# Operations on Numbers

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Learn to read, describe and write numbers. Learn to compare and order numbers. Learn to rename numbers using place value concepts. Also, learn to explain your thinking when solving a problem.
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CHAPTER 2
1 Exploring Greater Numbers
Goal Compare numbers to one million.
A bumblebee can flap its wings about 200 times At-Home Help
per second. A million is a number that is
1000 thousands.
A dragonfly can flap its wings about 38 times
1 000 000
per second.
To estimate an answer to a problem,
1. Predict how many times a dragonfly flaps its wings use numbers that are close to the
in 1000 s. values in the problem that are easier
to work with.
For example, if a problem involves
about 40 000 times
comparing times in weeks to years,
I rounded 38 to 40 to estimate. use about 50 weeks in a year.
40 x 1000 = 40 000
2. About how many hours would it take for a dragonfly to flap its wings
1 000 000 times? Show your work.
A dragonfly would have to flap its wings for about 25 000 s.
1 000 000 ÷ 40 = 25 000
60 s x 60 min = 3600 s in an hour
I’ll round 3600 s to 4000 s to estimate.
25 000 ÷ 4000 = about 6
A dragonfly would take about 6 h to flap it’s wings 1 000 000 times.
3. a) About how many times can a bumblebee flap its wings in 1000 min?
In 1 min the bee flaps its wings about 200 x 60 = 12 000 times.
So, in 1000 min the bee flaps its wings about 12 000 000 times.
b) How many 1000 thousands is your answer in part a)?
twelve 1000 thousands
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CHAPTER 2
2 Reading and Writing Numbers
Goal Read, write, and describe numbers greater than 100 000.
1. Write each number in standard and expanded form. At-Home Help
a) Millions Thousands Ones To read and write large numbers, we
Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones group the digits in periods. A period is
a group of hundreds, tens, and ones.
For example, the place value chart
204 010 below shows three periods.
200 000 + 4000 + 10 third period second period first period
Millions Thousands Ones
Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones
b) Millions Thousands Ones
Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones
The number modelled above is
384 400 in standard form.
1 329 027 In expanded form, the number is
1 000 000 + 300 000 + 20 000 + 9000 + 20 + 7 300 000  80 000  4000  400.
2. Write each number as a numeral in standard form. The number in words is three hundred
eighty-four thousand four hundred.
a) four hundred forty thousand twenty-six
440 026
b) twenty-two thousand eight
22 008
c) seven hundred thirty-one thousand three hundred five
731 305
3. Write the words for each number.
a) 304 000 three hundred four thousand
b) 21 000 twenty-one thousand
c) 12 600 twelve thousand six hundred
4. The sun in our solar system takes about 240 million years to orbit once around
the centre of the Milky Way galaxy. Write that number of years in standard form.
240 000 000
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CHAPTER 2
3 Comparing and Ordering Numbers
Goal Compare and order numbers to 1 000 000.
1. Compare each pair of numbers. At-Home Help
Use an inequality sign. When comparing and ordering
numbers to 1 000 000, compare the
a) 602 589 < 640 077
digits in this order:
• hundred thousands
b) 314 806 < 409 116 • ten thousands
• one thousands
c) 584 192 < 521 009 • hundreds
• tens
2. Order the numbers in Question 1 from least • ones
to greatest. You can also compare and order
314806, 409 116, 521 009, 584 192, 602 589, 640 077 numbers by their positions on a
number line.
Inequality signs  and  show
that one number is less than or
greater than another.
For example, 8  5 is read “eight is
greater than five.” 5  8 is read “five
3. List three numbers between 216 534 and 242 189. is less than eight.”
218 965, 234 567, 240 139
4. a) The number 5 8 206 is between 96 872 and 512 093. The two missing digits
are different. What might they be?
0 for 508 206
4 for 496 872
b) Order the numbers from part a) from least to greatest.
496 872, 508 206, 512 093
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CHAPTER 2
4 Renaming Numbers
Goal Rename numbers using place value concepts.
1. Complete each statement. At-Home Help
a) 4 625 239 is about 5 millions. When a number is used in a
measurement, the way the number
b) 276 081 is about 0.3 millions. is written depends on the size of
the unit.
c) 3 910 245 is about 4000 thousands.
