Story_of_Number_Jan_2011

8/7/2019 Story_of_Number_Jan_2011

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Story of Number, January 2011 2

FundamentalsProblem 1. In each of the following, nd the missing values:

1. 4 feet = .... inches

2. 60 inches = ..... feet

3. 2 ft. 3 in. = .... in.

4. 42 in. = ....ft.

5. 4.2 kg = ....g

6. 1970 mg = ....g

7. 15g 5 mg = ....g

8. 6804 m = ....km

9. 3.42 L = ...ml

10. 1 L 127 ml = ....L

Problem 2. Add the following (Hint: arrange columnwise before computing an-swer)

1. 5 lb. 7 oz. + 2 lb. 9 oz.

2. 3 hr. 45 min. + 2 hr. 30 min.

3. 2 ft. 9 in. + 2 ft. 8 in.

4. 3 yd. 6 in. + 2 yd. 11 in. + 4 yd. 9 in.

5. 12 min. 15 sec. + 15 min. 9 sec. + 26 min. 12 sec.

6. 5 yd. 2 ft. 10 in. + 3 yd. 1 ft. + 4 yd. 2 ft. 9 in.

Problem 3. Subtract the following (Recall hint in previous problem)

1. 6 qt. 1 pt. - 5 qt. 1 pt.

2. 8 gal. 2 qt. - 5 gal. 3 qt.


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3. 4 hr. 30 min. 15 sec. - 2 hr. 40 min. 30 sec.

4. 23 hr. 30 min. - 15 hr. 46 min.

5. 14 ft. 10 in. - 9 ft. 11 in.

Problem 4. Multiply

1. 6 ft. 5 in. by 4

2. 7 hr. 15 min. by 6

3. 3 gal. 3 qt. 1 pt. by 3

4. 2 yd. 8 in. by 7

5. 10 lb. 6 oz. by 3

6. 2 hr. 5 min. 30 sec. by 4

Problem 5. Divide

1. 12 hr. 30 min. by 5

2. 8 ft. 9 in. by 3

3. 4 qt. 1 pt. by 3

4. 7 yd. 24 in. by 4

5. 35 min. 12 sec. by 6

6. 6 hr. 30 min. 20 sec. by 5


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Self Exploration of Number picture to be inserted.


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Figure 1: Number, Arithmetic-Algebra Concept Map

Story of Number : Ratio and ProportionProblem 6. There are 24 M&Ms in a packet. You share them with your sister in the ratio 5: 3 (ve to three). How many will you get and how many will your sister?

Problem 7. You and your brother share a bundle of money in the ratio 3:4. Your brother is supposed to get $ 9,000. How much do you get?

Problem 8. Ali, Bala and Paul are aged 14, 16 and 18 years respectively. They get shares of money in the ratio of their ages. Given that Ali receives $ 1,050, calculatehow much each of the other two receive and the total amount shared


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Problem 9. Draw a number line, labelling the unit, and the numbers 2, 3, , 12. Indicate the following ratios,

(a) 1:1:1:1:1:1:1:1:1:1:1:1

(b) 8:4

(c) 6:6

(d) 1:11

(e) 4:4:4

(f) 4:2:1:5

(g) 5:7

Problem 10. A number is twice the second number. What is the ratio of the twonumbers to each other?

Problem 11. A number is a third of a second number. What is the ratio of the twonumbers to each other?

Number Types, name and pattern.

1. counting numbers 1, 2, 3, 2. 03. unit is the scale that is used to measure the distance between the numbers 0

and 1 when the two are placed on the number line

4. even numbers 2, 4, 6, 5. odd numbers 1, 3, 5, 6. prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 7. integers , 3, 2, 1, 0, 1, 2, 3 8. composite numbers


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The PRIME NUMBER PYRAMID below is constructed for help with multiplica-tion tables. One starts by enumerating 2, 3 etc at the bottom of the page. Whencontinuing with each successive counting number, one checks if it is a multiple of the numbers listed before and then inserts the new number into all columns of thepreceding number in the lowermost row of which it is a multiple. The pyramidcan be created by students upto 100 or so.

