In a Year 9 Entry mathematics test, the scope of basic assumed knowledge is wide.

While it used to be smaller, as the competition has grown, so has the scope.

In this checkpoint, we’ll get you up to date with all the basic ‘assumed’ knowledge that you’ll need in order to then to the harder maths in the following checkpoints.

This is a long checkpoint but it’s important to make sure you’re fully across it as they’ll likely appear in some form in more complicated questions.

Let’s go through them now.

Numbers

Numbers exists on a number line. Numbers greater than 0 are ‘positive’ numbers. Numbers less than 0 are ‘negative’ numbers.

Here are rules when dealing with negative and positive numbers:

• - and + = -
• - and - = +
• + and + = +
• If there are 2 signs near each other, change to 1 sign using above rules.
• If adding 2 numbers with the same sign, keep the sign and add the numbers.
• If adding numbers with opposite sign, keep the sign of the number furthest from zero then subtract the numbers ignoring the sign.
• To minus 2 numbers, change minus sign to addition and change the sign of the number being subtracted to its opposite
• Multiplying and dividing is the same and follows rules 1 to 3 in determining the end sign.

Ordering of operators

There is an order in which operations should be done which follows this rule.

B rackets O ver D ivisio n M ultiplication A ddition S ubtraction

This means that you don’t just calculate from left to right. You need to select parts to calculate first in the above order. If you don’t do things in the set order, you can end up with the wrong answer.

Let’s see how this works (and also how negative / positive number rules come into play) through the following examples below.

Example Question/s

Watch video for explanation of the following question/s:

-2 + -3 x 2 - -4 = ?

A -14 B 14 C -4 D 4

-(-4)^ 2

A -16 B -12 C 16 D 8 E None of these

Using the addition pyramid, the number in the box between 5 and 15 at the fourth layer will be

A 15 B 8 C 10 D 20

The number in the box between 1 and 2 at the second layer will be

A 0 B 1 C 2 D 3

Key Rules to remember

• Ordering of operators is: Brackets Over Division Multiplication Addition Subtraction
• Number exist on a number line – know the rules when dealing with negative and positive numbers.

In this checkpoint, we’ll go through fractions separately and then percentages and decimals.

Fractions

Fractions are parts of a whole and written as [X/Y] where the top number is the part (numerator) and the bottom number is the number of parts it's divided by (denominator)

There are different types:

Proper – top number is small than the bottom.

Improper – top number is bigger than the bottom.

Mixed number – whole number and then a proper fraction next to it.

We’ll go over the following ‘must-know’ list below at a high level using the 2 fractions below as an example (the video will show you how it's done):

[6/12] and [2/8]

‘Must-know’ List – See how you can learn to do the following:

• Simplifying – look at factors.
• Converting to same denominator – look at multiples.
• Adding – same denominator, keep denominator and add top numbers.
• Subtracting – same denominator, keep denominator and minus the top numbers.
• Multiplying – multiply across both top and bottom.
• Dividing – swap the denominator and numerator around for one of the fractions and then multiply.
• Comparing – same denominator and compare the top numbers.

Percentages and Decimals

Decimals and percentages are the same as fractions but they're expressed in different ways.

Percentages are expressed as fractions out of 100.

Here are some fractions: 1/100, 2/100, 10/100, 20/100, 17/100, 83/100

Here are their corresponding percentages: 1%, 2%, 10%, 20%, 17%, 83%

Let's look at decimals now… if it's a fraction, why do we need this? It's just a different form of expression. What is the most commonly seen decimal? It’s the expression of price e.g. $2.50

Let’s see how you can convert from one to another:

• To convert from fractions to decimal—Divide the top number by the bottom number. E.g. 10/20
• To convert a decimal to a fraction—use the place value of numbers in the decimal. Remember decimals go up by 10ths.

Common Fractions, Decimals and Percentages

Fraction	Decimal	Percent
1/20	0.05	5%
1/10	0.1	10%
1/5	0.2	20%
1/4	0.25	25%
2/5	0.4	40%
1/2	0.5	50%
3/5	0.6	60%
3/4	0.75	75%
4/5	0.8	80%
1, 2/2, 4/4	1.0 or just 1	100%
2, 4/2, 8/4	2.0 or just 2	200%

Decimal Movement

Here are some shortcuts:

The short cuts for multiplication

• If additional zeros added on (a) If nothing else changes, decimal point on answer moves right by the number of zeros that were added. E.g. 10 x 1.1 = 11; 100 x 1.1 = 110 (point moved one unit right due to extra zero, every time it moves right it will add a zero unless there is already a number there e.g. 10 x 0.123 = 1.23; 100 x 0.123 = 12.3)
• If zeros taken off (a) If nothing else changes, decimal point on answer moves left by the number of zeros that were added. E.g. 100 x 1.1 = 110; 10 x 1.1 = 11 (point moved one unit left due to zero taken away).

