Download Unit 4 The Number System: Rational Numbers

Unit 4 The Number System: Rational Numbers
Introduction
In this unit, students will compare, add, and subtract decimals and fractions, including positive
and negative numbers. Students will write answers to whole-number division questions as
fractions and decimals, including for multi-digit numbers divided by one-digit numbers.
Notation for adding integers. In Unit 2, we introduced adding and subtracting integers using
gains and losses. We used the notation + 3 − 4 to mean a gain of $3 followed by a loss of $4.
As students gain experience with integers, students learn to change the notation of adding
integers—+3 + (−4)—to this gains and losses notation, making the addition of integers easier for
students. Although it can be tempting to interpret + 3 − 4 as “adding 3 and subtracting 4,” we
teach students to think of it as “adding 3 and then adding −4.” Please continue this approach in
this unit.
Terminology. Even though mathematically the number 1 is considered a power of 10 (with
exponent 0), we are not introducing it as such in this unit.
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 (this notation will be introduced in Unit 5).
Like this: -
1
2
Not like this:
-1
2
NOTE: Even though fractions often appear in line with the text in our lesson plans (e.g., 1/2),
remember to always either stack fractions when you show them to your students (e.g.,
1
) or to
2
introduce the non-stacked notation.
Materials. We recommend that students always work in grid paper notebooks. Paper
with 1/4-inch grids works well in most lessons. In this unit, grid paper will be especially useful
when adding and subtracting multi-digit numbers. 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.
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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NS7-18
Decimal Fractions
Pages 99–101
Standards: preparation for 7.NS.A.1
Goals:
Students will add tenths, hundredths, and thousandths when written as fractions.
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
Can multiply whole numbers by 10
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. SAY: These numbers are called powers of 10. We will learn
about powers of other numbers later in the year.
Review multiplying powers of 10. SAY: Multiplying powers of 10 is easy because you just
write more zeros at the end of the number you are multiplying by.
Exercises: 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
Exercises: What do you multiply by?
a) 10 ×
= 1,000
b) 100 ×
Bonus:
d) 100 ×
= 10,000
e) 1,000 ×
= 1,000
c) 10 ×
= 100
= 10,000,000,000
Answers: a) 100; b) 10; c) 10; Bonus: d) 100; e) 10,000,000
Students who are struggling can write the number of zeros under each power of 10. For
example, in part a), write “1” under 10 and “3” under 1,000.
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Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
(MP.7) Introduce decimal fractions. Write on the board:
Decimal fractions
5
10
4
10
3
100
Not decimal fractions
425
1,000
1
2
2
5
4
17
9
20
289
3,000
Have volunteers suggest fractions. Have the rest of the class point their thumbs toward the
correct group to signal where each fraction should be placed. Have students guess the rule for
putting the fractions in each group.
Explain that a decimal fraction is a fraction whose denominator is a power of 10. Decimal
fractions are important because powers of 10 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:
SAY: The picture shows why three tenths equals 30 hundredths. The second square has 10
times as many shaded parts and 10 times as many parts altogether. Write on the board:
Exercises: Write an equivalent fraction with the denominator 100.
Answers: a) 70/100, b) 40/100, c) 90/100
Equivalent tenths, hundredths, and thousandths. Write on the board:
SAY: Now you have to decide what to multiply the numerator by to get an equivalent fraction.
You have to figure out what the denominator was multiplied by and then multiply the numerator
by the same thing. Have volunteers tell you what to multiply by; then have other volunteers fill in
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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the numerators. (30, 700, 50) SAY: To make an equivalent fraction, you just have to add the
same number of zeros at the end of the numerator and denominator.
Exercises: Write the missing numerator in the equivalent fraction.
Answers: a) 80; b) 300; c) 900; Bonus: 300,000
Adding tenths and hundredths. Draw on the board:
SAY: If you can add hundredths, and if you can change tenths to hundredths, then you can add
tenths and hundredths. Three tenths is 30 hundredths, and six more hundredths is 36 hundredths.
Remind students that they can change tenths to hundredths without using a picture:
Exercises: Add.
4
7
7
4
5
8
b)
+
c)
+
+
a)
10 100
10 100
10 100
Answers: a) 47/100, b) 74/100, c) 58/100, Bonus: 93/100
Bonus:
3
9
+
100 10
Adding tenths, hundredths, and thousandths. Write on the board:
3
9
6
+
+
10 100 1,000
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 when you make all
the denominators 1,000, the numerators are in expanded form.
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Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
Exercises: Add.
a)
4
3
9
+
+
10 100 1,000
b)
5
2
1
+
+
10 100 1,000
Bonus:
3
8
4
+
+
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:
4
9
+
=
10 1,000
ASK: How many thousandths are in 4/10? (400) So how many thousandths are there
altogether? (400 + 9 = 409) Write the answer:
4
9
409
+
=
10 1,000 1,000
SAY: 4 tenths, 0 hundredths, and 9 thousandths add to 409 thousandths. Have students add
more tenths and thousandths.
Exercises: Add.
a)
3
7
+
10 1,000
b)
9
1
+
10 1,000
c)
2
6
+
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
SAY: We might be tempted to write this as 049/1,000, but we do not write the zero at the
beginning of a number.
Exercises: Predict the sum. Then check by adding.
3
7
+
100 1,000
8
2
c)
+
10 1,000
a)
8
2
+
100 1,000
5
8
d)
+
10 1,000
b)
Bonus:
e)
8
2
+
1,000 10
f)
8
2
+
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
SAY: You might need to add thousandths or just hundredths.
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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Exercises: Add the decimal fractions. Write the answer as a decimal fraction.
a)
8
9
2
+
+
10 100 1,000
b)
8
6
+
10 1,000
c)
9
6
+
10 100
d)
3
5
+
100 1,000
Bonus:
e)
6
8
2
+
+
100 1,000 10
f)
5
5
+
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
2. (MP.1) a) Is there a largest power of 10?
(MP.3) 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. Find the missing number.
a)
4
9
+
=
10 10 10
b)
3
38
+
=
10 100 100
c)
e)
4
+
10
f)
10
+
5
45
=
100 100
Bonus:
d)
6
+
10
=
67
100
=
67
100
7
+
100
=
7
10
Answers: a) 5, b) 8, c) 4, Bonus: d) 7/100, e) 27/100, f) 63/100
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Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
NS7-19
Place Value and Decimals
Pages 102–104
Standards: preparation for 7.NS.A.1
Goals:
Students will identify the place value and the actual value of digits in whole numbers and
decimals.
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
Vocabulary: decimal, decimal fraction, decimal point, denominator, hundredth, place value,
placeholder, power of 10, represent, tenth, thousandth
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) 6,742
b) 9,017
Answers: a) 700; b) 7; c) 70; d) 7,000
c) 6,572
Exercises: Find the actual value of the digit 6 in the number.
a) 632
b) 5,632
c) 75,632
Answers: a) 600, b) 600, c) 600, d) 600
d) 7,904
d) 875,632
Large place values—ten thousands, hundred thousands, and millions. SAY: The next
three place values after thousands are ten thousands, hundred thousands, and millions. Write
on the board:
7,902,416
millions hundred
thousands
ten thousands
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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Point out that we write commas to separate groups of three digits, between hundreds and
thousands, hundred thousands and millions, and so on.
