Mathematics. By Examples. By Courtney A. Pindling Department of Mathematics - SUNY New Paltz. First Edition, Summer 2001 FOR ELEMENTARY TEACHERS

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1 Cover: Math for teachers Mathematics FOR ELEMENTARY TEACHERS By Examples By Courtney A. Pindling Department of Mathematics - SUNY New Paltz First Edition, Summer 2001 file:///c /HP/Math/Math_Teachers/Resource/example/cover.html [05/29/ :35:21 AM]

2.2 Whole Numbers, and Numeration 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.

2 2.2 Whole Numbers, and Numeration 2.2 Whole Numbers, and Numeration Example 4. Write the Roman numeral for 1999 Answers: 1999 = = M + CM + XC + IX = MCMXCIX Example 5. Historic representation of numerals (state equivalent Hindu-Arabic numerals): Roman numerals: MCCCXLIV = M + CCC + XL +IV = = 1344 Egyptian (2563) Mayan ( ) Babylonian ( = ) file:///c /HP/Math/Math_Teachers/Resource/example/numeration.html [05/29/ :38:01 AM]

3 2.3 Whole Numbers (Base ten) 2.3 Whole Numbers (Base ten) Example 6. Write 31,407 in expanded form. Answers: = 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, one hundred 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 multibase pieces. (a) (c) (d) Answers: (a) = 1(5 2 ) + 3(5 1 ) + 4(5 0 ) = 1(25) + 3(5) + 4(1) = 3(10 2 ) +2(10 1 ) + 1(10 0 ) = 3(100) + 2(10) + 1 (1) (c) = 1(2 3 ) +1(2 2 ) + 0(2 1 ) + 1(2 0 ) = 1(8) + 1(4) +0(2) +1 (d) = 6(8 2 ) +1(8 1 ) + 3(8 0 ) 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) (c) Answers: (a) = 1(8 2 ) + 3(8 1 ) + 4(8 0 ) = 1(64) + 3(8) + 4(1) = = 2(4 4 ) + 3(4 3 ) + 0(4 2 ) + 3(4) + 2 = = 718 file:///c /HP/Math/Math_Teachers/Resource/example/number_theory.html (1 of 2) [05/29/ :38:36 AM]

4 2.3 Whole Numbers (Base ten) (c) =1(2 5 ) + 1(2 4 ) (2 2 ) + 1(2) + 0 = = 54 Example 11. Convert the following base ten to requested bases: (a) 613 = base to base 20 Answers: (a) = Bases 8 4 = = = = = 1 Answer/Sum 8 0 1R101 1R37 4R x512 1x64 4x8 5x1 613 =512 =64 =32 =5 Answers: (a) = Bases 20 4 = 160, = 8, = = = 1 Answer/Sum R R50 2R x x400 2x20 10x =16000 =7200 =40 =10 file:///c /HP/Math/Math_Teachers/Resource/example/number_theory.html (2 of 2) [05/29/ :38:36 AM]

5 5.1 Prime Numbers 5.1 Prime Numbers Example 12. Express the following as product of primes: (a) Answers: (a) 2268 = 2 2 x 3 4 x = 2 2 x 3 x 5 x 7 Divide by successive primes from 2 on: (a) 2268 = 2 2 x 3 4 x 7 Factor / Divisor Dividend Quotient Remainder Divide by successive primes from 2 on: (a) 420 = 2 2 x 3 x 5 x 7 Factor / Divisor Dividend Quotient Remainder Example 13. Are the numbers (a) 598 and 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 file:///c /HP/Math/Math_Teachers/Resource/example/primes.html (1 of 2) [05/29/ :39:25 AM]

6 5.1 Prime Numbers 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 are divisible by: 2, 3, 4, 5, 6, 10 or 11: 275, 78, 840, Y Y 78 Y Y Y 840 Y Y Y Y Y Y 891 Y Y Y file:///c /HP/Math/Math_Teachers/Resource/example/primes.html (2 of 2) [05/29/ :39:25 AM]

7 5.2 GCF and LCM 5.2 GCF and LCM Example 15. Use both the Intersection of set and Prime Factorization methods to 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 => 7 2 x => 3 2 x 7 x => 3 x 29 2 Common prime factor to all is 29 Example 16. Use the difference theorem to find: (a) GCF (1847, 1421) GCF(2523, 1827) GCF(1847, 1427) = GCF( , 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( , 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 = 2 R 435 file:///c /HP/Math/Math_Teachers/Resource/example/gcf_lcm.html (1 of 3) [05/29/ :39:56 AM]

8 5.2 GCF and LCM 696 / 435 = 1 R / 261 = 1 R / 174 = 1 R / 87 = 2 R 0 GCF(2523, 1847) = 87 Example 18. Use both the Intersection of set and Prime Factorization methods 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, , 1722,..} D = {80, 160, 240, , 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 => 2 4 x 5 LCM = 15 => 2 4 x 3 x 5 x 7 = 1680 Example 19. If GCF(2523, 1827) = 87, Find LCM (2523, 1827). file:///c /HP/Math/Math_Teachers/Resource/example/gcf_lcm.html (2 of 3) [05/29/ :39:56 AM]

9 5.2 GCF and LCM (Theorem: GCF(a,b) x GCF(a, b) = ab) So: GCF(2523, 1827) x LCM(2523, 1827) = 2523 x 1827 = LCM (2523, 1827) = / 87 = file:///c /HP/Math/Math_Teachers/Resource/example/gcf_lcm.html (3 of 3) [05/29/ :39:56 AM]

10 6. Fractions 6. Fractions (leave all solutions in fractional form) Example 20. Show that the following fractions are equal: (a) Answers: (a) 31 x 245 = 49 x 155 = 7595 (cross products are equal) 1 x 10 = 2 x 5 = 10 Example 21. Simplify fraction (a) Answers: (a) (divided by 2 then 7) (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 = So file:///c /HP/Math/Math_Teachers/Resource/example/fraction.html (1 of 4) [05/29/ :40:36 AM]

11 6. Fractions Ordered from smallest to largest: Example 23. Sum and simplify: (a) Answers: (a) LCM is 120, So 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) file:///c /HP/Math/Math_Teachers/Resource/example/fraction.html (2 of 4) [05/29/ :40:36 AM]

12 6. Fractions (a) (c) (d) Answers: (a) (c ) (d) So x = 12 Example 27. Find quotients for the following: (a) (c) Answers: (a) (c) Example 28. If a rectangular shape represents a whole, then shade the following regions:: file:///c /HP/Math/Math_Teachers/Resource/example/fraction.html (3 of 4) [05/29/ :40:36 AM]

13 6. Fractions (a) (c) (d) Answers: (a) (c) So 6 shaded rectangles (d) file:///c /HP/Math/Math_Teachers/Resource/example/fraction.html (4 of 4) [05/29/ :40:36 AM]

14 7. Proportions 7. Proportions (Decimals, ratio, rates, percent) Example 29. Express in decimals: (a) (c) (d) Answers: (a) (c) (d) Example 30. Round the to nearest: (a) tenth hundredth (c) thousandth (d) ten thousandth Answers: (a) (c) (d) Example 31. Write the following sum in decimal form: Answers: Example 32. If 70% = 420 what is: (a) 150% 50% (c) 2 % (d) 75% Answers: 1% = 420/70 = 6 file:///c /HP/Math/Math_Teachers/Resource/example/proportion.html (1 of 4) [05/29/ :41:50 AM]

15 7. Proportions (a) 150% = 150 x 6 = % = 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) (c) Answers: (a) 4,345,678,320: (four billion, three hundred & forty-five, six hundred & seventy-eight, three hundred & twenty) 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, seven hundred & eighty-nine) Example 34: Use "<" or ">" to compare the following fractions: (a) 7/12 and 8/16 29/50 and 49/100 (c) 25/52 and 50/102 Answers: (a) (c) file:///c /HP/Math/Math_Teachers/Resource/example/proportion.html (2 of 4) [05/29/ :41:50 AM]

16 7. Proportions Example 29. Write each number in scientific notation: (a) (c) (d) (e) Answers: (a) 1.23 x x 10 4 (c) 2.13 x 10 4 (d) X 10 9 (e) 1.0 x 10 7 Example 35. Compute the following: (a) x divide 6000 by 1.7 (c) Answers: (a) (c) Example 36. Express the following as fractions: (a) (c) (d) (a) (c) (d) Example 37. Is the decimal expansion of 143 / 8,345,415,131 terminating or non terminating? How can you tell without computing the decimal expansion? (nontermin: denominator not divisible by 2 or 5) Answers: So non terminating file:///c /HP/Math/Math_Teachers/Resource/example/proportion.html (3 of 4) [05/29/ :41:50 AM]

17 7. Proportions Example 38. Express each fraction as a decimal: (a) 1 / / 10,000 (c) 1 / (c) 0.01/ 100 Answers: (a) (c) (d) Example 39. If T is proportional to C and there are 10 T's when there are 12 C'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 file:///c /HP/Math/Math_Teachers/Resource/example/proportion.html (4 of 4) [05/29/ :41:50 AM]

18 8. Integers: 8. Integers: Example 41. Solve the following: (a) 84 + (-17) + (-34) ( ) Answer: (a) 84 + (-17) + (-34) = =84-51 = = 630 Example 42. If a and b are integers, under what condition is the following true? \(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 - a (c) a + b (d) -(a + b) Answer: (a) a + b: smallest => -4 + (-4)= -8, largest => = 4 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 = x = -6 (c) x = -x (d) -7 - x = -5 Answer: (a) 1 - x = 5, 1-5 = x = x = -6, = x = 15 (c) x = -x, x + x = 0 = 2x = 0, x = 0/2 = 0 (d) -7 - x = -5, = x = -2 file:///c /HP/Math/Math_Teachers/Resource/example/integers.html (1 of 2) [05/29/ :41:14 AM]

19 8. Integers: Example 45. Order the following integer from smallest to largest: 2 4, (-3) 3, (-2) 5, (-5) 2, (-3) 4, (-2) 6 Answer: 2 4, (-3) 3, (-2) 5, (-5) 2, (-3) 4, (-2) 6 => 16, -27, -32, 25, 81, 64 So: (-2) 5, (-3) 3, 2 4, (-5) 2, (-2) 6, (-3) 4 Example 46. Find the quotients for the following: (a) -27 / 3-81 / (-9) (c) 125 / (-5) (d) ( ) / (-4) Answer: (a) -27 / 3 = / -9 = 9 (c) 125 / -5 = - 25 (d) 0 / -4 = 0 Example 47. Name the property of multiplication of integers that justify each equation: (a) (-3)(-4) = (-4)(-3) Answer (commutative) (-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) file:///c /HP/Math/Math_Teachers/Resource/example/integers.html (2 of 2) [05/29/ :41:14 AM]

20 Rational Numbers Rational Numbers Example 48. Compute the following and simplify: (a) (c) Answers: (a) (c) Example 49. Solve the following inequalities: (a) (c) Answers: (a) (c) Example 50. Illustrate the following properties of rational numbers with an example: (a) closure commutativity (c) associatively (d) identity (e) inverse: Answers: (a) closure: commutativity : (c) associatively : (d) identity : file:///c /HP/Math/Math_Teachers/Resource/example/rationals.html (1 of 2) [05/29/ :42:39 AM]

21 Rational Numbers (e) additive inverse : file:///c /HP/Math/Math_Teachers/Resource/example/rationals.html (2 of 2) [05/29/ :42:39 AM]

Decimal Addition: Remember to line up the decimals before adding. Bring the decimal straight down in your answer.

Summer Packet th into 6 th grade Name Addition Find the sum of the two numbers in each problem. Show all work.. 62 2. 20. 726 + + 2 + 26 + 6 6 Decimal Addition: Remember to line up the decimals before