problem solving with directed numbers

No products in the cart.

Directed Number Worksheets with Answers

With a focus on developing numeracy skills, this directed number worksheets with answers pack contains 1500 questions! Each worksheet contains 75 questions broken down into 5 smaller exercises. Each exercise gets progressively harder. All answers are provided at the bottom of each sheet.

There are 20 worksheets in this pack.

Download the Directed Number Worksheets with Answers Pack

You will need to be a member to download the resources. You can become a member by clicking on the button below.

Example Questions and Full Solutions

When answering these styles of questions it is important to be methodical in your approach. You can often simplify the question by replacing instances of 2 operators that are placed together with a single operator. The table below can be useful. Basically, the rule states that when you have 2 operators which are the same, replace them with a plus. When you have 2 operators which are different, replace them with a minus.

Below you will find a couple of example questions from the worksheets. There are 5 exercises on each worksheet and each exercise contains randomised questions. This provides you with plenty of opportunities to strengthen your skills. Remember, all answers are provided at the bottom of each sheet so that you can check your work.

$$ \begin{align} &\quad-4- \raise 5px {-} 7\\\\ &=-4+7\\\\ &=3 \end{align} $$

The first step is to simplify the expression by looking for any 2 operators next to each other and replacing them with a single operator (using the rules in the table above).

So, replace the 2 minus symbols with a plus:

$$ \begin{align} &\quad-4\color{red}{-} \raise 5px {\color{red}{-}} 7\\\\ &=-4+7\\\\ &=3 \end{align} $$

$$ \begin{align} &\quad-4- \raise 5px {-} 7 + \raise 5px {-} 3 \times 2 + \raise 5px {-} 8 \div 4\\\\ \end{align} $$

Step 1 is to go through and replace all instances of consecutively placed operators with a single operator:

$$ \begin{align} &\quad-4- \raise 5px {-} 7 + \raise 5px {-} 3 \times 2 + \raise 5px {-} 8 \div 4\\\\ &=- 4 + 7{} -{} 3 \times 2 {}-{} 8 \div 4\\\\ \end{align} $$

Now perform the multiplication and division operations:

$$ \begin{align} &\quad-4- \raise 5px {-} 7 + \raise 5px {-} 3 \times 2 + \raise 5px {-} 8 \div 4\\\\ &=- 4 + 7{} -{} 3 \times 2 {}-{} 8 \div 4\\\\ &=-4 + 7 {}-{} 6{}-{}2\\\\ \end{align} $$

Now perform the addition and subtraction calculations, left to right:

$$ \begin{align} &\quad-4- \raise 5px {-} 7 + \raise 5px {-} 3 \times 2 + \raise 5px {-} 8 \div 4\\\\ &=- 4 + 7{} -{} 3 \times 2 {}-{} 8 \div 4\\\\ &=-4 + 7 {}-{} 6{}-{}2\\\\ &=3{}-{}6{}-{}2\\\\ &=-3{}-{}2\\\\ &=-5 \end{align} $$

Tutoring Request

Teaching tips for directed number.

Directed numbers, also known as integers, are numbers that can be positive or negative. When learning directed numbers, students may face a few common issues:

• Understanding the concept of negative numbers: Negative numbers can be confusing for students because they are less familiar and do not have a physical representation in the real world.
• Grasping the concept of opposites: Students may have difficulty understanding that a positive number and a negative number of the same magnitude are opposites and that they cancel each other out when added together.
• Applying the rules of arithmetic: Students may have difficulty remembering the rules for performing arithmetic operations with directed numbers, such as the fact that the product of two negative numbers is positive and the product of a positive and a negative number is negative.
• Solving real-world problems: Students may have difficulty applying directed numbers to real-world situations, such as calculating the change in temperature or the difference in height between two points.

To help students overcome these challenges, it can be helpful to provide concrete examples and hands-on activities, such as using a number line or manipulating physical objects (such as chips or blocks) to represent positive and negative numbers. It can also be helpful to emphasise the importance of directed numbers in everyday life and to encourage students to practice solving problems involving directed numbers.

TIP 1: USE NUMBER LINES

Until students become comfortable operating on negative numbers, you should allow them to use number lines. Some students prefer horizontal number lines while others may prefer vertical number lines (these can be linked directly to temperature and analog thermometers).

TIP 2: USE A TABLE OF REPLACEMENT OPERATORS

Use the table as previously shown to get students to replace instances of consecutive operators with a single operator. This will often significantly simplify the expression.

