Download EE8-39 Ratios and Fractions

EE8-39
Ratios and Fractions
Pages 144–147
Standards: preparation for 8.EE.B.5
Goals:
Students will review part-to-part and part-to-whole ratios, different notations for a ratio, and
equivalent ratios.
Students will understand ratios as a number of one thing “for every” of another.
Prior Knowledge Required:
Can interpret fractions
Can produce fractions equivalent to a given fraction
Vocabulary: colon, equivalent ratios, fraction, part-to-part, part-to-whole, ratio
Review ratios. 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 many in the whole set of shapes are squares, I can use fractions, just as we
did with 3/5. But if I want to compare the squares to the circles, I can’t use fractions of a whole,
because the squares are not a part of the set of circles. If I want to compare one part to another
part, I need to use a ratio. Write on the board:
The ratio of squares to circles is 3 to 2 or 3 : 2.
Point to the colon (:) and remind students that this is a common way to write a ratio numerically.
SAY: The symbol is called a colon and we read it as “to.”
Exercises: Write the ratio for the set.
a) circles to squares
b) circles to triangles
c) triangles to squares
Bonus: circles to polygons
Answers: a) 1 to 4 or 1 : 4, b) 1 to 3 or 1 : 3, c) 3 to 4 or 3 : 4, Bonus: 1 to 7 or 1 : 7
Ratios can compare parts to wholes. SAY: You can also use ratios to compare parts to the
whole. Refer students to the previous example on the board (3 squares and 2 circles). SAY: In
this example, the ratio of squares to shapes is 3 to 5 or 3 : 5. Write on the board:
The ratio of squares to all shapes is 3 to 5 or 3 : 5.
The ratio of circles to all shapes is ___ to ___ or ___ : ___.
Point at the blanks and have students signal what number should be written in each. (2 to 5, 2 : 5)
F-2
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Exercises: Write the ratio for the set of eight shapes in the previous exercises. Use a colon (:)
in your answer.
a) squares to all shapes
b) circles to all shapes
c) triangles to all shapes
Bonus: polygons to all shapes
Answers: a) 4 : 8, b) 1 : 8, c) 3 : 8, Bonus: 7 : 8
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.
Before students do the following exercises, remind them that the letters a, e, i, o, u are vowels
and other letters are consonants.
Exercises: Find the ratio. Then say if the ratio is part-to-part or part-to-whole.
a) vowels in “cat” to all letters in “cat”
b) vowels in “cat” to consonants in “cat”
c) consonants in “California” to all letters in “California”
d) consonants in “California” to vowels in “California”
e) weekend days to all days in a week
f) days in January to days in December
g) days in January to days in a year
Answers: a) 1 : 3, part-to-whole; b) 1 : 2, part-to-part; c) 5 : 10, part-to-whole; d) 5 : 5,
part-to-part; e) 2 : 7, part-to-whole; f) 31 : 31, part-to-part; g) 31: 365, part-to-whole
Part-to-whole ratios as fractions. SAY: You can think of part-to-whole ratios as fractions. 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.
You can compare and order ratios just as you can fractions.
Go through the previous exercises as a class and have students write each part-to-whole ratio
as a fraction. (a) 1/3, c) 5/10, e) 2/7, g) 31/365)
Exercises: Write a ratio and a fraction.
a) large circles to circles = ____ : _____
_____ of the circles are large
b) white circles to circles = _____ : _____
_____ of the circles are white
c) small white circles to circles = ____ : _____
_____ of the circles are small and white
Answers: a) 3 : 10 and 3/10, b) 5 : 10 and 5/10, c) 4 : 10 and 4/10
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-3
Changing part-to-part ratios to part-to-whole ratios and fractions. Tell students that, if a set
has only two parts, such as circles and triangles or girls and boys, then you can find a part-towhole 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 the part-to-part ratio.
a) The ratio of girls to boys is 3 : 7.
b) The ratio of boys to girls is 4 : 5.
c) The ratio of girls to boys is 8 : 7.
d) There are 5 boys and 5 girls on the team.
Bonus: The ratio of a team’s wins to losses is 2 : 5. (Ties are not allowed.)
Answers: a) 3/10 of the students are girls, 7/10 are boys; b) 4/9 of the students are boys,
5/9 are girls; c) 8/15 of the students are girls, 7/15 are boys; d) 5/10 of the players are boys,
5/10 are girls; Bonus: 2/7 of the games were won, 5/7 were lost
Fill in the first column of the chart below as a class, then have students fill in the rest of
the chart.
Exercises: Each ratio in the first row describes a different school club. Complete the chart.
Ratio of boys to girls
5:7
6:8
8:6
3
4
Fraction of students who are girls
3
9
Fraction of students who are boys
Are there more boys or girls?
Answers:
Ratio of boys to girls
5:7
6:8
8:6
1:3
3:6
Fraction of students who are girls
7/12
8/14
6/14
3/4
6/9
Fraction of students who are boys
5/12
6/14
8/14
1/4
3/9
Are there more boys or girls?
girls
girls
boys
girls
girls
ASK: How can you tell from each ratio if there are more boys than girls? (if the first number is
bigger than the second number) How can you tell from each fraction if there are more girls than
boys? (if the fraction that is girls is more than half)
F-4
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
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 is 19 : 21 in City A and 12 : 11 in City B. Which city has more
schools than parks?
