Mental Thinking: Using the Target Number Strategy

RIC-6091 3.1/103

Mental thinking: Using the target number strategy

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Published by R.I.C. Publications® 2012 Copyright© Richard Korbosky 2012 ISBN 978-1-921750-81-6 RIC– 6091

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For all Australian Curriculum material except elaborations: This is an extract from the Australian Curriculum.

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Foreword Mental thinking: Using the target number strategy is a set of prepared classroom activities which give students the opportunity to use mental thinking strategies for addition, subtraction, multiplication, money, fractions, decimals, measurement and making combinations using four operations, brackets and indices.

Mental thinking develops flexibility and gives students the necessary skills to solve problem situations in their head or by manipulating numbers based on standard and non-standard partitioning.

Contents

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Teacher notes ...................................................... ii–v Curriculum links ..................................................v–ix

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Early Primary Addition combinations for 8 .................................... 1 Addition combinations for 10 ................................. 2 Addition combinations for 12 .................................. 3 Addition combinations for 15 .................................. 4 Addition combinations for 18 .................................. 5 Addition combinations for 20 .................................. 6 Addition combinations for 24 .................................. 7 Addition combinations for 25 .................................. 8 Addition combinations for 27 .................................. 9 Addition combinations for 28 ................................ 10 Addition combinations for 30 ................................ 11 Money combinations to make 35c......................... 12 Money combinations to make 50c......................... 13 Money combinations to make $1 .......................... 14 Number sentence – Balancing each side of an equation ...................................................... 15

Subtraction – Making a difference of 10 for numbers from 1–30 .......................................... 29 Multiplication – Using two numbers from 1–50, make 12, 18, 20 and 24 .................................... 30 Number sentence – Multiplication and division ...... 31 Money combinations to make $25 ........................ 32 Upper Primary Multiplication – Using two numbers from 1–50, make 28, 32, 36 and 48 .................................... 33 Combinations of operations – Addition, subtraction and multiplication.............. 34 Combinations of operations – Addition and multiplication ................................ 35 Recognising equivalent fractions and decimals for 0.75 ....................................... 36 Fractions operations – Addition and subtraction of fractions with unlike denominators ................. 37 Multiplication and division of fractions and indices ...................................................... 38 Using addition, subtraction, multiplication and division to make 12 .................................... 39 Using addition, subtraction, multiplication and division to make 24 .................................... 40 Prime, composite, square and triangular numbers .. 41 Linking decimals, fractions and percentages to make 1 ......................................................... 42 Combinations of operations .................................. 43 Using addition, subtraction, multiplication and division and decimals to make 8.25 ............ 44 Perimeter – Whole number.................................... 45 Perimeter – Decimal number................................. 46 Area – Square cm ................................................. 47 Area – Hectares .................................................... 48 Volume of a rectangular prism .............................. 49 Order of operations ............................................... 50 Positive and negative integers ............................... 51 Develop your own ‘make the target number’ activity ....................................... 52

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Middle Primary Addition combinations for 40 ................................ 16 Addition combinations for 50 ............................... 17 Addition combinations for 100 .............................. 18 Addition combinations for 200 .............................. 19 Addition combinations for 500 .............................. 20 Addition combinations for 750 .............................. 21 Addition combinations for 1000 ............................ 22 Number sentence – Balancing each side of an equation ........................................................... 23 Number sentence – Addition and subtraction ......... 24 Subtraction – Making a difference of 2 for numbers from 0–15 ..................................... 25 Subtraction – Making a difference of 5 for numbers from 5–27 ..................................... 26 Subtraction – Making a difference of 8 for numbers from 0–23 ..................................... 27 Subtraction – Making a difference of 10 for numbers from 1–20 ..................................... 28

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Mental thinking: Using the target number strat strategy

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Why mental thinking?

Mental mathematics is the main form of calculation used by most adults in society. It is also the simplest way of doing many calculations. Unfortunately, due to the emphasis on written computations in numerous classrooms, many students believe that the correct way to calculate a simple subtraction fact such as 400 – 198 is to do it in the written form.

This written form often complicates mathematics.

When students are taught a range of mental mathematics strategies, they can choose different ways to solve any given situation.

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Through regular experiences with mental thinking, students come to realise that many calculations are in fact easier to perform mentally. When

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students compute mentally, they adopt a problemsolving approach: they take steps to understand the problem,choose a strategy and use it. A good mental mathematics session helps students become aware that there is more than one way to perform most calculations. It encourages them to be flexible in seeking and adopting the simplest and most economical method. In addition, when using mental mathematics, students almost always use a method which they understand (unlike with written computation), and are encouraged to think fluently about relationships involving the particular numbers they are dealing with.

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In order to be effective, mental mathematics sessions should:

• Give students the opportunity to practise the mental thinking strategy required in a given situation.

• Encourage a problem-solving ‘having a go’ approach on the part of all students.

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• Promote oral communication. Give students the opportunity to discuss the method they used to solve the mathematical situation and justify why their way of thinking works.

• Allow students to see that there are many ways to arrive at a correct answer rather than one correct way.

• Build a wide variety of connections between numbers and number facts to improve students’ fluency.

• Include standard and non-standard partitioning activities.

Learning to calculate mentally has many benefits: • Calculating in your head is a practical life skill.

• Proficiency in mental mathematics can make written computations easier and quicker.

• Proficiency in mental mathematics contributes to increased skill in estimation.

• Mental calculation leads to a better understanding of place value, mathematical operations, basic number properties, patterns and algebra thinking in mathematics. ii

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Why mental thinking?

Aspects of mental mathematics: • Basic numbers facts, mathematics facts and efficient mental strategies form the section known as mental mathematics. This is recognised in the Australian Curriculum as the proficiency strand Fluency.