The short cuts for division

• Move decimal to the left if the 'number being divided' is reduced and/or if the 'number dividing by' increases and Move decimal to the right if the 'number being divided' is increased and/or if the 'number dividing by' decreases

Tips and Tricks

• A seemingly whole number is really a number with a decimal point at the end with zeros. E.g. 145 = 145.000000 = 145.000 = 145.0
• This is a reason why if moving a decimal point to the right/left and there is no existing number, we add a zero to it because there are already existing 'zero' numbers

Example Question/s

Watch video for explanation of the following question/s:

If 20 ÷ 2.50 = 8.0, then 20 ÷ 250 = ?

A 800 B 8 C 0.8 D 0.08

Paloma made 2000ml of lemonade by mixing 750ml of lemon juice with 250ml of syrup with the remainder being water. What fraction of the lemonade was just water?

A 3/4 B 4/4 C 3/8 D 4/8

Solve for x

[(x-7)/6] + [1/2] = [8/6]

A X = 12 B X = 1/3 C X = 1 1/3 D X = 5/6 E 8

The next 3 questions refer to the following information:

This multiplication table shows that, for example 50 x 0.5 = 25

Multiplied by	50	5	0.5	0.05
50	2500	250	25	2.5
5	250	25	2.5
0.5	25	2.5		0.025
0.05	2.5	0.25	0.025

Using the number pattern shown in the table

0.5 x 0.5= ?

A 2.5 B 25 C 0.25 D 250

From the information given in the table.

0.025 ÷ 0.5 = ?

A 0.5 B 2.5 C 250 D 0.05

From the information given in the table

500 x 0.5 = ?

A 250 B 25 C 25 D 500

If 0.4 x 0.6 = 0.24, what is 40 x 0.06 = ?

A 0.0024 B 0.024 C 2.4 D 24

Key Rules to remember

• Fractions, Decimals and Percentages are the same thing expressed differently.
• Know how to add, subtract, multiply, divide and convert.
• Follow the short cuts with decimal points.

In this checkpoint we’ll look at units and how to convert them and look at how to start visualising inequalities and equivalence to make it easier to solve problems. We’ll also look at approximations and how to address such questions.

Common units

Here are common units you need to know for your exam.

Distance units

Units that measure distance i.e. how long something is (the length) such as the length of a table, from smallest unit size to largest unit size are: millimetres (mm), centimetres (cm), metres (m) and kilometres (km).

Here are their conversions:

• 1cm (cent = 100) = 1/100 of a m = 10mm = 1/100,000 of a km.
• 1 m = 100 cm = 1/1000 km = 1000 mm
• 1 km (kilo = 1000) = 1000 m = 100,000 cm = 1,000,000 mm

Or expressed differently, it’s:

• 10 mm = 1 cm
• 100 cm = 1 m
• 1000 m = 1km

To convert larger units to smaller units, multiply. For 1km to be expressed in metres, do 1 x 1000 = 1000 m.

To convert small units to larger sized units, divide. For example, 1 metre to 1km, divide 1 by 1000, therefore, 1 m = 0.001 km (or 1 / 1000 km).

This rule also applies to mass, time units and volume units.

Mass units

Mass units show weight – e.g. heavy or light. The common units are grams and kilograms and they equate each other in the following way: 1 kg = 1000 g

Time units

Time units show…time. The common units are seconds, minutes and hours. They equate each other in the following way:

• 60 seconds = 1 minutes
• 60 minutes = an hour

Volume units

Volume units show the fill of something, usually in a container. The common units are millilitres and litres and they equate each other in the following way: 1 L = 1000 mL

Equivalence and Inequalities

An inequality compares 2 values so that one number is different from the other in the following way:

• greater than (>)
• less than (<)

When there are two values and they are the same as each other, they are equivalent.

For example:

2 + 6 = 3 + 4 + 1

If values can be equal to each other OR greater than or less than, they are expressed like this:

• greater than or equal to ≥
• less than or equal to ≤

In an exam, a good way to depict inequalities and equivalence is to do it visually.

For example, if a question asks you:

Container A holds 5 ml of water. Container B holds 25 ml of water. How much water needs to be poured into one jug so that they both have the same amount of water?

To answer this question, equivalence and inequality comes into play and visualising this allows you to clearly see what to do.

You can see from the image that to find out what to do, you need to add the amounts together (5 + 25) and then divide by 2 to get the ‘equal’ water amounts and then, to get the amount poured (the yellow block) you need to take away 5 from 15 (30 / 2) which is 10. So, you’d need to pour 10 ml into the other jug.

Visualising allows you to see what’s happening to make it easier for you to do problem solving.

Approximation questions

Approximation questions may ask you for:

• latest time
• earliest time
• nearest dollar
• nearest cent

It’s important that you just don’t round up or down based on the nearest number (for example, if it’s 9.4, it’s not always correct to say 9). Instead, you need to focus on the content of the question as that will determine whether the nearest approximate number should be higher or lower.

For example, there may be a question like this:

I am building a house and require bricks. If I have calculated the number of bricks that I require to be 743892.3, what is the approximate number of bricks that I have to buy in order to complete the house? I want to buy the lowest quantity possible to complete the house.

A 743800 B 743892 C 743900 D 743950

The answer is C. Even though 743892 is the closest number rounded down, you’ll need more than that (0.3 of a brick) to complete the house so you’ll need to get 743,900.

Example Question/s

Watch video for explanation of the following question/s:

A wooden box filled with apples weighs 10kg. When it is half full, the wooden box and apples together weigh 6kg.

How much does the wooden box weigh?

A 12 kg B 2 kg C 8 kg D None of these

This box weighs 4.2 tonnes. This cylinder weighs 50kg.
How many cylinders are needed to balance the box?

A 21000 cylinders B 300 cylinders C 84 cylinders D 85 cylinders

Minnie has 8 kg of cashew nuts and peanuts in her basket. A quarter of the nuts in her basket are peanuts. How many kilograms of cashew nuts does the basket contain?

A 2 kg B 4 kg C 5 kg D 6 kg E None of these

To fence one side of a plot, 28 posts placed 4 m apart are required. How many posts are required if the posts are placed 3 m apart?

A 39 B 36 C 37 D 40

Key Rules to remember

• To get from larger units to smaller units, multiply.
• To get from smaller units to larger units, divide.
• Use visualisation to problem solve with equivalence and inequalities.
• Approximations should be based on the details and requirements of the question, not just rounding up or down based on closest number.

Transposing and Solving Worded Questions

Sometimes the hardest part in a mathematics question is not the actual solving of the formula or calculation. It's figuring out what the question is and what needs to be calculated in the first place is. These worded problems require you to:

• Translate from English to Math.
• Then work out the calculation.

Here are some key words and their symbols.

• Plus Sign – sum, plus, added together, combined, increased by.
• Subtraction Sign – minus, less, difference, decreased by
• Division Sign – over, divided by, quotient, out of
• Multiplication Sign – product, multiplied by, twice (x2), thrice (x3)
• Equals Sign – equals, the same as, totals, equivalent to

If there is an unknown number, you can denote it/represent it with x, y, z or any other symbol you feel comfortable with, even "?"

Often, you have to change the worded question into a formula and then solve.

Other ways to solve these questions are to:

• Substitute in the 'answer' option numbers to your formulas e.g. start with the easy and small numbers and then solve.

Venn diagrams

Venn diagrams are used to illustrate reasoning and they can be very useful when dealing with worded problems.

Note that with Venn diagrams:

• The population is in the box.
• There is usually one attribute/thing represented by 1 circle and another by the other circle (and there could be more than 2 circles).
• If something isn’t in any of the circles but in the box, it means that it is still part of the population but doesn’t have any of the attributes.

Let’s discuss the following Venn diagrams below. Note in particular the Venn diagram that has ¬A ∧ ¬B (not A and not B)—the red shaded part means that there is part of the population that doesn’t have A and that doesn't have B. For example, A is that you have black hair and B is that you have red hair. There may be people in the population (in the box) that won’t be represented in any of the circles at all, and these people won’t have red hair and they won't have black hair (they may have hair that is another colour).

Example Question/s

Watch video for explanation of the following question/s:

Annie has a discount card at the mall. For every $25 cost she buys, she only pays $23. How much she needs to pay if she buys a cost of $300?

A $276 B $298 C $288 D $245 E None of these

The third of a number is decreased by itself and the result is -8. An equation to find the value of x would be:

A 3x – x = -8 B 3/x – x = -8 C x – 3x = -8 D [x/3] - x = -8 E [x / (3 – x)] = -8

The following Venn diagram below shows the total of number of students in a classroom and the subjects they study.

Answer the following questions:

• How many students are in the classroom in total?
• How many students do art?
• How many students do English?
• How many students do at least 2 subjects?
• How many students do not do any of the 4 subjects listed?

Key Rules to remember

Translate from English to formulas – find out what the question asking you to do exactly by:

• Highlighting key figures. Remember that:
• “Plus” sign – sum, plus, added together, combined, increased by
• Subtraction Sign – minus, less, difference, decreased by
• Division Sign – over, divided by, quotient, out of
• Multiplication Sign – product, multiplied by, twice (x2), thrice (x3)
• Equals Sign – equals, the same as, totals, equivalent to.

Venn diagrams:

• Show total population of a group that:
• Have certain attributes shown in circles or overlapping attributes (shown by overlap in circles)
• Do not have certain attributes by space outside of circle but within the box.

Reading Graphs

The purpose of graphs is to convey information, usually quantitative information.

There are different types of graphs such as:

• Pie
• Bar
• Line

Graphs have information that is plotted (2 dimensions) on a y-axis and on a x-axis. Some of the more difficult questions may have graphs with 2 y-axes (one on the left and the other on the right side).

Therefore, when dealing with questions involving graphs, you are generally looking to:

• Locate information from within the graph
• Work with that information to answer a question. The question may require further calculations in order to answer the question.

Common mistakes

The common mistakes made by students when reading graphs are:

• Not paying attention to the units. Sometimes, figures are expressed in thousands, hundreds etc… this affects the selection of the answer option.
• Confusing the x and y axis information.

How to deal with questions involving graphs

1. Read the question and clarify the requirement. What do you need to find in the graph? This way you focus your efforts when looking at the graph.
2. Read the graph – look at the heading, legend and labels on any axes. Noting down any key information e.g. units, what the graph is about. Does the graph have different layers of information? For example, there is often ‘general’ information (like trends) to be obtained from the graph and you can also find specific information. Are there any ‘outliers’? Always work starting with broader information and then ‘zoom in’ for the details.
3. Locate the 1st axis where your information should be found, and then look at any subsequent axes, one by one. You can trace your finger on the axis to visually confirm that you are looking at the correct corresponding data for the axis (as confusion between axis and data can occur).
4. Once located, do any required calculations and then, select your answer.

We're going to look at some graphs now and discuss the information that can be obtained for the following graphs. Please watch video.

Example Question/s

Watch video for explanation of the following question/s:

Use the following graph to answer the next 2 questions.

The graph shows the number of hours a year 8 group spent doing their household chores for one week.

How many students did household chores for more than 7 hours per week?

A 16 B 22 C 18 D 26 E None of these

How many students did household chores for less than 10 hours in the week?

A 18 B 24 C 30 D 16 E None of these

Use the graph to answer the next 3 questions.

The graph shows the price per piece of bath towel in different shops

Which shop provided the least value for money?

A Shop B B Shop C C Shop D D Shop E E Shop A

Which two shops charged the same price per piece?

A Shops B and C B Shops A and B C Shops A and D D Shops A and C E Shops B and E

At which shop would you get three times the price of bath towel for the same piece as shop A?

A Shop C B Shop D C Shop A D Shop B E None of these

Key Rules to remember

• Know the requirement of the question first to focus your efforts when looking at a chart or graph.
• The labels, legend and headings are important to understand the information being presented.
• Ensure you deal with one axis at a time to avoid confusion with data.
• Work at a high level and then ‘zoom in’ for more details.

Practice time!

Now, it's your turn to practice.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has two solutions videos after you complete the question. The first is a quick 60 second video that shows you how our expert answers the question quickly. The second video is a more in-depth 5-steps or less explainer video that shows you the steps to take to answer the question. It's really important that you review the second video because that's where you'll learn additional tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.

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