Answers: a) more girls, b) City A
Review equivalent ratios. Tell students that a recipe for an orange-banana smoothie calls for
3 oranges and a banana. Write on the board:
OOOB
Explain that you wrote an “O” for each orange and a “B” for each banana. Tell students that you
want to make lots of smoothies. ASK: If I use 2 bananas, how many oranges would I need? (6)
Write another row of letters on the board, as shown below:
OOOB
OOOB
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. 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 write another row of letters on the board. ASK: What ratio is showing now?
(9 oranges to 3 bananas) Continue writing on the board:
OOOB
OOOB
OOOB
oranges to bananas = 3 : 1
=6:2
=9:3
Point out that the ratio 3 to 1 doesn’t describe just a single situation, but a whole sequence of
possible situations. NOTE: We use the word “sequence” here to mean a group of ratios that can
continue by following a rule. Ask a volunteer to continue the picture to find the next possible
situation and ratio. (12 : 4) SAY: All these ratios are equivalent ratios.
Point out that, for any description, the order of numbers in a ratio matches the order of what
they represent. So, for example, if we describe a ratio of 3 oranges to 1 banana, we write it as
3 : 1, not 1 : 3. Similarly, if we give a ratio of 19 : 21 to compare parks to schools, we mean
19 parks for every 21 schools, not 19 schools for every 21 parks.
Exercises: Write two equivalent ratios for the picture.
a)
b)
circles to squares
stars to dots
= ___ : ___ = ___ : ___
= ___ : ___ = ___ : ___
Sample answers: a) 2 : 1 = 6 : 3, b) 2 : 3 = 8 : 12
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-5
Return to the example about the orange-banana smoothie. Challenge a volunteer to find the
next possible ratio of oranges to bananas without writing the letters. (15 : 5) ASK: How did
[volunteer] find this ratio? (from 12 : 4, add 3 more O’s and 1 more B) Point out that students
can skip count by 3s to continue the first column and by 1s to continue the second column in the
column of ratios.
Exercises: Skip count to write three more equivalent ratios.
a) 5 : 1
b) 1 : 4
c) 10 : 2
Bonus: 3 : 5,000
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
= ___ : ___
Answers: a) 10 : 2 = 15 : 3 = 20 : 4, b) 2 : 8 = 3 : 12 = 4 : 16, c) 20 : 4 = 30 : 6 = 40 : 8,
Bonus: 6 : 10,000 = 9 : 15,000 = 12 : 20,000
Solving word problems by making sequences of part-to-part ratios. Explain that you can
use sequences of ratios, as in the previous set of exercises, to solve word problems. For
example, a recipe for dumpling dough calls for 8 cups of flour and 3 cups of water. In a
restaurant, a cook will make lots of dough, so he will use 40 cups of flour. The cook needs to
figure out how much water to use. Write on the board:
Dumpling dough recipe: 8 cups of flour for every 3 cups of water
Cook uses: 40 cups of flour
How much water?
Explain that one way to solve this problem is to write a sequence of ratios, using skip counting.
Write on the board:
Flour : Water
8 : 3
16 : 6
24 : ____
____ : ____
____ : ____
Have students extend both columns. (24 : 9, 32 : 12, 40 : 15) SAY: The cook needs to use
40 cups of flour. ASK: When do you need to stop writing the numbers in the sequence? (when
you reach 40 in the flour column) How many cups of water will the cook use? (15)
Provide the headings for the ratios in the following exercises to help students who struggle.
SAY: Remember that you could keep writing the sequence forever, but you only want to find out
the answer to your question. One way to stay focused is to write headings over the columns and
circle the column that will give you the information you need.
F-6
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Exercises: Write a sequence of equivalent ratios to solve the problem.
a) A recipe for granola calls for 3 cups of oatmeal for every 4 cups of raisins. Tina uses 15 cups
of oatmeal. How many cups of raisins does she need?
b) There are 5 boys for every 6 girls in a class. There are 15 boys in the class. How many girls
are there?
c) A car uses 7 L of gas for every 100 km. The tank holds 35 L of gas. How far can the car drive
on a full tank of gas?
Bonus: Maria is traveling to Canada. She knows that there are about 8 kilometers in every
5 miles. A road sign shows that the distance to Montreal is 64 km. How many miles away
is Montreal?
Selected solution:
a) Oatmeal : Raisins
3:4
6:8
9 : 12
12 : 16
15 : 20
Tina needs 20 cups of raisins.
Answers: b) boys : girls = 5 : 6 = … = 15 : 18, there are 18 girls; c) gas (L) : distance (km)
= 7 : 100 = … = 35 : 500, the car can drive 500 km; Bonus: km : miles = 8 : 5 = … = 64 : 40,
40 miles to Montreal
NOTE: If you present the answers above on the board, either write them vertically to match the
oatmeal/raisins solution above, or explain to students that you can write equivalent ratios in a
row to save space, but must place equal signs between each pair of ratios to show that they are
equivalent, as in the other answers.
Write 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?
SAY: Ted found a solution. Write on the board:
Boys : Girls
3:2
6:4
9:6
12 : 8
There are 8 boys in the class.
(MP.3) ASK: Is Ted’s solution correct? (no) Why not? (he needs to continue the sequence of
ratios until he gets 12 in the “girls” column, not in the “boys” column) Have a volunteer correct
the mistake (add two rows to the table) and find the right answer. (18 boys) Emphasize how the
headings for the columns make it easier to check that we have reached the correct quantity in
the correct column.
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-7
(MP.6) Exercises:
a) There are 3 boys for every 4 girls in a class. There are 12 girls. How many boys are there?
b) i) A recipe calls for 2 lb of rice for every 3 lb of carrots. A cook wants to use 15 lb of carrots.
