Download Objective 1

Objective 1
The student will demonstrate an understanding of numbers, operations, and
quantitative reasoning.
For this objective you should be able to
●
use place value to read, write, compare, and order whole
numbers and decimals;
●
use fractions in problem-solving situations;
●
add, subtract, multiply, and divide to solve problems; and
●
estimate to determine reasonable results.
How Do You Read Large Numbers?
When you read a whole number, start with the digits on the left.
Use the commas to help you read the number.
A comma is used to
separate each group of
three digits. Look at the
number below:
Read the number 965,702,318,406. Look at the number in a
place-value chart.
Hundred Ten
Hundred
Ten
Hundred
Ten
Millions
Thousands Hundreds Tens Ones
Thousands Thousands
Billions Billions Billions Millions Millions
9
6
5
7
0
2
3
1
8
4
0
●
Read the three-digit number to the left of the first comma.
Then say what they represent.
nine hundred sixty-five billion
●
Read the three-digit number to the left of the second comma.
Then say what they represent.
seven hundred two million
●
Read the three-digit number to the left of the third comma.
Then say what they represent.
three hundred eighteen thousand
●
Read the last three-digit number.
four hundred six
6
Read the number 965,702,318,406 as nine hundred sixty-five billion,
seven hundred two million, three hundred eighteen thousand, four hundred
six.
11
2,167,986,450
Objective 1
Try It
Read the number 2,309,758,011.
Write the number in the place-value chart below.
Billions
Hundred
Millions
Ten
Millions
Hundred
Ten
Thousands Hundreds
Thousands Thousands
Millions
Tens
Ones
To read this number, say
_____________ billion, _____________________________________________________________________ million,
_________________________________________________________ thousand, _____________________________ .
Read the number 2,309,758,011 as two billion, three hundred nine million, seven hundred fifty-eight thousand,
eleven.
How Do You Compare and Order Whole Numbers?
Look at the place values of the digits to help you compare and order
whole numbers.
Look at these two numbers.
4,362,124,631
4,397,125,729
To determine which number is greater, look at them in a place-value
chart. Then compare the digits in each place value.
You can use these
symbols when you
compare numbers.
Symbol Meaning
5 is equal to
> is greater than
< is less than
Billions
Hundred
Millions
Ten
Millions
Millions
4
3
6
2
1
2
4
4
3
9
7
1
2
5
Hundred
Ten
Thousands Thousands
Thousands Hundreds
Tens
Ones
6
3
1
7
2
9
●
Look at the digits in the billions place first. Both numbers have
the digit 4 in the billions place.
●
Look at the digits in the hundred millions place next. Both
numbers have the digit 3 in the hundred millions place.
●
Look at the digits in the ten millions place next. Since 9 . 6,
the number with the digit 9 in the ten millions place is greater.
4,397,125,729 . 4,362,124,631
12
Objective 1
Try It
Use a place-value chart to order these numbers from greatest to least.
6,490,781,004
Hundred
Billions
Ten
Billions
63,758,902,449
Billions
Hundred
Millions
Ten
Millions
Millions
6
4
9
0
6,837,200,312
Hundred
Ten
Thousands Thousands
7
67,554,309,377
Thousands Hundreds
8
1
0
Tens
Ones
0
4
●
Write the numbers in the place-value chart. The first one has been done for you.
●
Look at the digits in the ten billions place.
The numbers ____________________________ and ____________________________
both have a _______ in the ten billions place. They are the two greatest numbers.
For these two numbers, compare the digits in the billions place.
Since _______ . _______ , the number ____________________________ . ____________________________ .
The greatest number is ____________________________ .
The next-greatest number is ____________________________ .
●
Look at the remaining two numbers. They both have a _______ in the billions place. So
compare the digits in the hundred millions place.
Since _______ . _______ , the number ____________________________ . ____________________________ .
The least number is ____________________________ .
The numbers in order from greatest to least are
_______________________ _______________________ _______________________ _______________________
The numbers 63,758,902,449 and 67,554,309,377 both have a 6 in the ten billions place. Since 7 . 3,
the number 67,554,309,377 . 63,758,902,449. The greatest number is 67,554,309,377. The next-greatest
number is 63,758,902,449. They both have a 6 in the billions place. Since 8 . 4, the number
6,837,200,312 . 6,490,781,004. The least number is 6,490,781,004. The numbers in order from
greatest to least are 67,554,309,377 63,758,902,449 6,837,200,312 6,490,781,004
13
Objective 1
What Are Decimals?