TIP 3: USE A RED PEN AND UNDERLINE THE NEXT OPERATION/CALCULATION

This is one that I use for students who tend to try to complete too many operations at once. I ask them to underline the first operation/calculation in red. They then have to work from left to right writing out the same line as the one above. They must stop when they hit the red line, perform the calculation, and then continue writing out the rest of the line.

For example:

$$ \begin{align} &\quad 6{}-{}\raise 5px{-}4\times{}-{}2\\\\ \end{align} $$

Get students to underline, in red, the first stage of the process (simplification of the operators):

$$ \begin{align} &\quad 6\color{red}{\underline{{}\color{black}{-}{}\raise 5px{\color{#464646}{-}}}} 4\times{}-{}2\\\\ &=6+4\times{}-{}2 \end{align} $$

The next operation is multiplication. Again, underline in red.

$$ \begin{align} &\quad 6\color{red}{\underline{{}\color{black}{-}{}\raise 5px{\color{#464646}{-}}}} 4\times{}-{}2\\\\ &=6+\color{red}{\underline{\color{#464646}{4\times{}-{}2}}}\\\\ &=6+{}-{}8 \end{align} $$

Now replace the two operators with a single operator:

$$ \begin{align} &\quad 6\color{red}{\underline{{}\color{black}{-}{}\raise 5px{\color{#464646}{-}}}} 4\times{}-{}2\\\\ &=6+\color{red}{\underline{\color{#464646}{4\times{}-{}2}}}\\\\ &=6\color{red}{\underline{\color{#464646}{{}+{}-{}}}}8\\\\ &=6{}-{}8\\\\ &=-2 \end{align} $$

Latest Resources

Integers on Number Lines

Basic Integration 20 Questions and Answers PDF

10 Random Revision Questions

Order of operations worksheets with answers, the golden ratio, leave a reply cancel reply.

Your email address will not be published. Required fields are marked *

MathsAcademy.com.au

• PRIVACY POLICY
• TERMS AND CONDITIONS
• TECHNOLOGY IN EDUCATION
• EDUCATION BLOG
• ONLINE COURSES
• QUESTION PACKS
• EXAM PAPERS AND SOLUTIONS

Directed Numbers

Related Pages Directed Numbers Worksheet More GCSE Maths Lessons Math Worksheets

These lessons help GCSE/IGCSE Maths students learn about directed numbers and how to perform operations on directed numbers.

How to add and subtract directed numbers? Rule 1: If the signs are the same then add the numbers and keep the original sign. Rule 2: If the signs are different then subtract: Big number minus small number. Keep the sign of the big number.

How to multiply and divide directed numbers? Rule 1: The answer is positive when all the numbers are positive or when there is an even number of negative signs in the problem. Rule 2: The answer is negative when there is an odd number of negative signs in the problem.

The following diagram gives the rules for the addition, subtraction, multiplication and division of directed numbers. Scroll down the page for more examples and solutions on the operations for directed numbers.

Directed Numbers Adding and Subtracting

Directed Numbers Multiplying and Dividing

Rules for adding, subtracting, multiplying and dividing positive and negative numbers

GCSE Maths - Negative Numbers (Adding, Subtracting,Multiplying, Dividing)

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Directed (Negative) Numbers

Two options for teaching directed numbers:

Using double-sided counters is a new lesson that shows clearly the four operations with directed numbers. Includes questions from VariationTheory.com . Use MathsBot manipulatives to support teaching and student tasks.

The number line lesson on directed numbers starts at the beginning with where we encounter them in real life. The PowerPoint features a video by Fuse School (found here if you have problems playing in the PowerPoint) with questions to keep students focused. Tasks on adding and subtracting, then multiplying and dividing, are differentiated and answers are included.

• printable worksheets
• cool math games
• fun math projects
• math tutorials
• games & puzzles

Need more practise?

There are over 100 printable high school math worksheets on Funmath.com

Browse our Numeracy Worksheets ...

Directed Numbers Math Review

math tutorials > directed numbers

Directed numbers are just numbers with a positive or negative sign in front of them!

Directed numbers are signed numbers i.e. numbers with a positive or plus sign (+) in front of them or a negative or minus sign (-) in front of them  e.g.  +6, -2, +7 etc.  All these whole numbers (and 0) grouped together are called integers .

These numbers can be represented on a number line.

A positive number signifies a movement to the right. A negative number signifies a movement to the left.

A number lacking a sign is positive.

