Unit 5 Ratios and Proportional Relationships:
Equivalent Ratios and Percents
This Unit in Context
In this unit, students will extend their study of ratios that began in 6.1 Unit 1 to include finding
equivalent ratios and calculating percentages (6.RP.A.3c). They will see the relationship
between decimals, fractions, and percentages.
In Grade 7, students will use proportional relationships to solve multistep ratio and percent
problems, including calculating simple interest, tax, markups and markdowns, gratuities and
commissions, fees, percent increase and decrease, and percent error.
Ratios and Proportional Relationships
F-i
RP6-11 Equivalent Ratios
Page 126
STANDARDS
6.RP.A.1, preparation for
6.RP.A.3
Vocabulary
equivalent ratio
ratio table
Goals
Students will understand that equivalent ratios are obtained through
multiplication, and they will find examples of equivalent ratios.
PRIOR KNOWLEDGE REQUIRED
Is familiar with equivalent fractions
Is familiar with ratios
Introduce equivalent ratios. Tell students that you have a pancake recipe
that calls for 6 cups of flour and 2 bananas. Draw the picture shown in the
margin on the board.
ASK: How many cups of flour do we need for only 1 banana? (3 cups) How
can you use that information to find the number of cups of flour you would
need for 5 bananas? (multiply 5 × 3) Emphasize that the number of cups of
flour is always 3 times the number of bananas, so if students know how many
bananas they have, they can deduce the number of cups of flour. Have
students perform this calculation for various numbers of bananas: 7, 6, 3.
(Answers: 21 cups of flour, 18 cups of flour, 9 cups of flour) Tell students
that they’ve just found many equivalent ratios:
3 : 1 = 6 : 2 = 21 : 7 = 18 : 6 = 9 : 3
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ASK: Why are these ratios called equivalent? (because they show the same
ratio) Tell students that to say the ratio of cups of flour to bananas is 3 to 1
is to say that for every 3 cups of flour, we need 1 banana. Draw the picture
shown in the margin on the board. ASK: If you have 9 cups of flour, how
many bananas do you need? (3) Draw 3 cups of flour and 1 banana and
repeat until you have 9 cups of flour. How many bananas did you draw? (3)
ASK: How many cups of flour would you need for 10 bananas? (30) So the
ratio 30 : 10 is equivalent to the ratio 3 : 1. Emphasize that 30 is 3 × 10, so
students can just multiply the number of bananas by 3 to get the number
of cups of flour.
Emphasize that in the example above, students compared numbers through
multiplication rather than through addition. If we had said, “The recipe calls
for 3 cups of flour and 1 banana, so it calls for 2 more cups of flour than the
number of bananas,” this would be comparing through addition rather than
through multiplication. ASK: If I want to make the recipe with 5 bananas,
should I use 7 cups of flour since 7 is 2 more than 5? Would the recipe work
if I used 2 cups of flours with no (0) bananas since 2 is 2 more than 0?
Will my pancakes turn out right? Explain to students that in this situation
(a recipe), 3 (cups of flour) is 3 times 1 (bananas) and it’s this 3 : 1 ratio we
want to preserve—the number of cups of flour should always be 3 times
the number of bananas.
Ratios and Proportional Relationships 6-11
F-1
Cups of
Flour
Number of
Bananas
3
1
6
2
9
3
12
4
Then show students how they can make a series of equivalent ratios by
repeatedly drawing 3 cups of flour and 1 banana:
3 : 1
=
6 : 2
9 : 3
=
=
12 : 4
15 : 5
=
SAY: I would like to use a table to show these ratios. ASK: What will be the
headings in my table? PROMPT: What are the different units? (cups of flour
and bananas) Draw the table shown in the margin on the board.
ASK: Is this a ratio table? (yes) How do you know? (because every row is
a multiple of the first row) SAY: Mathematicians call such tables ratio tables
because all rows shown are equivalent ratios. Ask a volunteer to come and
write the next row in the table. (the next row is 15 and 5)
Exercises
a) Fill in the ratio tables.
i)
3
5
6
10
ii)
2
iii)
7
4
5
iv)
8
3
8
3
b) Write the next four equivalent ratios in each sequence.
i) 3 : 5 = 6 : 10 =
:
=
:
=
:
ii) 2 : 7 =
iii) 4 : 5 =
iv) 8 : 3 =
a)i)
ii)
2
7
10
4
9
15
12
20
3
5
6
iii)
iv)
4
5
14
8
10
16
6
6
21
12
15
24
9
8
28
16
20
32
12
b) Use the rows in the tables.
Explain to students how to find the missing part of the ratio here:
1:4=
: 12.
Tell students that they should continue the sequence of equivalent ratios
until 12 is the second number, so we have 1 : 4 = 2 : 8 = 3 : 12. SAY: To
find the ratios 2 : 8 and 3 : 12 from 1 : 4 you can use skip counting or
multiplication.
Ask a volunteer how he or she would find the missing part of the ratio here:
5 : 6 = 15 :
. (Continue the sequence of equivalent fractions until 15 is
the first number.)
F-2
Teacher’s Guide for AP Book 6.1
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Answers
Exercises: Find the missing term in each pair of equivalent ratios.
a) 3 : 5 =
: 20
b) 3 : 4 =
:12
e) 3 : 5 =
: 15
d) 3 : 5 = 15:
c) 3 : 4 = 12 :
Answers: a) 12, b) 9, c) 16, d) 25, e) 9
Ask students to find many real-life examples of ratios:
a)For every
years is
b)For every
weeks is
months, there is 1 year, so the ratio of months to
: 1.
:
days, there is
.
c)For every
to items is
dozen, there are
:
.
d)For every
:
.
mm, there is
week, so the ratio of days to
items, so the ratio of dozens
cm, so the ratio of mm to cm is
Extensions
1.Use your fingers and hands to show that 1 : 2 and 5 : 10
are equivalent ratios.
(MP.4, MP.1)
2.The mass density or density of a material is its mass per unit volume.
In other words, the ratio of mass to volume is density. For example,
the density of water at 4°C is 1 gram per mL. That means 1 gram of 4°C
water has a volume of 1 mL; the ratio of mass to volume is 1 : 1.
a)Lillian has a full 500 mL bottle of water in the fridge (almost 4°C).
What is the mass of the water in the bottle?
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b)Lillian’s mother has a small scale in the kitchen and she can weigh
up to 5 kg. She has a star-shaped cake pan and wants to know
what volume of liquid can fit into the pan. How can she use 4°C water
and the scale to find the volume?
Answers: a) 500 grams, b) She can fill the pan with 4°C water, and then
weigh the water with the scale. The ratio of mass (in gram) to volume (in
mL) is 1 : 1, so if the mass of the water is, for example, 200 grams, then
the volume is 200 mL.
Ratios and Proportional Relationships 6-11
F-3
RP6-12 Finding Equivalent Ratios
Pages 127–128
Goals
STANDARDS
6.RP.A.1, 6.RP.A.3
Students will find equivalent ratios through multiplication rather
than through repeated addition. Students will solve word problems
using ratios.
Vocabulary
equivalent ratios
PRIOR KNOWLEDGE REQUIRED
Can identify equivalent ratios
Part-to-part problems where a part is given. Put 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?
3 : 2
boys girls
Have a volunteer write the first few terms of the sequence of equivalent ratios.
ASK: Which ratio in the sequence are we looking for? Which number needs
to be 12? (the second number) Continue the sequence: 3 : 2 = 6 : 4 = 9 : 6 =
12 : 8 = 15 : 10 = 18 : 12. So there are 18 boys if there are 12 girls.
Exercise: There are 4 boys for every 5 girls in a class. There are 20 boys
in the class. How many girls are in the class?
Answer: 25 girls
(MP.7)
Part-to-part problems where the total is given. Now change the
question slightly again: There are 3 boys for every 2 girls in a class.
There are 25 students in the class. How many boys are in the class?
Now, students are looking for the term in the sequence where the two
numbers add to 25. Write the sum of the two numbers under each ratio:
boys : girls 3 : 2 = 6 : 4 = 9 : 6 = 12 : 8 = 15 : 10
5 1015 20 25
So there are 15 boys in a class of 25. (We know the first number is
the number of boys because we are given the ratio of boys to girls,
not girls to boys.)
F-4
Teacher’s Guide for AP Book 6.1
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Change the original question slightly: There are 3 boys for every 2 girls
in a class. There are 12 boys in the class. How many girls are in the class?
ASK: Which number has to be 12, the first or the second? (the first) When
the first number is 12, what is the second number? (8) So there are 8 girls
when there are 12 boys.
Exercises: Have students solve the following problems using
a similar method.
a)There are 4 boys for every 7 girls in a class of 33 children.
How many girls are in the class?
b)There are 6 boys for every 5 girls in a class of 22 children.
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 of them are red?
Then take up the answers with the whole class. For example, for
part a), write on the board:
4 : 7 = 8 : 14 = 12 : 21
Have a volunteer circle the number in each ratio that represents the number
of girls, and have another volunteer write the total number of students
below each ratio:
Number
of Girls
Total
Number of
Students
7
11
14
22
21
33
Number
of Girls
Total
Number of
Students
7
11
4: 7 11 students
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×3
Total
Number of
Students
7
11
21
33
=
12 : 21
33 students
So in a class of 33 students, there are 21 girls.