Decimals are another way to write fractions with denominators such
as 10, 100, and 1,000. Decimals and fractions both name part of a
whole. A decimal names part of a whole that has been divided into
10, 100, 1,000, or more parts.
3
10
The fraction }} is written as the decimal 0.3.
7
100
The fraction }} is written as the decimal 0.07.
9
1,000
The fraction }} is written as the decimal 0.009.
Look at the decimal below:
Decimal
point
When a number with a
decimal is written in
words, the -ths ending
tells you that those
digits belong on the
right side of the
decimal point.
1.47
The decimal point separates the whole part of the number from the
fractional part of the number. There is a 1 to the left of the decimal
point, so there is one whole. There is a 47 to the right of the decimal
point. This means 47 out of 100 parts. The decimal point means and.
The number 1.47 is read: one and forty-seven hundredths.
Looking at decimals in a place-value chart can help you read and
understand them.
Read the decimal 12.34 in the place-value chart.
Tens
Ones
1
2
.
.
Tenths
Hundredths
3
4
●
Read the number to the left of the decimal point, twelve.
●
Say the word and to represent the decimal point.
●
Read the number to the right of the decimal point, thirty-four.
●
Then say the place-value name of the last digit on the right,
hundredths.
Read the number 12.34 as twelve and thirty-four hundredths.
14
Objective 1
Read the number 5.826 in the place-value chart.
Ones
5
.
.
Tenths
Hundredths
Thousandths
8
2
6
●
Read the number to the left of the decimal point, five.
●
Say the word and to represent the decimal point.
●
Read the number to the right of the decimal point,
eight hundred twenty-six.
●
Then say the place-value name of the last digit on the right,
thousandths.
Read the number 5.826 as
five and eight hundred twenty-six thousandths.
Here are 10 blocks put together to model thousandths. Each block is
divided into 100 small squares. There are a total of 1,000 small
squares. The number 1,000 is the denominator of the fraction. The
numerator of the fraction is the number of small squares that are
shaded. The model shows 64 small squares shaded.
The model represents the fraction 64 . The fraction 64 is read as
1,000
1,000
sixty-four thousandths. The equivalent decimal should be written to
the thousandths place.
64
5 0.064
1,000
The decimal 0.064 is read as sixty-four thousandths. The decimal
0.064 represents the fraction 64 .
1,000
15
Objective 1
How Do You Compare and Order Decimals?
Look at the values of the digits to help you compare and order decimals.
Compare these decimals.
2.9
Placing zeros at the
end of a decimal
numeral does not
change its value.
To determine which decimal is greater, look at the numbers in a
place-value chart. You can write zeros at the end of 2.9 until it has
the same number of digits to the right of the decimal point as 2.925.
The number 2.9 5 2.900.
Ones
.
.
.
2
2
5.3 5 5.30
12.84 5 12.840
2.925
Tenths
Hundredths
Thousandths
9
9
0
2
0
5
●
Look at the ones place. Both numbers have a 2 in the ones
place.
●
Look at the tenths place. Both numbers have a 9 in the tenths
place.
●
Look at the hundredths place. Since 2 . 0, then 2.925 .
2.900.
576.03 5 576.030
Try It
Use the place-value chart to order these decimals from greatest
to least.
1.19
3.417
3.6
1.1
Ones
Should you place
zeros at the end
of any of these
numbers?
.
.
.
.
.
Tenths
Hundredths
Thousandths
Write the numbers in the place-value chart. First look at the digits
in the ones place.
The numbers __________ and __________ both have a 3 in
the ones place. They are the two greatest numbers.
For these two numbers, compare the digits in the tenths place.
Since ______ . ______ , the number __________ . __________ .
The greatest number is __________ ; it should be written first.
The next-greatest number is __________; it should be written next.
16
Objective 1
Look at the remaining two numbers. Compare the digits. Both
numbers have a _______ in the ones place and the tenths place.
Look at the hundredths place.
Since ______ . ______ , the number __________ . __________ .
The least number is ___________ ; it should be written last.
Now, list the numbers in order from greatest to least.
