Download Unit 5 Number Sense

Unit 5 Number Sense
This unit gives careful attention to reviewing and reinforcing all the essential
understandings about decimals.
Meeting your curriculum
Many of the concepts studied in this unit have been studied in earlier
grades. However, it is essential to review the material—you can use the
workbook as a diagnostic tool—before proceeding. Fluency in relating,
comparing, and ordering fractions and decimals and in performing decimal
operations using a variety of strategies is pivotal to students’ success
in math in middle school and beyond, as well as in the sciences and in
daily life. Students were introduced to decimals in grade 5 and have been
gradually building their conceptual understanding and procedural skills.
Still, many if not most grade 7 students will not yet have fully consolidated
the concepts taught so far. The time spent on the lessons in this unit will
be well repaid by the ease with which students will work with decimals in
future units this year (and in future years). The knowledge gained in this
unit will be applied in most of the subsequent units in this course.
An understanding of long division, for example, is essential for
understanding how repeating decimals work (an expectation for Grade 7
in the WNCP curriculum and Grade 9 in the Ontario curriculum). Whether
students do repeating decimals this year or later, they will need to be
fluent with the long division algorithm in order to see patterns develop in
the digits of repeating decimals.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Problem solving
Students are often weak in solving word problems. In this unit, the
worksheets are primarily devoted to concept and skill development. But
the teacher’s guide includes a selection of word problems in most lessons.
These problems require students to apply the skills and concepts just
acquired in the lesson. Some are single-step problems and some are
multi-step problems. The problems involve concepts from measurement
and geometry as well as real-life scenarios. Try to have your students
do a few word problems each day. If they do, by the end of the unit they
should be significantly more confident in approaching word problems
and more successful in solving them.
Materials
Base ten models are used in a number of lessons in this unit. You do
not need base ten materials for these lessons—students are shown how
to draw the models. However, if you have base ten materials in your
classroom, by all means, make them available; they may be quite helpful
for some students.
Students will use grid paper to line up decimals when ordering
and computing.
Teacher’s Guide for Workbook 7.1
F-1
NS7-32 Decimal Fractions
Pages 127–128
Curriculum
Expectations
Ontario: 7m1, 7m3, 7m5,
7m6, essential for
7m11, essential for
7m27
WNCP: 5N8, 5N9; 7N2,
[CN, R, V]
Goals
Students will understand the concept of a decimal fraction and that
there is more than one way to represent a decimal fraction.
PRIOR KNOWLEDGE REQUIRED
Recognizes increasing and decreasing patterns
Can use grids to represent tenths and hundredths
Can write equivalent fractions
Can add fractions with like denominators
Vocabulary
decimal fraction
powers (first, second, third,
and so on)
power of 10
tenth, hundredth, thousandth
numerator
denominator
equivalent fraction
represent
NOTE: In this lesson, do not
use decimal notation, which
is introduced in the next
lesson. Focus on the concept
of a decimal fraction.
Process Expectation
Looking for a pattern
Introduce powers of 10. Write the following pattern on the board:
1
1
1
1
1
,…
,
,
,
,
10 100 1000 10 000 100 000
Read the fractions aloud: “one tenth, one hundredth, one thousandth, one
ten thousandth, one hundred thousandth.” Ask students to suggest how
you could explain to someone the type of fractions that are in this pattern.
Point out that all fractions in this pattern have denominators that are
multiples of 10. Have students determine the reverse of this statement—all
fractions with denominator a multiple of 10 are in the pattern. ASK: Is this
statement true? (no) Have students name a fraction with denominator a
multiple of 10 that is not in the pattern (EXAMPLE: 1/30). Repeat with other
true statements whose reverse is false, such as: All fractions in this pattern
have numerator 1. Tell students that you are looking for a way to describe
all fractions in the pattern so that the reverse is true too.
Tell students that there are really two patterns in this list of fractions—
there is a pattern in the numerators and another one in the denominators.
ASK: What is the pattern in the numerators? (they are always 1) What
is the pattern in the denominators? PROMPT: How can we get the next
denominator from the previous one? (multiply by 10 each time)
10 = _____
10 × 10 = _____
10 × 10 × 10 = ______
10 × 10 × 10 × 10 = ______
10 × 10 × 10 × 10 × 10 = ______
Tell students that these are called powers of 10. We’ll learn about powers
of other numbers in Grade 8. Explain that 10 is called the first power of 10;
10 × 10 = 100 is called the second power of 10; 10 × 10 × 10 = 1000 is
F-2
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Have students start with 10 and multiply by 10 repeatedly a few times,
recording the resulting pattern.
called the third power of 10. ASK: What is 10 × 10 × 10 × 10 = 10 000
called? (the fourth power of 10) Have students write the eighth power of
10 in their notebooks. (100 000 000)
Have students use powers of 10 to describe the type of fractions in the
pattern above. (The numerator is 1 and the denominator is a power of 10.)
Introduce decimal fractions. Explain that fractions whose denominators
are a power of ten are called decimal fractions. This is because tenths,
hundredths, thousandths, and so on are the place values for decimals.
Write a few decimal fractions on the board:
1
4
5
18
,
,
,
10 10 100 100
Ask students to suggest a few other decimal fractions.
Bonus
Write 1 as a decimal fraction. SAMPLE ANSWERS:
10 100
,
10 100
NOTE: Although mathematically, 1 is considered a power of 10 (the zeroth
power of ten, in fact), we are not introducing it as such at this time.
Write on the board: All decimal fractions have a denominator that is a
power of 10. Have students write the reverse. (All fractions that have a
denominator that is a power of 10 are decimal fractions.) ASK: Is the
reverse true? (yes) Explain that when a statement and its reverse are both
true, then the two parts of the statement mean the same thing—a decimal
fraction means a fraction with a denominator that is a power of 10.
Process Expectation
Representing, Visualizing
Process Expectation
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Representing
Using a grid to represent ones, tenths, and hundredths. Review using
a hundredth grid to model ones, tenths, and hundredths. There are
10 squares in a row and in a column, so the whole grid has 10 × 10 = 100
squares. Each small square is 1 hundredth of the grid. A full column or
10
1
=
row represents 10 hundredths or 1 tenth (
). The entire grid
100 10
100
=1 .
represents one whole, since
100
Shade 3 tenths on a hundredths grid. ASK: How many tenths are shaded?
(3) How many hundredths are shaded? (30) Recall that equivalent fractions
represent the same amount. ASK: Are 3 tenths and 30 hundredths
equivalent fractions? (yes) How do you know? (the same area of the grid
is shaded for each)
Return to the grid on which you shaded 3 tenths. Shade 6 hundredths
in another colour. ASK: How many hundredths did I just shade?
(6 hundredths) Now how many hundredths are shaded on the grid?
How do you know? (30 + 6 = 36, so 36 hundredths)
Process Expectation
Reflecting on other ways
to solve a problem
Number Sense 7-32
Have students suggest strategies for counting the shaded squares. For
example, you could count the tenths first, convert the tenths to hundredths,
then count and add the remaining hundredths. If most of the squares were
shaded, you might want to subtract the number of squares that weren’t
shaded from 100. (Tell students that when they can solve a problem in two
F-3
different ways and get the same answer, they can know that they’re right
without you to check the answer because they can check it themselves!)
Process assessment
Word Problems Practice:
7m1, 7m6, [CN, V]
Workbook Question 6
1.Helen used 2 tenths of a sheet of 100 stickers to decorate her math
binder. She used 6 hundredths of the stickers to decorate her French
binder. Shade a grid to show how many stickers she used. What
fraction of the stickers did she use?
26
ANSWER:
100
Relating equivalent decimal fractions. Recall how you can create an
equivalent fraction by multiplying or dividing both the numerator and
2
denominator by the same number. Write
on the board. Have students
10
4
multiply the numerator and denominator by 2 to get
.
20
ASK: Is four-twentieths a decimal fraction? Why or why not? (No, because
the denominator is not a power of 10.)
ASK: What type of number do you have to multiply or divide a decimal
fraction by to get another decimal fraction? (a power of ten) Have a
2
2 ×10
20
=
to hundredths (
volunteer show how to convert
).
10
10 ×10 100
Then have a volunteer multiply the numerator and denominator by 10 again.
Ask how you would say this fraction (“two hundred thousandths”). Have
students work through a few more such problems.
300 ÷ 100
3
=
EXAMPLE:
.
1000 ÷ 100 10
Word Problems Practice:
2.From a box of 1000 matches, 600 matches were used. How many
tenths of the matches were used? (6 tenths)
3.Seven tenths of a box of 100 matches have been used. How many
matches have been used? (70 matches) What if there had been
1000 matches in the box? (700 matches)
1.Write equivalent decimal fractions for the fractions below. Use the most
“reduced” decimal fraction you can.
1 1 2 3
6
a) , , ,
,
2 4 5 25 50
F-4
b)
35
40
210
,
,
500 200 300
c)
3 100
,
6 125
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Extensions
SAMPLE ANSWERS:
a)
5 25 4 12 12
,
, ,
,
10 100 10 100 100
b)
7 2 7
, ,
100 10 10
c)
5 8
,
10 10
2.Is there a smallest decimal fraction? Explain.
ANSWERS: No. There is no largest power of 10 because you can
multiply any power of 10 by 10 to get an even larger one. Since making
the denominator larger results in a lesser fraction, multiplying the
denominator of any decimal fraction by 10 will get you an even smaller
decimal fraction.
3.Confusion with naming decimal fractions.
a)Ask students to spell out these decimal fractions:
i) 2100/1000
ii) 20/100 000
iii) 2000/100 000
For example, i) could be:
twenty-one hundred thousandths or
two thousand one hundred thousandths or
twenty-one hundred one-thousandths or
two thousand one hundred one-thousandths.
One way to avoid confusion with ii) and iii) when reading the words
aloud is to insist on saying the “one” in “one thousandths,” since ii)
is twenty one-hundred-thousandths, which sounds the same as the
first option for i), and iii) is two thousand one-hundred-thousandths,
which sounds the same as the second option above. The only
difference is in the way the numbers are written—where the dashes
go; but to say it differently, you would need to stress saying the
dashed words together.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
b)Now write these fractions on the board and tell students that you
insist on saying the “one” in one-thousandth: i) 120/1000 and
ii) 100/21 000.
Have students spell the fractions out:
i) one hundred twenty one-thousandths
ii) one hundred twenty-one thousandths
In this case, adding the “one” in one-thousandths could cause
confusion. Point out that there is confusion when saying words
this way, whether you insist on the “one” in one-thousandths or
not. To avoid this when necessary, you could read 120/1000 as
“one hundred twenty over one thousand.” This would avoid the
confusion altogether.
Process assessment
7m1, [PS]
Number Sense 7-32
c)Have students find two numbers that could be confused with
350/1000 depending on whether or not you read the “one” in
one-thousandths. (ANSWERS: 300/50 000 if you don’t read the one;
300/51 000 if you do. Point out that reading this as “three hundred
fifty over one thousand” will avoid confusion entirely.)
F-5
NS7-33 Place Value and Decimals
Pages 129–130
Goals
Curriculum
Expectations
Students will understand decimal place value to thousandths.
Ontario: 6m12; 7m1, 7m3,
7m5, 7m6, 7m7,
essential for 7m11,
7m27
WNCP: 5N8, 5N9; 7N2,
[C, CN, R, V]
PRIOR KNOWLEDGE REQUIRED
Knows the definition of a decimal fraction
Understands place value for whole numbers and the use of zero
as a placeholder
Can write expanded form for whole numbers
Vocabulary
The place value system extends to include decimal fractions.
Write the following on the board:
ones
tenths
hundredths
thousandths
decimal point
place value
placeholder
expanded form
4444 = 4000 + 400 + 40 + 4
= 4 thousands + 4 hundreds + 4 tens + 4 ones
Recall how 4444 is a short way to write 4 thousands + 4 hundreds +
4 tens + 4 ones. ASK: What is 2365 short for?
(2 thousands + 3 hundreds + 6 tens + 5 ones)
Draw the following place value chart on the board. Recall how, in the place
value system, each place represents 10 times as much as the place to
the right: one ten equals 10 ones, one hundred equals 10 tens, and so on
(which is the same as saying that 10 ones make one ten, 10 tens make
one hundred, 10 hundreds make one thousand, and so on):
× 10
× 10
× 10
thousands
hundreds
tens
ones
1
1
1
1
ASK: How many tenths make a one? (10) Have students draw a picture
Process Expectation
× 10
× 10
× 10
× 10
× 10
thousands
hundreds
tens
ones
tenths
1
1
1
1
1
× 10
hundredths thousandths
1
1
Explain that a dot, called a decimal point, is placed between the ones and
the tenths to show where the fraction part of the number starts. Draw the
following place value chart on the board:
F-6
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
to show this. Repeat for how many hundreds make a tenth (10) and how
many thousandths make a hundredth (10). Extend the chart above to
thousandths, as shown below.
Connecting, Visualizing
ones
tenths
hundredths thousandths
4
2
4
7
9
4
+
+
+
,
, and
on the board, and have
10 10 100
10 100 1000
volunteers write the fractions in the place value chart:
Write
ones
tenths
hundredths thousandths
0
4
0
2
4
0
7
9
4
Have students add more examples to the chart in their notebooks. Then
bring their attention back to the examples on the board, and point out how
the digit 4 represents a different amount in each case:
4
4
4
,
,
, and respectively.
10 100 1000
Beside the chart, write each number as a decimal:
.4
.24
.794
Tell students that a number with a decimal point in it is called a decimal.
Mathematicians invented decimals as a convenient way to write decimal
fractions. Draw a rule under the numbers to show how the dot sits just
above the rule. Note how it would be easy to miss the little dot. Advise
students that when they are writing a fraction as a decimal, writing a zero to
the left of the decimal point, in the ones place, will make it easier to read,
and demonstrate this:
0.4
0.24
0.794
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Remind students that we never put a zero at the front of a whole number.
So, when students see a zero and a decimal point at the front of a number,
they will know right away that they are looking at a decimal fraction less
than 1. Point out that people often write the decimal without the zero in
front, and students should still be able to read it—it is just as mathematically
correct to write .794 as it is to write 0.794, but the 0 in front makes it a little
easier to read the number.
Write 0.145 on the board and SAY: This decimal shows 1 tenth,
4 hundredths, and 5 thousandths. ASK: What tenths, hundredths,
and thousandths do these decimals show: 0.973, 0.862, 0.364?
