Contents. 3 Introduction. 4 Basic Operations. 6 Order of Operations. 8 Basic operations with fractions. 16 Introduction to decimals

Transcription

1 Judyth Hayne 0

2 Contents Introduction Basic Operations 6 Order of Operations 8 Basic operations with fractions 6 Introduction to decimals 0 Basic operations with decimals Significant figures and rounding Percentages 8 General review 0 General review Sample test Sample test 0 Sample test

3 Introduction This set of notes has been arranged to assist Primary Education students revise the basic skills in arithmetic. All students are required to sit and gain at least 80% in a basic skills test in Mathematics in order to be enrolled in Primary Education Mathematics courses. If a student does not gain a mark of 80% or more they are required to attend a 0 hour course where the basic skills are taught. This course is run by the Maths Study Centre. As Mathematics is a key learning area it is most important that teachers have a good understanding of the basic concepts in Primary Arithmetic. These notes include topic by topic revision and sample exam papers.

4 Basic Operations. Addition and subtraction and multiplication. ) ) ) ) 7 77 ) Exercise: ) 9 ) )8 6 ) 06 8 ) Answers: )777 )8 ) 0067 ) 779 ) 6879

5 Division Exercise ) 7 )67 ) 96 8 ) 9 )86 8 6) ) 69 Answers: )6 )68 )66 )8 )0 6)60 8 7)9

6 Order of operations It may have been some time since you have thought about order of operations as the calculator automatically does this for you. (Financial calculators do not usually do this however)> + = + 0 = Notice the multiplication is done before addition and subtraction. The order of operations can be explained by BODMAS B O D Brackets must be done first Outside brackets Division and multiplication must be done next working from left to right M A Addition and subtraction must be done next working from left to right S Examples: ) + 7 = + 6 = )6 + = + = 8 )8 + ( + 8 ) = 8 + = 8 + = ) = + = ) + ( 8 ) = + 8 = + 0 = 6 6

7 Exercise: ) 8 7 ) 0 ) ) 0 )8 ( 0 + ) Answers: ) ) 8 ) 8 )0 ) 7

8 Basic Operations with Fractions Equivalent fractions One whole is represented by the rectangle below. The whole may be divided into equal parts = One whole= halves Dividing this shape again we have equal parts. = = = Shade half of the rectangle below. Note that of the equal parts have now been shaded. Hence = 8

9 X Dividing this shape again we have 8 equal parts. 8 = = = also 8 = = 8 X = = Shade half of the rectangle above. 8 Note that of the 8 equal parts have now been shaded. Hence = = 8 Using the pattern above equivalent fractions may be derived. )? = ( multiply by to get 9 on the denominator so multiply by also on the numerator) 9 )? = 8 ) =? 0 ) =? 9

10 ) 8? = 6) 0 = 60? 7) = 7? 8 Solutions: ) 6 ) 6 ) 6 ) 0 ) 6) 7) 0

11 Addition and Subtraction of fractions. (i) Like denominators Shade square which represents one eighth of the rectangle Shade squares which represent five eighths of the rectangle If you add these together you have now shaded six eighths of a rectangle. Hence: = = 6 8 Rule: When you add or subtract fractions with like denominators you add or subtract the numerators, then simplify your answer if necessary.

12 Exercise: 8) ) 9 9 6) ) 9 ) 9 9 ) 7 ) 7 7 ) Answers: 7 8) 9 9 7) 9 6 6) ) ) 9 ) 0 ) 7 6 ) = = = =

13 (ii) Addition and subtraction of fractions with unlike denominators. If the denominators are not the same the fractions must first be changed to have like denominators. + 8 = = 8 (Notice from line to line equivalent fractions have been used so that the fractions have like denominators) Exercise: ) + 6 ) ) 9 8 ) 9 + ) Answers: 7 0 ) ) = ) ) = ) 6

14 Further addition and subtraction of fractions. Example: = = + + = + Exercise: ) 8 ) 6 ) ) 8 8 ) Answers: ) 8 ) ) ) )

15 Multiplication of fractions. 9 Draw a rectangle and shade and shade four ninths of it. Now shade half of the shaded area. You will find that you have now shaded two ninths of the rectangle. You have shaded of 9. Hence 9 = 9 Notice this answer may also be found by cancelling the and the. (Note this may only be done when multiplying fractions) Exercise: ) 8 9 ) 7 ) 8 9 ) Answers: ) ) ) ) = =

16 Decimals In order to understand decimals you need to understand place value. Thousands Hundreds tens Units Number Decimals follow the same concept for place value. units. tenths hundredths thousandths Number Using the concepts developed in the table above, write the following as decimals: 0 = = = = =

17 ) Write as decimals: ( i) ( ii) ( iii) ( iv) Express as a fraction in its simplest form: 0.7 = = 0 = 0.08 = = ) Express as a fraction in its simplest form: ( i) 0. ( ii) 0.9 ( iii) 0.08 ( iv)

18 Express as a mixed number in its simplest form: 0 = =.7. = 0 = = 8 00 = 8 ) Write as a mixed number in its simplest form: ( i)6. ( ii).7 ( iii).8 ( iv). ( v).6 8

19 9 Solutions: ) ( ) ( ) ( ) ( )6 )( 00 ) ( 00 9 ) ( 00 9 ) ( ) )( )0.007 ( )0.09 ( )0.08 ( )0.9 )( v iv iii ii i iv iii ii i iv iii ii i

20 Basic operations with decimals. ) Addition and subtraction. ) ) ) When adding or subtracting decimals just keep the decimal points under each other and use the normal rules for addition and subtraction. Notice in example a zero has been inserted after the in 8.60 to ensure that the 9 underneath is subtracted from 0 (Keep the place value). Exercise: ) ) ).7.6 ). 8.8 ) Answers: )6. )6.067 )0.89 ).68 ). 76 0

21 Multiplying decimals. units. tenths hundredths thousandths Number Recall the expansion of a decimal.. = unit + tenths. = = 0 + = 0 + = 0 0 Notice that both of the digits have moved one place in place value space to the left. If we multiplied. by 00 the answer would be 0 as each would move places in place value to the left. (If this is confusing you need to go to a basic textbook on decimals to recall the facts) Exercise: ) ). 0 ) ).8 0 ) Answers: )6.8 ). )887 ).8 ). 69

22 Significant Figures and Rounding Round to the nearest ten: (the zero must be added to keep the place value) (the 8 becomes 9 because 7 is or more) 8 60 Round to the nearest hundred: Round to significant figures: 6 60 (count digits from the left) (the zero is included as it is in between the other digits) 6 00 (the zeros must be added to keep the place value).. (count digits from the left) (count from the first non-zero number)

23 Exercise: ) Round to the nearest 0: (i) 6 (ii) 6 (iii) (iv) 86 ) Round to the nearest hundred: (i) 7 (ii) 67 (iii) 768 (iv) ) Round to significant figures: (i) 6 (ii) 67 (iii) 0.6 (iv).7 ) Round to significant figures: (i) 6 (ii) 89 (iii).79 (iv) 7.06 (v) (vi) (vii) 0.00 Solutions: ) (i) 60 (ii) 0 (iii) 0 (iv) 860 ) (i) 00 (ii) 600 (iii) 800 (iv) 00 ) (i) 00 (ii) 6000 (iii) 0. (iv). ) (i) 00 (ii) 90 (iii). (iv) 7. (v) 0.08 (vi) (vii) 0.00

24 Percentages 0% means 0 per hundred. = 0 00 = 0% To change to a percentage just multiply by = 0% Similarly to change 0. to a percentage = = = 0% ) Complete the table: Decimal Percentage 0. 0% 0.7 a) 0.8 b) 0.0 c) 0.06 d). e) 0.7 f) ) Complete the table Fraction Percentage a)

25 b) c) 7 d) 0 8 e) 0 f) 0 ) Complete the following table: Decimal Fraction Percentage 0.08 a) b) c) d) 0 e) f) % g) h) 80% 0.6 i) j) 0. k) l) m) n) 6% 0. o) p) ) a) Find 0% of hour 0 of 60 minutes = 00 = min

26 b) Find 7% of m in cm c) Find 6% of m in cm d) Find 8% of km in m ) What percentage is: a) 0 minutes of hour? = 0% (Note: both measurements must be changed to the same name-both to minutes in this case) b) 86cm of m c) 0m of km 6) a) A car is bought for $000 and sold for $000. What is the loss as a percentage of the cost price? Loss=$000-$000 =$ Percentage= = 0% 000 b) Sen bought a table for $800 and sold it for $000. Find his profit as a percentage of the cost price. c) Max bought a campervan for $000 and sold it for $

27 Find his loss as a percentage of the cost price. 7) a) If John spent $0 on tickets and this was 0% of his allowance, what was his allowance? 0% allowance=$0 % allowance=$ ( 0 0 ) Allowance=$00 b) Amanda paid a 0% deposit of $0 on a ticket. What was the full price of the ticket? c) 8% of Jeff s allowance is $. What is Jeff s allowance? Solutions: : a) 70% b) 8% c) % d).6% e) % f).7% : a) 60% b) % c) 7% d) 70% e) 90% f) 6% : a) 6 b) 8% c) 0. d) 0% e) 0. f) g) 0.8 h) 6 i) 9 j) 6% k) 0 l) % m) 0.6 n) 0 0) p) 0% : b) 7cm c) 68cm d) 0m : b) % c) 7 % 6: b) % c) 0% 7

28 General Review () ) ) ) ) ) 00 + ( ) 6) ) Write as a number: Three million, eighteen thousand and five. 8) Multiply the sum of and by the difference of 9 and 6. 9) 7 =? Find the value of? 0) +? = 6 Find the value of? = ) ( 8?) + Find the value of? ) Write 8 to the nearest hundred. 8

29 ) ) ) 6) 8 0 7) ) ) ) Find of 70m. Solutions ) 69 ) 88 ) 700 ) ) 08 6) 0 7) ) 0 9) 6 0) ) ) 800 ) ) ) 6 6)8 7) 70 8) 907 9) 6 0) 80m 9

30 General Review () ) Find of $0. ) Find the value of? if =? 0 ) Find the value of? if = 7? 9 ) What is 8 of m in cm? ) Write as an improper fraction. 6) Write as a mixed number. 7) Which is largest 7 or 8? 8) Find the sum of and 7 9) Evaluate +. 0

31 0) Evaluate 6 ) Find the value of ) Find the product of 8 and 09. ) Find the difference between 00 and 68. ) Max gave one third of his pizza to Ben, one quarter of the pizza to Elias and he ate the rest himself. What fraction of the pizza did Max eat? Solutions: ) $8 ) ) 8 ) 7cm ) 6) 7) 8 8) 9) 7 6 0) ) ) 0 ) 7 )

32 Primary Mathematics Skills Test Sample test Pass mark: 0 Total marks available: 0 Time allowed: minutes The use of calculators is not permitted. Question ( marks) (i) Alvin paid $6 for ten tickets. How much would each ticket cost assuming they were all the same price? (ii) 08 6

33 (iii) If 6? = 7 find?. (iv) Write the number in words. (v) 7 7 (vi) 8 = 9? find?

34 Question ( marks) (i) Write these decimals in order of size - largest to smallest (ii) (iii).6-8.0

35 (iv) Round.08 to decimal places. (v) Round $8.7 to the nearest ten cents. (vi) Evaluate Question (0 marks) (i) Write 7 as an improper fraction.