For example, 233 848 bytes can be
2. Irene takes pictures with her digital camera. written as about 0.2 MB.
The file sizes of four of her pictures are: 1 MB is the same as 1 000 000 bytes.
3.2 MB 720 kB 21500 bytes 408350 bytes 1 kB is the same as 1000 bytes.
a) Write the first two file sizes as a number of bytes. 1.4 kB can be written as 1400 bytes.
3 200 000 bytes, 720 000 bytes 3.13 MB can be written as
3 130 000 bytes.
To compare measurements,
b) Estimate each file size, except for the first one, compare values using the same units.
as millions of bytes or megabytes.
For example, if you use bytes
0.7 MB, 0.02 MB, 0.4 MB
3 130 000 bytes  233 848 bytes
 1400 bytes
c) Which photo uses the most bytes?
3.2 MB
3. Write each number in another form.
a) 1.9 million  1 900 000 ones
b) 4.6 million  4600 thousands
c) 0.28 million  2800 hundreds
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CHAPTER 2
5 Communicate About Solving Problems
Goal Explain your thinking when solving a problem.
A city produced 183 million kilograms of landfill waste At-Home Help
in 2003. In 2004, a composting program reduced the When writing a solution to a
landfill waste to 45 million kilograms. About how much problem, first write a rough copy.
less waste was taken to the landfill each day in 2004? • If the problem does not ask for an
Explain how you solved the problem. exact answer, use estimation.
• You can use estimation or numbers
that are easier to work with.
reasonable.
I write both values on a place value chart. Then write a good copy explaining
Millions Thousands Ones
Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Remember to show all your work.
Communication Checklist
✓ Did you explain your thinking?
✓ Did you use correct math
language?
The difference is 138 million. ✓ Did you include enough detail?
There are 365 days in a year. I divide 138 million by 365 to
calculate how much less waste is taken to the landfill each day.
The problem asks “about how much” so my answer can be an estimate.
I use rounded numbers that are easy to calculate with.
160 000 000 ÷ 400 = 400 000
About 400 000 kg less waste was taken to the landfill each day in 2004.
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CHAPTER 2
6 Reading and Writing Decimal
Thousandths
Goal Read, write, and model decimals.
1. Write each fraction as a decimal. At-Home Help
29 503 790 Decimal numbers can be modelled
a)  b)  c) 
1000 1000 1000 using a grid.
0.029 0.503 0.790 For example, 203 thousandths can
be represented as
2. Colour a 1000ths grid to represent each fraction.
29 503
a)  b) 
1000 1000
203 thousandths is written as 0.203
in standard form.
3. Write a decimal for each number.
In expanded form, the decimal
a) fifty-two hundreths 0.52 number is 2 tenths  3 thousandths.
0.052 0.203 can also be written as
b) fifty-two thousandths
a fraction.
4. Write a decimal to fit each description. 203
0.203  
1000
a) one-tenth less than 6.302
6.202
b) one-thousandth greater than 6.302
6.303
c) one-hundredth greater than 6.302
6.312
5. Write each answer in Question 3 in expanded form.
a) 6 ones + 2 tenths + 2 thousandths c) 6 ones + 3 tenths +1 hundredth + 2 thousandths
b) 6 ones + 3 tenths + 3 thousandths
6. List two fractions that are equivalent to 0.400.
4 400
Suggested answer:  and 
10 1000
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CHAPTER 2
7 Rounding Decimals
Interpret rounded decimals and round decimals to the nearest
Goal tenth or hundredth.
1. Round each decimal to the nearest hundredth. At-Home Help
a) 0.526 c) 0.078 Decimal numbers can be rounded
to the nearest tenth or hundredth.
0.53 0.08
For example, 0.286 rounds up to
0.29 (decimal hundredth) and 0.3
b) 0.896 d) 3.006 (decimal tenth).
0.90 3.01
A number line helps with rounding.
0.286
2. Round each decimal to the nearest tenth.
a) 0.72 c) 2.462 0.280 0.290 0.300
0.7 2.5
b) 1.073 d) 0.98
1.1 1.0
3. Which numbers below round to the same hundredth?
2.417 2.423 2.024 2.400
Suggested answer: 2.417 and 2.423 both round to 2.42.