_______________ _______________ _______________ _______________
The numbers 3.417 and 3.600 both have a 3 in the ones place. Since 6 . 4,
the number 3.600 . 3.417. The greatest number is 3.6. The next-greatest
number is 3.417. Both numbers have a 1 in the ones place and the tenths
place. Since 9 . 0, the number 1.190 . 1.100. The least number is 1.1; it
should be written last. Listed from greatest to least, the numbers are:
3.6 3.417 1.19 1.1.
How Do You Add or Subtract Decimals?
To add or subtract decimals, vertically line up the decimal points of
the numbers. If necessary, put zeros after the last digits so that each
number has the same number of digits to the right of the decimal
point. This will not change the value of the number. Then add or
subtract as usual. Remember to regroup when necessary. Be sure to
include the decimal point in the answer.
Find the sum of 3.24, 6.7, and 0.29.
Write the numbers in a column. Line up the decimal points.
1 1
3.24
6.70
10.29
10.23
●
●
●
●
●
Put a zero at the end of 6.7 so that it
has the same number of digits.
Add the hundredths.
Add the tenths.
Add the ones.
Bring the decimal point straight down
in the answer.
The sum is 10.23.
17
Objective 1
In subtraction problems,
remember to check your
answer by adding. For
example,
1 1
2.85
+2.15
5.00
Hannah buys a magazine for $2.85. If she pays for it with a
five-dollar bill, how much change should she receive?
Use subtraction to find how much change Hannah should receive.
Write the numbers in a column and line up their decimal points.
A five-dollar bill can be written as $5.00.
9
4 10 10
●
$5.00
22.85
$2.15
●
●
●
●
●
●
Subtract the hundredths. You need to regroup, but
there are no tenths.
Regroup 5 ones as 4 ones and 10 tenths.
Regroup 10 tenths as 9 tenths and 10 hundredths.
Now subtract the hundredths.
Subtract the tenths.
Subtract the ones.
Be sure to place the decimal point in the answer.
Hannah should receive $2.15 in change.
Try It
Travis ran 1.5 kilometers on Monday, 2.25 kilometers on
Wednesday, and 0.9 kilometer on Friday. How many kilometers did
he run in all?
Use the operation of _____________________ to solve this problem.
Use the space below to find the answer.
Do you need to place
zeros at the end of the
decimal numbers? Will
you need to regroup?
_______ . _______ _______
_______ . _______ _______
1 _______ . _______ _______
_______ . _______ _______
The sum of the numbers is ____________ .
Travis ran a total of __________ kilometers.
Use the operation of addition to solve this problem.
1.50 1 2.25 1 0.90 5 4.65. The sum of the numbers is 4.65. Travis ran a
total of 4.65 kilometers.
18
Objective 1
When Do You Use Multiplication and Division to Solve Problems?
Use multiplication to combine groups of equal size. Use division to
separate a whole into groups of equal size.
A baker uses 325 pounds of flour each week to make pies and cakes.
How many pounds of flour does the baker use in 26 weeks?
Multiply to find the total number of pounds of flour.
13
●
Multiply the ones.
325
3 26
1950
●
Multiply the tens.
325
3 26
1950
6500
1
●
Finally, add the products.
325 3 26 5 8,450
The zero is a
place holder.
325
3 26
1950
16500
8450
The baker uses 8,450 pounds of flour.
The principal needs 200 tacos for lunch on field day. The tacos come
12 in a package. What is the least number of packages of tacos the
principal will need for the lunch? Divide to find how many packages
of tacos the principal will need.
1
12qw
2w
00
w
– 12
8
16
12qw
2w
00
w
– 12
80
– 72
8
●
●
●
●
●
●
●
Divide: 20 4 12
Multiply: 1 3 12
Subtract: 20 2 12
Bring down the zero
Divide: 80 4 12
Multiply: 6 3 12
Subtract: 80 2 72
200 4 12 5 16 R8
There is a remainder of 8. If the principal buys 16 packages of tacos,
she will not have enough. She will need 8 more tacos to have 200.
The principal will have to buy one more package to have enough
tacos. She will need to buy 17 packages.
19
Do you see
that . . .
Objective 1
Try It
There are 178 baseball players who want to play in the summer
league. There can be no more than 15 players on each team. What
is the least number of teams that can be formed for the summer
league?
Use the operation of _____________________ to find the number of
teams that can be formed.
_______ qw
1w
7w8
178 4 _______ = _______ with a remainder of _______. Since there
is a remainder of 13, _______ more team can be formed.
What does the
remainder tell you
to do?
_______ 1 1 = _______. The least number of teams that can be
formed for the summer league is _______.
