Online Investigations

Scroll down to browse the listor use the filter on the left to select a topic.

Welcome to hours of mathematical enjoyment!

(Ages 3-6): OLD Multiply by 5 (FREE!)

Investigate what happens when you multiply by 5. Start from the 5x table and investigate the link with the 10x table. Then explore the idea multiplying by 10 and then halving the answer.

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(Ages 5-8): Multiplying with Rectangles and Squares (FREE!)

The best way to understand multiplying is to arrange counters in a rectangle.

Start with 6 arranged in 2 rows of 3. Ask the questions: 'How many rows?' (2), 'How many in each row?' (3), 'How many altogether?' (6). Say together '2 rows of 3 is 6. 2 x 3 = 6.

Try now with 4 counters arranged in a 2 by 2 square. 'How many rows?' (2), 'How many in each row?' (2), 'How many altogether?' (4). Say together '2 rows of 2 is 4. 2 x 2 = 4.

Repeat with different numbers of counters! Keep asking and answering the 3 questions and writing down the sums you get.

Investigate rectangles with 3 rows. 3 rows of 1 = 3, 3 rows of 2 = 6, etc. Build number patterns: 3 x 1 = 3; 3 x 2 = 6; 3 x 3 = 9, etc.

Investigate rectangles with 4 rows in the same way. What about other numbers of rows?

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(Ages 5-8): Multiplication Rectangle Pairs (FREE!)

If you have used rectangles to introduce the idea of multiplying, then it is an easy step to explore the idea that multiplication facts comes in pairs.

Arrange 6 counters in 2 rows of 3. Ask the questions: 'How many rows?' (2), 'How many in each row?' (3), 'How many altogether?' (6). Say together '2 rows of 3 is 6. 2 x 3 = 6.

Now turn the rectangle round the other way. Ask the questions: 'How many rows?' (3), 'How many in each row?' (2), 'How many altogether?' (6). Say together '2 rows of 3 is 6, so 3 rows of 2 is 6!

Write down the paired facts: 2 x 3 = 6, 3 x 2 = 6.

Explore other rectangles in the same way.

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(Ages 5-8): 10x Table (FREE!)

Knowing your tables is REALLY important. But before you learn them, you have to understand where they come from.

Arrange 20 counters in a 2 by 10 rectangle. Say together: 2 rows of 10 is 20; 2 x 10 = 20. Do the same with 3 rows of 10: 3 rows of 10 is 30; 3 x 10 = 30.

Write out the pattern that you get when you count rows of 10 like this: Start with no rows: 0 x 10 = 0; then 1 row: 1 x 10 = 10; then 2 rows etc.

Once you have explored the pattern, the next thing is to try to work out the facts jumbled up. Use counters again to work out the ones you cannot remember.

The next step is to try working backwards. For example, if you have 30 cubes altogether and put them in rows of 10, how many rows would you get? ? x 10 = 30. What is the missing number?

Repeat with other numbers of cubes.

Finally, once you understand how it all works, you need to memorise the tables facts. to do this, play the Tap Say Turn game.

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(Ages 5-8): 2x table (FREE!)

Knowing your tables is REALLY important. But before you learn them, you have to understand where they come from.

Arrange four counters in a square. Say together: 2 rows of 2 is 4; 2 x 2 = 4. Do the same with 3 rows of 2: 3 rows of 2 is 6; 3 x 2 = 6.

Write out the pattern that you get when you count rows of 2 like this: Start with no rows: 0 x 2 = 0; then 1 row: 1 x 2 = 2; then 2 rows etc.

Once you have explored the pattern, the next thing is to try to work out the facts jumbled up. Use counters again to work out the ones you cannot remember.

The next step is to try working backwards. For example, if you have 6 cubes altogether and put them in rows of 2, how many rows would you get? ? x 2 = 6. What is the missing number?

Repeat with other numbers of cubes.

Finally, once you understand how it all works, you need to memorise the tables facts. to do this, play the Tap Say Turn game.

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(Ages 5-8): 5x table (FREE!)

Once you have learned the 2x and 10x tables, the next one to work on is the 5x table.

