How to Read Coordinates

Coordinates are 2 numbers written in brackets, separated by a comma. The first number is the x coordinate which when positive, indicates how far right the point is and when negative, indicates how far left the point is. The second number is the y coordinate which when positive, indicates how far up the point is and when negative, indicates how far down the point is.

The order in which coordinates are written matters. The first number is the x coordinate and the second number is the y coordinate.

For example, the coordinate (4, 9) has an x coordinate of 4 and a y coordinate of 9.

The x coordinate is 4. This is a positive number and so, it means to move 4 places right. This means we move 4 places right from where the axes cross over at (0, 0).

The y coordinate is 9. This means that after moving 4 places right, we move 9 places up. We know that we move upwards because 9 is a positive number.

To remember the order that the first number tells us to go right and the second number tells us to go up, we use the phrase ‘along the corridor, up the stairs’.

The ‘along the corridor’ represents us going along (to the right) and the ‘up the stairs’ represents going up.

This phrase is commonly used in schools when teaching coordinates because it helps to avoid the most common mistake of writing the two numbers in the wrong order.

To read coordinates from a grid, compare the position of the coordinate to the origin where the axes meet. First write the number of places the coordinate is to the right of the y-axis. If it is to the left, this number is negative. Now write a comma and then write the number of places the coordinate is above the x-axis. If it is below, this number is negative.

Coordinates are always measured from the origin. The origin is where the x and y axes cross over at the centre of the grid.

We compare the position of the coordinate to the origin to read the coordinate.

Rules of reading coordinates:

• Right is positive.
• Left is negative.
• Up is positive.
• Down is negative.

The point is 10 left of the origin. Since it is left, we write down -10. Negative x coordinates mean left.

The point is 4 below the origin. Since it is down, we write -4. Negative y coordinates mean down.

The coordinates are written (-10, -4). The numbers must be written in this order.

Write the x coordinate first, which is how far left or right the coordinate is. Then write the y coordinate second, which is how far up or down the coordinate is.

How to Plot Coordinates

To plot coordinates:

1. Read the first coordinate before the comma.
2. Along the x-axis, move this many places right if it is positive or left if it is negative.
3. Read the second coordinate after the comma.
4. Move this many places up if the number is positive or down if it is negative.
5. Draw a cross at this position.

For example, plot (-8, 2).

The first number, the x coordinate, is -8. It is negative, so we move 8 places left.

Now we read the second number, the y coordinate, which is 2. It is positive so we move 2 places up.

For example, plot the coordinate (6, -6).

The x coordinate is positive, so we move 6 places right.

The y coordinate is negative, so we move 6 places down.

For example, plot (3, 5).

The x coordinate is 3, so we move 3 places right.

The y coordinate is 5, so we move 5 places up.

The Order of Coordinates Matters

The order in which coordinates are written matters. The x coordinate must be written first and the y coordinate must be written second. The x coordinate describes the horizontal position and the y coordinate describes the vertical position. If the x and y coordinates are written in the wrong order, the coordinate will be describing the wrong position.

For example:

The coordinate (1, 2) means 1 to the right and 2 up.

The coordinate (2, 1) means 2 to the right and 1 up.

We can see that switching the order of the coordinates results in the point being located in a different position.

When learning how to write coordinates, the most common mistake is to write the x and y coordinates in the wrong order.

Use the phrase, ‘along the corridor and up the stairs’ to remember that the first coordinate describes how far along the point is and the second coordinate describes how far up or down the point is.

Reading Coordinates with Fractions and Decimals

Coordinates can be located at any point on the grid, including inside the squares of the grid. If a coordinate is located directly between two whole numbers, it will contain fractions of one half. If the coordinate is not directly in between two whole numbers, use the scale to estimate the location as a decimal.

For example, here is the coordinate (4   ¹/₂, 3   ¹/₂).

The coordinate is located directly between 4 and 5 on the x axis and so has an x coordinate of 4   ¹/₂.

The coordinate is located directly between 3 and 4 on the y axis and so has a y coordinate of 3   ¹/₂.

Here is an example of a coordinate that is not in the centre of a grid and so, it is nearer to one number than the other.

Here we estimate its position and we can write it as a decimal.

The coordinate is very close to 2 on the x axis but it is not quite on 2. It is at 1.9.

The coordinate is very close to 1 on the y axis but it is just above 1. It is at 1.1.

How to Read 3D Coordinates

To read coordinates in 3D, read the distance travelled in the direction of the x axis, y axis and z axis in that order. There are 3 numbers that make up a set of coordinates in 3D and they are always written in the form (x, y, z).

For example, Here is the point (1, 3, 2).

The point has an x coordinate of 1, a y coordinate of 3 and a z coordinate of 2.

Here is the 3D coordinate of (2, 0, 5).

The point has an x coordinate of 2, a y coordinate of 0 and a z coordinate of 5.

What is Bridging to 10?

Bridging to 10 is a method of adding two numbers that have an answer larger than 10. Count up to 10 and then add on the remainder. For example, to work out 7 + 4, firstly do 7 + 3 = 10 and then add the remainder of 1 to make 11.

The bridging to 10 strategy is often referred to as the bridging through 10 strategy or simply the bridging 10 strategy.