INSERT PYRAMID

Divisibility Tests

(a) Every even number is a multiple of 2

(b) Every number whose digts add to a multiple of 3 is a multiple of 3

(c) Every number that has either 0 or 5 in the units place is a multiple of 5

Fact : Every integer greater than 1 can be expressed as a product of primes (withperhaps only one factor).

Problem 12. Write each number as a product of its prime factors: 26, 64, 315,90, 32

Problem 13. Reduce the following to the simplest ratios by writing the prime factorization of each number of the ratio

(a) 42:84

(b) 33:55

(c) 39:51

(d) 33:57

(e) 44:66

(f) 80:180

(g) 36:60

(h) 210:270


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FRACTIONS GRID

0

1

0

1

12

0

1

13

23

0

1

14

24

34

0

1

15

25

35

45

0

1

16

26

36

46

56

0

1

17

27

37

47

57

67

0

1

18

28

38

48

58

68

7

8

0

1

19

29

39

49

59

69

79

89

0

1

110

210

310

410

510

610

710

810

910

0

1

111

211

311

411

511

611

711

811

911

1011

0

1

112

212

312

412

512

612

712

812

912

1012

1112

0

1

113

213

313

4

13

513

613

713

813

913

1013

1113

1213

0

1

114

214

314

414

514

614

714

814

914

1014

1114

1214

1314

0

1

115

215

315

415

515

615

715

815

915

1015

1115

1215

1315

1415

0

1

116

216

316

416

5

16

616

716

816

916

1016

1116

1216

1316

14

16

1516

0

1

117

217

317

417

517

617

717

817

917

1017

1117

1217

1317

1417

15

17

1617

0

1

118

218

318

418

518

618

718

818

918

1018

1118

1218

1318

1418

1518

1618

1718

0

1

119

219

319

419

519

619

719

819

919

1019

1119

1219

1319

1419

1519

1619

1719

1819


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Problem 16. The construction of the Fraction Grid can be seen as an applicationof the ratio.

(a) Note that the rst line segment is not divided at all, the second is divided into two equal pieces, the third into three equal pieces and so forth. If we

were to add one piece from the second last segment and one from the last linesegment, would it be easy to indicate the answer?

(b) What do you observe about the size of the subunits as one moves from left toright?

(c) What is the ratio of the subunit of the second to the subunit of the fourth linesegment?

(d) What about the ratio of the subunit of the second line segment to the subunit

of the fourth, eighth and sixteenth line segment?

Thanks to Peter Yom of Bronx CC for providing the beginnings of the Texcode to create FractionsGrid, as well as the pictures that appear in the contextof similarity of triangles, area and perimeter.

Problem 17. Divide the line segment on the grid below in the ratio 1:1

0 16

Problem 18. Divide the line segment on the grid below in the ratio 1:1:1

0 15

1. How many equal parts are there?

2. How would you divide the same line segment in the ratio 2:1?


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0 15

Problem 20. Divide the line segment on the grid below in the ratio 1:1:1:1

0 16

1. How many equal parts are there?2. How would you divide the same line segment in the ratio 2:2?

3. How would you divide the same line segment in the ratio 1:3?


0 16


0 16


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Problem 23. Divide the line segment on the grid below in the ratio 1:2:5

0 16

Denition : A proportion is an equality of ratios.

Direct and inverse proportion.

Problem 24. A 6-foot tall man casts a 4-foot shadow at the same time and placeas a tree casts a 20-foot shadow. How tall is the tree?

Problem 25. 7 builders complete 10 houses in a month. How many houses could 35 builders complete in the same time?

Problem 26. The ratio of a mans weight on Mars compared with his weight onearth is 2 to 5. How much would a 180-pound man weigh on Mars? What would your weight be on Mars?

Problem 27. A market is selling three cans of beets for 99 cents. How much will12 cans cost at the same rate?

