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Transcript
Math Fundamentals for
Statistics I (Math 52)
Homework Unit 4:
Multiplication
By Scott Fallstrom and Brent Pickett
“The ‘How’ and ‘Whys’ Guys”
This work is licensed under a Creative Commons AttributionNonCommercial-ShareAlike 4.0 International License
3rd Edition (Summer 2016)
Math 52 – Homework Unit 4 – Page 1
Table of Contents
This will show you where the homework problems for a particular section start.
4.1: Multiplication of Whole Numbers ......................................................................................................2
4.2: Multiplication of Integers ....................................................................................................................6
4.3: Rewriting Multiplication (Factoring).................................................................................................9
4.4: Using FLOF to Simplify Fractions ...................................................................................................11
4.5: Multiplication of Fractions ...............................................................................................................12
4.6: Multiplication of Decimals ................................................................................................................13
4.7: Number Sense and Multiplication ....................................................................................................14
4.8: Repeated Multiplication (Exponents) ..............................................................................................16
4.9: Multiplication with variables ............................................................................................................17
4.10: Multiplication on a Number Line ...................................................................................................19
4.11: 2-Dimensional Geometry and Multiplication ................................................................................20
4.12: Multiplication Applications ............................................................................................................23
4.13: Multiplication Summary ..................................................................................................................25
4.14: Multiplication Wrap-Up (Practice) .................................................................................................26
4.1: Multiplication of Whole Numbers
Vocabulary and symbols – write out what the following mean:





Factors
Product
Field
Commutative Property of Multiplication
Associative Property of Multiplication