The bridging 10 strategy is a useful strategy for mental addition.

How to Teach Bridging to 10

When learning the bridging to 10 strategy, it is first necessary to have a strong understanding of the number bonds to 10. Number lines help to teach bridging to 10 because they can be used to count up to the next ten. Furthermore, tens frames and counters can be used to help count up to the next ten.

For example, the number line below shows the bridging to 10 of the addition 7 + 5.

We can count on by 3 to reach ten and then see that 2 more must be added to add 5 in total.

It helps to know the number bonds to 10 very well before learning the bridging to 10 method. This way we know that 3 must be added to 7 to make 10.

Tens frames can be used to help teach the idea of number bonds to ten and they can also be used to teach the bridging to 10 process.

Tens frames allow us to easily see how many more must be added to a number in order to make ten. This helps us with the first step of the bridging ten process.

Addition by Bridging to 10

To add two numbers by bridging to 10:

1. Find the number that when added to the larger of the two numbers makes the next multiple of ten.
2. Find the remainder by subtracting this same number from the smaller number.
3. Add this remainder onto the multiple of ten found in step 1.

For example, add 46 + 7 by bridging to 10.

Step 1. Find the number that when added to the larger of the two numbers makes the next multiple of ten

46 is the larger of the two numbers. We can add 4 to 46 to make the next multiple of ten which is 50.

Step 2. Find the remainder by subtracting this same number from the smaller number

We subtract 4 from 7 to make 3. This means that the remainder is 3.

Step 3. Add this remainder onto the multiple of ten found in step 1

The remainder is 3. We add this to the multiple of ten found in step 1, which was 50.

50 + 3 = 53.

Therefore 46 + 7 = 53.

We bridged to the next ten along which was 50. Since we added 4, we needed to add 3 more to add a total of 7.

Subtraction by Bridging to 10

To subtract by bridging to 10:

1. Subtract the ones digit of the larger number from the larger number to make a multiple of ten.
2. Find how much more must be subtracted in order to subtract the amount required.
3. Subtract this extra amount from the multiple of ten to obtain the answer to the subtraction.

For example, calculate 42 – 7 using the bridging to ten subtraction strategy.

Step 1. Subtract the ones digit of the larger number from the larger number to make a multiple of ten

We subtract 2 from 42 to obtain 40.

Step 2. Find how much more must be subtracted in order to subtract the amount required

We need to subtract 7 and we have already subtracted 2.

We need to subtract 5 more because 2 + 5 = 7.

Step 3. Subtract this extra amount from the multiple of ten to obtain the answer to the subtraction

We subtract 5 more from 40 to make 35.

Therefore 42 – 7 = 35.

We subtracted 2 to bridge ten at 40 and then subtracted 5 more to subtract a total of 7.

What are Cube Numbers?

A cube number is formed when any whole number is multiplied by itself twice. The symbol for cubing a number is ³. For example, 2 cubed or 2³ means 2 × 2 × 2 which equals 8. Therefore 8 is a cube number. Cube numbers are called this because the volume of a cube is found by cubing its side length.

Below shows a cube with side lengths of 2 cm.

The volume of the cube is found by multiplying a side length by itself twice.

2 × 2 × 2 = 8 and so the volume of the cube is 8 cm³.

This means that the overall cube is made up of 8 smaller 1 cm³ cubes.

8 is a cube number because it is formed by multiplying a whole number by itself twice.

2 × 2 × 2 = 8

We can write this more simply as 2³ = 8.

The number is written to the power of 3, which is a shorter way of writing the full multiplication.

How to Find a Cube Number

To find a cube number, multiply any whole number by itself and then by itself again. The easiest way to do this is to do each multiplication separately. For example, 3 cubed is 3 × 3 × 3. The first multiplication is 3 × 3 = 9 and then the second multiplication is 9 × 3 = 27. Therefore the 3rd cube number is 27.

Here is another example of calculating 4 cubed.

4³ means 4 × 4 × 4.

The first step is to multiply 4 × 4 to get 16.

The next step is to multiply 16 by 4 again.

16 × 4 = 64.

64 is the 4th cube number.

List of Cube Numbers

The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.

Here is a complete list of the first 100 cube numbers:

Properties of Cube Numbers

Cube numbers have the following properties:

• An even number cubed is always even.
• An odd number cubed is always odd.
• A positive number cubed is always positive.
• A negative number cubed is always negative.
• Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit.
• The sum of the cubes of the first n natural numbers is equal to the square of their sum.

Here are some examples to illustrate these properties of cube numbers.

An even number cubed is always even

For example, 2 is an even number. If we cube it we get 8, which is an even answer.

2 × 2 × 2 = 8

This property works because if any number is multiplied by an even number at least once, the result is even. Cubing an even number means that we must multiply by an even number.

An odd number cubed is always odd

For example, 3 is an odd number. If we cube it we get 27, which is an odd answer.

3 × 3 × 3 = 27

This property works because if two odd numbers are multiplied together, the result is always odd. To make an even number, at least one of the numbers being multiplied will need an even factor. However, when an odd number is cubed, we have odd × odd × odd and so, no factor of two appears in the final result.

A positive number cubed is always positive

For example, 10 is a positive number. If we cube it we get 1000, which is also positive.