Use the operation of division to find the number of teams that can be
formed.
11
15qw
17w8
w
– 15
28
– 15
13
178 ÷ 15 5 11 with a remainder of 13. Since there is a remainder of 13,
1 more team can be formed. 11 1 1 5 12. The least number of teams that
can be formed for the summer league is 12.
What Are Factors?
Numbers multiplied together are called factors. For example, 6 and 5
are factors of 30 because 6 3 5 5 30. Some other factors and their
products are shown below.
Factor 3 Factor 5 Product
20
3
3
7
5
21
8
3
6
5
48
9
3
7
5
63
5
3
12
5
60
Objective 1
To find all the factors of a number, write down all the factor pairs for
that number. For example, the factor pairs for 15 are 1 3 15 and
3 3 5. So the factors, listed in order, are 1, 3, 5, and 15.
What Are Common Factors?
Factors shared by two or more numbers are called common factors of
those numbers. For example, 5 is a common factor of 10 and
25 because 5 is a factor of 10 and also a factor of 25.
Find all the common factors of 30, 45, and 50.
●
List the factors of each number.
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 50: 1, 2, 5, 10, 25, 50
●
Circle the factors that the three numbers have in common.
The common factors of 30, 45, and 50 are 1 and 5.
Try It
Find all the common factors of 27 and 36.
The factors of 27 are: _______ , _______ , _______ , and _______ .
The factors of 36 are: _______ , _______ , _______ , _______ , _______ ,
_______ , _______ , _______ , and _______ .
The common factors of 27 and 36 are: _______ , _______ , and
_______ .
The factors of 27 are: 1, 3, 9, and 27. The factors of 36 are: 1, 2, 3, 4, 6, 9,
12, 18, and 36. The common factors of 27 and 36 are: 1, 3, and 9.
21
Objective 1
How Are Decimals Related to Fractions?
Decimals and fractions both name part of a whole or part of a group.
A decimal shows part of a whole that has been divided into 10, 100,
1,000, or more parts. A fraction shows part of a whole that has been
divided into any number of parts.
The denominator tells
the number of equal
parts into which the
whole has been divided.
The numerator tells how
many of the equal parts
have been selected.
Models can represent both fractions and
decimals. Look at this model. The model is
divided into 100 equal parts, and 39 of the
parts are shaded. You can represent the
shaded part of the model in two ways.
●
As a fraction: Use 100 as the
denominator. Use 39 as the numerator.
39
100
The fraction is }}.
●
As a decimal: Write the decimal to the hundredths place. Use
39 as the number of hundredths. The decimal is 0.39.
39
100
Both the fraction }} and the decimal 0.39 are read as
thirty-nine hundredths.
39
100
The fraction }} and the decimal 0.39 are equivalent.
22
Objective 1
Try It
Each large square in this model represents 1 whole.
What fraction does the model represent? What decimal does the
model represent?
The model shows _______ large squares completely shaded. In the
third large square, _______ out of _______ equal parts are shaded.
Write the number as a fraction: _____________
as a decimal: _____________
In words, you read this number as ________________________________
__________________________________________________________________ .
The model shows two large squares completely shaded. In the third large
square, 71 out of 100 equal parts are shaded. Written as a fraction, the
71
number is 2 }} , and as a decimal, it is 2.71. The number is read as
100
two and seventy-one hundredths.
23
Objective 1
How Do You Find Equivalent Fractions?
Equivalent fractions are fractions that have the same value but are
written differently. Equivalent fractions name the same part of a whole
in different ways.
Look at the circles below:
1
2
1 2
2 4
2
4
4
8
4
8
The fractions }}, }}, and }} all name the same part of a whole.
Any fraction with the
same number in both
its numerator and
its denominator is
equal to 1.
3
}} 5 1
3
2
}} 5 1
2
To find an equivalent fraction, multiply or divide the numerator and
the denominator by the same number. This is the same as multiplying
or dividing by a fraction that is equal to 1.
To find an equivalent fraction with a larger denominator, multiply by a
4
6
fraction equal to 1. Here are some fractions that are equivalent to }}.
Original
Fraction
Fraction
Equal to 1
Multiply
Equivalent
Fraction
4
6
2
2
4325 8
6 3 2 5 12
8
12
4
6
3
3
4 3 3 5 12
6 3 3 5 18
12
18
4
6
10
10
4 3 10 5 40
6 3 10 5 60
40
60
To find an equivalent fraction with a smaller denominator, divide by a
fraction equal to 1. Look at these examples.