4. Name a decimal thousandth that could be rounded as described below.
a) up to 0.35 or down to 0.3
b) down to 2.12 or down to 2.1
5. Maya cut strips of fabric to make a quilted design. Each piece measured 0.365 m.
If she had measured to the nearest centimetre instead, what might the length of
fabric be?
0.37 m
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CHAPTER 2
8 Comparing and Ordering Decimals
Goal Compare and order decimals to thousandths.
1. Which decimal is greater? At-Home Help
a) 2.03 or 2.4 c) 0.526 or 1.004 To compare and order decimal
numbers to thousandths, compare
2.4 1.004
the digits in this order:
• ones
b) 5.7 or 3.99 d) 0.403 or 0.067
• tenths
5.7 0.403 • hundredths
• thousandths
2. Order these decimal numbers from least to greatest. You can also compare and order
decimals by their positions on a
a) 2.108 0.053 0.872 1.096 number line.
0.053, 0.872, 1.096, 2.108
b) 2.085 2.008 3.004 2.805
2.008, 2.085, 2.805, 3.004
3. Which measurement is greater?
a) 0.087 kg or 0.800 kg
0.800 kg
b) 4.312 km or 3567 m
4.312 km
c) 450 g or 1.088 kg
1.088 kg
4. List the numbers of the form . between 1.3 and 1.5 that are greater than
140 hundredths.
Suggested answer: 140 hundredths is the same as 1.4.
The numbers greater than 1.4 are 1.41, 1.42, 1.43, 1.44, 1.45, 1.46, 1.47, 1.48, and 1.49.
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CHAPTER 2
Test Yourself Page 1
Circle the correct answer.
1. Which statement is true?
A. 1 million  100 thousands C. 1 million  1000 ten thousands
B. 1 million  100 000 hundreds D. 1 million  10 hundred thousands
2. Which is the expanded form for 2 506 084?
A. 2 000 000  50 000  6000  80  4
B. 2 millions  5 hundred thousands  6 hundreds  84 ones
C. 2 000 000  500 000  6000  80  4
D. 2 millions  56 hundreds  84 ones
3. Which inequality is incorrect?
A. 206 354  216 089 C. 907 645  980 004
B. 706 821  799 035 D. 625 138  739 156
4. What is the correct order of the numbers below from least to greatest?
871 052, 86 304, 280 546, 901 034, 807 621
A. 86 304, 280 546, 871 052, 807 621, 901 034
B. 86 304, 280 546, 807 621, 871 052, 901 034
C. 86 304, 901 034, 871 052, 807 621, 280 546
D. 280 546, 871 052, 807 621, 901 034, 86 304
5. Which estimate is correct?
A. 1.7 MB is about 2 million bytes. C. 3 230 050 bytes is about 300 kB.
B. 0.4 kB is about 1 thousand bytes. D. 89 400 bytes is about 1 MB.
6. Which description fits for the number 87 640?
A. eighty-seven thousand sixty-four
B. eight hundred seven thousand sixty-four
C. eight hundred seven thousand six hundred forty
D. eighty-seven thousand six hundred forty
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CHAPTER 2
Test Yourself Page 2
7. Which math statement is incorrect?
52 79
A.   0.052 C.   0.790
1000 1000
206 358
B.   0.206 D.   0.358
1000 1000
28
8. Which decimal represents the fraction ?
1000
A. 0.028 B. 0.280 C. 0.208 D. 2.800
9. Which fraction represents the decimal 0.403?
43 43 403 430
A.  B.  C.  D. 
100 1000 1000 1000
10. Which numbers below round to the same hundredth?
4.806 3.987 4.813 4.811
A. 3.987, 4.806, 4.813 C. 4.806, 4.811, 4.813
B. 3.987, 4.811, 4.813 D. 3.987, 4.806, 4.811
11. Which number would be 2.065 rounded to the nearest tenth?
A. 2.0 C. 2.5
B. 2.1 D. 2.6
12. What is the order of the numbers below from least to greatest?
1.804, 2.053, 1.692, 0.982, 1.086
A. 0.982, 1.804, 1.086, 1.692, 2.053 C. 0.982, 1.086, 1.692, 1.804, 2.053
B. 0.982, 1.086, 1.804, 1.692, 2.053 D. 0.982, 1.692, 1.804, 1.086, 2.053