10 × 10 × 10 = 1000

This property works because a negative number can only be made from another negative. If we only multiply positive numbers, the result must be positive.

A negative number cubed is always a negative

For example, (-2) × (-2) × (-2) = -8.

When we cube a number, we multiply it by itself twice. If a negative number is cubed, we have three negative numbers multiplied together.

When three negative numbers are multiplied together, the result is always negative.

(-2) × (-2) = +4 and then 4 × (-2) = -8.

Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit

For example:

10³ = 1000. Both numbers end in 0.

21³ = 9261. Both numbers end in 1.

4³ = 64. Both Numbers end in 4.

15³ = 3375. Both numbers end in 5.

66³ = 287496. Both numbers end in 6.

19³ = 6859. Both numbers end in 9.

The sum of the cubes of the first n natural numbers is equal to the square of their sum

1³ + 2³ + 3³ + … + n³ = (1 + 2 + 3 + … + n)².

For example, for an n of 5:

1³ + 2³ + 3³ + 4³ + 5³ = 225.

(1 + 2 + 3 + 4 + 5)² = 225.

Cubing the consecutive numbers and then adding them up gives the same result as adding the numbers up and then squaring them.

For example, for an n of 10:

1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ + 10³ = 3025.

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)² = 3025.

Cube Numbers That Are Also Square Numbers

The cube numbers that are also square numbers are found by raising whole numbers to the power of 6. For example, 1⁶ = 1, 2⁶ = 64 and 3⁶ = 729. 1 is 1² and 1³, 64 is 8² and 4³ and 729 is 27² and 9³.

1³ = 1 = 1² 2³ = 8 3³ = 27 4³ = 64 = 8² 5³ = 125 6³ = 216 7³ = 343 8³ = 512 9³ = 729 = 27² 10³ = 1000

1, 64 and 729 are cube numbers and also square numbers.

Further cube numbers that are also square numbers are found by raising any integer to the power of 6:

1⁶ = 1 2⁶ = 64 3⁶ = 729 4⁶ = 4096 5⁶ = 15625 6⁶ = 46656 7⁶ = 117649 8⁶ = 262144 9⁶ = 531441 10⁶ = 1000000

What are Cuisenaire Rods?

Cuisenaire rods are coloured plastic bars of different lengths, which are used to represent different number sizes. These rods are used to demonstrate the comparative sizes of numbers and can be useful in teaching basic addition, subtraction, multiplication and division. It is also possible to use Cuisenaire rods to demonstrate equivalent fractions, decimal numbers and ratio.

Here is a colour chart showing the different Cuisenaire rod values.

Cuisenaire rods were invented by Belgian schoolteacher Emile-Georges Cuisenaire in the 1950s.

They are often used in primary schools for teaching number facts and therefore are commonly used with children between the ages of 4-7. However, they are often used with children older than this who struggle with mathematics as they can support their understanding of number size. Cuisenaire rods can also be used to teach older children between 11-16 concepts such as ratios and fractions.

How to Introduce Cuisenaire Rods

Cuisenaire rods can be introduced by asking for the rods to be placed in order of size and then asking if any smaller rods can be combined to make the same length as the larger rods. Then ask if further combinations of rods that are of equal lengths can be found. Can we make any number using only white or red rods? These activities will help build the idea of equivalences and familiarise children with the rod values.

Showing basic equivalences can be helpful such as this example of 2 + 3 = 5. Can your child find further combinations of rods that are equal in size?

Further introductory Cuisenaire rod ideas are:

• Can you make a combination of rods the same length as the blue rod using exactly 3 rods?
• Can you make a rod the same length as the orange rod using only red rods?
• How about using only green rods?
• What about using only red and green rods?
• What is larger, a black plus a white or a brown plus a purple?

  Games that ask children to create equivalent lengths and to play with the size of different rods are useful for introducing Cuisenaire rods.

  Why Use Cuisenaire Rods

  Cuisenaire rods are used because they are a simple and hands-on way to compare number sizes. They help to engage children in mathematics as they are brightly coloured and easy to stack together. Cuisenaire rods also have a number of purposes across many different branches of mathematics which makes them a versatile tool.

  Here are some of the advantages and disadvantages of teaching with Cuisenaire rods.

  Advantages of Cuisenaire Rods	Disadvantages of Cuisenaire Rods
  They provide tactile learning for students with maths difficulties	You cannot count up in ones on each block
  Helps to build comparative number sense	The rods can be limited to teaching smaller numbers
  Different coloured rods for each size	Lengths are not written on the rods
  Can be used in a variety of games/settings	Cannot represent all fractions easily
  Simple to use	Potentially limited in challenge
  Allows for trial and error without fear of making mistakes	Limited feedback on learning
  Time to explore numbers without rushing
  Can build stronger understanding of numbers using visual learning

  What are the Values of the Cuisenaire Rods?

