Download Pre-Algebra

Rounding and Estimating
Lesson 3-1
Pre-Algebra
Objectives:
1. To round decimals
2. To estimate sums and differences
Rounding and Estimating
Lesson 3-1
Pre-Algebra
Tip: ≈ means approximately equal to
Rounding and Estimating
Lesson 3-1
Additional Examples
a. Round 8.7398 to the nearest tenth.
tenths place
8.7398
less than 5
Round down to 7.
8.7
Pre-Algebra
Rounding and Estimating
Lesson 3-1
Additional Examples
(continued)
b. Round 8.7398 to the nearest integer.
nearest integer is ones place
8.7398
5 or greater
Round up to 9.
9
Pre-Algebra
Rounding and Estimating
Lesson 3-1
Pre-Algebra
Additional Examples
Estimate to find whether each answer is reasonable.
a. Calculation
$115.67
$ 83.21
+$ 59.98
$258.86
Estimate
$120
$ 80
+$ 60
$260
The answer is close to the estimate. It is reasonable.
Rounding and Estimating
Lesson 3-1
Pre-Algebra
Additional Examples
(continued)
b. Calculation
$176.48
–$ 39.34
$137.14
Estimate
$180
–$ 40
$140
The answer is not close to the estimate. It is not reasonable.
Rounding and Estimating
Lesson 3-1
Pre-Algebra
Additional Examples
You are buying some fruit. The bananas cost $1.32,
the apples cost $2.19, and the avocados cost $1.63. Use
front-end estimation to estimate the total cost of the fruit.
Add the
front-end digits.
+
The total cost is about $5.10.
.30
Estimate by
rounding.
.20
.60
1.10 = 5.10
Rounding and Estimating
Lesson 3-1
Pre-Algebra
Additional Examples
Estimate the total electricity charge:
March: $81.75; April: $79.56; May: $80.89.
3 months
The values cluster around $80.
80 • 3 = 240
The total electricity charge is about $240.00.
Estimating Decimal Products and Quotients
Lesson 3-2
Pre-Algebra
Objectives:
1. To estimate products
2. To estimate quotients
Estimating Decimal Products and Quotients
Lesson 3-2
Pre-Algebra
Tips:
On multiple choice questions, sometimes you can eliminate answers by
estimating.
Estimating Decimal Products and Quotients
Lesson 3-2
Pre-Algebra
Additional Examples
Estimate 6.43 • 4.7.
6.43
6
4.7
6 • 5 = 30
6.43 • 4.7
30
5
Round to the nearest integer.
Multiply.
Estimating Decimal Products and Quotients
Lesson 3-2
Pre-Algebra
Additional Examples
Joshua bought 3 yd of fabric to make a flag. The fabric
cost $5.35/yd. The clerk said his total was $14.95 before tax.
Did the clerk make a mistake? Explain.
5.35
5
5 • 3 = 15
Round to the nearest dollar.
Multiply 5 times 3, the number of
yards of fabric.
The sales clerk made a mistake. Since 5.35 > 5, the actual cost
should be more than the estimate. The clerk should have charged
Joshua more than $15.00 before tax.
Estimating Decimal Products and Quotients
Lesson 3-2
Pre-Algebra
Additional Examples
The cost to ship one yearbook is $3.12. The total cost
for a shipment was $62.40. Estimate how many books were in
the shipment.
3.12
3
Round the divisor.
62.40
60
Round the dividend to a multiple of 3
that is close to 62.40.
60 ÷ 3 = 20
Divide.
The shipment is made up of about 20 books.
Estimating Decimal Products and Quotients
Lesson 3-2
Pre-Algebra
Additional Examples
Is 3.29 a reasonable quotient for 31.423 ÷ 5.94?
5.94
31.423
6
Round the divisor.
30
Round the dividend to a multiple of 6
that is close to 31.423.
30 ÷ 6 = 5
Divide.
Since 3.29 is not close to 5, it is not reasonable.
Mean, Median, and Mode
Lesson 3-3
Pre-Algebra
Objectives:
1.To find mean, median, mode, and range of a set of data.
