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6
Simplifying by Combining Like Terms
Consider this expression: 6n + n + 8 + 2n + 4
The numbers (8 and 4) and the variables (6n, n, and 2n) are the terms of the expression.
The terms are separated by plus or minus signs. This expression has five terms.
6n, n, and 2n are like terms because they each have the same variable n.
8 and 4 are like terms because they do not have variables.
Like terms can be combined to simplify expressions.
Combine the terms with variables first, then combine the terms without variables.
Combine terms
with variables.
n + 6n + 2n = 9n
n by itself means
1n
n + 6n + 8 + 2n + 4
9n + 12
Combine terms
without variables.
8 + 4 = 12
The plus or minus sign always goes with the number following it.
4 + 5b + 8 – 3 – 2 b
5b – 2b
4+8–3
Simplified expression: 3b + 9
Combine like terms to simplify the expressions.
1. a. 10a – a + 4 + 3a + 7
5a + a + 6 – 2
5a + a
6–2
Simplified expression: 6a + 4
b. 2 + 12b + 6 – 3 – 7b
c. 8a – 4a + 6 + 2a
23
Lesson 6
We R e m e m b e r
The information on the chart shows
how many books are in circulation
in several libraries. Organize the
information on the pictograph.
Draw one for every 2,000,000
books. Give the pictograph a title
and a legend.
Number of Books in Circulation
Chicago
7,000,000
Brooklyn
11,000,000
Fairfax County
12,000,000
New York
15,000,000
San Francisco
Detroit
Buffalo & Erie County
6,000,000
1,000,000
8,000,000
2.
Substitute 4 for n. Simplify the expressions.
3. a. 7 • n + 6 – n
b. 8n – 5
c. 6(n)
Finish showing the properties using the variables a, b, and c.
4. The commutative property of multiplication: a • b =
5. The associative property of addition: (a + b) + c =
24
You will
need to show
half symbols
for some of
these.
d. 8n
Lesson 6
M astery D rill
6. 1 millennium =
years
7. a. 1 decade =
b. 1 century =
years
8. An isosceles triangle has
9. A scalene triangle has
10. An equilateral triangle has
years
congruent sides.
congruent sides.
congruent sides.
Convert each fraction to a decimal to the nearest hundredth. Then write
the percent.
11. a.
3
13
≈
b.
=
7
9
=
≈
? . . . M ental M ath
12.
2
5 of 25
× 67
+ 30
+ 900
÷ 100
+ 14
=
Solve this logic problem.
13. If a space probe traveled 25,000 miles per day toward Mars, it would reach the planet in
about 4 years. How many miles is Mars from Earth?
25
Lesson 6
+ -x S k i l l B u i l d e r s
÷
14. a.
3
2 16
÷
1
38
Write the
remainder with R.
1 8 7,4 1 6
b. – 9 , 4 8 9
=
c. 6 9 8 ) 1 , 4 1 7
Rewrite each decimal as a fraction or mixed number with a denominator of 10, 100,
or 1,000. Reduce to simplest form.
15. a. 2.2 =
b. 0.004 =
=
Solve and check.
16. a. n + 3 = 8
b.
Use the formula to find the circumference. Use
40 ft
17.
26
=
c. 14 = n + 5
22
7
for π.
c. 6.45 =
d.
=
Euclid’s ancient
geometry
textbook had
thirteen sections
including plane
geometry, ratios,
and solid
geometry.
Complete the chart.
18.
63
19.
Lesson 6
Base
Exponent
1
4
Math Sentence
Product
Write an expression for each, using an exponent.
20. a. Base 2, exponent 10
b. Ten to the twentieth power
Convert these fractions and mixed numbers to decimals. Annex zeros to finish.
Round a to the nearest thousandth.
5
21. a. 2 13 =
Combine integers.
22. a. -3 + 0 =
b.
5
8
=
b. -9 + 8 =
c. 8 + (-20) =
Write the digits that hold each place.
23. a. tenths
24. a. hundredths
25. a. ten thousands
Simplify the expressions.
26. a. 2a + a + 13 – 3
246,801,357,936,482.369
b. tens
c. billions
b. thousands
c. hundreds
b. trillions
b. 12b + 3 + 4 – 2b – 6
c. ten trillions
c. 8 + 3 + 7b + 5 – 2b
27
7
Prime Factor Division
You have been finding the prime factors of a number using factor trees. Another method is
division by primes. Here are some rules for dividing by primes.
1. Divide using upside-down short division.
2. Divide by the smallest prime you can. First divide by 2 as many times as you can,
then by 3, then by 5, and then by 7. Use divisibility rules to help you.
3. If your answer is not prime, draw another upside down division box and divide again
by the smallest prime you can. Continue dividing like this until your answer is prime.
4. The prime factors of the number are the divisors on the left and the last prime factor
at the bottom. They should be in order from least to greatest.