Adding and Subtracting.

 e.g.  +2 + -3 + 2

+2 + -3 + 2 = +1

Note:   A double negative changes to a single positive e.g.  +3 - -3 = +3 +3 = +6 A  negative sign with a positive sign changes to a single negative e.g.  3 + -4 = 3 – 4 = -1 3 + -4 = 3 – 4 = -1

Multiplying and Dividing.

The following rules apply:

The same rules apply for division

Math problems for you to try.

Note:  Remember the correct order of operations .

• 9 - (6 - 9)
• -11 - (3 + 4)

Solutions:   1)  -3    2)  +7    3)  +12    4)  +12    5)  -18    6)  +12

Printable Math Worksheets & Problems | Cool Math Games | Fun Math Projects | Math Review & Tutorials | High School Math Games & Puzzles | Shop

About this site | Terms of use | About our advertising and cookies

© Copyright 2000 to 2018 Funmaths.com. All rights reserved.

All effort has been made to source copyright material. If appropriate acknowledgement of copyright material has not been made we would like to rectify this. Please contact us .

The Educational Company of Ireland

• [email protected]
• +353 1 4500 611

Digging Deeper into … Directed Numbers (5th & 6th classes)

• February 25, 2018

Tags : Algebra digging deeper Directed Numbers

Category : Uncategorized

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of area, please check out the following post:  Dear Family, your Operation Maths Guide to Directed Numbers

In Operation Maths for 5th Class the children are formally introduced to the concept of directed numbers (from the strand of algebra) and are enabled to:

• Explore and identify directed numbers, i.e. numbers above and below zero, including zero
• Describe and record directed numbers as positive or negative
• Use directed numbers to represent real-life situations
• Compare and order directed numbers
• Solve problems involving directed numbers.

In Operation Maths for 6th class, this knowledge and understanding is revised and extended to include the addition of positive and negative numbers.

Directed numbers (formally known as integers, the set of positive and negative whole numbers including zero) can be a very difficult concept for some children for a number of reasons:

• Being able to explore numbers less than zero requires a solid understanding of zero, an abstract concept in itself since it indicates the absence of something
• Many children find it difficult to comprehend how something can be less than zero. Therefore, when ordering it can be difficult for many children to appreciate that 0 is greater than -1, -2, etc.
• Operations involving integers are particularly troublesome as having similar signs so close together can be very confusing.

Use real-life contexts

In fifth class, since this is the first time that the children have been formally introduced to negative numbers, it is essential that this happens through reference to real-life contexts, e.g. temperature, buttons in a lift, goal difference in soccer league tables, depths in a swimming pool, or any other context with which the children can readily identify. Other real-life contexts such as bank account balances, elevations, etc., can also be used.

Use ‘positive’ and ‘negative’ rather than ‘plus’ and ‘minus’

Whilst real-life contexts are useful, they can also be confusing, particularly when plus and minus are used to describe directed numbers. Insist that the children use the language of positive and negative, since that is more correct mathematically. Using positive and negative can also reduce complications in future classes when calculating with directed numbers, e.g. avoiding having to say three minus minus five for 3-(-5). This is particularly important when the children progress to adding positive and negative numbers: for example (–3) + (+9) should be read as ‘negative three add/plus positive nine’ rather than ‘minus 3 plus plus 9’. It is also important that the children recognise that positive numbers can be written either with, or without, the positive sign, therefore we can assume that any number without a sign is positive.

To further reduce complications with written forms of directed numbers, raised signs should be used for the positive and negative (as shown in image below) and brackets should be used to make it easier for the child to distinguish between the operations signs (i.e. + and -) and the directed numbers signs.

Representing directed numbers

Since negative numbers are so abstract, it is vital that opportunities to represent them concretely and pictorially are maximised. Many children consider only the cardinality of numbers, i.e. that they represent the number of objects in a set, whereas understanding of directed numbers relies on understanding the ordinality of numbers; this is why visual representations are so vital.

One way to do this, is to use a counting stick, if available. Start with the counting stick in a vertical position, asking the children that if the centre line is now representing zero, can they identify the points above and below that?

• Initially describe the positions as “one above zero”, “two below zero” etc
• Then, when appropriate, introduce positions as “positive three”, “negative four” etc
• Development: if each interval now represents two/ten/five as opposed to one, how does this change the identity of the positions?
• Finally, move the counting stick into a horizontal position, so that the negative numbers are now to the left of the centre/zero and the positive to the right (as the children see it).