How much rice does the cook need?
ii) The recipe for rice calls for 1 lb of onions for every 2 lb of rice. How many pounds of onions
does the cook need?
Bonus: 2 inches are about 5 cm long. 1 foot is 12 inches long. How many centimeters long is 1 foot?
Answers: a) 9 boys; b) i) 10 lb of rice, ii) 5 lb of onions; Bonus: about 30 cm
Using part-to-part and part-to-whole ratios to solve word problems. 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 the 1 is understood. Write on the board:
Boys : Girls
4:1
ASK: 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 club. Add a column for “Total.” ASK: If there are 4 boys and
1 girl, how many students are in the club? (5) Ask a volunteer to add 4 more boys and 1 more
girl to the chart. ASK: How many boys are there now? How many girls? (8 boys, 2 girls) How
many is that in total? (10) Tell students that you have to keep adding rows until you get 25 in the
Total column. Have more volunteers continue adding rows to the chart until the total is 25, as
shown below:
Boys : Girls
4:1
8:2
12 : 3
16 : 4
20 : 5
Total
5
10
15
20
25
ASK: How many boys are in the club? (20)
Exercises:
a) There are 4 boys for every 5 girls in a class of 36 students. How many girls are in the class?
b) There are 7 boys for every 5 girls in a class of 36 students. How many boys are in the class?
c) A bracelet is made of red and blue beads. There are 3 red beads for every 5 blue beads. If
there are 96 beads, how many of them are red?
Bonus: The ratio of girls to boys in a school is 12 : 13. If the school has 300 students, how
many girls are there?
Answers: a) 20, b) 21, c) 36, Bonus: 144
F-8
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Extensions
1. Describe the set in three different ways using the ratio 3 : 5.
Answer: 3 : 5 = circles to squares = large shapes to small shapes = gray shapes to white shapes
(MP.2) 2. Write the ratio of the lengths. Example: AB to CD = 4 : 5
4 cm
3 cm
5 cm
A
B
C
D
a) BC to CD
b) AB to AC
c) CD to AD
Bonus: Which ratios are part-to-whole ratios?
Answers: a) 3 : 5, b) 4 : 7, c) 5 : 12, Bonus: AB to AC and CD to AD
(MP.3) 3. Can there be a set of shapes with the ratio “triangles to polygons = 3 : 2”? Explain.
Answer: No, because triangles are polygons, so they are a part of the whole, so there cannot
be more triangles than polygons.
(MP.1) 4. Sun made 20 cups of green paint by mixing 1.25 cups of blue paint with every
3.75 cups of yellow paint.
a) How much blue paint and how much yellow paint did she use?
b) Sun meant to use 1.25 cups of yellow paint for every 3.75 cups of blue paint. She thinks she
can correct her mistake by adding some blue paint. How much blue paint does she need to
add? Does this idea make sense?
Answers:
a) Sun used 5 cups of blue paint and 15 cups of yellow paint.
b) Sun needs the ratio yellow : blue = 1.25 : 3.75, but she used 15 cups of yellow. If we use ? for
the number of cups of blue paint she needs, we get equivalent ratios y : b = 1.25 : 3.75 = 15 : ?.
The missing number ? is 45. This means she needs to have 45 cups of blue paint in total; she
used 5 cups of blue paint and 15 cups of yellow paint already, so she needs to add 40 more
cups to make 45 cups and will have 60 cups of paint in total. If she has a large project and
would need that large quantity of paint, her solution makes sense.
(MP.1) 5. Find two numbers that add to 12 and are in the given 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) 6. 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 and
the sum is always even.
7. Find two fractions that add to 1 and are in the ratio 3 : 2.
Answer: 3/5 and 2/5
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-9
EE8-40
Ratio Tables
Pages 148–149
Standards: preparation for 8.EE.B.5
Goals:
Students will create equivalent ratios using multiplication, and create and identify ratio tables.
Students will determine whether two quantities are proportional from a table comparing their values.
Prior Knowledge Required:
Can interpret fractions
Can produce fractions equivalent to a given fraction using multiplication and division
Understands a ratio in terms of “for every”
Can create a sequence of equivalent ratios
Vocabulary: colon (:), denominator, equivalent fractions, equivalent ratios, numerator, ratio,
ratio table, terms
Using multiplication to find equivalent ratios. SAY: Just as you can multiply the numerator
and the denominator of a fraction by the same number to get an equivalent fraction, you can
multiply both parts of a ratio by the same number to get an equivalent ratio. For example, I can
multiply both parts of a ratio 2 : 3 by 4 and get the equivalent ratio 8 : 12. Write both ratios with
an equal sign between them on the board. Explain that the parts of the ratio are called terms.
SAY: The first ratio has terms 2 and 3, and when we multiply both terms by 4, we get a ratio
with terms 8 and 12.
