Position and direction
By the end of this chapter you will be able to
- Carry on a pattern and say what comes next
- List the different ways to arrange or pair objects
- Say where an object is on a grid, using rows and columns
- Describe a move in a straight line and say where it ends
- Name a quarter, half or three-quarter turn, and count the right angles
- Say whether a turn goes clockwise or anticlockwise
You should already know
- Shapes and their corners (Chapter 8)
- Position words: above, below, left and right (Year 1, Chapter 11)
- Counting in twos and tens (Chapter 1)
- The clock face and the time to the hour (Year 1, Chapter 13)
Where this leads
- Right angles as a property of shapes, and lines (Year 3, Chapter 8)
- Position on a grid with coordinates (later years)
National Curriculum for England, Key Stage 1 mathematics, Year 2. Content: Order and arrange combinations of objects in patterns and sequences, Use mathematical vocabulary to describe position, direction and movement, including movement in a straight line, Distinguish rotation as a turn: quarter, half and three-quarter turns, clockwise and anticlockwise (right angles).
Exercise P Do you remember?
Non-calculatorMethod. Compare the rows. A smaller row number is higher up, nearer the top.
- Find the row of the flower. The flower is in row 2.flower: row 2
- Find the row of the bee. The bee is in row 1.bee: row 1
- Say where it is. Row 2 is compared with row 1, so the flower is below the bee.below
Check. Say it the other way: the bee is above the flower. ✓
Watch out. Row 1 is the top row. A smaller row number is higher up.
Method. Each number is 2 more than the one before.
- Look at the numbers. Each number goes up by 2.4, 6, 8
- Add 2 to the last number. The next number is 2 more than 8.8 + 2 = 10
- Write the answer. 10
Check. Count back in twos from 10: 10, 8, 6, 4. ✓
Watch out. Do not count the 8 again. Start from 8 and add 2.
Method. Count the cars once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6
- Read the last number. The last number is how many there are.6
- Write the answer. 6
Check. Count them again from the other end. You get 6 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Look at the shape. Its edge is round all the way round.
- Look at the edge. The edge has no straight sides.curved all the way round
- Match it to a name. A round flat shape with no corners is a circle.circle
Check. A circle has no straight sides and no corners. ✓
Watch out. A circle has no corners. Its edge is one curve.
Method. Count the corners of each square, then add them together.
- Count the corners of the first square. A square has 4 corners.4 corners
- Count the corners of the second square. The second square also has 4 corners.4 corners
- Add them together. Four and four make eight.4 + 4 = 8
- Write the answer. 8
Check. Count all 8 corners, one by one, to check. ✓
Watch out. Count each corner once. Do not count the same corner twice.
Patterns and sequences
A pattern repeats. The same objects come again and again, in the same order.
Find the part that repeats. Then you can say what comes next.
A sequence is a list of things in order. In a growing pattern, each row adds the same amount.
Key ideas
- A pattern repeats the same order again and again.
- Find the part that repeats, then carry the pattern on.
- In a growing pattern, each row has the same amount more than the row before.
- Count from the first object to find the place you need.
Method. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: star, ball.star ball ...
- Count the objects you can see. There are 5 objects, so the next one is number 6.next: number 6
- Carry the repeat on. Number 6 is the 2nd object in the repeat.ball
Check. The last 2 objects match the repeat, so the next object is a ball. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Count along the pattern, one object at a time.
- Number the objects in order. Each object gets one number as you count.1 star, 2 ball, 3 cube, 4 star, 5 ball, 6 cube, 7 star
- Find number 7. Number 7 is the 1st object in the repeat.7: star
- Write the answer. star
Check. The repeat is star, ball, cube, so number 7 is a star. ✓
Watch out. Count from the first object. The repeat starts again after each full set.
Method. Find how many more each row has than the row before.