• Some facts have to be learned.

• Mental calculations should be used in any calculation if it is appropriate.

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• Students need to be explicitly taught the mental techniques like we teach any other idea.

• Rather than teaching, in many mental mathematics situations, the students are being tested and not taught.

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• Some strategies are generic; e.g. partitioning or non-partitioning. If students at particular ages have not been introduced to them, then their flexibility to think about mathematics in a calculation or in a problem is diminished.

• Mental techniques need to be practiced. A technique every week, which includes an introduction to the mental thinking strategy and then practicing it could be employed.

Mental techniques should be taught explicitly so students have a range of mental strategies available to them to solve calculations.

This range of strategies is for Foundation–Year 2.

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C.Publ i at i 7.c Adding 9:o 10 n – 1s 8.o Adding 8:s 10 –o 2n •f orr evi ew pur p se l y• 2. Count on: only 1, 2 and 3 1. When adding, start with biggest number first (commutative)

9. Adding 11: 10 + 1

4. Combinations of ten: using the ten frame

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10. Skip counting: 2s, 5s, 10s and 3s

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3. Partitioning: breaking up any number; e.g.

10 is 7 and 3, or it could be 4 and 6 (partpart-whole thinking)

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5. Doubles: 1 + 1 up to 10 + 10

6. Adding near doubles: such as 6 + 7 is 6 + 6 + 1 or 7 + 7 – 1

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Mental thinking: Using the target number strat strategy

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Why mental thinking?

This range of mental thinking strategies for years 3–7 includes: 1. 28 + 27

5. Doubling and halving

Change 28 to 30 That means the 2 that you have added needs to be taken away so that the quantity is not changed when using this strategy.

Make sure the method is clearly shown to all the students.

36 x 25 18 x 50 9 x 100 = 900

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(28 + 2) + (27 – 2) = 30 + 25 = 55

6. Non-partitioning 25% of $360

25% is a quarter; a quarter is a half of a half

(364 + 2) – (198 + 2) = 366 – 200 = 166

3. Non-standard partitioning makes the combination easier to subtract.

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⁄2 of 360 is $180, 1⁄2 of $180 is $90

7. Non-partitioning 17.5% of $240

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2. 364 – 198

575 – 298

17.5% is made up of 10% + 5% + 21⁄2%

275 + 300 – 298 = 277

⁄10 of $240 is $24, half of that is $12 and half of that is $6 1

So 17.5% oft $240 is $24 © R. I .

C.Pub l i ca i on s+ $12 + $6 = $42 (45 x 2) x (14 2)o = 90 7e = 630 •÷f rxr vi ew pur posesonl y•

4. Doubling and halving

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45 x 14

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Teachers notes in the book are opened-ended problems which allow students from any level to respond to the target number according to their understanding of mathematics. The activities also focus on the ‘look for all possibilities’ problem-solving strategy.

This book is a set of prepared classroom activities which give students the opportunity to use mental thinking strategies for addition, subtraction, multiplication, money, fractions, decimals, measurement and making combinations using the four operations, brackets and indices.

In the classroom situation, when introducing this strategy, it has been found that working in pairs and then using the pairs-pair strategy supports the uptake of knowledge needed to be successful at this strategy.

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Teachers are asked to make sure that students are given the opportunity to be flexible when working with numbers. This flexibility gives them the necessary skills to solve problem situations in their head or by manipulating numbers based on standard and non-standard partitioning.

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Using actual mathematic manipulatives, such as counters, craft sticks, play money or an interactive whiteboard, has been found to be an important teaching strategy for students who have difficulties.

Reflecting on how students have used different numbers to reach the answer is an important learning strategy.

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The use of the ‘make the target number strategy’ is meant as a mental thinking activity.This is similar to a common mental activity called ‘Today’s number is’; the difference being that the students have a set of numbers in front of them from which to choose from to calculate the target number. The use of mathematical manipulative materials and calculators have been found to be very useful in allowing students to focus on the problem-solving strategy of ‘guess, check and improve’.

Whole class reflection at the end of a mental thinking session is an important metacognition strategy to support students’s learning.

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C.Publ i cat i ons The activities in this book can be used to introduce mental thinking, practice mental thinking or assess The activities are developmental and, therefore, • f o r r e v i e w p u r p o s eso n y • what the students know atl the end of the teaching teachers can set work at different levels to suit

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Assessing mental thinking How did you work it out? Did you …

• use manipulative maths materials?

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and learning process.

the needs of individuals or groups. The activities

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• use a calculator?

• calculate answers using pencil and paper?

• see a pattern?

This book has been organised to focus on aspects of: • standard and non-standard partitioning

• decimals

• addition

• measurement

• subtraction

• using combinations of the four operations, brackets and indices

• money • fractions R.I.C. Publications® ~ www.ricpublica www.ricpublic www.ricpublications.com.au www.ricpub www.ricpu

• positive and negative integers.

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Curriculum links Links to the Australian Curriculum have been made for Foundation to Year 6.

E

Activities are coded

for Early Childhood, M for Middle Primary and

U

for Upper Primary.

The activities in this book can suit any year level in the primary class.

Early Childhood Foundation–Year 2 It is suggested that the teacher gives students a set of counters/craft sticks so that they can manipulate the materials and either draw or symbolically write the different solutions to each of the questions asked.