Exercises:
1. Multiply both terms by 4 to make an equivalent ratio.
a) 2 : 5
b) 1 : 3
c) 4 : 7
d) 11 : 9
Bonus: 1,500 : 8
Answers: a) 8 : 20; b) 4 : 12; c) 16 : 28; d) 44 : 36; Bonus: 6,000 : 32
SAY: I would like to check that the ratios are, indeed, equivalent. ASK: How did we find
equivalent ratios in the previous lesson? (wrote a sequence of equivalent ratios using skip
counting) SAY: Let’s check that 2 : 5 and 8 : 20 are equivalent ratios. Have students find the
sequence of equivalent ratios using skip counting individually, then have a volunteer write the
sequence on the board, as shown below:
2:5
= 4 : 10
= 6 : 15
= 8 : 20
Repeat with other ratios from the previous exercises. ASK: Which method is more efficient,
multiplying or skip counting? (multiplying)
F-10
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Exercises:
1. What number is each term being multiplied by to produce the second ratio?
a)
3:7
b)
2:5
× ___
× ___
× ___
× ___
30 : 70
c)
× ___
10 : 25
9:4
× ___
72 : 32
Answers: a) 10, b) 5, c) 8, Bonus: 3
Bonus:
132 : 210
× ___
× ___
396 : 630
2. Multiply both terms by the same number to produce an equivalent ratio.
a)
5:7
b)
8:3
×2
×2
×5
×5
___ : ___
___ : ___
c)
9:2
×7
567 : 983
Bonus:
×7
× 10
× 10
___ : ___
___ : ___
Answers: a) 10 : 14; b) 40 : 15; c) 63 : 14; Bonus: 5,670 : 9,830
3. What number is the first term multiplied by? Multiply the second term by the same number to
produce an equivalent ratio.
a)
5 : 11
b)
9:7
× ___
× ___
× ___
× ___
30 : ___
45 : ___
c)
× ___
8:7
× ___
Bonus:
240 : 312
× ___
× ___
64 : ___
960 : ___
Answers: a) × 6, 66; b) × 5, 35; c) × 8, 56; Bonus: × 4, 1,248
4. Multiply the first term by the same number the second term was multiplied by.
a)
6 : 11
b)
9:8
× ___
× ___
× ___
× ___
___ : 77
___ : 72
Answers: a) ×7, 42; b) ×9, 81
Introduce ratio tables. Write on the board:
×2
×3
×4
3
6
9
2
4
6
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-11
Explain that you made this table by multiplying both numbers in the first row by the same number.
To fill in the second row, you multiplied both 3 and 2 by 2, so you produced an equivalent ratio.
To fill in the third row, you multiplied 3 and 2 by 3, and produced another equivalent ratio. Have
students tell you what numbers you need to write in the fourth row. (12, 8) SAY: This table is
called a ratio table. Emphasize that, in a ratio table, we get all the numbers in all the rows from
the first row, not from the row immediately above, by multiplying both terms in the first row by
the same number. SAY: All rows in a ratio table make equivalent ratios.
Exercises: Complete a ratio table for the ratio. Multiply the first row by 2, and then by 3.
a)
4:1
b)
3:4
c)
5:7
d)
9:2
4
1
Answers: a) 8 : 2, 12 : 3; b) 3 : 4, 6 : 8, 9 : 12; c) 5 : 7, 10 : 14, 15 : 21; d) 9 : 2, 18 : 4, 27 : 6
Point out that the ratio tables that students produced in the previous exercises are very similar
to the lists of equivalent ratios they produced in the previous lesson. The difference is in the way
they produced them (by multiplying, instead of by skip counting) and wrote them, not in the
ratios themselves.
Draw on the board:
4
8
11
33
SAY: In this ratio table, some numbers are missing. But we know that both ratios in the second
and third row are equivalent to the ratio in the first row, 4 : 11. ASK: What number do we need
to multiply by to get from 4 to 8? (2) SAY: This means the second row was made by multiplying
both numbers in the first row by 2. ASK: What number should be in the empty cell of the second
row? (22) How do you know? (because 2 × 11 = 22)
Repeat with the third row. (the first row was multiplied by 3 to give 33, so the missing number in
the third row is 12, because 4 × 3 = 12)
Exercises: Find the missing numbers in the ratio table.
a)
7
1
2
21
b)
6
13
c)
9
2
12
7
2
7
39
27
2
d)
0.5
1.0
2.4
3.6
Answers: a) 14, 3; b) 26, 18; c) 9, 21/2; d) 1.2, 1.5
F-12
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Ratio tables with two rows. Explain that multiplication is a more efficient way to produce a
ratio table than skip counting. Actually, in ratio tables, the rows do not always have to be
multiplied first by 2, then by 3, and so on. There might be only two rows, and the number you
need to multiply the first row by to get the second row can be any number. Draw on the board:
7
4
20
ASK: What number do you need to multiply 4 by to get 20? (5) How do you know? (20 ÷ 4 = 5)
SAY: If this is a ratio table, then the ratios in each row must be equivalent, so we can multiply
the other number in the first row by 5 to get the missing number and create a second, equivalent
ratio. ASK: What is 7, the first number in the first row, multiplied by 5? What number should be
in the empty cell? (35)
Exercises: What number is the first row being multiplied by to make the second row? Find the
missing number.
a)
7
3
b)
8
11
c) 10.1
9
5
100
Bonus:
21
99
81
4,000
Answers: a) × 7, 49; b) × 9, 72; c) × 9, 90.9; Bonus: × 40, 200
NOTE: Students who struggle with the exercises above will benefit from adding arrows to the
sides of the table and writing the number they use to multiply, as shown below:
×5
7
4
20
×5
Using ratio tables to solve word problems. Remind students that, in the previous lesson,
they solved problems by finding an equivalent ratio, and that they used a sequence of ratios to
do that. Today, they learned a more efficient method to find equivalent ratios, so they can use
ratio tables to solve the problems similar to those in the previous lesson. Write on the board:
3 T-shirts cost $10. How much do 15 T-shirts cost?
SAY: In this problem, we have two changing quantities, the number of T-shirts and the price.
Draw on the board:
Number of
T-shirts
Price ($)
ASK: What is the first ratio that I know from the problem? (3 T-shirts for every 10 dollars)
SAY: We write this ratio in the first row. Write “3” and “10” in the first row of the table. SAY: Then
we need to write the other number we are given, 15, somewhere in the table. ASK: Which
column should the number 15 go in? (Number of T-shirts) Why not in the Price column?