- Compare two rows next to each other. Each row has 1 more than the row before.2 - 1 = 1
- Add the same step to the last row. The next row keeps the same step.3 + 1 = 4
- Write the answer. 4
Check. The rows so far have 1, 2, 3 objects. The next row has 4. ✓
Watch out. Look at the step from one row to the next, not at how many rows there are.
Watch out
Watch out: check the whole part that repeats. Do not stop after the first two objects.
Exam tip
Grown-up tip. Make a row of buttons, pasta shapes or coins. Make a pattern, then ask: “What comes next?” Ask your child to point to the part that repeats.
Exercise 9A Patterns and sequences
Non-calculatorMethod. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: star, ball.star ball ...
- Count the objects you can see. There are 4 objects, so the next one is number 5.next: number 5
- Carry the repeat on. Number 5 is the 1st object in the repeat.star
Check. The last 2 objects match the repeat, so the next object is a star. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: cube, car.cube car ...
- Count the objects you can see. There are 5 objects, so the next one is number 6.next: number 6
- Carry the repeat on. Number 6 is the 2nd object in the repeat.car
Check. The last 2 objects match the repeat, so the next object is a car. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: star, ball, cube.star ball cube ...
- Count the objects you can see. There are 7 objects, so the next one is number 8.next: number 8
- Carry the repeat on. Number 8 is the 2nd object in the repeat.ball
Check. The last 3 objects match the repeat, so the next object is a ball. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Look at the objects on each side of the gap. Then carry the repeat into it.
- Look before the gap. The object before the gap is a star.before: star
- Look after the gap. The object after the gap is a cube.after: cube
- Carry the repeat into the gap. The repeat is star, ball, cube, so the gap holds a ball.ball
Check. Put a ball in the gap and the pattern carries on the same way. ✓
Watch out. Use the repeat to choose the missing object, not only the one next to the gap.
Method. Count along the pattern, one object at a time.
- Number the objects in order. Each object gets one number as you count.1 car, 2 bus, 3 car, 4 bus, 5 car, 6 bus
- Find number 6. Number 6 is the 2nd object in the repeat.6: bus
- Write the answer. bus
Check. The repeat is car, bus, so number 6 is a bus. ✓
Watch out. Count from the first object. The repeat starts again after each full set.
Method. Count along the pattern, one object at a time.
- Number the objects in order. Each object gets one number as you count.1 star, 2 ball, 3 cube, 4 star, 5 ball, 6 cube, 7 star, 8 ball, 9 cube, 10 star
- Find number 10. Number 10 is the 1st object in the repeat.10: star
- Write the answer. star
Check. The repeat is star, ball, cube, so number 10 is a star. ✓
Watch out. Count from the first object. The repeat starts again after each full set.
Method. Find how many more each row has than the row before.
- Compare two rows next to each other. Each row has 1 more than the row before.2 - 1 = 1
- Add the same step to the last row. The next row keeps the same step.4 + 1 = 5
- Write the answer. 5
Check. The rows so far have 1, 2, 3, 4 objects. The next row has 5. ✓
Watch out. Look at the step from one row to the next, not at how many rows there are.
Method. Find how many more each row has than the row before.
- Compare two rows next to each other. Each row has 2 more than the row before.4 - 2 = 2
- Add the same step to the last row. The next row keeps the same step.6 + 2 = 8
- Write the answer. 8
Check. The rows so far have 2, 4, 6 objects. The next row has 8. ✓
Watch out. Look at the step from one row to the next, not at how many rows there are.
Method. Check each object against the repeat, one place at a time.
- Write the repeat. The pattern repeats: apple, pear.1 apple, 2 pear, 3 apple, 4 pear, 5 apple, 6 cube, 7 apple
- Check each place. Place 6 should be a pear, but it is a cube.place 6: cube, should be pear
- Write the place number. That place breaks the pattern.6
Check. All the other places match the repeat. ✓
Watch out. The object that breaks the pattern does not match the repeat. Check every place.