Content description

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Year 1: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015)

• developing a range of mental strategies for addition and subtraction problems

Year 2: Explore the connection between addition and subtraction (ACMNA029)

• becoming fluent with partitioning numbers to understand the connection between addition and subtraction • using counting on to identify the missing element in an additive problem

Year 2: Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

• becoming fluent with a range of mental strategies for addition and subtraction problems, such as commutativity for addition, building to 10, doubles, 10 facts and adding 10 • modelling and representing simple additive situations

Year 2: Count and order small collections of Australian coins and notes according to their value (ACMNA034)

• identifying equivalent values in collections of coins or notes, such as two five-cent coins having the same value as one 10-cent coin • counting collections of coins or notes to make up a particular value

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• using a range of practical strategies for adding small groups of numbers, such as visual displays or concrete materials

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Foundation: Represent practical situations to model addition and sharing (ACMNA004)

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Curriculum links Middle Primary Years 3–4 It is suggested that some students will need the support of the materials, whilst others can work abstractly in solving problems presented.

Elaborations

Year 3: Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053)

• justifying choices about partitioning and regrouping numbers in terms of their usefulness for particular calculations

Year 3: Recognise and explain the connection between addition and subtraction (ACMNA054)

• demonstrating the connection between addition and subtraction using partitioning or by writing equivalent number sentences

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Content description

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• recognising that certain single-digit number combinations always result in the same answer for addition and subtraction, and using this knowledge for addition and subtraction of larger numbers

Year 3: Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)

• establishing multiplication facts using number sequences

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Year 3: Recall addition facts for singledigit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation (ACMNA055)

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Year 4: Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076)

• using known facts and strategies, such as commutativity, doubling and halving for multiplication, and connecting division to multiplication when there is no remainder

Year 4: Use equivalent number sentences involving addition and subtraction to find unknown quantities (ACMNA083)

• writing number sentences to represent and answer questions such as: ‘When a number is added to 23 the answer is the same as 57 minus 19. What is the number?’

• using partitioning to find unknown quantities in number sentences

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• recognising the relationship between dollars and cents

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Year 3: Represent money values in multiple ways and count the change required for simple transactions to the nearest five cents (ACMNA059)

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Curriculum links Upper Primary Years 5–6 Students at this level should be able to abstractly work out a range of combinations.

Elaborations

Year 5: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)

• using calculators to check the reasonableness of answers

Year 5: Use equivalent number sentences involving multiplication and division to find unknown quantities (ACMNA121)

• using relevant problems to develop number sentences

Year 5: Compare, order and represent decimals (ACMNA105)

• locating decimals on a number line

Year 5: Calculate the perimeter and area of rectangles using familiar metric units (ACMMG109)

• exploring efficient ways of calculating the perimeters of rectangles such as adding the length and width together and doubling the result • exploring efficient ways of finding the areas of rectangles

Year 6: Identify and describe properties of prime, composite, square and triangular numbers (ACMNA122)

• understanding that some numbers have special properties and that these properties can be used to solve problems

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Content description

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Year 6: Investigate everyday situations that use integers. Locate and represent these numbers on a number line (ACMNA124)

• understanding that integers are … –3, –2, –1, 0, 1, 2, 3, …

Year 6: Make connections between equivalent fractions, decimals and percentages (ACMNA131)

• connecting fractions, decimals and percentages as different representations of the same number, moving fluently between representations and choosing the appropriate one for the problem being solved

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• applying strategies already developed for solving problems involving small numbers to those involving large numbers

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Year 6: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)

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Year 6: Explore the use of brackets and order of operations to write number sentences (ACMNA134)

• appreciating the need for rules to complete multiple operations within the same number sentence

Year 6: Connect decimal representations to the metric system (ACMMG135)

• recognising the equivalence of measurements such as 1.25 metres and 125 centimetres

The above content descriptions and codes have been reproduced with permission from ACARA. © Australian Curriculum, Assessment and Reporting Authority 2012

Extension Extension activities are built into many of the problem situations.

Use of a calculator It is suggested that students are given the opportunity to use a calculator to solve problems, particularly when trying to work out combinations of complex numbers and when using more than one operation. viii

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Addition combinations for 8

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How many ways can you make the number 8 by adding any two numbers in the grid?

10

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Find the combinations of 8 and write or draw them below.

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Extension

• What three numbers can you put together to make 8? Write your numbers here or on a separate page.

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Mental thinking: Using the target number strategy

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Addition combinations for 10

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How many ways can you make the number 10 by adding any two numbers in the grid?

9

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Find the combinations of 10 and write or draw below.

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Extension

• What three numbers can you put together to make 10? Write your numbers here or on a separate page.

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Addition combinations for 12

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How many ways can you make the number 12 by adding any two numbers in the grid?

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Find the combinations of 12 and write them below.

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Extension

• What three numbers can you put together to make 12? Write your numbers here or on a separate page.

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Mental thinking: Using the target number strategy

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Addition combinations for 15

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How many ways can you make the number 15 by adding any two numbers in the grid?

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Extension

• What three numbers can you put together to make 15? Write your numbers here or on a separate page.

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Addition combinations for 18

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How many ways can you make the number 18 by adding any two numbers in the grid?

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C.Publ i ca i ons 18 •f orr evi ew pur posesonl y• Find the combinations of 18 and write them below.

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Extension

• What three numbers can you put together to make 18? Write your numbers here or on a separate page.

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Mental thinking: Using the target number strategy

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Addition combinations for 20

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How many ways can you make the number 20 by adding any two numbers in the grid?

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C.Publ i cat i ons Find the combinations ofi 20w and them below.

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Extension

• What three numbers can you put together to make 20? Write your numbers here or on a separate page.

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Addition combinations for 24

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How many ways can you make the number 24 by adding any two numbers in the grid?

8

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C.Publ i cat i ons Find the of and write them •combinations f orr evi e w24p ur p os esbelow. onl y•

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Extension

• What three numbers can you put together to make 24? Write your numbers here or on a separate page.

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Addition combinations for 25

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How many ways can you make the number 25 by adding any two numbers in the grid?