SAY: As in the previous lesson, we can use a ratio table to find the number
of girls in the class. In this case we have to start with the ratio of girls to the
total number of students, which is 7 : 11. Draw the ratio table shown in the
margin on the board.
Emphasize that the other rows can be obtained using skip counting or multi­
plication. Multiplication can be used as a shortcut so that we don’t even need
to find the whole table. SAY: We can find the missing number in 7 : 11 =
: 33 by finding the number we need to multiply 11 by to get 33 using
a ratio table. Then draw the table shown in the margin on the board.
33
Number
of Girls
8 : 14 22 students
=
SAY: Since 3 × 11 = 33, we multiply 3 × 7 = 21. Draw the arrows and write
21 in the empty cell, as in the margin.
Exercises: Find the missing number in each ratio table.
a)
×3
4
12
7
b)
3
5
15
c)
2
9
27
d)
12
14
36
Answers: a) 21, b) 25, c) 6, d) 42
Explain to students that in a ratio table arrows can point from bottom to top.
Emphasize that both arrows must have the same direction; if you multiply
from top to bottom in one column, you have to multiply from top to bottom
in the other column.
Ratios and Proportional Relationships 6-12
F-5
Exercises: Find the missing number in each ratio table.
a)
5
Number
of Tickets
Cost
5
9
20
?
Number
of Tickets
Cost
5
9
20
36
×4
b)
20
6
10
3
c)
2
d)
12
4
7
48
8
12
Answers: a) 24, b) 15, c) 21, d) 32
Show students how to solve word problems using equivalent ratios. Use
this problem: If 5 bus tickets cost $9, how much would 20 tickets cost?
Step 1: (make the ratio table) Draw a table with two columns and the
headings shown. Write the ratio of bus tickets to dollars (5 : 9) in the first
row. In the second row, write 20 in the “Number of Tickets” column and a
question mark (?) for the missing number in the “Cost” column.
Step 2: (find the missing number) Find the number being multiplied by in
the first column (in this example, 4). Then multiply by that number in the
second column to find the missing number.
×4
(MP.4)
Since 5 : 9 = 20 : 36, 20 tickets cost $36.
Exercises: Have students answer the following questions, allowing
volunteers to do the first few.
a) If 5 bus tickets cost $4, how much will 15 bus tickets cost?
b) Five bus tickets cost $6. How many can you buy with $30?
c)On a map, 3 cm represents 10 km. How many kilometers
does 15 cm represent?
d)Tanya gets paid $25 for 3 hours of work. How much would
she get paid for working 6 hours?
e)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?
f)There are 2 apples in a bowl for every 3 oranges. If there are
12 oranges, how many apples are there?
Bonus: A goalie stopped 18 out of every 19 shots. There were 38 shots.
How many goals were scored? (i.e., how many did she not stop?) Answer: 2
ASK: The ratio of boys to girls in a class is 4 : 7. What is the ratio of boys to
students? (4 : 11) The ratio of girls to boys in a class is 5 : 3. Are there more
girls or boys in the class? (girls)
Extensions
1.There are 6 boys for every 10 girls on a school trip. If there are 35 girls,
how many boys are there? (NOTE: To solve this question, you need to
reduce the ratio given to lowest terms.)
F-6
Teacher’s Guide for AP Book 6.1
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Answers: a) $12, b) 25, c) 50 km, d) $50, e) 40 km, f) 8
(MP.1)
2. a)Sindi is reading on her way to work. She reads 3 pages on the 1 km
bus ride. What is the ratio of pages read to kilometers travelled on
the bus? (3 : 1)
b)Sindi gets off the bus and gets on the subway train. She reads
6 pages on the 6 km subway ride. What is the ratio of pages read
to kilometers travelled on the subway? (6 : 6 = 1 : 1)
c)For each kilometer Sindi travels, what is the ratio of pages read on
the bus to pages read on the train? PROMPT: If Sindi travelled 6 km
on the bus, how many pages would she read? (The ratio of pages
read on the bus to pages read on the train is 3 : 1.)
d)Sindi reads at the same rate on the bus as on the train. Which
mode of transportation is faster, the bus or the train? How many
times faster? (The train is three times as fast as the bus.)
e)Anna is knitting on her way to work. She knits 120 stitches on the
2 km bus ride, switches to the subway train and then knits 450 stitches
on the 15 km subway ride. How much faster is the subway train than
Anna’s bus? (Answer: bus ratio is 60 stitches : 1 km, train ratio is
30 stitches : 1 km, so she gets twice as much done for each kilometer
on the bus than on the train. This means the subway train travels
twice as fast as Anna’s bus.) What assumption did you need to make?
(Anna knits at the same rate on the bus as the train.)
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f ) Whose bus travels faster, Anna’s or Sindi’s? (Anna’s bus is faster;
the subway train is only twice as fast as Anna’s bus, but it is three
times as fast as Sindi’s bus.) What assumption do you need to make?
(The subway travels at the same speed for both Anna and Sindi.)
Ratios and Proportional Relationships 6-12
F-7
RP6-13 Percents
Pages 129–130
Goals
STANDARDS
6.RP.A.3
Students will write given fractions as percents, where the given
fractions have a denominator that divides evenly into 100.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
percent
Can find equivalent fractions
Can reduce fractions to lowest terms
Can convert decimals to fractions
Can find a decimal equivalent to a fraction
(MP.6)
Percents as ratios. Ask students what the word “per” means in
these sentences:
Rita can type 60 words per minute.
Anna scores 3 goals per game.
John makes $10 per hour.
The car travels at a speed of up to 140 kilometers per hour.
Emphasize that “per” means “for each” or “for every.” Ask volunteers to
read the sentences with “for every” replacing “per.” Then write “percent”
on the board. ASK: What is a “cent” (it’s an amount of money; 100 cents
is a dollar) SAY: In French, cent means 100. Explain that “percent” means
“for every 100” or “out of every 100.” For example, a score of 84% on a
test would mean that you got 84 out of every 100 marks or points. Another
example: If a survey reports that 72% of people read the newspaper every
day, that means 72 out of every 100 people read the newspaper daily.
ASK: Sally got 84% on a test where there were 200 possible points. How
many points did she get? Then rephrase the question: A test has 200
possible points. Sally got 84 points for every 100 possible points. How
many points did she get?
=
200
possible points
Explain to students that a percent is a ratio that compares a number to 100.
Exercises: Have students rephrase the percents in these statements using
the phrases “for every 100
” or “out of 100
.”
a)52% of students in the school are girls.
b)40% of tickets sold were on sale.
c)Alejandra scored 95% on the test.
d)About 60% of your body weight is water.
Answers: a) For every 100 students, 52 are girls OR 52 out of every 100
students in the school are girls, b) For every 100 tickets sold, 40 were on
F-8
Teacher’s Guide for AP Book 6.1
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84 points
100 possible points
sale OR 40 out of every 100 tickets were on sale, c) For every 100 possible
points, Alejandra scored 95 points on the test OR Alejandra got 95 out
of every 100 points on the test, d) For every 100 kg of body weight,
about 60 kg is water OR 60 kg out of every 100 kg of body weight is
made up of water
Percents as fractions. Explain to students that a percent is just a short way
of writing a fraction with denominator 100. For example, you can write the
fraction 84/100 as 84%.
Exercises
1. Have students write each fraction as a percent:
a)28/100
b)9/100
c)34/100
d)
67/100e)
81/100f)
3/100
2. Have students write each percent as a fraction:
a)6%
b)19%
c)8%
d)
54%e)
79%f)
97%
Writing hundredths as percents. Now have students write each decimal
as a percent by first changing the decimal into a fraction with denominator
100. For example, 0.84 is 84/100, which is 84%.
Exercises
a) 0.74 b) 0.03 c) 0.12
d)0.83
e)0.91
f) 0.09
Changing fractions to percents when the denominator divides evenly
into 100. Write the fraction 3/5 on the board and have a volunteer find an
equivalent fraction with denominator 100 (60/100). ASK: If 3 out of every
5 students at a school are girls, how many out of every 100 students are
girls? (60) What percent of the students are girls? (60%) Write on the board:
3/5 = 60/100 = 60%
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Exercises
a)Have students find the equivalent fraction with denominator 100 and then
the equivalent percent for more fractions with denominator 5. Examples:
2/5, 4/5, and 1/5.
b)Repeat for fractions with various denominators. Examples: 4/10, 9/20,
3/4, 1/2, 29/50, 21/25, 17/25.
Using percents to order fractions. Point out that percents are easily
ordered because they are all fractions with the same denominator, 100.
Use the equivalent percents to put the above fractions in order from
least to greatest.
Writing decimal tenths as percents. Then have students write various
decimal tenths as percents by first changing the decimal to a fraction with
denominator 100. Examples: 0.2 (= 2/10 = 20/100 = 20%), 0.3, 0.9, 0.7, 0.5.