Identity Property of Multiplication
Multiplicative Identity Element
Additive Identity Element
Distributive Property of Multiplication
over Addition
Concept questions:
1. Using the definition of multiplication, what is another way to write 6 × 3 using addition?
2. Using the definition of multiplication, what is 3 × 0 using addition?
3. Using the definition of multiplication, what is 0 × 3?
4. What property allows us to change the order of the factors in a multiplication problem?
5. What property allows us to change the grouping of the factors in a multiplication problem?
6. What can we multiply any number by and get back the same number?
7. Is there another way to write 2(5 + 3)? What is it?
Math 52 – Homework Unit 4 – Page 2
Exercises:
8. Use the property listed to rewrite 2(3 + 4):
a. Commutative property of multiplication
b. Commutative property of addition
c. Distributive property of multiplication over addition
9. Use the property listed to rewrite 2 × (3 × 4):
a. Associative property of multiplication
b. Commutative property of multiplication
c. Commutative property of multiplication (different than previous)
d. Can we rewrite this expression using the distributive property of multiplication over addition?
Show how or explain why not.
10. Rewrite without any parenthesis:
a. 2x  5
e. 65x  7m
b. 3x  6
f.
c. 32 x  5
g. 220  3
d. 37 y  3
h. 4300  2
42 x  3m  y  8
11. What are the two ways to find the value of 25  3 ? Show both and see if they produce same values.
Which one did you like best?
12. Are there two ways to express 6x  5 without parenthesis or only one? Explain why this problem is
not the same as the previous.
13. Find the nth term in the arithmetic sequence and simplify the result: a n  a  n  1d .
a. Arithmetic sequence with a = 7 and d = 4.
b. Arithmetic sequence with a = 6 and d = 20.
c. Arithmetic sequence with a = 11 and d = 9.
d. Arithmetic sequence with a = 53 and d = 2.
e. Arithmetic sequence with a = 21 and d = 11.
14. Multiply using place value numbers:
a. 10010
e. 27  1,000
b. 1010
f. 158100
c. 1,000  100
g. 5510,000
d. 1,000 1,000
h. 10013,000
Math 52 – Homework Unit 4 – Page 3
15. Multiply using place value numbers and the associative property:
a. 70 × 3
f. 130 × 2,000
b. 90 × 200
g. 40 × 80
c. 80 × 300
h. 400 × 800
d. 80 × 900
i. 4,000 × 8,000
e. 500 × 3,000
j. 40,000 × 80,000
16. Show how to multiply these using the distributive property and find the product quickly.
a. 5 × 13
d. 27 × 4
g. 6 × 41
b. 25 × 6
e. 91 × 7
h. 43 × 11
c. 18 × 3
f. 34 × 5
i. 61 × 30
17. Practice multiplying without using a calculator. Try using different algorithms to see which is more
efficient for you.
a. 28 × 13
e. 78 × 89
b. 48 × 23
f. 103 × 24
c. 57 × 29
g. 111 × 25
d. 31 × 49
h. 281 × 47
18. For these problems, you only need to re-write as one multiplication (instead of two or more).
Do not use your calculator! When finished, find the product. Part (a) is done for you.
a. 91 × 7 + 91 × 3 = 91(7 + 3) = 91 × 10 = 910
f. 14 × 68 + 14 × 2
b. 31 × 9 + 31 × 11
g. 93 × 574 + 7 × 574
c. 42 × 16 + 42 × 4
h. 983 × 574 + 17 × 574
d. 47 × 23 + 13 × 23
i. 16 × 53 + 16 × 9 + 14 × 62
e. 22 × 11 + 22 × 39
j. 23 × 18 + 23 × 13 + 27 × 31
19. For these problems, re-write as one multiplication … but then find the product. Do not use your
calculator!
a. 91 × 7 + 91 × 3
f. 14 × 68 + 14 × 2
b. 31 × 9 + 31 × 11
g. 93 × 574 + 7 × 574
c. 42 × 16 + 42 × 4
h. 983 × 574 + 17 × 574
d. 47 × 23 + 13 × 23
i. 16 × 53 + 16 × 9 + 14 × 62
e. 22 × 11 + 22 × 39
j. 23 × 18 + 23 × 13 + 27 × 31
Math 52 – Homework Unit 4 – Page 4
20. Practice estimating the products without using a calculator unless you see the .
a. Estimate the value of 31 × 49 by rounding to the tens and then multiplying the rounded values.
Then use your calculator to compare your estimate to the exact value. Is your estimate higher or
lower than the actual?
b. Estimate the value of 37 × 42 by rounding to the tens and then multiplying the rounded values.
Then use your calculator to compare your estimate to the exact value. Is your estimate higher or
lower than the actual?
c. Estimate the value of 427 × 593 by rounding to the hundreds and then multiplying the rounded
values. Then use your calculator to compare your estimate to the exact value. Is your estimate
higher or lower than the actual?
d. Estimate the value of 422 × 561 by rounding to the hundreds and then multiplying the rounded
values. Then use your calculator to compare your estimate to the exact value. Is your estimate
higher or lower than the actual?
e. Estimate the value of 478 × 583 by rounding to the hundreds and then multiplying the rounded
values. Then use your calculator to compare your estimate to the exact value. Is your estimate
higher or lower than the actual?
f. Estimate the value of 428 × 513 by rounding to the hundreds and then multiplying the rounded
values. Then use your calculator to compare your estimate to the exact value. Is your estimate
higher or lower than the actual?
g. Create a rule: if we round both factors up, then the estimate of the product will be (Always,
Sometimes, Never) larger than the actual product. Explain your rule.
h. Create a rule: if we round both factors down, then the estimate of the product will be (Always,
Sometimes, Never) larger than the actual product. Explain your rule.
21. Practice estimating the products without using a calculator unless you are told to use it.
a. Estimate the value of 75 × 38 by rounding to the tens and then multiplying the rounded values.
Then use your calculator to compare your estimate to the exact value. Is your estimate higher or
lower than the actual?
b. What did you round 75 to in the previous problem? What other value is equally correct when