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-13
(15 is the number of T-shirts; we need to find the cost of those 15 T-shirts) Write “15” in the
second row, as shown below:
Number of
T-shirts
3
15
Price ($)
10
ASK: What number do we need to multiply the first row by to make this a ratio table? (5)
PROMPT: What number do we need to multiply 3 by to get 15? ASK: So, what is the missing
number? (50) Write “50” in the empty cell. SAY: This means that 15 T-shirts will cost $50.
Repeat with the following problem: 3 T-shirts cost $10. How many T-shirts can I buy for $70?
(21)
Exercises:
(MP.6) 1. There are 5 plums for every 2 apples. Write the quantity you know in the correct
column of the ratio table. Then find the quantity you don’t know.
a) There are 10 plums.
b) There are 10 apples.
How many apples are there?
How many plums are there?
Plums Apples
Plums Apples
5
2
5
2
c) There are 20 plums.
How many apples are there?
Plums Apples
5
2
d) There are 50 apples.
How many plums are there?
Plums Apples
5
2
Answers: a) 10 : 4, b) 25 : 10, c) 20 : 8, d) 125 : 50
2. Solve the problem by making a ratio table.
a) Roy earns $350 every week. How much money will he earn in 4 weeks?
b) Kathy saves $25 every 2 weeks. How long will it take her to save $150?
c) A motorcycle can travel 100 km on 3 L of gas. Its gas tank can hold 18 L. How far can the
motorcycle travel on a full tank of gas?
d) A recipe calls for 3 lb meat for 4 portions. A restaurant cook has 75 lb of meat. How many
portions can the cook make?
e) A recipe calls for 0.5 lb of carrots for 4 portions of salad. How many pounds of carrots do you
need for 12 portions of salad?
f) Marco saves $12.50 every 2 weeks. How much will he save in 6 weeks?
Bonus: Emma reads 7 pages in 10 minutes. Her book is 210 pages long. How many hours will
it take her to finish the book?
Answers: a) $1,400; b) 12 weeks; c) 600 km; d) 100 portions; e) 1.5 lb; f) $37.50;
Bonus: 300 minutes = 5 hours
F-14
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Extensions
(MP.3) 1. How can you tell immediately, without doing any calculations, that the table below is
not a ratio table?
3
5
36,225 24,525
Answer: 3 < 5, so to make an equivalent ratio, the first number in the second row should be
smaller than the second number in the second row. 36,225 > 24,525, so the table cannot be a
ratio table.
(MP.3) 2. Tessa and Sam are siblings. Tessa is 2 years old and Sam is 6 years old. Sam says
that he is three times as old as Tessa, so when Tessa is 5, he will be 15. Is that correct? Explain.
Answer: No, Sam will always be 4 years older than Tessa. So when Tessa is 5, Sam will be 9.
(MP.1) 3. The ratio between two numbers is 1 : 6. The sum is more than 50 but less than 60.
What are the numbers?
Answer: 8 and 48
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-15
EE8-41
Graphing Ratios
Pages 150–152
Standards: 8.EE.B.5
Goals:
Students will review coordinate grids.
Students will graph ratios by converting a ratio table into ordered pairs and plotting the
ordered pairs.
Prior Knowledge Required:
Can interpret fractions
Can name fractions equivalent to a given fraction
Can create a list of equivalent ratios
Vocabulary: axes, axis, colon (:), coordinates, equivalent ratios, first coordinate, ordered pair,
origin, plot, ratio, ratio table, second coordinate, skip counting, x-axis, x-coordinate, y-axis,
y-coordinate
Materials:
BLM Small Coordinate Grids (p. I-2), 2 to 3 copies per student
Review coordinates in the first quadrant. Draw the picture below on the board, but without
the arrows and labels for the axes and the origin. Point to the axes and tell students that axes is
the plural of axis. SAY: The horizontal line is called the x-axis and the vertical line is called the
y-axis. Tell students that the point at which the two axes intersect is called the origin. Label the
axes and the origin, as shown below:
y
4
y-axis
3
origin (0, 0)
2
x-axis
1
0
1
2
3
4
x
Mark point (4, 3) on the grid and use it to explain how to describe the location of a point on
the grid: From the point (4, 3), trace with your finger down to the x-axis and look at the number
that is directly below the point. SAY: This point has x = 4. Go back to the point and trace with
your finger left to the y-axis and look at the number on the y-axis directly to the left of the point.
SAY: This point has y = 3.
Repeat with the point (3, 4), where the order of the coordinates is reversed. Emphasize that the
numbers for the two points are the same, but the points are different. In other words, point out
F-16
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
that when you write the numbers in a different order, you get different points. SAY: The x and y
numbers for a point identify its unique location. They are like an address or locator for the point,
and they are called the coordinates of the point. The x number is called the x-coordinate, and
the y number is called the y-coordinate.
NOTE: Before students look at the conventions for writing an ordered pair, look at more points
and review how to locate them, as follows.
To practice identifying the x- and y-coordinates of various points on the grid, have students
copy the grid from the example on the board. (To save time, students can also use a grid from
BLM Small Coordinate Grids.) Have students mark and label any four points they choose on
the grid intersections, so that no two points are on the same horizontal or vertical grid line, and
all points are to the right and above the axes. Have them label the points and then identify the
x- and y-coordinates for each point.
