first
row is 1 : 3, then the unit ratio in the second row is also 1 : 3. Show this on the board:
×3
4
12
8
×3
ASK: What is 8 × 3? (24) Is that the same answer as we got the other way? (yes) Remind
students that finding the answer two different ways allows them to check their answers.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-58
(MP.1) Exercises: Find the missing number in two ways and show what you multiplied each
way. Make sure you get the same answer both ways.
Bonus:
a)
b)
7
35
3
12
2
8
6
12
7,000
Answers: a) 12 × 2 = 6 × 4 = 24, b) 8 × 6 = 12 × 4 = 48,
Bonus: 35 × 1,000 = 7,000 × 5 = 35,000
(MP.5) Choosing between strategies. Tell students that sometimes one of the two ways will
be easier, depending on the numbers in the table. Write on the board:
2
5
4
15
7
24
Point students’ attention to the first ratio table. Tell students that whenever possible you prefer
to work with whole numbers. SAY: I have two options. Write on the board:
× ______
2
5
15
× _____
ASK: Is 2 times a whole number equal to 5? (no) ASK: Is 5 times a whole number equal to 15?
(yes, 3) Then write 3 in the blank. SAY: So it’s easier to multiply the rows here. Repeat for the
second table above (this time, it is easier to multiply the columns because 4 × 6 is 24 but
4 times no whole number is 7).
Exercises:
(MP.5) 1. Find the missing number. Did you multiply the rows or the columns?
a)
7
8
35
b)
4
8
17
c)
13
4
d)
39
9
72
4
Answers: a) 40, columns; b) 34, rows; c) 12, rows; d) 32, columns
(MP.4, MP.5) 2. Make ratio tables. Find the missing number.
a) A recipe calls for 2 cups of flour for every 5 tbsp of water. How much flour is needed for 15
tbsp of water?
b) A recipe calls for 4 cups of flour for 12 muffins. How many muffins can you make with 5 cups
of flour?
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
B-59
c) Three centimeters on a map represents 20 km in real life. If a river is 80 km long, how long
will it appear on the map?
d) Three centimeters on a map represents 20 km in real life. If a lake is 6 cm long on the map,
what is its actual length?
Answers: (answers are in italics)
a)
c)
Cups flour
Tbsp water
2
b)
Cups flour
Muffins
5
4
12
6
15
5
15
Cm on map
Actual km
Cm on map
Actual km
3
20
3
20
12
80
6
40
d)
(MP.1) Multi-step ratio problems.
Exercises:
a) Fred has 3 nickels for every 5 pennies. He has $1.00 in nickels and pennies. How many
nickels and how many pennies does he have?
b) Emma has 3 quarters for every 2 dimes. She has $4.75 in quarters and dimes. How many
quarters and how many dimes does she have?
c) Kyle had 6 quarts of blue paint and 5 quarts of yellow paint. (A quart is 4 cups.) He made 30
cups of green paint from the ratio of blue paint to yellow paint = 2 : 1. How much of each color
paint is left over?
Answers: a) 15 nickels and 25 pennies, b) 15 quarters and 10 dimes, c) 4 cups of blue paint
and 10 cups of yellow paint is left over
Extensions
(MP.4) 1. A website provides estimated biking times from one point to another. Lynn takes 8
minutes to bike to school, but an online search tells her that biking to school along that same
route will take only 5 minutes. The same website tells her that biking to a friend’s place will take
only 20 minutes. How long will it actually take Lynn to bike to her friend’s place?
Answer: 32 minutes
2. There are 18 boys and 30 girls in a class. In another class, the ratio of boys to girls is the
same, but there are 35 girls. How many boys are there?
Solution: 18 : 30 = 3 : 5 = ? : 35, so ? = 21 boys
(MP.1) 3. The ratio of girls to boys to teachers in a school is 8 : 7 : 2, so there are 8 girls and 7
boys for every 2 teachers. There are 300 students at the school. How many teachers are at the
school? Challenge students to look for a shortcut way to solve the problem.
Solution: The ratio of students to teachers is 15 : 2 and there are 300 students. Since
300 = 15 × 20, the number of teachers is 2 × 20 = 40.
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
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4. (MP.4) a) Raj is reading on his way to work. He reads 3 pages on the 1 km bus ride. What is
the ratio of pages to km on the bus?
b) Raj gets off the bus and moves to the subway train. He reads 6 pages on the 6 km subway
ride. What is the ratio of pages to km on the subway?
c) For each km Raj rides, what is the ratio of pages read on the bus to pages read on the
subway?
d) Raj reads at the same rate on the bus as on the subway. Which is faster—the bus or the
subway? How many times faster?
e) Anna is knitting on her way to work. She knits 120 stitches on the 2 km bus ride, switches to
the subway, then knits 450 stitches on the 15 km subway ride. How much faster is the subway
train than Anna’s bus? What assumption did you need to make?
f) The subway speed is the same for both Anna and Raj. Whose bus is faster—Anna’s or Raj’s?
Answers: a) 3 : 1; b) 6 : 6 = 1 : 1; c) 3 : 1; d) the subway is three times as fast as the bus; e) the
subway is twice as fast as Anna’s bus—you needed to assume she knitted at the same rate on
the bus and the subway; f) Anna’s, because the subway is only twice as fast as her bus, but it is
three times as fast as Raj’s bus
Teacher’s Guide for AP Book 7.1 — Unit 1 Ratios and Proportional Relationships
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