Exercises: Write the place value of the digit 3 in the number.
a) 312,607
b) 453,207
c) 3,762,906
d) 7,401,235
e) 8,435,241
f) 8,004,312
Answers: a) hundred thousands, b) thousands, c) millions, d) tens, e) ten thousands,
f) hundreds
The place value system extends to include tenths. Write on the board:
thousands
hundreds
tens
ones
SAY: 10 hundreds fit into a thousand, 10 tens fit into a hundred, and 10 ones fit into a ten. Tell
students that you want to continue the place value system so that you can use place value for
fractions too. ASK: Ten of what make one whole? (tenths) To guide students, draw pictures of
10 equal parts fitting into one whole and shade 1/10:
ASK: Does anyone remember how to show 1/10 using place value? (0.1) Write on the board:
1
= 0.1
10
3
= 0.3
10
8
3
= 8.3
10
27
4
= 27.4
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 wholenumber 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 the number.
a)
5
10
b) 3
8
10
c) 74
6
10
Bonus: 800
3
10
Answers: a) 0.5, b) 3.8, c) 74.6, Bonus: 800.3
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
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tens
ones
tenths
hundredths
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
SAY: There is also symmetry in the values. Write on the board:
Ask volunteers to continue the place values in both directions. Show students how to write
decimals for one-digit hundredths and thousandths:
3
= 0.03
100
8
= 0.008
1,000
SAY: The next place value after tenths is hundredths. The next place value after that is
thousandths.
Exercises: Write the decimal.
a)
7
100
b)
4
1,000
c)
5
100
d)
9
1,000
e)
6
1,000
Answers: a) 0.07, b) 0.004, c) 0.05, d) 0.009, 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
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:
9+
6
7
+
= 9.67
10 100
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.”
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.
Exercises: Write the decimal.
a) 3 +
4
9
+
10 100
b) 8 +
5
3
+
10 100
c) 6 +
5
8
2
+
+
10 100 1,000
Answers: a) 3.49, b) 8.53, c) 6.582
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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Using 0 as a placeholder. Write on the board:
3
= 0.3
10
5+
3
= 5.3
10
3
= 0.03
100
5+
3
=
100
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
+
10 100
5
3
+
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 will have to write zeros in those positions.
Exercises: Write the decimal.
a) 2 +
3
100
b) 8 +
8
1,000
c) 5 +
7
1,000
d) 3 +
5
2
+
10 1,000
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 do the zeros hold 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
Extensions
(MP.1) 1. Write the correct decimal: $800,000 + $40 + 9¢ = $ _________________
Answer: $800,040.09
(MP.1, MP.8) 2. a) How many times as much as the second 3 is the first 3 worth?
i) 28,331
ii) 24,303
iii) 320,135
iv) 3,789,453
b) Which is worth more in each number, the 3 or the 6? How many times more?
i) 63
ii) 623
iii) 6,342
iv) 36
v) 376
vi) 3,006
Hint: Pretend the 6 is a 3, compare the first 3 to the second 3 and then solve the harder
problem. Example: In part ii), look at the number 323. The first 3 is worth 100 times as much as
the second 3. Since 6 is 2 times as much as 3, the 6 in 623 is worth 200 times as much as the 3.
c) How many times as much as the 5 is the 2 worth?
i) 25
ii) 253
iii) 2,534
iv) 342,580
v) 3,472,508
vi) 2,345
vii) 23,457
viii) 234,576
ix) 2,345,768
d) How much more is the 2 worth than the 5 in the decimal 0.324067568?
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Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
Answers:
a) i) 10, ii) 100, iii) 10,000, iv) 1,000,000;
b) i) 6, 20 times; ii) 6, 200 times; iii) 6, 20 times; iv) 3, 5 times; v) 3, 50 times; vi) 3, 500 times;
c) As long as the numbers are immediately next to each other, the 2 is always worth 4 times as
much as the 5; as long as the numbers are 3 digits apart, the 2 is always worth 400 times as
much as the 5. i) 4, ii) 4, iii) 4, iv) 4, v) 4, vi) 400, vii) 400, viii) 400, ix) 400;
d) 400,000
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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NS7-20
Positive and Negative Decimals
Pages 105–107
Standards: preparation for 7.NS.A.1
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
Vocabulary: decimal, decimal fraction, decimal point, hundredth, mixed number, negative,
positive, tenth, thousandth
Writing decimals as proper fractions. Write on the board:
0.206 = 2 tenths + 6 thousandths =
2
6
206
+
=
10 1,000 1,000
Point out that the numerator is the decimal without the “zero point” in front. The number of zeros
in the denominator is equal to the number of digits after the decimal point. Write on the board:
0.206 =
206
1,000
0.037 =
37
1,000
SAY: There are three digits after the decimal point, so there are 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.
Answers: a) 3/10; b) 56/1,000; c) 801/1,000; d) 437/1,000,000
Writing proper fractions as decimals. Write on the board:
34
1,000
Tell students that you want to write this fraction 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
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Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
decimal point. Write the digits “34” on the board, and point out that there are only two digits. So,
to make three digits after the decimal point, students need to add a 0 before the 3. Write on the
board:
0.034
Point out that 34 thousandths is the same as 3 hundredths and 4 thousandths, so 0 ones and 0
tenths makes sense. Write on the board:
3
100
Have students raise the correct number of fingers to signal the answer as you ASK: How many
digits would you put after the decimal point in the decimal? (2) How many digits are in the
numerator? (1) How many zeros do you need to write after the decimal point? (1) 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 on the board beside the
fraction:
3
= 0.03
100
In the exercises below, have students hold up the correct number of fingers to signal the
number of zeros.
Exercises: How many zeros do you need to write after the decimal point in the fraction?
a)
34
100
b)
7
1,000
c)
32
10,000
d)
8,405
100,000
Answers: a) 0 (closed fist), b) 2, c) 2, d) 1
Exercise: 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, for example,
instead of writing 0.4, they will write .4. SAY: Be careful not to miss the decimal point; you don’t
want to mistake .4 for 4.
Comparing tenths and hundredths. Write on the board:
0.63
0.9
ASK: How many hundredths are in 0.63? (63) How many hundredths are in 0.9? (90) Which 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.
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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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
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:
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.”
NOTE: Another correct way to read 5.28 is “five point two eight.”
Write on the board:
three
3
six hundredths
0.06
three and six hundredths
Ask a volunteer to write the missing decimal. (3.06)
Exercises: Write the decimal.
a) five and eight hundredths
b) thirty-five thousandths
c) seven and twelve thousandths
d) twenty and two thousandths
Answers: a) 5.08, b) 0.035, c) 7.012, d) 20.002
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
SAY: 20.04 is said “twenty and four hundredths” while 0.24 is said “twenty-four hundredths.”
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Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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
Exercises: 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 ____,
79
so
= ____.
10
so
608
= ____.
100
Have volunteers fill in the blanks. (7 R 9, so 7 9/10; 6 R 8, so 6 8/100)
Exercises: Write the improper fraction as a mixed number.
a)
43
10
b)
780
100
c)
3,524
1,000
d)
1,234
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:
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.
Exercises: Write the improper fraction as a mixed number, then as a decimal.
