Order of Operations with rational numbers. objective use the order of operation to simplify numerical expression containing rational numbers.

Order of Operations

with rational numbers

objective use the order of operation to simplify numerical expression containing rational numbers

Example 1. Simplify: (-0.4)3(-5/2)2

(-0.4)(-0.4)(-0.4)

Simplify each power before multiplying each factor.

Multiply like terms.

-0.064

(-5/2)(-.5/2)

(-0.4)3(-5/2)2

25/4

Multiply.

Multiply the factors.

Simplify.

-0.064

-1.6/4

-0.4

25/4

Example 2. Simplify: -6(2/3- 5/9) ÷ [(2.4 5)(-1)5]

To simplify expressions that contain more than one grouping symbol, begin computing with the innermost set

Begin computing within parentheses.

-6(2/3-5/9) ÷ [(2.4 5)(-1)5]

-6(1/9) ÷ [(12)(-1)5]

-6(1/9) ÷ [(12)(-1)]

Simplify the power.

Multiply.

Multiply.

-6/9 ÷ (-12)

Simplify.

Divide; multiply by the reciprocal.

-2/3

Simplify.2/36

= 1/18

-2/3 ÷ (-12)

-6(1/9) ÷ [(12)(-1)]

-1/12

52 – 7 2/10

The division bar is a grouping symbol. To work on an expression with a division bar, first simplify the numerator, then the denominator, and finally divide.

Subtract and rewrite the answer in simplest form.

Simplify the power.

+ 23(2 – 15)

52 – 7 2/10(2 – 15)

+ 23

25 – 7 2/10(2 – 15)

+ 23

Simplify:

Example 3

Subtract and rewrite the answer in simplest form.

Compute within parentheses.

25 – 7 2/10

(2 – 15)

+ 23

17 4/5

(2 – 15)

+ 23

17 4/5+ 23 -13

Simplify the power.

17 4/5

-13

+ 8Add.

17 4/5 -5

Divide.

17 4/5 ÷ (-5) Rewrite in horizontal form.

17 4/5

-5

Divide.

17 4/5 ÷ (-5) Rename as fractions.

89/5 ÷ -5/1 Multiply by the reciprocal to divide

89/5 x -1/5

-89/25Rename as a mixed number

-3 14/25

-12.5 + 0.5

The division bar is a grouping symbol. To work on an expression with a division bar, first simplify the numerator, then the denominator, and finally divide.

Rename 0.5 as 1/2.

Add

0.53/4

-12.5 + 0.53/4 0.5

-123/4 0.5

Simplify:

Example 4

Rename 0.5 as 1/2.-123/4 0.5 -123/4 1/2

Multiply.

-123/8

Simplify. Write in horizontal form.

-12 ÷ 3/8 Write as multiplication.

-12/1 x 8/3 Simplify.

-4/1 x 8/1 = -32/1 = -32

Homework

PB, p 147-148

x – 2 5/8 = 1 1/4

x – 2 5/8 = 1 1/4

Addition/Subtraction Equations With FractionsExample 3. Solve and check.

+ 2 5/8

+ 2 5/8

x = 1 2/8 + 2 5/8

x = 3 7/8

Substitute 3 7/8 for x to check.

x – 2 5/8 = 1 1/4

3 7/8 – 2 5/8 = 1 1/4

1 2/8 = 1 1/4

Simplify.

1 1/4 = 1 1/4; true 3 7/8 is a solution.

Homework

PB, p 149-150

-5/8 – 1/8 + n = 1

Addition/Subtraction Equations With FractionsExample 2. Solve and check.

Combine like terms.

-6/8 + n = 1

Simplify. Add 6/8 to both sides.+6/8 +6/8

n 1 6/8= Simplify the fraction

n = 1 3/4 Check the solution. Replace n with 1 3/4

-5/8 – 1/8 + 1 3/4 = 1 -6/8 + 1 3/4 = 1

Addition/Subtraction Equations With FractionsExample 2. Solve and check.

n = 1 3/4 Check the solution.

-5/8 – 1/8 + 1 3/4 = 1 -6/8 + 1 3/4 = 1

Combine

Simplify

-3/4 + 1 3/4 = 1

True, so 1 3/4 is a solution.

Replace n with 1 3/4. -5/8 – 1/8 + n = 1

Add.

1 = 1

Multiplication and division equations with

fractions

objective: apply the Multiplication Property of Equality

Text, pp 136-137

1/4 w + 2/4 w = 15

Example 1. Solve and check.