Problem 28. On a map, 12 inch represents 12 miles. How many miles would 214

inches represent?

Problem 29. Five workers, each working 8 hours per day, can dig a rectangular trench 600 m long, 0.9 m wide and 1 m deep in 3 days. Find the number of workersneeded to dig a trench 1.2 km long, 1.1 m wide, and 1.8 m deep in 3 days. Calculatethe labor cost given that each worker is paid $ 5 per hour. (Assume all the workersworked 8 hours per day for 3 days).

Problem 30. The cost of gas for a 240-km journey for a car which runs 12 km on

each liter of gas is $ 24. What would be the cost of gas for a 500-km journey for avan which runs 11 km on each liter of gas?

Problem 31. Twelve women working 7 hours a day can nish a piece of work in 8days. How many hours a day must 16 women work in order to nish the job in 14days?


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Problem 32. Is there a connection of the equation: 7128 = 84 8 = 672 =x 16 14 with any problem above?

Similarity

Similar triangles are those with the same shape but not necessarily the same size.Corresponding angles of similar triangles are equal, i.e. have the same measure.Corresponding sides of similar triangles are proportional.

Problem 33. Given ABC

RST . Find x and y.

A

BC

5 4

x

R

S

y 8

10

Problem 34. Triangle ABC is similar to triangle ADE . Find the lengths of thesides from the given lengths by using proportionality of sides of similar triangles.

A

B

C 8

x

D

E

9

6

Problem 35. In the given triangle, divide the base into half to create two similar triangles. Find the lengths of the sides of the new triangle.

x

6

4

Problem 36. Given AB CD , nd DE .


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t t

t t

t t

t

t t

t t

A 12E D

8

B

5

C

Average, Perimeter and AreaProblem 37. Find the perimeter and area of this gure:

% $

46

6

Problem 38. Find the perimeter of the shape.

12 cm

4 cm

5 cm

7 cm

Problem 39. Find the perimeter and area of this gure:

4

9

6

Problem 40. Find the average of the following numbers....


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Problem 41. Given the average of 5 numbers and given 4 of the numbers how dowe nd the fth number?


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Story of Number : Fraction

Fraction Denition A fraction (or rational number) is a number that can be writ-ten as ab , where a and b are integers and b is not zero. Thus, we have

(i) a pair of integers a and b, or written as ab and

(ii) a Whole.

The number b informs about the total number of equal parts the Whole is dividedinto; the number a informs about the number of those equal parts under consider-ation. b is the denominator and a is the numerator of the fraction ab .

Problem 42. Use the denition of fraction above to determine if the numbers are

rational numbers:34 , 5, 0, 6

Problem 43. (a) Note the dark line segment in the gure below indicates the sizeof the fraction 88 . The division mark and label

88 indicate the position of the

fraction 88 . What is another number representation of 88 ?

(b) In what ratio does the 88 divide the line segment of length 2 units?

(c) Locate the position of the following fractions 18 , 138 ,

118 ,

138 in the gure below.

3

0

88

Problem 44. In each of the grids below, locate the following numbers:


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(a) 125 ,75 ,

125

(b) 234 , 114 ,

114

Problem 45. Estimate the position of the following fractions 119 , 123 , 13 , 43 on thesame number line in the grid below:

0

99

2


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Problem 46. Estimate the position of the following fractions on the grid below:16 ,

23 ,

126 ,

123 .

Problem 47. A number line is given below with 75 indicated. Find where is 1 ,1

10 ,710 , 1910 , 1 310 . Extend the line and use the approprite labeling if needed.

E'0 7

5

Note: The problems below are meant to be done at the beginning of the explorationwith the FG while investigating patterns with it.

Problem 48. Locate the length determined by the following numbers on the Frac-tionsGrid:

(a) 12

(b) 13

(c) 14

(d) 34

(e) 217


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Arrange the above numbers in increasing order.