Method. Count the stars in the first 10 objects, one at a time.
- Write the first objects in order. Count from the first object.1 star, 2 ball, 3 ball, 4 star, 5 ball, 6 ball, 7 star, 8 ball, 9 ball, 10 star
- Find the stars. The stars are in places 1, 4, 7, 10.places 1, 4, 7, 10
- Count them. There are 4 stars.4
Check. The repeat is star, ball, ball, so each part has 1 star(s). ✓
Watch out. Count the places you listed. Do not count the other objects.
Method. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: star, ball.star ball ...
- Count the objects you can see. There are 5 objects, so the next one is number 6.next: number 6
- Carry the repeat on. Number 6 is the 2nd object in the repeat.ball
Check. The last 2 objects match the repeat, so the next object is a ball. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: cube, car, bus.cube car bus ...
- Count the objects you can see. There are 7 objects, so the next one is number 8.next: number 8
- Carry the repeat on. Number 8 is the 2nd object in the repeat.car
Check. The last 3 objects match the repeat, so the next object is a car. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: bus, bus, car.bus bus car ...
- Count the objects you can see. There are 7 objects, so the next one is number 8.next: number 8
- Carry the repeat on. Number 8 is the 2nd object in the repeat.bus
Check. The last 3 objects match the repeat, so the next object is a bus. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Find object number 10 in the repeat, then compare it with what is said.
- Find number 10. Number 10 in the pattern is a star.10: star
- Compare with what is said. The claim says it is a cube.star does not match cube
- Decide. Is the claim right?No
Check. Count the repeat again from the start to check number 10. ✓
Watch out. Count the repeat from the start, then find the place you need.
Arranging objects and making combinations
We can arrange objects in different orders. Each different order is a different arrangement.
We can also make combinations. A combination picks one thing from each group.
To find every one, make a list. Then count each one once.
Key ideas
- Each different order of the same objects is an arrangement.
- Make a list of every order, so that none is missed.
- A combination picks one thing from each group, such as one hat and one coat.
- Count each pair once. Do not count it twice.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.star, ball; ball, star
- Count the orders. There are 2 different orders.2 arrangements
- Write the answer. 2
Check. Each object can go at the front in 1 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 3 coats.hat 1 with coat 1; hat 1 with coat 2; hat 1 with coat 3
- Take each of the other hats. Pair each one with every coat too.hat 2 with coat 1; hat 2 with coat 2; hat 2 with coat 3
- Count every outfit once. There are 2 hats and 3 coats.6 outfits
- Write the answer. 6
Check. Count the outfits again, hat by hat: 6. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.ball, cube, apple; ball, apple, cube; cube, ball, apple; cube, apple, ball; apple, ball, cube; apple, cube, ball
- Count the orders. There are 6 different orders.6 arrangements
- Write the answer. 6
Check. Each object can go at the front in 2 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Watch out
Watch out: a ball then a cube is a different arrangement from a cube then a ball. The order matters.
Exam tip
Grown-up tip. Use three toys. Ask: “How many ways can we line them up?” Write each line on paper, then count the lines.
Exercise 9B Arranging objects and combinations
Non-calculatorMethod. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.cup, bag; bag, cup
- Count the orders. There are 2 different orders.2 arrangements
- Write the answer. 2
Check. Each object can go at the front in 1 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 2 coats.hat 1 with coat 1; hat 1 with coat 2
- Take each of the other hats. Pair each one with every coat too.there are no other hats
- Count every outfit once. There are 1 hats and 2 coats.2 outfits
- Write the answer. 2
Check. Count the outfits again, hat by hat: 2. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Pick two objects at a time. Write each pair once, then count the pairs.