Combine like terms.

Multiply both sides by 4/3.

3/4 w = 15

4/3 4/3

w = 60/3

Divide.

w = 20

Check. Substitute 20 for w.

1/4 (20) + 2/4 (20) = 15

Simplify.

5 + 10 = 15 True. So 20 is a solution

Homework

PB, p 151-152

Two-Step equations with fractions

objective: apply the properties of equality to simplify two-step equations with fractions

Text, pp 138-139

1/2 p –16 1/2 = 15

Example 1. Solve and check.

Add 16 ½ to both sides.

Multiply both sides by 2/1.

Check.

+16 1/2 +16 1/21/2 p

=

31 1/2

2/1

2/1

p =

31 1/2

2/1

Rename 31 1/2 as improper fraction

p =

63/2

2/1

=

63/1

=

63

1/2 p –16 1/2 = 15

Substitute 63 for p.

1/2 63 –16 1/2 = 15

Example 1. Solve and check. 1/2 p –16 1/2 = 15

Substitute 63 for p.

1/2 63 –16 1/2 = 15

Multiply.

31 1/2 –16 1/2 = 15

Subtract.

15 = 15

True, so 63 is a true solution

59

Example 2. Solve and check.

Rename 2 1/4 as a fraction

42

d=

2 1/4

(-17)

Simplify the grouping symbols.

59

=

d 2 1/4

+17

Subtract 17 from both sides.-17

-17

=

d 2 1/4

42

d 9/4

=

Multiply both sides by 4/9

9/4

9/4

42

9/4

=

d Multiply

Example 2. Solve and check.

42

9/4

=

d Simplify.21

2

9/2

21

=

d Multiply.

189/2

=

d Rename as mixed number.

94 1/2

=

d Check.

Use 94 1/2 in place of d.

59d

2 1/4

=

(- 17)

Example 2. Solve and check.

Use 94 1/2 in place of d.

59 d 2 1/4

=

(- 17)

5994 1/2 2 1/4

(- 17) Simplify the parentheses.

59 =

94 1/2 2 1/4

+ 17 Write the division in horizontal form.

59=

94 1/2 ÷ 2 1/4

+ 17

59 =

189/2 ÷ 9/4

+ 17

Write the division in horizontal form.

Rename as fractions.

Write as multiplication.

Example 2. Solve and check.

59 =

189/2 4/9

+ 17 Simplify.21

1 59 =

21/2 4/1

+ 17 Simplify.1

2

59 =

21 2

+ 17 Multiply

59 =

42 + 17 Add.

59 =

59 True. So 94 1/2 is a solution.

HomeworkPB, p 153-154

Class workPB, p 153

Customary units of measure

objective: rename customary units measure to a larger or smaller units

Text, pp 138-139

Customary units of length

1 foot (ft) = 12 inches (in)

1 yard (yd) = 3 ft or 36 in

1 mile (mi) = 5280 ft or 1760 yd

Customary units of capacity

1 cup (c) = 8 fluid ounces (fl oz)

1 pint (pt) = 2 c

1 quart (qt) = 2 pt

1 gallon (gal) = 4 qt

Customary units of weight

1 pound (lb) = 16 ounces (oz)

1 ton (T) = 2000 lb

Customary units of Measure

To rename larger units as smaller units, multiply by the conversion unit

To rename smaller units as larger units, divide by the conversion unit

Example. How many yards are there in 2 ½ miles?

Think. 2 1/2 mi = _________ yd1 mi = 1760 yd mi larger than yardlarger to smaller, multiply

2 1/2 mi 1760 yd

Rename as fraction. 5/2 mi 1760 yd Simplify.

880 1 5 mi 880 yd Multiply.

4400 yd

2 1/2 mi.

HomeworkPB, p 155-156

Class workPB, p 155

Problem solving strategy:

objective: solve word problems using the strategy “Make A Drawing”

Text, pp 138-139

Make a drawing

Sample Problem 1. The clock tower in Liberty Square, known for its accuracy, chimes its bell every hour on the hour at equal intervals. If the clock strikes 6 chimes in 6 seconds, how long would it take for the clock to strike 12 chimes at 12 o’clock?

(To complete the problem, assume that the chime itself takes no time) Hint: 12 seconds is not the answer.

ReadRead to understand what is being asked. (List the facts and restate the question.)

Facts: Chime occurs in equal intervals. 6 chimes strike in 6 seconds at 6 o’clock. The answer is not twelve seconds.

Question: How long would it take for the clock to strike 12 chimes at 12 o’clock.