28
728
793
7,845
63,457
b)
c)
d)
Bonus:
a)
1,000
10
10
100
100
Answers: a) 2 8/10 = 2.8, b) 72 8/10 = 72.8, c) 7 93/100 = 7.93, d) 78 45/100 = 78.45,
Bonus: 63 457/1,000 = 63.457
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-15
(MP.8) 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? (the number without the decimal
point) Where do you see the denominator in the answer? (the number of zeros in the
denominator is the number of digits after the decimal point)
Point out that the decimal can be obtained by writing the numerator and then making sure that
the number of digits after the decimal point is the same as the number of zeros in the
denominator.
(MP.7) Exercises: Convert into a decimal without writing the mixed number.
a)
654
10
b)
43,654
100
c)
3,756
10
d)
53,094
1,000
Bonus:
845,036,714
100,000
Answers: a) 65.4, b) 436.54, c) 375.6, d) 53.094, Bonus: 8,450.36714
NOTE: When writing decimals, the convention is to separate groups of three place values with
commas before the decimal point, but not after.
Negative decimals. Tell students that, just as 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 , so = -7.34
100
100
(MP.7) Exercises: Write the decimal for the negative number.
2
10
a) -61
b) -
382
100
c) -
407
100
d) -83
2
1,000
Answers: a) −61.2, b) −3.82, c) −4.07, d) −83.002
Extensions
1. Teach students to interpret whole numbers written in decimal format. Examples: 5.1 million
is 5,100,000; 3.7 thousand is 3,700.
2. Have students look for decimals in the media (e.g., news stories, billboards, advertisements,
posters) and write the decimals as mixed numbers.
3. Find the mistakes.
0.003 =
3
100
0.05 =
5
100
0.47 =
47
10
0.032 =
32
1, 000
Answers: 0.003 = 3/1,000, not 3/100; 0.47 = 47/100, not 47/10
E-16
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
NS7-21
Comparing Fractions and Decimals
Pages 108–109
Standards: preparation for 7.NS.A.1
Goals:
Students will place simple fractions and decimals on number lines.
Students will compare positive and negative decimals and simple fractions.
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
Vocabulary: decimal, decimal fraction, decimal point, equivalent decimals, equivalent fractions,
hundredth, negative, positive, tenth, thousandth
Materials:
BLM Number Lines from −2 to 2 and −0.2 to 0.2 (p. E-45)
BLM Hundredths Number Lines (p. E-46)
Decimals on number lines. Draw on the board:
A 0
B C 1
A=
D E 2
3
C=
Have volunteers write point A as a decimal (0.3) and a 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). Point out how the wholes and the tenths are shown in both ways of writing the
number.
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. 2.7 and 2 7/10
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-17
Positive and negative mixed numbers and decimals on a number line. Display the number
line below:
Point to the point A. ASK: How far from 0 is this? (3 tenths) Is it positive or negative? (negative)
Write “−.3” under A. Ask a volunteer to write the decimal for B. (−0.5 or −.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 write the decimal for D. (−1.8)
Exercises: Write a decimal and a fraction or mixed number for each point marked.
A B C D E −2
−1
0
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. 0.9, 9/10
Remind students that, on a number line, numbers on the left are less than numbers on the right.
Provide students with a strip from BLM Number Lines from −2 to 2 and −0.2 to 0.2.
Exercises: Write < or >.
6
10
a) -1
−.2
b) −1.4
6
-1
10
c) −0.8
1.1
Answers: a) <, b) >, c) <
Partial number lines divided into hundredths. Project on 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:
.2
.3
.20
.30
Point out that we can label .20 and .30 as tenths, .2 and .3. The length of this part of the number
line is 1 tenth and it is divided into 10 equal parts, so each part is a hundredth. Now mark the
two points and, pointing to each in turn, ASK: What number is this? (.21 and .25) Now draw on
the board:
E-18
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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, −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—but the numbers themselves get smaller.
Exercise: Write the points marked from least to greatest.
Answers: −0.49, −0.47, −0.44, −0.42
Exercises: Use the number lines from BLM Number Lines from −2 to 2 and −0.2 to 0.2.
Write < or >.
a) -
2
100
−.13
b) −0.1
0
c) −.13
−.2
d) 0.07
0.11
Answers: a) <, b) <, c) >, d) <
(MP.4) Word problems practice. Before giving part b) below, remind students that integers are
used to describe elevation, with 0 being sea level, anything higher being positive, and anything
lower being negative.
a) Which temperature, −3.6°F or −2.58°F, is warmer? (−2.58°F)
b) Which elevation, −14.2 m or −14.7 m, is higher? (−14.2 m)
Comparing decimal tenths to 1/2. Draw on the board:
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) 0.3
b) 0.6
c) 0.8
d) 0.4
e) 0.2
Answers: a) less, b) more, c) more, d) less, e) less
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 40
hundredths and 0.37 is only 37 hundredths. ASK: Which is greater, 0.56 or 0.5? (0.56)
PROMPTS: How many hundredths are in 0.56? (56) And in 0.5? (50) ASK: 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) 0.42
b) 0.87
c) 0.39
d) 0.51
Answers: a) less, b) greater, c) less, d) greater
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-19
Comparing decimal tenths to quarters. Add the marks for quarters (1/4 and 3/4) to the top
number line on the board and invite volunteers to label them.
Exercises: Write > or <.
a) 0.4
1
4
b) 0.6
3
4
3
4
c) 0.8
1
4
d) 0.3
Answers: a) >, b) <, c) >, d) >
Comparing negative fractions and decimals. Draw a double number line from −1 to 0, with
markings for decimal tenths and fourths. Invite volunteers to label all the increments.
Remind students that the order in the 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
d) −0.1
1
4
1
4
-
b) −0.6
e) −0.8
3
4
3
4
-
c) −.6
f) -
1
2
-
1
2
−0.4
Answers: a) <, b) >, c) <, d) >, e) <, f) <
Comparing decimal hundredths to quarters. Write on the board:
1
=
4 10
1
=
4 100
ASK: Can you multiply 4 by a whole number to get 10? (no) Cross out the first equation.
ASK: Can you multiply 4 by a whole number to get 100? (yes, 25) Show this on the board:
How can you write 1/4 as a decimal? (0.25) Repeat for writing 3/4 as a decimal (0.75); then
summarize on the board as follows:
1
= 0.25
4
1
= 0.5 = 0.50
2
3
= 0.75
4
SAY: You can use this to compare any number of hundredths to 1/4, 1/2, or 3/4.
Exercises: Which number is greater?
a) 0.12 or 1/4
b) .73 or 1/2
c) 0.73 or 3/4
d) .29 or 1/4
e) 0.45 or 3/4
f) .89 or 1/2
g) 0.87 or 3/4
h) .36 or 1/4
Answers: a) 1/4, b) .73, c) 3/4, d) .29, e) 3/4, f) .89, g) 0.87, h) .36
E-20
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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 −1/4
b) −.63 or −1/2
c) −0.67 or −3/4
d) −.59 or −1/2
e) −0.85 or −3/4
f) −.28 or −1/4
Answers: a) −.19, b) −1/2, c) −0.67, d) −1/2, e) −3/4, f) −1/4
Converting fractions to decimal fractions and decimals. Write on the board:
1
=
5 10
2
=
5 10
3
=
20 100
Ask volunteers to write the missing numerators. (2, 4, 15) ASK: How can you write 1/5 as a
decimal? (0.2) Repeat with 2/5 (0.4) and 3/20 (0.15).