SAY: Imagine you have to write the coordinates of 100 points. ASK: Would you want to write
x = ___ and y = ___ 100 times? What could you do to shorten the notation—in other words, how
could you write it faster? Students might suggest making a chart or writing pairs of numbers with
the x numbers first and y numbers second, which follows alphabetical order. Explain that
mathematicians have a convention, which is to express the place of a point with two numbers
in brackets and a comma in between. The x is always the first number, on the left, and the y is
always the second, on the right: (x, y). Point out to students that they already know that the
order is important and that (4, 3) and (3, 4) are different points. To emphasize that the order
is important, such a pair of numbers in brackets is called an ordered pair.
NOTE: Technically, the term “brackets” refers to square brackets, and coordinates are written in
parentheses. We will refer to “brackets” instead of parentheses, as the the latter word might
present a stumbling block for students, especially English language learners.
Have students rewrite the coordinates of the four points on their grids using conventional
notation. Explain that the x-coordinate is often called the first coordinate, because it is written
first, and the y-coordinate is therefore called the second coordinate.
Students can a grid from BLM Small Coordinate Grids to do part b) of the exercises below.
Exercises:
a) Write the coordinates of each point.
y
6
5
4
3
2
1
0
A
B
C
D
1 2 3 4 5 6
x
b) Mark the following points on a coordinate grid: E (1, 3), F (5, 6), G (2, 7), H (7, 4)
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-17
Answers:
a) A (2, 5), B (4, 4), C (6, 2), D (3, 1)
b) y
G
6
5
4
3
2
1
F
H
E
0
x
1 2 3 4 5 6
Coordinate grids can include fractions, decimals, and negative numbers. Explain that the
lines on the coordinate grid are drawn parallel to the axes, but the distances between them are
usually equal to a whole number. On the grids students have seen so far in this lesson, the lines
were one unit apart. SAY: However, we can use skip counting to label the axes—for example,
when we need to show points with very large coordinates. Use the grid from the previous
exercises to create the grid for the following exercises. Change the x-axis to skip count by 2s
and the y-axis to skip count by 5s. Tell students that certain points will need to be placed between
the grid lines. Plot the point (1, 1) on the new grid to demonstrate how to do it. Point out that the
x-axis uses skip counting by 2s, so the point with x = 1 is exactly halfway between 0 and 2. The
y-axis skip counts by 5s, so the point with y = 1 should be one fifth of the way from 0 to 5.
Explain that when you mark a point given by its coordinates on a coordinate grid, you plot the
point. When you have a list of points, it helps to cross out each ordered pair as you plot them to
make sure you do not miss any.
Exercises:
a) Write the coordinates of each point.
y
30
25
20
15
10
5
A
B
C
D
0
x
2 4 6 8 10 12
b) Plot the following points on a coordinate grid. Cross out the points as you plot them.
E (2, 15), F (3, 25), G (6, 7), H (11, 24)
Answers:
a) A (4, 25), B (8, 20), C (12, 10), D (6, 5)
b) y
30
25
20
15
10
5
0
F-18
F
H
E
G
2 4 6 8 10 12
x
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Remind students that fractions, decimals, and negative numbers can also appear on number
lines. SAY: This is true for coordinate grids as well; a coordinate grid can show numbers that
are fractions (for example, 1/2), decimals (for example, 2.5), and even negative values. For
negative values, we have to make sure that the area of the grid and the axes allow for the point
to be plotted. For example, point J (1.5, −1) will be halfway between lines for x = 1 and x = 2,
because it has x = 1.5, and on the line for y = −1. Draw on the board:
y
3
2
1
0
−1
−2
−3
1 2 3 4 5 6
x
J
Before students continue to plot points, look at the grids on BLM Small Coordinate Grids to see
how the grids vary and discuss which would be good to use in specific situations, such as with
negative numbers or fractions. Have students use the second grid on BLM Small Coordinate
Grids for the following exercises.
1
2
Exercises: Plot the points on a coordinate grid: K (2, –3), L (3, 1 ), M (4.5, –2), N ( 5
1
, 2.5)
2
Answers: y
3
2
1
0
−1
−2
−3
N
L
1 2 3 4 5 6
x
M
K
Review creating ratio tables. Remind students that to create a ratio table, they must start with
a given ratio in the first row, then produce any other row by multiplying both numbers in the first
row by the same number. SAY: Usually, we multiply by numbers in order: 2, 3, 4. As an
example, draw the ratio table for 1 : 3 on the board, as shown below:
1
2
3
4
3
6
9
12
Leave the ratio table on the board for later use.
Exercises: Make a ratio table for the ratio.
a) 1 : 2
b) 4 : 5
c) 5 : 4
d) 7 : 10
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-19
Answers:
a)
1
2
3
4
2
4
6
8
b)
4
8
12
16
5
10
15
20
c)
5
10
15
20
4
8
12
16
d)
7
14
21
28
10
20
30
40
Order matters in ratios. ASK: Is 1 : 3 the same ratio as 3 : 1? (no) Have students explain
how they know. Share the following illustration if something similar does not arise during the
discussion: If you mix 3 cups of red paint with 1 cup of yellow paint, the color you get will be
a deep orange that is close to red. If you mix 1 cup of red paint with 3 cups of yellow paint,
the resulting color will be a light orange that is close to yellow. So, red : yellow = 3 : 1 is very
different from red : yellow = 1 : 3.