36 (ii) Leif had a box of pencils. He gave of the pencils to his sister, 6 to his younger brother and of the pencils to his cousin. There were still pencils remaining in the box. How 9 many pencils were there in the box originally? (iii) Evaluate 6 8 6

37 (iv) In a swimming squad of people, three out of five were female. How many were female? (v) Write these fractions in order from greatest to least:

38 Question (0 marks) (i) Express 6.% as a decimal. (ii) Winnie was given a 0% discount on a coat costing $0. How much did she pay for the coat? (iii) Jason bought a bike for $0 and sold it for $70. Find his loss as a percentage of the cost price. 8

39 (iv) Write as percentages: a) b)0.08 (v) What percentage is 0 minutes of hours? Question (6marks) a)suyee wanted to paint one of her bedroom walls. Her wall measured.m by m. What was the area of this wall? 9

40 b) How much will it cost to paint the wall if the painter charges $ 0 / m? 0

41 Primary Mathematics Skills Test Sample test Pass mark: 0 Total marks available: 0 Time allowed: minutes The use of calculators is not permitted. Question ( marks) (i) Chloe bought pumpkins for $.0. How much would pumpkin cost? (ii) 06 67

42 (iii) If 9? = 8 find?. (iv) Write the number 00 0 in words. (v) 7

43 (vi) 7 6 =? find? Question ( marks) (i) Write these decimals in order of size - largest to smallest (ii)

44 (iii) (iv) Round.0876 to decimal places.. (v) Write down the multiples of between and 6. (vi) Evaluate

45 Question (0 marks) (i) Write as an improper fraction. 8 (ii) Tono won a box of CDs. He kept 8 of the CDs, gave to his younger brother and 6 of the CDs to his cousins. There were still CDs remaining in the box. How many CDs were there in the box originally?

46 (iii) Evaluate 8 6 (iv) Find two thirds of two and a half hours? (v) Write these fractions in order from greatest to least:

47 Question (0 marks) (i) Express.9% as a decimal. (ii) Byron paid a deposit of $ for his ticket. This was in fact 0% of the full price. What was the full price of the ticket? (iii) Jason bought a campervan for $0 000 and sold it for $ 000. Find his loss a percentage of the cost price. 7

48 (iv) Write as percentages: a) 0 b)0.07 (v) What percentage is minutes of hours? 8

49 Question (6marks) a) (i) Find the area of the following figure. 6 cm 6 cm (ii) Find the perimeter of the figure above. (ii) If all of the dimensions of the figure were doubled, calculate the new perimeter of the figure above. 9

50 Primary Mathematics Skills Test Sample test Pass mark: 0 Total marks available: 0 Time allowed: minutes The use of calculators is not permitted. Question ( marks) (i) Owen paid $6 for tickets to a movie. How much did each ticket cost assuming they were of equal value? (ii)

51 (iii) If 80? = 8 find?. (iv) Write the number in words. (v) 8 8 (vi) = 6? find?

52 Question ( marks) (i) Write these decimals in order of size - smallest to largest (ii) (iii)

53 (iv) Round.089 to decimal places. (v) (vi) Evaluate Question (0 marks) (i) Write as an improper fraction.

54 (ii) Tono won a box of CDs. He kept 6 of the CDs, gave to his younger brother and 8 of the CDs to his cousins. There were still CDs remaining in the box. How many CDs were there in the box originally? (iii) Evaluate 6 7 (iv) How many centimetres in of metres? 0

55 (v) Write these fractions in order from greatest to least: Question (0 marks) (i) Express 6.% as a decimal. (ii) Andy paid a deposit of $ for his ticket. This was in fact % of the full price. What was the full price of the ticket?

56 (iii) Jack bought a bike for $00 and sold it for $0. Find his loss a percentage of the cost price. (iv) Write as percentages: a) 9 0 b)0.6 (v) What percentage is minutes of hours? 6

57 Question (6marks) a) (i) Find the area of the following figure. m m (ii) Find the perimeter of the figure above. (ii) How much would it cost to put a fence around the area above at $0 per metre? 7