Exercises: Write the fraction as a decimal hundredth.
1
20
7
d)
50
a)
1
25
23
e)
25
b)
c)
1
50
Bonus:
24
300
Answers: a) 0.05, b) 0.04, c) 0.02, d) 0.14, e) 0.92, Bonus: 0.08
SAY: Writing the fractions as decimal hundredths is like converting them to the common
denominator 100, so ordering fractions becomes easy.
Exercise: Order the fractions from the preceding exercises from greatest to least.
Bonus: Write the fractions with denominator 1,000; then put them in order from least to
greatest.
35
500
16
200
28
7,000
Answers: 0.92, 0.14, 0.08, 0.05, 0.04, 0.02, so the fractions from greatest to least are 23/25,
7/50, 24/300, 1/20, 1/25, 1/50; Bonus: 35/500 = 70/1,000, 16/200 = 80/1,000,
28/7,000 = 4/1,000, so from least to greatest, the fractions are 28/7,000, 35/500, 16/200.
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 < 5, then
−2 > −3 > −5.
Exercises: Write the numbers from least to greatest.
a) -
6
10
−0.48
-
35
100
b) −0.5
-
7
100
-
7
10
Answers: a) −6/10, −0.48, −35/100; b) −7/10, −0.5, −7/100
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-21
SAY: Positive numbers are always greater than negative numbers.
Exercises: Write the numbers from least to greatest.
3
5
a)
−.45
-
1
4
b) −2.14
-3
1
100
89
10
89
100
Answers: a) −.45, −1/4, 3/5; b) −3 1/100, −2.14, 89/100, 89/10
(MP.4) Word problems practice.
a) Which elevation, −2.8 m or -2
3
3
m, is higher? ( -2 m)
5
5
4
5
b) Which temperature, -3 °C or −3.76°C, is warmer? (−3.76°C)
Extensions
(MP.7) 1. a) Write the missing number. Hint: Write the decimal as a decimal fraction.
i)
ii)
= 0.7
10
3
iv)
= 0.13
100
8
v) = 0.8
= 0.03
iii)
= 0.327
1,000
7
vi) = 0.0007
b) Use equivalent fractions to find the missing number.
i)
100
iv)
ii)
= 0.4
v)
= 0.6
5
1,000
25
= 0.3
= 0.24
iii)
vi)
100
4
= 3.27
= 1.25
Sample solutions: a) i) 0.7 = 7/10, so 7 is the missing number; b) i) 0.4 = 4/10 = 40/100, so the
missing number is 40.
Answers: a) ii) 13, iii) 327, iv) 100, v) 10, vi) 10,000; b) ii) 300, iii) 327, iv) 3, v) 6, vi) 5
2. Write any number that works.
3
4
>>10
100
10
58
47
b) <- <100
10
100
183
c) <1,000
10
a) -
Answers: a) any number between 31 and 39, b) 5, c) 1
(MP.1) 3. Use 10, 100, and 1,000 once each to make the statement true.
a) b)
27
1
>-
3
and -
4
4
>100
=
Answers: a) −27/1,000 > −3/100, −4/100 > −4/10; b) 1/10 = 100/1,000 or 1/100 = 10/1,000
E-22
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
(MP.1, MP.2) 4. Ray 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.
● 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.
Answers: coelacanth: −0.18 km, football fish: −0.9 km, deep sea angler: −1.8 km,
rattail: −0.025 km
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-23
NS7-22
Adding and Subtracting Multi-Digit Decimals
Pages 110–112
Standards: 7.NS.A.1c, 7.NS.A.1d
Goals:
Students will add and subtract positive and negative decimals.
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 add and subtract multi-digit numbers with or without regrouping
Uses addition to check subtraction
Vocabulary: absolute value, algorithm, decimal point, hundredth, regrouping, tenth, thousandth
Adding decimals. Write on the board:
21 + 14 = 35
21 14 35
+
=
10 10 10
21
14
35
+
=
100 100 100
Write the first equation in vertical format; then ask volunteers to write the other two equations as
decimals in vertical format:
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
b) 2.74 + 3.58
c) 192.8 + 15.4
e) .78 − .42
f) 0.37 − 0.29
g) 34.85 − .65
E-24
d) 4.186 + 1.234
h) 6.432 − 2.341
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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
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 decimals with different numbers of digits to the right of the decimal point. Write
on the board:
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)
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
would 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:
Exercises: Add.
a) 4 + 13.7
b) 16 + 2.3
Answers: a) 17.7, b) 18.3, c) 52.71
c) 38 + 14.71
SAY: You can check your answers by adding the fractions. Do this for the example on the
board:
7
7
32 + 4.7 = 32 + 4
= 36
= 36.7
10
10
(MP.1) Exercise: Check your answers to the exercises above by adding the fractions.
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-25
Subtracting decimals. SAY: You can subtract by lining up the decimal points too. You might
have to write zeros to make both decimals have the same number of digits after the decimal
point. Demonstrate as shown:
7 10
36.94
36.94
34.8
34.80
− 15.3
− 15.30
− 21.65
−21.65
21.64
13.15
NOTE: Writing zeros at the end of the bottom number is optional, since it is easy to subtract,
say, 4 − 0, even when the zero isn’t written. But writing zeroes at the end of the top number is
necessary because you will have to regroup, and it’s easier to regroup when you can write what
you’re doing.
Exercises: Subtract. Use grid paper.
a) 7.4 − 2.1
b) 6.93 − 4.52
c) 8.56 − 3.87
d) 6.5 − 3
e) 1.7 − .42
f) 20.37 − 5.294
g) 2 − 0.52
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) Mona made 0.6 L of milkshake by adding ice cream to 0.48 L of milk. How much ice cream
did she add? (0.6 L − 0.48 L = 0.12 L)
(MP.3) b) Len 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. (1.23 m + 2.13 m = 3.36 m; the bed will not fit)
c) Sara 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? (2.31 m)
Adding multi-digit negative decimals. Remind students that if they can add two positive
numbers, then they can add two negative numbers too. Write on the board:
− 3.50 − 4.10
SAY: If you lost $3.50, then lost $4.10, how much did you lose overall? ($7.60) Finish the
equation on the board:
− 3.50 − 4.10 = −7.60
SAY: You just had to add as though they were positive and then put the negative sign in front.
Exercises: Add.
a) − 5.4 − 4.17
b) − 3.12 − 4.25
Answers: a) −9.57, b) −7.37. c) −10.048
E-26
c) − 7.04 − 3.008
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
SAY: In these exercises, you will have to regroup.
Exercises: Add.
a) − 6.08 − 5.28
b) − 3.7 − 4.93
Answers: a) −11.36, b) −8.63, c) −12.72
c) − 5.8 − 6.92
Adding multi-digit positive and negative decimals. Remind students that if they know how to
subtract positive numbers, then they can add positive and negative numbers. Write on the
board:
− 3.4 + 2.1
ASK: If you lost 3.4 dollars and gained 2.1 dollars, did you gain more or lose more? (lose)
SAY: So the answer will be negative. How much did you lose? (3.4 − 2.1 = 1.3, so you lost 1.3
dollars) Write the answer on the board:
− 3.4 + 2.1 = −1.3
SAY: The answer is negative because the number with the greater absolute value is negative.