Operation Maths users can also use the counting stick eManipulative to model the counting stick in a horizontal position, as shown below (Hint: zoom the screen in to 150% to view the numbers better).

Number lines are also very useful. When using number lines (or a counting stick), emphasize that numbers get smaller in value as you move above/to the left of zero, and larger as you move below/to the right of zero . If possible, have a number line that includes negative numbers on permanent display in the classroom.

The Operation Maths MWBs can also be used to create a dynamic number line for ordering activities. For example, ask individual children to write a specific integer on their MWB. Then ask a group of children, each with different integers,  to stand out and put their MWBs in order. These numbers could even be hung from a ‘washing line’ using clothes pegs.

In sixth class, to better illustrate the processes involved with the addition of integers, teachers are encouraged to use the positive and negative chips, that accompany the sixth class books as part of the free ancillary resources (they’re on the same sheet as the images of the base ten block for modelling decimal numbers).

The teacher can also use the positive and negative chips built into the Sorting eManipulative to model calculations on the class IWB, as shown below (the positive and negative chips are located at the end of the numbers tab).

The children can then progress to using number lines to both reinforce their understanding of addition with integers and extend it to bigger numbers. A physical number line that the children can walk along would be ideal initially, as this suits the kinaesthetic learners and the bigger nature makes it easier for teachers to assess the children’s ability to use the number line before they move to smaller scale number lines in books and copies.

Further Reading and Resources

• Dear Family, your Operation Maths Guide to Directed Numbers includes practical suggestions for supporting children, and links to a huge suite of digital resources.
• Virtual Maths Manipulatives for Algebra: Lots of tools that can be used in many different ways to explore the Algebra concepts, including directed numbers.
• Operation Maths users don’t forget to use the Counting Stick eManipulative as mentioned earlier and to check out the  Maths Around Us video for this topic on  Edco Learning . Check out the first page of the Directed Numbers chapter in the Pupils Books for a quick synopsis of the suggested digital resources and then refer to the Directed Numbers chapter in the TRB for more detailed information.
• NRICH : selection of problems, articles and games for negative numbers. In particular, check out Number Lines in Disguise .
• Check out this  Pinterest Board of Algebra ideas

Subscribe to Blog via Email

Enter your email address to subscribe to this blog and receive notifications of new posts by email.

Email Address

Blog Archive

• February 2021
• January 2021
• November 2020
• September 2020
• August 2020
• February 2020
• January 2020
• November 2019
• September 2019
• February 2019
• January 2019
• December 2018
• November 2018
• October 2018
• September 2018
• August 2018
• February 2018
• January 2018
• December 2017
• November 2017
• October 2017
• September 2017
• August 2017
• February 2017
• January 2017

Office Hours Monday - Friday 8:45am - 4:45pm Lunch 1:00pm - 1:45pm

• Local Edco Representatives
• Cookie Policy
• Accessing Resources
• Junior Infants – Second Class
• Teacher Testimonial
• Third – Sixth Class

© 2019 The Educational Company of Ireland

Directed Numbers Diagnostic Questions

FREE DOWNLOAD

Help your students prepare for their Maths GCSE with these free Diagnostic Questions on Directed Numbers.

• Diagnostic questions are a quick and easy way of assessing your students’ knowledge and understanding of a particular topic
• There are 20 multiple choice questions, each designed to assess each of the key skills required to master directed numbers.
• Each question has one correct answer and three carefully chosen incorrect answers that are designed to identify and highlight fundamental misconceptions
• The questions include topics such as calculations with negative numbers, powers, fractions, real life worded problems and substitution.

Unlock access to download your free resource

To receive this printable resource and regular emails with more free resources, blog posts and other Third Space updates, enter your email address and click below.

To receive this resource and regular emails with more free resources, blog posts and other Third Space updates, enter your email address and click below.

• Job Title * Job title Headteacher Deputy Head Assistant Head Head of Maths Deputy Head of Maths Trust Leader Primary SLT Other SLT Secondary Maths Teacher Primary Teacher Teaching Assistant Tutor Parent Student Other
• Comments This field is for validation purposes and should be left unchanged.

You can unsubscribe at any time (each email we send will contain an easy way to unsubscribe). To find out more about how we use your data, see our privacy policy .

Raise maths attainment across your school with hundreds of flexible and easy to use GCSE maths worksheets and lessons designed by teachers for teachers.

Related worksheets

Money Problems Exam Questions

Place Value Exam Questions

Adding And Subtracting Negative Numbers Exam Questions

Reciprocal Exam Questions

Popular for gcse.