rounding (but not conventional)?
c. Estimate the value of 75 × 38 by rounding to the tens and then multiplying the rounded values.
This time round 75 differently than you did before.
d. Compare your estimates from both rounded values. Which one is closer?
22. Practice estimating the products without using a calculator unless you are told to use it.
a. Estimate the value of 85 × 32 by rounding to the tens and then multiplying the rounded values.
Then use your calculator to compare your estimate to the exact value. Is your estimate higher or
lower than the actual?
b. What did you round 85 to in the previous problem? What other value is equally correct when
rounding (but not conventional)?
c. Estimate the value of 85 × 32 by rounding to the tens and then multiplying the rounded values.
This time round 85 differently than you did before.
d. Compare your estimates from both rounded values. Which one is closer?
Math 52 – Homework Unit 4 – Page 5
23. Practice estimating the products without using a calculator unless you are told to use it.
a. Estimate the value of 94 × 38 by rounding to the tens and then multiplying the rounded values.
Then use your calculator to compare your estimate to the exact value. Is your estimate higher or
lower than the actual?
b. Create a rule: if we round one factor up and another down, then the estimate of the product will be
(Always, Sometimes, Never) larger than the actual product. Explain your rule.
24. Multiply these using the distributive property.
a. 408 × 11
e. 598 × 9
i. 7 × 297
b. 902 × 5
f. 301 × 12
j. 25 × 398
c. 899 × 5
g. 3 × 912
k. 21 × 299
d. 799 × 8
h. 5 × 297
l. 43 × 199
25. What property is used here.
a.
 447   7   440
d. 5  7  11  5  7  11
b.
 447   7  7   7 44
e. 5  7  11  7  11  5
c. 53x  4  15x  20
f.
5  7  1  5  7
Wrap-up and look back:
26. When estimating 31 × 84, Larry saw that 30 × 80 = 2,400. Is his estimate higher than the actual, or
lower than the actual? Explain why.
27. When estimating 31 × 87, Larry saw that 30 × 90 = 2,700. Is his estimate higher than the actual, or
lower than the actual? Can you tell quickly, or do you need to calculate the actual value? Explain.
28. When estimating 29 × 87, Larry saw that 30 × 90 = 2,700. Is his estimate higher than the actual, or
lower than the actual? Explain why.
29. How many 0’s will be in this product: 100,000 × 10,000,000? Explain why.
30. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.2: Multiplication of Integers
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. Are the products 4   7 and  4  7 the same or opposites? Why?
Math 52 – Homework Unit 4 – Page 6
2. Are the products 4   7 and  4 7 the same or opposites? Why?
3. If two factors have the same sign, will the product be positive or negative? Why?
4. If two factors have different signs, will the product be positive or negative? Why?
Exercises:
5. Use what you learned in this section to “prove” that  8 3 = 24. When you prove this, be sure to
put in the explanation not just the equations!
6. Use what you learned in this section to “prove” that  5 7 = 35. When you prove this, be sure to
put in the explanation not just the equations!
7. If I begin with  440  0 and rewrite it as  447   7  0 , what property was used?
8. If I rewrite  447   7  0 as  447   44 7  0 , what property was used?
9. If I rewrite  447   7  0 as  44 7  7  0 , what property was used?
10. If I rewrite  447   7  0 as 7   7 44  0 , what property was used?
11. Multiply the following without a calculator: be sure that your product has the appropriate “sign”:
a.
b.
c.
 87
d.
 913
 8 11
e.
f.
5 9
87
13 17
12. When dealing with integers, the sign of the result depends on the sign of the starting numbers. The
result could be always positive (P), always negative (N), or sometimes positive and sometimes
negative (S). Label each of the following expressions as P, S, or N. If the answer is P or N, explain
why. But if the answer is S, give one example that shows the product could be positive and one
example that shows the product could be negative. Note: some questions review previous topics as
well as multiplication.
Expression
Sign (circle one)
a.
neg – neg
P
S
N
b.
pos + neg
P
S
N
c.
neg × neg
P
S
N
d.
pos × pos
P
S
N
e.
neg – pos
P
S
N
Examples or Explanation
Math 52 – Homework Unit 4 – Page 7
f.
pos – neg
P
S
N
g.
neg + neg
P
S
N
h.
pos × neg
P
S
N
i.
neg × pos
P
S
N
j.
pos – pos
P
S
N
13. Write the following values without parenthesis.
a.  x  8
e.   13x  23 y   9
b.
 1x  11
f.
 173  5x  9 y 
c.
 1 3x  5
g.
 1 55  12x  93 y 
d.  14 x  57 y   1
h.  5x  2 y  6m  3  15 p 
14. Find the nth term in the arithmetic sequence and simplify the result: a n  a  n  1d .
a. Arithmetic sequence with a = 7 and d = – 4.
b. Arithmetic sequence with a = 6 and d = – 20.
c. Arithmetic sequence with a = 11 and d = – 9.
d. Arithmetic sequence with a = 53 and d = – 2.
e. Arithmetic sequence with a = 21 and d = – 11.
Wrap-up and look back:
15. Which of the operations seem the easiest to understand and explain (addition, subtraction, or
multiplication)? Why do you think it is easier?
16. When you see the value (– x), is it positive, negative, zero… or can you not tell? Explain.
17. Martin is confused with integers. One of his previous teachers said “two negatives make a positive,”
but when he performed  8   11 and wrote 19, he was marked wrong. Explain the confusion, why
he was marked wrong, and what the previous teacher should have said.
18. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
Math 52 – Homework Unit 4 – Page 8
4.3: Rewriting Multiplication (Factoring)
Vocabulary and symbols – write out what the following mean:





Factoring
Prime
Composite
Unique
Fundamental Theorem of Arithmetic
Concept questions:
1. Is there only one way to factor a number like 20? Explain.
2. Is there only one way to prime factor a number like 20? Explain.
3. Is the number 1 prime or composite (or something else)? Explain.
4. Is the number 23 prime or composite (or something else)? Explain.
Exercises:
5. Describe the following numbers as prime (P), composite (C), or neither (N) and when possible, give
one factor other than 1:
Number
Describe
A factor is…
Number
Describe
a.
40
P
C
N
b.
61
P
C
N
c.
29
P
C
N
d.
14,502
P
C
N
e.
15
P
C
N
f.
31
P
C
N
g.
1
P
C
N
h.
39
P
C
N
i.
19
P
C
N
j.
69,363
P
C
N
k.
85
P
C
N
l.
0
P
C
N
A factor is…
Math 52 – Homework Unit 4 – Page 9
6.
Determine if these are prime – use your calculator for the square root values. Also, find the primes
we would need to check to determine if the number is prime.
a. 71
b. 87
c. 91
d. 57
e. 183
f. 73
7. Determine if 2 is a factor of the following numbers – if it is, rewrite using 2 as a factor:
a. 76
c. 800
e. 764
b. 65
d. 89
f. 4,802
8. Determine if 5 is a factor of the following numbers – if it is, rewrite using 5 as a factor:
a. 37
c. 810
e. 525
b. 60
d. 75
f. 327
9. Determine if 3 is a factor of the following numbers – if it is, rewrite using 3 as a factor:
a. 48
c. 78
e. 318
b. 97
d. 326
f. 28
10. Let’s practice some factoring. For each number below, factor into prime numbers or state that it is
prime. Remember our tools: (1) short-cut tips for small primes, (2) square root to know how high to
check, and (3) calculator to check.
a. 36
g. 157
b. 50
h. 125
c. 70
i. 27
d. 100
j. 63
e. 53
k. 91
f. 84
l. 333
Wrap-up and look back:
11. Mary said 57 is prime because it is not divisible by 2 or 5. Is 57 prime? Is her reasoning correct?
12. Peter claims that 7 is prime because the only factors are 1 and itself. He says that 1 must be prime
because the only factors are 1 and itself. Is he correct?
13. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
Math 52 – Homework Unit 4 – Page 10
4.4: Using FLOF to Simplify Fractions
Vocabulary and symbols – write out what the following mean:

FLOF
Concept questions:
1. Hillary claims that
16 1
 because she cancelled out the 6’s. Is her answer correct? Is her reasoning
64 4
correct?
2. Bill looks at Hillary’s work and concludes that
15 1
 because he cancelled out the 5’s. Is he correct?
35 3
Is his reasoning correct?
1
28
2  8 2 1 3
3. If Raul wrote