Ratios and Proportional Relationships 6-13
F-9
(MP.4)
a)
b)
c)
Equivalent percent of a fraction. Explain to students that they can
find a percent of a figure just as they can find a fraction of a figure. Ask
students to decide first what fraction and then what percent of each
figure at left is shaded. (a) 4/10 or 40%, b) 1/4 or 25%, c) 7/20 or 35%)
Fractions that need to be reduced before changing the denominator
to 100. Write on the board the fraction 9/15. Tell students that you want to
find an equivalent fraction with denominator 100 so that you can turn it into
a percent. ASK: How is this fraction different from previous fractions you
have changed to percents? (The denominator does not divide evenly into
100.) Is there any way to find an equivalent fraction whose denominator
does divide evenly into 100? (Reduce the fraction by dividing both the
numerator and the denominator by 3.) Write on the board:
9/15 = 3/5 = 60/100 = 60%
Summarize the steps for finding the equivalent percent of a fraction.
Step 1: Reduce the fraction so that the denominator is a factor of 100.
Step 2: Find an equivalent fraction with denominator 100.
Step 3: Write the fraction with denominator 100 as a percent.
Example: Change 14/35 to a percent by first reducing it to lowest terms.
Solution:
14 ÷ 7 2 × 20
40
= 40%
= =
=
35 ÷ 7 5 × 20 100
Exercises: Write fractions as percents.
a) 3/12 b) 6/30 c) 24/30 e) 6/15 f) 36/48 g) 60/75
d) 3/75
Answers: a) 25%, b) 20%, c) 80%, d) 4%, e) 40%, f) 75%, g) 80%
ASK: How many degrees are in a circle? (360) If I rotate an object
90° counterclockwise, what fraction and what percent of a complete
360-degree turn has the object made? PROMPT: 90 out of 360 is what
out of 100? (90/360 = 1/4 = 25/100 = 25%)
Repeat for 180°, 18°, 126°, 270°, 72°, 216°.
F-10
Teacher’s Guide for AP Book 6.1
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Extension
RP6-14 Visual Representations of Percents
Pages 131–132
STANDARDS
6.RP.A.3
Goals
Students will visualize various percentages of different shapes,
including rectangles, squares, triangles, and line segments.
Vocabulary
percent
PRIOR KNOWLEDGE REQUIRED
Can write equivalent fractions
Understands the relationship between decimal tenths
and hundredths and fractions with denominator 100
39
, 0.39, 39%)
(
100
Percent of a shape. Draw a hundreds block on the board and have
students write what part of the block is shaded in three different ways:
as a fraction, a decimal, and a percent. See the margin for an example.
Have students find 25% of each shape at left in various ways:
Then draw shapes on the board and divide them into equal pieces,
the number of which divide evenly into 100:
ASK: What fraction of each shape is shaded? Have students change each
fraction to an equivalent fraction with denominator 100, and then to a
decimal and a percent.
Percent of a line. Show students a double number line with fractions
on top and percents on the bottom.
0
1 10
2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 1
10
0 10%20% 30%40%50% 60%70% 80%90%100%
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01
01
Exercises: Have volunteers add fractions and percents to each number
line in the margin. Hint: If there are 5 parts in a whole, each part is 1/5.
Bonus: If there are 5 parts in 1/2, how many parts are in a whole? (10) So
what is each part? (1/10)
0
1
2
1
01
2
Draw line segments of varying lengths and have volunteers estimate (visually)
and mark a different percent on each one. Use only multiples of 25.
Example:
50%
Ratios and Proportional Relationships 6-14
Answer:
F-11
Exercises
a)25%
b)75%
c)50%
d)75%
NOTE: Teach students the strategy of estimating 25% and 75% using 50%.
Once they’ve estimated and marked 50%, they can halve the left part of the
line to estimate 25% and halve the right part of the line to estimate 75%.
Draw on the board a line one meter long, and have students estimate the
percent of various marks on the number line (to the nearest 10%). Then,
using the meter stick, draw another line of the same length divided into
10 equal parts above or below the first line, so that students can check
their estimates.
Extending a line to 100%. Draw a line 2 cm long:
. Tell students
this line is 1/3 of a longer line segment; it is part of a whole. Have a volunteer
extend the line to make the whole. Point out that we need 3 equal parts and
we already have 1, so we need to add 2 more (see the margin).
Exercises
a)Draw a 3 cm line segment. It is 1/4 of a line segment.
Extend to make the whole.
b)Draw a 5 cm line segment. It is 1/2 of a line segment.
Extend to make the whole.
c)Draw a 4 cm line segment. It is 1/3 of a line segment.
Extend to make the whole.
Repeat for a 6 cm line segment that is 3/5 of a line segment. Now we
divide the line segment into 3 equal parts, and we need to draw 2 more
of those equal parts.
(MP.2)
Exercises
a)Draw a 6 cm line segment. It is 2/5 of a line segment.
Draw the whole line segment.
b)Draw an 8 cm line segment. It is 4/7 of a line segment.
Draw the whole line segment.
F-12
Teacher’s Guide for AP Book 6.1
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Now draw a 6 cm line segment. Then tell students it is 2/3 of a line segment.
SAY: This line segment is 2 out of 3 equal parts. How can we find what 1 of
the 3 equal parts looks like? (divide the line segment in two equal parts) Do
so and then SAY: Now we know what 1 part looks like. How many more of
those parts do we need to draw? (1) Have a volunteer draw the 1 extra part.
Notice that 50% = 1/2, so a given line segment that is 50% of the whole is
1 of 2 equal parts, and you can simply draw another equal part to make the
whole. Also, 40% = 4/10 = 2/5, so a given line segment that is 40% of the
whole is 2 of 5 equal parts. You can divide this line segment into two equal
parts and draw three more identical parts. Then have students extend lines
that are a percent of the whole (instead of a fraction, as in the previous
exercises). Students will have to change the percent to a reduced fraction
to turn these problems into problems they already know how to do.
Exercises
a)Draw a 12 cm line segment. It is 80% of a line segment.
Draw the whole line segment.
b)Draw a 5 cm line segment. It is 40% of a line segment.
Draw the whole line segment.
Estimating percents. Finally, have students estimate the percent of various
marks on a line segment (to the nearest 10%), and then superimpose a
number line of the same length divided into 10 equal parts so that students
can check their estimates.
Extension
Have students find 25% of a triangle like this one:
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Answer:
Ratios and Proportional Relationships 6-14
F-13
RP6-15
Comparing Decimals, Fractions,
Page 133
and Percents
STANDARDS
6.RP.A.2, 6.RP.A.3
Vocabulary
decimal
percent
Goals
Students will compare and order fractions, percents, and decimals.
PRIOR KNOWLEDGE REQUIRED
Understands percents as fractions with denominator 100
Can order fractions
Can find equivalent fractions
Knows the signs for less than (<) and greater than (>)
Review comparing and ordering:
• fractions with the same denominator (7/10 is greater than 4/10).
• percents (30% is greater than 24% because 30/100 is greater
than 24/100).
• fractions with different denominators (5/10 is greater than
6/20 = 3/10 ).
• fractions and decimals (3/5 is greater than 0.52 because
60/100 is greater than 52/100).
Comparing fractions and percents. Remind students of the signs for less
than (<) and greater than (>) and use them throughout the lesson. Teach
students how to compare fractions and percents by changing both to an
equivalent fraction with denominator 100.
Exercises
1. Which is larger?
a) 1/2 or 38%
b) 3/5 or 70%
d) 7/25 or 30%
e) 9/20 or 46%
c) 9/10 or 84%
Answers: a) 1/2, b) 70%, c) 9/10, d) 30%, e) 46%
2.Which is closer to 50%? Hint: Change the fractions to percents first.
b) 3/10 or 4/5
c) 3/5 or 1/4
Bonus: 2/5 or 3/5
Answers: a) 2/5, b) 3/10, c) 3/5, Bonus: the same
Comparing decimals and percents. Compare decimals and percents by
changing both to an equivalent fraction with denominator 100.
Exercises
1. Which is larger?
a) 0.9 or 10%
b) 0.09 or 10% c) 28% or 0.34 d) 4% and 0.3
Answers: a) 0.9, b) 10%, c) 0.34, d) 0.3
2.Change the numbers to a fraction with denominator 100, and then put
the numbers in order from least to greatest.
F-14
Teacher’s Guide for AP Book 6.1
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a) 1/4 or 2/5
a)0.28 42% c)
3
14
b)
23% 0.3
10
50
19
0.72 7%
25
d)
1
4% 0.4
4
Bonus: 13/20 0.6 66% 0.7 7% 16/25 3/50
Answers: a) 0.28, 3/10, 42%; b) 23%, 14/50, 0.3; c) 7%, 0.72, 19/25,
d) 4%, 1/4, 0.4; Bonus: 3/50, 7%, 0.6, 16/25, 13/20, 66%, 0.7
(MP.3)
Comparing fractions and percents when a denominator does not divide
evenly into 100. ASK: How can we compare 35% to 1/3? If we changed 35%
to a fraction, what would it be? (35/100) Do we have a way to compare 1/3 to
35/100 or are we stuck? We have two fractions with different denominators,
but 3 doesn’t divide evenly into 100. How can we give both fractions the same
denominator? (use denominator 300) Have volunteers change both fractions
to equivalent fractions with denominator 300 and ask the class to identify
which is greater, 35% or 1/3, and to explain how they know.
Repeat with various reduced fractions whose denominator does not divide
evenly into 100. Then have students compare more fractions and percents
independently, in their notebooks.
Exercises: Compare 5/6 and 85%, 3/7 and 42%, 2/9 and 21%.