  Cuisenaire rods always have the same fixed values:

  Cuisenaire Rod Colour	Colour Code	Value	Length
  White	w	1	1 cm
  Red	r	2	2 cm
  Lime Green	g	3	3 cm
  Purple	p	4	4 cm
  Yellow	y	5	5 cm
  Dark Green	d	6	6 cm
  Black	b	7	7 cm
  Tan (Brown)	t	8	8 cm
  Blue	B	9	9 cm
  Orange	o	10	10 cm

  

  How to Use Cuisenaire Rods to Teach Addition

  To teach addition using Cuisenaire rods, combine two smaller rods end to end and place them alongside a larger rod that is the same length as the two smaller rods combined. For example 2 white rods are the same length as 1 red rod. We can say that 1 + 1 = 2 because they are the same length. This can then be extended to larger numbers by combining two or more larger rods.

  For example, the red plus green rods are the same length as the yellow rod.

  Since the red = 2, the green = 3 and the yellow = 5, we can see that 2 + 3 = 5.

  

  Here is a larger example of 4 + 6 = 10 using the purple, dark green and orange rods.

  

  Cuisenaire rods are great for teaching the number bonds to 10. These are the pairs of numbers that add to make 10.

  Simply ask your child to find and list as many different combinations of rods that add to make the same length as the orange 10 rod.

  How to Use Cuisenaire Rods to Teach Subtraction

  To teach subtraction using Cuisenaire rods, find a combination of two smaller rods that are the same length as one larger rod. Place the two smaller rods alongside the larger rod to show that they are the same size. Then remove the rod that you are subtracting and the remaining rod is the answer. Cuisenaire rods help to reinforce that the subtraction gives us the difference between two numbers.

  For example, here is 7 – 4 = 3.

  The black rod is worth 7, the purple is worth 4 and the lime green is worth 3.

  

  To show the subtraction of 4, simply remove the purple 4 rod. The remaining green rod is the answer. This shows us that 3 is the difference between 7 and 4.

  Here is another example of subtraction with Cuisenaire rods. We have 9 – 2 = 7.

  

  The blue is worth 9, the red is worth 2 and the black is worth 7.

  Removing the red 2 rod leaves us with the red 7 rod. 7 is the difference between 9 and 2.

  How to Use Cuisenaire Rods to Teach Multiplication

  Cuisenaire rods can be used to teach multiplication by combining rods of the same size end-to-end to make a larger value. The number of rods combined together tell us how many lots of the base number we have. For example, 2 × 5 = 10 can be shown by lining up 5 lots of the red size 2 rods alongside an orange size 10 rod.

  

  We can see that 5 lots of 2 is equal to 10.

  Here is an introductory example of 1 × 4 = 4.

  Each white Cuisenaire rod is worth 1 and the purple Cuisenaire rod is worth 4. We can see that 4 lots of 1 is the same as 4.

  

  Cuisenaire rods can be used to build a conceptual understanding of multiplication as repeated addition.

  How to Use Cuisenaire Rods to Teach Division

  Cuisenaire rods can be used to teach division by first combining rods of the same size end-to-end to form a rod of a larger size. The smaller rods can then be counted to see how many go into the larger rod. The number of smaller rods is the answer to the division. For example, a blue size 9 rod can be made from 3 lime green size 3 rods. This demonstrates the division of 9 ÷ 3 = 3.

  

  There are 3 lots of the smaller lime green rods and so, the answer to the division is 3.

  Here is another example of division using Cuisenaire rods. We have 6 ÷ 1 = 6.

  The dark green rod is worth 6. The white rod is worth 1.

  There are 6 white rods and so, 6 ÷ 1 = 6.

  

  How to Use Cuisenaire Rods to Teach Fractions

  To teach fractions using Cuisenaire rods, a smaller rod can be used to represent the numerator and the larger rod can be used to represent the denominator. For example a size 2 red rod and a size 6 dark green rod represent the fraction ²/₆. We can see that the red rod is one third of the size of the dark green rod and so, ²/₆ is the same as ¹/₃.

  

  The number of times the smaller rod fits into the larger rod tells us the simplified fraction.

  Cuisenaire rods are useful to build basic fraction understanding, such as comparing a size 5 yellow with a size 10 orange rod can easily show that 5 is half of ten.

  How to Use Cuisenaire Rods to Teach Decimals

  Cuisenaire rods can be useful tools for explaining the comparative sizes of decimals. For example, labelling the size 10 orange rod as worth 1, then the size 1 white rod will be one tenth of this, which is 0.1. Then the red size 2 rod is worth 0.2 and so on. Decimals can be taught by combining Cuisenaire rods to make further decimal numbers.

  Here the orange rod is worth one whole. This means that every white size 1 rod is worth 0.1.

  We can combine 3 white rods to make 0.3.

  

  Alternatively, two red size 2 rods would be worth 0.4 because this is the same length as 4 of the 0.1 rods.

  Cuisenaire rods are very useful when teaching the comparative sizes of decimals. In the diagram above we can easily see that 0.4 is just under half of the size of 1 whole.

  We can also represent decimals in other ways if necessary. For example, this time the yellow size 5 rod is worth one whole.

  This means that each size 1 white rod is one fifth, or 0.2.

  Every time we add a white rod, we count up in 0.2s.

  

  How to Use Cuisenaire Rods to Teach Ratio

  Due to the ease in which Cuisenaire rod sizes can be compared, they are a very useful tool for teaching ratios. By allocating a value to the smallest white rod, larger values can be demonstrated. For example, if the white size 1 rod is worth 5, then the red size 2 rod is worth 10.