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C.Publ i cat i ons Find the combinations ofi 25w and them below.

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Extension

• What three numbers can you put together to make 25? Write your numbers here or on a separate page.

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Addition combinations for 27

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How many ways can you make the number 27 by adding any two numbers in the grid?

8

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Extension

• What three numbers can you put together to make 27? Write your numbers here or on a separate page.

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Addition combinations for 28

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How many ways can you make the number 28 by adding any two numbers in the grid?

27

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C.Publ i cat i ons Find the combinations of 28 and write them below.

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Extension

• What three numbers can you put together to make 28? Write your numbers here or on a separate page.

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Addition combinations for 30

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How many ways can you make the number 30 by adding any two numbers in the grid?

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11

24 16 0 20 4 © R. I .

C.Publ i cat i ons 30•f 15pur 6l or2 r evi ew p31 oseson y•17

w ww

. te

m . u

Find the combinations of 30 and write them below.

o c . che e r o t r s super

Extension

• What three numbers can you put together to make 30? Write your numbers here or on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

11

Money combinations to make 35c

E

How many different coin combinations can you use to make 35c?

20c

5c

5c

5c

5c

r o e t s Bo r e p 5c 10c ok 5c u S

10c 5c

ew i ev Pr

Teac he r

5c

10c

5c

5c 20c 10c © R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• Show your combinations of 35c by drawing the coins or writing

w ww

. te

12

m . u

the numbers below.

o c . che e r o t r s super

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Money combinations to make 50c

E

How many different coin combinations can you use to make 50c?

20c

5c

r o e t s Bo r e p 5c 10cok u S

10c

5c

5c

5c

5c 5c

ew i ev Pr

Teac he r

5c

10c

5c

5c 20c 10c © R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• Show your combinations of 50c by drawing the coins or writing

w ww

. te

m . u

the numbers below.

o c . che e r o t r s super

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

13

Money combinations to make $1

E

How many different coin combinations can you use to make $1?

20c

10c

5c

10c

5c

50c

10c

10c

20c

r o e t s Bo 5c r e 5c 10c p ok u S

5c

ew i ev Pr

Teac he r

5c

10c

10c

20c 50c ©R .

I .

C.Pub20c l i cat i ons20c •f orr evi ew pur posesonl y• Show your combinations of $1 by drawing the coins or writing

w ww

. te

m . u

the numbers below.

o c . che e r o t r s super

Extension

• Using any combination of 5c, 10c, 20c, 50c and $1, how many different ways can you make $2.50? Write your answers on a separate page.

14

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Number sentence – Balancing each side of an equation

E

Combinations of 12 12 11

+

1

7

+

5

6

+

6

r o e t + 2 s B r e oo p 9 + 3 k u S 8 + 4 Any three numbers

ew i ev Pr

Teac he r

10

w ww

1

8

3

1

9

2

. te 11 + 1 = 3 +

Solve these number sentences.

10 +

m . u

© R. I C.Pub i cat i ns 2.

7 l 3o •f orr evi w pu pose2sonl y• 4e 6r

o c . che e r o t r s su r =6+ 2p + e +3=4+6+2 7+5=4+

8+1+3=1+2+ Extension

• Make a grid using the number 15 and develop your own number sentences on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

15

Addition combinations for 40

M

How many ways can you make the number 40 by adding any two numbers in the grid?

31

26

37

7

6

Teac he r

17

9

21

3

2

12

1

25

18

32 38 34

ew i ev Pr

28

29

r o e t s 23 15o 13 B r e p ok u S30 19 14 22 5 22

33

27

11

39

© I .

C Pu8bl i ca t i ons 24R. 16.

20 4 36 •f orr evi ew pur posesonl y•

35

10

w ww

. te

m . u

Find the combinations of 40 and write them below.

o c . che e r o t r s super

Extension • What other ways can you use the numbers to make 40? Write your numbers on a separate page.

16

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Addition combinations for 50

M

How many ways can you make the number 50 by adding any two numbers in the grid?

31

26

48

27

29

21

3

32

45

7

22

12

23

15

13

43 34

44

r o e t s B r e 28 p 6 19 14 o 30 ok 5 u S 9 90 1 25 18 11

35

10

24

16

0

20

4

36

50

100

2

49

80

75

17

42

.

I .

C. Publ i cat i o s39 37© R 46 55 38 8n

47

41

40

ew i ev Pr

Teac he r

33

•f orr evi ew pur posesonl y•

w ww

. te

m . u

Find the combinations of 50 and write them below.

o c . che e r o What three numbers can you use to make 50?

Write t r s them below. super

Extension • What other ways can you use the numbers to make 50? Write your numbers on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

17

Addition combinations for 100

M

How many ways can you make the number 100 by adding any two numbers in the grid?

10

120

80

60

70

35

110

50

30

15

45

95

120

85

20

Teac he r

ew i ev Pr

50

r o e t s Bo 25 r e 65 40 p ok u S 75 30 90 55 5

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

Extension

m . u

Find the combinations of 100 and write them below.

o c . che e r o t r s super

On a separate page, write what:

• is the largest number you could make using two numbers from the grid?

• is the smallest number you could make using two numbers from the grid?

• is the the difference between the smallest and the largest number?

• three numbers from the grid could you use to make 100?

18

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Addition combinations for 200

M

How many ways can you make the number 200 by adding any two numbers in the grid?

100

110

155

130

120

135

105

125

95

Teac he r 45

115

165

75

145

35

100 60

ew i ev Pr

90

r o e t s B r e 140 80o p ok u S 70 85

55

65

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

Extension

m . u

Find the combinations of 200 and write them below.

o c . che e r o t r s super

On a separate page, write what: • is the largest number you could make using two numbers from the grid?