Let’s try finding the prime factors of 72 this way.
72 is an even number, so we divide first by 2.
36 is not prime. We can divide again by 2 to get 18
and then once more by 2 to get 9.
Nine is not prime. It divides by 3. The quotient is 3,
and 3 is a prime number, so the division is finished.
72 = 2 • 2 • 2 • 3 • 3
Here are some more examples.
84 =
2•2•3•7
2 ) 8 14
2 ) 4 12
3 ) 2 11
7
28
98 =
2 ) 9 18
7 ) 4 19
7
2•7•7
2 ) 7 12
2 ) 3 16
2 ) 7 12
2 ) 3 16
2 ) 1 18
3 ) 1 19
2 ) 7 12
2 ) 3 16
2 ) 1 18
3 ) 1 19
3
125 =
5•5•5
5 ) 1 2 25
5 ) 1 12 15
5
Use prime factor division to find the prime factors.
1. a. 280 =
1
1
21
21
1
b. 294 =
1
1
1
21
1
1
Lesson 7
c. 165 =
1
11
1
We R e m e m b e r
Simplify the expressions.
2. a. 8 + 6b + 2 – 3b
b. 8b + 3 – 2 + 2b + 7
c. 8a – a + 16 + 4a – 11
Find the sales tax and total bill for each amount below.
3.
4.
purchase
$4.24
a. 5% sales tax $
b. total bill
$
purchase
$10.50
a. 6% sales tax $
b. total bill
$
Euclid lived and studied
geometry in the time
between the
Old Testament and the
New Testament.
5. Helen went with her mother to a fabric sale. On the sale
table, fabric was marked 25% off. The regular price of a
certain bolt of fabric was $7.79 a yard.
a. Round to the nearest dollar and estimate the cost
of 4 yards at regular price.
b. Change the percent into a fraction to estimate the
discount and the sale price of 4 yards.
Discount:
Sale Price:
6. Now find the exact sale price of 4 yards. Find the cost of
4 yards at regular price first, then figure the discount on the
total.
29
Lesson 7
M astery D rill
7. a. Deca means
8. a. The decimal for
9. a. The decimal for
1
2
3
4
is
.
b. Hecto means
b. The decimal for
.
is
c. Milli means
.
b. 1 pint =
.
10. The four angles of a quadrilateral measure a total of
11. An obtuse angle measures between
and
°.
1
4
°.
is
cups
Use your ruler to draw the following. Fill in the blanks
with chord, diameter, or radius.
D
13. Draw a line from E to A.
What did you draw?
mL
B
E
14. Draw a line from D to B.
What did you draw?
15. a. 0.25 L =
.
A
12. Draw a line from A to C.
What did you draw?
Convert the metric units.
.
C
b. 350 g =
kg
c. 0.42 m =
cm
Answer the questions about the graph.
16. How many tons of cargo did
the port of Portland handle?
17. How many more tons of cargo
were handled by Indiana Harbor
than by Oakland?
18. Approximately how many tons
of cargo did Jacksonville handle
in 2001?
30
Cargo Handled in U.S. Ports in 2001
Jacksonville, FL
Oakland, CA
Portland, ME
Chicago, IL
Indiana Harbor, IN
LEGEND:
= 10,000 tons
Lesson 7
+ -x S k i l l B u i l d e r s
÷
19. a.
2
3
1
3
1
b. 8 4 ÷ 2 2 =
× 24 =
Write the
remainder with R.
Write the repeating
decimal with a bar.
20. a. 4 3 6 ) 2 , 9 6 3
b. 1 2 ) 3 4
Find the sale price.
21.
975
c. × 6 3 8
regular price $12.00
sale 20% off
a. sale price $
regular price $8.99
sale 40% off
b. sale price $
Use the formula to find the area. Use 3.14 for π.
13 cm
22.
Do the short division upside down.
23. a. 5 ) 1 6 5
b. 3 ) 1 6 5
c. 3 ) 2 5 5
d. 5 ) 2 5 5
31
Lesson 7
Plot these points on the grid.
24. a. M (2, 1)
b. N (7, 4)
4
c. O (7, 1)
3
Connect the points in the order given, then
back to the first one. Fill in the blank.
25. I drew a
y
2
1
triangle.
0
0
1
2
3
4
5
6
7
8
Draw and label these angles. Circle the vertex on each angle.
26. a. ∠WXY, 65°
b. ∠GHI, 90°
27. a. ∠ABC, 180°
b. ∠RST, 165°
Draw an angle congruent to ∠RST.
28.
Go back to Numbers 26 and 27. Classify each angle as acute, obtuse, right, or straight.
29. a. ∠WXY
b. ∠GHI
30. a. ∠ABC
b. ∠RST
Use prime factor division to list the prime factors.
31. a. 48 =
1
1
1
11
32
b. 69 =
1
c. 255 =
1
11