- List the pairs. Each pair is written only once.ball and cube; ball and apple; cube and apple
- Count the pairs. Count each pair once.3 pairs
- Write the answer. 3
Check. ball and cube is the same pair as cube and ball, so it is counted once. ✓
Watch out. The order of the two does not matter when you only pick them.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 2 coats.hat 1 with coat 1; hat 1 with coat 2
- Take each of the other hats. Pair each one with every coat too.hat 2 with coat 1; hat 2 with coat 2; hat 3 with coat 1; hat 3 with coat 2; hat 4 with coat 1; hat 4 with coat 2
- Count every outfit once. There are 4 hats and 2 coats.8 outfits
- Write the answer. 8
Check. Count the outfits again, hat by hat: 8. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.car, bus, bike; car, bike, bus; bus, car, bike; bus, bike, car; bike, car, bus; bike, bus, car
- Count the orders. There are 6 different orders.6 arrangements
- Write the answer. 6
Check. Each object can go at the front in 2 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.ball, cube; cube, ball
- Count the orders. There are 2 different orders.2 arrangements
- Write the answer. 2
Check. Each object can go at the front in 1 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.ball, cube, apple; ball, apple, cube; cube, ball, apple; cube, apple, ball; apple, ball, cube; apple, cube, ball
- Count the orders. There are 6 different orders.6 arrangements
- Write the answer. 6
Check. Each object can go at the front in 2 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 3 coats.hat 1 with coat 1; hat 1 with coat 2; hat 1 with coat 3
- Take each of the other hats. Pair each one with every coat too.there are no other hats
- Count every outfit once. There are 1 hats and 3 coats.3 outfits
- Write the answer. 3
Check. Count the outfits again, hat by hat: 3. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Count the balls in the first 9 objects, one at a time.
- Write the first objects in order. Count from the first object.1 star, 2 ball, 3 ball, 4 star, 5 ball, 6 ball, 7 star, 8 ball, 9 ball
- Find the balls. The balls are in places 2, 3, 5, 6, 8, 9.places 2, 3, 5, 6, 8, 9
- Count them. There are 6 balls.6
Check. The repeat is star, ball, ball, so each part has 2 ball(s). ✓
Watch out. Count the places you listed. Do not count the other objects.
Method. Count the ways to arrange a star and a ball, then compare the number with the claim.
- Count the ways to arrange a star and a ball. There are 2 of them.2
- Compare with the claim. The claim says 4.2 is not the same as 4
- Decide. Is the claim right?No
Check. Count them again to check: 2. ✓
Watch out. Count every one, then compare. Do not guess from the picture.
Method. Count the ways to arrange a bag and a hat, then compare the number with the claim.
- Count the ways to arrange a bag and a hat. There are 2 of them.2
- Compare with the claim. The claim says 1.2 is not the same as 1
- Decide. Is the claim right?No
Check. Count them again to check: 2. ✓
Watch out. Count every one, then compare. Do not guess from the picture.
Position on a grid
A grid is made of squares, in rows and columns.
A row goes across, from left to right. Row 1 is the top row.
A column goes up and down. Column 1 is the left column.
We say where things are by counting the rows down and the squares across.
Key ideas
- Row 1 is the top row. Count the rows down from the top.
- Column 1 is the left column. Count the squares across from the left.
- 2 squares to the right means you move 2 squares across.
- A smaller row number is higher up on the grid.
Method. Find the row first, then find the column. Read the object in the square.
- Count down to the row. Row 1 is 1 rows down from the top.row 1
- Count across to the column. Column 3 is 3 squares across from the left.column 3
- Read the object. It is in the square where the row and the column meet.ball
Check. Count 1 rows down and 3 squares across to check. ✓
Watch out. Count the rows from the top. Count the columns from the left.
Method. Start on the star and move one square at a time, as the words say.
- Find the square you start on. It is in row 1, column 1.start: row 1, column 1
- Move 2 squares down. Move one square at a time, in a straight line.now at row 3, column 1
- Read the object on the square you land on. car
Check. Walk back the same way from row 3, column 1 and you reach row 1, column 1. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Compare the columns. A smaller column number is further to the left.
