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Homework – Unit 4 –

Math Fundamentals for

Statistics (Math 52)

Homework Unit 4: Multiplication

Scott Fallstrom and Brent Pickett “The ‘How’ and ‘Whys’ Guys”


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?

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. 52 x

b. 63 x

c. 523 x

d. 373 y

e. mx 756

f. 8324 ymx

g. 3202

h. 23004


11. What are the two ways to find the value of 352 ? Show both and see if they produce same values.

Which one did you like best?

12. Are there two ways to express 56 x 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: dnaan 1 .

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. 10100

b. 1010

c. 100000,1

d. 000,1000,1

e. 000,127

f. 100158

g. 000,1055

h. 000,13100

15. Multiply using place value numbers and the associative property:

a. 70 × 3

b. 90 × 200

c. 80 × 300

d. 80 × 300

e. 500 × 3,000

f. 130 × 2,000

g. 40 × 80

h. 400 × 800

i. 4,000 × 8,000

j. 40,000 × 80,000

16. Show how to multiply these using the distributive property and find the product quickly.

a. 5 × 13

b. 25 × 6

c. 18 × 3

d. 27 × 4

e. 91 × 7

f. 34 × 5

g. 6 × 41

h. 43 × 11

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

b. 48 × 23

c. 57 × 29

d. 31 × 49

e. 78 × 89

f. 103 × 24

g. 111 × 25

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!

a. 91 × 7 + 91 × 3

b. 31 × 9 + 31 × 11

c. 42 × 16 + 42 × 4

d. 47 × 23 + 13 × 23

e. 22 × 11 + 22 × 39

f. 14 × 68 + 14 × 2

g. 93 × 574 + 7 × 574

h. 983 × 574 + 17 × 574

i. 16 × 53 + 16 × 9 + 14 × 62

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

b. 31 × 9 + 31 × 11

c. 42 × 16 + 42 × 4

d. 47 × 23 + 13 × 23

e. 22 × 11 + 22 × 39

f. 14 × 68 + 14 × 2

g. 93 × 574 + 7 × 574

h. 983 × 574 + 17 × 574

i. 16 × 53 + 16 × 9 + 14 × 62

j. 23 × 18 + 23 × 13 + 27 × 31

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. b. 31 × 49. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual? c. Estimate the value of 37 × 42 by rounding to the tens and then multiplying the rounded values. d. 37 × 42. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual? e. Estimate the value of 427 × 593 by rounding to the hundreds and then multiplying the rounded

values. f. 427 × 593. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual? g. Estimate the value of 422 × 561 by rounding to the hundreds and then multiplying the rounded

values. h. 422 × 561. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual? i. Estimate the value of 478 × 583 by rounding to the hundreds and then multiplying the rounded

values. j. 478 × 583. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual? k. Estimate the value of 428 × 513 by rounding to the hundreds and then multiplying the rounded

values. l. 428 × 513. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual? m. Create a rule: if we take round both factors up, then the estimate of the product will be (Always,

Sometimes, Never) larger than the actual product. Explain your rule. n. Create a rule: if we take 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 see the .


a. Estimate the value of 75 × 38 by rounding to the tens and then multiplying the rounded values.

b. 75 × 38. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual?

c. What did you round 75 to in the previous problem? What other value is equally correct when

rounding (but not conventional)?

d. 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.

e. Compare your estimates from both rounded values. Which one is closer?

f. Estimate the value of 85 × 32 by rounding to the tens and then multiplying the rounded values.

g. 85 × 32. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual?

h. What did you round 85 to in the previous problem? What other value is equally correct when

rounding (but not conventional)?

i. 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.

j. Compare your estimates from both rounded values. Which one is closer?

k. Estimate the value of 94 × 38 by rounding to the tens and then multiplying the rounded values.

l. 94 × 38. Compare your estimate to the exact value. Is your estimate higher or lower than the

actual?

m. Create a rule: if we take 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.