Original
Fraction
Fraction
Equal to 1
Divide
Equivalent
Fraction
4
6
2
2
4425 2
6425 3
2
3
9
15
3
3
943 53
15 4 3 5 5
3
5
12
28
4
4
12 4 4 5 3
28 4 4 5 7
3
7
24
Objective 1
6
What are two equivalent fractions for 12 ?
●
To find an equivalent fraction with a denominator larger than
2 3 4
12, multiply by a fraction equal to 1. You could use 2 , 3 , 4 , or
any other fraction equal to 1.
6 3 2 5 12
12 3 2 5 24
6
12
is a fraction equivalent to }}.
24
12
●
To find an equivalent fraction with a denominator smaller than
3
12, divide by a fraction equal to 1. You could use 3 because 3
is a factor of both 6 and 12.
643 5 2
12 4 3 5 4
2
6
is a fraction equivalent to 12 .
4
6 12
2
The fractions 12 , 24 , and 4 are equivalent fractions.
Try It
5
Find two equivalent fractions for 15 , one with a larger denominator
and one with a smaller denominator.
To find an equivalent fraction with a larger denominator,
4
____________ by a fraction equal to _______ . For example, use 4 .
534 5
15 3 4 5
5
is a fraction equivalent to 15 .
To find an equivalent fraction with a smaller denominator,
5
____________ by a fraction equal to _______ . For example, use 5
because 5 is a factor of both 5 and 15.
545 5
15 4 5 5
25
Objective 1
5
is a fraction equivalent to 15 .
5
The fractions 15 ,
, and
are _________________ fractions.
To find an equivalent fraction with a larger denominator, multiply by a fraction
equal to 1.
5
15
3
3
4
4
5 20
5 60
20
5
is a fraction equivalent to
.
60
15
To find an equivalent fraction with a smaller denominator, divide by a fraction
equal to 1.
5
15
4
4
5
5
5 1
5 3
1
5
is a fraction equivalent to
.
3
15
5 20
1
The fractions
,
, and are equivalent fractions.
15 60
3
How Do You Compare Fractions?
To compare fractions that have the same denominator, compare their
numerators.
3
6
5
6
Compare the fractions }} and }}.
●
Both fractions have the denominator 6.
●
Compare the numerators to find how many parts are in each
fraction.
3
6
5
6
Since 3 , 5, then }} , }} .
Do you see
that . . .
5
6
3
6
It is also true that 5 . 3. So you can also say that }} . }}.
26
Objective 1
To compare fractions that have different denominators:
●
First find a common denominator for the two fractions. A
common denominator is one that is the same in two or more
fractions. To find a common denominator for two fractions, list
the multiples of each denominator, then find one multiple that is
in both lists.
●
Rewrite each fraction as an equivalent fraction with the common
denominator.
●
Compare the numerators of the two rewritten fractions.
3
4
5
6
Compare the fractions }} and }}. Which fraction is greater?
●
The denominators 4 and 6 are different. Find a common
denominator by listing the multiples of 4 and 6. A number
found in both lists of multiples is a common multiple. It is also
a common denominator for the two fractions.
Multiples of 4: 4, 8, 12, 16, . . .
Multiples of 6: 6, 12, 18, 24, . . .
A common multiple of 4 and 6 is 12. A common denominator
is 12.
●
●
3
4
5
6
Rewrite }} and }} to have 12 as their denominators.
3
?
}} 5 }}
4
12
3 3 22 5 ?
4 3 22 5 12
333 5 9
}
}
}}
4 3 3 5 12
3
9
}} 5 }}
4
12
5
?
}} 5 }}
6
12
5 3 22 5 ?
6 3 22 5 12
5 3 2 5 10
}
}
}}
6 3 2 5 12
5
10
}} 5 }}
6
12
Now both fractions have a denominator of 12. Compare the
10
9
numerators. Since 10 . 9, then 12 . 12 .
10
5
9
3
5
3
5
Since 12 5 6 and 12 5 4 , then 6 . 4 . The fraction 6 is greater
3
than the fraction 4 .
27
A multiple of a number
is the product of that
number and any other
factor.
Objective 1
Try It
5
6
Bart and Pablo had small pizzas for lunch. Bart ate }} of his pizza.