Converting ratios to ordered pairs. ASK: What other mathematical object did we just learn
about that has two numbers and where the order matters? (ordered pairs, coordinates of a point)
Point out that we can create an ordered pair from each ratio, then draw a point on a coordinate
grid using the numbers in the ordered pair as the coordinates. Add a column to the 1 : 3 ratio table
on the board, and write the ordered pairs in the new column: (1, 3), (2, 6), (3, 9), and (4, 12).
Exercises: Add a column to each ratio table from the previous exercises. In the new column,
write the ordered pair for each row.
Answers: a) (1, 2), (2, 4), (3, 6), (4, 8); b) (4, 5), (8, 10), (12, 15), (16, 20);
c) (5, 4), (10, 8), (15, 12), (20, 16); d) (7, 10), (14, 20), (21, 30), (28, 40)
Plotting ratio tables on coordinate planes. Explain that, to plot a ratio on a coordinate grid,
you need to make a ratio table first, make a list of ordered pairs from that ratio table, and, finally,
plot the ordered pairs on the grid.
Students can use the grids from BLM Small Coordinate Grids for the exercises below. Point out
that, for the last two tables, they will need to use the largest grids on the BLM.
Exercises: Plot the ordered pairs from parts a), b), and c), of the previous exercises on a grid.
Answers:
a) y
b) y
c) y
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8
x
0
F-20
20
18
16
14
12
10
8
6
4
2
20
18
16
14
12
10
8
6
4
2
2 4 6 8 10 12 14 16 18 20
x
0
2 4 6 8 10 12 14 16 18 20
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
x
(MP.7) Equivalent ratios produce points on the same line through the origin. Ask students
to join, or connect, the points that come from the same ratio table on each graph. ASK: What do
you notice? (the points always fall onto the same line) Have them extend the lines so that they
intersect the axes. ASK: What happens to the lines? (they all pass through the point (0, 0))
What special name does this point have? (the origin)
Summarize the facts just discovered: points produced by ratio tables fall on the same line and
that line passes through the origin. Explain that these facts can be used to check whether a ratio
table has been plotted correctly. Students can check if the line passes through the origin when
continued by placing a ruler alongside it.
Students can use another copy of BLM Small Coordinate Grids to do the exercises below. To
save time, they can produce and plot just three points for each ratio.
Exercises: Create a graph that shows the ratio. Are the points on the same line? If so, does the
line pass though the origin?
a) 2 : 3
b) 3 : 5
c) 2 : 1
Answers:
a) points (2, 3), (4, 6), and (6, 9) should be connected with a line
b) points (3, 5), (6, 10), and (9, 15) should be connected with a line
c) points (2, 1), (4, 2), and (6, 3) should be connected with a line
All lines should pass through the origin.
Graphing ratios to solve problems. Explain that graphing a ratio can help solve certain
problems. SAY: For example, if a train travels at the speed of 70 miles per hour, this means that
it travels 70 miles every hour. We can make a ratio table for this train. Write on the board:
Time (h)
1
Distance (mi)
70
SAY: I know that the train travels 70 miles in 1 hour. ASK: How many miles will it travel in
2 hours? (140) How do you know? (2 × 70 = 140) Fill in the second row. ASK: How many miles
will the train travel in 3 hours? (3 × 70 = 210) Fill in the third row. Have students copy the table
into their notebooks.
Add a third column to the table for ordered pairs and have students fill in the third column. The
finished table should look like this:
Time (h)
1
2
3
Distance (mi)
70
140
210
Ordered Pair
(1, 70)
(2, 140)
(3, 210)
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-21
Draw axes on the grid on the board and explain that you have to use skip counting to mark the
vertical axis, because the third ordered pair in your table has the second coordinate y = 210.
Explain that you will skip count by 35s to label the axis, because 35 divides 70 and skip counting
like this allows you to make a graph that does not require too much vertical space. Fill in the
axes, as shown in the completed graph below.
Explain that the ratio we were given, 70 miles each hour, gives us more information than just
numbers. SAY: We know that the numbers on the horizontal axis are the number of hours, or
the time, that the train travels. Label the horizontal axis, as shown in the completed graph
below. ASK: What does the vertical axis show? (the number of miles the train travels, the
distance in miles) Label the vertical axis, as shown in the completed graph below.
Invite volunteers to plot the points from the table on the graph and join the points with a line.
The completed graph should look like this:
Distance (mi)
y
210
175
140
105
70
35
0
1
2
3
Time (h)
4
x
ASK: If I extend the line toward the axes, will it pass through the origin? (yes) Have a volunteer
check by extending the line.
SAY: The graph tells us more than the ratio table does. The graph has a line that joins the
points, and any point on that line has the same ratio. Draw a point on the line at (1.5, 105) and
ASK: What are the coordinates of this point? ((1.5, 105)) What is the meaning of this point?
(the train travels 105 miles in 1.5 hours) How do you know? (the x-coordinate is the time in
hours and x = 1.5, so the train travels for 1.5 hours; the y-coordinate shows the distance in
miles and y = 105, so the train travels 105 miles) PROMPTS: What does the x-coordinate
show? What does the y-coordinate show?
ASK: Which point on the graph shows how far the train travels in half an hour? (point (0.5, 35))
Have a volunteer draw the point on the graph and identify the coordinates. ASK: How far does
the train travel in half an hour? (35 miles) Which point on the graph shows how long it takes the
train to travel 175 miles? (point (2.5, 175)) Again, have a volunteer identify the point on the
graph and its coordinates. ASK: How long does it take the train to travel 175 miles? (2.5 hours)
Keep the graph on the board for later use.