Bonus: Make up your own question and have a partner solve it.
Finally, have students order lists of numbers (fractions, percents, and
decimals) in which the fractions do not have denominators that divide
evenly into 100.
Exercises: Order the numbers from least to greatest.
a) 1/6 , 0.17, 13% b) 0.37, 1/3 , 28% c) 5/7 , 71%, 0.68
Bonus: 7/9, .8, 4/7, 51%, .78, 62%
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Answers: a) 13%, 1/6, 0.17, b) 28%, 1/3, 0.37, c) 0.68, 71%, 5/7,
Bonus: 51%, 4/7, 62%, 7/9, .78, .8
Extensions
1. Ask students to name percents that indicate
• almost all of something,
• very little of something,
• a little less than half of something.
Ask students to explain their thinking.
2.Ask students to look for percents in newspapers, flyers, magazines, and
other printed materials, such as food packaging, trading cards, and
order forms. What kind of information is expressed as a percent? Ask
students to clip examples and to make a collage for a class display.
Ratios and Proportional Relationships 6-15
F-15
RP6-16 Long Multiplication (Review)
Pages 134–135
Goals
STANDARDS
preparation for 6.RP.A.3
Students will use the standard algorithm to solve 2-digit by 2-digit
multiplication problems.
PRIOR KNOWLEDGE REQUIRED
Understands order of operations
Can apply the distributive property (without using the terminology)
Review. Review 2-digit by 1-digit products using the standard algorithm.
Have students solve in their notebooks: 37 × 2, 45 × 3, 38 × 7, 32 × 5.
Review multiplying by multiples of 10. Have students solve, using mental
math: 37 × 20, 45 × 30, 38 × 70, 32 × 50.
Write the products shown in the margin on the board.
1
3
1
7
2
×
7
4
× 10
3
7
×
2
0
7
4
0
Students should work on grids to start, to help ensure correct alignment.
Discuss the similarities and differences between the two algorithms. Why can
we do the multiplication 37 × 20 as though it is 37 × 2 and then just add a 0?
Emphasize that adding together the number of tens in 7 × 20 and 30 × 20
is the same as adding together the number of ones in 7 × 2 and 30 × 2;
they are both 14 + 60 = 74, so while 37 × 2 is 74, 37 × 20 is 74 tens or 740.
Then have students solve another multiplication problem this way. Example:
23 × 40 = 3 × 40 + 20 × 40 = 12 tens + 80 tens = 92 tens = 920.
28 × 36 = (20 × 36) + (8 × 36) OR 28 × 36 = (28 × 30) + (28 × 6)
Ask students to multiply 28 × 6 and 28 × 30 using the methods they have
learned so far:
6
4
28 × 6
168
2
28
× 30
840
30
F-16
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Multiplying 2-digit numbers by 2-digit numbers. Write on the board:
28 × 36. ASK: How is this multiplication different from any we have done
so far? Challenge students to think of a way to split the problem into two
easier problems, both of which we know how to do. Ask students to list
the types of multiplication problems that they already know how to do.
Can we break this problem into two problems that we already know how
to do? Suggest that they change one of the two numbers, either 28 or 36.
The possibilities include:
ASK: How can we find 28 × 36 from 28 × 30 and 28 × 6? (add them
together) SAY: 6 twenty-eights plus 30 twenty-eights equals 36 twenty-eights,
then show the addition on the board:
168
+ 840
1008
Have students multiply several pairs of 2-digit numbers by using this method.
Exercises
a)43 × 27
b) 54 × 45
c) 36 × 44
Then show students how to use this interim notation (it helps to explain
where the standard notation comes from, later):
28
× 36
1
168
+ 840
1008
Then have students do the same for the problems above. Now show
students the standard algorithm:
4
2 4
24
28 28 × 36 × 36 168
168
+ 840 28
× 36
1
168
+ 840
1008
Have students practice the first step of the standard algorithm (that is,
multiplying the first number by the ones digit of the second number) for
several problems.
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Exercises
a)23 × 17
b) 46 × 19
c) 25 × 32
d) 42 × 26
Ensure that students leave enough space in their notebook to solve the
entire problem, not just the first step.
Example:
2
23
× 17
161 = 23 × 7
Ensure that any digits “carried” at this stage are put in the tens column
and that students can explain why this makes sense. When students are
finished the first step, have them practice the second step of the standard
algorithm (that is, multiplying the first number by the tens digit of the
second number) for the same problems they started above. Emphasize
that students erase all carrying before starting the second step.
Ratios and Proportional Relationships 6-16
F-17
When students are finished, have them complete the final step of the
standard algorithm (adding the two results to find the total answer).
Example:
2
23
× 17
161
230 = 23 × 10
NOTE: To explain in a different way why the multiplication algorithm works,
point out to students that a multiplication statement like 32 × 23 is merely a
short form for a sum of four separate multiplication statements, namely:
32 × 23 = (30 × 20) + (2 × 20) + (30 × 3) + (2 × 3)
You can demonstrate this fact by drawing an array of 32 by 23 squares:
32
30
23
2
20
3
30 × 20
2 × 20
30 × 3
2×3
From the pictures above you can see that the total number of squares
in the array is:
32 × 23 OR (30 × 20) + (2 × 20) + (30 × 3) + (2 × 3)
F-18
32 32 32 32
× 23 × 23 × 23 × 23
6 (= 2 × 3)
6
96
90 (= 30 × 3)
This is the
standard way
of showing the
multiplication
32 32 32
× 23 × 23 × 23
6
6
96
90
90 + 640
40 (= 20 × 2)
40
+ 600 (= 30 × 20)
This is the
standard way
of showing the
multiplication
Teacher’s Guide for AP Book 6.1
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The four products that appear above are hidden in the steps of the standard
multiplication algorithm:
The final sum is 96 + 640 = 736. You might assign students a few questions
and ask them to try multiplying in the non-standard way shown above (i.e.,
writing each of the four products separately).
Extensions
1.a)Which is larger: 28 × 6 or 26 × 8? How can you tell without actually
multiplying the numbers? Encourage students to do the calculations
and then to reflect back on how they could have known which was
larger before doing the calculations. For the pairs below, challenge
students to first decide which product will be larger and then do the
actual calculations to check their predictions.
i)34 × 5 or 35 × 4 ii) 27 × 9 or 29 × 7
iii)42 × 3 or 43 × 2 iv) 89 × 7 or 87 × 9
Hint: For i), compare both to 34 × 4.
Sample solution: For i) 34 × 5 is 34 more than 34 × 4, but 35 × 4
is 4 more than 34 × 4, so 34 × 5 is larger.
b) Have students investigate which is larger in each pair:
i)35 × 27 or 37 × 25
ii) 36 × 27 or 37 × 26
iii)46 × 25 or 45 × 26
iv) 49 × 25 or 45 × 29
Students should investigate by actually doing the calculations
and then reflecting back on how they could have solved the
problem using the least amount of effort possible. Hint: For i),
compare both to 35 × 25.
2. T
each students that any even number, when multiplied by a multiple
of 5, will result in a multiple of 10. The resulting multiple of 10 can be
determined as follows:
2 × 35 = 2 × 5 × 7 = 10 × 7 = 70
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Have students find the following products using this method:
a)4 × 45 b) 8 × 75 c) 6 × 35
d)12 × 15 e) 18 × 25 f) 22 × 45
Sample solution: a) 4 × 45 = 4 × 5 × 9 = 20 × 9 = 180
Students can use this strategy to multiply 3 or 4 numbers that include
at least one of each (an even number and a multiple of 5):
g)6 × 7 × 35 h) 4 × 55 × 3 j)22 × 3 × 15 k) 2 × 6 × 15 × 35
i) 14 × 15 × 4
Sample solution: g) 6 × 7 × 35 = 2 × 3 × 7 × 5 × 7 = 10 × 3 × 7 × 7 =
30 × 49 = 1,470
Ratios and Proportional Relationships 6-16
F-19
3. A
SK: Is it easier to find the product of two 2-digit numbers or the
product of a 1-digit number and a 2-digit number? Why? How much
more work is involved in finding the product of two 2-digit numbers?
Have students change the following products into products of a 2-digit
and 1-digit number by repeatedly doubling one factor and halving the
other. Then find the product. Emphasize that changing a harder problem
(in this case, multiplying two 2-digit numbers) into an easier problem
(in this case, multiplying a 2-digit number by a 1-digit number) is a
common problem-solving strategy.
a)14 × 17 b) 32 × 13 c) 36 × 15 d) 24 × 24
Sample solutions: a) 14 × 17 = 7 × 34 = 238, b) 32 × 13 = 16 × 26 =
8 × 52 = 4 × 104 = 2 × 208 = 416
Bonus: 28 × 26 (this one will become a 3-digit by a 1-digit number)
(MP.1)
4. Fill in the missing numbers.
a)
2
3 c)
4 b)
×
×
9
6
1
6
4 d)
5
×
5
1
2
3
9
1
×
6
1
4
3
0
0
4
4
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Answers: a) 4, b) 3, c) 64 × 3 = 192, d) 24 × 56 = 144 + 1,200=1,344
F-20
Teacher’s Guide for AP Book 6.1
RP6-17 Finding Percents
Page 136
Goals
STANDARDS
6.RP.A.3
Students will find multiples of 10% of a number.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can convert fractions to decimals and vice versa
Understands the relationship between percents and fractions
percent
MATERIALS
base ten materials
(MP.2)
Percents and base ten representations. Tell students that you will use one
thousands block to represent one whole. Given this information, ask students
to identify the decimal each model represents:
a)
b)
c)
d)
Answers: a) 1or 1.00, b) 0.3 or 0.30, c) 0.33, d) 0.05
Then tell students that you want to make a model of the number 1.6, again
using one thousands block as one whole. ASK: What do I need to make
the model? (1 thousands block, 6 hundreds blocks) How do you know?