22. Multiply these using the distributive property.

a. 408 × 11

b. 902 × 5

c. 899 × 5

d. 799 × 8

e. 598 × 9

f. 301 × 12

g. 3 × 912

h. 5 × 297

i. 7 × 297

j. 25 × 398

k. 21 × 299

l. 43 × 199

23. What property is used here.

a. 0447744

b. 44777744

c. 2015435 xx

d. 11751175

e. 51171175

f. 75175

Wrap-up and look back:


24. 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.


lower than the actual? Can you tell quickly, or do you need to calculate the actual value? Explain.


lower than the actual? Explain why.

27. How many 0’s will be in this product: 100,000 × 10,000,000? Explain why.

28. 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 74 and 74 the same or opposites? Why?

2. Are the products 74 and 74 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 38 = 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 75 = 35. When you prove this, be sure to

put in the explanation not just the equations!

7. If I begin with 0044 and rewrite it as 07744 , what property was used?

8. If I rewrite 07744 as 0744744 , what property was used?




11. Multiply the following without a calculator: be sure that your product has the appropriate “sign”:

a. 78

b. 139

c. 118

d. 95

e. 78

f. 1713

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) Examples or Explanation

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

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. 8 x

b. 111 x

c. 531 x

d. 15714 yx

e. 92313 yx

f. yx 95731

g. yx 9312551

h. pmyx 153625

14. Find the nth term in the arithmetic sequence and simplify the result: dnaan 1 .

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. Did you prefer the first type of “proof” or the alternative explanation? Describe why you liked one

over the other.

16. Which of the operations seem the easiest to understand and explain (addition, subtraction, or

multiplication)? Why do you think it is easier?

17. When you see the value (– x), is it positive, negative, zero… or can you not tell? Explain.

18. Martin is confused with integers. One of his previous teachers said “two negatives make a positive,”

but when he performed 118 and wrote 19, he was marked wrong. Explain the confusion, why

he was marked wrong, and what the previous teacher should have said.


back to class.

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 factor is…

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

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

b. 65

c. 800

d. 89

e. 764

f. 4,802


a. 37

b. 60

c. 810

d. 75

e. 525

f. 327



a. 48

b. 97

c. 78

d. 326

e. 318

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

b. 50

c. 70

d. 100

e. 53

f. 84

g. 157

h. 125

i. 27

j. 63

k. 91

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?


back to class.

4.4: Using FLOF to Simplify Fractions Vocabulary and symbols – write out what the following mean:

FLOF

Concept questions:

1. Hillary claims that 4

1

64

16 because she cancelled out the 6’s. Is her answer correct? Is her reasoning

correct?

2. Bill looks at Hillary’s work and concludes that 3

1

35

15 because he cancelled out the 5’s. Is he correct?

Is his reasoning correct?


3. If Raul wrote 7

3

25

12

165

82

165

82

2

1

, is he correct? Where is his mistake? What should the result

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. 15

10

b. 60

36

c. 21

16

d. 350

275

e. 24

16

f. 55

132

g. 40

37

h. 371

318

i. 19,393

12,341


5. Willie wrote 40

34 as

0

3. What is one big indicator that he made a mistake?

6. What operation is needed to remove common factors? Think of 3

5

83

85

… is this correct?

7. When we simplify fractions, can we stop when one of the numbers (numerator or denominator) is

prime? Explain and give some examples.

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)?


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:

a.

3

25 b.

9

42 c. 3

8

5

3. Simplify the product if you can. Rewrite as a single fraction without using addition.

a.

5

112 b.

6

57 c. 11

7

3

4. Create a picture with appropriate shading that will represent 5

2

3

1 … use the box to help.

5. Create a picture with appropriate shading that will represent 2

1

4

3 … use the box to help.

6. Use the definition of fraction multiplication to find the following (simplify the result):

a. 5

4

7

3

b. 9

5

11

6

c. 7

5

10

9

d.

21

20

5

3

e. 20

7

49

40

f.

20

9

81

36

7. Practice these multiplications – and please, for your own sake, “Simplify before you multiply!”

a. 15

20

40

32

b.

15

77

11

9

c. 25

4

34

35

d. 70

35

310

620

e.

7

3

33

28

16

55

f. 12

3

600

500

20

15

35

40


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.


back to class.