Exercises: Satellite A is moving through space, around Earth. It travels 25,000 km every hour.
a) Make a ratio table for the satellite’s movement.
b) Graph the ratio. Skip count by 12,500 km on the vertical axis.
c) Use the graph to tell how far the satellite moves in 2.5 hours.
d) Use the graph to tell how long it takes the satellite to travel 37,500 km.
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Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Distance (km)
25,000
50,000
75,000
c) 62,500 km
d) 1.5 hours
y
b)
Distance (km)
Answers:
a) Time (h)
1
2
3
75,000
62,500
50,000
37,500
25,000
12,500
0
1
2
3
x
Time (h)
Use graphs to compare ratios for everyday situations. Return to the graph showing the
distance a train travels. Label the graph itself (i.e., the line and the points) “Train A.”
SAY: Another train, Train B, travels 105 miles in 2 hours. I want to know which train travels
faster. I can plot the ratio for Train B on the same graph. Have students make a ratio table for
Train B. Point out that since the graph only shows a distance up to 210 miles, students do not
need to create too many rows in the table. When they reach 210 or a number larger than that,
they should stop. ASK: How many rows does your table have? (2) Have a volunteer write the
ratio table on the board. Have students write the ordered pairs in their notebooks, then have
another volunteer plot the ratio for Train B on the same graph as Train A. Have the volunteer
use a different color to plot the second ratio. The ratio table and graph should look like this:
y
Distance (mi)
105
210
Distance (mi)
Train B
Time (h)
2
4
Train A
210
175
140
105
70
35
0
1
2
3
Train B
4
x
Time (h)
ASK: How far does Train A travel in 2 hours? (140 mi) How far does Train B travel in 2 hours?
(105 mi) Which train travels farther in 2 hours, Train A or B? (Train A) How can we see that on
the graph? (the point that shows the train’s position at 2 hours is higher on the grid for Train A
than for Train B) SAY: The points closest to the top on the graph are on the same level.
ASK: What does this mean? (the trains both travelled 210 mi) How long does it take Train A
to travel 210 mi? (3 hours) How long does it take Train B to travel 210 mi? (4 hours) Which
train travels faster? (Train A)
(MP.4) Exercises: Satellite B travels 15,000 km in half an hour.
a) Make a ratio table to show how far Satellite B travels in 1 hour, 1.5 hours, and 2 hours.
b) Graph the ratio using the table from part a). Use the same coordinate grid you used in the
previous exercises, for Satellite A.
c) Compare Satellite A with Satellite B. Which satellite travels farther in 1 hour and in 2 hours?
d) Which travels faster, Satellite A or Satellite B?
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-23
Answers:
a) Time (h)
0.5
1
1.5
2
y
b)
Distance (km)
Distance (km)
15,000
30,000
45,000
60,000
c) Satellite B
d) Satellite B
75,000
62,500
50,000
37,500
25,000
12,500
0
Satellite A
Satellite B
1
2
Time (h)
3
x
Ask the class to compare the lines for the slower and the faster objects in the previous
exercises and examples (satellites, trains) on the grids. ASK: How can you see which object
goes faster from the graph? (the line for the object that goes faster is steeper)
Distance (km)
Extensions
1. Randi runs 2 km every 15 minutes.
a) Draw a graph to show Randi’s running. Skip count by 15s on the x-axis and count by 1s on
the y-axis.
b) At that same pace, how far will Randi run in 30 minutes? Which point on the graph shows
the answer?
c) How far will she run in 1 hour? Which point on the graph shows the answer?
Answers:
a) y
7
6
5
4
3
2
1
0
15
30
45
60
75
x
Time (min)
b) Randi will run 4 km in 30 minutes, (30, 4)
c) Randi will run 8 km in 1 hour, (60, 8)
2. Ethan earns $17 every 2 hours.
a) Draw a graph to show Ethan’s earnings. Show intervals of half an hour on the x-axis. Skip
count by $4.25 on the y-axis.
b) How much will Ethan earn in 30 minutes? How does the graph show that?
c) How much will he earn in 3 hours? How does the graph show that?
d) How long will it take him to earn $42.50? How does the graph show that?
F-24
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
Earnings ($)
Answers:
y
a)
46.75
42.50
38.25
34.00
29.75
25.50
21.25
17.00
13.75
9.50
4.25
0
1
2
3
4
Time (h)
5
x
b) Ethan earns $4.25 in 30 minutes; the line passes through point (1/2, 4.25).
c) Ethan earns $25.50 in 3 hours, because the line passes through point (3, 25.50).
d) It will take 5 hours to earn $42.50, because the line passes through point (5, 42.50).
(MP.1) 3. Tony and Lynn are saving money for a gift for their grandfather. They will not share
the cost equally.
a) Tony saves $15 every 2 weeks. Make a graph to show his savings.
b) Lynn saves $10 every week. Graph her savings on the same grid you used in part a).
c) How much money will they have together after 2 weeks? After 4 weeks? Show their
combined savings on the same grid.
d) The gift costs $87.50. Draw a horizontal line showing the price.
e) When will Tony and Lynn have enough money together to pay for the gift?
f) How much money will Tony pay? How much money will Lynn pay?
Answers:
parts a), b), c), d):
Savings ($)
y
100
90
80
70
60
50
40
30
20
10
0
Total savings
Gift price
Lynn’s savings
Tony’s savings
1 2 3 4 5 6
Time (weeks)
x
c) Tony and Lynn will save $35 after 2 weeks and $70 after 4 weeks
e) Tony and Lynn will have enough money after 5 weeks
f) Tony will pay $37.50 and Lynn will pay $50
Teacher’s Guide for AP Book 8.1 — Unit 5 Expressions and Equations
F-25