Remind students of the notation for absolute value. Write on the board:
|−3.4| = 3.4
|2.1| = 2.1
SAY: The number part is 1.3 because you had to subtract the absolute values of the two
numbers.
Exercises: Add the positive and negative numbers.
a) − 15.4 + 3.2
b) − 2.8 + 5.91
Answers: a) −12.2, b) +3.11 or 3.11, c) −1.37
c) − 3.97 + 2.6
SAY: The next problems will need regrouping.
Exercises: Add the positive and negative numbers.
a) − 6.83 + 5.14
b) − 15.3 + 17.15
Answers: a) −1.69, b) +1.85 or 1.85, c) +8.58 or 8.58
c) + 18.45 − 9.87
Remind students that if they know how to add positive and negative numbers, then they
automatically know how to subtract positive and negative numbers, because they can always
change a subtraction to an addition. Write on the board:
+(+) = +
+(−) = −
−(+) = −
−(−) = +
Exercises: Add or subtract.
a) −3.4 − (+6.87)
b) −3.4 + (+6.87)
c) −14.37 + (+15.8)
d) −11.5 − (−15.8)
e) +2.8 − (−3.41)
f) +9.01 − (+11.4)
Answers: a) −10.27, b) 3.47, c) 1.43, d) 4.3, e) 6.21, f) −2.39
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-27
Debits and credits. Remind students that the bank records a debit when you take money out of
your account and a credit when you put money in. Draw on the board:
Debit (−)
Credit (+)
Balance
$0
$3.45
$8.31
$7.12
Have volunteers calculate the balance after each transaction. ($3.45, −$4.86, $2.26) Tell
students that a negative balance means that they owe the bank. It is not a good idea, but it can
happen sometimes. Now add a row to the chart, and tell students that the debit of $8.31 was
actually a mistake by the bank. ASK: How can the bank correct its mistake? (add a credit of
$8.31) ASK: What is the new balance? ($10.57) Write on the board:
$2.26 − (−$8.31) = $2.26 + $8.31 = $10.57
SAY: You can think of correcting the mistake as taking away the negative amount or as adding
the positive amount, because they both do the same thing.
(MP.4) Exercises: Write an addition and a subtraction to find the corrected balance.
a) John’s bank account balance was $14.00. John noticed an incorrect debit to his account for
$15.81, so the bank now has to correct its mistake. What is the corrected balance?
b) Nancy’s bank account balance was $37.00. Nancy noticed an incorrect debit to her account
for $13.29. What is the corrected balance?
c) Tony’s bank account balance was −$14.23. Tony noticed an incorrect debit to his account for
$103.29. What is the corrected balance?
d) Wendy’s bank account balance was −$7.50. Wendy noticed an incorrect debit to her account
for $5.11. What is the corrected balance?
Answers: a) $14.00 − (−$15.81) = $14.00 + $15.81 = $29.81,
b) $37.00 − (−$13.29) = $37.00 + $13.29 = $50.29,
c) −$14.23 − (−$103.29) = −$14.23 + $103.29 = $89.06,
d) −$7.50 − (−$5.11) = −$7.50 + $5.11 = −$2.39
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: 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.
E-28
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
(MP.1) 3. Subtract 1.27 − 0.5 using the number line. Do you get the same answer by lining up
the decimal points?
(MP.3) 4. Abdul says that 0.91234 is the largest number less than 1 that can be added to
0.08765 without needing to regroup. Is that 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 is larger than
0.91234 and can be added to 0.08765 without needing to regroup.
(MP.1) 5. Place decimal points in each number to make the statement correct.
a) 327 + 148 = 1807
b) 534 − 812 = 4528
Sample answers: a) 3.27 + 14.8 = 18.07, b) 53.4 − 8.12 = 45.28
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-29
NS7-23
Division with Fractional and Decimal Answers
Pages 113–114
Standards: preparation for 7.NS.A.2
Goals:
Students will divide a whole number by a whole number and will write the answer as a fraction
and as a decimal.
Prior Knowledge Required:
Can divide whole numbers by whole numbers with remainder
Can convert an improper fraction to a mixed number
Can multiply a fraction by a whole number
Can draw pictures representing proper fractions, improper fractions, and mixed numbers
Can create equivalent fractions with a given denominator
Can write decimal fractions as decimals
Understands that multiplying both terms of a division by the same number does not change the
answer
Materials:
a calculator for each student (see Extension 5)
BLM Division, Fractions, and Decimals (Advanced) (p. E-47, see Extension 8)
Vocabulary: decimal, decimal point, denominator, division, fraction, improper fraction, mixed
number, numerator, remainder
Drawing pictures to show equal sharing. Write on the board:
4 people share 7 pies
Point out that you are drawing circles for pies. ASK: How many circles should I draw? (7) Draw
the 7 circles on the board and then SAY: In case some pies taste better than others, they decide
to share each pie equally. ASK: How many pieces should I divide each circle into? (4) Divide the
pies; then have a volunteer shade the amount that one person gets:
Write the multiplication below the picture:
7´
E-30
1 7
=
4 4
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
Exercises: Draw a picture to show how much one person gets. Then write the multiplication.
a) 4 people share 2 pies
b) 4 people share 5 pies
c) 2 people share 3 pies
d) 2 people share 5 pies
Selected solution:
Answers: b) 5 × 1/4 = 5/4, c) 3 × 1/2 = 3/2, d) 5 × 1/2 = 5/2
Using division for equal sharing when the answer is a fraction. Remind students that
division is used for equal sharing. ASK: If 2 people share 6 pies, how much does each person
get? (3 pies) Write on the board:
2 people share 6 pies
4 people share 5 pies
6÷2=3
________ ÷_________ = ________
Ask a volunteer to fill in the blanks (5 ÷ 4 = 5/4) and have students signal whether they agree
(thumbs up) or disagree (thumbs down). SAY: 4 people sharing 5 pies is another way of saying
that 5 pies are divided equally between 4 people, so the answer is 5 ÷ 4. Point out that the
number of objects being divided goes first, and the number of people sharing goes second. The
answer is how much each person gets. Students may be used to seeing 5 ÷ 4 = 1 R 1, but now
the answer includes a fraction: 5/4 or 1 1/4. Have another volunteer write the division equation
to show 5 people sharing 2 pies. (2 ÷ 5 = 2/5) SAY: When 2 pies are shared equally among 5
people, the answer is 2 ÷ 5.
Exercises: Write the division equation.
Answers: a) 3 ÷ 5 = 3/5, b) 4 ÷ 5 = 4/5, c) 3 ÷ 6 = 3/6, d) 4 ÷ 6 = 4/6
Dividing whole numbers without a picture. Write on the board:
5 ÷ 7 = __________ 4 ÷ 9 = __________ 3 ÷ 8 = __________
Challenge students to predict the answers to these questions without using a picture. Point out
that the first number in the division is the numerator (number on top of the fraction) and the
second number in the division is the denominator (number on the bottom of the fraction).
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-31
Exercises: Divide. Write your answer as a fraction.
a) 3 ÷ 10
b) 4 ÷ 7
c) 8 ÷ 9
Bonus: 13 ÷ 1,000
Answers: a) 3/10, b) 4/7, c) 8/9, d) 7/8, Bonus: 13/1,000
d) 7 ÷ 8
Have students check their answers by multiplication. For example, since 3 ÷ 10 = 3/10, they
should check that 3/10 × 10 = 3.