(NOTE: Students will learn to read 0.973 as “973 thousandths” in the
next lesson; for now, stick with expanded form.)
Number Sense 7-33
F-7
Process Expectation
Connecting
Write 608 on the board. ASK: What does the zero in the tens place mean?
(that there aren’t any tens) Have a volunteer write 608 in expanded form
(6 hundreds + 0 tens + 8 ones). Write 0.3 on the board. ASK: What fraction
does this decimal represent? (3 tenths) Write 0.03 on the board. ASK: What
place is the first zero holding in this decimal? (the ones place) What place is
the second zero holding? (the tenths place) What fraction does this decimal
represent? (3 hundredths) Repeat with 0.304.
Bonus
What place are the zeros holding in 0.3402? In 0.34206?
ANSWERS: thousandths place and ones place in 0.3402, ten-thousandths
place and ones place in 0.34206
ASK: Which is greater, a tenth or a hundredth? (a tenth) Which is smaller,
a hundredth or a thousandth? (a thousandth) Write 0.6, 0.21, and 0.403
on the board. Have volunteers underline the smallest place value in each
number (0.6, 0.21, and 0.403).
Have students complete Workbook Questions 1–11. Note that Question
11 c) is a teaser—it introduces a decimal greater than 1. Watch for any
students who are forgetting to use zero as a placeholder, and writing
7 hundredths as 0.7, for example.
Process assessment
7m6, [V]
To conclude the lesson, have students list all the ways they now
know to represent 47 hundredths (you could use the Scribe, Stand,
Share technique):
47
• as a fraction:
100
• by shading 4 columns (or rows) and 7 squares on a hundredths grid
4
7
+
• in expanded form:
or 4 tenths + 7 hundredths
10 100
• in a place value chart
• as a decimal: 0.47 (or just .47)
Process assessment
ASK: Which way helps you understand the value of the digit 7 best?
7m3, 7m7, [R,C]
Which way is quickest and easiest to write? Which way helps you to see
that the fraction is close to a half?
1.Two friends ate 6 tenths of a pizza. Write the fraction of the pizza they
ate as a decimal. (they ate 0.6 of the pizza)
2.A carpenter used 4 tenths of a box of 100 nails on Monday and
3 hundredths of the box on Tuesday. Write the total fraction of the
nails used as a decimal. (0.43 of the nails)
3.A carpenter used 0.5 of the nails in a box of 1000 nails. How many
nails did he use? (500 nails)
F-8
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Word Problems Practice:
Extension
Write 205 on the board. ASK: How many times more is the 2 worth than
the 5? (PROMPTS: How much is the 2 worth? (200) How much is the
5 worth? (5) How many times more is 200 than 5? (40)) Repeat for 0.205.
Emphasize that because each place value is 10 times the one to the right,
the relative values for 0.205 and 205 are the same. Continue with the
decimal 0.324 067 568. Because each place value is 10 times the one to
the right, the relative values here are the same as for 324 067 568. Because
we’re only interested in comparing the 2 and the 5, we can pretend the
5 is the ones digit and start at the 2, so that the number becomes 240 675.
The 5 is five places over from the 2, so we can make a T-chart to help us:
Number of places over
EXAMPLE:
How many times more?
1
25 = 20 + 5
4
2
205 = 200 + 5
40
3
2005 = 2000 + 5
400
4
20 005 = 20 000 + 5
4000
5
200 005 = 200 000 + 5
40 000
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
If the 2 were one place to the left of the 5, it would have a value of 20,
which is 4 times as much as the 5. The T-chart shows that the 2 in our
decimal is worth 40 000 times as much as the 5.
Number Sense 7-33
F-9
NS7-34 Fractions and Decimals
Pages 131–132
Goals
Curriculum
Expectations
Students will translate between fractional and decimal notation
(to thousandths).
Ontario: 6m11, 6m27; 7m1,
7m6, 7m7, essential for
7m11, 7m27
WNCP: 5N8, 5N9; essential
for 7N2, [C, CN, R]
PRIOR KNOWLEDGE REQUIRED
Can write decimal fractions and decimals
Can identify tenths, hundredths, and thousandths
Knows the value of tenths, hundredths, and thousandths
Can represent a decimal fraction on a hundredth grid
Vocabulary
equivalent decimals
Relating fractions and decimals. Draw a decimal place value
chart to hundredths on the board. Have students copy the chart into
their notebooks.
ones
tenths
hundredths
0
0
0
0
0
Draw on the board or show on an overhead transparency:
d)
Process Expectation
Representing
F-10
b)
c)
e)
For each grid, have students record the number of tenths in the tenths
place in the place value chart and the number of hundredths left over
(after they’ve counted the tenths) in the hundredths place. Tell students that
if there are 0 tenths, or 0 hundredths left over, they should write 0. When
students are done, have them write in their notebooks the fraction shown
on each grid, as a fraction with denominator 100 (a number of hundredths)
and as the equivalent decimal (e.g., 63/100 = 0.63)
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
a)
Remind students that hundredths are shown with two decimal places:
5 hundredths is written as 0.05, not 0.5, because 0.5 is how we write
5 tenths and we need to make 5 hundredths look different from 5 tenths.
Have students write a few decimal hundredths as fractions without
using grids:
a) 0.32
Process Expectation
Using logical reasoning
b) 0.77
c) 0.19
d) 0.02
ASK: When you wrote 0.32 as a fraction, how did you know that the
denominator of the fraction would be 100? See what reasons students
suggest. Then tell students that when there are two digits after the decimal
point, the fraction will be hundredths. Another way to know is that the digit
in the smallest place value is in the hundredths place. Note that 3 tenths
and 2 hundredths is 32 hundredths.
Have students write a few fraction hundredths as decimals:
12
65
4
51
b)
c)
d)
a)
100
100
100
100
Point out again that 12/100 is 12 hundredths, which is 1 tenth and
2 hundredths—0.12.
Write 0.4 on the board. ASK: If you write this decimal as a fraction, what will
the denominator be? (10) How do you know? What will the numerator be?
(4) How do you know?
Have students fill in the numerator of each fraction:
a) 0.6 =
10 d) 0.63 =
100
b) 0.9 =
10 e) 0.87 =
100
c) 0.2 =
10
f) 0.48 =
100
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Then tell students that you are going to make the problems a bit harder.
They will have to decide whether the denominator in each fraction is
10 or 100. Ask a volunteer to remind you how to decide whether the fraction
should have denominator 10 or 100. Emphasize that if there is only one
digit after the decimal point, the digit tells you the number of tenths, so the
denominator is 10; if there are two digits after the decimal point, the digits
tell you the number of hundredths, and the denominator is 100.
Have students write the fraction for each decimal in their notebooks:
a) 0.4
Bonus
Process Expectation
Connecting
Number Sense 7-34
b) 0.75
0.254, 0.009
c) 0.03
ANSWERS:
d) 0.33
e) 0.2
f) 0.38
254
9
,
1000 1000
Write 0.325 on the board. ASK: How many digits are there after the decimal
point? (3) What place is the last digit in? (the thousandths place) If you write
this decimal as a fraction, what will the denominator be? (1000) How do you
know? What will the numerator be? (325) How do you know?
F-11
SAY: Let’s check that 0.325 = 325 thousandths. Write the following
on the board:
3
2
5
0.325 =
+
+
10 100 1000
=
=
Process Expectation
7m7, [C]
1000
+
1000
+
5
1000
1000
Have students explain what the equivalent fractions are and how they
know. Have students explain what the sum of the thousandths is, and
how they know.
Equivalent decimals. Write 0.8 and 0.80 on the board. Tell students that
mathematicians call these equivalent decimals, and ask if anyone can
explain why they are equivalent. (They have the same value; the fractions
they are equivalent to are equivalent.) Have students write the fractions that
the decimals are equivalent to, in order to demonstrate that these fractions
are indeed equivalent. (8/10 and 80/100)
Process Expectation
Representing
Write 0.8 = 0.80 on the board. Tell students that saying “0.8 = 0.80”
is the same as saying “8 tenths is equal to 80 hundredths or 8 tenths
and 0 hundredths.” ASK: Is “3 tenths” the same as “3 tenths and
0 hundredths”? How many hundredths is that? Have a volunteer write
the equivalent decimals on the board (0.3 = 0.30)
Have students fill in the blanks:
3
=
= 0. _ _
a) 0.3 =
10 100
c) 0.4 =
10
=
100
b) 0. _ =
7
=
= 0.70
10 100
= 0. _ _
8
80
=
=
on the board. Have a volunteer write
10 100 1000
the equivalent fraction, and then the equivalent decimals. Repeat
300
=
=
.
with
1000 100 10
Have students rewrite these equivalent fractions as equivalent decimals:
9
90
900
5
50
500
60
600
6
=
=
=
=
=
=
a)
b)
c)
10 100 1000
10 100 1000
100 1000 10
Bonus
2
20
200
2000
=
=
=
10 100 1000 10 000
Put the following equivalencies on the board and SAY: I asked some
students to change decimals to fractions and these were their answers.
Which ones are incorrect? Why are they incorrect?
37
50
47
62
a) 0.37 =
b) 0.05 =
c) 0.047 =
d) 0.62 =
100
100
1000
1000
F-12
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Then write
(parts b and d are incorrect, in part b, the denominator is correct, but
the numerator should be 5; in part d, the numerator is correct, but the
denominator should be 100)
Process assessment
Word Problems Practice:
7m6, [CN]
Workbook Question 13
1.Josie and her family ate 60 of 100 cherries. How many tenths of the
cherries did they eat? Write your answer as a decimal. (0.6)
2.Michael is trying to watch all the films on a list of the 100 best films of
all time. He has watched 0.2 of the films so far. How many of the films
has he watched? (20 films)
3.Kai has a set of 1000 Lego pieces. There are 124 red pieces in his set.
What fraction of the Lego pieces are not red? Write your answer as a
decimal. (0.876)
Extensions
1. Challenge students to write decimals for the following fractions:
1
1
27
b) 100 000
c) 10 000
a) 10 000
ANSWERS: a) 0.0001 b) 0.00001 c) 0.0027
2. a) How would you change 0.004 206 to a fraction?
9823
b) How would you change 1 000 000 to a decimal?
4206
b) 0.009 823
ANSWERS: a) 1 000 000
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
3.Decimals are not the only numbers that can be read in different ways.
Show students how all numbers can be read according to place value.
The number 34 can be read as “34 ones” or “3 tens and 4 ones.”
Similarly, 7.3 can be read as “73 tenths” or “7 ones and 3 tenths.”
Challenge students to read the following numbers two ways:
Number Sense 7-34
a) 3 500
b) 320
c) 5.7
d) 1.93
e) 0.193
ANSWERS: Answers may vary; for example:
a) 3 thousands and 5 hundreds or 35 hundreds
b) 3 hundreds and 2 tens or 32 tens
c) 5 ones and 7 tenths or 57 tenths
d) 19 tenths and 3 hundredths or 193 hundredths
e) 19 hundredths and 3 thousandths or 193 thousandths
F-13
NS7-35 Decimals, Money, and Measurements
Page 133
Goals
Curriculum
Expectations
Students will relate tenths, hundredths, and thousandths of whole
numbers and of dollars and metres.
Ontario: 7m5, essential for
7m11, 7m27
WNCP: essential for 7N2,
[CN, V]
PRIOR KNOWLEDGE REQUIRED
Knows the value of pennies, dimes, and dollars
Can write decimals
Can write values such as 3 tenths and 5 hundredths as 35 hundredths
Vocabulary
currency
dollar, dime, penny
metre
NOTE: This lesson connects decimal concepts to the real world and
highlights the convenience of decimals. It could be used, in whole or part,
to close lesson NS7-34 or to introduce lesson NS7-36.
Connecting money and decimals. Display a dollar, a dime, and a
penny. Tell students that one dime is one-tenth of a dollar, and one cent
is one-tenth of a dime or one-hundredth of a dollar. ASK: What do I mean
when I say that a dime is one-tenth of a dollar? Does this mean I can take
a loonie and fit 10 dimes onto it? Does it mean 10 dimes weigh the same
as a loonie? Make sure students can articulate that you are referring not to
weight or area, but to value—a dime has one-tenth the value of a dollar,
it is worth one tenth the amount. Ask students what fraction of a dollar a
penny is worth (one hundredth).
Tell students that since a dime is worth a tenth of a dollar and a penny
is worth a hundredth of a dollar, they can write 5 dimes and 4 pennies as
5 tenths and 4 hundredths. ASK: How many hundredths of a dollar are
5 dimes and 4 pennies? (54)
Bonus
If a penny is a hundredth of a dollar, what is a penny a
thousandth of? ($10.00)
Real-world
Prefix
Decimal
kilo
thousand
hecto
hundred
deca
ten
one
deci
tenth
centi
hundredth
milli
thousandth
F-14
The metric system. Explain that a currency system is a measurement
system. Canada, like many countries in the world, uses the metric system
of measurement units. The metric system is a decimal system. Each type of
measurement has a base unit; for example, length is measured in metres
and volume is measured in litres.
Tell students that there are standard prefixes for various decimal multiples.
For example, “kilo” means “thousand,” so 1 kilometre = 1000 metres,
1 kilogram = 1000 grams, 1 kilowatt = 1000 watts, and so on.
Show students the chart in the margin.
Have students look at the chart to see that “deci” is the prefix for tenth.
ASK: What is one tenth of a metre? (a decimetre) On the board, write:
1 decimetre = 0.1 metre
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
connection
ASK: What fraction of a metre is 3 decimetres? (3 tenths) Have students
complete this statement:
3 decimetres = ____ metre (0.3)
ASK: What unit is one hundredth of a metre? (a centimetre) Have students
Process assessment
7m5, [CN]
Workbook Question 3
look at the chart to see that “centi” is the prefix for hundredth. Have
students complete this statement:
1 centimetre = _____ metre (0.01)
Have students complete Workbook Question 1.
ASK: What is the prefix of “millimetre”? (milli) What fraction of a metre is
a millimetre? (a thousandth) Have students complete this statement:
1 millimetre = _____ metre (0.001)
ASK: What are some other units that start with “milli”? (milligram, millilitre,
millisecond). What fraction of a second is a millisecond? (1 thousandth)
What fraction of a gram is 500 milligrams? (5 tenths, or one half)
Process Expectation
Visualization
Visualizing the relative sizes of ones, tenths, hundredths, and
thousandths. To reinforce how relatively small the fractions represented by
the digits after the decimal point are, show students a metre stick. Draw a
line 1 m long on the board. SAY: This line is 1 m long. Let’s draw a line that
is 0.473 of a metre. Write “0.473 m” on the board.