2. To choose the best measure of central tendency.
Mean, Median, and Mode
Lesson 3-3
Pre-Algebra
New Terms:
1 Measures of Central Tendency – mean, median, mode of a collection of data.
2. Mean – is the sum of the data values divided by the number of data values,
average.
3. Median – is the middle number when data values are written in order and there
is an odd number of data values. For an even number of data values, the median
is the mean of the two middle numbers.
4. Mode – is the data item that occurs most often. There can be one mode, more
than one, or none.
5. Range – the difference between the greatest and least values in the data set.
6. Outlier – a data value that is much greater or less than the other data values.
Mean, Median, and Mode
Lesson 3-3
Additional Examples
Pre-Algebra
Six elementary students are participating in a one-week
Readathon to raise money for a good cause. Use the graph. Find
the (a) mean, (b) median, and (c) mode of the data if you leave
out Latana’s pages.
a. Mean:
sum of data values
number of data values
=
40 + 45 + 48 + 50 + 50
5
=
233
5
= 46.6
The mean is 46.6.
Mean, Median, and Mode
Lesson 3-3
Pre-Algebra
Additional Examples
(continued)
b. Median: 40 45 48 50 50
Write the data in order.
The median is the middle number, or 48.
c. Mode: Find the data value that
occurs most often.
The mode is 50.
Mean, Median, and Mode
Lesson 3-3
Additional Examples
Pre-Algebra
How many modes, if any, does each have? Name them.
a. $1.10 $1.25 $2.00 $2.10 $2.20 $3.50
No values are the same, so there is no mode.
b. 1 3 4 6 7 7 8 9 10 12 12 13
Both 7 and 12 appear more than the other data values.
Since they appear the same number of times, there are two modes.
c. tomato, tomato, grape, orange, cherry, cherry, melon, cherry, grape
Cherry appears most often.
There is one mode.
Mean, Median, and Mode
Lesson 3-3
Pre-Algebra
Additional Examples
Use the data: 7%, 4%, 10%, 33%, 11%, 12%.
a. Which data value is an outlier?
The data value 33% is an outlier. It is an outlier because it is 21%
away from the closest data value.
b. How does the outlier affect the mean?
77
6
12.8
Find the mean with the outlier.
44
5
8.8
Find the mean without the outlier.
12.8 – 8.8 = 4
The outlier raises the mean by about 4 points.
Mean, Median, and Mode
Lesson 3-3
Additional Examples
Pre-Algebra
Which measure of central tendency best describes
each situation? Explain.
a. the monthly amount of rain for a year
Mean;
since the average monthly amount of rain for a year is not likely to
have an outlier, mean is the appropriate measure.
When the data have no outliers, use the mean.
b. most popular color of shirt
Mode;
since the data are not numerical, the mode is the appropriate measure.
When determining the most frequently chosen item, or when the data
are not numerical, use the mode.
Mean, Median, and Mode
Lesson 3-3
Additional Examples
Pre-Algebra
(continued)
c. times school buses arrive at school
Median;
since one bus may have to travel much farther than other buses, the
median is the appropriate measure.
When an outlier may significantly influence the mean, use the median.
Using Formulas
Lesson 3-4
Pre-Algebra
Objectives:
1. To substitute into formulas
2. To use the formula for the perimeter of a
rectangle
Using Formulas
Lesson 3-4
Pre-Algebra
New Terms:
1. Formula – an equation that shows a relationship between quantities that are
represented by variables.
2. Perimeter – the distance around a figure.
Using Formulas
Lesson 3-4
Pre-Algebra
Additional Examples
Suppose you ride your bike 18 miles in 3 hours.
Use the formula d = r t to find your average speed.
d = rt
Write the formula.
18 = (r )(3)
Substitute 18 for d and 3 for t.
18
3r
=
3
3
Divide each side by 3.
6=r
Simplify.
Your average speed is 6 mi/h.
Using Formulas
Lesson 3-4
Pre-Algebra
Additional Examples
Use the formula F = n + 37, where n is the number
4
of chirps a cricket makes in one minute, and F is the
temperature in degrees Fahrenheit. Estimate the temperature
when a cricket chirps 76 times in a minute.
n
F = 4 + 37
76
Write the formula.