Writing the answer as a mixed number. Now tell students that 3 people are sharing 5 pies.
ASK: How much does each person get? (5/3 pies) Remind students that we can write an
improper fraction as a mixed number. Ask a volunteer to shade 5/3 to find the mixed number.
Exercises: Divide. Write the answer as an improper fraction and as a mixed number. Show
your answer with a picture.
a) 9 ÷ 4
b) 7 ÷ 2
c) 6 ÷ 4
Bonus: 15 ÷ 8
Answers: a) 9/4 = 2 1/4, b) 7/2 = 3 1/2, c) 6/4 = 1 2/4 = 1 1/2, Bonus: 15/8 = 1 7/8
Point out that another way to get the answer to 9 ÷ 4 is to not break the pies into parts until you
have to. If you start by giving each person 2 whole pies, then you just have to divide the
remaining pie into fourths. So each person gets 2 1/4 pies. Have students solve the other
exercises the same way to verify their answers.
Review writing decimal fractions as decimals. Write on the board:
82
100
7
100
704
100
6,572
100
Remind students that the number of zeros in the denominator of the fraction is the number of
digits after the decimal point. SAY: Each of these fractions has denominator 100, so each of the
decimals will have two digits after the decimal point. Write on the board:
82
7
704
6572
Ask volunteers to finish writing the decimal for each fraction by writing the decimal point in the
correct place. (0.82 or .82, 0.07 or .07, 7.04, 65.72) SAY: You may need to write a 0 after the
decimal point.
E-32
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
Exercises: Write the fraction as a decimal.
38
100
4
e)
10
15
10
548
f)
10
a)
b)
Bonus:
7
1,000
3
g)
10
c)
476
100
91
h)
1,000
d)
4,503
1,000,000
Answers: a) 0.38, b) 1.5, c) 0.007, d) 4.76, e) 0.4, f) 54.8, g) 0.3, h) 0.091, Bonus: 0.004053
Division with decimal answers. Write on the board:
82 ¸100 =
82
100
SAY: This can also be written as a decimal. Have a volunteer write the decimal answer. (0.82)
Exercises: Divide. Write the answer as a fraction and a decimal.
a) 432 ÷ 100
b) 3 ÷ 1,000
c) 18 ÷ 10
d) 4 ÷ 100
e) 98 ÷ 100
f) 37 ÷ 1,000
Answers: a) 432/100 = 4.32, b) 3/1,000 = 0.003 or .003, c) 18/10 = 1.8, d) 4/100 = 0.04 or .04,
e) 98/100 = 0.98 or .98, f) 37/1,000 = 0.037 or .037
Connecting two ways of dividing. Tell students that when dividing by 10, 100, or 1,000, they
are just moving the decimal point one, two, or three places to the left. Write on the board:
82.0
So 82 ÷ 100 = 0.82.
SAY: There are two zeros in 100, so move the decimal point two places to the left.
Exercises: Divide by moving the decimal point the correct number of places. Make sure you get
the same answers as above.
a) 432 ÷ 100
b) 3 ÷ 1,000
c) 18 ÷ 10
d) 4 ÷ 100
e) 98 ÷ 100
f) 37 ÷ 1,000
Answers: a) 4.32, b) 0.003 or .003, c) 1.8, d) 0.04 or .04, e) 0.98 or .98, f) 0.037 or .037
Dividing by a number that is not a power of 10. Write on the board:
4÷5
Ask a volunteer to write the answer as a fraction. (4/5) Then ASK: Is there an equivalent fraction
with denominator 10? Write on the board:
4
=
5 10
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-33
Ask a volunteer to fill in the missing numerator. Then show the multiplicative relationship
between the numerators and denominators.
SAY: So 4 fifths = 8 tenths. Write on the board:
4 ÷ 5 = 0.8
Exercises: Divide. Write your answer as a fraction and as a decimal.
a) 2 ÷ 5
b) 3 ÷ 2
c) 6 ÷ 5
d) 7 ÷ 2
(MP.1) Bonus: Divide. Then divide your answer by 10. For each question, do you get the same
answer as you got in the exercises above?
e) 20 ÷ 5
f) 30 ÷ 2
g) 60 ÷ 5
h) 70 ÷ 2
Answers: a) 2/5 = 0.4; b) 3/2 = 1.5; c) 6/5 = 1.2; d) 7/2 = 3.5; Bonus: e) 4 ÷ 10 = 0.4, yes;
f) 15 ÷ 10 = 1.5, yes; g) 12 ÷ 10 = 1.2, yes; h) 35 ÷ 10 = 3.5, yes
Write on the board:
1¸ 4 =
1
4
ASK: Is there an equivalent fraction with denominator 10? (no) How do you know? (4 does not
divide into 10) Is there an equivalent fraction with denominator 100? (yes) How do you know? (4
divides into 100) Write on the board:
1
=
4 100
Ask a volunteer to write the missing numerator (25); then ask another volunteer to write the
decimal answer (0.25).
Exercises: Divide by making an equivalent fraction with denominator 10 or 100.
a) 3 ÷ 2
b) 2 ÷ 5
c) 9 ÷ 20
d) 31 ÷ 50
e) 3 ÷ 4
f) 7 ÷ 5
g) 9 ÷ 25
h) 7 ÷ 4
Bonus: Reduce the fraction if you can before making the denominator a power of 10.
i) 6 ÷ 8
j) 9 ÷ 60
k) 6 ÷ 15
l) 111 ÷ 200
m) 314 ÷ 500
n) 21,403 ÷ 500,000
Answers: a) 3/2 = 15/10 = 1.5, b) 4/10 = 0.4, c) 45/100 = 0.45, d) 62/100 = 0.62, e) 0.75, f) 1.4,
g) 0.36, h) 1.75, Bonus: i) 3/4 = 0.75, j) 3/20 = 0.15, k) 2/5 = 0.4, l) 0.555, m) 0.628, n) 0.042806
When doing Question 5 on AP Book 7.1 p. 114, students might write their answers with
different numbers of digits after the decimal point. For example, in part e), students could write
3/4 = 750/1,000 instead of 75/100 and get 0.750 instead of 0.75. Both answers are correct.
E-34
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
Extensions
1. Write the fact family for 3/5 × 5 = 3.
Answers: 3/5 × 5 = 3, 5 × 3/5 = 3, 3 ÷ 3/5 = 5, 3 ÷ 5 = 3/5
(MP.1, MP.2) 2. To divide 3 ÷ 5, replace 3 with 30 ÷ 10. Do you get the same answer? Hint: Use
the fact that 30 ÷ 5 ÷ 10 = 30 ÷ 10 ÷ 5. This is true because they are both equal to 30 ÷ (5 × 10).
Answer: Yes: (30 ÷ 10) ÷ 5 = (30 ÷ 5) ÷ 10 = 6 ÷ 10 = 6/10 = 0.6.
(MP.1) 3. Write 11 ÷ 2 as a decimal in two ways: first write the answer as an improper fraction
and convert the improper fraction to a decimal; then write the answer as a mixed number and
convert the mixed number to a decimal. Make sure you get the same answer.
