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Prep
Adding means putting two groups together and finding how many there are altogether. The quickest way for small numbers is to start at the bigger number and count on. For 6 + 3, hold 6 in your head and count on three more: 7, 8, 9. Because order doesn't change the total, 3 + 6 gives the same answer, so always start from the bigger number to do less counting.
Practise for five minutesYear 3
Multiplication is repeated groups. 6 × 4 means six groups of four. You can skip-count in fours six times — 4, 8, 12, 16, 20, 24 — or just recall the fact. Knowing your tables by heart makes everything later (division, fractions, area) much faster, so these are worth learning until they are instant.
Practise for five minutesYear 3
Dividing shares a number into equal groups. 48 ÷ 6 asks how many sixes fit into 48. Because division undoes multiplication, use your tables: 6 × 8 = 48, so 48 ÷ 6 = 8. Every division fact has a matching multiplication fact — find the one you know.
Practise for five minutesYear 5
Sometimes a number doesn't share equally. 23 ÷ 5: the most fives that fit is four (4 × 5 = 20), and 3 are left over. We write this as 4 remainder 3, or '4 r 3'. The remainder is always smaller than the divisor — if it isn't, another group fits.
Practise for five minutesYear 5
To multiply two 2-digit numbers, split the second into tens and units and multiply each part. For 23 × 14: 23 × 4 = 92, and 23 × 10 = 230. Add the partial products: 92 + 230 = 322. The tens row is a multiple of ten, so it always ends in zero — never forget that shift.
Practise for five minutesYear 6
When a calculation has more than one operation, there is an order: brackets first, then × and ÷, then + and −. For 3 + 4 × 2, do 4 × 2 = 8 first, then 3 + 8 = 11 — not left to right. Brackets override everything: (3 + 4) × 2 = 14.
Practise for five minutesYear 1
A two-digit number is built from tens and ones. In 35 the left digit counts tens and the right digit counts ones, so 35 is 3 tens (30) and 5 ones. When you hear 'thirty-five', the 'thirty' already fills the tens column, so the 5 sits in the ones column beside it — you do not write 30 and 5 next to each other. Say the tens digit first and write it on the left, then the ones digit on the right.
Practise for five minutesYear 3
Larger numbers use four columns: thousands, hundreds, tens and ones. Each spoken part names a column, not a new number to write in full. 'Two thousand and five' is 2 in the thousands column and 5 in the ones column, with zeros holding the empty hundreds and tens: 2005. An empty column still needs a 0 to keep the other digits in place — without it, 3040 collapses to 340 and every digit shifts.
Practise for five minutesYear 3
To compare two whole numbers, first count the digits: more digits means a larger number, so 100 beats 89. When two numbers have the same number of digits, compare from the left — the biggest place value decides. 61 beats 58 because 6 tens beats 5 tens; the ones digit only matters if the tens are equal. A big first digit alone, a big last digit, or a big digit total do not make a number larger.
Practise for five minutesYear 4
Rounding replaces a number with the nearest ten or hundred. Find the deciding digit — the ones digit when rounding to 10, the tens digit when rounding to 100. If it is 5 or more, round up; if it is less than 5, round down. 45 rounds up to 50 because the 5 rounds up, and 43 rounds down to 40 because 3 is under 5. Decide in one step: do not round to 10 first and then to 100.
Practise for five minutesYear 6
The number line runs on past zero into the negatives. Numbers get larger as you move right and smaller as you move left, so -1 is larger than -3 even though 3 looks bigger — -1 is closer to zero. With negatives, a bigger digit means a smaller number. You can also subtract past zero: start at 3 and count back 5, passing through zero to land on -2. Counting back does not stop at zero.
Practise for five minutesYear 4
A number is even if it can be shared into two equal whole groups, and odd if one is left over. You only need the ones digit to tell: 0, 2, 4, 6 and 8 are even, and 1, 3, 5, 7 and 9 are odd. So 34 is even because it ends in 4, and 25 is odd because it ends in 5. Zero counts as even, so numbers ending in 0 — like 40 — are even. Do not add the digits or look at the first digit.
Practise for five minutesYear 4
Two fractions are equivalent when they show the same amount, even though the numbers look different. You make an equivalent fraction by multiplying the numerator (top) and the denominator (bottom) by the very same number. For 2/3, multiplying both by 4 gives 8/12 — the same slice of the whole, just cut into more pieces. The golden rule is 'whatever you do to the bottom, do to the top', and you must multiply, not add. Ask 'what was the bottom multiplied by?' then do exactly that to the top.
Practise for five minutesYear 5
To compare fractions with different denominators, you cannot just look at the top or the bottom on its own. A bigger denominator means the pieces are smaller, so 1/5 is less than 1/3. The safe method is to rename both fractions with a common denominator, then compare the numerators. For 2/3 and 3/5, use fifteenths: 2/3 = 10/15 and 3/5 = 9/15. Now 10/15 is larger, so 2/3 is larger. Multiplying the two denominators together always gives a common denominator you can use.
Practise for five minutesYear 6
You can only add fractions when the pieces are the same size — that is, when the denominators match. For 1/2 + 1/3, rename both as sixths: 1/2 = 3/6 and 1/3 = 2/6. Now the pieces match, so add the numerators: 3/6 + 2/6 = 5/6. You never add the denominators. Multiplying the two denominators gives a common denominator, but remember to scale each numerator by the same factor you used on its denominator. Finish by writing the answer in simplest form.
Practise for five minutesYear 6
Simplifying means writing a fraction with the smallest possible numbers while keeping the same value. Find the highest number that divides both the top and the bottom, then divide both by it. For 8/12, the highest common factor is 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3, so 8/12 = 2/3. You must divide the top and the bottom by the same number. A fraction is in simplest form when no number bigger than 1 divides both parts — if it still can be divided, keep going.
Practise for five minutesYear 5
A longer decimal is not automatically larger. 0.5 is bigger than 0.45, even though 0.45 has more digits. The trick is to give both numbers the same number of decimal places by padding with zeros at the end: 0.5 becomes 0.50. Now compare digit by digit from the left — tenths first, then hundredths. 0.50 has 5 tenths and 0.45 has only 4 tenths, so 0.50 is larger. Remember the digits after the point are tenths and hundredths, not an ordinary whole number.
Practise for five minutesYear 6
When you add decimals, line up the decimal points, not the last digits. Write the numbers one above the other so the points sit in a straight line, then add each column from the right just like whole numbers. Bring the decimal point straight down into the answer. It helps to pad with zeros at the end so both numbers have the same length: 2.5 becomes 2.50, then 2.50 + 0.25 = 2.75. The extra zero always goes at the end, never straight after the point.
Practise for five minutesYear 3
The length of an object is how many centimetre spaces it covers on the ruler, not how many lines it touches. When one end sits exactly on the 0 mark, the number at the other end is the length. When the object does not start at 0, take the starting number away from the finishing number. From 2 cm to 8 cm is 8 − 2 = 6 cm, because there are six one-centimetre spaces in between.
Practise for five minutesYear 4
The perimeter is the total distance all the way around the outside of a shape. A rectangle has two lengths and two widths, so add all four sides. For a 5 cm by 3 cm rectangle: 5 + 3 + 5 + 3 = 16 cm, which is the same as 2 × (5 + 3). Only two measurements are printed, but each one is used twice. Multiplying the two sides together gives the area instead, not the perimeter.