• is the smallest number you could make using two numbers from the grid?

• is the the difference between the smallest and the largest number?

• three numbers from the grid could you use to make 200?

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

19

Addition combinations for 500

M

How many ways can you make the number 500 by adding any two numbers in the grid?

100

120

175

250

150

220

205

225

300

360

325

240

280

275

380

Teac he r

ew i ev Pr

260

r o e t s Bo 140 r e 250 400 p ok u S 295 200 300 105 295

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

Extension

m . u

Find the combinations of 500 and write them below.

o c . che e r o t r s super

On a separate page, write what: • is the largest number you could make using two numbers from the grid?

• is the smallest number you could make using two numbers from the grid?

• is the the difference between the smallest and the largest number?

• three numbers from the grid could you use to make 500?

20

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Addition combinations for 750

M

How many ways can you make the number 750 by adding any two numbers in the grid?

375

390

380

400

440

480

450

500

220

Teac he r 310

370

270

250

460

240

375 290

ew i ev Pr

510

r o e t s B r e oo 350 200 p u k S 550 360

530 300

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

Extension

m . u

Find the combinations of 750 and write them below.

o c . che e r o t r s super

On a separate page, write what:

• is the largest number you could make using two numbers from the grid?

• is the smallest number you could make using two numbers from the grid?

• is the the difference between the smallest and the largest number?

• three numbers from the grid could you use to make 750?

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

21

Addition combinations for 1000

M

How many ways can you make the number 1000 by adding any two numbers in the grid?

500

400

460

510

440

540

520

600

100

485

700

800

450

560

200

Teac he r

ew i ev Pr

490

r o e t s Bo 515 r e 475 300 p ok u S 900 500 550 525 480

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

Extension

m . u

Find the combinations of 1000 and write them below.

o c . che e r o t r s super

On a separate page, write what:

• is the largest number you could make using two numbers from the grid?

• is the smallest number you could make using two numbers from the grid?

• is the the difference between the smallest and the largest number?

• three numbers from the grid could you use to make 1000?

22

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Number sentence – Balancing each side of an equation

M

317 300

+

17

200

+

117

+

267

r o + 217 e t s B r e o p 150 + 167o u k S 250 + 67 50

Any three numbers 100

200

17

ew i ev Pr

Teac he r

100

w ww

. te

140

150

27

130

130

57

123

127

67

m . u

© R. I .

C.Publ i cat i ons 160 120 37 •f orr ev i e w p u r p o s e s o n l y • 145 125 47

o c = 123 + 127 + 67 .

2. 150 + ch e r e o rtheu st r pe Look at the combinations ins grid above and create your own 1. 200 + 117 =

+ 150 + 17

number sentence with one or two of the numbers missing.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

23

Number sentence – Addition and subtraction

M

Add and subtract number sentences.

1. If 3 + 5 = 8, then 5 = 8 – 3 and 3 = 8 – 5. Show this with counters.

2. If 4 + 9 = 13, then

r o e t s B r and =o 17 – . e p o u k S 20 = 13 – 4 and

4.

= 17 –

+ + + +

ew i ev Pr

Teac he r

3. If 6 + 11 = 17, then

= 13 – 9.

= 20 = 20 = 20 = 20

© R+.

I .

C.Publ i cat i ons= 20 •f orr ev ew pur poseson y• +i =l 20 = 20

m . u

+

w ww

5. Now write some number sentences that show the connection between addition and subtraction. (a)

. te

o c . c e (c) her r o t scould be written for the sup er 6. What other number sentences that balance (b)

number 20? For example: 9 + 11 = 17 + 3

24

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Subtraction – Making a difference of 2 for numbers from 0–15

M

How many different ways can you make 2 by subtracting any 2 numbers in the grid?

2

12

r o e t s Bo r e p ok u 5 S 13

15

6

14 10

7 8 © R. I .

C .Publ i ca t i ons 1 •f orr evi ew pur posesonl y• List the numbers that when subtracted make 2.

w ww

. te

m . u

0

9

11

ew i ev Pr

Teac he r

4

3

o c . che e r o t r s super

Extension • Use the same grid to work out combinations with a difference of 3, 4, 5, 6 and 7 on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

25

Subtraction – Making a difference of 5 for numbers from 5–27

M

How many different ways can you make 5 by subtracting any 2 numbers in the grid?

8

26

27

9

13

1

6

10

21

18

5

22 14

ew i ev Pr

r o e t s Bo r e p ok u 15S 20 12 19

Teac he r

25

23

16

w ww

. te

m . u

24 7 17 © R. I .

C.Pub l i cat i o ns 11 •f orr evi ew pur posesonl y• List the numbers that when subtracted make 5.

o c . che e r o t r s super

Extension • Use the same grid to work out combinations with a difference of 2, 3, 4, 5, 6 and 7 on a separate page.

26

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Subtraction – Making a difference of 8 for numbers from 0–23

M

How many different ways can you make 8 by subtracting any 2 numbers in the grid?

8

23

2

9

r o e t s Bo r e p ok u 20 12 19 S15

13

1

18

14

5

16

10 4 7 11 © R. I .

C. Publ i c at i on17 s •f orr evi ew pur posesonl y• List the numbers that when subtracted make 8.

w ww

. te

m . u

6

21

22

ew i ev Pr

Teac he r

3

0

o c . che e r o t r s super

Extension • Use the same grid to work out combinations with a difference of 2, 3, 4, 5, 6 and 7 on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

27

Subtraction – Making a difference of 10 for numbers from 1–20

M

How many different ways can you make 10 by subtracting any 2 numbers using the numbers 1–20?

r o e t s Bo r e p ok u S 15 6 16 5 17 10

7

18

3

12

19

9

2

13

20

8

1

14

4

ew i ev Pr

Teac he r

11

0

w ww

. te

m . u

© R. I .

