p4 victory maths · Ĵ multiplication and division (division of a whole number by another whole...

Preface

What is this series about?

The Math Booster Primary 1 to 6 is a series of practice books designed to develop pupils’ problem-solving skills using the various heuristics. Topics in these books closely follow those laid out in the syllabus set out by the Ministry of Education, Singapore. Emphasis is also given to mental calculations by using the heuristic, restate the problem.

Appendices at the end of each book include the ‘History of Mathematics’ games to extend pupils’ interest in Mathematics.

What are heuristics?

Heuristics are strategies that increase the likelihood of solving a problem. The Singapore Primary Math Syllabus focuses on using thinking skills and heuristics to solve problem sums. The following diagram shows four classes of heuristics. The heuristics that are used in this Primary 4 book are marked with *.

Represent

*Draw diagrams*Make a list

Form equations

Go through the process

*Act it out*Work backwardsBefore and after

*Logical Reasoning

Make a calculated guess

Guess and checkLook for patterns

*Make suppositions

Change the problem

*Restate the problem Simplify the problem

*Solve part of the problem

Heuristics

To Parents

and

Tutors

136_P4_Mathematics Booster_7thPass 20/05/15

ii

What do the heuristics mean and how are they used?

The meanings of the heuristics marked with * and examples of their uses are as follow:

✓✓ Draw diagrams: Make a pictorial description of the problem to help understand the problem better and use the given information to manipulate the data.

Find ♥.

2 × 4 – (2 × ♥ + 1) = 7

Heuristic: Drawing diagrams

2 × 4 – (2 × ♥ + 1) = 7

8 – (2 × ♥ + 1) = 7

8

2 × ♥ + 17

From the diagram: 2 × ♥ + 1 = 8 – 7 = 1

Let’s make a guess that ♥ is 0.

2 × 0 + 1 = 0 + 1 = 1

♥ = 0

✓✓ Work backwards: Work backwards from the end of the problem sum to the beginning by reversing the given conditions.

Example

Solution with explanation


iii

After giving 12

of his money to his sister, Alex has 45

as much money

as his sister. If Alex had $56 more than his sister at first, how much money did Alex give to his sister?

Heuristic: Work backwards and draw diagrams

A

S

End of A at first12

A

S

At first56

7 units = 56 1 unit = 56 ÷ 7 = 84 units = 4 × 8 = 32Alex gave his sister $32.

✓✓ Make a list: Organise the given data (e.g. numbers) into lists to assist in finding the required data or by identifying the pattern based on the given information.

There are marbles inside a jar. To count the number of marbles, Fellini uses his hand to take 8 marbles out each time. There are 5 marbles left over after he finishes counting. Guido, who has smaller hands, takes out 5 marbles each time and finds that there are 2 marbles left over. What is the smallest number of marbles in the jar?

Example


Example


iv

Heuristic: Make a list

Multiples of 8: 8, 16, 24, 32, 40, 48…

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40…

Given that:

There are 5 marbles left over if we count by 8.

There are 2 marbles left over if we count by 5.

Multiples of 8 + 5:

8 + 5 = 13, 16 + 5 = 21, 24 + 5 = 29, 32 + 5 = 37, 40 + 5 = 45, 48 + 5 = 53

Multiples of 5 + 2:

5 + 2 = 7, 10 + 2 = 12, 15 + 2 = 17, 20 + 2 = 22, 25 + 2 = 27, 30 + 2 = 32, 35 + 2 = 37, 40 + 2 = 42

The smallest number of marbles in the jar is 37.

✓✓ Act it out: Take on the role of people, things or process in the problem and trying to do what they do.

Can a pentagon be used to tessellate a plane? Explain your answer.

Heuristic: Act it out

There are gaps between the unit shapes. Therefore, the pentagon cannot be used to tessellate a plane.

✓✓ Restate the problem: Restate or rewrite the problem in simpler words to help understand the problem better.


Example



v

Which fraction is the greatest?12

34

35

Heuristic: Restate the problem

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

Multiples of 4: 4, 8, 12, 16, 20

Multiples of 5: 5, 10, 15, 20

Equivalent fractions of the above fraction with the same denominator 20:12

1 102 10

1020

34

3 54 5

1520

35

3 45 4

1220

= ××

=

= ××

=

= ××

=

The greatest fraction is 34

.

✓✓ Solve the problem in parts: Break the problem into parts and solve them.

Example



vi

The figure is made up of five squares which overlap one another such that the corner of one square lies on the centre of the other square. Find the perimeter and area of the figure.

4 cm

Heuristic: Solve parts of problem

Height of figure = 4 cm + 2 cm + 2 cm + 2 cm + 2 cm

= 12 cm

Length of figure = 4 cm + 2 cm + 2 cm + 2 cm + 2 cm

= 12 cm

Perimeter of figure = 12 cm + 12 cm + 12 cm + 12 cm

= 48 cm

Area of each square = 4 cm × 4 cm

= 16 cm2

Area of each overlapping part = 2 cm × 2 cm

= 4 cm2

Area of the figure

= area of 5 squares – area of 4 overlapping parts

= 5 × 16 cm2 – 4 × 4 cm2

= 80 cm2 – 16 cm2

= 64 cm2

The perimeter of the figure is 48 cm and its area is 64 cm2.

Example



Contents

Whole Numbers

Summary of the syllabus

✓Ĵ Numbers up to 10 000 (Comparing and ordering numbers, number patterns rounding numbers to nearest 10 or 100)

✓Ĵ Multiplication and Division (Multiplication of 4-digit and 1-digit numbers, 3-digit and 2-digit numbers, division of 4-digit by 1-digit number, solving up to 3 step word problems involving the 4 operations, estimation of answers)

✓Ĵ Factors and Multiples (Listing factors of a number up to 100, finding common factors of 2 numbers, listing the first 12 multiples of a 1-digit number, finding the common multiples of two 1-digit numbers)

Title Page No.

(A) Mental Calculations 1 (Addition, Subtraction, Multiplication and Division) 1Heuristic: Restate the problem

(B) Quest for Missing Numbers 9Heuristic: Draw diagrams

(C) Let’s suppose 13Heuristic: Supposition

(D) Factors and Multiples 17Heuristic: Make a list and draw diagrams

(E) Finding the Number of Ways 28Heuristic: Make a list and draw diagrams

Unit1


viii

Fractions


✓Ĵ Mixed numbers and improper fractions (Expressing improper fractions as mixed numbers and vice versa, expressing improper fractions and mixed numbers in simplest form)

✓Ĵ Addition and Subtraction (Addition and Subtraction of like fractions or related fractions)

✓Ĵ Multiplication (Multiplication of proper or improper fractions, solving 2-step word problems)

Title Page No.

(A) Comparing Fractions 37Heuristic: Restate the problem

(B) Part-Whole Relationships 46Heuristic: Draw diagrams

Decimals


✓Ĵ Decimals up to 3 decimal places (Comparing and ordering decimals, conversion of decimal to fraction and vice versa, rounding off decimals to the nearest whole number, 1 decimal place or 2 decimal places)

✓Ĵ Addition and Subtraction (Addition and Subtraction of decimals up to 2 decimal places)

✓Ĵ Multiplication and Division (Division of a whole number by another whole number with answer up to 2 decimal places, Multiplication and Division of decimals by a 1-digit whole number, solving 2-step word problems)

Title Page No.

(A) Problem sums involving Decimals 56Heuristics: Restate the problem or draw diagrams

Unit2

Unit3


ix

Measurement

Summary of syllabus

✓Ĵ Time (Time in seconds, 24-hour clock, solving word problems)

✓Ĵ Money (Multiplication and Division of money decimal notation, solving word problems involving money in decimal notation)

✓Ĵ Area and Perimeter (Finding area of composite figures made up of rectangles and squares, finding one dimension given the other dimensions and its area or perimeter, solving word problems)

Title Page No.

(A) Tree Planting 62Heuristic: Logical reasoning

(B) Quest for Starting Value 68Heuristic: Work backwards

(C) Area and Perimeter of Squares and Rectangles 73Heuristic: Solve part of the problem

Geometry

Summary of syllabus

✓Ĵ Perpendicular and Parallel lines (Drawing perpendicular lines and parallel lines using ruler and set squares, using “vertical” and “horizontal”)

✓Ĵ Angles (Measure angles, drawing angles using a protractor)

✓Ĵ Rectangle and Square (Identify symmetric figures, determine if a line is a line of symmetry, complete a symmetric figure, design and make patterns)

✓Ĵ Tessellation (Recognize shapes that can tessellate, identify the unit shape in a tessellation, make different tessellations with a given shape, draw a tessellation on dot paper, design and make patterns)

Unit4

Unit5


p4 victory maths · Ĵ multiplication and division (division of a whole number by another whole...

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