ASK: How many centimetres are there in a metre? (100 cm) How many
centimetres is a tenth of a metre? (10 cm) How many centimetres is
four-tenths of a metre? (40 cm) Have a volunteer measure and mark off
40 cm on the 1 m line you drew.
0.4 m
0.47 m
ASK: How many centimetres is a hundredth of a metre? (1 cm) How
many centimetres is seven hundredths of a metre? (7 cm) Have a volunteer
measure and mark off 7 more centimetres on the line.
ASK: What measurement is one thousandth of a metre? (1 mm) How
0.473 m
many millimetres is 3 thousandths of a metre? Have a volunteer measure
and mark off 3 more millimetres on the line.
ASK: Did the 3 thousandths add much to the length of the line? (no)
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Suggest that students can use the metre-stick model to help them
remember how very small thousandths are.
Extensions
1. What fraction of a kilometre is a centimetre?
1
ANSWER: 100 000
2.Students can research the metric system (also known as the
International System of Units, or SI, for the French Système
International). Name three countries that use the system and three
countries that do not.
Number Sense 7-35
F-15
NS7-36 Decimals and Fractions Greater Than 1
Pages 134–135
Ontario: 6m11, 6m27; 7m1,
7m5, 7m6, 7m7,
essential for 7m11,
7m27
WNCP: 5N8, 5N9; 7N4,
[C, CN, R]
Vocabulary
mixed number
Process Expectation
Connecting
Process Expectation
Communicating
Goals
Students will write mixed numbers as decimals and decimals greater
than 1 as mixed numbers, to thousandths.
PRIOR KNOWLEDGE REQUIRED
Can write mixed numbers and decimals
Can translate between decimal and fractional notation for numbers
less than 1
In this lesson, students connect the concepts taught in lessons NS7-32 to
NS7-34 to decimals greater than 1.
Money and decimals. Write a dollar amount such as $5.28 on the board.
SAY: This represents 5 dollars and 28 cents, or 5 dollars and 28 hundredths
of a dollar. ASK: Which digit or digits show whole dollars? (the 5) How do
you know? (the 5 is before the decimal point) Write 5 28/100 beside $5.28
on the board. Tell students that decimals can also be used to represent
mixed numbers. The whole-number part of the mixed number goes to the
left of the decimal point, and the fraction part goes to the right. Write 5.28
on the board. Tell students that just as we say “5 dollars and 28 cents,”
we read this decimal as “5 and 28 hundredths.” We read the decimal point
in a mixed number as “and.”
Mixed measurements in decimals. Recall that there are 100 cm in 1 m.
Write 3.45 m on the board and ask how you might say this measurement
(“3 metres and 45 centimetres”). Note that sometimes people will just say
“three point four five metres,” and that that’s okay too, so long as you
understand that the 0.45 part represents 45 hundredths of a metre,
or 45 cm.
Ask for a volunteer to write 3 4/10 as a decimal (3.4). Repeat with 6 15/100,
20 7/10, and 365 214/1000. Write 9 7/100 = 9.7 on the board. ASK:
Did I write the decimal correctly? Is 9.7 equal to 9 and 7 hundredths? (no)
Have a volunteer write the correct decimal (9.07). Remind students to
be careful to use zeros as placeholders as required when writing mixed
numbers as decimals.
Draw students’ attention to the title of this lesson, “Decimals and Fractions
Greater Than 1.” ASK: Could you ever have a mixed number that was
less than 1? (no) Why not? (because a mixed number always has a whole
number part that is at least 1) Have students identify which of the following
decimals is a mixed number, then write those decimals as mixed numbers:
Process Expectation
a) 3.06
b) 0.7785
c)7820.0
d) 200.004
e) 0.001 001
Representing
F-16
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Curriculum
Expectations
Process Expectation
Generalizing from examples
Relative size of decimals. Start students thinking about the relative sizes
of decimals in preparation for the next lesson. Write 0.__ on the board.
ASK: What is the largest number of tenths I could put in this tenths place?
(9) Is 9 tenths more than 1? (no) Write 0.__ __ on the board. ASK: What is
the largest number of hundredths that could go here? (99) Is 99 hundredths
more than 1? (no) Write 0.999 999 999 999 999 on the board. ASK: Is this
decimal greater than 1? (no—although it is very close to 1) Ask students to
think about that number. You could have a million digits to the right of the
decimal point, and they would still represent a number smaller than one.
More digits on the whole-number side of a decimal mean a larger number,
but more digits on the right hand side of a decimal don’t necessarily mean
a larger fraction; for example, 0.111 111 is much less than 0.9.
Process assessment
Word Problems Practice:
7m7, [C]
Workbook Question 10
1.Kiyoko bikes 2 3/10 km to school each day. Write this distance as
a decimal. (2.3)
2.Lined paper comes in packs of 100. Karl used 356 sheets of lined
paper. How many packs of paper did he use? Write your answer as
a mixed number and as a decimal. (3 56/100, 3.56)
3.The length of Ted’s room is 3 m and 12 cm. Write the length (in metres)
as a mixed number and as a decimal. Remember, 1 m = 100 cm.
(3 12/100 m, 3.12 m)
Extensions
1.Have students list decimal numbers that take exactly six words to
say. Students can use the Scribe, Stand, and Share technique (see
Introduction to Grade 7). Make it clear to students that two words
joined by a dash count as one word (e.g., “fifty-eight” is one word).
This exercise gives students practice writing number words for
decimals, being organized, and looking for patterns.
EXAMPLES:
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
403.08 (four hundred three and eight hundredths)
600 000.43 (six hundred thousand and forty-three hundredths)
9 000 080.09 (nine million eighty and nine hundredths)
3.542 (three and five hundred forty-two thousandths)
500.000 1 (five hundred and one ten thousandth)
Encourage students to find as many different types of numbers as they
can. For example, 570.1, 680.4, and 830.5 are all six-word numbers,
and they are the same type of number—the non-zero digits are all in
the same place values.
2.Teach students to interpret whole numbers written in decimal format
(EXAMPLE: 5.1 million is 5 100 000, 3.7 thousand is 3700)
3.Have students look for decimals in the media and write the decimals
as mixed numbers.
Number Sense 7-36
F-17
NS7-37 C
omparing and Ordering Tenths
and Hundredths
Pages 136–137
Goals
Curriculum
Expectations
Students will use number lines and benchmarks to compare and order
fractions and decimals to hundredths.
Ontario: 6m11; 7m3, 7m5,
7m6, 7m11
WNCP: essential for 7N7,
[CN, R, V]
PRIOR KNOWLEDGE REQUIRED
Can compare whole numbers, fractions, and mixed numbers
Can use number lines to compare numbers
Can relate fractions and decimals
Vocabulary
number line
scale
benchmark
greater than (>)
less than (<)
Using a number line to compare and order fractions and mixed
numbers. Students have used number lines to compare whole numbers.
Tell students that in this lesson they will be using number lines to compare
fraction tenths and hundredths. Recall that a number line has a scale. If a
number line is divided into tenths, the scale is tenths; if hundredths, it is
hundredths. Explain to students that the number they skip count by to mark
the number line is called the scale.
Have students look at the number line at the top of Workbook p. 136.
ASK: What number does this number line start at? (0) As you move to the
right along the line, do the numbers get larger or smaller? (larger) What
happens as you move left? (the numbers get smaller) How many equal
parts is the number line divided into between 0 and 1? (10) What is the
scale of this number line? (tenths)
Have students complete Workbook Question 1. Students with weaker
visualization skills may need extra support. Help them with prompts and
questions such as: Is the number more than 1 or less than 1? How do
you know? Is the number between 1 and 2 or between 2 and 3? How do
you know?
Process Expectation
Visualizing
Using benchmarks. Recall that a benchmark is a number used to estimate
the position of a number on a number line. Benchmarks are numbers that
are easy to visualize, such as 1/2. Draw the following number line and
points on the board:
A
F-18
B
1
0
2
1
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Discuss how you would place a mixed number on the number line; for
example, 2 3/10, as shown. First, look at the whole number part of the
mixed number, which in this case is 2. Go to 2 on the scale. Then look
at the fractional part of the mixed number, 3/10. Count over 3 tenths,
heading right, because 2 3/10 is greater than 2.
ASK: Which is greater, A or B? (B) How do you know? (B comes after 1/2,
whereas A comes before 1/2) Draw the following number line and points
on the board:
B A
0
1
2
1
ASK: Which is closer to 1/2, A or B? Which is greater, A or B? (A)
Have students complete Workbook Questions 2–4. ASK: How did you
Process Expectation
Reflecting on other ways
to solve a problem
decide where to place 3/4 in Question 4a)? Discuss possible strategies
(e.g., estimate visually; write as fraction hundredths and then compare
with 0.7 = 0.70)
ASK: Which decimal tenth is equal to one half? (0.5) Which decimal
hundredth is equal to one half? (0.50) Have students complete Workbook
Question 5.
Direct students’ attention to the four number lines in Question 6. ASK: What
is the same about these four number lines? (They all run from 0 to 1. They
are all the same length. They are lined up at 0.) What is different? (They
each have a different scale. The bottom two each have a decimal scale).
Process Expectation
Visualizing
Have students line a ruler up with 1/4 and use a pencil to mark where
the ruler edge crosses the tenths and hundredths lines. ASK: How many
tenths is 1/4? (between 2 and 3 tenths) How many hundredths is 1/4?
(25 hundredths)
Have students complete Workbook Questions 6–7. If students need
assistance writing 12/10 as a mixed number, show them that, when the
denominator is 10, there is an easy way to tell whether the improper fraction
is between 1 and 2 or between 2 and 3. Just look at the number of tens
in the numerator—that tells you how many ones are in the number.
For example, 26/10 is 2 ones and 6 tenths.)
Extensions
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
1.How would you place the following numbers on the number line in
Workbook Question 5? Explain how you decided where to mark each
point. Encourage students to think about hundredths that are close to
the number if they can’t find exact hundredths that equal the number.
2
1
b) 0.346
c)
a)
5
3
ANSWER: a) 2/5 should be placed at 0.40; b) 0.346 is close to
350 thousandths or 35 hundredths, so place this number close to 0.35;
c) 1/3 should be placed a little to the right of 0.33, since 1/3 = 33/99
is close to, but slightly greater than, 33/100.
Number Sense 7-37
F-19
2. Draw a 0–1 number line with thirds marked:
Process assessment
a)Place the following numbers on this number line:
0.1, 0.25, 0.35, 0.4, 0.55, 0.65, 0.75
b)Describe the strategy(ies) you used to place each number on this
number line.
c)Which numbers did you find easiest to place? Why? Which
numbers were harder to place? Why?
d)Do you think thirds are good benchmarks for decimals? Explain
your reasoning.
e)Would fifths or sevenths be better benchmarks for decimals?
Explain your reasoning.
ANSWERS
a) 7m3, [R, C]
F-20
1
0.1
0.25 0.35 0.4
0.55 0.65
0.75
1
0
2
3
3
1
b)For example, I marked a half and fourths (OR a tenths scale) on the
number line; I compared 1/3 to 0.35 = 35/100 by finding a common
denominator—1/3 = 100/300 and 0.35 = 105/300, so 0.35 is very
close to 1/3 but is slightly bigger; I know that 0.25 = 1/4 which is
less than 1/3.
c)Answers will vary depending on the strategy used. The student
may use visualization to mark 1/2, 1/4, and 3/4 on the line, but
this is essentially an admission that 1/3 and 2/3 aren’t very useful
benchmarks.
d)Thirds can’t be changed into decimal fractions, so they aren’t easy
to compare with decimals. Also, the numbers halfway between the
thirds aren’t helpful either; you know 1/6 is halfway between 0 and
1/3, but what decimal fraction is 1/6 close to?
e)Fifths would be better than sevenths, as fifths can be written as
equivalent decimal fractions.
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
1
2
0
3
3
NS7-38 O
rdering Decimals and Fractions
to Thousandths
Pages 138–140
Curriculum
Expectations
Ontario: 6m11, 6m27; 7m1,
7m2, 7m5, 7m7, 7m11,
7m27
WNCP: 7N7, [C, CN, R, V]
Vocabulary
greatest
least
common denominator
Goals
Students will compare and order decimals using place value and
equivalent fractions or decimals.
PRIOR KNOWLEDGE REQUIRED
Can order whole numbers
Can write equivalent fractions and decimals
Can order proper and improper fractions with the same denominator
Understands decimal place values
Can translate between fractions with denominator 10, 100, or 1000
and decimals
Many students have difficulty ordering decimals, or sets of fractions and
decimals. Students may not yet have really grasped that digits to the right
of the decimal point represent very small numbers. They may incorrectly
apply whole number concepts and identify decimals with more digits after
the decimal point as greater. While they may understand that decimals
represent decimal fractions, they may not appreciate that decimals can
also represent fractions that are not decimal fractions but are equivalent
to decimal fractions (e.g., 0.6 represents 6/10 but also the equivalent 3/5).
This lesson builds students’ understanding in three stages:
1.Students convert decimals to fractions with a common denominator
to order decimals.
2. Students use place value to compare and order decimals.
3.Students convert halves, fifths, and fourths to decimal fractions and
then to decimals.
Ordering fractions with a common denominator. Write the following
set of fractions on the board:
1 3 2
5 5 5 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ASK: What is the denominator in all three fractions? (5) Tell students that,
when two or more fractions have the same denominator, we say they have
a common denominator.
ASK: Which fraction is the greatest? (3/5) How do you know? (They are all
fifths, and three-fifths is more than one-fifth or two-fifths.) Remind students
that when a set of fractions has a common denominator, you just need to
compare the numerators.
Write the following set of fractions on the board:
7
4
3
10 5 10
Number Sense 7-38
F-21
ASK: How can you compare these fractions? (Rewrite them with a
common denominator; in this case, write 4/5 as 8/10.) On the board, write:
7
4
3
7
8
3
→
so ____, ____, ____
10 5 10
10 10 10
Have students tell you which of the original fractions is the greatest, the
next greatest, and the least, and write them in descending order:
7
10
4
5
3
7
→
10
10
8
10
3
10
8 7 3
, ,
10 10 10
so
Write a few sets of decimal fractions on the board and have students order
them from least to greatest by first writing the fractions with a common
denominator. Include some improper fractions. For example:
6
53
4
37
405
42
b)
a)
10 100 10 100 1000 1000
c)
720
100
83
10
90
100
d)
165
100
969
1000
Process Expectation
Write the following decimals on the board:
Changing into a known
problem
0.5
0.36
65
10
0.52
Show students how to write these decimals as fractions and then as
fractions with a common denominator to order the decimals from least
to greatest:
0.5
0.36
0.52 →
→
→
5
36
52
10 100 100
50
36
52
100 100 100
0.36, 0.5, 0.52
a) 0.75 0.8 0.63
b) 0.504 0.49 0.6
c) 2.3 3.02 2.373
d) 18.6 18.059 18.07
Ordering decimals by rewriting them to the smallest place value.
Write the following on the board:
7
=
=
10 100 1000
Have volunteers rewrite the fraction as hundredths and thousandths,
and then rewrite the three fractions as decimals:
7
7
70
700
=
=
→
=
=
→ 0.7 = 0.70 = 0.700
10
100
1000
10
100
1000
F-22
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Have students repeat the above procedure with a few more sets of
decimals, including decimals greater than 1. Be sure to include some
decimals that include zeros, as students need more practice interpreting
decimals that have zeros than those that do not. EXAMPLES:
Remind students that adding zeros to a decimal doesn’t change the
decimal’s value: 0.7, 0.70, and 0.700 are all equivalent decimals. Write a
decimal tenth, hundredth, and thousandth on the board. EXAMPLE:
0.4
0.56
0.392
Process Expectation
Have a volunteer add zeros to the decimals so that they are all thousandths:
Changing into a known
problem
0.4
0.56
0.392
0.400 0.560 0.392
ASK: How many thousandths do these decimals represent? (400, 560, 392)
Which of these decimals represents the most thousandths? (0.560) Have
students use the decimal thousandths to order the original decimals from
greatest to least (0.56, 0.4, 0.392). Repeat with a few more sets of decimals,
including decimals greater than 1. EXAMPLES:
a) 0.54, 0.346, 0.6
b) 0.295, 0.3, 0.29
c) 2.22, 2.1, 2.012
d) 6.25, 61.3, 62.104
Identifying a decimal between two decimals. Check students’
understanding of place values by having them identify a decimal between
two decimals. Write 0.4 and 0.9 on the board. Have students name a
decimal between these two numbers. Repeat with a pair of hundredths
(e.g., 0.25 and 0.34) and thousandths (e.g., 0.675 and 0.690). Then, make
it harder. Write on the board:
Process Expectation
Using logical reasoning
Process Expectation
Changing into a known
problem
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
480
4700
49
4.3
4.4
ASK: Are there any numbers between 4.3 and 4.4? (Yes, but see what
students say. Some may say no because they are only thinking about the
tenths.) Write the two decimals as hundredths: 4.30 and 4.40. ASK: Are
there any numbers between 4.30 and 4.40? (Yes) Have students identify
a few decimals between 4.30 and 4.40 (for example, 4.35). Have students
work through a few more problems of this sort; for example, identify a
decimal between:
a) 7.8 and 7.9
b) 0.25 and 0.26
c) 0.2 and 0.24
d) 6.3 and 6.37
Using place value to compare decimals: Write several whole numbers
on the board, lining them up at the ones (see the margin).
Process Expectation
ASK: Which number is the greatest? (4700) How do you know? (It has
Justifying the solution
ASK: Which of these decimals has the most digits? (0.473) SAY: Let’s add
0.48
0.473
0.4
a digit in the highest place value; it has the most digits.) Then write the
following decimals on the board, lining them up at the smallest place values
as shown in the margin.
zeros to compare these decimals:
0.48
0.473
0.4
Number Sense 7-38
0.480
0.473
0.400
F-23
Process Expectation
ASK: Which is the greatest of the three decimals? (0.48) Is this the decimal
Changing into a known
problem
0.48
0.473
0.4
ASK: How many tenths do these decimals all have? (4) Which of these
Process Expectation
Revisiting conjectures that
were true in one context
with the most digits? (no) Then write the same three decimals lined up at
the decimal point as shown in the margin.
decimals has the largest number in the hundredths place? (0.48) Which of
the decimals has the most hundredths? (0.48) Do you need to look at the
thousandths to know which of these decimals is the greatest? (no) Why
or why not? (You only need to compare the largest place value that has
different digits.) So, which of these three decimals is the greatest? (0.48)
Did I need to add zeros to the decimals to figure that out? (no) How should
you line up decimals to compare them? (at the decimal point).
Emphasize that the decimal with the most digits is not necessarily the
greatest number. Have students practise ordering sets of decimals that
are lined up at the decimal point, from least to greatest. Include decimals
greater than 1. EXAMPLES:
a) 0.6 b) 0.25 c) 2.3 d) 12.005
0.78 0.234 2.04 1.967
0.254 0.219 2.19512.92
Process Expectation
Looking for a pattern
Have students complete Workbook Questions 9–13. For each pattern in
Question 10, ASK: Is the pattern increasing or decreasing? Which place
value or place values are changing? By how much does the pattern
increase/decrease each time? For Question 13, ASK: To find decimals
between 1.32 and 1.33, do you have to look to the hundredths or to
the thousandths?
Ordering factions and decimals when the fractions are equivalent
to decimal fractions. Write the following on the board: 2 × ___ = 10.
ASK: What is the missing factor in this product? (5) Write:
1× 5
=
2
× 5 10
ASK: How many tenths is one-half? (5) How can you write one-half as
a decimal? (0.5) Write:
1× 2
=
5 × 2 10
a decimal? (0.2) Write:
2× 2
=
5 × 2 10
ASK: How many tenths is two-fifths? (4) How can you write two-fifths as
a decimal? (0.4) Tell students that halves or fifths can always be written
as decimal tenths because 2 and 5 are factors of 10. Write on the board:
4 × ___ = 10. ASK: Is there a whole number that multiplies by 4 to give 10?
(no) Is 4 a factor of 10? (no) So, can you write fourths as decimals tenths?
(no) Write: 4 × ___ = 100. ASK: What is the missing factor? (25) Write:
F-24
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ASK: How many tenths is one-fifth? (2) How can you write one-fifth as
1× 25
=
4 × 25 100
ASK: How many hundredths is one-fourth? (25) How can you write
one-fourth as a decimal? (0.25) Write:
3 × 25
=
4 × 25 100
How many hundredths is three-fourths? (75) How can you write
three-fourths as a decimal? (0.75)
Tell students that if the denominator of a fraction is a factor of 100, the
fraction can be written as a decimal hundredth. Have students practise
by writing the following as decimal hundredths:
1
1
1
7
23
13
b)
c)
d)
e)
f)
a)
20
25
50
50
25
20
Process assessment
Word Problems Practice:
7m1, 7m6, [V, R, CN]
Workbook Question 17
1.Tom ran 2.51 km, Jessie ran 2.405 km, and Kay ran 2.6 km. Who ran
the farthest? (Kay)
2.Three plants are 0.6 m, 0.548 m, and 0.56 m tall. Order the heights
of the plants from least to greatest. (0.548 m, 0.56 m, 0.6 m)
3.Three bags of flour weigh 1.04 kg, 1.2 kg, and 1.175 kg. Order the
masses from greatest to least. (1.2 kg, 1.175 kg, 1.04 kg)
Extensions
1.Find as many ways as you can to use the digits 1 to 5 each once
so that 0.__ __ > 0.__ __ __
SAMPLE ANSWERS: 0.51 > 0.243, 0.23 > 0.145
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
2.Use the digits 0, 1, and 2 each once to create as many different
decimals as you can that are:
a) more than 1.2
b) between 0.1 and 0.2
c) between 1.0 and 2.0
ANSWERS: a) 2.01, 2.10, 10.2, 12.0, 20.1, 21.0
c) 1.20 or 1.02
b) 0.12 or .102
3.Three squares have side lengths 2.56 cm, 3.1 cm, and 2.78 cm.
Which square has the greatest area? Explain your reasoning.
ANSWER: The square with side length of 3.1 cm, because the area
is the side length times the side length, and the longest side length
will give the greatest product.
Number Sense 7-38
F-25
NS7-39 Regrouping Decimals
Pages 141–142
Curriculum
Expectations
Ontario: 6m11, 6m20; 7m2,
7m5, 7m6, essential for
7m22
WNCP: 5N8; essential for
7N2, [CN, V]
Vocabulary
regrouping
Goals
Students will rename decimals by regrouping in various ways.
PRIOR KNOWLEDGE REQUIRED
Can regroup ones, tens, hundreds
Can use base ten materials to represent whole numbers
Understands decimal place values
This lesson prepares students for adding and subtracting decimals by
developing a base ten model for tenths and hundredths and providing
practice in regrouping tenths, hundredths, and thousandths.
A base ten model for decimals. On the board, draw the heading row of a
place value chart as shown below. Remind students that each place value is
ten times the value of the place to the right.
Process Expectation
Visualizing, Modelling
Recall how students used base ten materials in earlier grades to represent
whole numbers. Draw the base ten blocks beneath their traditional wholenumber places, as shown below. There is no need to draw all the little
divisions on each block—it is beneficial for students to see you model
making a simple sketch, without needless detail.
× 10
hundreds
× 10
tens
× 10
ones
× 10
tenths
hundredths
× 10
hundreds
F-26
× 10
tens
× 10
ones
× 10
tenths
hundredths
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Recall that a little cube represented 1. A rod is 10 times as big as a little
cube, and it represented 10 ones, or one 10. A flat is 10 times as big as a
rod, and it represented 10 tens, or 100. Then, move all the blocks one
place value to the right:
Process Expectation
Representing, Modelling,
Using logical reasoning
ASK: Does this still work? Can we use the little block to show tenths, the
rod to represent ones, and the flat to represent tens? Why or why not?
(Yes, it works, because each place value is 10 times the one before it).
Do this again, ending up with the cube in the hundredths place:
× 10
hundreds
× 10
tens
× 10
ones
× 10
tenths
hundredths
Tell students that in this lesson you will be using this latter model, where the
flat represents one, the rod represents a tenth, and the little cube represents
a hundredth.
Process Expectation
Representing, Modelling,
Visualizing
Representing decimals using base ten materials. Draw a regular
base ten model of 1.32 as above. Then make your sketch even simpler,
as shown.
SAY: I want to draw base ten models of some decimals, but I don’t want
to spend a lot of time drawing the blocks. I think I can just draw the rods
as lines and the little cubes as dots. What do you think? Will this be clear?
Then draw a simplified drawing of 2.56 and have students tell you how
many ones, tens, and hundreds you’ve drawn. (2 ones, 5 tenths, and
6 hundredths)
Process assessment
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
7m6, [V]
Number Sense 7-39
Have students draw models for more decimals. EXAMPLES: 1.32,
0.64, 4.04.
Regrouping using base ten materials. Then, draw a base ten model of
2 ones, 12 tenths, and 15 hundredths. SAY: I’m drawing a base ten model
now where we’re going to have to do some regrouping. I’m going to draw
the lines and dots in groups of five, to make them easy to count. ASK: How
many ones, tenths, and hundredths have I drawn? (3 ones, 12 tenths, and
15 hundredths)
F-27
ASK: How many tenths can you make from 15 hundredths? (1 tenth)
How many hundredths are left over? (5 hundredths) Circle 10 hundredths
and show them being moved over to be a tenth.
ASK: How many tenths do I have now? (13 tenths) How many ones can
you make from 13 tenths? (1 one) How many tenths are left over? (3 tenths)
Have a volunteer circle 10 tenths and show them being moved over to
be a one.
ASK: How many ones are there now? (4 ones) Write 4 on the board.
How many tenths are there? (3 tenths) Add .3 to the 4, for 4.3. How many
hundredths are there? (5 hundredths) Add a 5 to show 4.35.
ASK: What if I had started by regrouping the tenths instead of the
SAY: Let’s start by regrouping the tenths. ASK: How many tenths are
there? (19) How many ones can you make from 19 tenths? (1 one) How
many ones do we have now? (2 ones) How many tenths are left over?
(9 tenths). Now let’s look at the hundredths. How many hundredths are
there? (10) How many tenths can you make from 10 hundredths? (1 tenth)
Now how many tenths do we have? (10) How many ones can you make
from 10 tenths? (1 one) How many ones do we have now? (3 ones)
Are there any tenths or hundredths left over? (no) What number does this
base ten model represent? (3) How many times did we have to regroup?
(three times) How many times would we have had to regroup if we had
started at the hundredths? (two times) Demonstrate doing this, then
ASK: So, should you start at the left or the right when you’re regrouping?
Why? (at the right, because it saves steps)
F-28
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
hundredths? Would I have got the same answer? Would it have been just
as quick? (yes to both) Then draw the following on the board:
Have students complete Workbook Questions 1–3.
Process Expectation
Connecting (whole number
and decimal procedures)
1
17
+
5
22
Regrouping when adding and subtracting. Write the following sum
on the board:
17
+
5
SAY: Remember how you regroup when you add whole numbers. Here we
have 7 ones plus 5 ones. That’s 12 ones. We regroup 10 of the ones as a
10, like this (demonstrate the regrouping—see margin).
SAY: We also often have to regroup the other way when we subtract. We
1 12
−
22
5
17
have to turn a 10 into ones, or a 100 into tens, to have a big enough number
to subtract from, like this (demonstrate the regrouping—see margin).
Tell students that they will shortly be adding and subtracting decimals,
and will often have to regroup tenths, hundredths, or thousandths.
Have them practise regrouping by completing Workbook Questions 4–9.
For Question 7, emphasize that the decimal point is always immediately
after the ones digit, so a whole number can be assumed to have a decimal
point (for example, 43 = 43. = 43.0).
Extensions
Process assessment
7m7, [C]
1.Have students explain, using an example, why it is more efficient to
regroup from right to left than from left to right.
SAMPLE ANSWER: Suppose you had 2 ones, 29 tenths, and
15 hundredths. If you regrouped the tenths before the hundredths, you
would regroup 20 tenths as 2 ones. You would then regroup 10 ones as
1 tenth. This would give you 10 tenths, which you would regroup to gain
another one. If you had started with the hundredths, you would have
saved a step.
Process assessment
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
7m7, [C]
2.Have students use an example to explain why, when you subtract a
whole number from another whole number, you never need to regroup
more than one unit from a place value into 10 of the next lowest place
value (for example, regroup 1 hundred into 10 tens, not 2 hundreds
into 20 tens) .
SAMPLE ANSWER: The fewest ones you could have to subtract from
is 0, and the most you could need to subtract is 9. If you regroup
one ten as 10 ones, you have enough to subtract 9 from.
Number Sense 7-39
F-29
NS7-40 Addition Strategies for Decimals
Pages 143–144
Curriculum
Expectations
Ontario: 6m20; 7m1,
essential for 7m21
WNCP: 5N11; 7N2, [CN]
Vocabulary
regrouping
Goals
Students will add decimals to thousandths with regrouping.
PRIOR KNOWLEDGE REQUIRED
Can shade hundredths grids to represent decimal tenths and hundredths
Can sketch base ten materials to represent decimal tenths and
hundredths
Can write decimals in place value charts
Can rewrite decimals as tenths, hundredths, or thousandths
Can write decimals as fractions with a common denominator
Can line decimals up at the decimal point to compare decimals
Adding decimals. Tell students that today they will use concepts and skills
developed in previous lessons to add decimals. Draw a place value chart
on the board as shown below, with the decimal points after the tenths:
tens
ones
tenths
hundredths thousandths
SAY: Something is wrong with this place value chart; what is wrong?
(The decimal points are after the tenths instead of after the ones.)
Have a volunteer place the decimal points correctly.
SAY: The decimals are lined up at the decimal point. This makes it easy to
add the same place values together. As long as you line up the numbers at
the decimal points, you can add decimals just like you add whole numbers.
ASK: Should I start by adding the tens or the thousandths? Why? (the
thousandths, because it saves steps to start by adding the place value
at the right)
ASK: What is the greatest digit? (9) When will the sum of digits in a
place value need regrouping? (when the sum is greater than 9) Do the
thousandths need regrouping? Why or why not? Do the hundredths need
regrouping? Why or why not? And so on.
F-30
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Have volunteers suggest two decimals to write in the chart. Add a third
decimal; ensure that at least one place value will need regrouping when
adding all three decimals. Tell students that together you will now add these
three decimals. ASK: Does it matter what order we wrote the numbers in
the place value chart to add them? Would we get a different sum if we put
the numbers in a different order? (no, you can add numbers in any order
(the commutative principle of addition))
Have students complete the Workbook pages. Note that in Question 3 d),
students are asked to skip a step, and write 0.44 directly as thousandths
in fraction form.
Word Problems Practice:
1.Jodi is 1.37 m tall. Jessie is 1.58 m tall. If Jodi stands on a stool
that is 0.26 m tall, will she be taller than Jessie? (yes; 1.37 + 0.26 =
1.63 > 1.58)
2.A squirrel 0.16 m tall sits on a post that is 1.56 m tall. How far is it from
the ground to the top of the squirrel’s head? (1.72 m)
3.The side length of an equilateral triangle is 7.832 cm. What is the
perimeter of the triangle? (23.496 cm)
4.The side length of a square is 6.54 cm. What is the perimeter of the
square? (26.16 cm)
Extension
Add:
a) 0.346 + 0.507 + 0.239 + 0.098
b) 15.557 + 0.57 + 9.234 + 187.39
c) 7.89 + 15.569 + 129.406 + 0.045
d) 9.121 + 1.219 + 2.911 + 1.921
b) 212.751
c) 152.91
d) 15.172
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ANSWERS: a) 1.19
Number Sense 7-40
F-31
NS7-41 Adding and Subtracting Decimals
Pages 145–146
Curriculum
Expectations
Ontario: 6m20; 7m1, 7m2,
7m3, 7m5, essential
for 7m21
WNCP: 5N11; 7N2,
[CN, R, PS]
Vocabulary
standard algorithm
regrouping
Goals
Students will add and subtract decimals to thousandths with
regrouping, using the standard algorithms.
PRIOR KNOWLEDGE REQUIRED
Knows the standard algorithms for addition and subtraction of
whole numbers
Can add and subtract with regrouping
Can regroup decimals
Understands decimal place values
Materials
grid paper
Adding decimals using the standard algorithm. On the board, give a
simple example of adding whole numbers with regrouping:
1
39
+ 85
124
Remind students of how you work from right to left, and how you
mark regrouped values above the appropriate place value. Then show
the following:
Connecting
12.4
ASK: What is the same about this sum and the one before? (The numbers
have the same digits.) What is different? (The numbers are decimals; these
are ones and tenths, the others were tens and ones.) Tell students that,
so long as they are careful to line up the decimal points, they can add
decimals just as they would add whole numbers. Show students another
sum, this time incorrect:
20.6
+ 8.43
10.49
ASK: Is my answer correct? (no) PROMPTS: The first number is more than
20—can the answer be about 10? Why not? (you can’t add a number to
20 and get something close to 10) What did I do wrong? (didn’t line up the
F-32
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Process Expectation
1
3.9
+ 8.5
decimal points) Have a volunteer correct your mistake and calculate
the sum (29.03).
Suggest that adding on grid paper makes it easy to keep straight which
place value each digit has. Give students grid paper. Have students
complete Workbook Questions 1–5.
Subtracting decimals using the standard algorithm. On the board,
give a simple example of subtracting whole numbers with regrouping:
2 13
33
− 17
16
Remind students of how you work from right to left, and how you mark
regrouped values above the appropriate place value. Then have a volunteer
show how to calculate 3.3 − 1.7.
Process Expectation
Revisiting conjectures that
were true in one context
Recall how, when subtracting with whole numbers, the number with more
digits always went on top. Explain that that’s because if you subtract
a number from a smaller number, you will get a negative number. Tell
students that they will learn about some negative numbers later this year.
Tell students that when you subtract decimals, you still want to take the
number that is smaller away from the one that is larger. Write two numbers
on the board; for EXAMPLE:
4.58
3.62
ASK: Which of these two numbers is larger? (4.58) Which will you subtract
from which? (3.62 from 4.58) Write two more numbers on the board, with
the smaller number having more digits; for EXAMPLE:
2.123
3.2
ASK: Which of these two numbers is larger? (3.2) Which will you subtract
from which? (2.123 from 3.2) With decimals, is the number with the most
digits always the larger number? (no)
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Tell students that when the number with fewer digits is the larger number,
you should add a zero or zeros before you subtract, to keep things from
getting confusing:
3.200
− 2.123
Remind students that zero is a placeholder. It holds the place for digits that
are not there, and that you need for subtraction. Have a volunteer calculate
the difference:
9 10
1 10
1 10
3.200
3.200
− 2 . 1 2 3
−2.123
Number Sense 7-41
F-33
Have the students subtract more decimals. EXAMPLES:
a) 6.5 − 4.32
Bonus
b) 0.75 − 0.608
c) 2.89 − 1.234
2.825 6 − 1.030 37 ANSWER: 1.795 23
Always encourage students to check their answers with addition.
Process Expectation
Word Problems Practice
Reflecting on the
reasonableness of
an answer
1.The length of a hockey stick is 0.889 m. What is the length of three
of these hockey sticks? (2.667 m)
Process assessment
7m1, [PS]
2.Leesa ran 23.1 km and Pat ran 15.7 km. How much farther did
Leesa run? (7.4 km)
3.Karin is 1.54 m tall and Gary is 1.46 m tall. How many centimetres
taller is Karin than Gary? (8 cm)
4.The official dimensions for an NHL ice hockey rink are 25.91 m by
60.96 m. Pretend the rink is a rectangle. What is the perimeter of an
official rink? (173.74 m)
5.The perimeter of a rectangle is 31.2 cm. One side length is 12.6 cm.
What is the other side length? HINT: Draw a sketch to help you solve
the problem. (3 cm)
6.Tariq hikes 15.32 km west, and camps for the night. The next day he
hikes 4.8 km north to a scenic lookout, and then he hikes 16.05 km
back to where he started.
a) How far did he hike? (36.17 km)
b) How much farther did he hike on the second day than on the
first day? (5.53 km)
Extensions
1. Add:
a) 0.234 78 + 96.35
c) 234.087 29 + 0.5408 + 31.6754
e) 0.876 004 678 + 0.345 800 502
b) 34.007 008 + 89.005 006
d) 0.000 053 42 + 0.564 98
ANSWERS: a) 96.584 78
d) 0.565 033 42
b) 123.012 014
e) 1.221 805 180
c) 266.303 49
2. Subtract:
a) 96.35 − 0.234 78
c) 234.087 29 − 31.6754
e) 0.876 004 678 − 0.345 800 502
ANSWERS: a) 96.115 22
d) 0.564 926 58
F-34
b) 89.005 006 − 34.007 008
d) 0.564 98 − 0.000 053 42
b) 54.997 998
e) 0.530 204 176
c) 202.411 89
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
NS7-42 Rounding Review and
NS7-43 Rounding Decimals
Pages 147–149
Goals
Curriculum
Expectations
Students will round decimals to the nearest one, tenth, hundredth,
or thousandth. Students will round to the nearest whole number to
estimate sums, products, and quotients.
Ontario: 7m3, 7m4, 7m5,
7m7, essential for 7m22
WNCP: 5N11; essential for
7N2, [C, CN, ME, T]
PRIOR KNOWLEDGE REQUIRED
Understands the place value system
Understands the concept of “closer to”
Vocabulary
rounding
approximately equal (≈)
This lesson reviews rounding and prepares students for estimation of sums,
differences, products, and quotients.
The concept of rounding. Have a student measure the chalkboard or
some other object in the room with a metre stick (select something that is
not an exact whole number of metres in length). ASK: How many metres
and how many centimeters long is it? Is that closer to (for example) 2 m or
3 m? SAY: We can decide not to count the extra centimetres and just say
that the length is about 2 m. What we have done is rounded the length to
the nearest metre.
Process Expectation
Mental math and estimation
connection
Real World
Rounding whole numbers. Write on the board the population of your city
or the nearest city, rounded to the nearest thousand. The dialogue below
uses the population of Hamilton, Ontario: 650 000 people.
SAY: This was the population of Hamilton in 2006. Does this mean that
there were exactly 650 000 people living in Hamilton in 2006? (No, the
actual population has been rounded.) ASK: Why is it important to round
the population? Why not give the exact number? (because the population
changes very rapidly with births, deaths, immigrants, and emigrants)
What is the smallest place value in 650 000 that has a digit other than zero?
(the ten thousands place) Underline the 5 in 650 000:
650 000 people
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
SAY: The population was rounded to the nearest ten thousand. Write on
the board:
647 634 people
SAY: This was the actual population of Hamilton at the time in 2006 that the
population was counted. This is how it was rounded. First, underline the
digit in the place value you want to round to:
647 634 people
Then, look at the digit to the right of the underlined digit. If it is 5 or
more, round up. Here, it is 7, so we round up by adding 1 to the digit we
underlined and we make all the digits after that one zero:
Number Sense 7-42, 43
F-35
650 000 people
Suppose we had wanted to round the population to the nearest hundred:
643 634 people
ASK: Is the digit to the right of the hundreds less than 5? (yes) When the
digit to the right of the place you are rounding to is less than 5, you round
down by just making all the digits after the place zero:
643 600 people
Tell students that another city has population 649 632. ASK: How would you
round this to the nearest thousand?
649 632
Have a volunteer show that you would round the 9 thousands up to
10 thousands, which you would then regroup as 1 ten thousand:
650 000
Have students complete Workbook p. 147 Questions 1–3. Remind students
that when rounding you only need to look at the digit to the right of the
place value you are rounding to—any digits farther right should be ignored.
Process Expectation
Connecting
Rounding decimals. Tell students that you use the same rule to round
decimals as you do to round whole numbers. Write on the board:
0.635 m
SAY: Here we have 635 thousandths of a metre. ASK: What is a thousandth
of a metre? (1 mm) What is a hundredth of a metre? (1 cm)
SAY: Suppose we just wanted to write this measurement to the nearest
centimetre. We would round it to the nearest hundredth, because a
centimetre is a hundredth of a metre. Have a volunteer round 0.635 to the
nearest hundredth and explain how he or she knew whether to round up
or down:
0.64 m
write the measurement to the nearest decimetre. We would round it to the
nearest tenth, because a decimetre is a tenth of a metre. Have a volunteer
round 0.635 decimal to the nearest tenth and explain how he or she knew
whether to round up or down:
0.6 m
Have students complete Workbook pp. 148–149, Questions 1–9.
Process Expectation
Mental math and estimation,
Technology
Using rounding to estimate when solving problems. Tell students that
rounding is a useful way to estimate sums and differences, and products
and quotients. Write on the board:
F-36
25.64 + 20.09 ≈ ____
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ASK: What is a tenth of a metre? (1 dm) SAY: Suppose we just wanted to
Tell students that the wavy equal sign means “is approximately equal to.”
Have a volunteer underline the whole number part of each addend, and add
the underlined parts (25 + 20) to estimate the sum:
25.64 + 20.09 ≈ 45
Have students calculate the actual sum using calculators (45.73). Be sure
students understand how to enter a decimal point on a calculator. ASK:
Is the estimate close to the sum?
Write a product on the board, but with the decimal in the wrong place
in the answer:
Process Expectation
Reflecting on the
reasonableness of an answer
3.14 × 20.5 = 6.437
SAY: Let’s estimate to check if this answer is correct. Rewrite the product
as shown below, underlining the whole-number part of each factor:
3.18 × 20.4 ≈ ____
Process assessment
Have a volunteer estimate the product by using just the whole-number
part of each factor (3 × 20 = 60). ASK: Is 6.437 the correct answer? (no)
Have students calculate the actual product using calculators (64.37)
ASK: What mistake was made in the wrong answer? (The decimal point
is in the wrong place.)
7m3, 7m4, [ME, T]
Workbook p. 149 Question 11
Tell students that it is easy to make mistakes in calculations and that it is a
good idea to always estimate to check the ANSWERS of their calculations.
Have students complete Workbook p. 149 Questions 10–12.
Extension
Process assessment
7m2, [R, C]
Decide what place value it makes sense to round each of the following
to. Round to the place value you selected. Justify your decisions.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Height of person: 1.524 m
Height of tree: 13.1064 m
Length of bug: 1.267 cm
Distance between Toronto and Hong Kong: 12 545.99 km
Distance between Earth and the Moon: 384 403 km
Population of Kolkata, India, in 2010: 5 138 208 people
Floor area of an apartment: 27.91 square metres
Area of Ontario: 1 076 395 km2
Angle between two streets: 82.469°
Time it takes to blink: 0.33 s
Speed of a car: 96.560 639 km/h
ANSWERS: Answers will vary. The larger the number, the less important
the smaller place values become. The use to which the measurement will
be put is also a factor. For example, if measuring the time it takes to ski
a downhill course, more accuracy might be needed to determine a world
record than to keep training records.
Number Sense 7-42, 43
F-37
NS7-44 Estimating Decimal Sums and Differences
Pages 150–151
Goals
Curriculum
Expectations
Students will estimate sums and differences in a variety of ways and
evaluate the accuracy of their estimates. They will use estimation to
help them solve problems and check solutions.
Ontario: 6m20, 6m24; 7m1,
7m4, 7m5, 7m22
WNCP: 5N11; 7N2,
[CN, ME, PS, T]
PRIOR KNOWLEDGE REQUIRED
Can round to the nearest tenth, hundredth, or thousandth
Can estimate by rounding
Vocabulary
estimating
rounding
approximately equal to (≈)
Estimation. ASK: If you wanted to quickly count how much money you
had, would you count the bills or would you count the coins? (the bills)
Why? (The bills are worth much more than the coins.) Write 0.265 on the
board. ASK: Which digit has the most effect on the value of this number?
(the 2) Why? (The 2 tells how many tenths there are, while the 6 only tells
how many hundredths, and the 5 how many thousandths.) SAY: If you want
to make a quick estimate of a calculation, you want to focus on the digits
on the left. You may want to use the first digit or the first two digits—it all
depends how accurate you want to be.
Present the following estimation problem to students, making the sketches
shown below as you go:
SAY: Quarter round is a moulding that is put on the floor, below the
baseboards, around the outside of a room. It is shaped like a very thin log
cut in four lengthwise. Suppose I want to put quarter round around a room.
The length of the room is 3.62 m. The width of the room is 2.43 m.
3.62 m
The quarter round can be bought in 3 m lengths that each cost $8.00.
3 m for $8
Process Expectation
Mental math and estimation
Suppose I round to the ones to estimate the length of the perimeter
of the room:
4 m + 4 m + 2 m + 2 m = 12 m
It would be convenient to buy 12 m of quarter round because that’s exactly
four of the 3 m pieces:
F-38
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
2.43 m
12 m = 3 m × 4
so cost = $8 × 4 = $32
But will 12 m be enough? Let’s calculate the exact amount needed:
3.62 m + 3.62 m + 2.43 m + 2.43 m = 12.1 m
We would be short a tenth of a metre, or 10 cm, if we bought only 12 m
of the quarter round. But if you estimated 12 m and then bought five of the
3 m lengths of quarter round, you would be sure to have enough. It might
be worth spending an extra $8 to be on the safe side. Or perhaps you aren’t
worried that your estimate might be a little low (if there is a doorway that
won’t need quarter round, for example), so you might just buy four of the
3 m lengths of wood.
SAY: The important thing to keep in mind when you do estimates is that you
need a quick and easy answer that is as close as possible to the correct
answer. For different purposes, you might need a more or less accurate
estimate. It’s good to calculate the exact answer after you do an estimate to
help you think about how accurate your estimate was and what you could
have done to get a closer estimate (if you need a closer estimate).
Process Expectation
Technology, Mental math
and estimation
Process Expectation
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Problem solving, Selecting
tools and strategies
Tell students that now they will try a few different ways to estimate and
then they will investigate which way is the most accurate. Have students
complete Workbook Questions 1–5. When students have completed
Question 5, have them make up more addition questions and try estimating
using the three different methods—use whole number parts of the decimal
(Question 1), round to the nearest tenth (Question 2), use the digits of the
two highest place values (Question 3). Have students think about when
they would use which method, then discuss students’ answers as a class.
Have students apply their estimation skills to some problems. Work through
the following problem as a class:
Andy had $755.50 and spent $326.87. Which of the methods above
would you use to estimate how much money he has left? Explain
your choice.
Have students complete Workbook Questions 6–12 in their notebooks.
Suggest that students ask themselves, for each problem: Do I need to add
or subtract? Can I do this problem in one step, or will it take two steps?
What strategy will I use to do the addition or the subtraction? Would it help
me to make a sketch of the problem? Tell students to estimate the solution,
then calculate using a calculator, and then use their estimate to check
their solution.
Process assessment
7m4, 7m5, [CN]
Workbook Question 12
Number Sense 7-44
Extra Practice: Create your own decimal addition and subtraction
problems, then trade with a partner and solve the problems.
F-39
NS7-45 Multiplying Decimals by 10 and
NS7-46 Multiplying Decimals by 100 and 1000
Pages 152–153
Goals
Curriculum
Expectations
Students will multiply decimals by 10, 100, and 1000.
Ontario: 6m23; 7m1, 7m2,
7m3, 7m4, 7m5, 7m6,
essential for 7m19,
essential for 7m20
WNCP: 6N8; 7N2,
[CN, R, ME, PS, V]
PRIOR KNOWLEDGE REQUIRED
Knows the commutative property of multiplication
Understands multiplication as repeated addition
Can multiply whole numbers by 10, 100, and 1000
Understands decimal place value
Can create base ten models for decimal tenths and hundredths
Multiplying whole numbers by 10. Write on the board:
2 × 10 = 10 + 10 = 20
3 × 10 = 10 + 10 + 10 = 30
4 × 10 = 10 + 10 + 10 + 10 = 40
Point out that we are adding one more 10 each time, and have students
fill in the blanks for these problems:
9 × 10 = ________
10 × 10 = ________
11 × 10 = ________
Have students predict each of these:
243 × 10 = ________
204 390 × 10 = ______________
ASK: How do you multiply a whole number by 10? Students might say “add
a zero.” In this case, force them to be more articulate by writing an incorrect
statement, such as 34 × 10 = 304. (The zero has to be added at the end, so
that the ones digit becomes the tens digit and zero becomes the ones digit)
Discuss how this makes sense:
34 = 3 tens + 4 ones,
so 34 × 10 = 3 hundreds + 4 tens = 340.
Multiplying decimal numbers by 10. Write on the board: 0.4 × 10.
0.4 times, but we can add 0.4 ten times. Remind students that multiplication
is commutative:
2 × 10 = 10 × 2
10 + 10 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2
We can do the same thing with decimals:
F-40
0.4 × 10 = 10 × 0.4
= 0.4 + 0.4 + 0.4 + 0.4 + 0.4 + 0.4 + 0.4 + 0.4 + 0.4 + 0.4
= 4 tenths + 4 tenths + … + 4 tenths (with ten 4s)
= 10 × 4 tenths = 40 tenths = 4 ones = 4
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ASK: How can we multiply 10 by a decimal number? We can’t add ten
Since students know how to add decimals, and multiplying a decimal
with a whole number can be looked at as repeated addition of the decimal
number, students can then multiply a decimal with a whole number.
In particular, they can multiply the decimal number by 10. Now have
students do more examples of multiplying a decimal number by 10:
a) 0.5 × 10
b) 0.2 × 10
c) 0.6 × 10
d) 0.9 × 10
Connect the two processes for multiplying whole numbers and
decimals by 10. Write the answers to the questions above on the board:
0.5 × 10 = 5
0.2 × 10 = 2
0.6 × 10 = 6
0.9 × 10 = 9
Remind students that when multiplying whole numbers by 10, we just
added a 0 at the end of the number. ASK: Why can’t we do that to decimal
numbers? (because 0.50 is the same as 0.5—adding 0 to the end of the
number doesn’t change the value at all)
Tell students that you are still just changing the place value of each digit
though, just as you did with whole numbers: If the 5 is worth tenths, it
becomes worth ones (which is ten times more than tenths). Essentially,
you are moving the decimal point one place to the right.
0.5 × 10 = 5.0
Process Expectation
Connecting
Visualizing
0.6 × 10 = 6.0
0.9 × 10 = 9.0
When you multiply whole numbers by 10, you are also moving the decimal
point one place to the right:
35 × 10 = 350
Process Expectation
0.2 × 10 = 2.0
or
35.0 × 10 = 350.
SAY: You can multiply any number by 10 by moving the decimal point one
place to the right. The number can be a whole number or a decimal. Write
on the board: 0.1 × 10. SAY: We could calculate this by visualizing adding
up 10 tenths blocks. Imagine 10 tenths blocks. How much do they make
altogether? (1 whole) What is 10 tenths? (1) Write: 0.1 × 10 = 1. SAY:
Suppose we had three tenths that we were multiplying by 10. If one-tenth
times 10 is 1, what is three-tenths times 10? (3)
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
× 10 =
0.3
× 10
=3
SAY: We could also write the three with a zero after the decimal point:
0.3 × 10 = 3.0. ASK: In which direction does the decimal point move?
(to the right) How many places does it move (one place) Have students
practise multiplying a few more decimals by 10 by moving the decimal point
one place to the right:
Number Sense 7-45, 46
a) 0.2 × 10
b) 0.58 × 10
c) 0.216 × 10
d) 7.46 × 10
F-41
Remind students that you can multiply in any order (commutative principle
for multiplication). ASK: Does it matter what order I multiply in? (no) Will I
get the same answer for both of these products? (yes)
Process Expectation
Mental math
0.2 × 10 = ____
10 × 0.2 = ____
Have students practice multiplying a few more decimals by 10, this time
with 10 as the first factor:
a) 10 × 7.04
b) 10 × 36.5
c) 10 × 0.062
d) 10 × 12.04
Have students complete Workbook p. 152, Questions 1–2 and 4.
Process Expectation
Reflecting on other ways to
solve a problem
Another way to understand multiplication by 10 and shifting the
decimal point. Write the number 2.35 on the board. Together as a class,
fill in the blanks with the correct place values:
2.35 = 2 ________ + 3 ________ + 5 _________ (ones, tenths, hundredths)
10 × 2.35 = 2 _______ + 3 ________ + 5 ________ (tens, ones, tenths)
Emphasize that 2 ones × 10 = 2 tens, 3 tenths × 10 = 3 ones, and
5 hundredths × 10 = 5 tenths; this is because each place value is worth
10 times more than the place to the right. Now we can read the answer:
10 × 2.35 = 23.5.
Process Expectation
Visualizing, Organizing data
Show the place values in a chart to emphasize how moving each digit left
and keeping the decimal point in the same position puts the decimal point
one place to the right relative to the digits:
tens
2
ones
tenths
hundredths
2
3
5
3
5
Have students individually do more questions:
a) 32.41 = 3 ______ + 2 ______ + 4 ______ + 1 ______
So 10 × 32.41 = 3 ______ + 2 ______ + 4 ______ + 1 ______ = ______
b) 74.5 = 7 ______ + 4 ______ + 5 ______
So 10 × 74.5 = 7 ______ + 4 ______ + 5 ______ = ______
c) 800.304 = 8 ______ + 3 ______ + 4 ______
connection
Measurement
F-42
So 10 × 800.304 = 8 ______ + 3 ______ + 4 ______ = ______
10 is a measurement conversion factor for some units. ASK: How many
decimetres are there in a metre? (10) Write on the board:
1 m = 10 dm
2 m = ____ dm
3 m = ____ dm
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ASK: If I want to know the number of decimetres from the number of
metres, what do I multiply the number of metres by? (10) Have students
change 5 m into decimetres and 0.3 m into decimetres.
ASK: How many millimetres are there in a centimetre? (10) Write on
the board: 1 cm = ____ mm. ASK: If I want to change centimetres into
millimetres, what do I multiply by? (10) Have students change 4 cm into
millimetres and 0.7 cm into millimetres. Have students complete Workbook
p. 152, Questions 3 and 5.
Multiplying by 100 and 1000. Write on the board:
3 × 100 = 300
3.0 × 100 = 300.0
ASK: In which direction does the decimal point move when you multiply
3 by 100? (to the right) How many places does it move (two places)
Process Expectation
Visualizing
SAY: Imagine a single hundredths block. Now imagine 100 hundredth
blocks. Write 100/100 on the board. How much is 100 hundredths? (1)
SAY: So, when you multiply a hundredth by 100, you get 1:
0.01 × 100 = 1.0
ASK: In which direction did the decimal point move when you multiplied the
hundredth by 100? (to the right) How many places did it move? (two)
Remind students that a tenth multiplied by 10 is 1.
× 10
0.1 × 10 = 1.0
SAY: A hundred is 10 times 10, so we can multiply 1.0 by 10 to find a
tenth times 100:
Process Expectation
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Splitting into simpler
problems
0.1 × 100 = 0.1 × (10 × 10)
= (0.1 × 10) × 10
= 1.0 × 10 = _____
× 10 =
Process Expectation
Generalizing from examples
Number Sense 7-45, 46
ASK: How much is a tenth times 100? (10) How did the decimal point
move? (two places to the right) What is the rule for moving the decimal
point to multiply by 100? (move it two places to the right) Have students
practise multiplying a few decimals by 100 by moving the decimal point two
places to the right:
F-43
a) 3.62 × 100
b) 0.725 × 100
c) 1.673 × 100
d) 0.085 × 100
Tell students that sometimes you have to add some zeros when you
multiply a decimal by 10 or 100. Show an example:
Process Expectation
Mental math and estimation
3.6 × 100 = 3.60 × 100 = 360
Have students practise multiplying a few decimals by 100 where they have
to add a zero or zeros:
a) 0.4 × 100
Process Expectation
Looking for a pattern
b) 9.8 × 100
c) 23.6 × 100
SAY: The decimal point moves one place to the right when you multiply by
10, and two places to the right when you multiply by 100. How many places
do you think the decimal point moves when you multiply by 1000? (3)
Have students practise multiplying a few decimals by 1000 by moving the
decimal point three places to the right, adding zeros where necessary:
a) 0.462 × 1000
b) 11.24 × 1000
c) 1.9 × 1000
d) 0.14 × 1000
Process assessment
Word Problems Practice
7m1, [PS]
1.A necklace has 100 beads. Each bead has a diameter of 1.32 mm.
How long is the necklace? (132 mm)
2.A regular decagon has side length 7.45 cm. What is the perimeter of
the decagon? (74.5 cm)
3.A Canadian banknote is 0.153 m long. How long are 1000 twenty-dollar
bills set end to end? (153 m)
4.A clothing-store owner wants to buy 100 coats for $32.69 each.
How much will the coats cost? ($3269.00)
5.There are 1000 m in a kilometre. How many metres are there in
30.75 km? (30 750 m)
a)How long are 100 cats and 100 guinea pigs standing in a row?
(76 m)
b)How much longer is a row of 100 cats than a row of 100 guinea
pigs? (16 m)
7.One marble weighs 3.5 g. The marble bag weights 10.6 g. How much
does the bag weigh with 100 marbles in it? (360.6 g)
F-44
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
6.The average length of a cat is 0.46 m. The average length of a
guinea pig is 0.3 m.
NS7-47 Dividing Decimals by 10 and 100 and
NS7-48 Multiplying and Dividing by Powers of 10
Pages 154–155
Goals
Curriculum
Expectations
Ontario: 6m23; 7m1, 7m2,
7m4, 7m6, essential
for 7m19, essential
for 7m20
WNCP: 6N8; 7N2,
[R, ME, PS, V]
Students will multiply and divide decimals by any power of 10 by
shifting the decimal point.
PRIOR KNOWLEDGE REQUIRED
Division
Can multiply and divide whole numbers by powers of ten
Can multiply decimals by 10, 100, and 1000
Understands decimal place value
Can create base ten model for decimal tenths and hundredths
Dividing by 10. Write on the board: 50 ÷ 10 = ____. ASK: What is
50 divided by 10? (5) How do you know? (10 × 5 = 50, so 50 ÷ 10 = 5;
I shifted the decimal left; you can make 10 equal groups of 5 out of 50)
SAY: We can write 50 and 5 as decimals, like this: 50.0 ÷ 10 = 5.0. ASK:
In which direction did the decimal point move when we divided 50 by 10?
(to the left) Is this the same direction the decimal point moves when we
multiply by 10? (no, it’s the opposite direction) How many places did the
decimal point move when we divided 50 by 10? (one place) Is this the same
number of places the decimal point moves when we multiply by 10? (yes)
Process Expectation
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Visualizing, Modelling
÷ 10 =
1.0 ÷ 10 = 0.1
÷ 10 =
0.1 ÷ 10 = 0.01
Number Sense 7-47, 48
SAY: Division undoes multiplication. When you multiply by 10, you move
the decimal point to the right one place. To undo multiplying by 10, you
move the decimal point back one place in the opposite direction. So, to
divide by 10 you move the decimal point one place to the left. You can
divide any number by 10 by moving the decimal point one place to the left.
The number can be a whole number or a decimal. Write on the board:
1.0 ÷ 10 and label the base tens blocks on the board, with a flat being the
ones block, a rod the tenths block and a cube the hundredths block.
SAY: Visualize dividing a ones block into 10 equal parts. Imagine a ones
block. If you divide a ones block into 10 equal parts, how big is each part?
(1 tenth) Draw this on the board, as shown in the margin.
SAY: Visualize dividing a tenths block into 10 equal parts. Imagine a tenths
block. If you divide a tenths block into 10 equal parts, how big is each part?
(1 hundredth) Draw this on the board, as shown in the margin.
ASK: How did the decimal point move when we divided 1 by 10? (one place
to the left) How did the decimal point move when we divided a tenth by 10?
(one place to the left)
Dividing by 100. Point out that 1 ÷ 10 = 0.1 and that 0.1 ÷ 10 = 0.01.
ASK: What is 1 ÷ 100? (0.01) How do you know? (because to divide by
100, you can just divide by 10, and then divide by 10 again) Show this on
the board as follows:
F-45
1 ÷ 100 = 1 ÷ (10 × 10)
= 1 ÷ 10 ÷ 10
= 0.1 ÷ 10
= 0.01
Point out that when dividing by 10, we move the decimal point one place
left, and when we divide by 10 again, we move the decimal point another
place left, so to divide by 100, we move the decimal point two places left.
Mental math and estimation
Have students calculate 4.0 ÷ 10 by shifting the decimal point (4.0 ÷ 10 =
0.4). Then have students calculate 0.4 ÷ 10 by shifting the decimal point
(0.4 ÷ 10 = 0.04). ASK: What is 4 ÷ 100? (0.04) How did you move the
decimal point? (two places left)
Process Expectation
ASK: How would you shift the decimal point to divide by 1000? (3 places
Process Expectation
Looking for a pattern
to the left)
Bonus
To divide by 10 000 000? (7 places to the left)
Another way to understand division by 10 and shifting the decimal
point. Write the number 432.15 on the board. Together as a class, fill in
the blanks with the correct place values:
432.15 =
4 ______ + 3 ______ + 2 ______ + 1 ______ + 5 ______
(hundreds, tens, ones, tenths, hundredths)
432.15 ÷ 10 =
4 ______ + 3 ______ + 2 ______ + 1 ______ + 5 ______ = ______
(tens, ones, tenths, hundredths, thousandths, 43.215)
Notice that by making each digit worth ten times less, we are moving the
decimal point one place to the left.
Process Expectation
Mental math
• I think of a simple whole-number example and see what happens
with that.
• When you multiply by 10, 100, 1000, … , the result is a bigger
number, and if you move the decimal point to the right, the number
gets bigger. When you divide by 10, 100, 1000, … , the result is a
smaller number than you started with, and that’s what happens when
you move the decimal point to the left.
Give students grid paper and have them practise multiplying and dividing
by powers of 10 by completing Workbook p. 155.
Bonus
F-46
Find 31 498.765 32 ÷ 1 000 000 ANSWER: 0.031 498 765 32
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Strategies for remembering which way to move the decimal point.
Ask students how they will remember which way to move the decimal point
when they multiply and when they divide by powers of 10, such as 10, 100,
1000. Some possible responses are:
Process assessment
Word Problems Practice
7m1, [PS]
1.In 10 months, a charity has raised $26 575.80 through fundraising.
How much did they raise each month on average? ($2657.58)
2.A stack of 100 cardboard sheets is 13 cm high. How thick is a sheet of
the cardboard? (0.13 cm)
3.A thousand people attended a “pay what you can” event. The total
money paid was $5750. How much did each person pay, on average?
($5.75)
4.A hundred walruses weigh 121.5 tonnes (1 tonne = 1000 kg).
How much does one walrus weigh on average, in kilograms? (1 215 kg)
5. A box of 1000 nails cost $12.95.
a) How much did each nail cost, to the nearest cent? (1¢)
b)A hundred of the nails have been used. What is the cost for the nails
that are left, to the nearest cent? NOTE: Use the actual cost of a nail
in your calculations, not the rounded cost from part a). ($11.66,
not $11.95)
6. Ten large vegetarian pizzas cost $248.50 altogether.
a) How much does each pizza cost? ($24.85)
b) How much would 1000 large vegetarian pizzas cost? ($24 850)
Extensions
1.A Canadian penny has a diameter of 19.05 mm. How long would a line
of 10 000 Canadian pennies laid end-to-end be, in mm, cm, m, and km?
ANSWER: 190500 mm, 19050 cm, 190.5 m, 0.1905 km
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
2. a) Ten of an object laid end-to-end have a length of 48 cm. How long
is the object? What might the object be? Think about this equation:
× 10 = 48 cm.
b) 100 of an object laid end-to-end have a length of 2.38 m. How long
is the object, in centimetres? What might the object be?
c) 1000 of an object laid end-to-end have a length of 274 m. How long
is the object, in centimetres? What might the object be?
ANSWERS: Answers may vary; for example:
a) 4.8 cm; an eraser b) 2.38 cm; a quarter (coin)
c) 27.4 cm; a shoe
3.Create your own problems that require multiplying and/or dividing
decimals by powers of 10. Then, trade with a partner and solve
the problems.
4.Find the mass of 1 bean by weighing 100 or 1000 beans. Use a
calculator to determine how many beans are in a 2000 g package.
Number Sense 7-47, 48
F-47
NS7-49 Multiplying Decimals by Whole Numbers
NS7-50 Multiplying Decimals Using
Different Strategies
Pages 156–157
Curriculum
Expectations
Ontario: 6m21; 7m1, 7m4,
7m5, 7m19, 7m20
WNCP: 6N8; 7N2,
[ME, CN, PS]
Goals
Students will discover the connection between multiplying decimals
by whole numbers and multiplying whole numbers by whole numbers.
Students will use a variety of mental strategies to solve problems
involving the multiplication of decimals by whole numbers.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
distributive law
Process Expectation
Modelling
Understands decimal place value
Can multiply 3- and 4-digit numbers by 1-digit numbers
Knows the distributive law
Multiplying decimals without regrouping. Using the hundreds block as
1 whole, have volunteers show:
a) 1.23 and then 2 × 1.23
c) 3.12 and then 3 × 3.12
b) 4.01 and then 2 × 4.01
a)
4.12 =
2 × 4.12 =
ones +
ones +
tenths +
tenths +
hundredths
hundredths =
b)
3.11 =
3 × 3.11 =
ones +
ones +
tenths +
tenths +
hundredths
hundredths =
c)
1.02 =
4 × 1.02 =
ones +
ones +
tenths +
tenths +
hundredths
hundredths =
Process Expectation
Have students multiply mentally:
Mental Math and estimation
a) 4 × 2.01 Bonus
b) 3 × 2.31 c) 3 × 1.1213
d) 2 × 1.114312
e) 3 × 1.1212231
Multiplying decimals with regrouping. Have students solve the following
problems by regrouping when necessary:
F-48
a) 3 × 4.42=
=
=
ones +
ones +
tenths +
tenths +
hundredths
hundredths
b) 4 × 3.32 =
=
=
ones +
ones +
tenths +
tenths +
hundredths
hundredths
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Have students individually solve the following problems by multiplying
each digit separately:
c) 3 × 3.45 =
=
=
=
ones +
ones +
ones +
tenths +
tenths +
tenths +
hundredths
hundredths
hundredths
Then have students solve the following problems by regrouping
when necessary:
Process Expectation
Connecting
a) 3 × 442 =
=
=
hundreds +
hundreds +
tens +
tens +
ones
ones
b) 4 × 332 =
=
=
hundreds +
hundreds +
tens +
tens +
ones
ones
c) 3 × 345 =
=
=
hundreds +
hundreds +
tens +
tens +
ones
ones
Discuss with students the similarities and differences between these
problems and solutions. Remind students about the standard algorithm
for multiplying 3-digit by 1-digit numbers and ask students if they think
they can use the standard algorithm for multiplying decimal numbers.
Emphasize that none of the digits in the answer changes when the question
has a decimal point; only the place value of the digits changes. The key to
multiplying decimals, then, is to pretend the decimal point isn’t there, and
then add it back in at the end. The tricky part is deciding where to put the
decimal point at the end.
Demonstrate using 442 × 3 and 4.42 × 3:
1
1
442 4.42
×
3
×
3
1326 1326
Process Expectation
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Mental Math and estimation
Tell students that now you need to know where to put the decimal point.
Will the answer be closer to 1 or 13 or 132 or 1326? ASK: How many whole
ones are in 4.42? How many ones are in 3? About how many ones should
be in the answer? (4 × 3 = 12) What is closest to 12: 1, 13, 132, or 1326?
Have a volunteer guess where the decimal point should go and ask the
class to explain why the volunteer chose the answer or to agree or disagree
with the choice.
Repeat with several problems. (EXAMPLES: 3.35 × 6, 41.31 × 2,
523.4 × 5, 9.801 × 3)
Bonus
834 779.68 × 2, 5 480.63 × 7
Have students complete Workbook p. 156.
Number Sense 7-49, 50
F-49
More decimal multiplication strategies. Tell students there are many
different strategies they can use to multiply a decimal by a whole number.
Tell them the next worksheet introduces a couple more.
Process Expectation
Mental Math and estimation
Recall that you can break a multiplication problem into a simpler problem
by multiplying in parts (the distributive law). Show students an example with
whole numbers; for example:
5 × 4 = (3 + 2) × 4 = (3 × 4) + (2 × 4) = 12 + 8 = 20
Tell students that you can use this method to multiply one place value
at a time. Work through a couple of examples, first using whole numbers,
then decimals:
15 × 3 = (10 + 5) × 3 = (10 × 3) + (5 × 3) = 30 + 15 = 45
1.5 × 3 = (1.0 + 0.5) × 3 = (1.0 × 3) + (0.5 × 3) = 3.0 + 1.5 = 4.5
54 × 3 = (50 + 4) × 3 = (50 × 3) + (4 × 3) = 150 + 12 = 162
0.54 × 3 = (0.5 × 3) + (0.04 × 3) = 1.5 + 0.12 = 1.62
Tell students that you don’t have to separate out each place value—you
could pick some other way to break up the decimal that looks like it will
make things easier; for example:
2.31 × 4 = (2.2 + 0.11) × 4 = (2.2 × 4) + (0.11 × 4) = 8.8 + 0.44 = 9.24
1.29 × 5 = (1.2 + 0.09) × 5 = (1.2 × 5) + (0.09 × 5) = 6.0 + 0.45 = 6.45
Have students complete Workbook p. 157. Encourage them to use
rounding to estimate and check their solutions. Note that estimates
involving multiplication often tend to be farther from the actual values
than estimates with addition and subtraction because of the nature of the
operations. A possible strategy to reduce this effect is to round one factor
up and the other factor down.
Word Problems Practice
1.One can holds 0.367 L. How much do five of the same can hold?
(1.835 L)
a) How far is it to school and back? (6.96 km)
b)Matt walks to school and back each school day. How far does he
walk in five days? (34.8 km)
(Note that part b) can be done in two ways—multiply the answer from
a) by 5, or multiply the given distance by 10)
NOTE: Before students do Questions 3–5 below, remind them of the
words pentagon (5-sided polygon), hexagon (6 sides), and octagon
(8 sides). Also point out that in a regular polygon, all sides have the
same length. (It is also true, but not relevant for these questions, that
all angles in a regular polygon have the same size.)
3.One side of a regular pentagon is 0.835 m. What is the perimeter?
(4.175 m)
F-50
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
2. The distance from Matt’s home to his school is 3.48 km.
4.One side of a regular hexagon is 14.5 cm. What is the perimeter?
(87.0 cm or 87 cm)
5.One side of a regular octagon is 2.68 m. What is the perimeter?
(21.44 m)
6.A rectangle has side lengths of 3 m and 0.96 m. What is the area
of the rectangle? (2.88 m2)
Extension
each students that just as they can take fractions of whole numbers, they
T
can take decimals of whole numbers. Ask students whether they get the
same answer when they take a quarter of a number (say 8) by dividing
the number into 4 equal parts and when they take a quarter of the same
number by multiplying the number by 0.25. Tell your students that 1/4
of the people in this class is the same as 0.25 of the people in this class.
They can find 0.25 of a number by multiplying that number by 0.25.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Ask students to find a context in which 0.25 represents a small amount and
one in which it represents a large amount.
Number Sense 7-49, 50
F-51
NS7-51 Dividing Decimals Using Different Strategies
Page 158
Goals
Curriculum
Expectations
Students will connect dividing decimals by whole numbers and
dividing whole numbers by whole numbers. Students will use a variety
of mental strategies to solve problems involving the division of decimals
by whole numbers.