- Find the column of the car. The car is in column 1.car: column 1
- Find the column of the bee. The bee is in column 3.bee: column 3
- Say which side. Column 1 and column 3 compared: the car is on the left of the bee.left
Check. Say it the other way: the bee is on the right of the car. ✓
Watch out. Left means towards column 1. Right means towards the last column.
Watch out
Watch out: count the rows from the top, not from the bottom. Row 1 is always the top row.
Exam tip
Grown-up tip. Draw a grid of 4 by 5 squares on paper. Put a sticker or a coin in some squares. Ask: “Which row is it in? Which column? What is one square to the right?”
Exercise 9C Position on a grid
Non-calculatorMethod. Find the row first, then find the column. Read the object in the square.
- Count down to the row. Row 1 is 1 rows down from the top.row 1
- Count across to the column. Column 2 is 2 squares across from the left.column 2
- Read the object. It is in the square where the row and the column meet.apple
Check. Count 1 rows down and 2 squares across to check. ✓
Watch out. Count the rows from the top. Count the columns from the left.
Method. Find the row first, then find the column. Read the object in the square.
- Count down to the row. Row 4 is 4 rows down from the top.row 4
- Count across to the column. Column 5 is 5 squares across from the left.column 5
- Read the object. It is in the square where the row and the column meet.tree
Check. Count 4 rows down and 5 squares across to check. ✓
Watch out. Count the rows from the top. Count the columns from the left.
Method. Find the row first, then find the column. Read the object in the square.
- Count down to the row. Row 3 is 3 rows down from the top.row 3
- Count across to the column. Column 1 is 1 squares across from the left.column 1
- Read the object. It is in the square where the row and the column meet.car
Check. Count 3 rows down and 1 squares across to check. ✓
Watch out. Count the rows from the top. Count the columns from the left.
Method. Count the rows down from the top, one row at a time.
- Find the boat. It is in column 4.boat: column 4
- Count down to the row. Row 1 is the top row.row 3
- Write the answer. 3
Check. Count 3 rows down from the top to reach the boat. ✓
Watch out. Count the rows from the top, not from the bottom.
Method. Start on the apple and move one square at a time, as the words say.
- Find the square you start on. It is in row 1, column 2.start: row 1, column 2
- Move 1 square down. Move one square at a time, in a straight line.now at row 2, column 2
- Read the object on the square you land on. cube
Check. Walk back the same way from row 2, column 2 and you reach row 1, column 2. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the boat and move one square at a time, as the words say.
- Find the square you start on. It is in row 3, column 4.start: row 3, column 4
- Move 1 square up. Move one square at a time, in a straight line.now at row 2, column 4
- Read the object on the square you land on. hat
Check. Walk back the same way from row 2, column 4 and you reach row 3, column 4. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the cube and move one square at a time, as the words say.
- Find the square you start on. It is in row 2, column 2.start: row 2, column 2
- Move 1 square right. Move one square at a time, in a straight line.now at row 2, column 3
- Read the object on the square you land on. duck
Check. Walk back the same way from row 2, column 3 and you reach row 2, column 2. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the bus and move one square at a time, as the words say.
- Find the square you start on. It is in row 3, column 5.start: row 3, column 5
- Move 2 squares left. Move one square at a time, in a straight line.now at row 3, column 3
- Read the object on the square you land on. bee
Check. Walk back the same way from row 3, column 3 and you reach row 3, column 5. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Compare the columns. A smaller column number is further to the left.
- Find the column of the car. The car is in column 1.car: column 1
- Find the column of the boat. The boat is in column 4.boat: column 4
- Say which side. Column 1 and column 4 compared: the car is on the left of the boat.left
Check. Say it the other way: the boat is on the right of the car. ✓
Watch out. Left means towards column 1. Right means towards the last column.
Method. Count the squares that lie between the star and the ball, in the same row.