  The red rod is always double the value of the white rod, no matter what the white rod is worth.

  

  Here is another example of teaching ratio using Cuisenaire rods. Here the lime green size 3 rod is worth 30.

  Because 3 white rods make 1 lime green rod, the white rod must be worth 10.

  

  We can then see that the yellow rod must be worth 50 because we have 5 lots of the white rods, which are worth 10 each.

  Here we have a purple rod worth 80 and we want to know the value of the red rod. We first see that 4 white rods go into the purple rod, so each white rod must be worth 20.

  The red rod is always double the white rod and so, it is worth 40.

  

  Cuisenaire Rod Activities

  Here are a list of short Cuisenaire rod activities:

  • Put the Cuisenaire rods in order of size from smallest to largest to make a ‘staircase’.
  • Count how many white rods are needed to make the other size rods.
  • Find which colour rods can be make using only the red rods.
  • Find as many combinations of two smaller rods which are the same length as one larger rod.
  • Find as many combinations of three smaller rods which are the same length as one larger rod.
  • Find as many different pairs of rods that have a difference in size of 1.
  • Find a pair of rods that have a difference of exactly 6.
  • Use the rods to find and list all different combinations of rods that add to make 10.
  • Cuisenaire rod snap – take it in turns to draw a different rod from your pile. If the two rods add to make 10, the person who says snap first wins the rods.

What Does it Mean to Rotate a Shape?

Rotating a shape means to change its direction by turning it. The size and shape do not change during rotation. A shape is often rotated about a specific point called the centre of rotation. We also need to know what angle and direction the shape is rotated in.

For example, this triangle has been rotated 90° counter-clockwise about the point.

You can tell that a shape has been rotated because it is not facing the same direction as it was originally. The original shape is called the object and the rotated shape is called the image.

When rotated, a shape remains the same distance away from the centre of rotation. It is just in a different direction.

Rules for Rotating a Shape About the Origin

The rules for rotating shapes using coordinates are:

How to Rotate a Shape by 90 Degrees

To rotate shape 90° clockwise about the origin, all original coordinates (x, y) becomes (y, -x). To rotate a shape 90° counter-clockwise about the origin, the coordinates (x, y) become (-y, x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.

For example, use the rule (x, y) to (y, -x) to rotate the shape 90° clockwise.

To use this rule, simply switch the (3, 1) to (1, 3) and then make the 3 negative to get (1, -3).

How to Rotate a Shape by 180 Degrees

To rotate a shape by 180° clockwise or counter-clockwise, the rule is to replace the (x, y) coordinates with (-x, -y). For example, a coordinate at (3, 1) will move to (-3, -1) after a 180° rotation.

Simply multiply each coordinate by -1 to rotate a shape 180°.

If a coordinate is negative, it will become positive after a 180° rotation. For example, the coordinate (-1, -4), will move to (1, 4) after a 180° rotation.

How to Rotate a Shape by 270 Degrees

To rotate shape 270° clockwise about the origin, all original coordinates (x, y) becomes (-y, x). To rotate a shape 270° counter-clockwise about the origin, the coordinates (x, y) become (y, -x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.

For example, to rotate the point (3, 1) 270° clockwise, it becomes (-1, 3). Simply switch the x and y coordinates to get (1, 3) and then multiply the 1 by -1 to get (-1, 3).

How to Rotate a Shape Using Tracing Paper

To rotate a shape using tracing paper:

1. Place the tracing paper over the shape and draw around the shape.
2. Draw an arrow from the centre of rotation pointing upwards.
3. Keep the pen over the centre of rotation and rotate the tracing paper.
4. Stop when the arrow is facing either right (for 90° CW / 270° CCW turn), down (for 180° turn) or left (for 270° CW / 90° CCW turn).
5. Draw the shape in this new position below the tracing paper.

The easiest way to rotate a shape is to use tracing paper.

For example, use tracing paper to rotate the shape 90° clockwise about the point.

After a 90° clockwise rotation, the upwards arrow is facing right. Use the grid lines of the paper to help line up the arrow correctly, ensuring that it is completely horizontal.

Then draw in the shape below.

Here is another example. Rotate the shape 270° clockwise about the point using tracing paper.

Every lot of 90° is equivalent to a quarter turn. 270° is 3 lots of 90° and so, 270° is equivalent to three-quarters of a turn.

The upward-facing arrow will be facing to the left after a 270° clockwise rotation.

We draw the shape in below.

How to Rotate a Shape Without Using Tracing Paper

To rotate a shape without tracing paper, draw horizontal and vertical arrows from the centre of rotation to each corner of the shape. The new corner can be found by rotating each of these arrows according to the following rules:

For example, rotate the shape 90° clockwise without using tracing paper.

The first step is to draw horizontal and vertical arrows connecting the centre of rotation to a corner on the shape.

The corner selected below is one square right and one square up from the centre of rotation.

Using the rules above, a right-facing arrow will be facing down following a 90° clockwise rotation. An upwards-facing arrow will be facing right after a 90° clockwise rotation.

Instead of the corner being one right and one up, it will now be one down and one right.

Using the new position of this corner, the rest of the shape can be drawn in. The same process can be repeated for all corners to check the result.

Here is another example. Rotate the shape 270° clockwise without using tracing paper.

Draw horizontal and vertical arrows from the centre of rotation to each corner.

After a 270° clockwise rotation all upwards-facing arrows will be facing left and all left-facing arrows will be facing down.

Once all corners are drawn in their new positions, the rotated shape can be drawn by connecting these together.

What is Subitizing?

Subitizing is the ability to recognise the total number of objects in a group without counting them. For example, when rolling a die we know the number without counting the dots on each face.

Here are the dot patterns on a standard dice.

When we roll a dice, we do not usually count the dots on each face. Instead we recognise the pattern.

Being able to recognise the number without counting it is known as subitizing.

The word subitize is pronounced ‘soo-bit-ize’. It derives from the Latin word ‘subitus’, which means sudden. This is because when subitizing, we know the total ‘suddenly’. We do not need to take time to count the total.

Why Subitizing is Important

Subitizing is an important mathematical skill where a total can be recognised without counting. It helps us to develop number sense, counting on, skip counting, grouping and partitioning. It also helps to develop pattern recognition.

Once we can subitize smaller numbers, it is possible to look for these patterns when subitizing larger numbers.

For example, to subitize 6, we would notice the two patterns of 3 and then recognise that this makes 6.

Subitizing helps us to spot pairs or groups of three. This can help us to count up in twos, threes or fives more easily and it builds the skills needed for skip counting.

How to Teach Subitizing

The following ideas will help with teaching subitizing:

• Dice games
• Domino games
• Five and ten frames
• Flashcard games
• Using manipulatives

The process of teaching subitizing involves exposing children to images of collections of objects in order to increase their familiarity with the image of each number. To improve your ability to subitize, increase your exposure to collections of numbers in different situations and forms.

Subitizing with Dice

Subitizing is the ability to recognise the number on a dice without counting the dots. To practise subitizing with a dice, play lots of games involving dice and saying the number. It helps to view the whole face as a pattern or image that can be remembered.

Dice games, such as snakes and ladders or Yahtzee familiarise children with subitizing in a fun way.

Subitizing with Dominoes

Subitizing is the ability to recognise the number on a domino without counting the dots. To practise subitizing with dominoes, play lots of games involving dominoes, particularly ones that involve matching two of the same number. This helps to reinforce the image of each number.

Dominoes also display numbers in a similar arrangement to dice. Matching common numbers on two different dominoes is another way to practise subitizing.

Subitizing with Ten Frames

Ten frames are grids arranged in 2 rows of 5 squares. The familiarity of the frame helps us to better recognise the numbers being represented and hence, improve our subitizing skill. When first learning to subitize, it is useful to start with five frames which are just one row of the tens frame.

Ten frames are a simple way to represent numbers. Children may find it useful to use ten frames when subitizing because the longer the number, the larger it is. This can help speed up the subitizing process. Seeing a full row of 5 may be an easier way to visualise 5 compared to a randomly allocated set of objects.

Tens frames can allow us to more easily subitize the numbers 6 to 10. Since we can subitize 1 and 2 easily, if we see 1 or 2 next to a complete row of 5, we know that we have 6 or 7.

We can subitize 10 easily on a tens frame because it is a completely full two rows.

We can also subitize 8 and 9 easily on a tens frame because we can subitize the missing squares in the frame. If we are missing one square, we must have 9 and if we are missing 2 squares, we must have 8.

Subitizing Flashcards

Subitizing flashcards contain an image of 1-5 objects on the front of the card and the corresponding number printed on the back. The purpose is to practise subitizing the image of the objects and then turning the cards over to see if it was correct. Subitizing cards can be used in games such as snap or memory games.

Flashcards can be the easiest way to practise subitizing. Simply print the flashcards above or use the interactive online subitizing game above.

Subitizing with Manipulatives

Manipulatives are physical objects that are used to represent amounts. Manipulatives are particularly useful for practising subitizing because children can move them around and see totals represented in a variety of different forms rather than in the same fixed pattern as seen on dice or ten frames.

Games involving manipulatives are another great way to practise subitizing. Using lego blocks, counters or balls of plasticine are great for children to use as they can move these items around and see the collection of objects in many different forms.

Subitizing Tally Marks

Subitizing is the ability to recognise a number without counting. Tally marks are an example where subitizing may be used. For example, when we see a collection of 4 lines with a diagonal across it, we immediately know we have a tally of 5 and we do not need to count each individual line.

As soon as we see a tally with a diagonal line striking through it, we know it represents 5. This allows us to see the collection as an image and hence, subitize. We would not count the individual lines to see that there are 5. Using tally marks encourages us to subitize.

Subitizing Large Numbers

Many people can subitize numbers 1-5. To subitize larger numbers, visualise them as groups of the numbers 1-5. For example, 6 can be seen as two groups of 3. Then 7 can be subitized by considering it as a group of 6 with 1 more. To subitize larger numbers, recognise the combinations of smaller numbers.

Subitizing One

Here is a subitizing flashcard for one.

Subitizing Two

Here is a subitizing flashcard for two.

Subitizing Three

Here is a subitizing flashcard for three.

Subitizing Four

Here is a subitizing flashcard for four.

It may be useful to think of four as two groups of two. If you can subitize two, then look for two groups of two when subitizing four.

Subitizing Five

Here is a subitizing flashcard for five.

The largest number that most people can subitize is 5. We can think of 5 as a group of 3 next to a group of 2, so if we can subitize to see a group of 3 and a group of 2, then we can remember that this makes 5.

Alternatively, it can be helpful to start by displaying five in same arrangement shown on a dice face. If we recognise a square of four dots with one in the middle, this can be a familiar image to use when subitizing 5.

Subitizing Six

Here is a subitizing flashcard for six.

6 may be subitized by recognising two groups of 3. It is particularly helpful to recognise two rows or columns of three if the objects are lined up alongside another in pairs.

Subitizing Seven

Here is a subitizing flashcard for seven.

To subitize 7, we can look for a group of 4 alongside a group of 3. If we can subitize 4 and 3, we can recognise that both together make 7.

Alternatively, we can subitize 7 as a group of 6 with one extra.

Subitizing Eight

Here is a subitizing flashcard for eight.

Since 4 is easily subitized, we can subitize 8 as two groups of 4. The groups of 4 as a square might be easier to use when subitizing 8.

Alternatively, we can subitize 8 as being 2 less than a complete tens frame. This is because 8 is 2 less than 10 and 2 is easier to subitize. If we can see 2 gaps on a tens frame, we know we have 8.

Subitizing Nine

Here is a subitizing flashcard for nine.

We can subitize nine as a group of 4 and a group of 5 together.

Alternatively, we can more easily subitize 9 as one less than 10 if using a tens frame. If we see that one is missing from a complete tens frame, then we know we have 9.

Subitizing Ten

Here is a subitizing flashcard for ten.

Ten can be subitized by visualising it as two groups of 5. It can be easier to visualise each group of 5 as a square of four with a single dot in between.

How to Represent Decimals on a Number Line

Decimals on a number line are written depending on the number of parts between each whole number.

First count the number of parts in between each whole number.

Then use the table above to decide what each increment increases by.

Count up by this amount as you go from one line to the next.

For example, this number line contains 5 parts between each whole number. This means that the increments increase by 0.2 each time.

The digit after the decimal point will increase by 2 each time.

The marked decimals on the number line are 25.4 and 26.8.

Negative Decimals on a Number Line

To represent negative decimals on a number line, first mark the negative numbers on the line to the left of zero. Divide each whole number up into the required number of parts. The negative decimals are a mirror image of the positive decimal numbers with the numbers decreasing as we move left from zero.

Here is an example of plotting the negative decimals on a number line between -0.1 and -0.2.

Between -0.1 and -0.2 is one tenth. We have divided this tenth into ten further parts and therefore the decimals go up and down in hundredths.

We have -0.11, -0.12 and so on as we move left.

Here is another example of plotting negative decimals.

We have the negative decimals shown between -2.5 to -2.7. Each tenth is divided into five parts and so we go up and down by 0.02.

We go from -2.5 to -2.52 to -2.54 etc. until we reach -2.6. We then have -2.62, -2.64 until we reach -2.7.

Fifths on a Number Line

Whenever there are 5 equal parts between each whole number on a number line, we count up in fifths. Each fifth is worth 0.2. To count up in fifths on a number line, increase the digit after the decimal point by 2 each time.

One fifth is double the size of one tenth.

One fifth is 2 tenths so it is worth 0.2.

We can see that 5 fifths make a whole. Adding 5 lots of 0.2 makes 1.

For example, find the missing decimals marked on the number line below.

Because each whole number is divided into 5 parts, we know we have fifths, increasing by 0.2 each time.

Counting on in 0.2s from 1, we have the first marked decimal as 1.6 and the second marked decimal as 2.4.

Tenths on a Number Line

Tenths are written on a number line by counting up by 0.1. Tenths are shown whenever there are ten equal parts between each whole number. To count up in tenths on a number line, increase the digit after the decimal point by 1 each time.

For example, if there are 10 parts separating 14 and 15 on a number line then we count up in tenths. The lines represent 14.1, 14.2, 14.3 etc.

Once we reach 14.9, the next tenth up is just 15.

Hundredths on a Number Line

Hundredths are shown on a number line whenever there are 100 parts between each whole number or if there are 10 equal parts between each tenth We count up by 0.01 when counting up in hundredths.

For example, here there are ten parts separating 0.1 and 0.2, so we count up in hundredths.

We can think of 0.1 as 0.10 and 0.2 as 0.20. This allows us to see that between 0.10 and 0.20 we have 0.11, 0.12, 0.13 and so on.

Since the tenths were divided into ten more parts, we now formed hundredths. Each decimal increases by 0.01.

Here is another example of representing hundredths on a number line.

We have the tenths of 1.1, 1.2 and 1.3 marked on the number line. Each tenth is divided into ten further parts and so, we have hundredths.

Each decimal increases by 0.01 each time.

Teaching Decimals on a Number Line

When teaching decimals on the number line, first identify any key numbers on the number line, such as whole numbers or halves. Count the number of parts in between each key number so we can identify how much to increase by each time.

Use the following chart to decide how much we increase by each time.

What are Lines of Symmetry?

A line of symmetry is a line that divides a shape exactly in half. They are drawn with a dashed line. Lines of symmetry can be found by folding a shape in half or using a mirror. Each side of the line of symmetry will look exactly the same. For example, a square has 4 lines of symmetry.

Lines of symmetry look like dashed lines. The shape on each side of the line looks exactly the same.

How to Find Lines of Symmetry

To find a line of symmetry use any of these tests:

1. If both sides of a shape look exactly the same when it is folded, this fold line is a line of symmetry.
2. If the reflection of the shape in a mirror looks the same as the shape in front of the mirror, the mirror is on a line of symmetry.
3. If a corner of the shape is the same distance from a line as the corresponding corner is on the other side of the line, then this line may be a line of symmetry.

Here is an example of using folding to find a line of symmetry.

The rectangle can be folded in half so that each half looks identical. This is a line of symmetry.

It can be easier to cut the shape out first so that the outsides of the shape can be matched up. This is easier for checking if the outline is identical.

Tracing paper can be used to find lines of symmetry. First, draw around the shape on the tracing paper, then fold it to find a line that divides it in half.

Here is an example of using a mirror to find a line of symmetry.

Looking into the mirror, the reflection looks identical to the part of the shape directly in front of the mirror. Therefore this is a line of symmetry.

The mirror can be moved around on the shape until the reflection matches up with the other half of the shape.

Here is an example of measuring the distance of the corners on each side of the line of symmetry. This a line of symmetry because the corners are equally far from each side of the line.

Shapes with Lines of Symmetry

Here is a list of shapes and the number of lines of symmetry:

We can see a large range in the number of lines of symmetry that each shape has.

Shapes with No Lines of Symmetry

Parallelograms are an example of a shape with no lines of symmetry. Any shape that contains sides that are all different lengths also has no lines of symmetry. For example, right-angled trapezoids or scalene triangles have no lines of symmetry.

Lines of Symmetry of Triangles

Equilateral triangles have 3 lines of symmetry, which each pass through each corner to the middle of the opposite side. Isosceles triangles have 1 line of symmetry, which is directly between the two equal sides and equal angles. Scalene triangles have no equal sides and so, they have no lines of symmetry.

Lines of Symmetry of a Square

A square has 4 lines of symmetry. There are 2 lines of symmetry passing from each corner to the opposite corner. There are a further 2 lines of symmetry passing through the middle of each side to the middle of the opposite side.

Lines of Symmetry of a Rectangle

A rectangle has 2 lines of symmetry. These lines pass from the middle of each side to the middle of the opposite side. There are no lines of symmetry passing through the diagonals of the rectangle.

The diagonals of a rectangle are not lines of symmetry.

We can see that the diagonals of a rectangle are not lines of symmetry.

Lines of Symmetry of a Rhombus

A rhombus has 2 lines of symmetry. These lines of symmetry pass through the diagonals of the rhombus, from each corner to the opposite corner.

Here are the 2 lines of symmetry of a rhombus.

Lines of Symmetry of a Parallelogram

A parallelogram has 0 lines of symmetry. This is because the diagonals of a parallelogram are not symmetrical. If we fold a parallelogram along its diagonals, it will not fold exactly in half without overlap.

Lines of Symmetry of a Kite

Every kite has one line of symmetry. This line of symmetry passes vertically down the centre of the kite.

Lines of Symmetry on a Trapezium

Trapeziums have no lines of symmetry unless they are isosceles trapeziums which have 1 line of symmetry. This means that a trapezium only has a line of symmetry if both of its diagonal sides are the same length. In this case, the line of symmetry passes directly between the 2 diagonal sides.

Lines of Symmetry of a Pentagon

A regular pentagon has 5 lines of symmetry. Each line of symmetry passes from each of the 5 corners, through the centre of the pentagon to the middle of the opposite side.

Here are the 5 lines of symmetry shown on a regular pentagon.

Lines of Symmetry of a Hexagon

A regular hexagon has 6 lines of symmetry. 3 lines of symmetry pass from each of the corners to the opposite corner. The other 3 lines of symmetry pass from the middle of each side to the middle of the opposite side.

Here are the lines of symmetry of a regular hexagon.

Lines of Symmetry on an Octagon

A regular octagon has 8 lines of symmetry. 4 lines of symmetry pass from each of the corners to the opposite corner. The other 4 lines of symmetry pass from the middle of each side to the middle of the opposite side.

Here are the lines of symmetry on a regular octagon.

Lines of Symmetry of a Circle

A circle has infinite lines of symmetry. This is because the circle looks the same in any direction. Each line of symmetry is the diameter of the circle, passing from one side to the other through the centre. There are infinite diameter lines that can be drawn and so, there are infinite lines of symmetry on a circle.

Here are just some of the lines of symmetry of a circle.

Letters with Lines of Symmetry

• The letters of F, G, J, L, N, P, Q, R, S and Z do not have any lines of symmetry.
• The letters of A, B, C, D, E, K, M, T, U, V, W and Y have 1 line of symmetry.
• The letters of H, I, O and X have 2 lines of symmetry.

Here are the capital letters of the alphabet on a worksheet. Draw the lines of symmetry on these capital letters.

Here are the answers to this worksheet with the lines of symmetry of the letters shown.