7
Sometimes it is necessary
to list more than the
first four multiples.
Pablo ate }} of his pizza. Who ate more of his pizza?
8
5
7
Find a common denominator for }} and }}.
6
8
List the first four multiples of the two denominators.
Multiples of 6: ________ , _______ , _______ , _______
Multiples of 8: ________ , _______ , _______ , _______
A common multiple of 6 and 8 is _______ .
A common denominator is _______ .
5
7
Find fractions equivalent to }} and }} with _______ as their common
6
8
denominator.
?
5
5
24
6
5 3
6 3
5
5
24
7
?
5
8
24
7 3
8 3
5
5
24
5
7
6
8
Since _______ . _______ , the fraction ______________
is greater.
(circle one)
______________ ate more of his pizza.
The first four multiples of 6 are: 6, 12 ,18, and 24. The first four multiples
of 8 are: 8, 16, 24, and 32. A common multiple of 6 and 8 is 24. A
5
7
common denominator is 24. Find fractions equivalent to }} and }} with
6
8
24 as their common denominator.
534
}}
634
733
}}
833
5
5
5
5
20
24
21
24
7
Since 21 . 20, the fraction }} is greater. Pablo ate more of his pizza.
8
28
Objective 1
What Are Mixed Numbers and Improper Fractions?
Mixed numbers and improper fractions are two ways to name fractions
greater than 1. A mixed number includes a whole number and a
1
fraction. For example, 4 3 is a mixed number. The whole number is 4,
1
and the fraction is 3 .
Numerator
1
Whole number
43
Denominator
An improper fraction has a numerator that is greater than or equal to
9
its denominator. The fraction 4 is an improper fraction because the
numerator 9 is greater than the denominator 4.
Numerator
9
4
Denominator
A mixed number and an improper fraction can both name the same
amount.
1
24 =
9
4
1
The mixed number 4 3 can be written as an improper fraction.
●
Multiply the denominator 3 by the whole number 4.
3 3 4 5 12
●
Add the numerator to this product.
12 1 1 5 13
●
13 becomes the numerator of the improper fraction. Keep the
original denominator of 3.
13
3
1
13
The mixed number 4 3 is equivalent to the improper fraction 3 .
29
Do you see
that . . .
Objective 1
22
The improper fraction 7 can be written as a mixed number.
● Divide the numerator 22 by the denominator 7.
3 R1
7qw
2w2w
– 21
1
22 4 7 5 3 with a remainder of 1
●
The 3 in the quotient becomes the whole number in the mixed
number.
●
The remainder can be written in fraction form using the
remainder as the numerator and keeping the original
denominator.
1
37
22
1
The improper fraction 7 is equivalent to the mixed number 3 7 .
Try It
14
Write the improper fraction 5 as a mixed number.
______________ the numerator _______ by the denominator _______.
qw
1w
6w
6w
The quotient _______ is a whole number in the mixed number.
The remainder 4 becomes the _________________ of the fraction.
The _________________ is 5.
14
The improper fraction 5 is equivalent to the mixed number _______.
Divide the numerator 14 by the denominator 5. The quotient 2 is the whole
number in the mixed number. The remainder 4 becomes the numerator of
14
the fraction. The denominator is 5. The improper fraction
is equivalent to
5
4
the mixed number 2 .
5
30
Objective 1
How Do You Add or Subtract Fractions with Common Denominators?
To add fractions with common denominators, add the numerators.
Then write the sum over the common denominator.
1
2
3
112
}} 1 }} 5 }} 5 }}
6
6
6
6
1
6
2
6
+
3
6
=
To subtract fractions with common denominators, subtract the
numerators. Then write the difference over the common denominator.
7
4
3
724
}} 2 }} 5 }} 5 }}
12
12
12
12
7
12
–
4
12
=
1
8
3
12
5
8
Rasheed bought }} pound of green onions and }} pound of carrots.
1
lb
8
5
lb
8
How many pounds of vegetables did Rasheed buy altogether?
Find the sum of the two fractions. Since the denominators are the
same, add the numerators.
1
5
6
1 8 5 8
8
6
Rasheed bought 8 pound of vegetables.
31
Objective 1
Try It
A cook measures lemon juice and orange juice in two measuring
cups. How much more lemon juice than orange juice does she have?
1 cup
3
4
1 cup
3
4
cup
1
2
1
4
cup
1
2
cup
1
4
cup
cup
cup
Orange juice
Lemon juice
The drawing shows
cup of lemon juice.