Do you have students who need additional support to achieve their target GCSE maths grade?

There will be students in your class who require individual attention to help them succeed in their maths GCSEs. In a class of 30, it’s not always easy to provide.

Help your students feel confident with exam-style questions and the strategies they’ll need to answer them correctly with personalised online one to one tutoring from Third Space Learning

Lessons are selected to provide support where each student needs it most, and specially-trained GCSE maths tutors adapt the pitch and pace of each lesson. This ensures a personalised revision programme that raises grades and boosts confidence.

Privacy Overview

Resources you can trust

Directed numbers

All reviews

Have you used this resource?

Resources you might like

Or search by topic

Number and algebra

• The Number System and Place Value
• Calculations and Numerical Methods
• Fractions, Decimals, Percentages, Ratio and Proportion
• Properties of Numbers
• Patterns, Sequences and Structure
• Algebraic expressions, equations and formulae
• Coordinates, Functions and Graphs

Geometry and measure

• Angles, Polygons, and Geometrical Proof
• 3D Geometry, Shape and Space
• Measuring and calculating with units
• Transformations and constructions
• Pythagoras and Trigonometry
• Vectors and Matrices

Probability and statistics

• Handling, Processing and Representing Data
• Probability

Working mathematically

• Thinking mathematically
• Mathematical mindsets
• Cross-curricular contexts
• Physical and digital manipulatives

For younger learners

• Early Years Foundation Stage

Advanced mathematics

• Decision Mathematics and Combinatorics
• Advanced Probability and Statistics

Resources tagged with: Positive and negative numbers

There are 23 NRICH Mathematical resources connected to Positive and negative numbers , you may find related items under Place value and the number system .

Vector Gem Collector

Use vectors to collect as many gems as you can and bring them safely home!

Number Lines in Disguise

Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

Making Sense of Positives and Negatives

This article suggests some ways of making sense of calculations involving positive and negative numbers.

Up, Down, Flying Around

Play this game to learn about adding and subtracting positive and negative numbers

Strange Bank Account

Imagine a very strange bank account where you are only allowed to do two things...

Vector Racer

The classic vector racing game.

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Adding and Subtracting Positive and Negative Numbers

How can we help students make sense of addition and subtraction of negative numbers?

The picture shows a lighthouse and some underwater creatures. Can you work out the distances between some of the different creatures?

Tug Harder!

In this game, you can add, subtract, multiply or divide the numbers on the dice. Which will you do so that you get to the end of the number line first?

Minus One Two Three

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

Consecutive Negative Numbers

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

First Connect Three

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

Connect Three

In this game the winner is the first to complete a row of three. Are some squares easier to land on than others?

Swimming Pool

In this problem, we're investigating the number of steps we would climb up or down to get out of or into the swimming pool. How could you number the steps below the water?

Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Being Collaborative - Primary Number

Number problems at primary level to work on with others.

First Connect Three for Two

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

Strange Bank Account (part 2)

Investigate different ways of making £5 at Charlie's bank.

Positive about Negative Numbers

In this article for teachers, Liz Woodham describes resources on NRICH that can help primary-aged children get to grips with negative numbers.

The History of Negative Numbers

This article -useful for teachers and learners - gives a short account of the history of negative numbers.

Negative Numbers

A brief history of negative numbers throughout the ages

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

Multiplying and Dividing Negatives Textbook Exercise

Click here for questions.

multiplication, subtraction, negative numbers

GCSE Revision Cards

5-a-day Workbooks

Primary Study Cards

Privacy Policy

Terms and Conditions

Corbettmaths © 2012 – 2024

• International
• Education Jobs
• Schools directory
• Resources Education Jobs Schools directory News Search

Maths: Directed number puzzles

Subject: Mathematics

Age range: 11-14

Resource type: Game/puzzle/quiz

Last updated

20 January 2015

• Share through email
• Share through twitter
• Share through linkedin
• Share through facebook
• Share through pinterest

Creative Commons "Sharealike"

Your rating is required to reflect your happiness.

It's good to leave some feedback.

Something went wrong, please try again later.

Empty reply does not make any sense for the end user

My 6th graders enjoyed the break from working problems.

rebeccalessing

A fun resource with different types of activity

mathsteacher42

Report this resource to let us know if it violates our terms and conditions. Our customer service team will review your report and will be in touch.

Not quite what you were looking for? Search by keyword to find the right resource: