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.

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## 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.

TWO OPERATORS | REPLACE WITH |

+ + | + |

– – | + |

+ – | – |

– + | – |

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.

**EXAMPLE 1:**

$$

\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}

$$

**EXAMPLE 2:**

$$

\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.

TWO OPERATORS | REPLACE WITH |

+ + | + |

– – | + |

+ – | – |

– + | – |

### 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}

$$