Arrange 10 counters in a 2 by 5 rectangle. Say together: 2 rows of 5 is 10; 2 x 5 = 10. Do the same with 3 rows of 5: 3 rows of 5 is 15; 3 x 5 = 15.

Write out the pattern that you get when you count rows of 5 like this: Start with no rows: 0 x 5 = 0; then 1 row: 1 x 5= 5; then 2 rows etc.

Notice that the numbers alternate with 0 and 5: 0, 5, 10, 15, 20, 25, 30 etc.

Once you have explored the pattern, try to work out the facts jumbled up. Use counters again to work out the ones you cannot remember.

Next try working backwards. For example, if you have 15 cubes altogether and put them in rows of 5, how many rows would you get? ? x 5 = 15. What is the missing number?

Repeat with other numbers of cubes.

Finally, once you understand how it all works, you need to memorise the tables facts. to do this, play the Tap Say Turn game.

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(Ages 5-8): Double and Half Small Numbers (FREE!)

Explore doubling first by looking at the idea of 2 lots of (p2), at then at multiplying by 2 (p6). Next look at halving (p10) and how this is the same as dividing by 2 (p14). Along the way, practise each of the four skills using numbers within 20, including missing number problems.

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(Ages 6-9): Factors of 10 and 20 (FREE!)

Factor rainbows are a lovely way of exploring the different multiplication facts you can make with a particular number of counters.

Start with 10 counters and investigate how to arrange them in different rectangles = 2 rows of 5, 5 rows of 2, 1 row of 10, 10 rows of 1. Write down the corresponding multiplication facts: 2 x 5 = 10, 5 x 2 = 10, 1 x 10 = 10, 10 x 1 = 10.

Then draw the factor rainbow. Put the four numbers 1, 2, 5, 10 in order and join together the numbers that multiply to make 10. Discuss how this links with the rectangles you have made and the facts that you have written.

Repeat with 20 counters.

Further investigation: Which other numbers make good factor rainbows?

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(Ages 6-9): 4x table (FREE!)

Once you have learned the 2x, 10x and 5x tables, you can work on the 3s and 4s.

For the 4x table, arrange 8 counters in a 2 by 4 rectangle. Say together: 2 rows of 4 is 8; 2 x 4 = 8. Do the same with 3 rows of 4: 3 rows of 4 is 12; 3 x 4 = 12.

Write out the pattern that you get when you count rows of 4 like this: Start with no rows: 0 x 4 = 0; then 1 row: 1 x 4 = 4; then 2 rows etc.

Notice the repeating pattern in the ending digits: 0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 0. Then use the Counting Caterpillar game to learn to count forwards and backwards in 4s by heart.

Once you have explored the pattern, try to work out the facts jumbled up. Use counters again to work out the ones you cannot remember.

Next try working backwards. For example, if you have 20 cubes altogether and put them in rows of 4, how many rows would you get? ? x 4 = 20. What is the missing number?

Repeat with other numbers of cubes.

Finally, once you understand how it all works, you need to memorise the tables facts. to do this, play the Tap Say Turn game.

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(Ages 6-9): 3x table (FREE!)

Once you have learned the 2x, 10x and 5x tables, you can work on the 3s and 4s.

For the 3x table, arrange 6 counters in a 2 by 3 rectangle. Say together: 2 rows of 3 is 6; 2 x 3 = 6. Do the same with 3 rows of 3: 3 rows of 3 is 9; 3 x 3 = 9.

Write out the pattern that you get when you count rows of 3 like this: Start with no rows: 0 x 3 = 0; then 1 row: 1 x 3= 3; then 2 rows etc. Then use the Counting Caterpillar game to learn to count forwards and backwards in 3s.

Once you have explored the pattern, try to work out the facts jumbled up. Use counters again to work out the ones you cannot remember.

Next try working backwards. For example, if you have 15 cubes altogether and put them in rows of 3, how many rows would you get? ? x 3 = 15. What is the missing number?

Repeat with other numbers of cubes.

Finally, once you understand how it all works, you need to memorise the tables facts. to do this, play the Tap Say Turn game.

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(Ages 6-9): Counting in 10s (FREE!)

See how far you can get counting in tens. Explore what happens when you oount over 100, 200, 300, 1000.

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(Ages 6-9): Double and Half with Multiples of Ten (FREE!)

Explore how to double multiples of ten building on knowledge of simple doubles facts. Extend to halving where the answer is a multiple of ten.

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(Ages 6-9): Counting in 25s (FREE!)

Explore what happens when you chop a metre stick into four pieces. Learn that 4 25s make a 100 and investigate other multiples of 25 by counting.

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(Ages 6-9): Counting in 20s (FREE!)

Investigate how to divide a metre stick into 5 equal pieces. Count in 20s beyond 100 and explore multiples of 20.

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(Ages 6-9): Double Multiples of 5 (FREE!)

Investigate the patterns for doubling multiples of 10 and then explore the in-between numbers to establish patterns for multiples of 5.

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(Ages 6-9): Multiply and Divide by 10 (FREE!)

Revise the ideas introduced in the Counting section that (eg) 8 tens = 80 (p2) and recap on the 10x table (p3).

Revise the inverse link between division and multiplication (p6) and explore the 10x table division facts (p7).

Recap on the idea that 10 tens (100) + 3 tens (30) = 13 tens (130) (p9) and use this to explore beyond the table - 10 x 13 = 130 etc (p11). Investigate the corresponding division facts (p13).

Extend to facts such as 36 tens = 360 etc (p17).

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(Ages 6-9): Half Multiples of Ten (FREE!)

Revise halving multiples of twenty and then explore what happens when you try to halve a multiple of ten like 50. Extend to halving numbers like 170.

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(Ages 6-9): Multiplication Triangles (FREE!)

Investigate multiplication triangles and the linked multiplication and division facts.

Investigate what happens if you try to divide by zero!

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(Ages 6-9): Multiplying and Dividing with Rectangles (FREE!)

See how many different ways you can divide up 12 objects. Explore the use of the division sign and how division relates to rows and columns in a rectangle.

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(Ages 6-9): Inverse Arrows - Multiply and Divide (FREE!)

Learn the links between multiplication and division.

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(Ages 7-10): 9x Table Patterns (FREE!)

Explore the amazing patterns in the 9x table and then investigate digital roots.

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(Ages 7-10): Double Tens and Hundreds (FREE!)

Discover how, knowing the 2x table facts, you can use these to double numbers like 20, 300 etc.

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(Ages 7-10): Something out of Twelve as a Fraction (FREE!)

Explore how to find what fraction has been selected when you select a number of objects from a larger group. Investigate the link with finding fractions of things.

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(Ages 7-10): Multiply Teens Numbers by 5 (FREE!)

Use rectangles to discover that, for example, 14 rows of 5 is the same as ten rows plus four rows. Use this to work out 14 x 5 mentally etc.

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(Ages 7-10): Multiply and Divide with Tens by 5 (FREE!)

Explore the patterns in multiplying tens by 5.

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(Ages 7-10): Double Two-digit Numbers (FREE!)

Investigate how doubling questions are easiest when both digits are less than 5, harder when one digit is greater than 5 and trickiest when both digits are greater than 5.

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(Ages 7-10): Multiply and Divide beyond the 5x Table (FREE!)

Explore multiplication and division within the 5x table and use this knowledge to tackle problems with bigger numbers.

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(Ages 7-10): Half Two-digit Even Numbers (FREE!)

Investigate how to half two digit even numbers by halving the tens, halving the units and combining the answers.

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(Ages 7-10): Multiply and Divide beyond the 3x Table (FREE!)

Explore multiplication and division using the 3x table. Begin with multiplying single digits and then investigate multiplying multiples of ten by 3.

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(Ages 7-10): Multiply and Divide beyond the 4x Table (FREE!)

Explore multiplication and division within the 4x table and use this knowledge to tackle problems with bigger numbers.

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(Ages 7-10): Multiply and Divide by 10 and 100 (FREE!)

Investigate how to multiply and divide different numbers by 100.

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(Ages 7-10): Factors of 100 (FREE!)

Discuss where you would cut a metre stick to chop it into two equal pieces (p3). Make the link with halves (p4) and factor pairs (p5).

Next work out where to cut if you want to chop a metre stick into four equal pieces (p8). Again, make the link with quarters (p7) and factor pairs (p8). Repeat for five equal pieces (p9) and then ten (p12).

Finally, draw the factor rainbow for 100 (p15) and then build the factor pair pattern (p16).

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(Ages 7-10): 20x table (FREE!)

Discover how easy the 20x table is when you see the link with the 2x table.

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(Ages 7-10): Multiply Single Digits by Tens (FREE!)

Learn how to multiply by a multiple of 10.

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(Ages 8-11): Factors of 12, 24 and 60 (FREE!)

Discuss the fact that the numbers 12, 24 and 60 are used a lot when measuring time (p1).

Investigate the factors of 12 (p3), draw the factor rainbow (p5) and build the factor pair pattern (p6). Explore the link with months (p7).

Repeat for 24 (p8), exploring the link with hours (p12). Then repeat with 60 (p13), exploring the link with minutes (p17).

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(Ages 8-11): Multiply and Divide beyond the 6x Table (FREE!)

Investigate how to multiply and divide different numbers by 6 and use this knowledge to tackle problems with bigger numbers.

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(Ages 8-11): Patterns Multiplying by 5 (FREE!)

Explore the link between multiplying by 10, dividing by 2 and multiplying by 5 and establish key facts and quick methods for tackling questions like ? x 5 = 80 and 170 x 5 = ?

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(Ages 8-11): Patterns Dividing by 5 (FREE!)

Investigate the link between dividing by 10, multiplying by 2 and dividing by 5 and explore how this can be used to tackle problems like 120 ÷ 5 = ?

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(Ages 8-11): Divide and Multiply by 10 with Decimals - Part 1 (FREE!)

Explore what happens if you try to divide a number by 10. Learn the rule for moving numbers relative to the decimal point.

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(Ages 8-11): Divide and Multiply by 10 with Decimals - Part 2 (FREE!)

Explore more about what happens if you try to divide a number by 10. Learn more about the rule for moving numbers relative to the decimal point.

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(Ages 8-11): Double 3-digit Multiples of 10 (FREE!)

Explore how to use your knowledge of doubling 2-digit numbers such as 74 to double 3-digit numbers like 740 etc.

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(Ages 8-11): Half 3-digit multiples of 10 (FREE!)

Explore how to use the skill of halving 2-digit numbers such as 74 to halve 3 digit numbers such as 740.

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(Ages 8-11): Multiply and Divide beyond the 8x Table (FREE!)

Explore multiplication and division within the 8x table and use this knowledge to tackle problems with bigger numbers.

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(Ages 8-11): A Quick Way to Multiply by 9 (FREE!)

Learn the quick method for multiplying by 9 (find ten lots and take away one lot) and then practise on numbers of your own choice. Self-checking and self-marking.

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(Ages 8-11): Multiply and Divide beyond the 7x Table (FREE!)

Investigate how to use the 7x table to multiply and divide all sorts of numbers by 7.

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(Ages 8-11): Multiply Big Numbers by 7 (FREE!)

If you think you know your tables then apply your knowledge to multiplying larger numbers by a single digit. Use the 7x table because this one is the hardest!

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(Ages 8-11): Multiply 2 Digits by Multiples of 10 (FREE!)

Explores how to multiply by numbers like 30, 40, 60 etc

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(Ages 8-11): How do Square Numbers Work? (FREE!)

Learn about square numbers.

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(Ages 9-12): Multiply and Divide Negative Numbers (FREE!)

Explore how multiplication and division work with negative numbers. Explore patterns in number facts to deduce well known relationships and explain why they work.

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(Ages 9-12): Multiply and Divide Decimals by 10, 100, 1000 (FREE!)

Build understanding step by step from simple examples (6 x 10) to more complicated ones (0.22 x 1000).

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(Ages 9-12): Factors of 360 (FREE!)

Discuss the fact that for thousands of years people have measured angles in a circle by dividing the circle into 360 equal degrees (p1). Speculate as to why the number 360 might have been chosen (365 days in a year, but 365 does not divide up nicely - 360 is close to 365 and does divide up nicely).

Explore the idea that if there are 360 degrees in a full revolution, there will be 180 degress in half a revolution (p2) and 90 in a right angle (p4).

Consider a compass rose with 8 points, work out the angle between the points (p6) and investigate the link with factor pairs (p7).

Next consider hours on a clock (p8), angles in the six equilateral triangles in a hexagon (p10), a three armed windmill (p12) and then minutes on a clock (p14), investigating the angles each time and relating them to factor pairs.

See how many other factors you can find for 360 (p16) and then build the factor pairs pattern (p17). Finally draw the factor rainbow (p19).

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(Ages 9-12): How Many Times Bigger? (FREE!)

Learn how to solve problems such as 'How many times bigger is London than Edinburgh?' by exploring the inverse relationship between multiplication and division.

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(Ages 9-12): Getting Division the Right Way Round (FREE!)

Which is which, '8 divided by 5' or '5 divided by 8'?

Sort out this confusion once and for all!

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(Ages 10-13): Fractions, Division and Decimal Equivalents (FREE!)

Discover the amazing fact that ½ = 1 ÷ 2 = 0.5, ¾ = 3 ÷ 4 = 0.75 and go on to explore what happens with other fractions.

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(Ages 10-13): The Connection between Of and Times (FREE!)

Explore the idea that multiplying by (eg) ¼ is the same as finding ¼ of something which, in turn, is the same as dividing by 4. This investigation introduces the fraction triangle, a powerful little tool for exploring this idea.

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(Ages 10-13): Expressing Things as a Fraction or Percentage: Part 1 (FREE!)

Discuss the mark 17 out of 20 in a test. Agree that this would be 17 twentieths of the total mark (p2).

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(Ages 10-13): Fraction, Decimal, Percentage Triangle Magic (FREE!)

Explore the link between finding a % of something and expressing something AS a percentage.

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(Ages 10-13): Multiply a Fraction by a Whole Number (FREE!)

Investigate how to multiply fractions by whole numbers.

Get pupils exploring 1 x ¼, 2 x ¼, 3 x ¼ etc and they will quickly discover the rules that for multiplication you multiply the numerator and for division you multiply the denominator.

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(Ages 10-13): Divide a Fraction by a Whole Number (FREE!)

Investigate how to divide fractions by whole numbers.

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(Ages 10-13): Negative Number Reflections (FREE!)

Explore how multiplying and dividing by different positive and negative numbers moves things around on the number line. Start by investigating reflections and then find out how to make numbers move closer to or further away from zero.

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(Ages 10-13): How Can Multiplication Make Things Smaller? (FREE!)

Investigate how multiplying by certain numbers (eg 0.5, 0.1, 0.2, a half, a tenth, a fifth etc) makes a number smaller.

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(Ages 10-13): How Can Division Make Things Bigger? (FREE!)

Investigate why some divisions give answers that are larger than the starting number.

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(Ages 10-13): Multiply and Divide by a Decimal (FREE!)

Weird things start to happen when you multiply and divide by decimals. Things that should get bigger get smaller!

Investigate how to multiply and divide by 0.1 and by 0.01. Then go on to investigate multiplying and dividing by other easy decimals such as 0.2, 0.02, 0.4, 0.04, 0.5, 0.05

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(Ages 11-14): Recurring Decimals and Fractions (FREE!)

Investigate the patterns of digits in recurring decimals arising through division and how this links to fractions decimal equivalences.

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(Ages 11-14): Multiply by a Fraction (FREE!)

Investigate how to multiply a whole number or a fraction by a unit fraction (such as one quarter or one fifth). Then investigate multiplying by other fractions.

Multiplying by a fraction is where things get interesting. Pupils are used to thinking that multiplying a number by something will make it bigger. But when fractions are around, all that changes!

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(Ages 11-14): Divide by Any Fraction (FREE!)

Explore how to divide a whole number or fraction by any fraction using the idea of reciprocals.

Discover how dividing by 5 eighths is the same as multiplying by 8 fifths and vice versa.

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