MathematicsFOR ELEMENTARY TEACHERS

ByExamples

By Courtney A. Pindling

Department of Mathematics - SUNY NewPaltz

First Edition, Summer 2001

Cover: Math for teachers

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2.2 Whole Numbers, and Numeration

Example 4. Write the Roman numeral for 1999

Answers: 1999 = 1000 + 900 + 90 + 9 = M + CM + XC + IX

= MCMXCIX

Example 5. Historic representation of numerals (state equivalentHindu-Arabic numerals):

Roman numerals:MCCCXLIV = M + CCC + XL +IV = 1000 + 300 + 40 + 4 = 1344

Egyptian (2563)

Mayan (31781148) Babylonian (1 603 + 57 602 + 46 60 + 40 =424000 )

2.2 Whole Numbers, and Numeration

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2.3 Whole Numbers (Base ten)

Example 6. Write 31,407 in expanded form.

Answers: 31407 = 31(1000) + 4(100) + 0(10) + 7(1)

Example 7. Write the numeral 43,762,123,504,931 as words.

Answers: forty-three trillion, seven hundred and sixty-two billion, onehundred and twenty-three million, five hundred and four thousand, nine hundred and thirty-one

Example 8. Express each numeral as an expansion of its base or with multibasepieces. (a) 1345 (b) 32110 (c) 11012 (d) 6138

Answers: (a) 1345 = 1(52) + 3(51) + 4(50) = 1(25) + 3(5) + 4(1)

(b) 32110 = 3(102) +2(101) + 1(100) = 3(100) + 2(10) + 1 (1)

(c) 11012 = 1(23) +1(22) + 0(21) + 1(20) = 1(8) + 1(4) +0(2) +1

(d) 6138 = 6(82) +1(81) + 3(80)

Example 9. Write the first 20 base 3 terms

Answers: 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121,122, 200, 201, 202, 210

Example 10. Convert the following to base ten:

(a) 1348 (b) 230324 (c) 1101102

Answers: (a) 1348 = 1(82) + 3(81) + 4(80) = 1(64) + 3(8) + 4(1) = 68

(b) 230324 = 2(44) + 3(43) + 0(42) + 3(4) + 2 = 512+192+12+2 = 718

2.3 Whole Numbers (Base ten)

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(c) 1101102 =1(25) + 1(24) + 0 + 1(22) + 1(2) + 0 = 32+16+4+ 2 = 54

Example 11. Convert the following base ten to requested bases:

(a) 613 = base 8 (b) 23250 to base 20

Answers: (a) 61310 = 11458

Bases 84 =4096

83 =512

82 =64

81 =8

80 =1

Answer/Sum

8 0 1R101 1R37 4R5 5 11458

10 0 1x512

=512

1x64

=64

4x8

=32

5x1

=5

613

Answers: (a) 2325010 = 2 18 2 1020

Bases 204 =160,000

203 =8,000

202 =400

201

= 20200

= 1Answer/Sum

20 0 2R7250 18R50 2R10 10 2 18 2108

10 0 2x8000

=16000

18x400

=7200

2x20

=40

10x1

=10

23250

2.3 Whole Numbers (Base ten)

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5.1 Prime Numbers

Example 12. Express the following as product of primes:

(a) 2268 (b) 420

Answers: (a) 2268 = 22 x 34 x 7 (b) 420 = 22 x 3 x 5 x 7

Divide by successive primes from 2 on: (a) 2268 = 22 x 34 x 7

Factor /

Divisor

2 2 3 3 3 3 7

Dividend 2268 1134 567 189 63 21 7

Quotient 1134 567 189 63 21 7 1

Remainder 0 0 0 0 0 0 0

Divide by successive primes from 2 on: (a) 420 = 22 x 3 x 5 x 7

Factor /

Divisor

2 2 3 5 7

Dividend 420 210 105 35 7

Quotient 210 105 35 7 1

Remainder 0 0 0 0 0

Example 13. Are the numbers (a) 598 and (b) 823 primes?

(a)

Check to see if primes 2 to 23 is a divisor of 598

or {2, 3, 5, 7, 11, 13, 17, 19, 23} | 598 ? Since answer is yes (13 | 598), 598 is not a prime

5.1 Prime Numbers

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(b)

Check to see if primes 2 to 23 is a divisor of 823

or {2, 3, 5, 7, 11, 13, 17, 19, 23} | 598 ? Since answer is no, 823 is a prime.

Example 14. Use Number Theorems to state whether the following aredivisible by: 2, 3, 4, 5, 6, 10 or 11: 275, 78, 840, 896.

2 3 4 5 6 8 9 10 11

275 Y Y

78 Y Y Y

840 Y Y Y Y Y Y

891 Y Y Y

5.1 Prime Numbers

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5.2 GCF and LCM

Example 15. Use both the Intersection of set and Prime Factorizationmethodsto find GCF (1421, 1827, 2523):

Intersection of set method: GCF (largest factor which divides both)

Prime Factorization method: GCF (lowest prime common to all)

1421 => 72 x 29

1827 => 32 x 7 x 29

2523 => 3 x 292 Common prime factor to all is 29

Example 16. Use the difference theorem to find:

(a) GCF (1847, 1421) (b) GCF(2523, 1827)

GCF(1847, 1427) = GCF(1827-1421, 1421) = GCF(1421, 406) = GCF(1015, 406)

GCF(1015, 406) = GCF(609, 406) = GCF(203, 203) = 203

GCF(1847, 1427) = 203

GCF(2523, 1827) = GCF(2523-1847, 1827) = GCF(1827, 696) = GCF(1131, 696)

GCF(1131, 696)=GCF(696, 435) = GCF(435, 261) = GCF(261, 174) = GCF(174, 87)

GCF(87, 87) = 87

GCF(1847, 1427) = 87

Example 17. Use the reminder theorem to find:

(a) GCF (2523, 1847)

GCF(2523, 1847) => 2523 / 1847 = 1 R 696

1847 / 696 = 2 R 435

5.2 GCF and LCM

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696 / 435 = 1 R 261

435 / 261 = 1 R 174

261 / 174 = 1 R 87

174 / 87 = 2 R 0

GCF(2523, 1847) = 87

Example 18. Use both the Intersection of set and Prime Factorizationmethods to find LCM (15, 35, 42, 80):

Intersection of set method: LCM (smallest multiple of all)

Where A = {15, 30, 45, ....,1680, 1695, ....}

B = {35, 70, 105, ...,1680, 1715, ...}

C = {42, 84, 126, .... 1680, 1722, ..}

D = {80, 160, 240, .. 1680, 1760, ..}

Prime Factorization method: LCM (highest exponent of primes in set)

15 => 3 x 5

35 => 5 x 7

42 => 2 x 3 x 7

80 => 24 x 5

LCM = 15 => 24 x 3 x 5 x 7 = 1680

Example 19. If GCF(2523, 1827) = 87, Find LCM (2523, 1827).

5.2 GCF and LCM

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(Theorem: GCF(a,b) x GCF(a, b) = ab)

So: GCF(2523, 1827) x LCM(2523, 1827) = 2523 x 1827 = 4609521

LCM (2523, 1827) = 4609521 / 87 = 52983

5.2 GCF and LCM

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6. Fractions (leave all solutions in fractional form)

Example 20. Show that the following fractions are equal:

(a) (b)

Answers: (a) 31 x 245 = 49 x 155 = 7595 (cross products are equal)

(b) 1 x 10 = 2 x 5 = 10

Example 21. Simplify fraction (a) (b)

Answers: (a) (divided by 2 then 7)

(b) (divided by: 2, and then 3)

Example 22. Arrange in order from smallest to largest:

(strategy, express all in terms of LCM)

LCM = 2 x 5 x 9 x 11 x 31 = 30690

So

6. Fractions

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Ordered from smallest to largest:

Example 23. Sum and simplify:

(a) (b)

Answers: (a) LCM is 120, So

(b) LCM is 240, So

Example 24. Compute and simplify:

Answers::

Example 25. Compute and simplify:

Answer:

Example 26. Solve for x: (strategy isolate x on one side of equation)

6. Fractions

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(a) (b) (c) (d)

Answers: (a)

(b)

(c )

(d)

So x = 12

Example 27. Find quotients for the following:

(a) (b) (c)

Answers: (a)

(b)

(c)

Example 28. If a rectangular shape represents a whole, then shade thefollowing regions::

6. Fractions

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(a) (b) (c) (d)

Answers: (a)

(b)

(c) So 6 shaded rectangles

(d)

6. Fractions

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7. Proportions (Decimals, ratio, rates, percent)

Example 29. Express in decimals:

(a) (b) (c) (d)

Answers: (a)

(b)

(c)

(d)

Example 30. Round the 3.14678238 to nearest:

(a) tenth (b) hundredth (c) thousandth (d) ten thousandth

Answers: (a) 3.1 (b) 3.15 (c) 3.147 (d) 3.1468

Example 31. Write the following sum in decimal form:

Answers:

Example 32. If 70% = 420 what is:

(a) 150% (b) 50% (c) 2 % (d) 75%

Answers: 1% = 420/70 = 6

7. Proportions

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(a) 150% = 150 x 6 = 900

(b) 50% = 50 x 6 =300

(c) 2 % = 2 x 6 = 12

(d) 75% = 75 x 6 = 450

Example 33. Write the following numbers in words:

(a) 4345678320 (b) 12345.678 (c) 123456789

Answers: (a) 4,345,678,320:

(four billion, three hundred & forty-five, six hundred & seventy-eight,three hundred & twenty)

(b) 12,345,678:

(twelve million, three hundred & forty-five, six hundred &seventy-eight)

(c) 123,456,789:

(one hundred & twenty-three thousand, four hundred & fifty-six, sevenhundred & eighty-nine)

Example 34: Use "<" or ">" to compare the following fractions:

(a) 7/12 and 8/16 (b) 29/50 and 49/100 (c) 25/52 and 50/102

Answers: (a)

(b)

(c)

7. Proportions

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Example 29. Write each number in scientific notation:

(a) 123 (b) 45671 (c) 21300 (d) 4345678320 (e) 10000000

Answers: (a) 1.23 x 102

(b) 4.5671 x 104

(c) 2.13 x 104

(d) 4.3456.. X 109

(e) 1.0 x 107

Example 35. Compute the following:

(a) 154.63 x 1.571 (b) divide 6000 by 1.7 (c) 154.63 + 1.571

Answers: (a) 242.92373

(b)

(c) 156.201

Example 36. Express the following as fractions:

(a) (b) 0.1818181818.. (c) 0.47121212... (d) 0.45315961596..

(a) (b) (c) (d)

Example 37. Is the decimal expansion of 143 / 8,345,415,131 terminating ornon terminating?How can you tell without computing the decimal expansion?

(nontermin: denominator not divisible by 2 or 5)

Answers: So non terminating

7. Proportions

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Example 38. Express each fraction as a decimal:

(a) 1 / 100 (b) 1 / 10,000 (c) 1 /1000000 (c) 0.01/ 100

Answers: (a) 0.01 (b) 0.0001 (c) 0.000001 (d) 0.0001

Example 39. If T is proportional to C and there are 10 T's when there are 12C's,How many C must there be when there are 69 T's?

Answers:

Example 40. It takes 1.5 dozen egg whites to bake a Super D Egg Nog;how many eggs is needed to make 12 Super D?

Answers: 1.5 doz. eggs or 1.5 x 12 = 18 eggs = 1 Super D

7. Proportions

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8. Integers:

Example 41. Solve the following: (a) 84 + (-17) + (-34) (b) -121 +(625 + 126)

Answer: (a) 84 + (-17) + (-34) = 84-17-34=84 - 51 = 33

(b) -121 + 751 = 630

Example 42. If a and b are integers, under what condition is the followingtrue?\(a) ( a -b ) - c = ( a - c ) - b

Answer: (a) (a - b) - c = a - b - c = a - c - b = a - b - c (always)

Example 43. If a is an element of {-4, -2, -1, 0, 1, 2, 3} and b is an element of{-4, -3, -1, 0, 1} find the smallest and largest values of:

(a) a + b (b) b - a (c) |a + b| (d) -(a + b)

Answer: (a) a + b: smallest => -4 + (-4)= -8, largest => 3 + 1 = 4

(b) b - a: smallest => -4 - (-1) = -5, largest => 3 - (-4) = 7

(c) |a + b|: smallest => |0+0| = 0, largest => |-4 +(-4)|= 8

(d) -(a+b): smallest => -(3+1) = -4, largest => -(-4+(-4))=8

Example 44. Find the integer that satisfies the equations (i.e. solve for x):

(a) 1 - x = 5 (b) 9 - x = -6 (c) x = -x (d) -7 - x = -5

Answer: (a) 1 - x = 5, 1 - 5 = x = -4

(b) 9 - x = -6, 9 + 6 = x = 15

(c) x = -x, x + x = 0 = 2x = 0, x = 0/2 = 0

(d) -7 - x = -5, -7 + 5 = x = -2

8. Integers:

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Example 45. Order the following integer from smallest to largest:

24, (-3)3 , (-2)5 , (-5)2, (-3)4, (-2)6

Answer: 24, (-3)3, (-2)5, (-5)2, (-3)4, (-2)6=> 16, -27, -32, 25, 81, 64

So: (-2)5, (-3)3 , 24, (-5)2, (-2)6, (-3)4

Example 46. Find the quotients for the following:

(a) -27 / 3 (b) -81 / (-9) (c) 125 / (-5) (d) (-15 + 15) / (-4)

Answer: (a) -27 / 3 = -9

(b) -81 / -9 = 9

(c) 125 / -5 = - 25

(d) 0 / -4 = 0

Example 47. Name the property of multiplication of integers that justify eachequation:

(a) (-3)(-4) = (-4)(-3) Answer (commutative)

(b) (-5)[2(-7)]=[(-5)(2)](-7)] Answer (associative)

(c) (-5)(-7) is an integer Answer (closure)

(d) (-8) x 1 = -8 Answer (identity)

(e) If (-3)n = (-3)7, then n = 7 Answer (cancellation)

8. Integers:

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Rational Numbers

Example 48. Compute the following and simplify:

(a) (b) (c)

Answers: (a) (b)

(c)

Example 49. Solve the following inequalities:

(a) (b) (c)

Answers: (a)

(b) (c)

Example 50. Illustrate the following properties of rational numbers with anexample:

(a) closure (b) commutativity (c) associatively (d) identity (e) inverse:

Answers: (a) closure:

(b) commutativity :

(c) associatively :

(d) identity :

Rational Numbers

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(e) additive inverse :

Rational Numbers

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