Problem 49. Find a collection of fractions from the FractionsGrid that have thesame size as the ones indicated below: (Hint: Find the equivalent fractions for theones given below)

(a) 12

(b) 14

(c) 13

(d) 34

(e) 23

In each case what operation is needed to transform the given fraction into anequivalent one you found? The fractions are called equivalent because they rep-resent the same length or size written in different fractional units.

Problem 50. On the FractionsGrid, nd all fractions equivalent to 1015

Problem 51. Which of the following pairs of fractions are equivalent, i.e., of thesame size?

(a)46 and

1015

(b) 2128 and 1216

(c) 2048 and 3584

(d) 516 and 34

111

Problem 52. Which of the two fractions is smaller, and why? Hint: Use the Frac-tionsGrid, if needed.

(a) 410 ,9

10

(b) 116 ,14

(c) 512 ,13


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Problem 53. Complete the following, by observing the patterns in the FG if needed:

(a) 35 =?

20

(b) 30100 =3?

(c)4

13 =?

169

(d) 250750 =1?

(e) 79 =105

?

(f) 179 =?

99

Problem 54. Demonstrate by geometry the equality below:

(a)1

2 =3

6

(b) 13 =39

(c) 25 =6

15

Problem 55. Which of the three fractions is largest?

(a) 17 ,67 ,

47

(b) 12, 1

4, 1

3

(c) 34 ,78 ,

316

(d) 415 ,23 ,

59

Problem 56. Arrange the following fractions in ascending order:

(a) 1112 ,58 ,

34

(b) 23, 4

9, 5

6

(c) 13 ,47 ,

12

(d) 711 ,56 ,

23

Problem 57. Arrange the following in descending order:


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(a) 34 ,56 ,

59 ,

712

(b) 34 ,45 ,

710 ,

1112

(c) 23 ,5

12 ,12 ,

58

(d) 79 ,56 ,

1318 ,

23

Problem 58. Estimate the position of 34 , 214 ,

104 on the interval [0,3] of the number

line, shown below. Note that 2 and 3 are indicated on the number line. Describethe process through which you estimated the positions of fractions and mixed num-bers.


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0

2

Problem 59. A number line is given below with 75 indicated. Find where is 1 ,15 .

E'0 7

5

Problem 60. On a well-drawn number line, position the following:

(a) 14 ,24 ,

34 ,

44 , 1,

12 . Explain what you understand about the number

34 . What is

its relationship with the numbers 3 and 4 geometrically? Make sure you payattention to the unit, and the spacing of the numbers on the number line withrespect to the size of the unit.

(b) 13 ,33 ,

63 , 2,

93 ,

123 , 4

(c) 13 ,16 ,

112

Problem 61. Draw a diagram to show that

(a) 13

= 39

(b) 25 =6

15

Problem 62. Explain in a sentence or two, how one determines whether a fractionis smaller than another fraction. Think about explaining to third graders.


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00 0

1 1 1

14 1

5

25

114 25 =

Problem 69. Compute 234 using the grids below. First explain clearly to yourself

the meaning of all the components of the diagram below. In particular, make surethat you pay attention to the operation indicated in the problem.

0

1

0

1

0

14

24

34

0

14

24

34

1

Problem 70. Compute 314 125 using the grids below. Set up the problem correctly

and check that you have set it up right before proceeding? Write the process in your own words.


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0

1

0

1

0

1

0

14

24

34

1

0

1

0

15

25

35

45

1

Steps

(a) 14 >15 , but

14 25

(c) 14

and 15

are two different sized pieces. We need a common measuring piece.Which grid would be useful to use as a measuring device to measure both 14and 15 .

(d) That is, which grid is useful to us that has a tick mark at the same height/levelas 14 and also as

25 ?

(e) The rst such grid is the twentieths grid.

(f) 14 =5

20 and15 =

420 ,

25 =

820 .

(g) 114 = 54 = 2520

(h) 114 25 =

2520

820 =

1720

(i) The given problem was

314 125


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=

214 25

=

12520 8

20

=11720

Problem 71. Compute 2 58 using the grids below.

0

1

0

1

0

18

28

38

48

58

6

8

78

1


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Problem 72. Compute 334 245

Problem 73. Compute 3 5

12

Problem 74. (a) How many fourths are in 12 ?

(b) How many eighths are in 12 ?

(c) How many sixths are in 13 ?

(d) How many sixths are in 23 ?

(e) How many eights are in 34 ?

(f) How many tenths are in 25 ?

Writing Exercise 1. (a) What is the sequence of thinking steps you have to carryout to determine how many sixths are there in 124 ?

(b) What are the thinking steps you have to carry out to determine how manysixths are in the fraction a6b?

Problem 75. Find the least common multiples of the following:

(a) 4 and 5

(b) 3 and 4

(c) 6 and 4

Problem 76. How many fths are there in 34 ?

Problem 77. Label the following

0 0

1 1


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Problem 83. Compute 214 + 113

Denition The fraction ba is the reciprocal of the fractionab . Reciprocal fractions

always fulll ab ba = 1Example: 52 is the reciprocal of

25 and

52

25 = 1

Problem 84. (a) What is the reciprocal of 34 ?

(b) What is the reciprocal of 212 ?

(c) Is 35 a reciprocal of 56 ? Explain.

Problem 85. (a) What is one-fourth of 2?

(b) What is three-fourth of 8?

(c) What is 43 rd of 8?

(d) What is 43 rd of 34 of 8?

Problem 86. (a) What is one-third of 14 ?

(b) What is 16 of 23 ?

(c) What is 115 of 334 ?

Problem 87. Critical Thinking Exercise: Beginnings of Division of Fractions

(a) How many or what part of a halves/f are/is in 14 ? How would you express thisas a division?

(b) How many halves are in 18 ?

(c) How many thirds are in 16 ?

Note: By the above exercises we see that

(i) 12 (12 ) =

14

(ii) 14 (12 ) =

18

(iii) 12 (16 ) =

13


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Question 1. What is the whole? What fraction of the whole are we interested in? What is the relationship of the whole across different representations? Howdoes the fraction of the whole appear in the different representations? How caninformation be translated across different representations?

We consider 4 representations:

(1) Part of a whole (1) represented via Fractions

(2) Part of a whole (1) represented by Decimals

(3) Part of a whole (100% ) represented by Percents

(4) Part of a whole ( 360) represented by a Pie Graph or Circle graph

To easily compare across representations, each of the above 4 is shown on a verti-cal line. Note a circle has a total of 360.


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FRACTION DECIMAL PERCENT CIRCLE MEASURE

0

15

25

35

45

1

0

0.2

0.4

0.6

0.8

1

0

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0

72

144

216

288

360


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Decimal Fractional notation Common fraction

5.3 5 + 310 53

10 or53

100

0.02 0 + 010 +2

1002

100

2.0103 2 + 010 +1

100 +0

1000 +3

10 ,000 2103

10 ,000 OR20 ,10310 ,000

3.6 (3 +6

10 ) 36

10 OR 3610

Complete the following table:

Percent Fraction

100%

25%

50%

75%

3313 %

6623 %

10%

1%


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Problem 97. Complete the following table:

Decimal Percent Fraction

99.44%

658

36.5%

34%

1.27

0.21

130

25

One-step problems

Problem 98. What number is 88% of 1000?

Problem 99. What percent of 436 is 87.2?

Problem 100. Which number is 35% of 126?

Two-step problems

Problem 101. The cost of a birthday celebration at McDonalds is $ 40.50. A 15%tax is added to the cost. What is the total cost?

Problem 102. By decreasing the length of the side by one unit, the area of asquare decreased from 100 m2 to 81m2 . Find the percent decrease in the area.(Think what is the base here, what is the amount in the formula.)

Problem 103. Find the original price of a pair of shoes if the sales price is $ 78after a 25% discount.