- Find the star and the ball. Both are in row 1.star: column 1, ball: column 3
- Count the squares in between. Do not count the two objects themselves.apple
- Write the answer. 1
Check. Count the squares from the star to the ball as well: 2 moves, so 1 in between. ✓
Watch out. Do not count the objects at the ends. Count only the squares in between.
Method. Find the row first, then find the column. Read the object in the square.
- Count down to the row. Row 3 is 3 rows down from the top.row 3
- Count across to the column. Column 5 is 5 squares across from the left.column 5
- Read the object. It is in the square where the row and the column meet.bus
Check. Count 3 rows down and 5 squares across to check. ✓
Watch out. Count the rows from the top. Count the columns from the left.
Method. Compare the columns of the bee and the car.
- Find the bee and the car. The bee is in row 3, column 3. The car is in row 3, column 1.column 3 and column 1
- Say which side the bee is on. The bee is on the right of the car.the right side
- Decide. The claim says the bee is on the right of the car.Yes
Check. The bee is on the right of the car, so the claim is right. ✓
Watch out. Left means towards column 1. Check the columns, not the sizes of the objects.
Movement in a straight line
A straight line keeps going the same way. It does not turn.
Moving right or left keeps you in the same row. Moving up or down keeps you in the same column.
When you turn a corner, you start a new straight line.
Key ideas
- Move one square at a time. Count each square you move onto.
- Right and left move you across a row.
- Up and down move you along a column.
- A move in a straight line does not turn.
Method. Start on the hat and move one square at a time, as the words say.
- Find the square you start on. It is in row 4, column 1.start: row 4, column 1
- Move 3 squares right. Move one square at a time, in a straight line.now at row 4, column 4
- Read the object on the square you land on. car
Check. Walk back the same way from row 4, column 4 and you reach row 4, column 1. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the flag and move one square at a time, as the words say.
- Find the square you start on. It is in row 5, column 5.start: row 5, column 5
- Move 3 squares up. Move one square at a time, in a straight line.now at row 2, column 5
- Read the object on the square you land on. duck
Check. Walk back the same way from row 2, column 5 and you reach row 5, column 5. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Count the squares from the flower to the star, one move at a time.
- Find the flower and the star. The flower is in row 1, column 1. The star is in row 1, column 4.row 1, column 1; row 1, column 4
- Count the moves. They are in a straight line, across the same row.3 squares apart
- Write the answer. Count each move once. Do not count the starting square.3
Check. Walking back 3 squares from the star reaches the flower. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Watch out
Watch out: a move in a straight line does not turn. A second move in a new direction is a new line.
Exam tip
Grown-up tip. Lay a row of coins on the floor as a path. Walk a toy along it and ask: “Are we still going in a straight line?” Then turn the toy and ask again.
Exercise 9D Movement in a straight line
Non-calculatorMethod. Start on the star and move one square at a time, as the words say.
- Find the square you start on. It is in row 1, column 4.start: row 1, column 4
- Move 2 squares down. Move one square at a time, in a straight line.now at row 3, column 4
- Read the row number of the square you land on. 3
Check. Walk back the same way from row 3, column 4 and you reach row 1, column 4. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the cake and move one square at a time, as the words say.
- Find the square you start on. It is in row 5, column 2.start: row 5, column 2
- Move 3 squares up. Move one square at a time, in a straight line.now at row 2, column 2
- Read the object on the square you land on. bird
Check. Walk back the same way from row 2, column 2 and you reach row 5, column 2. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the hat and move one square at a time, as the words say.
- Find the square you start on. It is in row 4, column 1.start: row 4, column 1
- Move 3 squares right. Move one square at a time, in a straight line.now at row 4, column 4
- Read the object on the square you land on. car
Check. Walk back the same way from row 4, column 4 and you reach row 4, column 1. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the flower and move one square at a time, as the words say.