F = 4 + 37
Replace n with 76.
F = 19 + 37
Divide.
F = 56
Add.
The temperature is about 56°F.
Using Formulas
Lesson 3-4
Pre-Algebra
Additional Examples
Find the perimeter of a rectangular tabletop with a
length of 14.5 in. and width of 8.5 in. Use the formula for the
perimeter of a rectangle, P = 2 + 2w.
P = 2 + 2w
Write the formula.
P = 2(14.5) + 2(8.5)
Replace
P = 29 + 17
Multiply.
P = 46
Add.
with 14.5 and w with 8.5.
The perimeter of the tabletop is 46 in.
Solving Equations by Adding or Subtracting Decimals
Lesson 3-5
Pre-Algebra
Objectives:
1. To solve one-step decimal equations involving addition
2. To solve one-step decimal equations involving subtraction
Solving Equations by Adding or Subtracting Decimals
Lesson 3-5
Pre-Algebra
Additional Examples
Solve 6.8 + p = –9.7.
6.8 + p = –9.7
6.8 – 6.8 + p = –9.7 – 6.8
p = –16.5
Check:
Subtract 6.8 from each side.
Simplify.
6.8 + p = –9.7
6.8 + (–16.5)
–9.7
–9.7 = –9.7
Replace p with –16.5.
Solving Equations by Adding or Subtracting Decimals
Lesson 3-5
Pre-Algebra
Additional Examples
Ping has a board that is 14.5 ft long. She saws off a
piece that is 8.75 ft long. Use the diagram below to find the
length of the piece that is left.
x + 8.75 = 14.5
x + 8.75 – 8.75 = 14.5 – 8.75
x = 5.75
Subtract 8.75 from each side.
Simplify.
The length of the piece that is left is 5.75 ft.
Solving Equations by Adding or Subtracting Decimals
Lesson 3-5
Pre-Algebra
Additional Examples
Solve –23.34 = q – 16.99.
–23.34 = q – 16.99
–23.34 + 16.99 = q – 16.99 + 16.99
–6.35 = q
Add 16.99 to each side.
Simplify.
Solving Equations by Adding or Subtracting Decimals
Lesson 3-5
Pre-Algebra
Additional Examples
Alejandro wrote a check for $49.98. His new account
balance is $169.45. What was his previous balance?
Words previous balance minus check is new balance
Let p = previous balance.
Equation
p
–
49.98
=
169.45
p – 49.98 = 169.45
p – 49.98 + 49.98 = 169.45 + 49.98
p = 219.43
Add 49.98 to each side.
Simplify.
Alejandro had $219.43 in his account before he wrote the check.
Solving Equations by Multiplying or Dividing Decimals
Lesson 3-6
Pre-Algebra
Objectives:
1. To solve one-step decimal equations involving multiplication
2. To solve one-step decimal equations involving division
Solving Equations by Multiplying or Dividing Decimals
Lesson 3-6
Pre-Algebra
Additional Examples
Solve –6.4 = 0.8b.
–6.4 = 0.8b
–6.4
0.8b
=
0.8
0.8
–8 = b
Check:
Divide each side by 0.8.
Simplify.
–6.4 = 0.8b
–6.4
0.8(–8)
–6.4 = –6.4
Replace b with –8.
Solving Equations by Multiplying or Dividing Decimals
Lesson 3-6
Pre-Algebra
Additional Examples
Every day the school cafeteria uses about 85.8
gallons of milk. About how many days will it take for the
cafeteria to use the 250 gallons in the refrigerator?
Words
daily milk
consumption
times
number
of days
is
250 gallons
=
250
Let x = number of days.
Equation
85.8
•
x
Solving Equations by Multiplying or Dividing Decimals
Lesson 3-6
Pre-Algebra
Additional Examples
(continued)
85.8x = 250
85.8x
250
=
85.8
85.8
Divide each side by 85.8.
x = 2.914 . . . Simplify.
x
3
Round to the nearest whole number.