Problem 104. Given numbers in fractional, decimal form, arrange in increasingorder


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Story of Number in Abstract

Solving Equations

Solve the following equations

(a) x + 5 = 8

(b) x 7 = 2(c) 2x = 8

(d) x3 = 4

(e) x6 =

4

(f) 3.04 = x 2.96(g) 25 = 8x 15(h) 9x + 2 = 1(i) 5 8x = 5(j) 0.3x 1.8 = 5.04(k) 3x 4 = 8 x(l) 7 4x = 4x 9

(m) 3x + 15 6x 10 = 9x 6 3x + 5The square of a number is the product of the number with itself.

Problem 105. Fnd the square of each of the following numbers: 1, 2, 3, 4, 0, -1,

-2 Arrange in the table:

Number Square of Number


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The square root of a number x is that number which when multiplied by itself gives us the number x whose square root we want. The square root of the numberx is denoted symbolically as xProblem 106. Find the square root of 0, 4, 9, 16, 25 and arrange in the table

Number Square root of Number

Problem 107. Evaluate if possible:

1. 642. 1003. 25 + 494. 16 + 81

To nd the square root of primes is not possible using integers. We approximatethe square root to a degree of accuracy.

Problem 108. How would you estimate the square root of 2?

The cube of a number is the product of the number with itself three times. Thecube of x is x x x and is denoted by x 3 .Problem 109. Evaluate if possible:

1. 3642. 310003. 3125 + 3644. 38 + 327


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Rules of Exponents

1. a n a m = a n + m2. a n a m = a nm

3. (an

)m

= anm

4. a 0 = 1

5. a1n = na

6. an = 1a nProblem 110. Evaluate if possible:

1.3

64 + 42

33

+ 1442. 31000 + 4625 36 + 22 33 + 4 23. 3125 464 + 3644.

323 + 3335.

326 + 339

Note The nth power of a number x is denoted by xn . The power n is also calledas the exponent and x is referred to as the base.

Notation x1n is the same as nx . Thus 4 12 = 4 = 2, or 9 12 = 9 = 3.

Problem 111. Evaluate if possible

(a) 64(b)

19

(c) 38(d) 7128(e)

81


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(f) 481(g) 664(h)

585(i)

696(j) 3 164(k) 3 827(l)

4x 8(m)

4a 8 b4(n) 8

y40

(o) 9 y36(p) 36x 8 y4(q) 49x 2 y8(r)

38a 6 b15(s) 3125x 30

Problem 112. Write each of the following in simplest form using only positiveexponents:

1. ( 12 )3 (

12 )

7

2. ( 12 )9 (

12 )

6

3. ( 23 )5 (

49 )

2

4. (3

5 )7

(3

5 )7

5. ( 35 )7 (53 )

4

6. [(56 )7 ]3

7. 327


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8. 3a 3a 2

Problem 113. Combine like terms

(a) 8x 3 + 4 x 3

(b) 2x2

+ 4 x2

(c) 5x 2x + 4(d) 14x 3 8x 3 + x(e) 10x 2 21x 2 + x + 1(f) 4x 5 3x 2 + 4 x 2 + 1

(g) 4x 5

5x 2 + 5 x

10

(h) 4 3x 2 + 5 x 10(i) 4(x 2 5) + 5( x 2 + 1)(j) 2(x 2 + x) 3(x x 2 )

Addition and subtraction of polynomials

Multiplication and division of polynomials

Problem 114. Find the indicated products

(a) 5x 2 multiplied with 3x

(b) (x 5)(x + 5)(c) (x + 4)( x 4)(d) (2x + 1)(2 x

1)

(e) (ax + by)(ax by)(f) (2x + x 2 + x 3 )(1 x)


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Recall prime factorization of numbers

6 = 2 3Prime factorization of polynomials In each of the following where do you seethe use of the prime factorization?