Answer: 11 ÷ 2 = 11/2 = 55/10 = 5.5 or 11 ÷ 2 = 5 1/2 = 5 5/10 = 5.5
(MP.1) 4. (1 + 2) ÷ 3 + (4 + 5) ÷ 6 = (7 + 8) ÷ ?
Answer: 6
(MP.1, MP.3) 5. a) Write the decimal answers to 1/4 and 5/4. Make sure that the decimal for 5/4
is exactly 1 greater than the decimal for 1/4. Why does this make sense?
b) How do you expect the decimals for 1/3 and 4/3 to compare? Check on a calculator.
Answer: a) The decimals are 0.25 and 1.25, and indeed 0.25 + 1 = 1.25. This makes sense
because the fraction 5/4 is 1 1/4; b) I expect the decimal for 4/3 to look exactly like the decimal
for 1/3, but with 1 in front. My calculator says 1/3 = 0.333333333 and 4/3 = 1.333333333.
(MP.1) 6. a) Predict which divisions will have an answer greater than 1; then check by writing
the answer as a decimal.
i) 9 ÷ 10
ii) 11 ÷ 10
iii) 3 ÷ 4
iv) 5 ÷ 4
v) 6 ÷ 5
vi) 5 ÷ 8
b) Estimate where B ÷ A is on the number line below.
A B
0
1
2
3
Answers: a) ii), iv), and v) have answers greater than 1. The answers are i) 0.9, ii) 1.1, iii) 0.75,
iv) 1.25, v) 1.2, vi) 0.625; b) slightly greater than 1
(MP.7) 7. a) Write the missing numerators; then change the fractions to decimals.
3¸2 =
3
=
=
2 10
6¸4 =
6
=
=
4 100
b) Are the answers equivalent decimals? Explain why this makes sense.
c) What property of division does this show?
Answers: a) 15 and 1.5, 150 and 1.50; b) The answers are equivalent decimals. This makes
sense because the fractions they are made from, 3/2 and 6/4, are equivalent fractions. c) This
shows the property that multiplying both numbers in a division by the same number keeps the
answer the same.
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-35
(MP.1) 8. After completing Extension 7, provide students with BLM Division, Fractions, and
Decimals (Advanced). As students work on the questions, they will make connections between
properties of division and properties of fractions.
Answers: 1. b) 4/4 = 1, c) 7/7 = 1, d) a/a = 1; 2. a) 4 + 3 = 7 and 14 ÷ 2 = 7, so
(8 ÷ 2) + (6 ÷ 2) = (8 + 6) ÷ 2; b) 24 + 8 = 32 and 48 ÷ 8 = 6, so (48 ÷ 2) + (48 ÷ 6) ≠ 48 ÷ (2 + 6);
3. a)
48
(8 + 6)
8 6
48 48
+ and
, b)
+
and
; 4. yes, a); 5. no, b); 6. a) 6 ÷ 3 = 2,
(2 + 6)
2
2 2
2
6
b) (3 × 5) ÷ 5 = 3, c) (7 × 4) ÷ 7 = 4, d) 3 ÷ 4 = (3 × 2) ÷ (4 × 2), e) 3 ÷ 4 + 2 ÷ 4 = 5 ÷ 4,
f) 12 ÷ 8 = (12 ÷ 4) ÷ (8 ÷ 4)
(MP.3) 9. Mandy says 2 R 1 = 2 1/4 because 9 ÷ 4 = 2 R 1 and 9 ÷ 4 = 2 1/4. Will says 2 R 1 = 2
1/3 because 7 ÷ 3 = 2 R 1 and 7 ÷ 3 = 2 1/3. Explain why their reasoning is incorrect.
Answer: 2 R 1 is not a number. The division sign is being used in two different ways.
NOTE: The two ways of using the division sign are essentially different. One way is an
operation using whole numbers that can only have whole-number answers (with remainder).
The other way is an operation using any numbers, including fractions, that can have any kind of
answer.
E-36
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
NS7-24
Long Division
Pages 115–118
Standards: preparation for 7.NS.A.2
Goals:
Students will use the standard algorithm for long division to divide 2-digit, 3-digit, and 4-digit
whole numbers by 1-digit whole numbers.
Prior Knowledge Required:
Understands remainders
Understands division as finding the number in each group
Can use base ten materials to represent numbers
Vocabulary: dividend, divisor, quotient, remainder, standard algorithm
Materials:
Base ten materials (optional)
BLM Hundreds Charts (p. E-48, see Extension 2)
Long division: three-digit by one-digit. Using pictures of base ten materials, explain why the
standard algorithm for long division works.
SAY: Let’s divide 726 into 3 equal groups.
Step 1. Make a model of 726 units.
7 hundreds blocks
2 tens blocks
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
6 ones blocks
E-37
Step 2. Divide the hundreds blocks into 3 equal groups.
Keep track of the number of units in each of the 3 groups, and the number remaining, by using
the long-division algorithm.
2
3 7 2 6
2 hundreds blocks, or 200 units, have been divided into each group.
6
600 units (200 × 3) have been divided.
1 2
1 hundred and 2 tens still need to be divided.
Point out how the long-division algorithm keeps track of the place values because hundreds are
put in the hundreds column, tens in the tens column, and ones in the ones column.
Exercises: Carry out Steps 1 and 2 of long division.
a) 2 512
b) 3 822
c) 2 726
d) 4 912
Students should show their work using actual base ten materials or a model sketched on paper
for at least one problem (but some students will need more practice).
Step 3. Divide the remaining hundreds block and the 2 remaining tens blocks equally among
the 3 groups. To do this, you need to regroup the hundreds block as 10 tens blocks.
There are 120 units in total, so 40 units can be added to each group from Step 2.
E-38
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
SAY: You can keep track of Step 3 using the standard long-division algorithm. Write on the
board, using a different color for the new step:
24
3 726
4 tens (or 40 new units) have been placed into each group.
6
12
12
06
12 tens (4 tens × 3) have been divided altogether.
6 units still need to be divided.
Emphasize that the 1 is in the hundreds column and the 2 is in the tens column because
together they represent 120.
Exercises: Carry out Step 3 of long-division on the problems you started above.
b) 3 822
c) 2 726
d) 4 912
a) 2 512
Exercises: Do Steps 1–3 on these problems.
b) 4 792
c) 2 538
d) 2 956
a) 3 612
Students should show their work using either base ten materials or a model drawn on paper for
at least one problem (but some students will need more practice).
Step 4. Divide the 6 remaining blocks among the 3 groups equally.
SAY: There are now 242 units in each group, so 726 ÷ 3 = 242.
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-39
SAY: You can keep track of Step 4 using the standard long-division algorithm. Write on the
board, using a different color for the new step:
242
3 726
6
12
12
06
6
0
2 new units (or 2 ones) have been placed into each group.
6 (or 2 × 3) new units have been divided.
There are no units left to be divided.
Emphasize that the 2 and the 6 are in the ones column because they both represent ones.
Encourage students to check this answer by multiplying 242 × 3.
Exercises: Do Step 4 on the problems you started above.
a) 2 512
b) 3 822
c) 2 726
d) 4 912
Answers: a) 256, b) 274, c) 363, d) 228
Exercises: Divide.
a) 2 856
b) 2 536
c) 3 816
d) 3 735
Answers: a) 428, b) 268, c) 272, d) 245
Students should show their work using either base ten materials or a model sketched on paper
for at least one problem (but some students will need more practice).