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ASK: How can I show 1/10 of 1.6? (one tenth of a thousands block is a
hundreds block and one tenth of a hundreds block is a tens block, so I
need 1 hundreds block and 6 tens blocks to make 1/10 of 1.6)
What number is 1/10 of 1.6? (0.16) PROMPT: What do the base ten
materials show? Do a few more examples together:
a) 1/10 of 1
b) 1/10 of .1
c) 1/10 of 0.01
d) 1/10 of 2.3
e) 1/10 of 0.41
f) 1/10 of 5.01
ASK: How can you find 1/10 of any number? (move the decimal point one
place to the left) PROMPT: What do you do to the decimal point? Refer
students to the answers above. Remind students that when they move the
decimal point one place to the left, each digit becomes worth 1/10 as much,
so the entire number becomes 1/10 of what it was before they moved the
decimal point. Examples:
4 is 1/10 of 40
.1 is 1/10 of 1
4.1 is 1/10 of 41
Ratios and Proportional Relationships 6-17
F-21
ASK: How else can I find 4.1 from 41—what is this like dividing by? (10)
Emphasize that to find 1/10 of anything, you divide it into 10 equal parts;
to find 1/10 of a number, you divide the number by 10. ASK: What decimal
is the same as 1/10 ? (0.1 or .1) What percent is the same as 1/10 ? (10%)
Write on the board:
1/10 = 10/100 = 10%
SAY: So to find 10% of a number you can divide the number by 10 or simply
move the decimal point one place to the left.
Exercise: Find 10% of each number by just moving the decimal point.
a) 40 b) 4
c) 7.3 e) 408 f) 3.07 g) 432.5609
d) 500
Answers: a) 4, b) 0.4, c) 0.73, d) 50, e) 40.8, f) 0.307, g) 43.25609
Finding percents with a number line. Show the number line below but
without the bold numbers. Have a volunteer fill in the missing numbers on
the number line.
0
3
6
9
1215 1821 242730
0 10%20% 30%40%50% 60%70% 80%90%100%
Then ask volunteers to look at the completed number line and identify: 10%
of 30, 40% of 30, 90% of 30, 70% of 30.
Exercise: Repeat the exercise for a number line from 0 to 21. (Start at 0
and add 2.1 each time.)
Answer
0
2.1
4.2
6.3
8.4 10.5 12.614.7 16.818.921
ASK: If you know 10% of a number, how can you find 30% of that number?
(multiply 10% of the number by 3) Tell students that you would like to find
70% of 12. ASK: What is 10% of 12? (1.2) If I know that 10% of 12 is 1.2,
how can I find 70% of 12? (multiply 1.2 × 7). You can review multiplying a
decimal by a whole number with an example.
Exercise: Using this method, have students find the following:
a) 60% of 15
b) 40% of 40 c) 60% of 4
d) 20% of 1.5
e) 90% of 8.2
f) 70% of 4.3 g) 80% of 5.5
h) 30% of 3.1
Answers: a) 6 × 1.5 = 9, b) 4 × 4 = 16, c) 6 × 0.4 = 2.4, d) 2 × 0.15 = 0.30,
e) 9 × 0.82 = 7.38, f ) 7 × 0.43 = 3.01, g) 8 × 0.55 = 4.4, h) 3 × 0.31 = 0.93
F-22
Teacher’s Guide for AP Book 6.1
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0 10%20% 30%40%50% 60%70% 80%90%100%
Emphasize that to find 1/100 of a number, you divide the number by 100.
Explain to students that taking 1% of a number is the same as dividing the
number by 100. (The decimal shifts two places to the left.)
Exercise: Find 1% of each number by shifting the decimal point two
places to the left.
a)27
b)3.2
c)773 d)12.3
e)68.2
Answers: a) 0.27, b) 0.032, c) 7.73, d) 0.123, e) 0.682
Extension
(MP.7)
Have students compare:
a) 20% of 60 and 60% of 20 b) 30% of 50 and 50% of 30
c) 40% of 20 and 20% of 40 d) 70% of 90 and 90% of 70
e) 80% of 60 and 60% of 80 f) 50% of 40 and 40% of 50
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What pattern do students see?
Ratios and Proportional Relationships 6-17
F-23
RP6-18 Finding Percents Using Multiplication
Page 137
Goals
STANDARDS
6.RP.A.3
Students will solve word problems involving percents.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can reduce fractions
Can multiply decimals
Knows the standard algorithm for multiplying
percent
MATERIALS
BLM Percent Strips (p. F-42)
Finding fractions of whole numbers. Review finding fractions of
whole numbers:
2
Example:
of 9 = (9 × 2) ÷ 3 = 18 ÷ 3 = 6
3
Exercises: Find the fraction of the number.
2
3
2
1
of 30
d)
a)
of 15
b)
of 35
of 20
c)
6
3
4
5
Answers: a) 10, b) 15, c) 5, d) 14
(MP.8)
Finding percents using multiplication. ASK: How can we calculate 53% of
12 using what we know about finding fractions of whole numbers? Students
should notice that 53% is an abbreviation for a fraction (53/100), so they
can find the percent by first changing the percent to a fraction:
53
53% of 12 =
× 12 = (53 × 12) ÷ 100
100
Remind students that they can find products like 53 × 12 by using long
multiplication (or mentally, if it is an easy product). Also, remind them that
dividing by 100 shifts the decimal two places to the left. So,
Students can use estimation to check whether their answers to percent
problems are reasonable. For instance, they can round a given percent to
the nearest multiple of ten and use the rounded percent to estimate the
answer. ASK: How can we tell if 6.36 is a reasonable answer to 53% of 12?
Is there a percentage of 12 that is close to 53% and easy to calculate?
(yes, 50%) Will the estimate be lower or higher than the actual answer?
(50% of 12 is 6, which is lower than the actual answer because 50% is less
that 53%. But 6 is close to 6.36, so the answer seems reasonable.) Students
might also round the number as well as the percent: 12 is close to 10 and 53
is close to 50. So to estimate 53% of 12, find 50% of 10, which is easy to
calculate. ((50 × 10) ÷ 100 = 5)
F-24
Teacher’s Guide for AP Book 6.1
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(53 × 12) ÷ 100 = 636 ÷ 100 = 6.36
Exercises: Find the percents.
a) 68% of 33
b) 5% of 42
c) 76% of 85
d) 55% of 21
Answers: a) 22.44, b) 2.1, c) 64.6, d) 11.55
Students can use BLM Percent Strips to check their answers for the
exercise. Each of the four numbers in the exercise has been placed on
a number line (all of the same length but with a different scale for each).
A fifth number line, divided into a hundred parts to represent percents,
is at the bottom of the BLM. To estimate 68% of 33, students can find the
number 68 on the percent number line and then locate the number that
is in the same position on the number line for 33.
(MP.3)
After students have finished their calculations for the exercise, ask them
what method they would use to estimate 68% of 33 (to see if their answer
is reasonable). Encourage them to give a variety of answers. For instance:
“68% is close to 70% and 33 is close to 30. I found 70 × 30 mentally (2100)
and then shifted the decimal place two places to the left. So my estimate is
21.” Or: “68% is close to 75%, which is the same as 3/4 (75/100 = 3/4). And
33 is close to 32. I know 1/4 of 32 is 8, so 3/4 of 32 is 24. So my estimate is 24.”
Extensions
1.Sara says that to find 10% of a number, she can divide the number by
10, so to find 5% of a number, she can divide the number by 5. Is she
right? Explain. (No. 5% of a number is 5/100 or 1/20 of the number, so
to find 5%, or 1/20, of the number, she should divide it by 20.)
2. Continue the extension from RP6-17. Have students compare:
a) 36% of 24 and 24% of 36
c) 29% of 78 and 78% of 29
b) 17% of 35 and 35% of 17
d) 48% of 52 and 52% of 48
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Ask students to predict a rule and make up another example to check
that the rule works. Challenge them to figure out why this pattern holds.
(because 36 × 24 is equal to 24 × 36)
3.Discuss: Does it make sense to talk about 140% of a number? What
does 140% of a number mean? Lead the discussion by referring to
fractions greater than 1. Discuss what 100% and 40% of a number
mean separately. Could 50% of a number be obtained by adding 20%
and 30% of that number?
Could 140% be obtained by adding 100% and 40% of the number?