- Find the square you start on. It is in row 1, column 1.start: row 1, column 1
- Move 3 squares right. Move one square at a time, in a straight line.now at row 1, column 4
- Read the object on the square you land on. star
Check. Walk back the same way from row 1, column 4 and you reach row 1, column 1. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the car and move one square at a time, as the words say.
- Find the square you start on. It is in row 4, column 4.start: row 4, column 4
- Move 2 squares up. Move one square at a time, in a straight line.now at row 2, column 4
- Read the row number of the square you land on. 2
Check. Walk back the same way from row 2, column 4 and you reach row 4, column 4. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the duck and move one square at a time, as the words say.
- Find the square you start on. It is in row 2, column 5.start: row 2, column 5
- Move 3 squares left. Move one square at a time, in a straight line.now at row 2, column 2
- Read the object on the square you land on. bird
Check. Walk back the same way from row 2, column 2 and you reach row 2, column 5. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the hat and move one square at a time, as the words say.
- Find the square you start on. It is in row 4, column 1.start: row 4, column 1
- Move 3 squares up. Move one square at a time, in a straight line.now at row 1, column 1
- Read the object on the square you land on. flower
Check. Walk back the same way from row 1, column 1 and you reach row 4, column 1. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Count the squares from the flower to the star, one move at a time.
- Find the flower and the star. The flower is in row 1, column 1. The star is in row 1, column 4.row 1, column 1; row 1, column 4
- Count the moves. They are in a straight line, across the same row.3 squares apart
- Write the answer. Count each move once. Do not count the starting square.3
Check. Walking back 3 squares from the star reaches the flower. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the star and move one square at a time, as the words say.
- Find the square you start on. It is in row 1, column 4.start: row 1, column 4
- Move 1 square down. Move one square at a time, in a straight line.now at row 2, column 4
- Move 1 square right. Move one square at a time, in a straight line.now at row 2, column 5
- Read the object on the square you land on. duck
Check. Walk back the same way from row 2, column 5 and you reach row 1, column 4. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the flag and move one square at a time, as the words say.
- Find the square you start on. It is in row 5, column 5.start: row 5, column 5
- Move 3 squares up. Move one square at a time, in a straight line.now at row 2, column 5
- Move 3 squares left. Move one square at a time, in a straight line.now at row 2, column 2
- Read the object on the square you land on. bird
Check. Walk back the same way from row 2, column 2 and you reach row 5, column 5. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the apple and move one square at a time, as the words say.
- Find the square you start on. It is in row 3, column 3.start: row 3, column 3
- Move 1 square left. Move one square at a time, in a straight line.now at row 3, column 2
- Move 2 squares down. Move one square at a time, in a straight line.now at row 5, column 2
- Read the object on the square you land on. cake
Check. Walk back the same way from row 5, column 2 and you reach row 3, column 3. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Follow each move from the hat, one square at a time.
- Start on the hat. It is in row 4, column 1.start: row 4, column 1
- Follow the moves. 2 squares right then 2 squares left.now at row 4, column 1
- Decide. Is it back on the start square?Yes
Check. Compare row 4, column 1 with row 4, column 1. ✓
Watch out. Two moves in opposite directions cancel out only when they are the same number of squares.
Quarter, half and three-quarter turns
A turn is when something goes round. It can go part of the way round, or all the way.
A quarter turn is a quarter of the way round. It makes one right angle.
A right angle is a square corner, like the corner of a book.
A half turn is 2 quarter turns, so it makes 2 right angles. A three-quarter turn makes 3.
Four quarter turns make a whole turn. It ends back where it started.
Key ideas
- A quarter turn makes one right angle.
- A half turn is two quarter turns, so it makes two right angles.
- A three-quarter turn is three quarter turns.
- A whole turn is four quarter turns, back to the start.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.2 quarter turns
- Count the right angles. Each quarter turn makes one right angle, so there are 2.2 right angles
- