Download Unit 4 The Number System: Decimals

Unit 4 The Number System: Decimals
Students will extend the place value system to decimals, and extend
to decimals their knowledge of comparing, adding, and subtracting
multi-digit numbers. Students will also multiply fractions and decimals
by whole numbers.
Terminology. Even though mathematically 1 is considered a power of 10
(with exponent 0), we are not introducing it as such in this section.
There is some confusion in naming decimal fractions. We use the convention
that 1/100, for example, is one hundredth, not one one-hundredth. When
this causes confusion, as in 350/1,000 (three hundred fifty thousandths)
compared to 300/50,000 (three hundred fifty thousandths), always clarify
by showing the fraction you are referring to. Note that 350/1,000 would be
confused with 300/51,000 if we did read the “one” in “one-thousandths,”
therefore, doing so would not eliminate the confusion.
Do not shorten “decimal point” to “decimal.” This creates confusion
between two different concepts: decimal (a number) and decimal point
(the symbol separating parts of the number). Make sure students use
proper terminology.
When writing negative fractions, be sure to write the negative sign in front
of the fraction, not in front of the numerator.
1
-1
Not like this:
2
2
We often use slashes for fractions (such as 1/2 or −1/2) to save time and
space. Do not display fractions to students this way.
Like this: -
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Materials. We recommend that students always work in grid paper
notebooks. Paper with 1/4-inch grids works well in most lessons. Grid
paper is very helpful when comparing decimals, drawing base ten blocks
or number lines, and performing operations with multi-digit numbers and
decimals using standard algorithms. If students do not use grid paper
notebooks in general, you will need to have lots of grid paper on hand
throughout the unit. If students who have difficulties in visual organization
will be working without grid paper, they should be taught to draw a grid
before starting to work on a problem.
The Number System
E-1
NS6-24 Decimal Fractions
Pages 82–84
STANDARDS
6.NS.B.3, preparation for
6.NS.C.7
Goals
Students will represent decimal fractions in expanded form.
PRIOR KNOWLEDGE REQUIRED
Recognizes increasing and decreasing patterns
Can use grids to represent tenths and hundredths
Can write equivalent fractions
Can add fractions with the same denominator and fractions with
different denominators
Vocabulary
decimal fraction
denominator
equivalent fraction
hundredth
numerator
power of 10
represent
tenth
thousandth
Introduce powers of 10. Write on the board:
10 = 10 × 10 = 10 × 10 × 10 = 10 × 10 × 10 × 10 = Have volunteers fill in the blanks. Tell students that these numbers
are called powers of 10. We’ll learn about powers of other numbers
later in the year.
Review multiplying powers of ten. SAY: Multiplying by powers of ten
is easy because you just write zeros at the end of the number.
Exercise: Multiply.
a)10 × 100
b) 10 × 10
c) 1,000 × 10
d) 100 × 100
Answers: a) 1,000, b) 100, c) 10,000, d) 10,000
Exercise: What do you multiply by?
a)10 ×
= 1,000
Bonus: d) 100 ×
b) 100 ×
= 1,000
= 10,000 e) 1,000 ×
c) 10 ×
= 100
= 10,000,000,000
Struggling students can write the number of zeros under each power of ten.
For example, in part a), write “1” under 10 and “3” under 1,000.
Answers: a) 100, b) 10, c) 10, Bonus: d) 100, e) 10,000,000
(MP.7)
E-2
Introduce decimal fractions. Write the fractions 5/10, 4/10, 3/100, and
425/1,000 on one side of the board and the fractions 1/2, 2/5, 4/17, 9/20,
and 289/3,000 on the other side of the board. Write the heading “Decimal
Fractions” over the first group and “Not Decimal Fractions” over the second
group. Have volunteers suggest fractions, and have the rest of the class
point the thumbs towards the correct group to signal in which group each
fraction should be placed. Have students guess the rule for putting the
fractions in each group.
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
NOTE: Do not use decimal
notation until the next lesson.
Focus on the concept of a
decimal fraction.
Explain that a decimal fraction is a fraction whose denominator is a power
of ten. They’re important because powers of ten are easy to work with. Point
out that while some of the denominators in the “not decimal fractions” group
are multiples of 10, they are not powers of 10. Also, some fractions (such as
1/2 and 2/5) are equivalent to decimal fractions but are not decimal fractions.
Review equivalent tenths and hundredths. Draw on the board the two
squares shown in the margin. SAY: The picture shows why three tenths
equals thirty hundredths. The second square has ten times as many shaded
parts and ten times as many parts altogether. Write on the board:
3 × 10
30
=
100
10 × 10
Exercises: Write an equivalent fraction with the denominator 100.
Show your work.
7 × 10
9
4
=
=
b)
c)
a)
=
×
10
10
10
10
100
100
100
Answers: The numerators are: a) 70, b) 40, c) 90
Equivalent tenths, hundredths, and thousandths. Write on the board:
3 ×
7 ×
5 ×
=
=
=
100 ×
10 ×
10 ×
100
1,000
1,000
SAY: Now you have to decide what to multiply the numerator by to get an
equivalent fraction. You have to decide what the denominator was multiplied
by, then multiply the numerator by the same thing. Have volunteers tell
you what to multiply by, then have other volunteers fill in the numerators
(30, 700, 50). SAY: To make an equivalent fraction, you just have to add
the same number of zeros to the numerator and denominator.
Exercises: Write the missing numerator in the equivalent fraction.
3
9
8
a)
b)
=
c)
=
=
100
10
10
1,000
1,000
1,000
3
Bonus:
=
10
1,000,000
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 80, b) 300, c) 900, Bonus: 30,000
+
3
10
+
=
6
100
=
36
100
Adding tenths and hundredths. Draw on the board the picture in the
margin. SAY: If you can add hundredths, and if you can change tenths to
hundredths, then you can add tenths and hundredths. Three tenths is thirty
hundredths, and six more hundredths is thirty-six hundredths. Remind
students that they can change tenths to hundredths without using a picture:
3 × 10
6
30
6
36
+
+
=
= 100 100 100
100
10 × 10
Exercises: Add. Show your work.
4
7
7
4
5
8
a)
+
b) +
c) +
10 100
10 100
10 100
Bonus:
3
9
+
100 10
Answers: a) 47/100, b) 74/100, c) 58/100, Bonus: 93/100
The Number System 6-24
E-3
Adding tenths, hundredths, and thousandths. Write on the board:
3
9
6
+
+
100
1, 000
10
ASK: How can you change the fractions to make them easier to add?
(change all denominators to 1,000) Write underneath:
+
+
=
1, 000 1, 000 1, 000 1, 000
Have volunteers complete the equation:
300/1,000 + 90/1,000 + 6/1,000 = 396/1,000. Point out how adding
fractions with denominators 10, 100, and 1,000 is easy because it’s
just using expanded form.
Exercises: Add.
4
3
9
5
2
1
3
8
4
+
+
+
+
+
a)
Bonus:
b) +
10 100 1, 000
10 100 1, 000
100 1, 000 10
Answers: a) 439/1,000, b) 521/1,000, Bonus: 438/1,000
Adding decimal fractions with missing tenths or hundredths. Write on
the board the following equation, but without the answer shown in italics:
4
9
409
+
=
10 1, 000 1,000
ASK: How many thousandths are in 4/10? (400) So how many thousandths
are there altogether? (400 + 9 = 409) Write the answer, then SAY: 4 tenths,
0 hundredths, and 9 thousandths add to 409 thousandths. Have students
add more tenths and thousandths.
Exercises: Add.
3
7
9
1
2
6
+
+
c)
a) 10 + 1,000 b)
10 1,000
10 1,000
Answers: a) 307/1,000, b) 901/1,000, c) 206/1,000
Repeat the process with 4/100 + 9/1,000. Then write on the board:
0 tenths + 4 hundredths + 9 thousandths = 49 thousandths
Exercises: Predict, then check by adding.
8
2
8
2
5
8
3
7
+
+
b)100 + 1,000 c)10 + 1,000 d)
a)
10 1,000
100 1,000
8
2
8
2
+
+
f)
1, 000 10
100 10, 000
Answers: a) 37/1,000, b) 82/1,000, c) 802/1,000, d) 508/1,000,
Bonus: e) 208/1,000, f) 802/10,000
Bonus: e)
SAY: You might need to add thousandths or just hundredths.
E-4
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
SAY: We could be tempted to write this as 049/1,000, but we do not write
the zero at the beginning of a number.
Exercises: Add the decimal fractions. Write the answer as
a decimal fraction.
8
9
2
8
6
9
6
+
+
a)
+
b) +
c)
10 100 1, 000
10 1, 000
10 100
3
5
6
8
2
5
5
+
+
+
+
f)
d)
Bonus: e)
100 1, 000
100 1, 000 10
10 100, 000
Answers: a) 892/1,000, b) 806/1,000, c) 96/100, d) 35/1,000,
Bonus: e) 268/1,000, f) 50,005/100,000
Extensions
1. Write 1 as a decimal fraction.
Sample answers: 10/10, 100/100
(MP.1)
(MP.3)
2. a) Is there a largest power of 10?
b) Is there a smallest decimal fraction? How do you know?
Answers: a) No, because you can multiply any power of 10 by 10 to
get an even larger one; b) No, because you can make the fraction
smaller by making the denominator a larger power of 10.
3. Solve the equation for x.
4
9
3
38
x
5
45
x
x
a)
+
=
b)+
=
c) +
=
10 10 10
10 100 100
10 100 100
Bonus
d)
6
67
4
67
7
4
+x=
e)+ x =
f) + x =
10
100
10
100
100
10
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 5, b) 8, c) 4, Bonus: d) 7/100, e) 27/100, f) 33/100
The Number System 6-24
E-5
NS6-25 Place Value and Decimals
Pages 85–87
STANDARDS
preparation for 6.NS.C.7
Vocabulary
decimal
decimal fraction
decimal point
denominator
hundredth
place value
placeholder
power of 10
represent
tenth
thousandth
Goals
Students will represent decimals in expanded form.
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
Can write equivalent fractions
Can add fractions with the same denominator and fractions with
different denominators
Review the place value system. Write on the board:
5,834 = 5,000 + 800 + 30 + 4
SAY: We use place value to write numbers. That means that where a digit is
placed in the number tells you its value. Because the 5 is in the thousands
place, it is worth 5,000.
Exercises: What does the 7 represent?
a)
6,742b)
9,017c)
6,572d)
7,904
Answers: a) 700, b) 7, c) 70, d) 7,000
The place value system extends to include tenths. Write on the board:
SAY: The place values get ten times smaller: 10 ones fit into a ten, 10 tens fit
into a hundred, and 10 hundreds fit into a thousand. Tell students that you
want to continue the place value system so that you can use place value for
fractions too. ASK: What is ten times smaller than 1? PROMPT: Ten of what
make one whole? (tenths) To guide students, draw pictures of ten equal
parts fitting into one whole (see margin). Tell students that there is a way to
show 1/10 that uses place value. Write on the board:
3
3
1
4
= 0.1 = 27.4
= 8.3 27
= 0.3 8
10
10
10
10
SAY: We call these numbers decimals. The dot between the whole numbers
and the number of tenths is called a decimal point. Decimals are similar to
mixed numbers. There’s a whole-number part to the left of the decimal point
and a fractional part to the right. But when the number is less than 1 whole,
we write 0 as the whole-number part.
Exercises: Write the decimal for these numbers:
6
5
8
b)3
c)74
10
10
10
Answers: a) 0.5, b) 3.8, c) 74.6, Bonus: 800.3
a)
E-6
Bonus: 800
3
10
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
thousands hundreds tens ones
Extending the place value system beyond tenths. Write on the board:
hundreds tens ones tenths
ASK: What should the next place value be? (hundredths) PROMPT: Ten
of what fit into a tenth? Point out that there is symmetry in the place value
names with the ones as the center of reflection:
hundreds tens ones tenths hundredths
100 10 1 1
1
10
100
SAY: There is also symmetry in the values. Write on the board the picture
shown in the margin. Ask volunteers to continue the place values in both
directions.
Show students how to write decimals for one-digit hundredths and
thousandths:
3
8 = 0.008
= 0.03 100
1, 000
SAY: The next place value after tenths is for hundredths. The one after
that is for thousandths.
Exercises: Write the decimal.
7
4
5
8
6
b) c) d) e)
a)
100
1, 000
100
1, 000
1, 000
Answers: a) 0.07, b) 0.004, c) 0.05, d) 0.008, e) 0.006
Tell students that there are two ways to read 0.03 out loud: “zero point zero
three” or “three hundredths.” SAY: We write 0.03 as “three hundredths”
when we use words to write it on paper.
Exercises: Write the decimal in words.
a)0.04
b)0.8
c)0.009
d)0.07
e)0.003
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) four hundredths, b) eight tenths, c) nine thousandths,
d) seven hundredths, e) three thousandths
More than one non-zero digit in decimals. Write on the board:
6
7
9+ +
= 9.67
10 100
NOTE: Reading 9.67 as
“nine point sixty-seven” is
incorrect and should be
discouraged. It may create
the misconception that 9.67 is
greater than 9.8, since 67 > 8.
Read the place values in the decimal to show how they correspond to the
expanded form: 9 ones, 6 tenths, and 7 hundredths. Tell students that they
can read 9.67 out loud as “nine point six seven.”
Exercises: Write the decimal.
5
8
2
9
3
4
5
6+
+
+
a) 3 +
+
b) 8 +
+
c)
10 100 1, 000
10 100
10 100
Answers: a) 3.49, b) 8.53, c) 6.582
Using 0 as a placeholder. Write on the board:
The Number System 6-25
3
3
3
3
= 0.3 5 +
= 5.3 = 0.03 5 +
=
100
100
10
10
E-7
Ask a volunteer to write the last decimal. (5.03) Point out that because
there are no tenths, the tenths place has a zero. Write on the board:
5
3
5
3
+
+
10 100 10 1, 000
Ask volunteers to write the decimals. (0.53, 0.503) Point out how the
denominator tells you how many places the digit goes after the decimal
point: tenths go one place after the decimal point, hundredths go two
places, and thousandths go three places. SAY: You have to be careful
because some place values might be missing. You’ll have to write zeros
in those positions.
Exercise: Write the decimal.
a) 2 +
8
7
5
2
3
8+
5+
3+
+
b)
c)
d)
1, 000
1, 000
10 1, 000
100
Answers: a) 2.03, b) 8.008, c) 5.007, d) 3.502
Exercises: Write the value of the 6 as a fraction or whole number.
a) 0.642
b) 0.063
c) 0.603
d) 26.453
e)13.456
Bonus: What places are the zeros holding in 0.3402? In 0.34206?
Answers: a) 6/10, b) 6/100, c) 6/10, d) 6, e) 6/1,000, Bonus: ones place
and thousandths place in 0.3402, ones place and ten-thousandths place
in 0.34206
Dimes, pennies, tenths, and hundredths. Write on the board:
$0.56 = 56 cents = 5 dimes and 6 pennies
SAY: We use decimal notation for money because a dime is a tenth
of a dollar and a penny is a hundredth of a dollar. Write on the board:
0.56 = 56 hundredths = 5 tenths 6 hundredths
ASK: How many hundredths are in 0.7? (70) Write on the board:
Exercises: Write the amount as hundredths and as mixed units of tenths
and hundredths.
a)
0.34b)
0.68c)
0.90d)
0.5
Answers: a) 34 hundredths = 3 tenths 4 hundredths, b) 68 hundredths
= 6 tenths 8 hundredths, c) 90 hundredths = 9 tenths 0 hundredths,
d) 50 hundredths = 5 tenths 0 hundredths
ASK: How many hundredths are in 0.63? (63) How many hundredths are in
0.9? (90) What is greater, 0.63 or 0.9? (0.9) Remind students that comparing
fractions is easier when both fractions have the same denominator. So it’s
convenient to change 9 tenths to 90 hundredths.
E-8
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
0.7 = 0.70 = 70 hundredths = 7 tenths 0 hundredths
Exercises: Write both decimals as hundredths. Which one is greater?
a) 0.5 and 0.42
b) 0.6 and 0.78
c) 0.3 and 0.05
Answers: a) 0.50 > 0.42, b) 0.60 < 0.78, c) 0.30 > 0.05
Extensions
(MP.8)
1. How much more is the 2 worth than the 5 in the decimal 0.324067568?
Answer: Since each place value is 10 times the one to the right, the
relative values between the 2 and the 5 are the same as for 240,675.
The 2 is worth 200,000 and the 5 is worth 5. How many times as much
as 5 is 200,000? Make the table below.
Number
How many times as much as 5?
20
200
2,000
20,000
200,000
4
40
400
4,000
40,000
So the 2 is worth 40,000 times as much as the 5.
(MP.1)
2. Write the correct decimal:
$700 + $80 + 9¢ = $ . COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer: $780.09
The Number System 6-25
E-9
NS6-26 Positive and Negative Decimals
Pages 88–90
STANDARDS
6.NS.C.7
Vocabulary
decimal
decimal fraction
decimal point
hundredth
mixed number
negative
positive
tenth
thousandth
Goals
Students will write mixed numbers as decimals and decimals as mixed
numbers, including negative numbers.
PRIOR KNOWLEDGE REQUIRED
Can write mixed numbers and decimals
Can add fractions with different denominators
Writing decimals as proper fractions. Write on the board:
6
206
2
+
=
0.206 = 2 tenths + 6 thousandths =
1
,
000
1
,
000
10
Point out that the numerator is the decimal without the “zero point” in front.
The number of zeros in the denominator is the number of digits after the
decimal point. Write on the board:
0.206 = There are three
digits after the
decimal point.
206
1, 000
0.037 = So there are
three zeros in
the denominator.
three digits
after the
decimal point
37
1, 000
so three
zeros in the
denominator
Exercises: Hold up the correct number of fingers to signal how many
zeros you would put in the denominator.
a)0.3
b)0.056
c)0.801
d)0.000437
Answers: a) 1, b) 3, c) 3, d) 6
Exercises: Write the fraction for each decimal in the previous exercise.
Writing proper fractions as decimals. Write on the board the fraction
34/1,000. Tell students that you want to write it as a decimal. SAY: The
numerator tells you what digits to write. The denominator has three zeros,
so you have to put three digits after the decimal point. Write the digits “3
4” on the board, and point out that there are only two digits. So, to make
three digits after the decimal point, they need to add a 0 before the 3. Write
“0.034” on the board. Point out that thirty-four thousandths is the same
as three hundredths and four thousandths, so zero ones and zero tenths
makes sense. Write the fraction 3/100 on the board. ASK: How many digits
would you put in the decimal? (2) How many digits are in the numerator?
(1) How many zeros do you need to add after the decimal point? (1) Have
students raise the correct number of fingers to signal the answer to the
questions above. How do you know? (The number of zeros you need to
add is the number of zeros in the denominator minus the number of digits
in the numerator.) Finally, write the decimal: 3/100 = 0.03.
E-10
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 3/10, b) 56/1,000, c) 801/1,000, d) 437/1,000,000.
Exercises: How many zeros do you need to add after the decimal point
in each fraction? Hold up the correct number of fingers to signal the
number of zeros.
34
7
32
8, 405
a)
b)
c) d)
100
1, 000
10, 000
100, 000
Answers: a) 0 (closed fist), b) 2, c) 2, d) 1
Exercises: Write the fractions from the previous exercise as decimals.
Answers: a) 0.34, b) 0.007, c) 0.0032, d) 0.08405
Tell students that some people don’t write the 0 in front of the decimal point,
so some people will write .4 for 0.4. SAY: Be careful not to miss the decimal
point; you don’t want to mistake .4 for 4.
Reading decimals. Remind students that decimals can be used to
represent mixed numbers. The whole-number part of the mixed number
goes to the left of the decimal point, and the fractional part goes to the
right. Write on the board:
28
= 5.28
100
Tell students that we read the decimal the same way we read the mixed
number, as “5 and 28 hundredths.” Point out that the decimal point is read
as “and.” Write on the board:
5
NOTE: Another correct
way to read 5.28 is “five point
two eight.” However, “five point
twenty-eight” is incorrect and
should be discouraged. It
may create the misconception
that 5.28 is greater than 5.3,
since 28 > 3.
three six hundredths three and six hundredths
30.06
Ask a volunteer to write the missing decimal (3.06)
Exercises: Write the decimal.
a) five and eight hundredths
c) seven and twelve thousandths
b) thirty-five thousandths
d) twenty and two thousandths
Answers: a) 5.08, b) 0.035, c) 7.012, d) 20.002
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
SAY: Remember—look at the number of digits after the decimal point
to tell you whether the decimal is tenths, hundredths, or thousandths.
Exercises: Write tenths, hundredths, or thousandths.
a)4.7 = four and seven b)8.03 = eight and three c)5.13 = five and thirteen d)3.028 = three and twenty-eight Answers: a) tenths, b) hundredths, c) hundredths, d) thousandths
Emphasize how important the “and” can be when reading decimals.
Write on the board:
20.04 0.24
The Number System 6-26
E-11
SAY: 20.04 is said “twenty and four hundredths” while 0.24 is said “twentyfour hundredths.”
Writing decimals as mixed numbers. Write on the board: 45.891. Remind
students that the decimal point separates the whole-number part from
the fractional part. ASK: What is the whole-number part? (45) What is the
fractional part? (891 thousandths) Write the equivalent mixed number on
the board (45 891/1,000).
Exercise: Write the decimal as a mixed number.
a) 25.4 b) 1.73 c) 20.07 d) 5.303 Bonus: 123,456.9
Answers: a) 25 4/10, b) 1 73/100, c) 20 7/100, d) 5 303/1,000,
Bonus: 123,456 9/10
Review converting improper fractions into mixed numbers using
division. Remind students that they can use division with remainders to
convert improper fractions to mixed numbers. For example, 37 ÷ 10 = 3
R 7, so 37/10 = 3 7/10. Point out that this makes sense, because
37 tenths = 3 ones and 7 tenths. Write on the board:
79 ÷ 10 = R ,608 ÷ 100 = R ,
so
79
= 10
so
608
= 100
Have volunteers fill in the blanks.
Exercises: Write the improper fraction as a mixed number.
a)
43
780
3, 524
1, 234
b)
c)
d)
10
100
1, 000
100
Answers: a) 4 3/10, b) 7 80/100, c) 3 524/1,000, d) 12 34/100
Converting improper fractions to decimals. Write on the board:
Exercises: Write the improper fraction as a mixed number,
then as a decimal.
28
728
793
7, 845 Bonus: 63, 457
a)
b) c) d)
10
10
100
1, 000
100
Answers: a) 2 8/10 = 2.8; b) 72 8/10 = 72.8; c) 7 93/100 = 7.93;
d) 78 46/100 = 78.46; Bonus: 63 457/1,000 = 63.457
(MP.8)
E-12
Ask students to compare the improper fractions and the decimals. ASK:
Is there a shorter way to find the answer, without converting the improper
fraction to a mixed number? PROMPTS: Where do you see the numerator
in the answer? Where do you see the denominator in the answer?
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
37
7
=3
= 3.7
10
10
SAY: Once you can change an improper fraction to a mixed number,
you can change it to a decimal.
Point out that the decimal can be obtained by writing the numerator, then
making sure that the number of digits after the decimal point is the same
as the number of zeros in the denominator.
Exercise: Convert into a decimal without writing the mixed number.
3, 756
654
43, 654
53, 094
a)
b)c) d)
10
100
1, 000
10
845, 036, 714
Bonus:
100, 000
NOTE: When writing
decimals, the convention
is to write commas between
every third place value
before the decimal point,
but not after.
Answers: a) 65.4, b) 436.54, c) 375.6, d) 53.094, Bonus: 8,450.36714
Negative decimals. Tell students that just like fractions can be negative,
decimals can be negative too. When two numbers are equal, their
opposites are equal too. Write on the board:
734
734
= -7.34
= 7.34 so 100
100
Exercises: Write the decimals for the negative numbers.
2
382
407
2
a) -61
b)
c)
d)
-83
100
100
1, 000
10
Answers: a) −61.2, b) −3.82, c) −4.07, d) −83.002
Extensions
1.Have students list at least five decimal numbers that take exactly six
words to say. 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 in writing number words for
decimals, being organized, and looking for patterns.
Sample answers:
600,000.43 (six hundred thousand and forty-three hundredths)
9,000,080.09 (nine million eighty and nine hundredths)
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
−700.8 (negative seven hundred and eight tenths)
2.Teach students to interpret whole numbers written in decimal format
(Example: 5.1 million is 5,100,000, 3.7 thousand is 3,700)
3.Have students look for decimals in the media and write the decimals
as mixed numbers.
(MP.3)
4. Find the mistakes.
32
47
5
3
0.05 =
0.003 =
0.47 =
0.032 =
1
,
000
100
100
10
Answers: 0.003 = 3/1,000, not 3/100; 0.47 = 47/100, not 47/10
The Number System 6-26
E-13
NS6-27 Equivalent Fractions and Decimals
Pages 91–92
STANDARDS
6.NS.C.6, 6.NS.C.7
Vocabulary
decimal
decimal fraction
decimal point
equivalent decimals
equivalent fractions
hundredth
negative
positive
tenth
thousandth
Goals
Students will write positive and negative numbers in many different,
but equivalent, forms.
PRIOR KNOWLEDGE REQUIRED
Can write mixed numbers as decimals
Can write expanded form for whole numbers and decimals
Understands that opposite numbers are the same distance from 0,
but in opposite directions
Can produce decimal fractions equivalent to a given decimal fraction
Understands that equivalent fractions are equal and at the same
location on a number line
Introduce equivalent decimals. Write on the board:
3
30
=
=
10 100 1, 000
SAY: Three tenths is the same as thirty hundredths. ASK: How many
thousandths is that? (300) Fill in the missing numerator, then ask volunteers
to write the decimals for each of the fractions on the board. (0.3, 0.30, 0.300)
Explain that these are called equivalent decimals, because the fractions they
are equal to are equivalent.
Exercises: Write the equivalent decimals from the equivalent fractions.
2
20
200
4
40
7
70
=
=
=
b)
=
c)
a)
10 100 1, 000
10 100
10 100
Tell students that saying “0.3 = 0.30” is the same as saying “3 tenths is
equal to 30 hundredths or 3 tenths and 0 hundredths.” ASK: How many
hundredths is 8 tenths equal to? (80) Have a volunteer write the equivalent
decimals on the board. (0.8 = 0.80) Write on the board five ways to write
seven tenths:
7
0.70 0.7 seven tenths seventy hundredths
10
Ask students to write more ways. Then ask volunteers to show ways on
the board. (sample answers: 0.7, .70, .700, 0.700, 70/100, 700/1,000,
seven hundred thousandths)
Exercises: Which of these equations are incorrect? How do you know?
47
37
50
62
a)0.37 =
b)0.05 =
c)0.047 =
d) 0.62 =
1, 000
100
1, 000
100
Answers: Parts b) and d) are incorrect. In part b), the numerator should
be 5; in part d), the denominator should be 100.
E-14
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 0.4 = 0.40, b) 0.7 = 0.70, c) 0.2 = 0.20 = 0.200
Review writing decimals in expanded form. Write on the board:
0.837 = 8 tenths + 3 hundredths + 7 thousandths
7
8
3
=
+
+
10
100
1, 000
Tell students that writing 0.837 like this as a sum of decimal fractions is
called expanded form.
Exercises: Write the decimal in expanded form.
a) 0.672 b)0.45
c)0.023 d)0.809 Bonus: 0.200002
Answers: a) 6/10 + 7/100 +2/1,000, b) 4/10 + 5/100, c) 2/100 + 3/1,000,
d) 8/10 + 9/1,000, Bonus 2/10 + 2/1,000,000
Review writing decimals as fractions. SAY: The number of digits after
the decimal point tells you the denominator of the fraction. You can get
the numerator by writing the decimal without the “zero point” in front.
Exercises: Write the decimal as a fraction.
a)0.00043
b)0.6002
c)0.0035
d)0.000502
Answers: a) 43/100,000, b) 6,002/10,000, c) 35/10,000, d) 502/1,000,000
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Review opposite numbers. Teach in greater depth why equal numbers
have equal opposites. Remind students that opposite numbers are the same
distance from 0, but on opposite sides. Draw a number line from −3 to 3 on
the board, and ask students what the opposite number to 2 is. (−2) Now
draw on the board:
-1
3
0
10
1
-1
00.3
1
Ask a volunteer to place −3/10 on the number line and another volunteer to
place −0.3 on the number line. Point out that because 3/10 and 0.3 are at
the same place on the number line, so are their opposites. Then SAY: So
if you know how to find positive fractions and decimals that are equivalent,
then you know how to find negative fractions and decimals that are equivalent.
(MP.7)
Exercises: Write the equivalent fractions.
a)0.304 =
b)0.27 =
1,000
100
so −0.304 =
so −0.27 =
c) −0.031d)
−0.89e)
−0.906
Bonus: −0.5050505
Answers: a) 304, −304/1,000, b) 27, −27/100, c) −31/1,000, d) −89/100,
e) −906/1,000, Bonus: - 5,050,505/10,000,000
The Number System 6-27
E-15
Extensions
1. Finish writing the equivalent fractions.
7
30
8
80, 000
=
=
a)
c)
b) =
10 100, 000
1, 000 100, 000
1, 000
d)
700
7
=
1, 000, 000
Answers: a) 70,000, b) 3,000, c) 10,000,000, d) 10,000
2. Fill in the blanks.
40 hundredths = 4 = 4,000 = millionths
Answers: tenths, ten thousandths, 400,000
(MP.7)
3.Convert the decimals to fractions with the same denominator.
Then find the next two terms in the pattern.
a) 0.05, 0.2, 0.35, 0.5, , b 0.2, 0.325, 0.45, 0.575, , c) 0.9, 0.75, 0.6, 0.45, , COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 0.65, 0.8, b) 0.7, 0.825, c) 0.3, 0.15
E-16
Teacher’s Guide for AP Book 6.1
NS6-28 Ordering Decimals
Pages 93–94
STANDARDS
6.NS.C.7a
Vocabulary
common denominator
decimal
decimal fraction
decimal point
decreasing order
equivalent decimals
equivalent fractions
increasing order
negative
positive
Goals
Students will compare positive and negative 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 1,000,
and decimals
Understands the concept of an opposite number
Is familiar with < and > signs
MATERIALS
play money (dimes and pennies)
Comparing decimals by comparing their equivalent fractions. Write on
the board:
2
7
10
10
0.2 0.7
ASK: Which fraction is greater? (7/10) So which decimal is greater? (0.7)
SAY: You can compare decimals by comparing the fractions they are
equivalent to.
Exercises: Write the decimals as fractions. Which decimal is greater?
a) 0.4 or 0.3
b) 0.35 or 0.27
c) 0.8 or 0.9
d) 0.76 or 0.84
Answers: a) 0.4, b) 0.35, c) 0.9, d) 0.84
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Comparing different place values. Write on the board:
0.5 0.36
36
5
10
100
50
36
100
100
Have students signal which number is the largest in each pair (e.g., by
pointing their thumbs left or right). Start with the pairs at the bottom, and
work up. SAY: You can compare decimals by writing them as fractions
with the same denominator.
The Number System 6-28
E-17
NOTE: Students who
struggle with comparing
decimals with 1 and 2 decimal
places (e.g., saying that
0.17 > 0.2) can use play
dimes and pennies.
Exercises: Write the decimals as fractions with the same denominator.
Then decide which decimal is greater.
a) 0.3 and 0.24
b) 0.57 and 0.614
Bonus: c) 0.009 and 0.0045 d) .0004 and .00005
Answers: a) 0.3, b) 0.614, Bonus: c) 0.009, d) .0004
Ordering decimals by rewriting them to the smallest place value. Write
on the board:
0.7 = 0.70 0.64
SAY: I want to compare 0.7 to 0.64. Writing them both as hundredths makes
comparing them easy—70 is more than 64. Write the “>” sign between the
decimals.
Exercises: Write both decimals as hundredths. Then compare them.
a)0.4 0.51
b)0.5 0.47
c)0.3 0.28
Answers: a) 0.40 < 0.51, b) 0.50 > 0.47, c) 0.30 > 0.28
Write on the board:
0.5 0.487
Ask a volunteer to write 0.5 as thousandths, in decimal form. (0.500) ASK:
Which is greater—500 thousandths or 487 thousandths? (500 thousandths)
Finish the inequality: 0.5 > 0.487.
Exercises: Make both decimals have the same number of digits after the
decimal point. Then compare them.
a) 0.35 and 0.4
b) 0.006 and 0.03
c) 0.786 and 0.31
Bonus: 0.24 and 0.01904
Comparing decimals greater than 1. SAY: When the whole-number parts
are the same, you just have to compare the decimal parts. When the wholenumber parts are different, you only need to compare the whole numbers.
Write on the board:
0.5 > 0.36, so 4.5 > 4.36 5 > 3, so 5.16 > 3.247
Exercises: Compare the decimals.
a)3.4 3.067
b)8.56 17.001
c)2.012 20.05
d)0.54 0.346
e)0.3 0.295
f)61.3 62.104
Bonus: 8.4444444444 88.4
Sample solution: a) 3.400 > 3.067
E-18
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 0.35 < 0.40, b) 0.006 < 0.030, c) 0.786 > 0.310,
Bonus: 0.24000 > 0.01904
Answers: b) 8.56 < 17.001, c) 2.012 < 20.05, d) 0.54 > 0.346,
e) 0.3 > 0.295, f) 61.3 < 62.104, Bonus: 8.4444444444 < 88.4
(MP.6)
Contrast the importance of digits after the decimal point with the
importance of digits before the decimal point. Write on the board:
37.9999999999 364.3
NOTE: Many students
have difficulty comparing
and ordering decimals. A
common mistake is thinking
decimals as larger when
they have more digits after
the decimal point.
ASK: Which number is greater? (364.3) How do you know? (because 364
is greater than 37) Point out that more digits on the whole-number side of a
decimal mean a larger number, but more digits on the fractional side of a
decimal don’t necessarily mean a larger number.
Identifying a decimal between two decimals. Write 0.4 and 0.9 on the
board. Have students name a decimal between these two numbers. Write
on the board 4.3 and 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). SAY: 4.35 is between
4.30 and 4.40. Is it also between 4.3 and 4.4? (yes) SAY: If there are no
decimal tenths between the numbers, you can always change them to
tenths. If there are no hundredths between them, you can always change
them to thousandths.
Exercises: Find a decimal between the numbers.
a) 7.8 and 7.9
b) 0.25 and 0.26 c) 0.2 and 0.24 d) 6.3 and 6.37
Bonus: e) 3.789 and 3.79 f) 21.9000099 and 21.90001
Sample answers: a) 7.83, b) 0.251, c) 0.22, d) 6.34, Bonus: e) 3.7896,
f) 21.90000995
Using place value to compare decimals. Write on the board:
0.48 0.473
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
SAY: You can compare 0.48 to 0.473 by comparing 480 to 473, since 0.48
is equivalent to 480 thousandths and 0.473 is 473 thousandths. But you can
compare 480 to 473 by using place value. Write on the board:
4 8 0
0.4 8
4 7 3
0.4 7 3
Pointing to the whole numbers, SAY: They both have 4 hundreds, but 8 tens
is more than 7 tens. You can compare the decimals the same way. They
both have 4 tenths, but 8 hundredths is more than 7 hundredths, so 0.48
is greater than 0.473.
Point out how lining up the place values made it easy to find the first
place value where they are different. Then SAY: You can line up the place
values by lining up the decimal points because the decimal point is always
between the ones and the tenths.
The Number System 6-28
E-19
Write on the board:
0.703 0.619413.41
0.71 0.61853.42
For each pair, ASK: What is the largest place value in which the decimals
are different? (hundredths, thousandths, tens) Which decimal is greater?
(0.71, 0.6194, 13.41)
Exercises: Order the decimals from least to greatest by lining up the
decimal points.
a) 0.6, 0.78, 0.254 b) 0.25, 0.234, 0.219
c) 2.3, 2.04, 20.1, 2.195
Make sure students align the place values, not only the decimal points.
Encourage them to write each digit in its own cell of grid paper.
Answers: a) 0.254, 0.6, 0.78; b) 0.219, 0.234, 0.25; c) 2.04, 2.195, 2.3, 20.1
(MP.7)
Comparing negative decimals. Display a number line from −5 to 5.
Remind students that there is a way to compare negative numbers using
their opposites: 1 < 3, so −1 > −3.
Exercises: Compare the decimals.
a)1.3
b)13.5
c) −0.043
1.35, so −1.3
1.35, so −13.5
−1.35
−1.35
−0.05d)
−0.67
−0.6e)
−1.67
−2
Answers: a) <, >; b) >, <; c) >; d) <; e) >
Ordering positive and negative decimals. Write on the board the
following decimals: −2.35, −2.335, −2.25, −2.5. SAY: To order the negative
decimals, start by putting their positive opposites in order. Have students
order the decimals individually (2.25 < 2.335 < 2.35 < 2.5), then have a
volunteer demonstrate. SAY: Then reverse the order to order the negative
decimals: −2.5 < −2.35 < −2.335 < −2.25.
a) −0.6, −0.48, −0.654 b) −0.35, −0.344, −0.319
c) −5.2, −5.04, −50.4, −5.085d)
−2.1, −21.4, −21.15, −2.098
Answers: a) −0.654, −0.6, −0.48; b) −0.35, −0.344, −0.319;
c) −50.4, −5.2, −5.085, −5.04; d) −21.4, −21.15, −2.1, −2.098
SAY: Negative numbers are always smaller than positive numbers.
Exercises: Order the decimals from least to greatest.
a) −0.16, 0.61, 0.6, −0.616b)
−30.45, 30.34, 3.987, −30.9
Answers: a) −0.616, −0.16, 0.6, 0.61; b) −30.9, −30.45, 3.987, 30.34
E-20
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Exercises: Order the decimals from least to greatest.
Extensions
(MP.1)
1.Find as many ways as you can to use the digits 1 to 5 once
each so that ...
a)0. > 0. b)
−0. > −0. Sample answers: a) 0.31 > 0.245 b) −0.1234 > −0.5
(MP.1)
2.Use the digits 0, 1, and 2 once each to create as many different
decimals as you can that are:
a) larger than 1.2 b) between .1 and .2 c) between 1.0 and 2.0
Hint: 0.201 can be also written as .201; 2.1 can be written as 2.10.
Answers: a) 2.01, 2.10, 10.2, 12.0, 20.1, 21.0, 102, 120, 201, 210;
b) 0.12, .102, .120; c) 1.20, 1.02
3. a)Draw two squares so that 0.2 shaded in one is more than 0.3
shaded in the other.
b)Research to find what is worth more, 0.3 Brazilian Reals or
0.2 US dollars.
(MP.1)
4. Find the number halfway between each pair.
a) 0.4 and 0.7
d) 0.56 and 0.57
b) −0.2 and −0.3c)
−0.2 and + 0.3
e) −1.35 and −1.36f)−0.2 and −0.36
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 0.55, b) −0.25, c) + 0.05, d) 0.565, e) −1.355, f ) −0.28
The Number System 6-28
E-21
NS6-29 C
omparing Decimal Fractions
Pages 95–96
and Decimals
STANDARDS
6.NS.C.6, 6.NS.C.7
Goals
Students will place decimal fractions and decimals on number lines.
Students will compare positive and negative decimals and decimal
fractions.
Vocabulary
decimal
decimal fraction
decimal point
equivalent decimals
equivalent fractions
hundredth
negative
positive
tenth
thousandth
PRIOR KNOWLEDGE REQUIRED
Can order and compare fractions
Can order and compare decimals
Can write equivalent fractions and decimals
Understands that opposite numbers are the same distance from 0,
on the opposite side of 0
Is familiar with < and > signs
Is familiar with number lines, including negative decimals and fractions
MATERIALS
BLM Number Lines from −2 to 2 (p. E-69)
BLM Hundredths Number Lines (p. E-70)
BLM Number Lines from −0.2 to 0.2 (p. E-71)
Decimals on number lines. Draw on the board:
AB
C
D
E
012 3
Exercises: Write a decimal and a fraction or mixed number for
points B, D, and E.
Answers: B. 0.8 and 8/10, D. 2.4 and 2 4/10, E. 3.1 and 3 1/10
Negative decimals on number lines divided into tenths. Draw on the
board a number line from −1 to 1 divided into tenths, with only −1, 0, and 1
marked. SAY: We can mark the positive tenths by starting at 0 and moving
right. Demonstrate doing so. Then ASK: How can we mark the negative
tenths? (start at 0 and move left) Mark −.1 and −.2. Point out how much
easier it is to write on the number line when we don’t put the 0 in front of
the decimal point. Challenge a volunteer to finish writing the negative
decimals as quickly as they can, using this approach:
E-22
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Have volunteers write point A as a decimal (0.3) and fraction (3/10). Then
SAY: Point C is 6/10 more than 1. Demonstrate counting the increments after
the 1 to verify this. SAY: So it is one whole and six tenths. Have volunteers
write the number as a mixed number (1 6/10) and as a decimal (1.6).
SAY: This is one whole (pointing to the 1 in 1 6/10) and 6 tenths (pointing
to the 6/10), and this is one whole (pointing to the 1 in 1.6) and 6 tenths
(pointing to the 6).
-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3-.2 -.1 0.1.2.3.4.5.6.7.8.9 1
Exercises: Name the numbers marked.
-101
Answers: −0.7, −0.2, 0.4, 0.9
Positive and negative mixed numbers and decimals on a number line.
Display the number line below.
D
C
BA
-2
-10
Point to point A. ASK: How far from 0 is this? (3 tenths) Is it positive or
negative? (negative) Write “−0.3” under A. Ask a volunteer to mark the point
for B. (−0.5) Now point to C, and ASK: How far from 0 is this? (1 and 4 tenths)
Is it positive or negative? (negative) Write “−1.4” under C. Ask a volunteer
to mark the point for D. (−1.8)
Exercises: Write a decimal and a fraction or mixed number for each
point marked.
AB
-2
C
D
E
-10 1
Answers: A. −1.9, −1 9/10; B. −1.3, −1 3/10; C. −0.5, −5/10; D. 0.3, 3/10;
E. 1.1, 1 1/10
Remind students that, on a number line, numbers on the left are less than
numbers on the right.
Exercises: Use BLM Number Lines from −2 to 2. Write < or >.
6
6
a) -1
−.2b)
−1.4
-1
c) −0.8
1.1
10
10
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) <, b) >, c) <
NOTE: Students who have
trouble locating the negative
decimals on number lines
can benefit from repeating
the Activity from Lesson
NS6-4 with folding number
line strips through the 0 mark.
They can use the number
lines on BLM Number Lines
from −2 to 2.
The Number System 6-29
Partial number lines divided into hundredths. Project onto the board a
number line from BLM Hundredths Number Lines. Circle the interval from
0.20 to 0.30 and tell students you want to enlarge this section. Draw on the
board the number line shown, but without the two points marked:
0.20.3
0.200.30
Point out that we can label 0.20 and 0.30 as tenths, 0.2 and 0.3. The length
of this part of the number line is 1 tenth and it is divided into 10 equal parts,
so it makes sense that each part is a hundredth. Now mark the two points
and, pointing to each in turn, ASK: What number is this? (0.21 and 0.25)
Now draw on the board:
E-23
-0.5
-0.4
-0.50
-0.40
Have students point to which side of the number line 0 will be. (point right)
ASK: How do you know? (because the numbers are negative) Which
decimal number, −0.5 or −0.4, is farther from 0? (−0.5) Point out that as
you go left from 0, the numbers without the minus signs get bigger—0.5 is
bigger than 0.4.
Exercises: Write the points marked from least to greatest.
Answers: −0.49, −0.47, −0.44, −0.42
Exercise: Identify the decimal and the fraction for the marked points.
AB
C
D
E
-0.100.1 0.2
Answers: A −0.09, - 9/100; B −0.03, - 3/100; C 0.05, 5/100;
D 0.13, 13/100; E 0.21, 21/100
Exercise: Use the number lines from BLM Number Lines from −0.2 to 0.2.
Write < or >.
2
a) −.13 b) −0.1 0
c)−.13 −.2 d)0.07 0.11
100
Answers: a) <, b) <, c) >, d) <
(MP.4)
Word problems practice.
a) Which temperature, −3.6°C or −2.58°C, is warmer?
b) Which elevation, −14.2 m or −14.7 m, is higher up?
Answers: a) −2.58°C, b) −14.2 m
Extensions
a) -
4
47
3
58
>><- <b)
10
100
10
100
10
100
c) -
183
<1, 000
10
Answers: a) any of 31 to 39, b) 5, c) 1
(MP.1)
2. Use 10, 100, and 1,000 once each to make the statements true.
1
4
4
> - b) =
100
Answers: a) −27/100 > −3/10, −4/100 > −4/1,000;
b) 1/10 = 100/1,000 or 1/100 = 10/1,000
a) -
(MP.1, MP.2)
E-24
27
>-
3
and -
3.Sarah saw four fish at different elevations: −0.025 km, −0.18 km,
−0.9 km, −1.8 km. Use the information below to decide which fish
was seen at which elevation.
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1. Write any number that works.
The coelacanth lives between 150 m and 400 m below sea level.
The football fish lives between 200 m and 1 km below sea level.
The deep sea angler lives between 250 m and 2 km below sea level.
The rattail lives between 22 m and 2.2 km below sea level.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: coelacanth: −0.18 km; football fish: −0.9 km; deep sea
angler: −1.8 km; rattail: −0.025 km
The Number System 6-29
E-25
NS6-30 Comparing Fractions and Decimals
Pages 97–98
STANDARDS
6.NS.C.7
Vocabulary
decimal
decimal fraction
decimal point
equivalent decimals
equivalent fractions
hundredth
negative
positive
tenth
thousandth
Goals
Students will compare positive and negative decimals and fractions.
PRIOR KNOWLEDGE REQUIRED
Can order and compare fractions
Can order and compare decimals
Can write equivalent fractions and decimals
Understands the concept of an opposite number
Is familiar with < and > signs
Is familiar with number lines, including negative decimals and fractions
Comparing decimal tenths to 1/2. Write on the board:
01
01
Invite volunteers to mark the missing fraction on the top number line (1/2)
and the decimal increments on the bottom number line. ASK: What decimal
does one half represent? (0.5) What is the decimal fraction for 0.5? (5/10)
Remind students that 1/2 is equivalent to 5/10, so it makes sense that they
are at the same place on the number line.
Exercises: Is the decimal more than half or less than half? (Students can
signal thumbs up for more than half and thumbs down for less than half.)
a)
0.3b)
0.6c)
0.8d)
0.4e)
0.2
Comparing decimal hundredths to one half. Remind students how to
compare decimal tenths to decimal hundredths. For example, 0.4 is greater
than 0.37 because 4 tenths is more than 3 tenths and 7 hundredths. ASK:
Which is greater, 0.56 or 0.5? (0.56) Which is greater, 0.38 or 0.5? (0.5)
Students can signal the answers to the exercises below.
Exercises: Is the decimal greater than or less than one half?
a)
0.42b)
0.87c)
0.39d)
0.51
Answers: a) less, b) greater, c) less, d) greater
Comparing decimal tenths to quarters. Add the marks for quarters
to the number line on the board and invite volunteers to label them.
Exercises: Write > or <.
1
b)0.6
a)0.4
4
3
4
c)0.8
3
4
d)0.3
1
4
Answers: a) >, b) <, c) >, d) >
E-26
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) less, b) more, c) more, d) less, e) less
Comparing negative fractions and decimals. Draw a double number line
from −1 to 0, with decimal tenths and with fourths, and invite volunteers to
label all the increments. Have students label −1/2 two ways: −2/4 = −1/2.
Remind students that the order in negative numbers is the opposite of the
order in the positive numbers, so if we know that 1/2 > 0.2, then we also
know that −1/2 < −0.2. ASK: How does the number line show that? (−0.2
is to the right of −1/2 on the number line)
(MP.7)
Exercises: Write > or <.
a) −0.3
−1/4b)
−0.6
−3/4c)
−.6
d) −0.1
−1/4e)
−.8
−3/4f)
−1/2
−1/2
−0.4
Answers: a) <, b) >, c) <, d) >, e) <, f) <
Comparing decimal hundredths to quarters. Write on the board:
1
1
= =
4
4
10
100
ASK: Can you multiply 4 by a whole number and get 10? (no) Cross out
the first equation. ASK: Can you multiply 4 by a whole number and get 100?
(yes, 25) Show this on the board:
1 × 25
25
× 25 =
100
4
Draw a picture, as in the margin, to show why this makes sense. You
can project grid paper onto the board. ASK: How many hundredths
are shaded? (25) How can you write one fourth as a decimal? (0.25)
Ask a volunteer to shade three quarters of the square. ASK: How many
hundredths is three fourths? (75) How can you write three fourths as a
decimal? (0.75) Show the multiplication.
3 × 25
75
= 0.75
× 25 =
100
4
Exercises: Which number is greater?
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a) 0.12 or 1/4
e) 0.45 or 3/4
b) .73 or 1/2
f) .89 or 1/2
c) 0.73 or 3/4
g) 0.87 or 3/4
d) .29 or 1/4
h) .36 or 1/4
Answers: a) 1/4, b) .73, c) 3/4, d) .29, e) 3/4, f) .89, g) .87, h) .36
Remind students that the order is reversed when you compare negative
fractions. For example, 0.12 < 1/4, so −0.12 > −1/4.
(MP.7)
Exercises: Which number is greater?
a) −.19 or −1/4b)
−.63 or −1/2c)
−0.67 or −3/4
d) −.59 or −1/2e)
−0.85 or −3/4f)
−.28 or −1/4
Answers: a) −.19, b) −1/2, c) −0.67, d) −1/2, e) −3/4, f) −1/4
The Number System 6-30
E-27
Converting fractions to decimal fractions and decimals.
Write on the board:
1
2
3
=
= = 20
5
5
100
10
10
Ask volunteers to write the missing numerators. ASK: How many tenths is
one fifth? (2) How can you write one fifth as a decimal? (0.2) Repeat with
2/5 (4 tenths = 0.4) and 3/20 (15 hundredths = 0.15).
Exercises: Write the fraction as a decimal hundredth.
1
1
7
23
1
b) c) d) e) a)
20
25
50
50
25
Bonus:
24
300
Answers: a) 0.05, b) 0.04, c) 0.02, d) 0.14, e) 0.92, Bonus: 0.08
(MP.1)
When students have finished the exercise above, have them order the
fractions from greatest to least. Point out that writing the fractions as
decimal hundredths is like converting them to the common denominator
100, so ordering fractions becomes easy.
Solution: decimals from least to greatest are 0.92, 0.14, 0.08, 0.05, 0.04,
0.02; so the fractions from least to greatest are 23/25, 7/50, 24/300, 1/20,
1/25, 1/50
Bonus: Write the fractions with denominator 1,000, then put them in order
from least to greatest.
28
35
16
7, 000
500
200
Answers: 70/1,000, 80/1,000, 4/1,000; from least to greatest: 4/1,000,
70/1,000, 80/1,000
Ordering positive and negative decimals and fractions. Remind students
that if they can order positive numbers, then they can order negative
numbers too. Since 2 < 3, then −2 > −3.
SAY: Positive numbers are always larger than negative numbers.
3
1
89 89
1
c)
−2.14, -3
,
−.45 - d)
,
100 10 100
5
4
Answers: a) −6/10, −0.48, −35/100, b) −7/10, −0.5, −7/100,
c) −.45, −1/4, 3/5, d) −3 1/100, −2.14, 89/100, 89/10
(MP.4)
Word problems practice.
3
a) Which elevation, −2.8 m or − 2 m, is higher up?
5
4
b) Which temperature, - 3 °C or −3.76°C, is warmer?
5
3
Answers: a) −2 m, b) −3.76°C
5
E-28
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Exercises: Write the numbers from least to greatest.
6
7
35
7
−0.48 b) −0.5 a) 100
100
10
10
Extensions
(MP.7)
1. a) Write the decimal as a decimal fraction. Then solve for n.
n
n
n
i)
= 0.7 ii)
= 0.13 iii) = 0.327
1, 000
10
100
3
8
7
= 0.03 v)
= 0.0007
= 0.8 vi)
n
n
n
b) Use equivalent fractions to solve for n.
n
n
n
i)
= 0.4 ii) = 0.3 iii)= 3.27
1, 000
100
100
iv)
iv)
n
n
n
= 0.6 v)
= 0.24 vi)
= 1.25
5
25
4
Sample solutions:
n
7
n
4
40
a) i)
=
, so n = 7, b) i)
= 0.4 =
=
, so n = 40
10 10
100
10 100
Answers: a) i) 7, ii) 13, iii) 327, iv) 100, v) 10, vi) 10,000; b) i) 40, ii) 300,
iii) 327, iv) 3, v) 6, vi) 5
2.Write the most reduced decimal fraction that is equivalent to each
fraction. Hint: Reduce the fraction first.
546
35
210
3
b)
c)
d)
a)
7
, 000
6
125
300
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 5/10, b) 28/100, c) 7/10, d) 78/1,000
The Number System 6-30
E-29
NS6-31 Multi-Digit Addition
Pages 99–101
STANDARDS
preparation for 6.NS.B.3
Goals
Students will add multi-digit numbers, regrouping where necessary.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Understands place value
Can use base ten materials to represent numbers
Can add multi-digit numbers without regrouping
algorithm
carrying
regrouping
Using base ten materials and place value charts for addition.
Write on the board:
27 + 15 +
Tens
2
1
Ones
7
5
Regroup:
27
15
Demonstrate making 27 and 15 with tens and ones blocks. Point out that
you can add the tens and ones separately. ASK: How many ones are there
in total? (12) How many tens? (3) Point out that you can replace 10 ones
with 1 tens block, and demonstrate putting together 10 ones blocks to
make 1 tens block. ASK: Now how many ones do we have? (2) And how
many tens? (4) Complete the last two rows of the chart. ASK: What number
do we have altogether? (42)
Exercises: Draw the place value charts and add.
a)46 + 36
b) 39 + 28
Bonus: 29 + 11 + 34
Standard notation for addition with regrouping—2 digits. SAY: When you
use a tens-and-ones chart, you add the tens and ones first, then regroup.
When you add the sum directly using standard notation, you regroup right
away: 9 + 8 = 17, which is 1 ten + 7 ones, so you put the 7 in the ones
column and add the 1 to the tens column. Demonstrate the first step of
writing the sum of the ones digits as shown in the margin.
Exercises: Add using the standard notation.
a)56 + 29
b) 39 + 45
e)85 + 28
f) 99 + 15
Bonus: 29 + 74 + 63
c) 76 + 14
g) 78 + 9
d) 37 + 48
Answers: a) 85, b) 84, c) 90, d) 85, e) 113, f) 114, g) 87, Bonus: 166
Some students may, at first, need to do only the first step for all the
problems, then only the second step. Be sure that, eventually, students
can do both steps together.
E-30
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 82, b) 67, Bonus: 74
Make sure students align the place values properly. Encourage struggling
students to write each digit in its own cell of the grid.
Using expanded form to add 3-digit numbers. Show how to add
152 + 273 using the expanded form:
152
+273
hundred
hundreds
+
+
tens
tens
+
+
ones
ones
After regrouping:
hundreds
hundreds
+
+
tens
tens
+
+
ones
ones
Have students signal the answers for each blank. Point out that sometimes
students will need to regroup a second time if one place value still has a
2-digit number. Exercises:
a)349 + 229
d)186 + 596
b) 191 + 440
e) 159 + 242
c) 195 + 246
f) 869 + 237
Answers: a) 578, b) 631, c) 441, d) 782, e) 401, f) 1,106
1
+
1
5
2
2
7
3
4
2
5
Standard notation for addition with regrouping—3 digits. Now
demonstrate using the standard algorithm alongside the place value chart
for the first example you did together (152 + 273). Point out that after
regrouping the tens, you add the 1 hundred that you carried over from the
tens at the same time as the hundreds from the two numbers, so you get
1 + 1 + 2 = 4 hundreds.
Have students rewrite any two of the addition statements above, this time
using the standard algorithm. Exercises: Add using the standard notation.
a)358 + 217
b) 475 + 340
e)695 + 258
f) 487 + 999
Bonus: 427 + 382 + 975 + 211
c) 643 + 847
g) 658 + 247
d) 978 + 791
h) 675 + 325
Answers: a) 575, b) 815, c) 1,490, d) 1,769, e) 953, f) 1,486, g) 905,
h) 1,000, Bonus: 1,995
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Struggling students can use place value charts alongside the standard
algorithm. Exercises: Add 4-, 5-, and 6-digit numbers.
Not correct:
a)1,358 + 7,217
b) 1,235 + 7,958
c) 4,658 + 8,347
d)94,358 + 18,647
e) 862,595 + 198,857 f) 394,348 + 415,656
Bonus: g) 3,875 + 5,827 + 2,132 h) 15,891 + 23,114 + 36,209
841,765
+ 832,491,780
Answers: a) 8,575, b) 9,193, c) 13,005, d) 113,005, e) 1,061,452,
f) 810,004, Bonus: g) 11,834, h) 75,214
Correct:
SAY: You have to make sure the place values are lined up, the ones with the
ones, tens with tens. This can be tricky when the numbers have a different
number of digits, but you just have to make sure the ones digits are aligned
and the commas are aligned. Exercises:
Example:
841,765
+ 832,491,780
a)32,405 + 9,736
b) 789,104 + 43,896
d)94,358 + 8,647
e) 652,722 + 798
Bonus: 17,432 + 946 + 3,814 + 568,117
The Number System 6-31
c) 999,678 + 1,322
f) 5,973 + 297,588
E-31
Answers: a) 42,141, b) 833,000, c) 1,001,000, d) 103,005, e) 653,520,
f) 303,561, Bonus: 590,309
Word problems practice.
a)Ron ran 1,294 km one year and 1,856 km the next. How many
kilometers did he run altogether?
b)In an election between three candidates, the candidate who won
received 567,802 votes. The other two received 213,435 votes and
342,095 votes. Did the candidate who won get more than the other
two combined?
Answers: a) 3,150 km, b) Yes; the other two combined received
only 555,530 votes
Extension
(MP.1)
A palindrome is a number whose digits are in the same order when
written from right to left as when written from left to right. (747 is a
palindrome, 774 is not)
a)Which numbers are palindromes?
33, 12, 512, 515
Answers: 33, 515
b)I am a 2-digit palindrome, and 200 more than me is also a palindrome.
What number am I?
Answer: 22, 22 + 200 = 222
c)A reverse of a number is the number in which the digits are in the
opposite order. Example: A reverse of 13 is 31; a reverse of
23,567 is 76,532.
Write the reverse of each number and add it to the number itself.
i)
35ii)
21iii)
52iv)
435
v)
1,428
Answers: i) 88, ii) 33, ii) 77, iv) 969, v) 9669; the numbers are all
palindromes
d)Find a 2-digit number for which you don’t get a palindrome by adding
it to its reverse. Then add the resulting number to the reverse. Did you
get a palindrome? If not, add the resulting number to its reverse. Repeat
until you get a palindrome.
Tell students that most numbers will eventually become palindromes,
but that mathematicians have not proven whether all numbers will. Over
2,000,000 steps have been tried (using a computer, of course) on the
number 196, but mathematicians have still not found a palindrome.
E-32
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
What do you notice?
NS6-32 Multi-Digit Subtraction
Pages 102–104
STANDARDS
preparation for 6.NS.B.3
Goals
Students will subtract multi-digit numbers, regrouping where necessary.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Understands place value
Can use base ten materials to represent numbers
Understands subtraction as taking away
Can subtract multi-digit numbers without regrouping
Can add multi-digit numbers with or without regrouping
Knows that addition can be used to check subtraction
algorithm
regrouping
MATERIALS
base ten blocks for demonstration
Subtracting 2-digit numbers with regrouping. Tell students there is a
vending machine that takes only dimes and pennies. You have 4 dimes and
6 pennies, and you want to buy something that costs 19 cents. The vending
machine only takes exact change. ASK: What can I do? (wait for someone
to pass by and see whether they will give me 10 pennies for 1 dime) Show
this on the board:
Before regrouping
After regrouping
-
dimes
4
1
pennies
6
9
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
-
dimes
3
1
2
pennies
16
9
7
Finally, demonstrate the standard notation for regrouping, as shown in margin.
3
16
4
6
Exercises: Subtract using the standard algorithm.
2
9
a)66 − 39 1
7
Answers: a) 27, b) 43, c) 9, d) 23
b) 81 − 38 c) 74 − 65 d) 40 − 17
Checking answers for subtraction. Have students perform addition
with regrouping to check the answers for the problems they did above.
For example, in a), add 39 + 27. Is your answer 66?
Bonus: What other strategies can you use to check your answer?
(For example, in a), subtract 69 − 39 = 30 and subtract 3 to get 27.
Regrouping when necessary.
Exercises: Is regrouping required? Students can signal the answers.
a)58 − 19
b) 34 − 13
c) 85 − 27
d) 66 − 8
Answers: a) yes, b) no, c) yes, d) yes
The Number System 6-32
E-33
Struggling students can figure out whether they need to regroup by thinking
of a purchase from the vending machine that only takes dimes and pennies.
Do they have exact change or do they need to regroup? Is the number of
ones in the larger number enough?
Exercises: Subtract. Regroup when necessary.
a)58 − 29
e)48 − 18
b) 36 − 13
f) 74 − 8
c) 82 − 27
g) 69 − 25
d) 39 − 27
h) 91 − 46
Answers: a) 29, b) 23, c) 55, d) 12, e) 30, f) 66, g) 44, h) 45
-
2
16
3
6
7
1
9
2
1
7
5
Using the standard algorithm to subtract 3-digit numbers with
regrouping. Remind students that when you have more digits, you
might regroup not only 1 ten as 10 ones, but also 1 hundred as 10 tens,
or 1 thousand as 10 hundreds, and so on. Do the example in the margin
together as a class before having students work individually.
Exercises: Subtract, then check by adding.
a) 358 − 129
b) 346 − 183
c) 862 − 257
d) 309 − 127
Answers: a) 229, b) 163, c) 605, d) 182
14
-
7
4
12
8
5
2
4
5
9
3
9
3
Do the example in the margin together as a class. Emphasize that you write
the second regrouping above the first one, not over the first regrouping, so
that you can see each step easily.
Exercises: Subtract, then check by adding.
a) 563 − 175 b) 541 − 273
c) 422 − 358
d) 542 − 289
Answers: a) 388, b) 268, c) 64, d) 253
4 10
5 0 3
- 1 8 4
9
4 10 13
5 0 3
- 1 8 4
Borrowing from zero. Present (as vertical subtraction) a case in which the
ones need to be regrouped, but the tens digit in the minuend is 0: 503 − 184.
ASK: Do I have enough ones to subtract? (no) What do I need to do?
(regroup 1 ten as 10 ones) What is my problem? (there are no tens to take
from) Explain that, in this case, we need to regroup 1 hundred as 10 tens,
then we can easily regroup 1 ten as 10 ones. Show how to record the
process (see margin).
Then subtract each place value to get 319. Remind students to line up
the place values properly in the next exercises.
Exercises: Subtract using the standard algorithm.
a) 402 − 169
b) 501 − 223
c) 402 − 36
d) 500 − 289
Answers: a) 233, b) 278, c) 366, d) 211
Using the standard algorithm to subtract 4-, 5-, or 6-digit numbers.
In each case below, solve the first problem as a class, then have students
practice individually. Be sure students line up the place values.
E-34
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Bonus: Make up your own subtraction question that requires regrouping
twice. Ask a partner to solve your question.
Exercises: Subtract.
Only one digit needs to be regrouped.
a)4,562 − 417 b) 62,187 − 41,354
c) 207,122 − 24,001
Two digits need to be regrouped, but without borrowing from zero.
d)54,137 − 28,052
e) 9,319 − 6,450
f) 207,145 − 178,321
Borrowing from zero.
g)4,037 − 2,152
h) 90,319 − 6,405
i) 145,207 − 1,128
Several consecutive digits need to be regrouped.
j)3,695 − 1,697
k) 1,000 − 854
l) 10,000 − 4,356
m)45,683 − 1,487
n) 33,116 − 13,435
o) 101,363 − 7,907
Answers: a) 4,145, b) 20,833, c) 183,121, d) 26,085, e) 2,869, f ) 28,824,
g) 1,885, h) 83,914, i) 144,079, j) 1,998, k) 146, l) 5,644, m) 44,196,
n) 19,681, o) 93,456
Word problems practice.
a)Construction of the Statue of Liberty began in France in 1881. When it
was completed, the statue was shipped to the United States and it was
rebuilt there in 1886. How long ago was it built in France? How long
ago was it rebuilt in the United States?
b)In 1810, the population of New York City was 96,373. In 2010, the
population of New York City was 8,175,133. How much did the
population grow in those 200 years?
Answers: a) Depends on current year, e.g., 2013 − 1881 = 132 years ago,
and 2013 − 1886 = 127 years ago, b) 8,078,760
Extensions
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.1, MP.3)
1.Use the digits 1, 2, 3, and 4 once each to write two numbers with the
smallest possible difference. Repeat with 1 through 6, 1 through 8,
and 0 through 9.
Solution: The numbers need to be as close as possible on the
number line, so the tens digits need to be adjacent: 1 and 2, 2 and 3,
or 3 and 4. To make the difference as small as possible, the larger
number needs the smallest number of ones possible, and the smaller
number needs the largest number of ones possible. This gives three
pairs: 23 − 14 = 9, 31 − 24 = 7, and 41 − 32 = 9. The equation
31 − 24 = 7 is the smallest difference.
Answers: 1 through 6: 412 − 365 = 47; 1 through 8: 5,123 − 4,876 = 247;
0 through 9: 50,123 − 49,876 = 247
(MP.8)
The Number System 6-32
2.
A fast method for subtracting from powers of 10 without regrouping.
Do you need to regroup when you subtract from a number whose digits
are all 9?
E-35
To subtract a number from a power of 10, subtract 1, then subtract without
regrouping. Add 1 back to the answer. Examples:
99 − 42 = 57, so 100 − 42 = 58
999 − 423 = 576, so 1,000 − 423 = 577
Use this method to subtract.
a)1,000 − 768
b) 10,000 − 3,892
c) 100,000 − 56,381
Answers: a) 232, b) 6,108, c) 43,619
(MP.8)
3.Pretend that there is a vending machine that only takes one-dollar bills,
dimes, and pennies, and only takes exact change. You have 503¢:
5 one-dollar bills and 3 pennies. You want to buy an item that costs
184¢. What would you trade for, in one step?
Answer: Trade a dollar for 9 dimes and 10 pennies.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Tell students that they did the two regrouping steps in one step.
They traded a hundred for 9 tens and 10 ones instead of a hundred
for 10 tens, then a ten for 10 ones.
E-36
Teacher’s Guide for AP Book 6.1
NS6-33 Adding and Subtracting Decimals
Pages 105–108
STANDARDS
6.NS.B.3
Goals
Students will add and subtract positive decimals.
Vocabulary
algorithm
decimal point
hundredth
regrouping
tenth
thousandth
1 whole
PRIOR KNOWLEDGE REQUIRED
Understands place value
Can represent decimals using base ten materials
Can add fractions with the same denominator
Can tell how many hundredths are in a number with two decimal places
Can tell which number has a given number of hundredths
Can add and subtract multi-digit numbers with or without regrouping
Uses addition to check subtraction
Using a hundreds block as a whole. Tell students that they can use
a hundreds block as one whole. Draw on the board the picture in the
margin. SAY: Tenths and hundredths work just like other place values—
ten times each one is the next one over to the left. So you can regroup
them the same way.
Exercises: Use base ten blocks to regroup so that each place value
has a single digit.
1 tenth
1 hundredth
a) 3 tenths + 12 hundredths
b) 7 ones + 18 tenths
c) 7 ones + 15 tenths + 14 hundredths
Answers: a) 4 tenths + 2 hundredths, b) 8 ones + 8 tenths,
c) 8 ones + 6 tenths + 4 hundredths
SAY: You can regroup thousandths the same way too.
Exercises: Regroup without using base ten blocks.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a) 7 hundredths + 13 thousandths
b) 1 hundredth + 35 thousandths
c) 3 tenths + 25 thousandths
You may need to regroup twice.
d) 6 tenths + 14 hundredths + 13 thousandths
e) 5 tenths + 34 hundredths + 26 thousandths
Answers: a) 8 hundredths + 3 thousandths, b) 4 hundredths + 5 thousandths,
c) 3 tenths + 2 hundredths + 5 thousandths, d) 7 tenths + 5 hundredths +
3 thousandths, e) 8 tenths + 6 hundredths + 6 thousandths
SAY: When you subtract, you sometimes need to regroup a larger place
value as ten of a smaller place value. For instance, to subtract 52 − 27,
you need to regroup 1 ten as 10 ones. When subtracting decimals, you
might also need to regroup a larger place value as a smaller place value.
The Number System 6-33
E-37
Exercises: Trade 1 tenth for 10 hundredths or 1 hundredth for
10 thousandths.
a) 7 tenths + 3 hundredths
b) 8 tenths + 0 hundredths
c) 9 hundredths + 0 thousandths
Answers: a) 6 tenths + 13 hundredths, b) 7 tenths + 10 hundredths,
c) 8 hundredths + 10 thousandths
Adding decimals. Write on the board:
21 14 35
21
14
35
21 + 14 = 35 +
=
+
=
10 10 10
100 100 100
Write the first equation in vertical format, then ask volunteers to write the
other two equations as decimals in vertical format:
21
+ 14
35
2.1
+ 1.4
3.5
0.21
+ 0.14
0.35
Explain that you can add and subtract decimals the same way you add
whole numbers—line up the place values—but, instead of adding or
subtracting ones and tens, you’re adding or subtracting tenths and ones
or hundredths and tenths.
Exercises: Add or subtract by lining up the place values. Use grid paper.
a)3.4 + 1.5 b) 4.6 − 2.1 c) 8.53 + 1.26
Bonus: 134.3 + 245.5
Answers: a) 4.9, b) 2.5, c) 9.79, Bonus: 379.8
SAY: You might need to regroup the same way you do with whole numbers.
Exercises: Add or subtract. Use grid paper.
a)23.5 + 1.8
e).78 − .42
b) 2.74 + 3.58
f) 0.37 − 0.29
c) 192.8 + 15.4 d) 4.186 + 1.234
g) 34.85 − .65 h) 6.432 − 2.341
SAY: The answer to d) is 5.420 or just 5.42. ASK: What other answer can
be written shorter? (part g) can be written as 34.2)
Adding and subtracting decimals with different numbers of digits
to the right of the decimal point. Write on the board:
23.7
+ 2.15
SAY: It’s the place values that need to be lined up, not the last digits. You
can make sure the place values are lined up by lining up the decimal points,
because the decimal point is always between the ones and tenths. Have a
volunteer add: 23.7 + 2.15. (25.85)
E-38
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 25.3, b) 6.32, c) 208.2, d) 5.420 or 5.42, e) .36, f) 0.08,
g) 34.20 or 34.2, h) 4.091
Exercises: Add.
a).78 + .4 b) 0.37 + 0.495 c) 34.85 + 65.1 d) 1.43 + 2.904
Answers: a) 1.18, b) 0.865, c) 99.95, d) 4.334
Adding whole numbers and decimals. Write on the board:
32 + 4.7
ASK: How can you line up the decimal points when 32 has no decimal
point? PROMPT: Where should the decimal point go in 32? (after the 2)
SAY: You can look at 32 as 32.0, or 32 and 0 tenths. Now you can line up
the decimal points and add. Have a volunteer do so:
32.0
+ 4.7
36.7
Exercises: Add.
a)4 + 13.7
b) 16 + 2.3
c) 38 + 14.71
Answers: a) 17.7, b) 18.3, c) 52.71
SAY: You can check your answers by adding the fractions. Write on the board:
7
7
= 36
= 36.7
32 + 4.7 = 32 + 4
10
10
Exercises: Check your answers by adding the fractions.
Subtracting decimals. SAY: You can subtract by lining up the decimal
points too. You might have to add zeros to make both decimals have the
same number of digits after the decimal point. Demonstrate as shown:
36.94 36.94 34.8 34.80
− 21.65
− 21.65
− 15.3 − 15.30
21.64 13.15
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
NOTE: Adding zeros to the bottom number is optional, since it is easy to
subtract, for example, 4 − 0, but adding zeros to the top number is necessary.
Exercises: Subtract. Use grid paper.
a)7.4 − 2.1
e)1.7 − .42
b) 6.93 − 4.52 c) 8.56 − 3.87
f) 20.37 − 5.294 g) 2 − 0.52
d) 6.5 − 3
h) 10 − 2.413
Answers: a) 5.3, b) 2.41, c) 4.69, d) 3.5, e) 1.28, f) 15.076, g) 1.48, h) 7.587
Students can check their answers using addition.
(MP.4)
Word problems practice.
a)Tianna made 0.6 L of milkshake by adding ice cream to 0.48 L of milk.
How much ice cream did she add?
(MP.3)
The Number System 6-33
b)Mario placed a table 1.23 m long along a wall 3 m long. If his bed is
2.13 m long, will it fit along the same wall? Explain.
E-39
c)Natalie cut 0.86 m of wood board to make a shelf. The leftover piece is
1.45 m long. How long was the board before she cut off the shelf?
Answers: a) 0.6 L − 0.48 L = 0.12 L, b) 1.23 m + 2.13 m = 3.46 m,
the bed will not fit, c) 2.31 m
Extensions
1. a) Add mentally.
i) 2.6 + 3.4 ii) 0.8 + 19.2
iii) 5.7 + 5.3
b)Add the two numbers that are easiest to add first. Then find the
total: 4.7 + 7.9 + 5.3
(MP.5)
c)Would you use pencil and paper to add, or would you add
mentally?
i)3.5 + 4.5
ii) 3.69 + 2.74
iii) 7.63 + 2.37
Answers: 1. a) i) 6, ii) 20, iii) 11, b) 4.7 + 5.3 = 10 and 10 + 7.9 = 17.9,
c) i) mentally, ii) pencil and paper, iii) mentally
2.Make up two decimals that add to 4.53. Check your answer by adding
them.
(MP.1)
3.Subtract 1.27 − 0.5 using the number line. Do you get the same answer
by lining up the decimal points?
0 .1.2.3.4.5.6 .7.8.911.1
1.21.3
4.Aziz says that 0.91234 is the largest number less than 1 that can be
added to 0.08765 without needing to regroup. Is he correct?
Hint: 0.08765 = 0.087650
Answer: No, 0.912341 + 0.08765 also does not need regrouping, but
0.912341 is larger than 0.91234. In fact, any other decimal produced by
adding digits to the right of the 4 are all larger than 0.91234, and can be
added to 0.08765 without needing to regroup.
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Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.3)
NS6-34 Rounding
Pages 109–110
STANDARDS
preparation for 6.NS.B.3
Vocabulary
approximately equal to
sign (≈)
estimate
round number
rounding
Goals
Students will round whole numbers to the nearest ten, hundred, or
thousand, and estimate sums and differences using rounding.
PRIOR KNOWLEDGE REQUIRED
Can determine which multiple of ten, a hundred, or a thousand a number
is between
Can find which multiple of ten, a hundred, or a thousand a given number
is closest to
Can regroup when adding whole numbers
Rounding 2-digit numbers to the nearest ten. Draw a number line from
10 to 30, with 10, 20, and 30 in a different color than the other numbers.
101112131415161718192021222324252627282930
Circle the numbers 13, 18, 21, and 26, one at a time, and ask volunteers
to draw an arrow showing which ten is closest. Tell students that we often
want to pretend a number is equal to its closest ten, because multiples of
ten are nice round numbers and easier to work with. That process is called
rounding to the nearest ten.
Exercises: Round to the nearest ten.
a)14
b)19 c)27
d)22
Answers: a) 10, b) 20, c) 30, d) 20
Write on the board:
37
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ASK: How many tens are in 37? (3) SAY: The number 37 is between 3 tens
and 4 tens. That means it’s between 30 and 40. Is 37 closer to 30 or to 40?
(40) Repeat with 94. (94 is between 90 and 100, and is closer to 90)
Exercises: Round to the nearest ten.
a)97 b)34 c)46 d)72 e)81 f) 69 g)3
h)7
Answers: a) 100, b) 30, c) 50, d) 70, e) 80, f) 70, g) 0, h) 10
Rounding 3-digit numbers to the nearest hundred. Make a table with two
headings: “Closer to 300” and “Closer to 400.” Name several numbers
(342, 356, 312, 385, 352, 331, 327, 390, 309, 351) and ask students to signal
whether the number will be closer to 300 (thumbs down) or to 400 (thumbs
up). Place the numbers in their correct column as students answer.
ASK: What digit are you looking at to decide? (the tens digit) SAY: When the
tens digit is 0, 1, 2, 3, or 4, you round down. When the tens digit is 5, 6, 7,
8, or 9, you round up.
The Number System 6-34
E-41
Exercises: What is the nearest hundred?
a)
457b)
612c)
908d)
792e)
729
Answers: a) 500, b) 600, c) 900, d) 800, e) 700
Tell students that choosing the closest hundred is called rounding to the
nearest hundred. Numbers less than 350 are rounded down to 300 and
numbers more than 350 are rounded up to 400. ASK: Is 350 closer to
300 or to 400? (neither, it is the same distance from both) SAY: I want to
pick 300 or 400 anyway, and I only want to have to look at the tens digit
to decide. ASK: Where are all the other numbers with tens digit 5? (in
the Closer to 400 column) SAY: When a number is equally close to both
hundreds, you round up.
Exercises: Round to the nearest hundred.
a)
250b)
50 c)
850d)
950e)
650
Answers: a) 300, b) 100, c) 900, d) 1,000, e) 700
SAY: Do the same thing when rounding to the nearest ten.
Exercises: Round to the nearest ten.
a)25
b)45
c)95
d)5
e)55
f)75 g)35
Answers: a) 30 , b) 50, c) 100, d) 10, e) 60, f) 80, g) 40
Rounding multi-digit numbers to any place value. Show students
how numbers can be rounded in a grid. Follow the steps shown below.
Example: Round 12,473 to the nearest thousand.
Step 1: Underline the digit you are rounding to.
1 2 4 7 3
Step 2: Put your pencil on the digit to the right of the one you are rounding to.
Step 3: B
eside the grid, write “round up” if the digit under your pencil is
5, 6, 7, 8, or 9, or “round down” if the digit is 0, 1, 2, 3, or 4.
round down
1 2 4 7 3
Step 4: R
ound the underlined digit up or down according to the instruction
you have written. (Write your answer in the grid.)
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1 2 4 7 3
2
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1 2 4 7 3
Exercises: Round the underlined digit up or down as indicated.
a) round down
b) round up
1 6 4 7 3
2 0 7 5 2
c) round down
5 1 2 1 5
Answers: Round the underlined digit to: a) 6, b) 1, c) 5
Step 5: Change all digits to the right of the rounded digit to zeros.
1 2 4 7 3
2 0 0 0
Step 6: Copy all digits to the left of the rounded digit as they are.
1 2 4 7 3
1 2 0 0 0
SAY: So 12,473 to the nearest thousand is 12,000. That makes sense
because the number is between 12,000 and 13,000, but is closer to 12,000
than to 13,000.
Exercises: Round to the underlined place value.
a)35,623
c)12,943
b)12,871
d)9,587
Answers: a) 36,000, b) 12,900, c) 12,940, d) 9,600
Rounding with regrouping. Write on the board:
round up
1 7 9 7 8
10 0 0
SAY: The 10 hundreds need to be regrouped as 1 thousand. Add it to the
7 thousands to get 8 thousands. Then copy the remaining digits to the left:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1 7 9 7 8
1 8 0 0 0
SAY: This makes sense because the number is between 179 hundreds
and 180 hundreds.
Exercises: Round to the stated place value. Use grid paper.
a) 39,673, thousands
c) 12,993, tens
b) 12,971, hundreds
d) 9,987, hundreds
Answers: a) 40,000, b) 13,000, c) 13,000, d) 10,000
Estimating sums and differences by rounding. Write on the board:
475 + 321 500 + 300
The Number System 6-34
E-43
ASK: Which of these additions is easier? (the second one) Do you think the
answers will be close? (yes) Why? (because 500 is close to 475 and 300 is
close to 321) Have students calculate both sums to check their prediction.
(796 and 800) SAY: Sometimes you don’t need an exact answer, just an
answer that is close. We call this estimating. Tell students that there is a
symbol that looks almost like an equal sign to say that two numbers are
almost equal. Write on the board:
475 + 321 ≈ 500 + 300 = 800
SAY: The symbol that looks like a squiggly equal sign means “almost
equal.” In mathematics we say approximately equal, and we call this sign
“approximately equal to” sign.
Exercises: Estimate by rounding each number to the stated place value.
a)421 + 159 (tens)
c)3,652 + 4,714 (hundreds)
e)13,891 − 11,990 (thousands)
b) 4,501 − 1,511 (hundreds)
d) 7,980 + 1,278 (thousands)
f) 51,456 − 23,512 (hundreds)
Bonus
g) 8,541 + 972 + 37,218 (thousands) h) 6,730 + 9,050 − 612 (hundreds)
Answers: a) 580, b) 3,000, c) 8,400, d) 9,000, e) 2,000, f) 28,000,
Bonus: g) 47,000, h) 47,000
(MP.5)
Using estimation to check whether a calculated sum or difference
is reasonable. Write on the board:
273 + 385
Tell students that Daniel added these two numbers and got the answer 958.
ASK: Does the answer seem reasonable? (no) How can you tell? (the answer
will be much less than 900) Point out that even rounding both numbers up
will get only 700, so the sum cannot be more than 900.
Exercises: Is the answer reasonable?
b) 30,417 + 6,685 = 97,267
Solution: a) yes, the answer is 1,245 + 683 ≈ 1,200 + 700 = 1,900;
b) no, the answer is 30,417 + 6,685 ≈ 30,000 + 7,000 = 37,000,
not about 100,000
Extensions
1.Round 365,257 to the nearest ten, hundred, thousand,
and ten thousand.
Answers: nearest ten: 365,260; nearest hundred: 365,300; nearest
thousand: 365,000; nearest ten thousand: 370,000
(MP.3)
E-44
2.Carm says that 347 rounds to 350 and 350 rounds to 400, so 347
rounds to 400. Is she correct? Explain.
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)1,245 + 683 = 1,928
Answer: No. Carm needs to think about which place value she is
rounding to. If she is rounding to the hundreds, the number 347 rounds
to 300 because it is less than 350 and is closer to 300 than to 400.
If she is rounding to the tenths, 347 rounds to 350 because it is closer
to 350 than to 340.
3. Write two numbers that can be rounded to 20,000, 17,000, and 17,400.
Sample answers: 17,357, 17,432
(MP.8, MP.3)
4. Use the number line to round each negative number to the nearest ten.
-30 -29 -28 -27 -26 -25 -24 -23 -22 -21 -20
−29 ≈ −22 ≈ −26 ≈ −24 ≈ How is rounding negative numbers similar to rounding positive numbers?
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: −29 ≈ −30, −22 ≈ −20, −26 ≈ −30, −24 ≈ −20; Round
as though the decimals are positive, then put back the negative sign.
The Number System 6-34
E-45
NS6-35 Rounding Decimals
Pages 111–113
Goals
STANDARDS
preparation for 6.NS.B.3
Students will round decimals to the nearest one, tenth, hundredth,
or thousandth. Students will round to the nearest whole number to
estimate sums and differences.
Vocabulary
approximately equal to (≈)
rounding
PRIOR KNOWLEDGE REQUIRED
Can round whole numbers to any place value, including regrouping
Can regroup when adding decimals
Rounding decimals. Tell students that you use the same rule to round
decimals as you use to round whole numbers.
Example: Round 2.365 to the nearest tenth.
Step 1: Underline the digit you are rounding to.
2 3 6 5
Step 2: Put your pencil on the digit to the right of the one you are rounding to.
2 3 6 5
Step 3: B
eside the grid, write “round up” if the digit under your pencil is
5, 6, 7, 8, or 9, or “round down” if the digit is 0, 1, 2, 3, or 4.
round up
2 3 6 5
2 3 6 5
4
Step 5: Change all digits to the right of the rounded digit to zeros.
2 3 6 5
4 0 0
Step 6: Copy all digits to the left of the rounded digit as they are.
2 3 6 5
2 4 0 0
SAY: So 2.365 to the nearest tenth is 2.4. That makes sense because
the number is between 2.3 and 2.4, but is closer to 2.4 than to 2.3.
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Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Step 4: R
ound the underlined digit up or down according to the instruction
you have written. (Write your answer in the grid.)
Exercise: Round to the underlined place value.
a)13.451
b)38.479
c)612.389
d)804.749
Answers: a) 13.5, b) 38, c) 612.39, d) 804.7
Rounding with regrouping. Write on the board: “2.965.” Demonstrate
rounding 2.965 to the nearest tenth.
round up
2 9 6 5
2 10
3 0
SAY: 2.965 rounded to the nearest tenth is 3.0, or just 3.
Exercises: Round to the stated place value. Use grid paper.
a) 43.698, hundredths
c) 59.517, ones
b) 74.953, tenths
d) 84.09971, thousandths
Answers: a) 43.70 or 43.7; b) 75.0 or 75; c) 60.0 or 60; d) 84.100 or 84.1
Estimating sums and differences by rounding to the nearest whole
number to check for reasonableness. Tell students that they can round
to the nearest whole number to check whether the answers to sums and
differences are reasonable. Write on the board:
162.34 + 16.234 ≈ 162 + 16 = 178
Exercises: Somebody punched these numbers into a calculator and
got these answers. Are they reasonable?
a)162.34 + 16.234 = 178.574
b) 387.52 − 53.31 = 5.21
Answers: a) yes, the answer is about 178; b) no, 387.52 − 53.31 ≈
388 − 53 = 335
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Extensions
1.Place the decimal point by estimating rather than carrying
out the operation.
a)27.21 + 832.5 = 8 5 9 7 1
b) 57.23 − 2.5 = 5 4 7 3
Answers: a) 859.71, b) 54.73
2.Without calculating the sum, how can you tell whether the sum
is greater than or less than 435?
9.5 + .37 + 407.63
Answer: The sum is less than 10 + 1 + 410 = 421,
so it is less than 435.
The Number System 6-35
E-47
(MP.3, MP.6)
3.Decide what place value it makes sense to round each of the following
to. Round to the place value you selected. Justify your decisions.
Height of person: 1.524 m
Height of tree: 13.1064 m
Length of bug: 1.267 cm
Distance between Washington, DC, and Hong Kong: 13,116.275 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: 973.91 ft2
Area of New York State: 141,299 km2
Angle between two streets: 82.469°
Time it takes to blink: 0.33 s
Speed of a car: 66.560639 mi/h
Time it takes to ski a downhill course: 233.81 s
Answers: Answers will vary. The larger the number, the less important
the smaller place values become. The way the measurement will be
used is also a factor. For example, when measuring the time it takes to
ski a downhill course, more accuracy might be needed to determine a
world record than it would be to keep training records.
(MP.6)
4.Estimate the value of 14.502 − 13.921 by rounding both numbers
to the nearest:
a)ten
b)one
c)tenth
d)hundredth
Answers: a) 10 − 10 = 0, b) 15 − 14 = 1, c) 14.5 − 13.9 = 0.6,
d) 14.50 − 13.92 = 0.58, rounding to the nearest ten or one makes
estimating the fastest, but rounding to the nearest hundredth makes
it most accurate. Point out to students that there is always a trade-off
between speed and accuracy.
E-48
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Which place value made estimating the difference the fastest? Which
place value made estimating the difference the most accurate?
NS6-36 Decimals Review
Pages 114–115
STANDARDS
6.NS.B.3, 6.NS.C.7
Vocabulary
algorithm
decimal
hundredth
negative
opposite
positive
regrouping
tenth
thousandth
Goals
Students will review concepts learned to date and use them to solve
word problems using positive and negative decimals.
PRIOR KNOWLEDGE REQUIRED
Can solve word problems with whole numbers
Can add and subtract whole numbers, fractions, and decimals
Can order and compare positive and negative decimals
Can extend patterns
Can locate positive and negative decimals on number lines
MATERIALS
BLM Always, Sometimes, Never True (Decimals) (p. E-72)
This lesson is mostly a cumulative review. Question 7 part b) on AP Book
6.1 p. 114 is not a review of concepts learned in this unit, but instead
provides an opportunity for students to move beyond the expectation.
Students can be encouraged to draw a number line to solve the problem.
Here are some additional problems that you can use for cumulative review.
a)Two friends ate 6 tenths of a pizza. Write as a decimal the fraction
of the pizza they ate. (0.6 of the pizza)
b)A carpenter used 4 tenths of a box of 100 nails on Monday and
3 hundredths of the box on Tuesday. Write as a decimal the total
fraction of the nails used. (0.43 of the nails)
c)A carpenter used 0.5 of the nails in a box of 1,000 nails.
How many nails did he use? (500 nails)
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
d)Tom ran 2.51 km, Jessie ran 2.405 km, and Kay ran 2.6 km.
Who ran the farthest? (Kay)
e)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)
f)Which of the deep sea fish lives at the greatest depth?
At the least depth?
viperfish: −1.345 km
fangtooth: −4.8 km
deep sea dragonfish: −1.49 km
deep sea angler: −0.9 km
(the fangtooth is at the greatest depth, the deep sea angler is
at the shallowest depth)
The Number System 6-36
E-49
g)Write a decimal between the two given decimals. There are several
correct answers.
i) 45.79 and 45.8
ii)
−211.7 and −211.8
Sample answers: i) 45.791, ii) −211.75
ACTIVITY
(MP.3)
Give each student an index card and a card from BLM Always,
Sometimes, or Never True (Decimals). Have students decide whether
the statement on the card is always true, sometimes true, or never true.
They should write on the index card reasons for the answer, such as an
explanation for always true or never true statements, and two examples
(one true, one false) for the statements that are sometimes true. They can
then glue the card with the statement to the other side of the index card.
Have students pair up. Partners exchange cards and verify each
other’s answers. Then players exchange cards and seek a partner
with a card they have not yet seen.
Answers: In the table below, “S” stands for sometimes, “A” for always,
and “N” for never. The correct answers, in order, are:
A
A
N
A
A
A
S
A
A
S
S
A
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
S
S
A
N
S
S
E-50
Teacher’s Guide for AP Book 6.1
NS6-37 Fractions of a Whole Number
Pages 116–117
STANDARDS
preparation for 6.NS.A.1,
preparation for 6.NS.B.2,
preparation for 6.G.A.2
Goals
Students will find fractions of whole numbers.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can name and model fractions
Can convert mixed numbers to improper fractions and vice versa
fraction of a whole number
What can you take a fraction of? As a class, brainstorm things that
you can take a fraction of: Alternate between you and the class listing
ideas. Examples: a triangle, a meter, a foot, an angle, a pizza, a cup
of juice, a cookie.
Introduce fractions of a number. Tell students that you can take a fraction
of a whole number too. ASK: If there are six friends and half of them are
girls, how many are girls? If the distance to a store is six miles, how far away
is the halfway point? If I want to finish a race in six hours, when should I be
at the halfway point? Explain that since all of these questions have the same
numeric answer, we can say that the number 3 is half of the number 6.
Use pictures to show half. Explain that if you want to eat half a pizza, you
would divide the pizza into two equal parts and eat one of them. Draw a
pizza divided into two equal parts to illustrate this. Similarly, if you wanted to
eat half of six cherries, you would divide six cherries into two equal groups
and eat one of the two groups. Again, draw a picture to illustrate. There are
three cherries in each group, so 3 is half of 6.
Exercises: Draw pictures to show half of the number.
a)4
b)10
Bonus: 16
Answers: a) 2, b) 5, Bonus: 8
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Use pictures to find other fractions of whole numbers. ASK: If I wanted
to eat only one third of the cherries, how many groups should I make? (3)
Draw the picture in the margin and SAY: When you divide six cherries into
three equal groups, and take one of those groups, you are taking two of
the cherries. So 1/3 of 6 cherries is 2 cherries. In fact, 1/3 of 6 of anything
is 2 of anything.
ASK: What would two thirds be? Explain that you still need to make three
groups, but now you take two of the groups instead of just one. Circle two
of the three groups in the picture above to illustrate (see margin).
Exercise: Draw pictures to find the fractions of numbers.
2
3
3
2
a)
of 10 c)
of 12 e)
of 12
of 9
b)
3
5
4
3
f) 5
of 12
6
Answers: a) 6, b) 6, c) 9, d) 8, e) 10
The Number System 6-37
E-51
Using division to find fractions (with numerator 1) of whole numbers.
Draw on the board:
SAY: When you divide 15 objects into three equal groups, the size of each
group is 15 ÷ 3 = 5, so one third of 15 is 5. Write on the board:
1
of 15 = 15 ÷ 3 = 5
3
Exercises: Write a division statement to find the fraction of a number.
1
1
1
1
1
of 12 b)
of 10 c)
of 16 d)
of 12 e)
of 12
a)
3
5
4
4
2
1
of 12
3
Answers: a) 12 ÷ 3 = 4, b) 10 ÷ 5 = 2, c) 16 ÷ 4 = 4, d) 12 ÷ 4 = 3,
e) 12 ÷ 2 = 6
Find any fraction of a whole number using multiplication and division.
ASK: If I know 1/3 of 12 is 4, what is 2/3 of 12? (8) Draw a picture to help
explain that 2/3 of 12 is twice as many as 1/3 of 12. If 1/3 of 12 is a group of
4, then 2/3 of 12 is 2 groups of 4, or 2 × 4 dots. So 2/3 of 12 is 8.
2
of 12
3
Exercises: Use division and multiplication to find the fraction of a number.
2
3
3
4
8
of 20 b)
of 15 d)
a)
of 14 c)
of 35 e)
of 36
5
7
5
7
9
Answers: a) 20 ÷ 5 = 4, and 2 × 4 = 8, b) 6, c) 9, d) 20, e) 32
(MP.4)
Solving word problems. Tell students that each mathematical word in a
word problem can be replaced by a symbol. ASK: What number or symbol
would you use to replace each of the following: more than (>), is (=), half
(1/2), three quarters (3/4), and (+). Write on the board:
Calli’s age is half of Rob’s age.
Rob is twelve years old.
How old is Calli?
Calli’s age is half of Rob’s age.
Calli’s age =
1
of
2
12
Exercises: Write the data using mathematical symbols. Keep what you
don’t know in words.
a)Mark gave away three quarters of his 12 stamps.
How many did he give away?
b)John won three fifths of his five sets of tennis. How many sets did he win?
c)If Ruby studied math for 2/5 of an hour and then history for 1/3
of an hour, how long did she study for altogether?
Bonus
d)How many hours are in 5/8 of a day? Hint: Replace “a day” with 24 hours.
Answers: a) Mark gave away 3/4 of 12 stamps, b) John won 3/5 of 5 sets,
c) Ruby studied for 2/5 of 60 minutes + 1/3 of 60 minutes, d) There are 5/8
of 24 hours in 5/8 of a day.
E-52
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Teach students to replace each word they do know with a math symbol and
just keep what they don’t know (see margin).
Exercises: Solve the problems a) to d) above.
Answers: a) 9, b) 3, c) 24 + 20 = 44 minutes, d) 15
Mixed numbers of a whole number. SAY: There are 12 months in a year.
ASK: How many months are in half a year? (6) How many months are in
1 1/2 years? (12 + 6 = 18)
Exercises: How many months are in:
3
1
b) 2 years
a)1 years
6
4
c) 1
2
years
3
Answers: a) 21, b) 26, c) 20
Extensions
(MP.4)
1.By weight, about 1/5 of a human bone is water and 1/4 is living tissue.
If bone weighs 120 g, how much of the bone’s weight is water and how
much is tissue? (24 g is water, 30 g is tissue)
(MP.8)
2. a) Find the fractions of a number.
2
3
8
7
8
of 15 of 11
of 3 of 5 of 9 15
11
3
5
9
b) Use the pattern to predict 354/502 of 502.
Answers: a) 2, 3, 8, 7, and 8, b) 354
(MP.1)
3.Find the fractions of 20. Use the answers to write the fractions in order
from least to greatest.
a)3/4
b)7/10
c)3/5 d)4/5
Point out that you cannot use 2/3 of 12 and 3/5 of 15 to compare 2/3 to
3/5 because you have to use the same whole to compare fractions.
Answers: a) 15/20, b) 14/20, c) 12/20, d) 16/20; ordered from least to
greatest: 3/5, 7/10, 3/4, 4/5
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.4)
4.
A small fraction of a large number can still be a large number.
Show students the model of 1/60 (see margin).
a) Is 1/60 a large fraction or a small fraction?
b)If 1/60 of the total number of deaths in the US in 2009 was due to
lack of health insurance, and there were 2,436,000 deaths, how
many were due to a lack of health insurance? (40,600) Is that a
large number or a small number?
Answers: a) small, b) large
Explain that people who want to convince you that a problem isn’t very
important might quote the fraction of people rather than the number of
people. Students should be aware of this.
The Number System 6-37
E-53
NS6-38 Multiplying Fractions by Whole Numbers
Pages 118–119
STANDARDS
preparation for 6.NS.A.1,
6.NS.B.3
Goals
Students will multiply fractions and whole numbers.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
fraction of a whole number
Can find a fraction of a whole number
Can convert mixed numbers to improper fractions and vice versa
MATERIALS
rulers with inches and centimeters
Multiplication as a short form for addition. Remind students that 3 × 4 is a
short form of writing 4 + 4 + 4, so 3 × 1/4 is a short form of 1/4 + 1/4 + 1/4.
Exercises: Rewrite the product as a sum.
1
2
3
a)3 × b)3 × c)4 × 5
5
7
d)2 × 5
13
Answers: a) 1/5 + 1/5 + 1/5, b) 2/5 + 2/5 + 2/5, c) 3/7 + 3/7 + 3/7 + 3/7,
d) 5/13 + 5/13
Exercises: Rewrite the sum as a product.
1 1 1 1
3
3
3
3
3
a)
+ + + b)
+ + + +
3 3 3 3
11 11 11 11 11 c)
4 4 4
+ +
9 9 9
Answers: a) 4 × 1/3, b) 5 × 3/11, c) 3 × 4/9
Point out that the addition involves all identical fractions and, hence, like
denominators, so the addition itself is quite simple.
d)4 × 2
5
The rule for multiplying a whole number by a fraction.
Write on the board:
2 6
3× =
7 7
ASK: How did you get the 6? (2 + 2 + 2) SAY: But that’s 3 × 2.
Write on the board:
3
5× =
8
8
ASK: How would you get the numerator? (multiply 5 times 3)
Write 15 as the numerator.
Exercises: Multiply.
2
1
b) 7 × a)3 × 12
5
E-54
c) 6 × 3
8
d) 2 × 5
16
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Exercises: Multiply by adding.
2
3
2
c)5 × b)3 × a)3 × 7
5
7
Answers: a) 6/7, b) 9/5, c) 10/7, d) 8/5
Answers: a) 3/12, b) 14/5, c) 18/8, d) 10/16
Review converting improper fractions into mixed numbers. Use division
(21 ÷ 8 = 2 R 5, so 21/8 = 2 5/8). Remind students that some improper
fractions might represent whole numbers. For example, when you try to
convert 24/3 to a mixed number, 24 ÷ 3 = 8 R 0 so there is no fractional part.
Exercises: Write the number as a mixed number or whole number.
15
22
20
7
10
b)
e)
a)
c)
d)
2
3
4
5
4
Answers: a) 3 1/2, b) 3 3/4, c) 4 2/5, d) 3 1/3, e) 5
“Of” can mean multiply. Tell students that the word “of” can mean multiply.
For example, with whole numbers, 2 groups of 3 means 2 × 3 objects.
“Of” can mean multiply with fractions too: 1/2 of 6 means 1/2 of a group
of 6 objects, or 1/2 × 6.
Remind students that they learned how to find a fraction of a whole number.
For example, 2/3 of 9 is 2 × (9 ÷ 3).
Exercises: Multiply by finding the fraction of the whole number.
3
4
2
3
7
× 20b) × 14c) × 15d) × 35e) × 36
a)
5
7
5
7
9
Answers: a) 12, b) 8, c) 6, d) 15, e) 28
Multiplication of a whole number and a fraction commutes (i.e., order
does not matter). SAY: We know that order does not matter in multiplication;
for example, 3 × 5 = 5 × 3.
Exercises: Multiply the same numbers in different orders. Do you get the
same answer both times? If not, find your mistake.
2
2
3
3
5
5
× 10 and 10 × b) × 12 and 12 × c) × 8 and 8 × a)
5 5
4
4
4
4
Answers: a) 4, b) 9, c) 10
Multiplying mixed numbers by whole numbers. Remind students how
to convert mixed numbers to improper fractions:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
3 11
2 =
4
4
(2 × 4) + 3
Then tell students they can now multiply mixed numbers by whole numbers.
Just change the mixed number to an improper fraction.
Exercises: Multiply.
1
a)3 × 10
5
b) 2
2
× 9
3
c) 4
1
× 6
3
Answers: a) 32, b) 24, c) 26
The Number System 6-38
E-55
(MP.4)
Word problems practice.
a)Tina drinks 5/6 of a bottle of water each day. How many bottles of water
does she drink in seven days?
Answer: 5/6 × 7 = 35/6 = 5 5/6 bottles
b) i)One lane of a swimming pool is 3/8 the length of an Olympic
swimming pool. If Orlando swims the lane five times, how many
lengths of an Olympic swimming pool does he swim?
Answer: 3/8 × 5 = 15/8 = 1 7/8 lengths of an Olympic
swimming pool
ii)An Olympic swimming pool is 50 m long. How many meters did
Orlando swim?
Answer: 50 m × 15/8 = 750/8 m = 93 6/8 m = 93 3/4 m
ACTIVITY
Give students a ruler that has both inches and centimeters. Use inches
to multiply whole numbers by halves, fourths, and eighths, and use
centimeters to multiply whole numbers by halves, fifths, and tenths.
Example: 7 × 5/8
012345
So 7 × 5/8 = 35/8 = 4 3/8
Extensions
3 × 6
4×6
18192021222324
1.Use distance to multiply a mixed number by a whole number. For
example, to find 3 1/2 × 6, find the number that is halfway between
3 × 6 and 4 × 6. So 3 1/2 × 6 is 21. Similarly, 3 1/3 × 6 is one third
of the way from 3 × 6 to 4 × 6, so 3 1/3 × 6 is 20.
Students can then convert the mixed number to an improper fraction
to make sure they get the same answer using this method as they did
during the lesson.
1
2
1
b) 2 × 9
c) 4 × 6
a)3 × 10
5
3
3
Sample solution: a) the product is between 3 × 10 = 30 and 4 × 10 =
40 and the answer will be 1/5 of the distance from 30 to 40, or 32.
Answers: b) 24, c) 26
2. a)Is 5/8 of three pizzas more than, less than, or the same amount
as 3/8 of five pizzas?
b)Is 435/8 of 789 more than, less than, or the same amount as 789/8
of 435? Explain without computing either amount.
E-56
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
halfway point
Answers: a) same, b) same,
435
435 × 789
435/8 of 789 =
× 789 =
8
8
789
789 × 435
× 435 =
8
8
Multiplication commutes, so the numerators are the same
in both fractions.
789/8 of 435 =
(MP.8)
3. a) Check that each equation is true.
3
4
5
3
4
5
3 + = 3×
4 + = 4×
5 + = 5×
2
2 3
3 4
4
b) Find a fraction that makes the equation 7 + a = 7 × a true.
c) Find a fraction that does not make the equation 7 + a = 7 × a true.
Answers: a) all true, b) 7/6, c) Any fraction that is not equal to 7/6 works.
NOTE: The reason for the answer in part c) is:
7×a=a+a+a+a+a+a+a=7+a
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
So a + a + a + a + a + a = 7 or 6 × a = 7, and then a = 7/6.
The Number System 6-38
E-57
NS6-39 Multiplying Decimals by Powers of 10
Pages 120–122
Goals
STANDARDS
preparation for 6.NS.B.3
Students will multiply decimals by 10, 100, and 1,000.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Knows the commutative property of multiplication
Can multiply whole numbers by 10, 100, and 1,000
Understands decimal place value
Can regroup ten small place values as one large place value
Can write decimals in expanded form
Can read decimals in terms of smallest place value
decimal point
hundredth
tenth
thousandth
MATERIALS
small card with a large dot, prepared ahead
(MP.6)
Review multiplying whole numbers by 10. Ask students to describe how
they can 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 because each place value gets replaced
by the place value that is 10 times greater:
34 = 3 tens + 4 ones, so
34 × 10 = 3 hundreds + 4 tens = 340
Using place value to multiply decimals by 10. Write on the board:
0.4 × 10 = 4 tenths × 10
ASK: What place value is 10 times the tenths? (ones) Write on the board:
Draw the picture below to remind students of the connection between
place values:
×10
×10
×10
×10
tens ones tenths hundredths thousandths
Exercises: Multiply the place value by 10.
a)hundredths × 10
c)tenths × 10
b) ones × 10
d) thousandths × 10
Bonus: tens × 10
Answers: a) tenths, b) tens, c) ones, d) hundredths, Bonus: hundreds
E-58
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
= 4 ones = 4
Exercises: Use place value to multiply the number by 10.
a) 3 hundredths × 10 c) 5 ones × 10 b) 4 tenths × 10
d) 7 thousandths × 10
Answers: a) 3 tenths, b) 4 ones, c) 5 tens, d) 7 hundredths
Write on the board:
0.005 × 10 = 0.05
SAY: If 5 is in the thousandths position, then after multiplying by 10,
it will be in the hundredths position.
Exercises: Multiply using place value.
a)0.5 × 10 b) 0.02 ×10 c) 0.006 × 10 d) 0.09 × 10
Answers: a) 5 b) 0.2 c) 0.06 d) 0.9
Use expanded form to multiply decimals by 10. Show students how to
represent what they are doing using expanded form. For example, write
on the board:
4.36 = 4 + 0.3 + 0.06
SAY: To multiply by 10, you can multiply each place value by 10. So:
4.36 × 10 = 40 + 3 + 0.6 = 43.6
Exercises: Use expanded form to multiply by 10.
a)5.4 × 10 e)3.12 × 10 b) 60.3 × 10 f) 84.06 × 10 c) 3.004 × 10 g) 3.294 × 10 d) 5.81 × 10
h) 7.806 × 10
Sample solution:
f) 84.06 = 80 + 4 + .06, so 84.06 × 10 = 800 + 40 + .6 = 840.6
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 54, b) 603, c) 30.04, d) 58.1, e) 31.2, f) 840.6, g) 32.94,
h) 78.06
5 4 6 0 3
5 4 6 0 3
Move the decimal point to multiply decimals by 10. Ahead of time, draw
a large decimal point on a card. Write on the board the numbers from the
exercises above (multiplying by 10) and tape the card to the board so it
acts as a decimal point. The first two numbers are shown in the margin.
Ask volunteers to move the decimal point to show the answer for all
8 questions. The first two are shown in the margin.
Ask the rest of the class to look for a pattern in how the decimal point is
being moved. (always moved one place to the right)
Exercises: Move the decimal point one place to the right to multiply by 10.
a)3.2 × 10 b) 0.58 × 10 c) 10 × 0.216
d) 10 × 7.46
Bonus: 98,763.60789 × 10
Answers: a) 32, b) 5.8, c) 2.16, d) 74.6, Bonus: 987,636.0789
The Number System 6-39
E-59
(MP.1)
Move the decimal point to multiply decimals by 100 and by 1,000.
Write on the board:
3.462 × 100 = 3.462 × 10 × 10
SAY: Move the decimal point once to multiply by 10 and then again to
multiply by 10 again. Show this on the board:
3 . 4
6
2
So 3.462 × 100 = 346.2
SAY: To multiply by 100, move the decimal point two places to the right.
Exercises: Move the decimal point two places to the right to multiply by 100.
a)3.62 × 100
b) 0.725 × 100 c) 1.673 × 100 d) 0.085 × 100
Answers: a) 362, b) 72.5, c) 167.3, d) 8.5
(MP.8)
SAY: We moved the decimal point once to multiply by 10, and twice to
multiply by 100. ASK: How many times do we move the decimal point
to multiply by 1,000? (3 times) Show this on the board:
2 . 4
6
7
So 2.467 × 1,000 = 2,467
Exercises: Move the decimal point to multiply by 1,000.
a)0.462 × 1,000
b) 11.241 × 1,000
c) 9.32416 × 1,000
Answers: a) 462, b) 11,241, c) 9,324.16
3 4 2
3 4 2
Using zero as a placeholder when multiplying decimals. Write on the
board 3.42 × 1,000 in a grid. Use the card with a large dot for the decimal
point so it can be moved, as shown.
ASK: How many places do I have to move the decimal point? (3) Move
the decimal point three times, as shown. ASK: Are we finished writing the
number? (no) Why not—what’s missing? (the zero)
5 2 4
Encourage struggling students to write each place value in its own cell
of the grid for the exercises below, and to draw arrows to show how they
moved the decimal point (see margin for part b)).
Exercises: Multiply by 1,000.
a)0.4 × 1,000
b) 5.24 × 1,000 c) 23.6 × 1,000 d) 0.01 × 1,000
Answers: a) 400, b) 5,240, c) 23,600, d) 10
Exercises: Multiply by 10, 100, or 1,000.
a)0.6 × 100 b) 7.28 × 10
e)21.9 × 1,000
c) 25.6 × 1,000 d) 1.8 × 100
f) 326.3 × 1,000
g) 0.002 × 10
Bonus: 2.3 × 10,000
E-60
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
SAY: Each digit is worth 1,000 times as much as it was. Pointing to each digit
in the first grid, SAY: The number was 3 ones, 4 tenths, and 2 hundredths.
Pointing to each digit in the second grid, SAY: Now it is 3 thousands,
4 hundreds, and 2 tens. So the number is 3,420.
Answers: a) 60, b) 72.8, c) 25,600, d) 180, e) 21,900, f) 326,300, g) 0.02,
Bonus: 23,000
(MP.1)
3 4 0
3 4 0
Connect multiplying whole numbers by 10 to multiplying decimals by 10.
Write “34” on the board, leaving enough space between the digits to place
the card with the decimal point. ASK: What is 34 × 10? (340) SAY: We can
also multiply 34 × 10 by moving the decimal point. Write on the board “34.0,”
but place the card in the position of the decimal point. Move it one place to
the right and point out that this is the same answer you get the other way.
SAY: Multiplying whole numbers is really the same method we use to
multiply decimals.
(MP.4)
Word problems practice.
a)Massimo makes $12.50 an hour mowing lawn. How much does he
make in 10 hours?
b)A clothing-store owner wants to buy 100 coats for $32.69 each. How
much will the coats cost?
c) A dime is 0.135 cm thick. How tall would a stack of 100 dimes be?
d)A necklace has 100 beads. Each bead has a diameter of 1.32 mm.
How long is the necklace?
Answers: a) $125.00, b) $3,269.00, c) 13.5 cm, d) 132 mm
Extensions
1. Fill in the blanks.
a) × 10 = 38.2
b) × 100 = 6.74
c) 42.3 × = 4,230
d) 0.08 × = 0.8
Answers: a) 3.82, b) 0.0674, c) 100, d) 10
(MP.7)
2. Complete the patterns.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a) 0.0007, 0.007, 0.07, , b) 3.895, 38.95, 389.5, , Answers: a) 0.7, 7; b) 3,895, 38,950
(MP.2)
3.Switch the numbers around to make the product easier to find.
Then find the product.
a)(3.2 × 5) × 20
b) (6.73 × 2) × 50
c) (7.836 × 5) × (25 × 8)
Answers: a) 320, b) 673, c) 7,836
(MP.4)
4.Create a word problem that requires multiplying by 1,000. Have a
partner solve it.
(MP.4)
5.One marble weighs 3.5 g. The marble bag weighs 10.6 g. How much
does the bag weigh with 100 marbles in it?
Answer: 360.6 g
The Number System 6-39
E-61
NS6-40 Multiplying and Dividing by Powers of 10
Pages 123–124
STANDARDS
preparation for 6.NS.B.3
Vocabulary
decimal point
hundredth
tenth
thousandth
Goals
Students will multiply and divide decimals by 10, 100, and 1,000 by
shifting the decimal point.
PRIOR KNOWLEDGE REQUIRED
Can multiply whole numbers and decimals by 10, 100, and 1,000
Knows that multiplication and division are opposite operations
Understands decimal place value
Can write decimals in expanded form
MATERIALS
card with a large dot
Dividing by 10 using base ten materials. Draw on the board:
1.0
0.1 0.01
Tell students that you will represent one whole with a big square, so one
tenth is a column or row and one hundredth is a little square. Draw several
picture equations (below) on the board and have volunteers write the
decimal equations (shown in brackets):
÷ 10 =
(2.0 ÷ 10 = 0.2)
(0.5 ÷ 10 = 0.05)
÷ 10 =
(3.1 ÷ 10 = 0.31)
Exercises: Divide by drawing pictures on grid paper, or
using base ten blocks.
a)3.0 ÷ 10
E-62
b) 0.4 ÷ 10
c) 3.4 ÷ 10
d) 2.7 ÷ 10
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
÷ 10 =
Answers: a) 0.3, b) 0.04, c) 0.34, d) 0.27
Dividing by 10 by inverting the rule for multiplying by 10. Using the
card with a large dot from the previous lesson to show the decimal point,
write on the board:
3 4 2 5
Invite a volunteer to move the decimal point to multiply by 10. (342.5)
Write on the board:
34.25 × 10 = 342.5, so 342.5 ÷ 10 = _______
ASK: What number goes in the blank? (34.25) How do you know? (the
multiplication and division equations are in the same fact family) Now write
on the board:
3 4 2 5
Have a volunteer move the card with the decimal point in 342.5 to get the
answer for 342.5 ÷ 10. (move it one place to the left) SAY: Division is the
opposite of multiplication. When you multiply by 10, you move the decimal
point one place to the right. When you divide by 10, you move the decimal
point one place to the left.
Exercises: Divide by 10.
a)14.5 ÷ 10
b) 64.8 ÷ 10
c) 9.22 ÷ 10
d) 0.16 ÷ 10
Answers: a) 1.45, b) 6.48, c) 0.922, d) 0.016
Dividing by 100. Write on the board:
5.831 × 100 = 583.1, so 583.1 ÷ 100 = ______
Ask a volunteer to fill in the blank. (5.831) Point out that the equations are in
the same fact family, so knowing how to multiply by 100 also tells us how to
divide by 100. ASK: How do we move the decimal point to divide by 100?
(two places to the left) Point out that you had to move it two places to the
right to multiply 5.831 by 100, then to get 5.831 back, you need to move it
to the left two places.
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Exercises: Divide by 100.
a)14.5 ÷ 100
b) 464.8 ÷ 100 c) 9.22 ÷ 100
d) 0.6 ÷ 100
Answers: a) 0.145, b) 4.648, c) 0.0922, d) 0.006
Dividing whole numbers by 10 and 100. Write on the board the number 67,
again leaving room for the card between the digits. Tell students you want
to know how much is 67 ÷ 10. Then SAY: I would do the division by moving
the decimal point, but I don’t see any decimal point here. What should I do?
(add the decimal point to the right of the ones, because 67 = 67.0) Do so,
using the card with the big dot, then invite a volunteer to move the decimal
point one place to the left to get 67 ÷ 10 = 6.7 Repeat the process with
18 ÷ 100 (0.18) and 1,987 ÷ 100 (19.87).
The Number System 6-40
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Exercises: Divide by 10 or 100.
a)236 ÷ 10
b) 573 ÷ 100
c) 1,230 ÷ 100 d) 14,889 ÷ 10
Answers: a) 23.6, b) 5.73, c) 12.3, d) 1,488.9
Dividing by 1,000. ASK: How would you shift the decimal point to divide
by 1,000? (3 places to the left) Show an example done on a grid.
4 5
0 4 5
So 45 ÷ 1,000 = 0.045.
Exercises: Divide by 1,000.
a)2,934 ÷ 1,000 b) 423 ÷ 1,000 c) 18.9 ÷ 1,000 d) 1.31 ÷ 1,000
Bonus: 423 ÷ 100,000
Answers: a) 2.934, b) 0.423, c) 0.0189, d) 0.00131, Bonus: 0.00423
(MP.1)
Strategies for remembering which way to move the decimal point. SAY:
Remember: Multiplying by 10, 100, or 1,000 makes the number bigger, so
the decimal point moves to the right. Dividing makes the number smaller,
so the decimal point moves to the left.
If students have trouble deciding which direction to move the decimal
point when multiplying and dividing by 10, 100, or 1,000, one hint that
some students might find helpful is to use the case of whole numbers
as an example. Which way is the decimal point moving when multiplying
by 34 × 10 = 340? (to the right)
Exercises: Multiply or divide.
a)78,678 ÷ 1,000
d)1.31 × 1,000
g)0.2 ÷ 100
b) 2.423 × 100
e) 6 ÷ 100
h) 5.1 × 100
c) 18.9 ÷ 10
f) .082 × 10
i) .31 × 1,000
Answers: a) 78.678, b) 242.3, c) 1.89, d) 1,310, e) 0.06, f) 0.82, g) 0.002,
h) 510, i) 310, Bonus: j) 0.03149876532, k) 31,498,765,320
Remind struggling students to write each place value in its own cell on grid
paper when multiplying or dividing decimals by powers of 10.
(MP.4)
Word problems practice.
a)In 10 months, a charity has raised $26,575.80 through fundraising.
How much did they raise each month on average? ($2,657.58)
b)A stack of 100 cardboard sheets is 13 cm high. How thick is one sheet
of the cardboard? (0.13 cm)
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Teacher’s Guide for AP Book 6.1
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Bonus
j) 31,498.76532 ÷ 1,000,000 k) 31,498.76532 × 1,000,000
c)A thousand people attended a “pay what you can” event. The total
money paid was $5,750. Kim paid $0.60. Did he pay more or less than
average? (less, the average was $5.75)
(MP.2)
d)A hundred walruses weigh 121.5 metric tonnes (1 metric tonne = 1,000 kg).
How much does one walrus weigh on average, in kilograms? (1,215 kg)
(MP.6)
e) A box of 1,000 nails costs $12.95.
i) How much did each nail cost, to the nearest cent? (1¢)
ii)A hundred of the nails have been used. What is the cost for the nails
that are left, to the nearest cent? Hint: Use the actual cost of a nail
in your calculations, not the rounded cost from part i). ($11.66)
Extensions
1.A penny has a width of 19.05 mm. How long would a line of
10,000 pennies laid end-to-end be, in mm, cm, m, and km?
Answers: 190,500 mm, 19,050 cm, 190.5 m, 0.1905 km
(MP.2)
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?
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)1,000 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?
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Answers: a) 4.8 cm; sample answer: an eraser; b) 2.38 cm; sample
answer: a quarter (coin); c) 27.4 cm, sample answer: a shoe
(MP.4)
3.Create your own word problems that require multiplying and/or
dividing decimals by powers of 10. Then, trade with a partner and
solve the problems.
(MP.4)
4.Find the mass of one bean by weighing 100 or 1,000 beans. Use a
calculator to determine how many beans are in a 2 lb (908 g) package.
(MP.3)
5.How would you shift the decimal point to divide by 10,000,000? Explain.
Answer: move the decimal 7 places (because there are 7 zeros in
10,000,000) to the left (because I am dividing)
The Number System 6-40
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NS6-41 Multiplying Decimals by Whole Numbers
Page 125
STANDARDS
preparation for 6.NS.B.3
Vocabulary
decimal point
hundredth
round number
tenth
thousandth
Goals
Students will multiply decimals by 1-digit whole numbers and round
numbers by using place value.
PRIOR KNOWLEDGE REQUIRED
Can multiply and divide whole numbers and decimals by powers of 10
Can multiply and divide whole numbers by 1-digit numbers using the
standard algorithm
Understands decimal place value
Can regroup decimals
Can write decimals in expanded form
Multiplying decimals without regrouping. Draw on the board:
1.0
0.1 0.01
Ask a volunteer to draw a model for 1.23. Then extend the model yourself
to show 2 × 1.23 (see margin). ASK: What number is this? (2.46) Write
on the board:
2 × 1.23 = 2.46
SAY: This is 2 ones, 4 tenths, and 6 hundredths.
Exercises: Draw models to multiply.
a)2 × 4.01
b) 3 × 3.12
Answers: a) 8.02, b) 9.36
4.01 = 4 ones + 0 tenths + 1 hundredth
2 × 4.01 = 8 ones + 0 tenths + 2 hundredths = 8.02
Point out how each digit is multiplied by 2.
Exercises: Multiply mentally.
a)4 × 2.11 b) 3 × 2.31 c) 3 × 1.1213
Bonus: d) 2 × 1.114312 e) 3 × 1.1212031
Answers: a) 8.22, b) 6.93, c) 3.3639, Bonus: d) 2.228624, e) 3.3636093
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Point out that what students did is the same as writing the decimal in
expanded form and multiplying each place value separately. For example:
Multiplying decimals with regrouping by using expanded form. Write on
the board:
Number
3.8
2 × 3.8 = ??
Tens
Ones
3
6
Tenths
8
16
Hundredths
Ask a volunteer to regroup the ones and tenths to find the number that
equals 2 × 3.8. (7 ones and 6 tenths = 7.6) Have students copy the chart
into their notebook to do the exercises below.
Exercises: Multiply by regrouping when necessary.
a)3 × 2.4 d)3 × 4.42
b) 4 × 3.2
e) 4 × 3.32 c) 3 × 2.04
f) 3 × 3.45
Answers: a) 7.2, b) 12.8, c) 6.12, d) 13.26, e) 13.28, f) 10.35
Compare multiplying decimals to multiplying whole numbers. Have a
volunteer multiply 2 × 38 using the chart above. (6 tens + 16 ones =
7 tens + 6 ones = 76)
Discuss with students the similarities and differences between the two
problems and solutions. (The digits are the same, the regrouping is the
same, only the place values are now ten times bigger, tens instead of ones
and ones instead of tenths.) SAY: 2 × 38 is ten times more than 2 × 3.8
because 38 is ten times more than 3.8.
1
3
7
2
6
a)3.35 × 6 1
8
2
×
Recording multiplication with the standard method. Write the multiplications
in the margin on the board. 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. Even the regrouping looks the same. They just have to copy
the decimal point in the answer directly under where it is in the decimal.
Exercises: Multiply.
6
3
×
7
8
b) 41.31 × 2 c) 523.4 × 5
d) 9.801 × 3
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Bonus: e) 834,779.68 × 2 f) 5,480.63 × 7
Answers: a) 20.10 or 20.1, b) 82.62, c) 2,617.0 or 2,617, d) 29.403,
Bonus: e) 1,669,559.36, f) 38,364.41
(MP.5)
Explain that you can use estimation to check the answer. Suppose you’ve
done a multiplication 3.32 × 4 and got 132.8. Have a volunteer multiply
332 × 4 = 1,328. SAY: I know all the digits are correct. But is the answer
reasonable? (no) Why not? (3.32 is about 3, so 3.32 × 4 should be about 12.
So 132.8 cannot be a reasonable answer.)
Exercises: Estimate to make sure your answers to the exercises
above are reasonable.
Answers: Round to the leading place value in each case. a) 3 × 6 = 18,
b) 40 × 2 = 80, c) 500 × 5 = 2,500, d) 10 × 3 = 30,
Bonus: e) 800,000 × 2 = 1,600,000, f) 5,000 × 7 = 35,000
The Number System 6-41
E-67
Multiplying decimals by round numbers. Write on the board:
30 × 5
Ask a volunteer for the answer. (150) SAY: This problem is easy to do in
your head because you can separate the 30 out as being 10 × 3. Show this
on the board:
30 × 5 = 10 × 3 × 5
= 10 × 15
= 150
Write on the board:
40 × 3.2 = 10 × 4 × 3.2
ASK: What is 4 × 3.2? (12.8) What is 10 × 12.8? (128) SAY: Because you
know how to multiply 4 × 3.2, you can also multiply 40 × 3.2. Just multiply
your result by 10.
Exercises: Rewrite the product to multiply.
a)30 × 2.6
d)20 × 3.6
b) 200 × 5.4
e) 400 × 2.4
c) 70 × 2.71
f) 60 × 2.32
Answers: a) 78, b) 1,080, c) 189.7, d) 72, e) 960, f) 139.2
Remind students that order doesn’t matter when multiplying, so it doesn’t
matter whether the round number or the decimal number comes first.
Exercises: Multiply.
a)2.1 × 50
b) 500 × 5.7
c) 80 × 12.01
d) 3,000 × 0.781
e)1.51 × 700
f) 23.7 × 500
g) 2,000 × 2.05 h) 90 × 10.604
Answers: a) 105, b) 2,850, c) 960.8, d) 2,343, e) 1,057,
f) 11,850, g) 4,100, h) 954.36
(MP.1)
1.Check that you get the same answers to the exercises above by
multiplying the decimal by 10, 100, or 1,000 first, then by the 1-digit
number. For example, in part a) above, multiplying 2.1 × 10 first
and then multiplying by 5 (21 × 5 = 105) gets the same answer as
multiplying 2.1 × 5 first and then multiplying by 10 (10.5 × 10 = 105).
2.Which products do you expect to be greater than 10?
Check your prediction.
0.3 × 30
1.2 × 11
2.8 × 5
0.04 × 19
Answer: 1.2 × 11 and 2.8 × 5 are greater than 10
(MP.2)
3. Multiply: (0.8 × 5) × (0.2 × 3) × (0.5 × 200)
Answer: 4 × 0.6 × 100 = 2.4 × 100 = 240, or 4 × 60 = 240
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Extensions