Practise for five minutesYear 5
The area is the amount of surface a shape covers, measured in square units. A rectangle can be split into rows of unit squares, so the area is the number of rows times the number in each row — length × width. A 6 cm by 4 cm rectangle holds 6 × 4 = 24 square centimetres. Adding the sides gives only 10, and going around the edge gives the perimeter; area always needs a multiplication and covers the whole inside.
Practise for five minutesYear 4
Elapsed time is how long passes between a start time and a finish time. Because an hour has 60 minutes, not 100, you cannot just subtract the digits. Count up in two hops instead: first to the next o'clock, then on to the finish. From 3:50 to 4:10, it is 10 minutes up to 4:00 and 10 minutes more to 4:10, which is 20 minutes altogether. When the minutes seem to go backwards, it is a sign the hour has rolled over.
Practise for five minutesYear 6
Metric units go up in steps of 10, 100 or 1000 — not always 10. Changing a larger unit into a smaller one makes the number bigger, so you multiply by the fact for those units: 1 metre = 100 centimetres, 1 kilometre = 1000 metres, 1 kilogram = 1000 grams, 1 litre = 1000 millilitres. So 3 metres = 3 × 100 = 300 centimetres. Ask first whether the number should get bigger or smaller, then use the exact fact for the two units.
Practise for five minutesYear 3
Every 3D solid is built from three kinds of part. A face is a flat surface you could paint. An edge is the straight line where two faces meet. A vertex is a corner point where edges meet (the plural is vertices). A cube, for example, has 6 faces, 12 edges and 8 vertices. Count one kind at a time — trace every line for edges, poke every corner for vertices — and remember the parts you cannot see behind the solid still count.
Practise for five minutesYear 2
A quadrilateral is any shape with four straight sides, but the family has special members. A square has four equal sides and four right angles. A rectangle has four right angles with opposite sides equal. A rhombus has four equal sides but no right angles. A parallelogram has two pairs of parallel sides. A trapezium has exactly one pair of parallel sides, and a kite has two pairs of equal sides next to each other. To name a shape, check the sides and the angles — not how it is turned. Turning a shape never changes its name.
Practise for five minutesYear 3
An angle measures the amount of turn between two arms, not how long the arms are. Two arms can be long or short — what matters is how far one is turned from the other. We measure the turn in degrees with a protractor. Line one arm up with the 0° mark and read the scale that starts from there. An acute angle is less than 90°, so its measurement must be a small number. Watch the two rows of numbers on a protractor: follow the same row you started from, or you will read the angle's supplement by mistake.
Practise for five minutesYear 4
A line of symmetry is a fold line that makes the two halves of a shape land exactly on top of each other. To count them, imagine folding the shape every possible way and keep only the folds where the halves match. A rectangle has 2 lines (side to side and top to bottom) — its diagonals do NOT work. A square has 4, because it can also fold corner to corner. A parallelogram has none, because its slant tips one half off the other. Looking balanced is not the same as being symmetrical: only the fold test decides.
Practise for five minutesYear 5
A coordinate pair tells you where a point sits on a grid. It is written as (across, up) — the first number is how far to move right along the x-axis, and the second is how far up the y-axis. Always start at the origin, the corner where the axes cross, which counts as 0, not 1. Then count the spaces you move, not the lines you pass. So (3, 2) means 3 across and 2 up. Going up before across, or counting the first line as 1, lands the point in the wrong place.
Practise for five minutesYear 7
The mean is the 'even share' average. Add every value to get the total, then divide by how many numbers there are. For 4, 7, 3, 10, 6 the total is 30 and there are 5 numbers, so the mean is 30 ÷ 5 = 6. Every value counts — a zero is still a number to divide by, and one big number pulls the mean up. If you stop at the total you have only done half the job; the sharing step is what makes it an average.
Practise for five minutesYear 7
The median is the middle value once the numbers are in order. Always sort first — the middle of the unsorted list is not the median. With an odd count there is one middle number. With an even count there are two middle numbers, and the median is halfway between them: add them and divide by two. The median ignores how extreme the end values are, so one very large number does not drag it, which is what makes it useful.
Practise for five minutesYear 7
The mode is the value that appears most often — the number that repeats, not how many times it repeats, and not the largest number. The range measures how spread out the data is: subtract the smallest value from the largest. Range is a distance, so it is one number that describes the gap, never the biggest value on its own and never the two ends added together. Tally the values to find the mode; find the highest and lowest to find the range.
Practise for five minutesYear 4
Before reading any bar or point, read the scale. Graphs often go up in 2s, 5s or 10s, so each gridline is worth more than 1. To find a value, count how many gridlines the bar reaches and multiply by what each gridline is worth. If a scale goes up in 2s and the bar reaches the 4th line, that is 4 × 2 = 8, not 4. Count the lines from zero and stop exactly where the bar ends.
Practise for five minutesYear 3
We describe chance with words: impossible, unlikely, even chance, likely, certain. Compare how many outcomes you want to how many there are altogether. If a bag is all red, drawing red is certain; if there is no red, it is impossible. When reds and blues are equal it is an even chance. More reds than blues makes red likely; fewer makes it unlikely. Count the actual objects, not the number of colours — three reds and one blue is not 50/50, because there are far more reds.
Practise for five minutesYear 6
Per cent means 'out of 100', so a percentage is a share of an amount. To find 25% of 80, turn the percent into hundredths and multiply: 25 ÷ 100 × 80 = 20. A quick way is to find 1% first (divide by 100) and then multiply by the percent, or use friendly benchmarks — 10% is a tenth, 25% is a quarter, 50% is a half. The answer is always a part of the original amount, so it can never be bigger than what you started with.
Practise for five minutesYear 5
A fraction, a decimal and a percentage can all describe the same amount. To write a fraction as a percentage, multiply by 100: 1/4 = 1 × 100 ÷ 4 = 25, so 1/4 = 25%. It helps to learn the common ones by heart — 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/10 = 10%. Both numbers in the fraction matter: the bottom number tells you how many equal parts, and the top number tells you how many of them you have.
Practise for five minutesYear 7
To increase or decrease an amount by a percentage, do it in two steps. First find that percentage of the amount — the change. Then combine it: add for an increase, subtract for a decrease. To increase 60 by 10%: 10% of 60 = 6, then 60 + 6 = 66. To decrease 60 by 10%: 60 − 6 = 54. The change on its own is not the answer — you must apply it to the starting amount, and check the direction word 'increase' or 'decrease' before you add or subtract.
Practise for five minutesYear 6
A discount is a percentage taken off a price. Work it out in two steps: find the discount (that percentage of the price), then subtract it to get the new price. A $40 item at 25% off: 25% of $40 = $10 saved, so the new price is $40 − $10 = $30. The saving and the new price are two different amounts — read carefully which one the question wants. 'Off' always makes the price smaller, so the sale price should be lower than the original.
Practise for five minutesYear 7
A ratio compares two quantities. If a bag holds 6 red counters and 4 blue counters, the ratio of red to blue is 6 to 4, written 6:4. The order matters: 'red to blue' puts the red count first, so 6:4 and 4:6 are not the same. A ratio compares one part with the other part, not a part with the whole group — the 4 is the number of blue counters, not the 10 counters altogether. You read the two counts straight off and write them in the order the question names them.
Practise for five minutesYear 7
Simplifying a ratio means writing it with the smallest whole numbers while keeping the same comparison. Find the highest number that divides both terms, then divide both by it. For 8:12, the highest common factor is 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3, so 8:12 = 2:3. You must divide both terms by the same number — different divisors change the ratio. A ratio is fully simplified when no number bigger than 1 divides both terms; if both are still even, or share any factor, keep going.
Practise for five minutesYear 7
Two ratios are equivalent when they compare quantities in the same way, even though the numbers differ. You make an equivalent ratio by multiplying both terms by the same number. In 3:5 = 6:?, the first term goes from 3 to 6, which is × 2, so the second term is 5 × 2 = 10, giving 6:10. The key is to find the scale factor from the pair you know, then apply that same factor to the other term. You multiply by the factor — you never add the difference, and you never leave a term unchanged.
Practise for five minutesYear 7
To share an amount in a ratio, first add the terms to find how many equal parts there are. Sharing $60 in the ratio 2:3 gives 2 + 3 = 5 parts. Divide the total by the number of parts to find one part: $60 ÷ 5 = $12. Then multiply by how many parts each share has: the smaller share is 2 × $12 = $24 and the larger share is 3 × $12 = $36. Always do both steps — one part on its own is not the answer, and a ratio is not an equal split.
Practise for five minutesYear 7
A unit rate is the cost (or amount) for exactly one. If 3 pens cost $6, then one pen costs $6 ÷ 3 = $2. Finding 'how much for one' is always a sharing question, so it is a division: total cost divided by the number of items. The answer for one item is smaller than the total, so if your answer is bigger than the total you have multiplied by mistake. Once you know the cost of one, you can find the cost of any number by multiplying back up.
Practise for five minutesYear 4
A function machine takes a number in, does something to it, and sends a new number out. This one has a two-step rule: first multiply, then add. To find the output, put the input through both steps in order. For In 8 with the rule × 2 + 1: first 8 × 2 = 16, then 16 + 1 = 17, so Out is 17. Always do the steps in the order the machine shows them, and remember to do BOTH — the multiply and the add.
Practise for five minutesYear 6
In a growing pattern like 2, 5, 8, 11, each term is a fixed amount bigger than the one before — here it goes up by 3 each time. That fixed amount is the common difference. To find a later term without writing them all out, notice there is one fewer jump than the term number: the 6th term is 5 jumps of 3 added to the start. So 2 + 5 × 3 = 17. Count the GAPS, not the terms, and never forget to add the starting number.
Practise for five minutesYear 6
An equation says two things are equal, and a letter like x stands for the number we do not yet know. To find x we get it on its own by undoing what has been done to it — using the inverse operation and doing the same to both sides. In x + 7 = 12, something is being added to x, so we subtract 7 from both sides: x = 12 − 7 = 5. Always check by putting your answer back in: 5 + 7 = 12. ✓ Adding when you should subtract is the classic slip.
Practise for five minutesYear 7
A two-step equation like 3x + 5 = 20 does two things to x: it multiplies by 3, then adds 5. To solve it, undo those steps in reverse order. First subtract 5 from both sides: 3x = 15. Then divide both sides by 3: x = 5. Check by substituting: 3 × 5 + 5 = 20. ✓ The order matters — take the added number off first, then divide. Stopping after the subtraction (giving 15) or dividing before subtracting are the two most common mistakes.
Practise for five minutesYear 7
Substituting means swapping a letter for a number and working out the value. Remember that a number written next to a letter means multiply: 3n means 3 × n, not 3 + n and not the digits stuck together. So if n = 4, then 3n + 2 means 3 × 4 + 2 = 12 + 2 = 14. Do the multiplying first (following the order of operations), then add the extra number. Say '3n' out loud as 'three lots of n' to keep the times sign in mind.
Practise for five minutesYear 3
Change is the money you get back after paying. Work it out by subtracting the price from the amount you handed over. If you pay with $10.00 for something that costs $6.50, the change is $10.00 − $6.50 = $3.50. Keep the dollars under the dollars and the cents under the cents so the 50 cents isn't missed. The change is always less than what you paid.
Practise for five minutesYear 4
When you buy several of the same thing, multiply the price of one by how many you buy. 'Each' means every item costs the same. Three items at $4.25 each is 3 × $4.25 = $12.75 — the same as $4.25 + $4.25 + $4.25. Multiply the whole cents amount, including the cents, not just the dollars.
Practise for five minutesYear 6
GST in Australia is 10%, which means one tenth of the price. To add GST, find 10% of the price and add it to the original. For $80, 10% is $8, so the price with GST is $80 + $8 = $88. Finding 10% is easy: divide by 10. Remember the two steps — work out the GST, then add it, so the total is bigger than the start.
Practise for five minutesYear 7
The best value isn't always the cheapest pack — a bigger pack costs more but might be cheaper for the same amount. To compare fairly, work out the unit price: how much it costs per 100 g. Divide the price (in cents) by how many 100 g lots the pack holds. A 500 g pack for 400 c is 400 ÷ 5 = 80 c per 100 g. Whichever pack has the lower cents per 100 g is the better value.
Practise for five minutesPrep
Point to each object once and say the next number. The last number you say tells you how many there are. In a ten-frame, a full top row is five, so you can count on from five. When objects are scattered, start at one side and work across so none are missed. If a picture has two kinds of shape, count only the kind the question asks for.
Practise for five minutesPrep
Ten can be split into two parts. When one part is shown, the missing part is the empty spaces in a ten-frame. Knowing the pairs that make ten also answers subtraction: if 7 and 3 make 10, then 10 − 3 = 7 and 10 − 7 = 3.
Practise for five minutesPrep
Subtraction can mean taking some away. Start with the whole group, remove the amount in the question, then count what is left. Without a picture, count back: from 8, take away 3 by saying 7, 6, 5. When the missing number is the start, put back what was taken away.
Practise for five minutesYear 1
When an addition crosses ten, make ten first. Start with the larger number, find how many more make ten, and split the other number into that amount and the rest. Then add the rest to ten. For a missing number, add the jump to ten and the jump from ten.
Practise for five minutesYear 1
Take away in two steps. First take away the ones to reach ten, then take away what is still left. To subtract 6 from 14, take away 4 to reach 10, then take away the other 2 to reach 8. For a missing number, add the jump down to ten and the jump from ten.
Practise for five minutesYear 1
A double is two copies of the same amount. Halving is the opposite: share into two equal groups. Doubles also help with near doubles. For 6 + 7, double 6 is 12 and 7 is one more than 6, so the answer is one more: 13.
Practise for five minutesYear 1
Look at the gap between neighbouring numbers. Keep adding that same amount to continue the pattern. If the numbers get smaller, take the jump away. A pattern can start on any number: counting in tens from 3 gives 3, 13, 23, 33, where the ones digit stays the same.
Practise for five minutesYear 3
Write the numbers in columns so ones are under ones and tens are under tens. Add the ones first. If a column makes 10 or more, write the ones digit in that column and carry the ten to the next column, where it is added with the other digits. Work left, one column at a time. With three numbers a column can make 20 or more; then carry 2 tens.
Practise for five minutesYear 3
Write the bigger number on top, with ones under ones. Start with the ones. If the top digit is smaller than the bottom digit, don't swap them: regroup one ten from the next column into ten ones. Cross out that ten's digit and make it one less. If the next digit is 0, regroup from the place after it first, so the 0 becomes 10 and then 9.
Practise for five minutesYear 3
Split strategy: break numbers into tens and ones. For 47 + 36, add the tens (70), add the ones (13), then combine: 83. For subtraction, take away the tens and then the ones. Compensation: when a number is close to a round number, use the round number and then fix it. For 46 + 29, add 30 to get 76, then take away the extra 1: 75.
Practise for five minutesYear 3
Addition and subtraction undo each other, and so do multiplication and division. Three numbers make a fact family: 34 + 25 = 59 also tells you 59 − 25 = 34 and 59 − 34 = 25. To find a missing number, undo what was done: for ? − 38 = 45, add 38 back. For a puzzle with two steps, work backwards and undo the last step first.
Practise for five minutesYear 3
A missing-number fact asks which number makes the sentence true. For 4 × ? = 24, ask 'how many fours make 24?' and use the division fact 24 ÷ 4 = 6. For a division with a missing number, use the matching multiplication fact. For bigger numbers, take out ten groups first, then find how many groups are left.
Practise for five minutesYear 4
You can add or multiply two numbers in either order: 4 × 7 = 7 × 4 (a turnaround). When multiplying three numbers, you can also choose which two to multiply first. Look for a friendly pair that makes 10 or 100, such as 2 × 5 or 4 × 25. You can also split a factor to make a friendly pair: 16 × 25 = 4 × 4 × 25 = 4 × 100.
Practise for five minutesYear 4
Split each multiple of ten into a basic fact and a power of ten: 30 × 40 is 3 × 4 = 12, and there are two zeros, so 1200. Keep any zero the basic fact makes: 50 × 60 is 5 × 6 = 30 with two more zeros, so 3000. To find a missing factor, take the same number of zeros off both numbers and divide.
Practise for five minutesYear 4
To multiply a bigger number, split it into parts that are easy to multiply, multiply every part, then add the answers. 7 × 36 = 7 × 30 + 7 × 6 = 210 + 42 = 252. An area model shows why: the rectangle is split into two smaller rectangles. A number just below a round number can be done as round number minus a bit: 6 × 49 = 6 × 50 − 6 × 1.
Practise for five minutesYear 4
An estimate is a quick, close answer. Round each number to the place the question asks, then add or subtract the rounded numbers in your head. To round, look at the digit just after that place: 5 or more rounds up, 4 or less rounds down. For the leading place, round each number to the place of its first digit.
Practise for five minutesYear 4
Draw a bar model before calculating. Draw the whole as one long bar and mark the parts under it. Put a question mark on the part you need. Then decide: to find a whole, add the parts; to find a part, take the other parts from the whole. For equal groups, draw equal units and multiply. When one amount is more than another, take away the extra to make equal units.
Practise for five minutesYear 5
Round each factor to a friendly number, then multiply the rounded numbers using a basic fact and the zeros. For 38 × 7, 38 rounds to 40, and 40 × 7 = 280. The estimate tells you roughly how big the exact answer should be, so you can check your working.
Practise for five minutesYear 5
Write the number inside the bus stop and the divisor outside. Divide each digit from left to right. If a digit doesn't share exactly, write how many fit above it and carry the remainder to the next digit, where it becomes tens of that digit. If the divisor doesn't fit, write 0. Whatever is left at the end is the remainder. In a story, decide what the remainder means.
Practise for five minutesYear 6
Multiplying by 10 makes every digit ten times bigger, so each digit moves one place to the left. Multiplying by 100 moves digits two places, and by 1000 three places. Dividing moves the digits to the right. The decimal point stays still; use zeros to fill empty places. Adding a zero on the end only works for whole numbers.
Practise for five minutesYear 6
Long division repeats four steps: divide, multiply, subtract, bring down. First write some multiples of the divisor to help. Find how many times the divisor fits into the first part of the number, write that digit above, multiply and subtract. Bring down the next digit beside what is left and repeat. If the divisor doesn't fit, write 0. What is left at the end is the remainder.
Practise for five minutesYear 2
A half is one of 2 equal parts. Halve a half and you get a quarter: one of 4 equal parts. Halve a quarter and you get an eighth: one of 8 equal parts. The parts must be the same size. To find a quarter of a group, halve it and then halve again. To find the whole from a part, keep doubling.
Practise for five minutesYear 3
To name a fraction of a shape, first count how many equal parts make the whole. That is the bottom number (the denominator). Then count the parts you are asked about. That is the top number (the numerator). If the parts are not all the same size, measure them with equal units first, because fractions only work with equal parts.
Practise for five minutesYear 4
On a number line a fraction is a distance from 0. Count the equal spaces between 0 and 1: that is the denominator. Then count the spaces from 0 to the point: that is the numerator. Count spaces (jumps), not marks. Past 1 the counting keeps going, giving improper fractions like 5/4. If 1 is not labelled, use a labelled mark to work out what one space is worth.
Practise for five minutesYear 4
Look at the two labelled numbers and count the equal spaces between them. The gap shared by the number of spaces tells you what one space is worth. From 0 to 1 in 10 spaces, each space is 0.1 (one tenth). From 0.3 to 0.4 in 10 spaces, each is 0.01 (one hundredth). Then count the spaces from the start to the arrow.
Practise for five minutesYear 5
A mixed number like 2 3/4 has whole ones and a fraction. Each whole is made of the same parts: 1 whole = 4 quarters. So 2 3/4 is 2 × 4 + 3 = 11 quarters, written 11/4. To go the other way, divide: 17/5 is 17 ÷ 5 = 3 remainder 2, so 3 wholes and 2 fifths, written 3 2/5. The fraction part of a mixed number is always less than 1, and the denominator never changes.
Practise for five minutesYear 6
To find 3/4 of 20, draw a bar for the 20 and split it into 4 equal boxes. One box is 20 ÷ 4 = 5. Three boxes are 3 × 5 = 15. Divide by the denominator to find one part, then multiply by the numerator. If you know a part and need the whole, work backwards: find one box first, then all the boxes.
Practise for five minutesYear 6
Find the place you are rounding to. Look only at the digit straight after it. If that digit is 5 or more, round up; if it is less than 5, round down. Don't round in steps from the end, and when a 9 rounds up it becomes 0 and carries 1 to the left, like 3.96 → 4.0.
Practise for five minutesYear 6
Rename the decimal as tenths or hundredths so it becomes a whole number. 0.4 is 4 tenths, so 0.4 × 3 is 12 tenths, which is 1.2. 2.35 is 235 hundredths, so 2.35 × 4 is 940 hundredths, which is 9.4. Estimate first to check where the point goes: 2.35 × 4 is a bit more than 2 × 4 = 8.
Practise for five minutesYear 6
Rename the decimal in tenths or hundredths, divide, then rename back. 2.4 is 24 tenths, and 24 tenths ÷ 3 = 8 tenths = 0.8. If the digits run out, add a zero: 3.2 = 3.20 = 320 hundredths, and 320 ÷ 5 = 64 hundredths = 0.64. Use a 0 to hold any empty place, as in 0.36 ÷ 4 = 0.09.
Practise for five minutesYear 6
A whole number times a fraction means that many lots of the fraction: 4 × 2/5 is 4 lots of 2 fifths, which is 8 fifths (8/5). The denominator stays the same. To multiply two fractions, multiply the numerators together and the denominators together, then simplify: 2/3 × 3/4 = 6/12 = 1/2. Change mixed numbers to improper fractions first.
Practise for five minutesYear 7
Dividing asks how many of one thing fit into another. How many quarters fit in 3? Each whole has 4 quarters, so 3 ÷ 1/4 = 12. Dividing a fraction by a whole number shares it: 3/4 ÷ 2 = 3/8. For any division by a fraction, keep the first number, change ÷ to ×, and flip the second: 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9.
Practise for five minutesYear 7
A fraction is a division: 2/3 means 2 ÷ 3. Some divisions never end, because the same remainder keeps coming back, so the digits repeat forever: 2/3 = 0.666… These are recurring decimals. To use one, round it: 0.666… is 0.67 to 2 decimal places. To find a digit far along, use the length of the repeating block and the remainder.
Practise for five minutesPrep
Ordinal numbers tell us position: first, second, third, fourth and so on. Before counting, find where 'first' is: the child next to the flag, or the winner of a race. Then count along, saying a position word for each person, up to and including the person you want. If 3 runners finish ahead of you, they are first, second and third, so you are fourth. Counting from the back is counting from the other end of the same line.
Practise for five minutesYear 2
When you count on by one and the ones digit is 9, the ones are full: they go back to 0 and the tens go up by one. After 109 comes 110. Counting back from a 0, you exchange: one ten becomes ten ones, so before 400 comes 399. Ten more and ten less change the tens digit, and if the tens pass 9 or 0 the hundreds change too.
Practise for five minutesYear 2
A number can be split into parts in many ways. 46 is 40 + 6, but if you break one ten into ones it is also 30 + 16. Nothing is lost: the parts still add to 46. In the same way 352 = 200 + 150 + 2, and because every hundred is 10 tens, 352 is 35 tens and 2 ones. Check by adding the parts back together.
Practise for five minutesYear 2
Each part of a number word tells you a digit and its column. 'Three thousand and forty' is 3 in the thousands, 4 in the tens, and nothing in the hundreds or ones, so we write zeros there: 3040. Teen words say the ones first ('four-teen') but the ten is still written on the left: 14. To write a numeral in words, say each column's part, skipping zeros, with 'and' before the last tens and ones.
Practise for five minutesYear 5
Big numbers are grouped in threes: ones, thousands, millions. Each column is worth ten times the column to its right. A digit's value is the digit times its column: in 4 725 316 the 7 is in the hundred thousands, so it is worth 700 000. The digits after it are replaced by zeros. To add 10 000, change the ten thousands digit; if it is 9, it becomes 0 and the next column goes up by one.
Practise for five minutesYear 5
To round to the nearest 1000, 10 000 or 100 000, find the two multiples the number sits between, then look at the deciding digit: the one just right of the rounding place. If it is 5 or more, round up; if it is less than 5, round down. Every digit after the rounding place becomes 0. Look at the deciding digit only once: do not round one place at a time.
Practise for five minutesYear 5
A multiple of 6 is in the 6 times table: 6, 12, 18 and so on. A factor of 36 divides 36 exactly. To find all the factors, list factor pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. Every number has 1 and itself as factors. In a square number the middle pair repeats a factor, so count it once.
Practise for five minutesYear 6
A prime number has exactly two factors: 1 and itself, like 2, 3, 5, 7 and 11. A composite number has more than two factors, like 12. The number 1 is neither, because it has only one factor. 2 is the only even prime. Being odd does not make a number prime: 91 = 7 × 13. To test a number, try dividing by 2, 3, 5 and 7.
Practise for five minutesYear 7
Every whole number bigger than 1 can be written as a product of prime numbers in exactly one way. Split the number into any factor pair, then keep splitting each branch until every end is prime. Collect all the primes at the ends, including repeats, and write them smallest first: 60 = 2 × 2 × 3 × 5. Index notation groups repeats: 2 × 2 × 2 is 2³, which means 2 multiplied by itself three times.
Practise for five minutesYear 7
The highest common factor (HCF) of two numbers is the biggest number that divides both. List the factors of each and find the largest on both lists. The lowest common multiple (LCM) is the smallest number both divide into. List multiples of the larger number until you reach one the smaller number also divides. HCF answers 'largest equal groups' questions; LCM answers 'when will they happen together again' questions.
Practise for five minutesYear 6
A square number makes a square of dots: 5 rows of 5 is 5 × 5 = 25. A triangular number makes a triangle: rows of 1, 2, 3, 4 add to 10. Square numbers go 1, 4, 9, 16, 25. Triangular numbers go 1, 3, 6, 10, 15. To get the next triangular number, add a new bottom row one longer than before. To get the next square, add a new row and a new column.
Practise for five minutesYear 4
Roman numerals use letters: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000. Usually the symbols go from largest to smallest and you add them: XXVII = 27. When a smaller symbol comes just before a larger one, take it away: IV = 4, IX = 9, XL = 40, XC = 90. To write a number, split it by place value and write each part.
Practise for five minutesYear 6
Not every pattern adds the same amount each time. Write the jumps between the numbers underneath the pattern. If the jumps grow by the same amount (+1, +2, +3, +4), keep that going. If the jumps are not steady, check whether each number is multiplied: 3, 6, 12, 24 doubles. Square numbers 1, 4, 9, 16 have jumps of 3, 5, 7: growing by 2. Always check your rule on every number shown.
Practise for five minutesPrep
A flat shape with straight sides is named by how many sides it has. Touch each side once as you count round, and stop when you get back to the side you started on. 3 sides make a triangle, 5 a pentagon and 6 a hexagon. A shape with 4 sides and square corners is a square if every side is the same length, and a rectangle if two sides are longer. A circle has one curved edge and no corners. Turning, stretching or denting a shape does not change its name: a triangle is still a triangle when it points down, and an L-shape with 6 sides is a hexagon.
Practise for five minutesYear 5
Every prism and pyramid is built on a base shape, and the base tells you everything. A prism has two matching ends joined by rectangles. If its base has n sides, it has n + 2 faces, 3 × n edges and 2 × n vertices. A pyramid has one base and triangles that meet at a point. If its base has n sides, it has n + 1 faces, 2 × n edges and n + 1 vertices. So a hexagonal prism (base 6) has 8 faces, 18 edges and 12 vertices. You can also work backwards: a prism with 15 edges has 15 ÷ 3 = 5 sides on its base.
Practise for five minutesYear 6
For any solid with flat faces and straight edges (a polyhedron), the faces (F), vertices (V) and edges (E) are linked: F + V − E = 2. Check it on a cube: 6 + 8 − 12 = 2. So if you know two of the counts, you can find the third. To find edges, add the faces and vertices, then take 2. To find faces, take the vertices from the edges, then add 2. To find vertices, take the faces from the edges, then add 2. Always check by putting all three numbers back into the formula: the answer must be exactly 2.
Practise for five minutesYear 5
Triangles can be sorted two ways, so read which one the question asks. By sides: equilateral has all three sides equal, isosceles has exactly two equal sides, and scalene has no equal sides. Matching tick marks show equal sides. By angles: acute-angled has every angle less than 90°, right-angled has one angle of exactly 90° (shown by a small square), and obtuse-angled has one angle more than 90°. Turning a triangle never changes its type. If only two angles are given, find the third first: the three angles always add to 180°.
Practise for five minutesYear 5
To give a quadrilateral its best name, check three things: equal sides (matching tick marks), parallel sides (matching arrows) and right angles (small squares). Square: 4 equal sides and 4 right angles. Rectangle: 4 right angles, sides not all equal. Rhombus: 4 equal sides, no right angles. Parallelogram: 2 pairs of parallel sides. Trapezium: exactly 1 pair of parallel sides. Kite: 2 pairs of equal sides next to each other. Choose the most exact name that fits. On a dot grid, count the steps along each side: matching steps mean equal sides, and steps in the same direction mean parallel sides.
Practise for five minutesYear 6
Quadrilaterals belong to families, and one shape can belong to several. A rectangle is any quadrilateral with 4 right angles, so every square is a rectangle. A rhombus is any quadrilateral with 4 equal sides, so every square is a rhombus too. Rectangles and rhombuses are both parallelograms. To test 'Is every A a B?', write down what a B needs, then ask whether every A always has it. Careful: the reverse is often false. Every square is a rectangle, but a long thin rectangle is not a square. Some facts force others: a parallelogram with one right angle has four.
Practise for five minutesYear 4
A shape has rotational symmetry if it looks exactly the same after turning part of the way round its centre. The order of rotational symmetry is how many times it fits onto itself in one full turn. Count the fits during the turn, including the final one when it is back at the start, but do not count the start as well. A square fits after 90°, 180°, 270° and 360°, so its order is 4. A shape that only fits at the end of the full turn has order 1. With coloured patterns, the colours must match too, not just the outline.
Practise for five minutesYear 5
A transformation moves a shape. To find where a corner goes, follow that one corner. Translation slides the shape: add for right and up, subtract for left and down. Reflection flips it in a mirror line: count how far the corner is from the line, then go the same distance on the other side. Rotation turns it about a centre point P: note how far the corner is from P across and up, then turn those moves. For a quarter turn clockwise, 'right' becomes 'down' and 'up' becomes 'right'. For a half turn, both moves reverse. The image of A is written A′.
Practise for five minutesYear 6
An enlargement makes a shape bigger without changing its shape. The scale factor tells you how many times bigger: with a scale factor of 3, every length is multiplied by 3. It is never 'add 3'. To find the scale factor, pick a pair of matching sides and divide: new length ÷ original length. To find a missing length on the image, first find the scale factor from a pair you know, then multiply the matching original side by it. The angles stay the same; only the lengths grow.
Practise for five minutesYear 5
A net is a solid cut open and laid flat. To name the solid, look at the faces. Six equal squares make a cube; six rectangles in matching pairs make a rectangular prism. Two matching end shapes joined by a strip of rectangles make a prism, named after the end shape: two triangles and three rectangles make a triangular prism. One base with triangles all round it makes a pyramid, named after its base. Two circles and a rectangle make a cylinder; one circle and a curved fan make a cone. Name the solid, not a flat face.
Practise for five minutesYear 4
An angle is the amount of turn between two arms. Compare it with a right angle (a square corner, 90°) and a straight line (180°). Smaller than a right angle is acute. Exactly a square corner is a right angle. Between a right angle and a straight line is obtuse. A straight line is a straight angle. More than a straight line is reflex: the arc goes the long way round. Arm length never changes the type, and neither does turning the picture.
Practise for five minutesYear 5
Put the centre of the protractor on the corner of the angle and line the baseline up with one arm. That arm points to 0 on one of the two scales. Follow that same scale round to the other arm and read the number. Then check: if the angle is smaller than a right angle, the answer must be less than 90. If neither arm is on 0, read both arms on the same scale and find the gap between the numbers.
Practise for five minutesYear 6
A straight line is half a turn, so the angles that sit along one side of it add up to 180°. To find a missing angle, add the angles you know and take the total away from 180. The diagram is not drawn to scale, so work from the numbers, not by measuring. When two angles are marked with the same letter they are equal: share what is left between them.
Practise for five minutesYear 6
Angles all the way round a point make a full turn, 360°. Add the angles you know and take the total away from 360. A small square in a corner means a right angle, 90°, even though no number is written. The diagram is not drawn to scale, so use the numbers. Equal angles marked with the same letter share what is left.
Practise for five minutesYear 6
When two straight lines cross they make an X with four angles. The angles across from each other are vertically opposite, and they are equal. Angles next to each other sit on a straight line, so they add up to 180°. First decide which kind of pair you have. With more lines, carry a known angle across to its opposite, then use a straight line.
Practise for five minutesYear 7
The three angles inside any triangle add up to 180°. If you tear off the corners of a paper triangle, they fit together on a straight line. To find a missing angle, add the two you know and take the total from 180. A small square means 90°. In an isosceles triangle, two sides are equal, so the two angles at the ends of the third side are equal too.
Practise for five minutesYear 7
Any quadrilateral can be cut corner to corner into two triangles, so its four angles add up to 2 × 180 = 360°. Add the angles you know and take the total away from 360. Count a small square as 90°. When two angles have the same letter they are equal, so share what is left between them. The diagram is not drawn to scale.
Practise for five minutesPrep
Position words tell us where things are. Above means higher up and below means lower down. Left and right depend on where you look from, so look at the shelves from the front. Your left hand makes an L shape with its thumb and first finger. First find the thing named in the question, then look in the direction the words tell you.
Practise for five minutesYear 3
A full turn brings you back to facing the same way. A half turn faces you the opposite way. A quarter turn is a square corner of turn. Clockwise is the way a clock's hands move; anticlockwise is the other way. Count quarter turns one at a time from what you face now. In degrees, a quarter turn is 90°, a half turn 180°, a three-quarter turn 270° and a full turn 360°.
Practise for five minutesYear 4
The four main compass points go clockwise: north, east, south, west (Never Eat Soggy Worms). On most maps north is at the top, so east is right, south is down and west is left. Between them are north-east, south-east, south-west and north-west. Neighbouring points on an eight-point compass are 45° apart, so a 90° turn moves two points. A direction 'from the camp' means stand at the camp and look out.
Practise for five minutesYear 4
A grid map has letters along the top for columns and numbers down the side for rows. A grid reference gives the letter first, then the number: C3 means column C, row 3. To find a square, go along to the letter, then down to the number. To name a square, read the letter at the top of its column, then the number at the side of its row. Moving east or west changes the letter; north or south changes the number.
Practise for five minutesYear 5
A map is much smaller than real life. The scale tells you how far one square is in real life, like 1 square = 50 m. Count the squares along the path, counting the steps from one place to the next, not the squares the places sit in. Then multiply by the scale. If the path turns a corner, count both parts. To find the scale, divide the real distance by the number of squares.
Practise for five minutesYear 7
A bearing gives a direction as an angle. Stand at the starting point, face north and turn clockwise until you face the other point: the amount you turn is the bearing. Bearings always have three figures, so 65° is written 065. East is 090, south 180 and west 270. The bearing of B from A starts at A. To go back the other way, turn half a turn: add or subtract 180°.
Practise for five minutesPrep
To compare lengths, line the objects up so they all start at the same place, then see which one reaches furthest. If they do not start together, look at both ends. To compare how much jugs hold, pour each one into the same-sized glass and compare the water levels: a tall jug is not always the one that holds most. To compare masses, use a balance. The heavier side goes down. With two balances, put all the objects in order to find the heaviest or lightest.
Practise for five minutesYear 1
You can measure length with things that are all the same size, like paper clips. Lay them in a straight line, end to end, starting exactly at one end of the object. Leave no gaps and no overlaps. Then count how many units fit from one end of the object to the other. Only count the units beside the object, not any that stick out past it. If the units have gaps, the object is really longer than the count; if they overlap, it is really shorter.
Practise for five minutesYear 4
Area is how much surface a shape covers. On a grid of 1 cm squares, count the squares the shape covers, row by row, so none are missed. Count only the shaded squares, not the edges around the shape (that is the perimeter). Two half squares make one whole square, so count the halves and share them into pairs. A right-angled triangle is half of the rectangle drawn around it.
Practise for five minutesYear 6
Every triangle is half of a rectangle (or parallelogram) with the same base and height. So the area is ½ × base × height. The height must be at right angles to the base: it is the straight-up distance, often drawn as a dashed line. Never use a slanted side as the height. In an obtuse triangle, the height falls outside the shape and the base is extended with a dashed line, but the extension is not part of the base.
Practise for five minutesYear 6
Cut the triangle off one end of a parallelogram and slide it to the other end: you get a rectangle with the same base and height. So the area of a parallelogram is base × height. The height is the perpendicular height, at right angles to the base, not the slanted side. There is no ½, because a parallelogram is two whole triangles. If you know the area and one length, divide to find the other: height = area ÷ base.
Practise for five minutesYear 5
A composite shape is made of simpler shapes joined together. For an L-shape, make one straight cut to split it into two rectangles. Find the area of each rectangle (length × width) and add them. Make sure the rectangles do not overlap. Another way is to find the area of the big rectangle around the shape and take away the missing corner. If some sides are not labelled, work them out first from the sides you know.
Practise for five minutesYear 5
The perimeter is the total distance around the outside of a shape. For a shape with right-angled corners, first make sure you know every side. A missing side can be worked out from the others: opposite directions must balance, so the short sides across the top add up to the long side across the bottom. Then start at one corner and add every side as you go round, ticking each one off. Don't multiply: that gives the area.
Practise for five minutesYear 5
Volume is how much space a solid takes up. We can measure it by counting unit cubes. In a solid cuboid some cubes are hidden at the back and in the middle, so don't count only the ones you can see. Count the cubes in the top layer (rows × cubes in each row), then multiply by the number of layers. Count cubes, not the square faces you can see: a corner cube shows three faces but is only one cube.
Practise for five minutesYear 6
A rectangular prism (a box shape) can be filled with layers of centimetre cubes. The bottom layer holds length × width cubes, and there are as many layers as the height. So volume = length × width × height, measured in cubic centimetres (cm³). Adding the lengths or finding the faces' areas does not give the volume. To find a missing length, multiply the two you know, then divide the volume by that answer.
Practise for five minutesYear 6
Volume (cubic centimetres) and capacity (millilitres and litres) are linked. A 1 cm cube holds exactly 1 mL of water, so the number of cm³ equals the number of mL. A litre is 1000 mL, so 1000 cm³ = 1 L. To change cm³ into litres, divide by 1000. To find how many litres a tank holds, first find its volume with length × width × height, then divide by 1000.
Practise for five minutesYear 6
1 kilogram = 1000 grams and 1 litre = 1000 millilitres. To change the small unit into the big unit, divide by 1000, because you need fewer of the bigger unit. To change the big unit into the small one, multiply by 1000. For a mixed amount like 2 kg 50 g, change the kilograms first (2000 g) and then add: 2050 g. As a decimal, 2050 g is 2.05 kg: the zero holds the tenths place.
Practise for five minutesYear 1
A clock has two hands. The short hand shows the hour and the long hand shows the minutes. Read the short hand first: the hour is the number it points to, or the number it has just passed. Then read the long hand. Each number on the clock stands for 5 minutes, so count in fives from the 12: 1 is 5 minutes, 3 is 15, 6 is 30. Half past means 30 minutes, quarter past means 15, and quarter to means 45 minutes past, 15 minutes before the next hour. For exact minutes, count the small marks after the last number: each one is 1 minute.
Practise for five minutesYear 5
In 12-hour time the day is split in two: am is from midnight to midday, and pm is from midday to midnight. In 24-hour time the hours keep counting after midday: 1 pm is 13:00, 2 pm is 14:00. So for a pm time, add 12 to the hour to get 24-hour time, and take 12 away to go back. Two hours are special. The hour after midnight is 12 am, written 00 in 24-hour time. The hour after midday is 12 pm, which stays as 12. 24-hour times never use am or pm.
Practise for five minutesYear 4
Time does not work in tens and hundreds like metres or money. Learn the facts: 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day and 7 days in a week. To change to a smaller unit, multiply: 3 hours = 3 × 60 = 180 minutes. To change to a bigger unit, divide. With mixed units, change the big unit first, then add the extra part. Be careful with decimals: 1.5 hours means one and a half hours, which is 90 minutes, not 1 hour 50 minutes.
Practise for five minutesYear 2
A calendar shows one month. The columns are the days of the week and the numbers are the dates. To find the day of the week for a date, find the date and look straight up its column to the heading. Each week is 7 days, so the same day of the week comes every 7 dates. To count the days from one date to another, count the jumps, not the start day: from 5 to 19 is 19 − 5 = 14 days. To go into the next month, use how many days this month has, then keep counting.
Practise for five minutesYear 6
In a timetable each row is a stop and each column is one bus or train. To find when a bus arrives somewhere, find its starting time, then keep your finger in that column and move down to the stop you want. To find how long a trip takes, count up from the leaving time to the o'clock, then on to the arriving time. To catch the next bus, cross out the times at your stop that are earlier than when you get there. The first time left is your bus.
Practise for five minutesYear 6
Australia has three standard time zones. Eastern (AEST, UTC+10) covers Qld, NSW, Vic, Tas and the ACT. Central (ACST, UTC+9:30) covers SA and the NT. Western (AWST, UTC+8) covers WA. The sun rises in the east first, so eastern clocks are ahead. Find the difference from the UTC numbers, then add it going east or take it away going west. In summer NSW, Vic, Tas, the ACT and SA use daylight saving: their clocks go forward 1 hour. Qld, the NT and WA do not change.
Practise for five minutesYear 4
Most scales only label some of their marks. Before reading one, work out what each small mark is worth. Find two labels next to each other, take the smaller from the larger, and count the spaces between them (not the lines). Divide to find the value of one space. Then start at the last label before the pointer and count on the marks, adding that value each time. On a jug labelled every 100 mL with 4 spaces between labels, each mark is 25 mL.
Practise for five minutesYear 4
A thermometer is a scale that stands up. The labels usually go up by 10 degrees. Count the spaces between two labels to find what each mark is worth: 10 spaces means 1 degree each, 5 spaces means 2 degrees each. Find the top of the liquid, start at the label just below it and count up. Below zero the numbers are negative and get colder as you go down: the mark just under 0 is −1 or −2, not −9. Count down from 0 to read them.
Practise for five minutesYear 3
First decide what is being measured. Length (how long) uses millimetres, centimetres, metres or kilometres. Mass (how heavy) uses grams or kilograms. Capacity (how much something holds) uses millilitres or litres. Then use a benchmark to picture the size of one unit: a paperclip is about 1 gram, a bag of sugar is 1 kilogram, a milk carton holds 1 litre, a teaspoon holds about 5 millilitres, a big step is about 1 metre. Ask whether the object is about as big as a few of the benchmark, or hundreds of them.
Practise for five minutesYear 4
A timeline shows events in order, from earlier on the left to later on the right. Like any scale, only some years are labelled. Work out what each small mark is worth from the labels: 10 years split into 5 spaces is 2 years a mark. To find how long between two events, count on from the earlier year to the later one. To place an event that happened some years later, start at the first event and move right.
Practise for five minutesPrep
A repeating pattern uses the same group of shapes again and again. First find the part that repeats: say the pattern aloud and listen for where it starts again. Then keep the repeat going. If a shape is missing, read from the start and see what the repeat needs in the gap. To find a shape far along, count in whole repeats first, then count on.
Practise for five minutesYear 6
A table of values pairs each input with an output made by the same rule. Look at how much the outputs go up each time the input goes up by 1: that is the number to multiply by. Then check the first column to see if anything is added. Use the whole rule on any input, even one far along the table. To find an input from an output, undo the rule: take away, then divide.
Practise for five minutesYear 4
The equals sign means both sides have the same value. To find a missing number, use the inverse operation: undo adding with subtracting, and undo multiplying with dividing. Check by putting your number in the box and reading the sentence. When there are numbers on both sides, work out the side you can, then make the other side match it.
Practise for five minutesYear 5
A balanced scale shows two sides with the same mass, just like an equation. You can take the same amount off both sides and it stays balanced. Take off everything beside the boxes until the boxes are on their own. If there are several boxes, share what is left equally between them. If there are boxes on both sides, take the same number of boxes off each side first.
Practise for five minutesYear 6
An expression uses a letter for a number we don't know. Translate the words one instruction at a time. '5 less than n' means start with n and take 5 away: n − 5. '3 times n' is written 3n. Multiplying and dividing happen before adding, so 'multiply n by 3, then add 2' is 3n + 2. If the adding must happen first, use brackets: 'add 4 to n, then multiply by 3' is 3(n + 4).
Practise for five minutesYear 4
The sign < means 'is less than', > means 'is greater than' and = means 'is equal to'. The open end of < and > always faces the bigger number. Work out the value of each side, then choose the sign. Sometimes you can compare without calculating: if both sides start with the same number, only compare what changes, and remember that taking away more leaves less.
Practise for five minutesYear 6
In 3(x + 2) = 21, the number outside the brackets multiplies everything inside. To solve it, undo the steps in reverse order. First divide both sides by 3 to get x + 2 = 7, then take 2 from both sides to get x = 5. Check by putting 5 back: 3 × 7 = 21. When x is on both sides, expand the brackets and take the smaller x term from both sides.
Practise for five minutesYear 6
A scale like 1 : 100 means 1 cm on the drawing stands for 100 cm in real life. To find a real length, multiply the drawn length by the scale number. The answer is in the same unit as the drawing, usually centimetres, so convert it: 100 cm = 1 m and 100 000 cm = 1 km. To go from real life to the map, convert to centimetres first, then divide by the scale number.
Practise for five minutesYear 6
Every serve of a recipe needs the same amount, so amounts scale by multiplying, never by adding. If the new number of serves is a multiple of the old one, multiply by that many. Otherwise use the unitary method: find the amount for one serve, then multiply by the serves you need. You can also find how many times bigger the new recipe is, such as 6 ÷ 4 = 1.5 times.
Practise for five minutesYear 1
Australia has six coins. The silver coins are 5c, 10c and 20c, which get bigger as they are worth more, and the 50c, which has twelve sides. The gold coins are $1 and $2, and the $2 coin is the smaller one. Size does not tell you value: both gold coins are worth more than any silver coin. Notes have different colours: $5 pink, $10 blue, $20 red, $50 yellow and $100 green.
Practise for five minutesYear 2
Sort the money and start with the notes or coins worth the most. Count on by each coin's value, changing the size of your jumps when the coins change. Dollars and cents are different units: $1 is 100 cents, and cents always take two places after the point. To make an amount with the fewest coins, use the largest coin that fits, then the next largest, until the amount is made.
Practise for five minutesYear 6
A budget is the money you have to spend. Add up the prices of what you buy, then take the total away from the budget to find what is left. Include the cents. To see how many more things you can buy, use only the money left, and remember you can only buy whole items. When choosing items, check every total against the budget: you must not go over.
Practise for five minutesYear 2
A tally chart records counts with marks. Four upright marks with a fifth across them make a bundle of five, so count bundles in fives and then add the single marks. A picture graph uses a picture for each child, or for a group of children. Always read the key first: if each picture stands for 5, skip count in fives along the row. To find how many more chose one thing than another, work out both rows and subtract.
Practise for five minutesYear 4
Before reading any column, look at the scale on the side. It may go up in 1s, 2s, 5s or 10s. Go across from the top of the column to the scale to read its value. If the top stops halfway between two gridlines, the value is halfway between their numbers. To compare, read both columns and subtract. A side-by-side column graph shows two groups for each category: use the key to tell the groups apart.
Practise for five minutesYear 6
A dot plot puts a dot above a number line for each result. The number under a stack is the value; the number of dots is how many times it happened. The mode is the value with the tallest stack. 'More than 2' leaves out the dots above 2. The range is the largest value with a dot take away the smallest value with a dot. For the median, count the dots from the left to the middle one and read its value.
Practise for five minutesYear 6
A stem-and-leaf plot keeps every value in order. The stem on the left is the tens digit and each leaf on the right is a ones digit, so 3 | 4 8 shows 34 and 38. Use the key to check. Each leaf is one value, even when a leaf repeats. The lowest value is the first leaf in the top row and the highest is the last leaf in the bottom row. For the median, count leaves from the start to the middle value.
Practise for five minutesYear 5
A line graph shows how something changes over time. Find the time on the bottom axis, go up to the point, then across to the scale. Check where the scale starts and what each gridline is worth. The change between two times is the difference between their values. A steeper segment means a faster change. An average change per week is the total change shared equally between the weeks: count the gaps between points, not the points.
Practise for five minutesYear 6
A pie chart shows how a whole is shared. When the circle is split into equal parts, count the parts in a sector and write them over the total number of parts: 3 of 8 parts is 3/8. For a percentage, the whole is 100%, so each part is 100% ÷ the number of parts. To find how many people chose something, share the total number of people into the parts, then take as many parts as the sector covers.
Practise for five minutesYear 4
A good survey question matches your goal, is fair, and lets people give a real answer. A leading question pushes people towards one answer. A yes/no question cannot tell you which choice is the favourite. The people you ask matter too: friends, or a group gathered in one place, often think alike, and a very small group can easily be unusual. Before believing a claim about everyone, ask who was surveyed.
Practise for five minutesYear 6
Probability goes from 0 (impossible) on the left to 1 (certain) on the right, with one half in the middle. If the scale is split into equal steps, count steps from 0: 3 steps of 4 is 3/4. The same position can be written as a decimal, so 7 steps of 10 is 0.7 and 1/4 is 0.25. To place an event, count the outcomes where it happens out of all the equally likely outcomes, then find that point.
Practise for five minutesYear 5
An outcome is one possible result. For one spinner, list the different colours it could land on: sections of the same colour give the same outcome, and a colour with only one section still counts. For two steps, such as a coin and a spinner, every outcome of the first can go with every outcome of the second, so a table has rows × columns cells. For a condition like 'a total of 5', count every cell that fits, in both orders.
Practise for five minutesYear 5
When every outcome is equally likely, the probability of an event is the number of outcomes you want over the total number of outcomes. A spinner with 8 equal sections, 3 of them red, gives P(red) = 3/8. Count sections or counters, not colours. For numbered cards, list the numbers that fit: 'greater than 6' does not include 6. Fractions can be simplified: 4/10 = 2/5.
Practise for five minutesYear 6
Theoretical probability comes from counting equally likely outcomes. Experimental probability comes from what actually happened: the number of times the event happened over the number of trials. To predict, multiply the theoretical probability by the number of trials: a fair dice rolled 60 times should give about 60 ÷ 6 = 10 sixes. Real results wobble around the expected number, and the wobble matters less when there are more trials.
Practise for five minutesYear 6
An event and its complement (the event not happening) always add up to 1. If P(rain) = 3/10, then P(no rain) = 1 − 3/10 = 7/10. For two separate events, such as tossing a coin and rolling a dice, draw a table: 2 × 6 = 12 equally likely outcomes. Count the cells where both happen. 'Not both' is every other cell, so take the 'both' cells away from all the cells.
Practise for five minutes