Ontario: 6m21; 7m1, 7m4,
7m20
WNCP: 6N8; 7N2,
[R, V, ME, PS]
PRIOR KNOWLEDGE REQUIRED
Understands decimal place value
Can divide 3- and 4-digit numbers by 1-digit numbers
Knows the distributive principle
Vocabulary
quotient
divisor
dividend
Process Expectation
Modelling
Modelling decimal division. To remind students how we are using
base ten materials, draw on the board and label the blocks (see margin). Then draw the division statement model from Workbook Question 1 on
the board.
tenths
hundredths
Process Expectation
Changing into a known
problem
Process Expectation
Mental math
ASK: What division statement does this model show? First, how much is
the dividend (the part that is being divided)? (9 tenths and 5 hundredths)
What is that as a decimal? (0.95) How many equal groups are being made?
(4) The number of equal groups is the number you divide by, and it is
called the divisor. What number is the divisor? (4) How many tenths and
hundredths are in each group? (2 tenths and 3 hundredths) How many
hundredths is this? (23 hundredths). The quotient is 23 hundredths.
Were any tenths left over? (no) How many hundredths were left over
(3 hundredths) Draw base ten models for several more division statements
on the board and have students write the corresponding division statements
individually.
Some strategies for decimal division. Show a few ways to turn a decimal
division problem into a simpler problem. The first method involves rewriting
the decimal as a smaller place value so that it is easier to divide it into
groups. The result is a division problem that is essentially a whole number
division problem; for example: 0.4 ÷ 5. Note that you can’t divide 4 into
5 equal groups, but you can divide 40 into 5 equal groups. So, change 4
tenths into 40 hundredths. You know that 40 ÷ 5 = 8 from your math facts,
so it is easy to divide and get an answer of 8 hundredths:
F-52
0.4 ÷ 5 = 4 tenths ÷ 5
= 40 hundredths ÷ 5
= 8 hundredths
= 0.08
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ones
Another example:
Process Expectation
Splitting into simpler
problems
0.12 ÷ 8 = 12 hundredths ÷ 8
= 120 thousandths ÷ 8
= 15 thousandths
= 0.015
Recall that you can break a division problem into simpler problems by
dividing in parts (the distributive property). Show students an example with
whole numbers; for example:
84 ÷ 2 = (80 + 4) ÷ 2 = (80 ÷ 2) + (4 ÷ 2) = 40 + 2 = 42
Tell students that you can use this method to divide decimal numbers one
place value at a time:
8.4 ÷ 2 = (8 + 0.4) ÷ 2 = (8 ÷ 2) + (0.4 ÷ 2) = 4 + 0.2 = 4.2
0.96 ÷ 3 = (0.9 ÷ 3) + (0.06 ÷ 3) = 0.3 + 0.02 = 0.32
Tell students that you don’t have to separate out each place value—you
could pick some other way to break up the decimal that looks like it will
make things easier; for example:
9.2 ÷ 4 = (8 + 1.2) ÷ 4 = (8 ÷ 4) + (1.2 ÷ 4) = 2 + 0.3 = 2.3
10.8 ÷ 3 = (9 + 1.8) ÷ 3 = (9 ÷ 3) + (1.8 ÷ 3) = 3 + 0.6 = 3.6
Have students complete Workbook p. 158. Encourage them to use
rounding to estimate and check their solutions. Note that estimates
involving division tend to be farther from the actual values than those with
addition and subtraction. When dividing, round both the dividend and
divisor up or both down rather than rounding only one.
Word Problems Practice
1.Stefan has saved $651.75 over the past three years. How much has he
saved each year, on average? ($217.25)
NOTE: Before students do Questions 2–4 below, remind them of the
words pentagon (5-sided polygon), hexagon (6 sides), and octagon
(8 sides). Also point out that in a regular polygon, all sides have the
same length.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
2.The perimeter of a regular pentagon is 45.5 m. What is the length of a
single side? (9.1 m)
3.The perimeter of a regular hexagon is 110.4 cm. What is the length of
a single side? (18.4 cm)
4.The perimeter of a regular octagon is 38 m. What is the length of a
single side? (4.75 m)
5.The area of rectangle is 115.38 m2. The length of one side is 6 m.
What is the length of the other side? (19.23 m)
Number Sense 7-51
F-53
NS7-52 Long Division and
NS7-53 Dividing Decimals by Whole Numbers
Pages 159–164
Curriculum
Expectations
Ontario: 6m21; 7m1, 7m3,
7m4, 7m18, 7m20
WNCP: 6N8; 7N2,
[ME, R, V]
Goals
Students will use the standard algorithm for long division to divide
3- and 4-digit whole numbers by 1-digit whole numbers, and to divide
decimal numbers by 1-digit whole numbers.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
standard algorithm
remainder
quotient
dividend
divisor
tenths
hundredths
Can use the standard algorithm for long division to divide 2-digit
whole numbers by 1-digit whole numbers
Understands remainders
Understands division as finding the number in each group
Can use base ten materials to represent numbers
Understands decimal place value up to hundredths
Long division: 3-digit by 1-digit. Using pictures of base ten materials,
explain why the standard algorithm for long division works.
EXAMPLE: Divide 726 into 3 equal groups.
Process Expectation
Modelling
Step 1. Make a model of 726 units.
7 hundreds blocks
2 tens 6 ones
blocks
blocks
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.
200
3 726
− 600
126
F-54
2 hundred blocks, or 200 units, have been divided into
each group
600 units (200 × 3) have been divided
126 units still need to be divided
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
NOTE: Step 2 is equivalent to the following steps in the standard
long division algorithm.
2 3 726 −6 1
2 3 726
−6
12
Students should practise Steps 1 and 2 from both the modified and the
standard algorithms on the following problems.
2
512
3
822
2
726
4
912
Students should show their work using actual base ten materials or a
model drawn on paper.
Step 3. Divide the remaining hundreds block and the 2 remaining
tens blocks among the 3 groups equally.
There are 120 units in total, so 40 units can be added to each group
from Step 2.
Group 1
Group 2
Group 3
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Keep track of this as follows:
40
40 new units have been divided into each group
200
3 726
− 600
126 120 (40 × 3) new units have been divided
− 120
6
6 units still need to be divided
NOTE: Step 3 is equivalent to the following steps in the standard
long division algorithm.
24
3 726
3
− 6
−
12
12
Number Sense 7-52, 53
24
726
6
12
12
06
F-55
Students should carry out Step 3 using both the modified and standard
algorithms on 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.
Group 1
Group 2
Group 3
There are now 242 units in each group; hence 726 ÷ 3 = 242.
2
2 new units have been divided into each group
40 200
3 726
− 600 126 − 120 6
−6
6 (2 × 3) new units have been divided
0
there are no units left to divide
NOTE: Step 4 is equivalent to the following steps in the standard long
242242
3 726
3 726
− 6 − 6
12 12
− 12 − 12
06 06
6 6
0
Encourage students 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.
F-56
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
division algorithm.
Long division when the divisor is greater than the dividend’s
hundreds 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 groups):
5 27 Ask: How many tens blocks are in 27? Into how many groups do they
need to be divided into? Are there enough tens blocks to place one in each
group? How is this different from the problems you just did? Illustrate that
in the answer by writing a zero above the dividend’s tens digit. Then ask:
What is 5 × 0? Write:
5
0
27
0
27
Number of ones blocks (traded from tens blocks) 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 group
Number of ones blocks (traded from tens blocks) to be placed
Number of ones blocks placed
Number of ones blocks left over
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:
4 37
5 39
8 63
8 71
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).
Long division: 4-digit by 1-digit. Start by using base ten materials and
going through the steps of the recording process as before. Some students
may need to practise one step at a time, as they did with 3-digit by 1-digit
long division.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ASK: How long is each side if an octagon has a perimeter of…
a) 952 cm
b) 568 cm
c) 8104 cm
d) 3344 cm
Assessment tip. If some students are having problems with long division,
but you suspect it is really the multiplication tables that are holding them
back, try this: Have students copy and complete the 8 times table into
their notebook, at the top of a page, or have them find the times table
themselves by adding 8 successively each time:
Number Sense 7-52, 53
8
× 1
8
8
× 2
16
8
× 3
24
8
× 4
32
8
× 5
40
8
× 6
48
8
× 7
56
8
× 8
64
8
8
× 9 × 10
72
80
F-57
Then have students solve several long division problems that require
dividing by 8, only. EXAMPLES: )
d) 8)6149 (768 R 5)
a) 8 375 (46 R 7)
)
e) 8)2651 (331 R 3)
b) 8 973 (121 R 5)
)
f) 8)743 156 (92 894 R 4)
c) 8 654 (81 R 6)
Then have students write the 6 times tables and solve these problems:
a) 993 ÷ 6 b) 765 ÷ 6
c) 891 ÷ 6
d) 1 743 ÷ 6
e) 3 842 ÷ 6 f) 54 387 ÷ 6 g) 94 716 ÷ 6
h) 38 427 ÷ 6
i) 45 327 ÷ 6 j) 5 908 297 522 ÷ 6
Such problems will help students remember the times tables by repeatedly
using them, will allow students to focus more on learning the division, and
will allow you to assess the division alone. Have students use multiplication
and addition to check their answers (EXAMPLE: if the answer to b) was
127 R 3, find 6 × 127 + 3—do you get 765?).
Process Expectation
Modelling
1.0
0.1
0.01
1.0
0.1
0.01
Dividing decimals by 1-digit whole numbers. Tell students that a
hundreds block represents one whole and draw and label the base ten
blocks as shown in the margin (a line for a rod and a dot for a square).
Then remind students about the easier drawing possibility. Demonstrate
drawing the base ten model for: 5.43. Then have students do similar
problems in their notebooks. Examples: 8.01, 0.92.
Now tell students that you would like to find 6.24 ÷ 2. Have a volunteer
draw the base ten model for 6.24. Then draw 2 circles on the board and
have a volunteer show how to divide the base ten materials evenly among
the 2 circles. What number is showing in each circle? What is 6.24 ÷ 2?
Repeat for other numbers where each digit is divisible by 2 (or 3).
Process Expectation
Reflecting on what makes
a problem easy or hard
EXAMPLES: 46.2 ÷ 2, 3.63 ÷ 3, 4.02 ÷ 2, 6.06 ÷ 2, 6.06 ÷ 3.
Then write on the board 3.54 ÷ 2. ASK: In what way is this problem different
from the previous problems? (each digit is not divisible by 2) Have a
volunteer draw the base ten model for 3.54.
Process Expectation
Mental Math
Then lead students through the steps of long division. Compare the steps
for 3.54 ÷ 2 and 354 ÷ 2. Emphasize that students can just pretend that the
decimal point does not exist and then put the decimal point in the correct
place at the end. For example, students will see that 354 ÷ 2 = 177. To
find 3.54 ÷ 2, they can round 3.54 to 4 and estimate that 3.54 ÷ 2 is about
4 ÷ 2 = 2. Where should they put the decimal point so that the answer is
close to 2? (ANSWER: 1.77) Have students do several problems where they
figure out the answer this way, first by long division of whole numbers and
then by estimating to find where to put the decimal place.
Then, to ensure students are doing this last step correctly, give problems
where students do not have to divide but only have to add the decimal point:
F-58
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Draw 2 circles on the board and ask students why you chose to draw
2 circles rather than a different number of circles.
a)856.1 ÷ 7 = 1 2 2 3 (ANSWER: 122.3 since the answer will be close
to but more than 100)
b) 8922.06 ÷ 6 = 1 4 8 7 0 1 (ANSWER: 1487.01 since the answer will
be close to but more than 1000)
To ensure that students do not simply count decimal places, but actually
estimate their answers, give some examples where you add an extra zero
to the answer. For example, 23.28 ÷ 6 = 3 8 8 0 has answer 3.880 or
just 3.88.
Encourage students to discover their own rule regarding where to put the
decimal point when they have finished the long division (put the decimal
point above the decimal point, as shown on the worksheet). You can explain
why you line up the decimal points when doing long division as follows:
If you divide 3.42 by 2 the answer will be 100 times smaller than the answer
to 342 ÷ 2. Just as I need 100 times fewer 2s to make 8 as I need to make
800, I need 100 times fewer 2s to make 3.42 as I need to make 342. Notice
that when the decimal in the dividend (3.42) shifts 2 left, the decimal in the
quotient (1.71) shifts 2 left, so the decimals in the dividend and the quotient
line up.
Extensions
Process Expectation
Reflecting on other ways to
solve a problem
1.Teach students the Egyptian method of long division. The Egyptians
did not have the times tables available to them, but they knew how to
double numbers. For example, to divide 659 ÷ 7, the Egyptians would
find 7 × 1 = 7, 7 × 2 = 14, 7 × 4 = 28, 7 × 8 = 56, 7 × 16 = 112,
7 × 32 = 224, 7 × 64 = 448, and 7 × 128 = 996, by successive
doubling. They know that 996 is too high, so they can stop.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
7 × 64 = 448
7 × 16 = 112
7 × 8 = 56
7 × 4 = 28
7 × 2 = 14
659
− 448
211
− 112
99
− 56
43
− 28
15
− 14
1 Remainder
7 × (64 + 16 + 8 + 4 + 2) = 7 × 94
= 448 + 112 + 56 + 28 + 14 = 658, so 659 ÷ 7 = 94 R 1.
2.Teach students to divide decimals by single-digit decimals (EXAMPLE:
86.4 ÷ 0.9) by treating both the dividend and the divisor as whole
numbers (864 ÷ 9 = 96) and then estimating by rounding each number
to the nearest whole number (86 ÷ 1 is about 90) to decide where to
put the decimal point.
Number Sense 7-52, 53
F-59
NS7-54 Decimals Review
Page 165
Curriculum
Expectations
Ontario: 7m1, 7m4, 7m5,
7m19, 7m20, 7m21,
7m22
WNCP: 7N2, 7N7,
[CN, ME, PS, R, T, V]
Process assessment
Students will review concepts in decimals.
PRIOR KNOWLEDGE REQUIRED
Understands decimal place values
Can add and subtract decimals
Can multiply and divide decimals by 1-digit whole numbers
This worksheet is a review and can be used as an assessment. Remind
students to use estimation to check their answers when solving problems.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
[CN], 7m5
Workbook Question 7
Goals
F-60
Teacher’s Guide for Workbook 7.1