Ratios and Proportional Relationships 6-18
F-25
RP6-19 Percents: Word Problems
Page 138
Goals
STANDARDS
6.RP.A.3
Students will solve word problems involving percents.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can convert fractions to decimals
Can calculate the percent of a number
Can compare decimals, fractions, and percents
percent
Using percents to compare fractions. Tell students that Maria got 17/25
on her math test and 14/20 on her science test. What percent of the points,
or marks, did she get on each test? (68% in math, 70% in science) On which
test did she do better? Even though Maria got more marks on her math test
than on her science test (17 instead of 14), she got a higher percentage of
marks on the science test than the math test (70% instead of 68%). So she
did better on the science test. ASK: Is it easier to compare test scores when
they are given as fractions or when they are given as percents? Is it easier to
compare 17/25 to 14/20 or 70% to 68%? Why doesn’t the test have to have
100 marks in order for the result to be expressed as a percent? (We can
convert any fraction to a percent by changing the denominator to 100.) Tell
students that this is one application of percents: we can compare two test
scores easily, even when the total number of marks in each test is different.
Exercise: Convert Sally’s test scores to percents and decide which was
her best test and which was her worst:
Math: 17/20
Social Studies: 36/40
Science: 22/25
Language: 43/50
(MP.4)
Using percents of numbers to solve real-world problems. Tell students
that Olga has collected 50 stamps from various countries: 31 from the United
States, 14 from Canada, and 5 from elsewhere. Ask students to calculate
what percentage of Olga’s stamp collection is from the United States, what
percentage is from Canada, and what percentage is from elsewhere.
Rita, on the other hand, has collected 3,000 stamps: 1,020 from the United
States, 840 from Canada, and 1,140 from elsewhere. Ask students to
calculate what percentage of Rita’s stamp collection is from the United States,
what percentage is from Canada, and what percentage is from elsewhere.
ASK: Who has more stamps from the United States? Who has a greater
percentage of stamps from the United States? (Rita has more US stamps
than Olga, but Olga has a greater percentage of US stamps in her collection
—62% of Olga’s stamps are from the United States but only 34% of Rita’s
F-26
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Hint: One of the scores will need to be reduced before it can be expressed
as a fraction with denominator 100.
are from the US.) How do percentages help us to compare stamp
collections even when one has many more stamps than the other?
The whole is always 100%. Tell students that Anna’s stamp collection has
this distribution: 41% from the United States, 26% from Canada, and an
unknown percent from elsewhere. ASK: What percent of Anna’s collection is
from somewhere other than the United States or Canada? Emphasize that
percentages must add to 100 because the whole amount of anything is 100%.
(MP.4)
Word problems involving percents, fractions, and decimals. Jennifer
has stamps from all over the world. In her collection, 2/5 of the stamps
are from the United States and 36% are from Canada. What percentage of
Jennifer’s stamps are from neither the United States nor Canada? Solve this
problem with the class. (change 2/5 to 40%, then add 40% + 36% = 76%,
so the stamps from neither place make up 24% of Jennifer’s collection).
Exercise: Have students find the missing percentages of other stamps in
each collection:
a)
b)
c)
d)
e)
USA: 40% Mexico: 25% USA: 17/25 China: 13/20 Italy: 0.32
Canada: 1/2 USA: 3/5 Jamaica: 19% Japan: 0.31 USA: 45%
Other:
Other:
Other:
Other:
Other:
Tell students that Sayaka has spent 500 days travelling the world. ASK: If
she spent 60% of her days in Europe and 3/10 of her days in Africa, how
many days did she spend in each place? How many days did she spend in
neither Africa nor Europe? Draw a chart on the board and have volunteers
complete the chart.
Fraction of Trip
Percentage of Trip
Number of Days
Africa
Europe
Other
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers
F
P
A
3/10
30%
E 6/10 (or 3/5) 60%
O
1/10
10%
N
150
300
50
Extensions
1.Five people—2 adults and 3 children—attend a hockey game. What
percentage of the group do the children represent? Describe a group
of a different size with the same percentage of children.
2. Mr. Bates buys
• 5 single-scoop ice cream cones for $1.45 each
• 3 double-scoop ice cream cones for $2.65 each
Ratios and Proportional Relationships 6-19
F-27
A tax of 10% is added to the cost of the cones. Mr. Bates pays with a
20-dollar bill. How much change does he receive? Show your work.
3.The chart shows the fraction or percentage of stamps that children
have collected from various countries.
Brian’s Collection
Faith’s Collection
Andrew’s Collection
USA
23%
3/4
1/2
England
3/5
7%
30%
Other
Which child has the greatest percentage of stamps from
other countries?
(MP.3)
4. a)Discuss: Sally got 171/200 on a national math test. Can this mark
be written as a percent? The answer (yes, it can be written as a
decimal percent) is not as important as the discussion that should
arise. Leading questions you might use include: Is this mark better
or worse than 80%? How do you know? Is it better or worse than
90%? Than 85%? Than 86%? Is it closer to 85% or to 86%? (It is
halfway between them.) Is there a number halfway between 85 and
86? (yes, 85.5) Tell students that even though we said that percents
are just fractions with denominator 100, percents are actually even
better than fractions with denominator 100—you can’t write 85.5/100
as a fraction, but you can write 85.5%. (You could tell students that
they won’t learn about decimal percents until grade 8, but this class
is smart enough to know about them in grade 6.)
b)Teach students a strategy for estimating 12.5%. First they need to
find 50%, then by halving the left part they can estimate 25%, and
by halving the left part again they can estimate 12.5%.
12.5% 25%
50%
(MP.4)
5.Investigate on the Internet: What percentage of car passengers wear seat
belts in Canada? In the United States? In other countries? (Emphasize
that even though the United States has many more people than Canada,
a meaningful comparison can still be made in terms of percentages.)
6.Which has a greater percentage of water by volume, your body or
planet Earth? (Emphasize that although Earth has much more water
than your body, your body has a greater percentage of water than
does the Earth.)
7.Which has a greater percentage of water by surface area, the
United States or Canada? The United States or Finland? The
United States or Russia?
F-28
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Now ask them to estimate 37.5%, 62.5%, and 87.5%.
RP6-20 Fractions, Ratios, and Percents
Pages 139–141
Goals
STANDARDS
6.RP.A.2, 6.RP.A.3
Students will solve word problems involving fractions, ratios, and
percents.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
fractions
percents
ratios
Can compare fractions, ratios, and percents
Can convert among fractions, ratios, and percents
Recognize part-to-part and part-to-whole ratios. Ask students how many
boys are in their class, how many girls, and how many children altogether:
b:
g:
c:
ASK: Did you count everyone one by one or was there an easier way once
you found the number of boys and girls?
Ask students to fill in the numbers of boys, girls, and children given various
pieces of information:
a)
b)
c)
d)
7 girls and 8 boys
6 girls in a class of 20
12 boys in a class of 30
17 girls in a class of 28
b:
b:
b:
b:
g:
g:
g:
g:
c:
c:
c:
c:
Then have students determine the numbers of boys, girls, and children,
the fraction of girls, and the fraction of boys in these classes:
a) There are 6 boys and 5 girls.
b) There are 14 boys in a class of 23.
c) There are 15 girls in a class of 26.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Now have students write the fraction of girls and boys in these classes:
a)
b)
c)
d)
e)
f)
g)
h)
There are 3 boys and 4 girls in a class.
There are 7 boys and 13 children in a class.
There are 8 girls and 19 children in a class.
The ratio of boys to girls is 1 : 2.
The ratio of girls to boys is 2 : 3.
The ratio of boys to girls is 12 : 11.
The ratio of boys to girls is 11 : 12.
The ratio of girls to boys is 11 : 12.
Which two of the last three questions have the same answer? (parts f) and h)
have the same answer) Can you find a question that has the same answer
as part d)? (The ratio of girls to boys is 2 : 1.)
Then have students determine the number of girls and boy in these classes:
a) There are 30 children in a class and 3/5 are girls.
b) There are 36 children in a class and 4/9 are girls.
c) There are 21 children in a class and 4/7 are boys.
Ratios and Proportional Relationships 6-20
F-29
d) There are 18 children in a class. The ratio of boys to girls is 7 : 2.
e) There are 18 children in a class. The ratio of girls to boys is 2 : 7.
f) There are 18 children in a class. The ratio of boys to girls is 2 : 7.
Pause here to discuss which two of the last three questions have the same
answer (parts d) and e)) and how to find another question with the same
answer as part f).
g) There are 30 children in the class and 60% are girls.
h) There are 45 children in the class and 40% are girls.
Exercises
(part-to-whole)
1.A baseball player got a hit 2 out of every 3 times at bat. She was at bat
9 times. How many hits did she have?
(part-to-part)
2. Have students do the following problems in their notebooks.
a)The ratio of girls to boys in a school is 12 : 13. If the school has
200 students, how many girls are there?
b)The ratio of red marbles to green marbles in a jar is 4 : 11. If there
are 60 marbles in the jar, how many green marbles are there?
c)There are 2 apples in a bowl for every 3 oranges. If there are
15 fruits in the bowl, how many apples are there?
Answers: 1. 6, 2. a) 96, b) 44, c) 6
Extension
(MP.2)
To estimate a fraction or ratio, you can change one or both parts slightly.
Example A: 5 out of 11 is close to 5 out of 10, which is close to 1/2 or 50%.
The chart shows the lengths of calves and adult whales (in feet).