 , is he correct? Where is his mistake? What should the result
5  16 5  16 5  2 7
2
have been?
Exercises:
4. Simplify the following fractions. Only use your calculator when the symbol appears. Some of these
may already be simplified if they share no common factors.
a.
10
15
d.
275
350
g.
37
40
b.
36
60
e.
16
24
h.
318
371
c.
16
21
f.
132
55
i.
12,341
19,393
Wrap-up and look back:
5. Willie wrote
34
3
as . What is one big indicator that he made a mistake?
40
0
58 5
 … is this correct?
38 3
7. When we simplify fractions, can we stop when one of the numbers (numerator or denominator) is
6. What operation is needed to remove common factors? Think of
prime? Explain and give some examples.
Math 52 – Homework Unit 4 – Page 11
8. When we simplify fractions, can we stop when both of the numbers (numerator and denominator) are
prime? Explain and give some examples.
9. When will the calculator stop recording correct fraction representations (how big is the denominator)?
10. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.5: Multiplication of Fractions
Vocabulary and symbols – write out what the following mean:


Simplify before you multiply
Fraction Multiplication Box
Concept questions:
1. Using the definition of multiplication, what is another way to write 6 × 3 using addition?
Exercises:
2. Rewrite with addition and then find the result:
2
a. 5   
3
4
b. 2   
9
5
c.    3
8
3. Simplify the product if you can. Rewrite as a single fraction without using addition.
1
a. 12   
5
5
b. 7   
6
3
c.   11
7
4. Create a picture with appropriate shading that will represent
1 2
 … use the box to help.
3 5
5. Create a picture with appropriate shading that will represent
3 1
 … use the box to help.
4 2
6. Use the definition of fraction multiplication to find the following (simplify the result):
a.
3 4

7 5
d.
3  20 
 
5  21 
b.
6 5

11 9
e.
40 7

49 20
c.
9 5

10 7
f.
 36   9 
  
 81   20 
Math 52 – Homework Unit 4 – Page 12
7. Practice these multiplications – and please, for your own sake, “Simplify before you multiply!”
a.
32 20

40 15
d.
620 35

310 70
b.
9  77 
 
11  15 
 55  28  3 
e.        
 16  33  7 
c.
35 4

34 25
f.
40 15 500 3
 

35 20 600 12
Wrap-up and look back:
8. Why do we simplify before we multiply?
9. When simplifying fractions, are we looking for any common numbers, or looking for something else?
Explain your answer.
10. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.6: Multiplication of Decimals
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. In 1.5  0.3 , how many decimal places will the product have?
2. If you multiply a number with 2 decimal places by a number with 3 decimal places by hand, will you
always get 5 decimal places? Explain with some examples.
3. If you multiply a number with 2 decimal places by a number with 3 decimal places using the
calculator, will you always get 5 decimal places? Explain with some examples.
Exercises:
4. Write out the multiplication problem using fractions. Convert back to decimals when finished.
Confirm using your calculator.
a. 1.7  0.3
c. 0.05  0.3
b. 2.3  0.04
d. 0.07  0.002
Math 52 – Homework Unit 4 – Page 13
5. Find the product and show your work. Perform all multiplication by hand unless the calculator symbol
is included.
a. 3.02  0.3
b. 0.7   5.1
d.
e.
f.
c. 0.011 0.6
 8.3  0.04
17.312  24.3
8.47   0.00983
Wrap-up and look back:
6. When calculating 0.98  0.05 , Larry was expecting 4 decimal places. His calculator didn’t show 4
decimal places. Explain to Larry what happened.
7. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.7: Number Sense and Multiplication
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. With what operations (addition, subtraction, multiplication) can we take two negative numbers and get
back a positive? Explain with examples.
2. With what operations (addition, subtraction, multiplication) can we take one positive number and one
negative number and get back a negative? Explain with examples.
3. Are there any operations (addition, subtraction, multiplication) where two positive numbers would
result in a negative number? Explain with examples.
4. If we multiply A by 1.25, will the result have a size that is bigger than A, less than A, or equal to A?
Explain.
Exercises:
5. Circle the appropriate information about the multiplication problem listed.
If the problem is…
a.
8 0.056
c.
1 2

5 3
11 57
d.
0.89 14.2
b.
Then the product will be …
Greater
Less
Greater
Less
Greater
Less
Greater
Less
than 0.056
1
5
than 11
than
than 0.89
Greater
Less
than 8
Greater
Less
than
Greater
Less
Greater
Less
2
3
than 57
than 14.2
Math 52 – Homework Unit 4 – Page 14
6. Determine the correct responses based on the problem A × B.
If the size of A is…
Then the size of the product (A × B) will be …
a.
0
Greater than B
Less than B
Equal to B
Zero
b.
Between 0 and 1
Greater than B
Less than B
Equal to B
Zero
c.
1
Greater than B
Less than B
Equal to B
Zero
d.
Greater than 1
Greater than B
Less than B
Equal to B
Zero
7. Determine the size and sign of the product without performing actual multiplication.
Product
Size (absolute value) is…
Sign
a.
8
  5
3
Pos
Neg
Zero
b.
3 2

4 3
Pos
Neg
Zero
Greater than
2
3
Less than
2
3
c.
 1.21   7 
Pos
Neg
Zero
Greater than
7
9
Less than
7
9
d.
11 0.057
Pos
Neg
Zero
Greater than 11
e.
57.94  0
Pos
Neg
Zero
Greater than 57.94
Less than 57.94
f.
 0.92    3 
Pos
Neg
Zero
Greater than 0.92
Less than 0.92
g.
 6.92  4.95
Pos
Neg
Zero
Greater than 6.92
Less than 6.92
h.
9,135  0.15
Pos
Neg
Zero
Greater than 0.15
Less than 0.15
 9
 7
Greater than 5
Less than 5
Less than 11
Math 52 – Homework Unit 4 – Page 15
Wrap-up and look back:
8. Thelma was multiplying two numbers and ended up with 0. What could you say must be true about the
numbers?
9. Thelma then multiplied two other numbers and ended up with 6. Does this mean that one of the two
numbers is 6? Explain your answer.
10. Louise multiplied A by B and ended up with a product having larger size than A. What does this tell
you about B?
11. When does multiplication result in something that is “bigger”?
12. When does multiplication result in something that is “smaller”?
13. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.8: Repeated Multiplication (Exponents)
Vocabulary and symbols – write out what the following mean:


Base

Exponent
Power
Concept questions:
1. In 6 3 , what number is the base and what number is the exponent?
2. In  6 3 , what number is the base and what number is the exponent?
3. In  63 , what number is the base and what number is the exponent?
4. Does  2 3 have the same value as  2 ? How are the two expressions similar and how are they
3
different? Use the words ‘base’ and ‘exponent’ in your explanation.
Exercises:
5. Rewrite the following using exponents – you don’t have to go any further.
a. 0.7  0.7  0.7
g. 12  12  12  12
b. 2.8  2.8
h. 5  5  5  5  5  5  5
3 3 3 3
c.           
7 7 7 7
i.
 2  2  2 
      
 9  9  9 
 8 8 8 8 8
j.
 88888
k.
 1 1 1 1
l.
 13  13  13  13  13 
d.
e.  2  2  2  2  2  2
f.
ab ab ab 
2
2
2
Math 52 – Homework Unit 4 – Page 16
6. Use your calculator to find the following values.
a. 5 4
b.
c.
 7

d.  

4.113
6
2

3
e.
 92
f.
 92
4
7. Determine if the following are negative or positive – no need to determine the value, just the sign. No
calculator.
a.
 67
c.
 513
e.
 2567
g.   2567
b.
 88
d.
 148
f.
 2566
h.   356
8. Without a calculator, determine the following values. When in doubt… write it out!
a.
 52
f.
 17
b.
 82
g.   23
c.  7 2
h.  32
d.  2 3
i.
  7 
2
j.
  2
4
e.
 13
Wrap-up and look back:
9. What are the similarities and differences between 5 × 3 and 3 5 ?
10. Why is multiplication similar to exponents?
11. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.9: Multiplication with variables
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
 
1. How could we rewrite x A
B
?
2. How could we rewrite x A  x B ?
3. How could we rewrite xy  ?
A
Math 52 – Homework Unit 4 – Page 17
Exercises:
4. Simplify the following.
a.
x 5  x11
 
b. x 7 x 4
c.
x 5  x11
d. m 44  m 9
g.
e. b 24  b13  y 3
h. x 32 x 6 x11
f.
y 23  y 41
  
x8  x 2  x  x9
x 7  x18  x 5  x
i.
5. Simplify the following.
a.
b.
c.
x 
x 
x 
5 4
d.
8 9
3 4
e.
x
f.
x   x
x  x 
x   x 
7 3
5
10 5
6 2
9 3
g.
h.
5 2
i.
x   y   x
x  x  x 
m  m   m
4 2
9 5
5 2
3 3
2 3
5
 y 45
3 5
14 2
6. Simplify the following.
a.
b.
c.
d.
e.
f.
4x 2
g.
8m 
  3x 
  2x 
  9x 
x y 
4 2
h.
5 2
i.
8 3
j.
11 2
3
k.
2 5
l.
x y m
x y 
x y m
 4x   x
x y   x
x y    3x
2
7
6
31 12 4
4
21
3
3 2
2
4
5 7
11
11
1 5
11
y7

2
7. Find the following values based on our properties.
a. 10 6  10 5
b. 10 2  10 3
c. 10 5 1011
d. 10 6  10
2
8. Find 3 2  4 2 . Compare this to 7 2 . Would you say that 3 2  4 2  3  4 ? Why or why not?
Wrap-up and look back:
   10 . Is she correct? Explain why or why not.
10. Maryam claims that 5 5   5 . Is she correct? Explain why or why not.
9. Momo claims that 10 2
5
11
7
2
22
Math 52 – Homework Unit 4 – Page 18
  
  
11. Lucas says 5 2 5 2  5 4 because you multiply the exponents and 2  2  4 . Is 5 2 5 2  5 4 true? Is the
reasoning correct? Explain why or why not.
12. How does 100 × 1,000 = 100,000 relate to exponents? Explain.
13. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.10: Multiplication on a Number Line
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. If A and B are between 0 and 1, then A × B will be where?
2. If A is bigger than 1 and B is between 0 and 1, then will A × B be bigger than A or less than A?
Explain.
3. If A is bigger than 1 and B is between 0 and 1, then will A × B be bigger than B or less than B?
Explain.
4. If A is left of 0 on the number line and B is left of 0, then where is A × B located? Explain.
5. If A is left of 0 on the number line and B is right of 0, then where is A × B located? Explain.
6. If A is left of 0 on the number line, then where is A2 located? Explain.
7. If A is left of 0 on the number line, then where is – A2 located? Explain.
Exercises:
8. Practice more with the number lines. Find the region that best approximates…
a. 2C.
e. D × C.
b. C 2 .
f. A × B.
i. D × B.
c. B 2 .
g. A × C.
j.
d.   C2 .
h. C × B.
k. B 2  2 .
A
R1
B
R2
R3
0
C
R4
1
C 2  1.
D
R5
R6
Math 52 – Homework Unit 4 – Page 19
Wrap-up and look back:
14. Marcy multiplied A × 2 and the result was negative. What does this tell you about the location of
A on the number line? Why?
15. Did you have any questions remaining that weren’t covered in class? Write them out and bring
them back to class.
4.11: 2-Dimensional Geometry and Multiplication
Vocabulary and symbols – write out what the following mean:





Circle
Center (of a circle)
Radius
Diameter
Formulas for shapes








Area
Circumference
Chord
π
Rectangle
Square
Parallelogram
Triangle
Concept questions:
1. Is the area of a circle found with 2  r or   r 2 ?
2. What is π and how does it relate to a circle?
3. William sees a square with 5 inches on each side and says that the area is 5 in2. Is he correct? Explain.
4. Wendy sees a square with 5 inches on each side and says that the area is 25 in2. Is she correct?
Explain.
5. Winona sees a square with 5 inches on each side and says that the perimeter is 20 in2. Is she correct?
Explain.
6. If you are calculating something with a formula and end up with yards, could this be an area or
perimeter? How can you tell?
Exercises:
7. Compute the circumference of the circle given the information below. State the exact value and then
use the calculator to approximate the value to 4 decimal places.
a. diameter is 4 feet
e. radius is 8 inches
b. diameter is 21 centimeters
f. radius is 23 centimeters
c. diameter is 9 yards
g. radius is 11 centimeters
d. diameter is 2.3 inches
h. radius is 43 centimeters
Math 52 – Homework Unit 4 – Page 20
8. Find the areas of the shapes using the formulas we have found to this point. Draw a sketch of the
shape as well.
a. Rectangle with height of 4 feet and base of 16 feet.
b. Rectangle with height of 7 yards and base of 3 yards.
c. Square with sides of 9 cm.
d. Square with sides of 11 m.
e. Circle with radius of 7 inches (round to 2 decimal places).
f. Circle with radius of 4.6 inches (round to 2 decimal places).
g. Triangle with base of 10 inches and height of 9 inches.
h. Triangle with base of 40 inches and height of 80 inches.
i. Triangle with base of 2.9 inches and height of 6.8 inches.
j. Circle with diameter of 8 cm (round to 2 decimal places).
k. Circle with diameter of 14 cm (round to 2 decimal places).
l. Parallelogram with base of 9 feet and height of 7 feet.
m. Parallelogram with base of 20 feet and height of 11 feet.
n. Square with lengths of
5
m.
7
o. Square with lengths of
3
in.
4
p. Circle with circumference of 10 mm (round to 2 decimal places).
q. Circle with circumference of 40 inches (round to 2 decimal places).
9. For the following problems, find the area of the shaded region.
a. The shape has a triangle with height of 12 inches and base of 8 inches, and the square has 3
inch sides – find the area in square inches.
b. A brownie pan had a slice cut out. If the pan was 9 inches by 11 inches, and the slice was a
rectangle measuring 3 inches by 4 inches, find the area in square inches.
c. The shape has a circle with diameter 9 yards, while the triangle has 3 foot height and 9 foot
base – find the area in square feet. Approximate to 2 decimal places.
Math 52 – Homework Unit 4 – Page 21
d. This is a concrete circular driveway, where the inner radius is 40 feet, and the outer radius is
55 feet. Find the area in square feet. Approximate to 2 decimal places.
e. Think of a rectangular pool that is 12 feet by 15 feet. The owner wants to put a sidewalk
around the pool that is 2 feet wide (around the whole pool). Draw a picture of this shape and
then calculate the area of the sidewalk.
When finding the amount of area under a particular curve, we use
a table similar to the one below. The table finds the area from – ∞
to the z-value. The possible pictures look like these (first one is a
z-score less than 0, and the second has a z-score greater than 0).
For this curve, half the area is below z = 0, which is why the area for z = 0.00 is 0.5000 (like 50%).
z
– 3.0
– 2.7
– 2.4
– 2.1
– 1.8
– 1.5
– 1.2
– 0.9
– 0.6
– 0.3
– 0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0013
0.0035
0.0082
0.0179
0.0359
0.0668
0.1151
0.1841
0.2743
0.3821
0.5000
0.6179
0.7257
0.8159
0.8849
0.9332
0.9641
0.9821
0.9918
0.9965
0.9987
0.0013
0.0034
0.0080
0.0174
0.0351
0.0655
0.1131
0.1814
0.2709
0.3783
0.5040
0.6217
0.7291
0.8186
0.8869
0.9345
0.9649
0.9826
0.9920
0.9966
0.9987
0.0013
0.0033
0.0078
0.0170
0.0344
0.0643
0.1112
0.1788
0.2676
0.3745
0.5080
0.6255
0.7324
0.8212
0.8888
0.9357
0.9656
0.9830
0.9922
0.9967
0.9987
0.0012
0.0032
0.0075
0.0166
0.0336
0.0630
0.1093
0.1762
0.2643
0.3707
0.5120
0.6293
0.7357
0.8238
0.8907
0.9370
0.9664
0.9834
0.9925
0.9968
0.9988
0.0012
0.0031
0.0073
0.0162
0.0329
0.0618
0.1075
0.1736
0.2611
0.3669
0.5160
0.6331
0.7389
0.8264
0.8925
0.9382
0.9671
0.9838
0.9927
0.9969
0.9988
0.0011
0.0030
0.0071
0.0158
0.0322
0.0606
0.1056
0.1711
0.2578
0.3632
0.5199
0.6368
0.7422
0.8289
0.8944
0.9394
0.9678
0.9842
0.9929
0.9970
0.9989
0.0011
0.0029
0.0069
0.0154
0.0314
0.0594
0.1038
0.1685
0.2546
0.3594
0.5239
0.6406
0.7454
0.8315
0.8962
0.9406
0.9686
0.9846
0.9931
0.9971
0.9989
0.0011
0.0028
0.0068
0.0150
0.0307
0.0582
0.1020
0.1660
0.2514
0.3557
0.5279
0.6443
0.7486
0.8340
0.8980
0.9418
0.9693
0.9850
0.9932
0.9972
0.9989
0.0010
0.0027
0.0066
0.0146
0.0301
0.0571
0.1003
0.1635
0.2483
0.3520
0.5319
0.6480
0.7517
0.8365
0.8997
0.9429
0.9699
0.9854
0.9934
0.9973
0.9990
0.0010
0.0026
0.0064
0.0143
0.0294
0.0559
0.0985
0.1611
0.2451
0.3483
0.5359
0.6517
0.7549
0.8389
0.9015
0.9441
0.9706
0.9857
0.9936
0.9974
0.9990
10. Find the areas listed using the table above.
a. ** The area below z = 1.57.
d. The area below z = – 0.38.
b. ** The area between z = – 0.94 and
z = 1.29.
e. The area between z = – 2.46 and
z = – 1.89.
c. The area below z = 2.11.
f. The area between z = – 1.82 and z = 1.56.
Math 52 – Homework Unit 4 – Page 22
Wrap-up and look back:
11. Lane found the area of a parallelogram by multiplying the side lengths. Is this approach ever correct?
12. Describe how much area 10 square feet would represent? (use at least two examples)
13. Imagine a room that is 10 feet by 12 feet. You want to put wallpaper over all the walls, which are 9
feet high. How much wallpaper would you need (in square feet)? Explain your result.
14. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.12: Multiplication Applications
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. Some jobs have overtime pay and holiday pay. If the holiday pay is “double time,” and your base pay
is $11 per hour, how much would you earn in a 6 hour shift on Thanksgiving day? Would you do this
or not?
2. If you had the chance to work 20 more hours over a weekend (Saturday/Sunday) and you were paid
time-and-a-half, would you do it? (Assume the base pay is $20 per hour)
Exercises: The exercises are broken into different types based on the concepts involved.
3. Money Applications:
a. If someone earns $10.25 per hour, and works 30 hours per week, how much would they earn in
one week?
b. If someone earns $14 per hour for regular pay, and “time and a half” for overtime, what is the rate
of pay for overtime hours? [Overtime hours are counted for more than 40 hours per week]
c. The person earning $14 per hour works 46 hours per week, how much did they earn that week?
d. Someone working 55 hours per week with a wage of $22 per hour, how much did they earn that
week?
e. Who earns more over the course of one week? Alicia earns $22 per hour and works 40 hours per
week. Luisa earns $19 per hour and works 45 hours per week.
f. For a wage of $34 per hour, with weekly hours of 40 hours per week, how much money would the
person earn over …
a. 4 weeks?
b. 1 year?
Math 52 – Homework Unit 4 – Page 23
4. Time Applications:
a. How many days are there in 8 weeks? Would you use addition or multiplication for this?
b. How many total days are there in the months of May, June, and July?
c. How many months are in 18 years?
d. How many months are in 40 years?
e. How many days are in 13 years?
f. How many total days are there in the months of April, May, and June?
g. How many days are in 30 years?
5. Sports Applications:
a. In a season, a football player ran for an average of 5.7 yards per carry, and had 217 carries; how
many yards did the player rush for that season?
b. In a recent game, an LSU running back rushed for 218.8 yards per game through 4 games. How
many yards did the player rush for total?
c. In a recent game, an LSU running back rushed for 218.8 yards per game through 4 games. How
many yards would the player have total if he kept that rate for 15 games, how many total yards
would he have? Is it enough to break the record of 2,628 yards (Barry Sanders)?
d. A NASCAR driver had averaged 147.94 miles per hour for 2 hours before crashing. How far had
the driver traveled?
e. A NASCAR driver had averaged 147.94 miles per hour for 2.5 hours before crashing. How far had
the driver traveled?
f. A NASCAR driver had averaged 147.94 miles per hour for 2 hours, 20 minutes before crashing.
How far had the driver traveled?
6. Other Applications:
a. 12 groups of 12 is known as a gross. If a shipment of pencils included 14 gross and 6 dozen, how
many total pencils were there in the shipment?
b. If a shipment of pencils included 60 gross, 8 dozen, and 5 pencils, how many total pencils were
there in the shipment?
c. If a large table holds 10 people, and small table holds 6 people, how many people can be seated at
a table in a room with 13 large tables and 21 small tables?
d. If a large table holds 10 people, and small table holds 6 people, can a wedding party with 85
people be seated at a table in a room with 6 large tables and 4 small tables?
Math 52 – Homework Unit 4 – Page 24
e. Each page of a Case Logic DVD binder holds 8 DVDs, and each binder holds 40 pages. A
personal collection has 7,023 DVDs and the owner purchased 24 binders. Does the owner have
enough cases to fit all the DVDs? If not, how many more binders must be purchased?
f. Each page of a Case Logic DVD binder holds 8 DVDs, and each binder holds 40 pages. A
personal collection has 9,023 DVDs and the owner purchased 28 binders. Does the owner have
enough cases to fit all the DVDs? If not, how many more binders must be purchased?
g. From the previous example, if each binder cost $36.98 on amazon with free shipping, how much
would it cost to purchase all the binders needed for the collection?
h. If your math teacher has 16 shirts, 8 pairs of shorts, and 3 different pairs of shoes. How many
different outfits are possible if he picks one shirt, one pair of shorts, and a pair of shoes?
i. If your math teacher has 12 shirts, 5 pairs of shorts, and 2 different pairs of shoes, how many
different outfits are possible if he picks one shirt, one pair of shorts, and a pair of shoes?
j. Will the teacher in the previous problem be able to wear a different outfit each day for a whole
term?
Wrap-up and look back:
7. Which examples did you like best?
8. Create a problem that involves overtime pay. Then solve the problem.
9. Marcia is looking at her paycheck and she worked 48 hours. Her base pay is $10 per hour, and
overtime is time-and-a-half. She calculates her pay to be 48 × 15 = $720. Is she correct or not? If she
is wrong, how can you help her?
10. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
4.13: Multiplication Summary
Explain/Summarize the following addition properties:

Commutative Property of Multiplication

Associative Property of Multiplication

Identity Property of Multiplication

Distributive Property of Multiplication over Addition
Math 52 – Homework Unit 4 – Page 25
4.14: Multiplication Wrap-Up (Practice)
1. Practice rewriting without any parenthesis:
a. 123x  4m
b. 72 x  6m
c. 54 x  3 y  6
d. 52 x  6 y  11
2. Multiply the following:
a. 70  4
b. 312  300
c. 13  4,000
3. Rewrite as one multiplication.
a. 19  17  5  17
b. 231  39  8  39  107  39
c. 9  17  4  17
d. 372  56  5  56  110  56
4. Multiply using any algorithm.
a. 78  59
b. 136  98
c. 58  36
d. 129  85
5. Multiply, and be sure that your product has the appropriate “sign”:
a.  312
b.  6 12
e. 4 8
f.  5 13
c. 9 12
d.  89
6. Write the following values without parenthesis. Interpret the meaning.
c.  2 x  34 y   46
d.   2 x  9 y   6
a.  11x  8
b.  5x  8
7. Describe the following numbers as prime (P), composite (C), or neither (N) and give a factor other
than 1:
Number
Describe
A factor is…
Number
Describe
a.
42
P
C
N
b.
17
P
C
N
c.
23
P
C
N
d.
1
P
C
N
e.
32
P
C
N
f.
27
P
C
N
A factor is…
Math 52 – Homework Unit 4 – Page 26
8. Simplify the following fractions. When indicated by the symbol, use the calculator to simplify.
a.
40
15
c.
48
60
e.
1,692
2,021
b.
36
60
d.
20
15
f.
1,786
2,021
9. Use the definition of fraction multiplication to find the following (simplify the result):
 50   18 
e.       
 27   25 
 40   36 
f.       
 27   55 
3 4

7 5
9 20
d.

10 27
3 2

7 5
3 14
b.

10 27
a.
c.
10. Find the product and show your work. Perform all multiplication by hand unless the calculator symbol
is included.
b. 0.07   0.19
a. 1.8  0.04
0.648   0.0023
c.
11. Without a calculator, determine the following values:
a.  5
b.  10 2
2
c.
 23
e.
d.  9 2
f.
 82
 33
12. Determine the size and sign of the product without performing actual multiplication.
Product
Size (absolute value) is…
Sign
a.
1
  5
3
Pos
Neg
Zero
Greater than 5
Less than 5
b.
 11   2.45
Pos
Neg
Zero
Greater than 11
Less than 11
c.
6.92    3 
Pos
Neg
Zero
Greater than 6.92
Less than 6.92
d.
 6.92  49.17
Pos
Neg
Zero
Greater than 6.92
Less than 6.92
Pos
Neg
Zero
Greater than 5
Less than 5
Pos
Neg
Zero
Greater than 11
Less than 11
 7
f.
5
  5
3
 11  0.45
g.
6.92    11 
Pos
Neg
Zero
Greater than 6.92
Less than 6.92
h.
 6.92   0.17
Pos
Neg
Zero
Greater than 6.92
Less than 6.92
e.

9
Math 52 – Homework Unit 4 – Page 27
13. Compute the circumference of the circle given the information below. State the exact value and then
use the calculator to approximate the value to 4 decimal places.
a. diameter is 9.2 feet
b. radius is 25 inches
c. diameter is 2.3 inches
d. radius is 0.27 inches
14. Simplify the following using the rules on exponents.
a.
x 23  x14  x11 
b.
x   x 
14 2
5 9

c.
 4x 

d.
x
 x18 
5 3
2
y6

7
15. Is it true that a 2  b 2  a  b2 ? Explain why it is true, or find some numbers to show it is NOT true.
16. Describe the numbers as prime (P), composite (C), or neither (N) and give a factor other than 1:
Number
Describe
A factor is…
Number
A factor is…
Describe
a.
32
P
C
N
b.
29
P
C
N
c.
27
P
C
N
d.
1
P
C
N
17. Practice more with the number lines.
a.
b.
c.
d.
e.
Find the region that best approximates
Find the region that best approximates
Find the region that best approximates
Find the region that best approximates
Find the region that best approximates
A
R1
B
R2
R3
B × D.
A + B.
D2 .
B × C.
D – 2C.
0
C
1
R4
D
R5
R6
18. For the following problems, find the area of the shaded region.
a. The shape has a circle with diameter 9 feet, while the triangle has 3 foot height and 8 foot base –
find the area in square feet.
Math 52 – Homework Unit 4 – Page 28