Plan Select a strategy.

Guess and test. Organize data. Find a pattern.

Problem-Solving Strategies

Make a drawing. Reason logically Work backward

Solve a simpler problem. Adopt a different point of view. Account for all possibilities. Consider extreme cases.

Using the strategy “Make a Drawing” will help you understand the situation.

Solve Apply the strategy.

First make a drawing that help you understand the situation. Use dots to show the chimes that occur at 6 o’clock.

654321

1 2 3 4 5

The 6 chimes occur in 6 seconds. There are 5 intervals in those 6 chimes, therefore each interval must be 6/5 seconds. Think: 6/5 5 = 6.

6 sec

Solve Apply the strategy.

Now make a drawing to show the situation at 12 o’clock. Use dots also to show the chimes.

98765

1 2 3 4 5

There are 11 intervals between the 12 chimes at 12 o’clock. If an interval is 6/5 of a second, then 6/5 11 will give us what it will take for the 12 chimes the clock will make at twelve.

1 2 3 4 10 11 12

6 7 8 9 10 11

Solve Apply the strategy.

There are 11 intervals between the 12 chimes at 12 o’clock. If an interval is 6/5 of a second, then 6/5 11 will give us the it will take for the 12 chimes the clock will make at twelve.

6/5 11 = 66/5 = 13 1/5 seconds

The clock takes13 1/5 seconds to strike 12 chimes

CheckCheck to make sure your answer makes sense

There are twice as many chimes, so it ought to take twice as long. It appears to be so.

Check Check to make sure your answer makes sense

There are twice as many chimes, so it ought to take twice as long. It appears to be so.

• The 6 chimes occur in 6 seconds.•The 12 chimes occur in 13 1/5 seconds.• There are 5 intervals between the 6 chimes.• There are 11 intervals between the 12 chimes.

There are more than twice as many intervals, so it ought to take more than twice as long. It appears to be so.

Sample Problem 2. There are 240 seven graders at Kingston Middle School. Of these students, 1/6 walk to school. Of those who do not walk, 3/4 take the bus to school. Of those who do not walk or take the bus half ride their bikes. How many seventh graders ride their bikes to school?

ReadRead to understand what is being asked. (List the facts and restate the question.)

Facts: There are 240 seventh graders in all 1/6 walk to school. 3/4 of those who do not walk take the bus 1/2 of those who do not walk or take the bus ride their bike.

Question: How many seventh graders ride their bike to school?.

Plan Select a strategy.

This problem has a lot of information. To make this information easier to understand, you can use the strategy “Make a Drawing”.

Solve Apply the strategy.

Draw a rectangle to represent the entire seventh grade. Divide the rectangle to show those who walk and those who do not.

240

40

1/6walks

do not walk

200

Think. 1/6 of 240 is 40.

Divide the section representing those who do not walk into fourths.

do not walk

20040

50

50

50

50

Divide the section representing those who do not walk into fourths.

walks

Think. 1/4 of 200 is 50

40

50

50

50

50

walks

Divide the remaining fourth into two.

25 25

do not walk or take the bus

So 25 students ride their bikes to school.

Think. 1/2 of 50 is 25.

CheckCheck to make sure your answer makes sense.

Look back at the final drawing. Make sure the numbers that represent each section satisfy the condition in the problem.

The total is 40 + 50 + 50 + 50 + 25 + 25 = 240. 40 students walk. This is 1/6 of 240 students. 150 students ride the bus. This is 3/4 of the 200 students who do not walk. 25 students ride their bikes. This is 1/2 of the 50 who do not walk or ride the bus.

Different Ways to find GCF

objective: use two other ways of finding the GCF of two numbers.

Text, pp 144

Method 1. division

Example. Find the GCF of 72 and 56.

72 and 56. Divide the higher number by the lower number.

72 ÷ 56 = 1r16 If the remainder is 0, the lower number is the GCF. If not divide the divisor by the remainder. Continue this process until the remainder is 0. The last divisor is the GCF.

56 ÷ 16 = 3r 8

16 ÷ 8 = 2r 0

Method 2. Subtraction

Example. Find the GCF of 72 and 56.

72 – 56 = 16. Subtract the lower number from the higher number.

56 – 16 = 40 Compare the three numbers.Subtract the lowest from the next lowest. Continue the process until the last two numbers in the sentence are the same. That number is the GCF.

40 – 16 = 24

24 – 16 = 8

16 – 8 = 8

HomeworkPB, p 159

Class workPB, p 144