4.6: Multiplication of Decimals Vocabulary and symbols – write out what the following mean:

None

Concept questions: 1. In 3.05.1 , 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. 3.07.1

b. 04.03.2

c. 3.005.0

d. 002.007.0

5. Find the product and show your work. Perform all multiplication by hand unless the calculator symbol is included.

a. 3.002.3

b. 1.57.0

c. 6.0011.0

d. 04.03.8

e. 3.24312.17

f. 00983.047.8

Wrap-up and look back: 6. When calculating 05.098.0 , 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… Then the product will be …

a. 056.08 Greater Less than 0.056 Greater Less than 8

b. 3

2

5

1 Greater Less than

5

1 Greater Less than

3

2

c. 5711 Greater Less than 11 Greater Less than 57

d. 2.1489.0 Greater Less than 0.89 Greater Less than 14.2

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 Sign Size (absolute value) is…

a. 53

8 Pos Neg Zero Greater than 5 Less than 5

b. 3

2

4

3 Pos Neg Zero Greater than

3

2 Less than

3

2

c.

9

721.1 Pos Neg Zero Greater than

9

7 Less than

9

7

d. 057.011 Pos Neg Zero Greater than 11 Less than 11

e. 094.57 Pos Neg Zero Greater than 57.94 Less than 57.94

f.

7

392.0 Pos Neg Zero Greater than 0.92 Less than 0.92

g. 95.492.6 Pos Neg Zero Greater than 6.92 Less than 6.92

h. 15.0135,9 Pos Neg Zero Greater than 0.15 Less than 0.15


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”?


back to class.

4.8: Repeated Multiplication (Exponents) Vocabulary and symbols – write out what the following mean:

Base Exponent

Power

Concept questions:

1. In 36 , what number is the base and what number is the exponent?



4. Does 32 have the same value as 32 ? How are the two expressions similar and how are they

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. 7.07.07.0

b. 8.28.2

c.

7

3

7

3

7

3

7

3

d. 88888

e. 222222

f. 222 ababab

g. 12121212

h. 5555555

i.

9

2

9

2

9

2

j. 88888

k. 1111

l. 1313131313

6. Use your calculator to find the following values.

a. 45

b. 311.4

c. 67

d. 4

3

2

e. 29

f. 29

7. Determine if the following are negative or positive – no need to determine the value, just the sign. No

calculator.

a. 76

b. 88

c. 135

d. 481

e. 5672

f. 5662

g. 5672

h. 563

8. Without a calculator, determine the following values. When in doubt… write it out!

a. 25

b. 28

c. 27

d. 32

e. 31

f. 71

g. 32

h. 23

i. 27

j. 42


9. What are the similarities and differences between 5 × 3 and 53 ?

10. Why is multiplication similar to exponents?


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 BAx ?

2. How could we rewrite BA xx ?

3. How could we rewrite Axy ?

Exercises: 4. Simplify the following.

a. 115 xx

b. 47 xx

c. 115 xx

d. 944 mm

e. 31324 ybb

f. 928 xxxx

g. 4123 yy

h. 11632 xxx

i. xxxx 5187

5. Simplify the following.

a. 45x

b. 98x

c. xx 43

d. 537 xx

e. 26510 xx

f. 2539 xx

g. 4555924 yxyx

h. 533325 xxx

i. mmm 21432

6. Simplify the following.

a. 24x

b. 248m

c. 253x

d. 382x

e. 2119x

f. 523 yx

g. 762 myx

h. 41231 yx

i. 2134 myx

j. 11234 xx

k. 11752 xyx

l. 2711514 3 yxyx


7. Find the following values based on our properties.

a. 56 1010

b. 32 1010

c. 115 1010

d. 10106

8. Find 22 43 . Compare this to 27 . Would you say that 222 4343 ? Why or why not? Wrap-up and look back:

9. Momo claims that 752 1010 . Is she correct? Explain why or why not.

10. Maryam claims that 22211 555 . Is she correct? Explain why or why not.

11. Lucas says 422 555 because you multiply the exponents and 422 . Is 422 555 true? Is

the reasoning correct? Explain why or why not.