The school will take about 3 days to use 250 gallons of milk.
Solving Equations by Multiplying or Dividing Decimals
Lesson 3-6
Pre-Algebra
Additional Examples
Solve –37.5 =
–37.5 =
–37.5(–1.2) =
c
.
–1.2
c
–1.2
c
(–1.2)
–1.2
45 = c
Multiply each side by –1.2.
Simplify.
Solving Equations by Multiplying or Dividing Decimals
Lesson 3-6
Pre-Algebra
Additional Examples
A little league player was at bat 15 times and had
a batting average of 0.133 rounded to the nearest thousandth.
hits (h)
The batting average formula is batting average (a) =
.
times at bat (n)
Use the formula to find the number of hits made.
h
a= n
h
0.133 = 15
Replace a with 0.133 and n with 15.
Solving Equations by Multiplying or Dividing Decimals
Lesson 3-6
Pre-Algebra
Additional Examples
(continued)
h
0.133(15) = 15 (15)
1.995 = h
2
h
Multiply each side by 15.
Simplify.
Since h (hits) represents an integer,
round to the nearest integer.
The little league player made 2 hits.
Using the Metric System
Lesson 3-7
Pre-Algebra
Objectives:
1.To identify appropriate metric measures
2. To convert metric units
Using the Metric System
Lesson 3-7
Additional Examples
Pre-Algebra
Choose an appropriate metric unit. Explain your choice.
a. the width of this textbook
Centimeter; the width of a textbook is about two
hands, or ten thumb widths, wide.
b. the mass of a pair of glasses
Gram; glasses have about the same mass as
many paperclips, but less than this textbook.
c. the capacity of a thimble
Milliliter; a thimble will hold only a small amount of water.
Using the Metric System
Lesson 3-7
Additional Examples
Pre-Algebra
Choose a reasonable estimate. Explain your choice.
a. capacity of a drinking glass: 500 L or 500 mL
500 mL; a drinking glass holds less than a quart
of milk.
b. length of a hair clip: 5 m or 5 cm
5 cm; the length of a hair clip would be about 5
widths of a thumbnail.
c. mass of a pair of hiking boots: 1 kg or 1 g
1 kg; the mass is about one half the mass of your
math book.
Using the Metric System
Lesson 3-7
Pre-Algebra
Additional Examples
Complete each statement.
a. 7,603 mL =
L
7,603 ÷ 1,000 = 7.603
To convert from milliliters
to liters, divide by 1,000.
7,603 mL = 7.603 L
b. 4.57 m =
cm
4.57  100 = 457 cm
4.57 m = 457 cm
To convert meters to
centimeters, multiply by 100.
Using the Metric System
Lesson 3-7
Pre-Algebra
Additional Examples
A blue whale caught in 1931 was about 2,900 cm
long. What was its length in meters?
Words
length in
centimeters
÷
centimeters
per meter
=
length in
meters
Equation
2,900
÷
100
=
29
The whale was about 29 m long.
Problem Solving Strategy: Act It Out
Lesson 3-8
Pre-Algebra
Objectives:
1.To solve complex problems by first solving
simpler cases
Problem Solving Strategy: Act It Out
Lesson 3-8
Pre-Algebra
Additional Examples
Marta gives her sister one penny on the first day of October,
two pennies on the second day, and four pennies on the third day. She
continues to double the number of pennies each day. On what date will
Marta give her sister $10.24 in pennies?
Days after
the first
Number of
pennies
Amount
0
1
2
3
4
5
1
2
2•2= 4
4•2= 8
8 • 2 = 16
16 • 2 = 32
$0.01
$0.02
$0.04
$0.08
$0.16
$0.32
Problem Solving Strategy: Act It Out
Lesson 3-8
Additional Examples
(continued)
You can tell from the pattern in the chart that you just need to
count the number of 2’s multiplied until you reach 1,024, which
is $10.24 in pennies.
2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 1024
10 twos = 10 days after the first penny is given
Marta will give her sister $10.24 in pennies on October 11.
Pre-Algebra