(a) x 2 5x + 6 = ( x 3)(x 2)(b) x 2 a 2 = ( x a )(x + a)(c) x 2 + 4 x + 4 = ( x + 2) 2

(d)

Algebraic Fractions

Problem 115. (a) 1x+5 +3

x+5

(b) 8x+32x1 3(c) a + 3 + 23a5(d) a 2 +

32 a +1

(e) 2x+5 +1

x2

(f) y5

y+2 2

y4 y1Problem 116. Simplify

(a) 21 x3 y

14 x 2 y2

(b) 4 x2

6x 2 8 x(c) 2y

2

y159y2

Meaning of 1 as 2 x12 x1Problem 117. Solve the equations if possible and check your solution.

(a) 2x +3

2x =76


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(b) 53x + 2 =1x

(c) 1x 3 =4x

(d) 4x2 = 1

x+3

(e) 5xx 2

1 +7

x+1 =6x

(f) 12x1 +2

x5 = 22

2 x 2 11 x+5Problem 118. Solve each equation if possible and check your solution:

(a) x 2 = 7(b) 2x + 5 = 5(c) x = 6x + 7(d) 7x + 8 = x(e) y y 3 = 5(f) 2y 4 + 2 = y

(g) y + 1 1 = y(h) x 5 = 2 + x + 3(i) 2x + 9 x + 1 = 2

Recall as many uses of zero as you can. Then answer the following:

1. 0 is called the additive identity because 0 + a = a, for any number a.Explain what is the meaning of additive identity.

2. -a is called the additive inverse of a because a + (-a) = 0. Explain thisstatement. What is the additive inverse of -3? of 5? of 0?

Instead of using particular numbers, we will let a,b,c be any real numbers.The rules below are true for all real numbers.

Rules for Addition


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1. a + b = b + a Addition is commutative

2. (a + b) + c = a + ( b + c) Addition is associative

3. a + 0 = a = 0 + a 0 is the additive identity

4. a + (

a) = 0 =

a + a Additive inverse of a is

a

Be careful. Students make mistakes with the order of operations. The conven-tion is that multiplication and division dominates addition and subtraction. If there are parenthesis they acquire the highest priority.

1. 5 + 10 3 =2. +912 + 5 =

3. 7 + 4 + 5 3 =4. 4(+3) =5. +7( 2) =6. 5(8) =7. +6(+9) =

8.

7(+4) + 6 =

9. +6(+8)( 3) =

Rules of Multiplication

1. a b = b a Multiplication is commutative2. (a b) c = a (b c) Multiplication is associative

3. a 1 = a = 1 a 1 is the multiplicative identity4. a 1a = 1 =

1a a Multiplicative inverse of a is

1a . Here a is a non-zero

real number.

1. +65 .1789.03 =


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2. 6x(1.2)=

Distributivity of multiplication over addition

1. a (b + c) = a b + a c2. (b + c) a = b a + c a

Simplify:

1. 2(3x + y)

2. (1 8x) 4

Operations on fractions

Here b and d are non-zero real numbers.

1. ab +cd =

ad + bcbd Addition of fractions.

2. ab cd = adbcbd Subtraction of fractions.3. ab cd = acbd Multiplication of fractions.4.

ab

cd =

adbc Division of fractions.

Difference of two squares

a 2 b2 = ( a b)(a + b)Problem 119. Factor

(1) x2

y2

(2) 4x 2 y2(3) 9x 2 25y2(4) 1 x 2


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(5) x 2 1(6) 16 81y2(7) 2 x 2(8) 14

y2

(9) x y(10) 49 x

2 2516 y

2

Graphing by Discovery

Note that the square of a number means the number multiplied by itself.

Complete the table below.

Number Square of Number

Note that the square root of a number A means the number which when multiplied by itself will give the number A.

Complete the table below.

Number Square Root of Number

This is a conditional statement: If A is true then B. This means we must rst check whether A is true. If our answer is afrmative then we can assume that B is true.

Use this concept when completing the next table.Number Number, if Number is non-negative and negative of number if number is neg


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Graph the different tables.

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