C.Publ i cat i ons List the numbers that when subtracted make 10.

•f orr evi ew pur posesonl y•

o c . che e r o t r s super

Extension • Use the same grid to work out combinations with a difference of 2, 3, 4, 5, 6, 7 and 8 on a separate page.

28

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Subtraction – Making a difference of 10 for numbers from 1–30

M

How many different ways can you make 10 by subtracting any 2 numbers using the numbers 1–30?

11

30

10

9

12

r o e t s Bo8 r 28 e p ok u S 14 7 26 13

Teac he r 25

16

5

24

4

23

18

1

15 6

ew i ev Pr

27

29

17

22

w ww

. te

m . u

3.

21 20 2 ©R I .

C.Pu bl i cat i o ns •f orr evi ew pur posesonl y• List the numbers that when subtracted make 10.

19

o c . che e r o t r s super

Extension • What other numbers when subtracted will make ten? (Hint: The numbers do not have to be in the grid.) Write your answers on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

29

Multiplication – Using two numbers from 1–50, make 12, 18, 20 and 24 31

26

48

27

29

21

3

32

45

7

22

r o e t s B15o r 23 e p ok u S33 28 6 12

14

30

5

34

9

17

1

25

16

44

18

40R. 10 © I .

C.35 Publ i cat i ons 24 •f or r evi ew p r pos esonl y50 • 20 4u 36

42

w ww

46

19

2

6

41

38

8

39

. te

37

m . u

11

13

ew i ev Pr

Teac he r

43

30

M

47

o c . che e r o t r s super

1. Using two numbers, how many ways can you get 12?

3. Using two numbers, how many ways can you get 20?

2. Using two numbers, how many ways can you get 18?

4. Using two numbers, how many ways can you get 24?

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Number sentence – Multiplication and division

M

Look at the multiplication grid for 24.

24 6

multiplied by

4

4

6 r o e t s B3o r by e multiplied p ok u 8 S multiplied by multiplied by

8

12

multiplied by

2

2

multiplied by

12

24

multiplied by

1

1

multiplied by

24

= 24 = 24 = 24

ew i ev Pr

Teac he r

3

= 24

= 24 = 24 = 24 = 24

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

1. If 6 x 4 = 24, then 24 divided by 4 equals 6 and 24 divided by 6 equals 4.

2. If 8 x 3 = 24, then 24 divided by

equals 8 and 24 divided by

w ww

3. If 12 x 2 = 24, then 24 divided by

equals 12 and 24 divided by

equals 2.

. te

m . u

equals 3.

o c .

Therefore 6 x c= x 2. e her r o t s sup er 5. What other number sentences could you write that balance?

4. If 6 x 4 = 24 and 12 x 2 = 24, then 6 x 4 = 12 x 2.

Extension • Complete the same strategy with the number 30 on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

31

Money combinations to make $25

M

How many different combinations of coins can you use to make $25?

$15.50

$11

$14

$12.50

$15

$11.50

$12.25

$2

ew i ev Pr

$13.50

$12.50

r o e t s $40 Bo r e p ok u S $5

Teac he r

$12.75

$1.75

$9.50

$23.75

w ww

. te

m . u

© R. I .

C.Publ i cat i ons What money combinations did you use to make $25? Write them below.

•f orr evi ew pur posesonl y•

o c . che e r o t r s super

Extension • Can you think of any other ways to make $25? (Hint: You can use the same number twice and the numbers do not have to be in the grid.)

Write your answers on a separate page.

32

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Multiplication – Using two numbers from 1–50, make 28, 32, 36 and 48

U

31

26

48

27

29

21

3

32

45

7

22

r o e t s Bo15 23 r e p ok u S 33 28 6 12

Teac he r 14

30

5

34

9

17

1

25

19

ew i ev Pr

43

13

44

18

©R .

I .

C.Pu bl i cat i o ns 40 35 10 24 •f orr evi ew pur posesonl y•

11

4

36

50

42

2

6

41

37

38

8

39

46

. te

m . u

20

w ww

16

o c . che e r o t r s super

47

1. Using two numbers, how many ways can you get 28?

3. Using two numbers, how many ways can you get 36?

2. Using two numbers, how many ways can you get 32?

4. Using two numbers, how many ways can you get 48?

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

33

Combinations of operations – Addition, subtraction and multiplication

U

How many different ways can you get 12, by using 2 numbers to add, subtract or multiply them together?

5

9

0

2

7

1

12

14

3

11

6

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• What numbers did you add together to get a total of 12?

w ww

2. What numbers did you use to find a difference of 12?

. te

m . u

1.

r o e t s Bo r e p8 ok 10 u S

15

ew i ev Pr

Teac he r

13

4

o c . che e r o t r s super

3. What numbers did you multiply together to get 12?

Extension • What three numbers when multiplied together give you a product of 12? (Hint: You can use the same number twice and the numbers do not have to be in the grid.)

Write your answers on a separate page.

34

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Combinations of operations – Addition and multiplication

U

How many different ways can you get 24, by using 2 numbers to add or multiply them together?

20

8

r o e t s Bo r e p ok u 6 15 S

12

14

9

3

4

ew i ev Pr

Teac he r

16

18

10

© R. I .

C .Publ i ca t i ons 2 12 1 •f orr evi ew pur posesonl y• 24

w ww

. te

m . u

1. What numbers did you add together to get a total of 24?

o c . che e r o t r s super

2. What numbers did you multiply together to get a product of 24?

Extension • What three numbers when multiplied together will give you 24?

• Can you find the combinations of three numbers that, when added together, will give you a total of 24?(Hint: You can use the same number twice and the numbers do not have to be in the grid.)