Exercises: Divide. Then check your answer by multiplication.
a) 842 ÷ 2
b) 952 ÷ 4
c) 528 ÷ 3
d) 518 ÷ 7
Answers: a) 421, check: 421 × 2 = 842; b) 238, check: 238 × 4 = 952; c) 176,
check: 176 × 3 = 528; d) 74, check: 74 × 7 = 518
Remind students that they can check the answer to a division with remainder by multiplying and
then adding. Write on the board:
7 ÷ 3 = 2 R 1, so 7 = 3 × 2 + 1
Exercises: Divide. Then check your answer using multiplication and addition.
a) 3 941
b) 2 713
c) 2 639
d) 4 935
Answers: a) 313 R 2, check: 3 × 313 + 2 = 941; b) 356 R 1, check: 2 × 356 + 1 = 713;
c) 319 R 1, check: 2 × 319 + 1 = 639, d) 233 R 3, check: 233 × 4 + 3 = 935
E-40
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
Long division when the divisor is greater than the dividend’s leading digit. Begin by
dividing a two-digit number by a one-digit number where the divisor is greater than the
dividend’s tens digit (i.e., there are fewer tens blocks available than the number of groups):
5 27
ASK: How many tens blocks are in 27? (2) Into how many groups do they need to be divided?
(5) Are there enough tens blocks to place one in each group? (no) Illustrate that in the answer
by writing a 0 above the dividend’s tens digit. Then ASK: What is 5 × 0? (0) Write:
Have a volunteer finish this problem; then ask if the 0 needs to be written at all. Explain that the
algorithm can be started by assuming that the tens blocks have already been traded for ones
blocks:
Emphasize that the answer is written above the dividend’s ones digit because it is the answer’s
ones digit.
Have students complete several similar problems:
4 37
5 39
8 63
8 71
Then move to three-digit by one-digit long division, where the divisor is more than the dividend’s
hundreds digit (examples: 324 ÷ 5, 214 ÷ 4, 133 ÷ 2). Again, follow the standard algorithm
(writing 0 where required); then introduce the shortcut (omit the 0).
Long division: four-digit by one-digit. Write on the board:
4 3215
4 9123
4 4312
SAY: You start with the first place value that has at least as many as the number of groups. In
the examples above, you need a place value of at least 4. There are not 4 thousands in 3,215,
but there are 32 hundreds, so you can divide those among the 4 groups. In 9,123, there are at
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
E-41
least 4 thousands, and the same with 4,312. Have volunteers circle the first place value that has
enough to divide among the groups:
2 1234
2 7135
8 8146
8 7129
Exercises: Divide using long division.
b) 2 7094
c) 5 3124
a) 2 1352
5 6124
d) 3 1250
e) 3 2146
Answers: a) 676, b) 3,547, c) 624 R 4, d) 416 R 2, e) 715 R 1
Introduce long division where the quotient has a 0 digit. Start the division for 936 ÷ 9:
ASK: How many times does 9 go into 3? (none) What is the tens digit of the quotient? (0) Write
it in and then ASK: What is 9 × 0? (0) Have a volunteer show where to write the answer to 9 × 0;
then demonstrate finishing the long division:
Point out that it is important that students remember to write the 0 in the quotient. The answer is
not 14, but 104!
Exercises: Divide using long division.
a) 4 835
b) 5 512
c) 3 924
f) 3 2,185
g) 4 3,629
h) 5 6,947
d) 3 817
e) 4 817
i) 3 7,418
Bonus: 3 742,163
Answers: a) 208 R 3, b) 102 R 2, c) 308, d) 272 R1, e) 204 R 1, f) 728 R 1, g) 907 R 1,
h) 1,389 R 2 , i) 2,472 R 2, Bonus: 247,387 R 2
(MP.4) Exercises: How long is each side if an octagon has a perimeter of
a) 952 cm
b) 568 cm
c) 8,104 cm
d) 3,344 cm
Answers: a) 119 cm, b) 71 cm, c) 1,013 cm, d) 418 cm
E-42
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
Assessment tip. If some students are having problems with long division, but you suspect it is
really the multiplication tables that are holding them back, try this: Have students copy and
complete the 8 times table at the top of a page in their notebook, or have them find the times
table themselves by adding 8 successively each time:
Then have students solve several long division problems that require dividing by 8, only.
Exercises: Divide.
b) 8 973
a) 8 375
c) 8 654
d) 8 6,149
e) 8 2,651
f) 8 743,156
Answers: a) 46 R 7, b) 121 R 5, c) 81 R 6, d) 768 R 5, e) 331 R 3, f) 92,894 R 4
Students can also write the 6 times tables and solve problems involving dividing by 6 only.
Exercises: Divide.
a) 993 ÷ 6
b) 765 ÷ 6
c) 891 ÷ 6
d) 1,743 ÷ 6
e) 3,842 ÷ 6
f) 54,387 ÷ 6
g) 94,716 ÷ 6
h) 38,427 ÷ 6
i) 45,327 ÷ 6
Bonus: 5,908,297,522 ÷ 6
Answers: a) 165 R 3, b) 127 R 3, c) 148 R 3, d) 290 R 3, e) 640 R 2, f) 9,064 R 3, g) 15,786,
h) 6,404 R 3, i) 7,554 R 3, Bonus: 984,716,253 R 4
Such problems will help students remember the times tables by repeatedly using them, will
allow students to focus more on learning the division, and will allow you to assess the division
alone. Have students use multiplication and addition to check their answers. (Example: if your
answer to b) was 127 R 3, find 6 × 127 + 3—do you get 765?)
Extensions
1. Teach students the Egyptian method of long division. The Egyptians did not have the times
tables available to them, but they knew how to double numbers. For example, to divide 659 ÷ 7,
the Egyptians would find 7 × 1 = 7, 7 × 2 = 14, 7 × 4 = 28, 7 × 8 = 56, 7 × 16 = 112,
7 × 32 = 224, 7 × 64 = 448, and 7 × 128 = 896, by successive doubling. They knew that 896 is
too high, so they stopped.
Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System
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659 = 448 + 112 + 56 + 28 + 14 + 1
= 7 × 64 + 7 × 16 + 7 × 8 + 7 × 4 + 7 × 2 + 1
= 7 × (64 + 16 + 8 + 4 + 2) + 1
= 7 × 94 + 1
So 659 ÷ 7 = 94 R 1
2. On one of the hundreds charts from BLM Hundreds Charts, shade all the multiples of 3.
What is the sum of the digits in a multiple of 3? Predict which of these numbers is a multiple of
3; then check using long division.
a) 3,142
b) 8,037
c) 2,953
d) 3,681
Answers: a) no, b) yes, c) no, d) yes
3. A number is a multiple of 3 exactly when the sum of its digits is a multiple of 3. To see this for
a four-digit number abcd:
1,000a + 100b + 10c + d = 999a + 99b + 9c + a + b + c + d
a multiple of 3
Explain how this also shows that a four-digit number is a multiple of 9 precisely when the sum of
its digits is a multiple of 9.
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Teacher’s Guide for AP Book 7.1 — Unit 4 The Number System