Approximately what fraction and what percent of each adult’s length
is the calf’s length? Do you need to know how long a foot is to answer
this question? (no)
Type
Calf Length (feet)
Adult Length (feet)
Killer
7
15
Humpback
16
50
Narwhal Fin-backed
5
22
15
70
Sei
16
60
Answers: Killer 1/2 and 50%, Humpback 1/3 and 32%, Narwhal 1/3 and 30%,
Fin-backed1/3 or 30%, Sei 1/4 and 25%.
F-30
Teacher’s Guide for AP Book 6.1
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Example B: 9 out of 23 is close to 10 out of 25, which is 2/5 (10/25 = 2/5),
which is 40%.
RP6-21 Long Division (Review)
Pages 142–144
STANDARDS
preparation for 6.RP.A.3
Vocabulary
dividend
remainder
Goals
Students will use long division to divide 2-digit numbers by 1-digit
numbers.
PRIOR KNOWLEDGE REQUIRED
Is familiar with ones and tens blocks
Can do simple division (directly from times tables)
Materials
base ten materials
Long division notation. Write on the board:
3 6
5 10
4 12
5 20
6 18
9 18
). If they know what
Ask students if they recognize the symbol used (
the symbol means, have them solve the problems. If they don’t recognize
the symbol, have them guess its meaning from the other students’ answers
to the problems. If none of the students recognize the symbol, solve the
problems for them. Write more problems to increase the chances of
students being able to predict the answer:
2 8
2 6
4 16
5 15
4 8
6 12
3
3 R1
Explain that 2 6 is another way of expressing 6 ÷ 2 = 3, and that 2 7
is another way of expressing 7 ÷ 2 = 3 Remainder 1.
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Ask students to express the following statements using the new notation
learned above:
a)14 ÷ 3 = 4 Remainder 2
b)26 ÷ 7 = 3 Remainder 5
c)819 ÷ 4 = 204 Remainder 3
3 63
Exercises: To ensure that they understand the long division symbol, ask
them to solve the following problems. Students can do the problems by
skip counting.
a)2 11
b) 4 18
c) 5 17
Bonus
f)6 45
g) 4 37
h) 4 43
d) 4 21
e) 3 16
Using base ten materials to divide. Then demonstrate division using base
ten materials, as in the margin.
Ratios and Proportional Relationships 6-21
F-31
Exercises: Have students solve the following problems using
base ten materials.
a)2 84
b) 2 48
c) 3 96
d) 4 88
Then challenge students to solve 3 72 , again using base ten materials,
but allow students to trade tens blocks for ones blocks as long as the value
of the dividend (72) remains the same. ASK: Can 7 tens blocks be equally
placed into 3 circles? Can 6 of the 7 tens blocks be equally placed into
3 circles? What should be done with the leftover tens block? How many
ones blocks can it be traded for? How many ones blocks will we then have
altogether? Can 12 ones blocks be equally placed into 3 circles? Now,
what is the total value of blocks in each circle? (24) What is 72 divided by
3? Where do we write the answer? Explain that the answer is always written
above the dividend (the number that you want to divide up), with the tens
digit above the tens digit and the ones digit above the ones digit.
24
3 72
Exercises
1. Have students solve several problems using base ten materials.
a)4 92
b) 4 64
c) 4 72
d) 3 45
e) 2 78
2. The following problems will have remainders.
a)4 65
b) 3 82
c) 5 94
d) 2 35
e) 4 71
Tell students that they are going to learn to solve division problems without
using base ten materials. SAY: We will do problems with base ten materials
to see if we can find a way to record what we are doing, so that we don’t
even need the base ten materials anymore. The solutions to the following
problems have been started using base ten materials. Can students
determine how the solution is written?
2
4 92
8
Tell students to divide just the tens blocks and record the information for
these five examples, as shown:
1
5 72
5
F-32
2
4 91
8
3
3 95
9
4
2 87
8
2 78
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
3
2 64
6
Challenge students to illustrate several more problems (by drawing or
using base ten materials) and to write the solution. When all students are
recording the information correctly, ask them how they determined where
each number was written. Which number is written above the dividend?
(the number of tens equally placed into each circle) Which number is
written below the dividend? (the number of tens placed altogether)
Then, using the illustrations from the two previous sets of problems, ask
students what the circled numbers shown below express:
1
5 75
-5 2
2
4 91
-8 1
3
3 95
-9 0
4
2 87
-8 0
3
2 78
-6
1
ASK: What does the number express in relation to its illustration? (the
number of tens not equally placed into circles) Why does the subtraction
make sense? (the total number of tens minus the number of tens equally
placed into circles results in the number of tens blocks left over)
Have students do more problems by just imagining the base ten materials,
if possible (some students may still need to use actual base ten blocks).
Remind students that the number above the dividend’s tens digit is the
number of tens placed in each circle. For example, if there are 4 circles and
9 tens, as in 4 94 , the number 2 is written above the dividend to express
that 2 tens are equally placed in each of the 4 circles. Explain that the number
of tens placed altogether can be calculated by multiplying the number of
tens in each circle (2) by the number of circles (4); the number of tens placed
altogether is 2 × 4 = 8.
Ask students to say if the following algorithms have been started correctly
or not. Encourage them to illustrate the problems with base ten materials,
if it helps.
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3
2 85
6 2
1
3 87
3 5
2
3 84
6 2
2
4 95
8 1
1
4 95
4
5
Explain that the remaining number of tens blocks should always be less
than the number of circles, otherwise more tens blocks need to be placed
in each circle. The largest number of tens blocks possible should be
equally placed in each circle.
Display the multiplication facts for 2 times 1 through 5 (2 × 1 = 2,
2 × 2 = 4, etc.), so that students can refer to it for the following set
of problems. Then write on the board:
2 75
ASK: How many circles should we use? If 1 tens block is placed in each
circle, how many tens blocks will be placed altogether? (2 × 1 = 2) What if
2 tens blocks are placed in each circle? (2 × 2 = 4) What if 3 tens blocks
Ratios and Proportional Relationships 6-21
F-33
are placed in each circle? (2 × 3 = 6) And finally, what if 4 tens blocks
are placed in each circle? (2 × 4 = 8) Can we place 4 tens blocks in each
circle? (no, that would require us to have 8 tens blocks and we have only 7)
Then explain that the number of circles we need is the greatest multiple of
2 not exceeding the number of tens in the dividend. Have students perform
these steps for the following problems.
3
2 75
6 1
3 tens in each circle
2 65
2 38
2 81
2 59
3 × 2 = 6 tens place
1 tens block left over
Then display the multiplication facts for 3 times 1 through to 3 times 5 and
repeat the exercise for the problems below. Demonstrate the steps for the
first problem.
2
3 75
6 1
3 tens in each circle
3 65
3 38
3 81
3 59
3 × 2 = 6 tens place
1 tens block left over
Emphasize that the number above the dividend’s tens digit is the greatest
multiple of 3 not exceeding the number of tens.
Then, using the illustrations already drawn to express leftover tens blocks
(see the previous page), explain the next step in the algorithm. ASK: Now
what do the circled numbers express?
The circled number expresses the amount represented by the base ten
materials that have not yet been placed in the circles.
2
4 91
-8 11
3
3 95
-9 05
4
2 87
-8 07
3
2 78
-6
18
Using base ten materials, challenge students to start the process of long
division for 85 ÷ 3 and to record the process (the algorithm) up to the point
discussed so far. Then ask students to trade the remaining tens blocks for
ones blocks, and to circle the step in the algorithm that expresses the total
value of the ones blocks. Ensure that students understand the algorithm up
to the step where the ones blocks are totalled with the remaining (if any)
tens blocks.
2
4 94
8 14
3 65
3 38
3 81
3 59
14 ones to be place
Illustrate all of the placed tens and ones blocks and the finished algorithm,
and then ask students to explain the remaining steps in the algorithm.
Perform this for the examples already started. For example,
F-34
Teacher’s Guide for AP Book 6.1
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1
5 75
-5 25
23
4 94
-8
14
-12
2 Remainder
So, 94 ÷ 4 = 23 Remainder 2.
Ask students to explain how the circled numbers are derived. How is the 3
derived? (the number of ones blocks in each circle) The 12? (there are 12
ones blocks in the circles altogether) The 2? (14 − 12; these are the ones
blocks left over because 12 of the 14 blocks have been placed)
Then challenge students to write the entire algorithm. Ask them why
the second subtraction makes sense. (The total number of ones blocks
subtracted by the number of ones blocks placed into circles equals the
number of ones blocks left over.)
Using base ten materials, have students complete several problems and
write the entire algorithms. Show one complete example (with or without a
remainder—the example below has no remainder).
25
3 75
6
15
15
0
5 ones in each circle
3 65
3 38
3 81
3 59
ones to be placed
5 × 3 = ones placed
Remainder (no ones left over)
NOTE: Some students will need to have all the previous steps done so that
they can focus on this one.
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Have students finish the examples they have already started and then
complete the problems below from the beginning. They should use base
ten materials only to verify their answers. Exercises:
a)2 39
b) 3 39
f)3 94
g) 2 94
c) 2 57
d) 4 57
e) 2 85
With practice, students will learn to estimate the largest multiples that
can be used to write the algorithms. When they are comfortable with the
algorithm, introduce larger divisors. Exercises:
a)4 69
b) 5 79
f)7 94
g) 8 94
c) 6 87
d) 4 57
e) 6 85
Note that in all the problems to this point, the dividend’s tens digit has been
greater than the divisor.