Write your answers on a separate page.

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

35

Recognising equivalent fractions and decimals for 0.75 ⁄20

2

15

75 cm

⁄3

⁄100

$0.75

0.75

r o e t s r ⁄ 7 ⁄B e oo p u k S 24

75

1

32

⁄8

6

750⁄100

75⁄10

⁄25

75⁄100

5

⁄4

3

ew i ev Pr

Teac he r

U

⁄100

⁄5

75

3

⁄5

⁄16

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• 12

75c

⁄24

18

w ww

What numbers in the grid represent 0.75? Write them below.

. te

36

m . u

3

4

o c . che e r o t r s super

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Fractions operations – Addition and subtraction of fractions with unlike denominators 1

⁄2

⁄10

1

⁄20

⁄8

1

⁄8

3

⁄20

19

4

⁄10

7

⁄5

1

5

10

⁄20

3

⁄5

4

3

10

⁄20

11

ew i ev Pr

Teac he r

⁄4

3

r o e t s Bo⁄ r 1 ⁄e 1⁄ p ok u S 1

1

⁄5

3

U

⁄20

⁄5

17

2

⁄8

⁄20

5

7

© R. I .

C.Publ i cat i ons ⁄ •f ⁄ i orr ev ew pu⁄r poses⁄onl y• ⁄

9

7

20

8

13

20

9

1

10

4

w ww

. te

m . u

1. What combination of fractions could you use to make a total of 1?

o c . che e r o t r s super

2. What fraction combinations could you use to make 2.55?

R.I.C. Publications® ~ www.ricpublications.com.au

Mental thinking: Using the target number strategy

37

Multiplication and division of fractions and indices

U

How many different ways can you make 36 using these numbers?

⁄2

32

1

180

r o e t s 4B 144 r e o p ok u S 288 ⁄ 2 ⁄10

Teac he r

1

3

3

9

2

ew i ev Pr

23

⁄4

1

⁄ © R18 .

I .

C.Publ i c at i ons 72 •f orr evi ew pur posesonl y• 1

⁄12

1

5

6

432

m . u

42

2

1

12

360

108

w ww

Using two numbers, how many ways can you get 36? Write them below.

. te

o c . che e r o t r s super

Extension • Using combinations of addition, subtraction, multiplication and division, can you make 36 any other ways? Write your answers on a separate page.

38

Mental thinking: Using the target number strategy

R.I.C. Publications® ~ www.ricpublications.com.au

Using addition, subtraction, multiplication and division to make 12

U

How many different ways can you make 12 using these numbers?

2.25

0.25

r o e t s Bo r e p 0.75 6 o u k S 81⁄2

3.75

2

3

4

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• What numbers did you add together to get a total of 12?

w ww

. te

7.5

8.25

22

m . u

3

1.

31⁄2

ew i ev Pr

Teac he r

4.5

23

o c . che e r o t r s super

2. What numbers did you multiply together to get the product of 12?

Extension • What other calculations could you use to make 12? (Hint: You can use the four operations, combinations of operations, brackets or indices.) Write your answers on a separate page.

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Mental thinking: Using the target number strategy

39

Using addition, subtraction, multiplication and division to make 24

U

How many different ways can you make 24 using these numbers?

1.5

23

r o e t s r 6B e oo p u k S

3

36

2

11.5

15

12

4

ew i ev Pr

9

9

17.25

6.75

Teac he r

24

72

24

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C.Publ i cat i ons 48 8I 12.5 •f orr evi ew pur posesonl y•

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1. What numbers did you add together to get a total of 24?

o c . che e r o t r s super

2. What numbers did you multiply together to get the product of 24?

Extension • What other calculations could you use to make 24? (Hint: You can use the four operations, combinations of operations, brackets or indices.) Write your answers on a separate page.

40

Mental thinking: Using the target number strategy

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Prime, composite, square and triangular numbers 1

13

7

43

4

33 29 r o e t s B r e oo p u k S 6 25 9 27

66

21

45

15

55

49

28

ew i ev Pr

Teac he r

10

U

48

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C.Publ i cat i ons 64 23 16 36 3 •f orr evi ew pur posesonl y•

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Prime

. te

Composite

Square

m . u

List the numbers in their correct category.

Triangular

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What did you find out?

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Mental thinking: Using the target number strategy

41

Linking decimals, fractions and percentages to make 1

U

How many different ways can you make 1?

20%

65%

87.5%

⁄2

1

4

⁄5

12.5%

55%

45%

50%

35%

1

⁄4

3

ew i ev Pr

r o e t s r ⁄ 50%B e oo p u k S 1

Teac he r

0.73

0.27

⁄10

8

© R. I .

C.Publ i cat i ons 0.75 40% ⁄ 80% •f orr evi ew pur posesonl y• 4

5

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m . u

What combinations can you find that will equal 1?

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Extension • What combinations can you find that will give you a number greater than 1 but less than 2? Write your answers here or on a separate page.

42

Mental thinking: Using the target number strategy

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Combinations of operations

U

Think about the sign.

You can use up to three of the four operations to make these number sentences balance. Remember to use the order of mathematical operations.

10

24

8

r o e t s Bo r e p ok u S 6

5

3

=

14

=

15

4 7 12 ©R .

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C.Pub l i cat i on s = •f orr evi ew pur posesonl y•

30

m . u

10

5

ew i ev Pr

Teac he r

3

24

w ww

15

. te

6

1

2

6

=

o c . che e r o 12 8 t r s 3 = super

6

Create your own.

=

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Mental thinking: Using the target number strategy

43

Using addition, subtraction, multiplication and division to make 8.25

U

How many different ways can you make 8.25 using these numbers?