12. How does 100 × 1,000 = 100,000 relate to exponents? Explain.


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.

a. Find the region that best approximates 2C.

b. Find the region that best approximates 2C .

c. Find the region that best approximates 2B .

d. Find the region that best approximates 2C .

e. Find the region that best approximates D × C.

f. Find the region that best approximates A × B.

g. Find the region that best approximates A × C.

h. Find the region that best approximates C × B.

i. Find the region that best approximates D × B.

j. Find the region that best approximates 1C2 .

k. Find the region that best approximates 2B2 .

A B 0 C 1 D


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

Area Circumference Chord π

Rectangle Square Parallelogram Triangle

Formulas for shapes

1 2 3 4 5 6


Concept questions:

1. Is the area of a circle found with r2 or 2r ?

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

b. diameter is 21 centimeters

c. diameter is 9 yards

d. diameter is 2.3 inches

e. radius is 8 inches

f. radius is 23 centimeters

g. radius is 11 centimeters

h. radius is 43 centimeters

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 7

5m. o. Square with lengths of

4

3in.

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.

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 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 – 3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010

– 2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026

– 2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064

– 2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143

– 1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294

– 1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559

– 1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985

– 0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611

– 0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451

– 0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483

– 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

10. Find the areas listed using the table above.

a. ** The area below z = 1.57. b. ** The area between z = – 0.94 and

z = 1.29. c. The area below z = 2.11.

d. The area below z = – 0.38. e. The area between z = – 2.46 and

z = – 1.89. f. The area between z = – 1.82 and z = 1.56.

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.



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?

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?

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?


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

4.14: Multiplication Wrap-Up (Practice) 1. Practice rewriting without any parenthesis:

a. mx 4312

b. mx 627

c. 6345 yx

d. 11625 yx


2. Multiply the following:

a. 470

b. 300312

c. 000,413

3. Rewrite as one multiplication.

a. 1751719 b. 3910739839231

c. 174179 d. 5611056556372

4. Multiply using any algorithm.

a. 5978 b. 98136

c. 3658 d. 85129

5. Multiply, and be sure that your product has the appropriate “sign”:

a. 123

b. 126

c. 129

d. 98

e. 84

f. 135

6. Write the following values without parenthesis. Interpret the meaning.

a. 811 x

b. 85 x

c. 46342 yx

d. 692 yx

7. Describe the following numbers as prime (P), composite (C), or neither (N) and give a factor other than 1:


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

8. Simplify the following fractions. When indicated by the symbol, use the calculator to simplify.

a. 15

40

b. 60

36

c. 60

48

d. 15

20

e. 021,2

692,1

f. 021,2

786,1


9. Use the definition of fraction multiplication to find the following (simplify the result):

a. 5

2

7

3

b. 27

14

10

3

c. 5

4

7

3

d. 27

20

10

9

e.

25

18

27

50

f.

55

36

27

40

10. Find the product and show your work. Perform all multiplication by hand unless the calculator symbol is included.

a. 04.08.1

b. 19.007.0

c. 0023.0648.0

11. Without a calculator, determine the following values:

a. 25

b. 210

c. 32

d. 29

e. 28

f. 33

12. Determine the size and sign of the product without performing actual multiplication.

Product Sign Size (absolute value) is…

a. 53

1 Pos Neg Zero Greater than 5 Less than 5

b. 45.211 Pos Neg Zero Greater than 11 Less than 11

c.

7

392.6 Pos Neg Zero Greater than 6.92 Less than 6.92

d. 17.4992.6 Pos Neg Zero Greater than 6.92 Less than 6.92

e. 53

5

Pos Neg Zero Greater than 5 Less than 5

f. 45.011 Pos Neg Zero Greater than 11 Less than 11

g.

9

1192.6

Pos Neg Zero Greater than 6.92 Less than 6.92

h. 17.092.6 Pos Neg Zero Greater than 6.92 Less than 6.92


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. 111423 xxx

b. 95214 xx

c. 354x

d. 18762 xyx

15. Is it true that 222 baba ? Explain why it is true, or find some numbers that show it is NOT true.

16. Describe the numbers as prime (P), composite (C), or neither (N) and give a factor other than 1:


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. Find the region that best approximates B × D. b. Find the region that best approximates A + B. c. Find the region that best approximates D2. d. Find the region that best approximates B × C. e. Find the region that best approximates D – 2C.

A B 0 C 1 D

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.

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