Ratios and Proportional Relationships 6-21
F-35
Extension
Teach students to check their answers using multiplication. For example:
17
2
4 69 17
4 ×4
29 68
28
1R
68 + 1 = 69
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Ask students to check their answers for the last exercise.
F-36
Teacher’s Guide for AP Book 6.1
RP6-22 Long Division and Unit Rates
Pages 145–146
Goals
STANDARDS
6.RP.A.3
Students will use long division to divide 3-digit numbers by 1-digit
numbers and also to find unit rates.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
Is familiar with ones and tens blocks
Can do 2-digit by 1-digit division
Understands the concept of a unit rate
Materials
base ten materials
(MP. 8, MP.2)
Long division: 3-digit by 1-digit. Teach students to use long division
to divide 3-digit numbers by 1-digit numbers. Using base ten materials,
explain why the standard algorithm for long division works.
Example: Divide 726 into 3 equal groups.
Step 1: Make a model of 726 units.
7 hundreds blocks
2 tens
blocks
6 ones
blocks
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Step 2: Divide the hundreds blocks into 3 equal groups.
Keep track of the number of units in each of the 3 groups, and the number
remaining, by slightly modifying the long division algorithm.
22
3 726
3 726
-6
-6
112
200
3 726
-600 126
2 hundred blocks, or 200 units, have been divided into 3 groups
600 units (200 × 3) have been divided
126 units still need to be divided
NOTE: Step 2 is equivalent to the steps at left in the standard
long division algorithm.
Exercises: Students should practice Steps 1 and 2 from both the modified
and the standard algorithms in the following problems.
a)2 512
Ratios and Proportional Relationships 6-22
b) 3 822
c) 2 726
d) 4 912
F-37
Students should show their work using actual base ten materials or a model
drawn on paper.
Step 3: D
ivide the remaining hundreds block and the 2 remaining tens
blocks among the 3 groups equally.
There are 120 units, so 40 units can be added to each group from Step 2.
See the margin.
Keep track of this as follows:
Group 1
Group 2
Group 3
40
200
3 726
-600
126
-120
6
40 new units have been divided into each group
126 (40 × 3) new units have been divided
6 units still need to be divided
NOTE: Steps 1 through 3 are equivalent to the following steps in the
standard long division algorithm.
222424
3 726
3 726
3 726
3 726
-6
-6
-6
-6
1121212
-12
06
Group 1
Group 2
Group 3
Students should carry out Step 3 using both the modified and standard
algorithms in the problems they started above. Then give students new
problems and have them do all the steps up to this point. Students should
show their work using either base ten materials or a model drawn on paper.
Step 4: Divide the 6 remaining blocks among the 3 groups equally.
There are now 242 units in each group; hence 726 ÷ 3 = 242.
2 new units have been divided into each group
6 (2 × 3) new units have been divided
There are no units left to divide
NOTE: Step 4 is equivalent to the last of the following steps in the standard
long division algorithm.
F-38
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
2
40
200
3 726
- 600
126
- 120
6
-6 0
242242242
3 726
3 726
3 726
-6
-6
-6
12
12
12
-12 -12 -12
06 06 06
6
6
0
Students should be encouraged to check their answer by multiplying 242 × 3.
Students should finish the problems they started. Then give students
new problems to solve using all the steps of the standard algorithm. Give
problems where the number of hundreds in the dividend is greater than
the divisor. (Examples: 842 ÷ 2, 952 ÷ 4) Students should show their work
(using either base ten materials or a model drawn on paper) and check
their answers using multiplication.
(MP.2)
5 27
Long division when the divisor is greater than the dividend’s leading
digit. When students are comfortable dividing 3-digit numbers by 1-digit
numbers, introduce the case where the divisor is greater than the dividend’s
leading digit. Begin by dividing a 2-digit number by a 1-digit number where
the divisor is greater than the dividend’s tens digit (i.e., there are fewer tens
blocks available than the number of circles).
ASK: How many tens blocks are in 27? Into how many circles do they need
to be divided? Are there enough tens blocks to place one in each circle?
How is this different from the problems you just did? Illustrate that there are
no tens by writing a zero above the dividend’s tens digit. Then ASK: What is
5 × 0? Write:
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0
5 27
0
27
Number of ones blocks (traded from tens block) to be placed
Have a volunteer finish this problem, and then ask if the zero needs to be
written at all. Explain that the algorithm can be started on the assumption
that the tens blocks have already been traded for ones blocks.
5
5 27
25 2
Number of ones blocks in each circle
Number of ones blocks (traded from tens block) to be placed
Number of ones blocks placed
Number of ones blocks leftover
Emphasize that the answer is written above the dividend’s ones digit
because it is the answer’s ones digit.
Have students complete several similar problems. Exercises:
a)4 37
Ratios and Proportional Relationships 6-22
b) 5 39
c) 8 63
d) 8 71
F-39
Then move to 3-digit by 1-digit long division where the divisor is more
than the dividend’s hundreds digit. (Examples: 324 ÷ 5; 214 ÷ 4; 133 ÷ 2)
Again, follow the standard algorithm (writing 0 where required) and then
introduce the shortcut (omit the 0).
When students are comfortable with all cases of 3-digit by 1-digit long
division, progress to 4-digit by 1-digit long division. Start by using base ten
materials and going through the steps of the recording process as before.
Some students may need to practice one step at a time, as with 3-digit by
1-digit long division.
Write the 8 times table on the board, and then ASK: How long is each side
if an octagon has a perimeter of…
a) 952 cm
b) 568 cm c) 8,104 cm
d) 3,344 cm
Finding unit rates by division. SAY: To find a unit rate you can use
division. ASK: If 5 tickets cost $20, how much does one ticket cost? ($4)
How do you know? (by dividing 20 by 5) Then solve some more problems.
Example: Jim works 5 days per week and his salary is $750 per week.
How much does Jim earn each day? (750 ÷ 5 = 150, so he earns $150
per working day)
Exercises: Find the unit rate.
a) 3 jackets cost $135
1 jacket costs
b) 225 miles in 5 hours
miles in 1 hour
c) 124 students for 4 buses
students per bus
Answers: a) $45, b) 55 miles, c) 31
(MP.2)
Have students solve division problems using the method discussed in the
last lesson. If students do not know their multiplication tables, have students
do the 8 times tables by skip counting or by using the 4 times tables and
doubling (or use the 2 times tables and double twice). Discuss the advantages
of the doubling strategy over skip counting: students do not need to find all
the previous 8 times tables to find 8 × 7. If they know 2 × 7 = 14, then they
can find 4 × 7 = 28 = 20 + 8 and then 8 × 7 = 40 + 16 = 56.
8
×1
8
×2
8
×3
8
×4
8
×5
8
×6
8
×7
8
×8
8
×9
Tell students that you want to divide 519 by 8. Since 519 is between 80 and
800, 519 ÷ 8 will be between 10 and 100 and will have 2 digits, so we just
have to find those 2 digits. What is the first digit? From the 8 times tables,
we know that 8 × 60 = 480 and 8 × 70 = 560, so 519 is between 8 × 60
and 8 × 70. This tells us that the quotient is between 60 and 70 and so has
tens digit 6. This is much better than knowing just that it is between 10 and
100, but we still need to know more. Now, we need the ones digit.
F-40
Teacher’s Guide for AP Book 6.1
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Extension
64
8 519
480 480 ÷ 8 = 60
39
39 ÷ 8 = 4 R 7
32
7
1
8 993
800
193
800 ÷ 8 = 100
519 ÷ 8 = 480 ÷ 8 + 39 ÷ 8. But 39 ÷ 8 = 4 R 7, from looking at the times
tables, so 519 ÷ 8 = 60 + 4 R 7 = 64 R 7.
Have students do several more problems (like the one in the margin) of this
sort and show them how using the long division algorithm helps them to
keep track of the steps they are doing.
Include examples where the quotient has 3 digits (Example: 993 ÷ 8). Since
993 is between 800 and 8,000, students can see that the quotient is between
100 and 1,000 and so has 3 digits. We need to find the hundreds, tens, and
ones digits. Start by finding the hundreds digit: Since 993 is between 800
and 1,600, the quotient is between 100 and 200, so the hundreds digit is 1
(see the margin).
We know that 993 ÷ 8 = 800 ÷ 8 + 193 ÷ 8, so now we need to find
193 ÷ 8. Emphasize to students that now we just need to do long division
on 8 193 so we have reduced the problem into one that we have done
before. The answer now will have 2 digits:
2
8 193
160 33
193 is between 80 and 800 and so the quotient has 2 digits.
Furthermore, 193 is between 160 and 240, the tens digit is 2.
We are now left with finding 33 ÷ 8.
4
8 33
32
1 Remainder
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Combining all of these into one long division, we find
1 2 4 124
8 993
8 193
8 33
3 993
800 160 32
800
193 33 1 Remainder193
160
33
32
1 Remainder
Have students individually solve the following problems using this method,
by writing the 6 times table first. Exercises:
a)993 ÷ 6 d)1,743 ÷ 6 Ratios and Proportional Relationships 6-22
b) 765 ÷ 6 e) 3,842 ÷ 6
c) 891 ÷ 6
F-41