2.375

2

r o e t s Bo 5.875 r 1.25 e p ok u S

3.75

16.5

3

5.5

0.5

1.65

ew i ev Pr

4.125

7

4

5

Teac he r

4.5

2.75

6.25

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C.Pub4.125 l i cat i ons 9.25 •f orr evi ew pur posesonl y•

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1. What numbers did you add together to get a total of 8.25?

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2. What numbers did you multiply together to get the product of 8.25?

Extension • What other calculations could you use to make 8.25? (Hint: You can use the four operations, combinations of operations, brackets or indices.) Write your answers on a separate page.

44

Mental thinking: Using the target number strategy

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Perimeter – Whole number

U

The perimeter of the desk top is 24 metres.

What could be the size of the length and the width?

1m

2.5 m

3m

9.5 m

r o e t s B r e o 8 m 1.5 m 8.5 p omk u S

6m

ew i ev Pr

Teac he r

8.25 m

7m

4m

5m

1.2 m

4.5 m

6m

3.75 m

9m

7.5 m

5.5 m

11 m

m . u

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C.Publ i cat i ons •f orr evi ew p ur pose6.5 so nl y•2 m 10 m 3.5 m 10.5 m m

w ww

Write your answers below. You may use a calculator to help you.

. te

o c . che e r o t r s super

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Mental thinking: Using the target number strategy

45

Perimeter – Decimal number

U

The perimeter of the desk top is 3.4 metres.

What could be the size of the length and the width?

1.1 m

35 cm

0.3 m

r o e t s B r e 85 cm 125 cm 85 o cmo 65 cm p u k S

0.4 m

45 cm

1.2 m

145 cm

25 cm

0.8 m

75 cm

125 cm

95 cm

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C.Publ i cat i ons •f o rr evi ew1.4p ur pos esonl y • 135 cm m 0.6 m 1.3 cm

w ww

Write your answers below. You may use a calculator to help you.

. te

46

0.5 cm

m . u

1m

0.9 m

ew i ev Pr

Teac he r

0.85 m

0.7 m

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Mental thinking: Using the target number strategy

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Area – Square cm

U

The area of the desk top is 5400 cm2.

What could be the size of the length and width?

120 cm

60 cm

110 cm

85 cm 125 cm r o e t s Bo r e p o u 67.5 cm 45 cm 72k cm S

ew i ev Pr

Teac he r

54 cm

25 cm

80 cm

75 cm

100 cm

135 cm

90 cm

w ww

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m . u

© R. I .

C.Pu bl i cat i on s Length Width Area •f orr evi ew pur posesonl y•

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Mental thinking: Using the target number strategy

47

Area – Hectares

U

r o e t s Bo r e p ok u S

What could be the size of the length and width?

10 m

6m

60 m

300 cm

200 cm

0.3 m

0.6 m

100 cm

75 m

© R. I .

C. Publ i ca t i ons 100 m 5m 120 m 80 m •f orr evi ew pur posesonl y• Length

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48

50 m

Width

Area

m . u

60 cm

ew i ev Pr

Teac he r

The area of a soccer field is recommended to be 0.60ha for players under 14 years olds.

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Mental thinking: Using the target number strategy

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Volume of a rectangular prism 2m

1m

16 m

8m

r o e t s Bo r e p ok u S 5m 96 m 10 m

48 m

24 m

1.6 m

1.5 m

3m

12 m

ew i ev Pr

Teac he r

4m

6m

U

0.5 m

© R. I .

C.Publ i cat i ons •f orr evi e w pur po sesonl yVolume • Length Width Height

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The volume of a rectangular prism is 24 cm3. What could its length, width and height be?

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Extension • Continue the table on a separate page. Can you find any more solutions?

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Mental thinking: Using the target number strategy

49

Order of operations

U

3+2x6

3+3x5

2.4 x 5 + 6

8x2–1

9x2+6

6x3–3

7x3–3

5x6–6

2.5 x 5 + 2.5

4x3+3x4

7 x 4 – 22

0.5 x 20 + 8

5.5 x 6 – 32

5 x 4 – 10 ÷ 2

20 – 1 x 5

32 x 3 – 3

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C.Publ i cat i ons What combinations make 15?

•f orr evi ew pur posesonl y•

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2. What combinations make 18?

m . u

1.

ew i ev Pr

Teac he r

12 x 2 – 6

r o e t s Bo r e p 4x5 + 12 ÷ 3 5 x 3 + 9 ÷ 3 o 4 x 3 + 12 ÷ 4 u k S

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3. What combinations make 24?

50

Mental thinking: Using the target number strategy

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Positive and negative integers

U

(+12) + (+3)

(+8) – (–4)

(+15) + (+5)

(+7) – (–13)

(+15) + (–3)

(+19) + (–4)

(–4) x (–3)

(+18) – (+3)

Teac he r

(+11) – (–4)

(+9) + (+3)

(–3) – (–18)

(–3) x (–4)

(+7) – (–5)

(–6) – (–18)

(–4) x (–5)

(+9) – (–6)

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C.Publ i cat i ons What combinations make 12?

•f orr evi ew pur posesonl y•

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2. What combinations make 15?

m . u

1.

(+9) – (–3)

ew i ev Pr

r o e t s Bo r e ok (+32) + (–12) p (+12) – (–3) (+15) – (–5) u S

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3. What combinations make 20?

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Mental thinking: Using the target number strategy

51

Develop your own ‘make the target number’ activity

U

1. Use all the operations to make this chosen number.

2. Write results in a table.

r o e t s Bo r e p ok u S

ew i ev Pr

Teac he r

3. Make up your own activity.

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4. What questions would you ask?

52

m . u

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Mental thinking: Using the target number strategy

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