The drawing shows
cup of orange juice.
The number sentence
2
5
shows how much
more lemon juice than orange juice the cook has.
3
1
The drawing shows }} cup of lemon juice and }} cup of orange juice. The
4
4
3
1
2
number sentence }} 2 }} 5 }} shows how much more lemon juice than orange
4
4
4
juice the cook has.
When Should You Estimate an Answer?
When you do not need an exact answer to a problem, you can estimate
to find an answer that is close to the exact answer. For example, some
problems ask about how many or approximately how much. Use
estimation when solving such problems.
One way to estimate is to round numbers in a problem before working
it out. You can round decimals in the same way you round whole
numbers. A number line or a set of rounding rules can help you. For
example, to round a decimal to the nearest tenth, look at the digit in
the hundredths place.
●
If the digit in the hundredths place is 0, 1, 2, 3, or 4, leave the
digit in the tenths place the same.
●
If the digit in the hundredths place is 5, 6, 7, 8, or 9, round the
digit in the tenths place to the next-higher value.
32
Objective 1
Jill and Simon are walking along a trail that is shaped like a triangle.
The first side of the trail is 2.76 kilometers long. The second side
of the trail is 2.39 kilometers long. The third side of the trail is
0.63 kilometer long. About how long is this trail?
Jill and Simon use different methods to round the numbers.
Jill’s method:
●
Jill rounds the length of each side of the trail to the nearest
tenth of a kilometer.
2.76 rounded to the nearest tenth is 2.8.
2.39 rounded to the nearest tenth is 2.4.
0.63 rounded to the nearest tenth is 0.6.
●
Jill adds the rounded numbers to find the approximate length
of the trail.
2.8
2.4
1 0.6
5.8
●
Jill says the trail is about 5.8 kilometers long.
Simon’s method:
●
●
Simon rounds the length of each side of the trail to the nearest
kilometer. This is the same as rounding to the nearest whole
number or ones place.
2.76 rounded to the nearest whole number is 3.
2.39 rounded to the nearest whole number is 2.
0.63 rounded to the nearest whole number is 1.
Simon adds the rounded numbers to find the approximate
length of the trail.
3121156
●
Simon says the trail is about 6 kilometers long.
Both methods show correct rounding. Even though Jill and Simon
rounded the numbers to different places, both estimates are
reasonable.
33
Do you see
that . . .
Objective 1
Another way to estimate is by using compatible numbers. Compatible
numbers are numbers that are easy to add, subtract, multiply, or divide.
Using compatible numbers makes the computation easier.
Walter is estimating the total number of marbles that he and five of
his friends have. The number of marbles each person has is as
follows:
125
73
145
90
120
50
To estimate the sum, you can use compatible numbers. For this
example, group together numbers that equal about 200.
● 125 1 73 is close to 200.
●
50 1 145 is close to 200.
●
90 1 120 is close to 200.
●
200 1 200 1 200 5 600.
Walter estimates that he and his friends have about 600 marbles
altogether.
When using compatible numbers to estimate a product or quotient,
change the numbers to other numbers that form a basic fact. This can
help you solve the problem in your head.
In the fall, 86 students played soccer. If there were 9 teams, about
how many players were on each team?
To find about how many, estimate the answer. Use compatible
numbers, or numbers that are easy to divide, to make the problem
easier. Use basic facts to help you.
●
Find a number close to 86 that you can divide by 9 in your
head.
86 is close to 90, and 90 divides easily by 9
●
Divide to find about how many players were on each team.
90 ÷ 9 = 10
There were about 10 players on each team.
34
Objective 1
Try It
Suzanne borrowed $358 from her mother to buy a computer. In
order to pay back the entire $358, Suzanne gave her mother the
same amount of money each month for 1 year. About how much
money did Suzanne give her mother each month?
Use ____________ to find about how much money Suzanne will give
her mother each month.
There are ______ months in a year.
Find a number close to 358 that divides easily by 12. Use the basic
fact 36 ÷ 12 = 3 to help.
$358 is close to __________.
360 4 12 5 ______
Suzanne gave her mother about ________ each month.
Use division to find about how much money Suzanne will give her mother
each month. There are 12 months in a year. $358 is close to $360.
360 4 12 5 30. Suzanne gave her mother about $30 each month.
Now practice what you’ve learned.
35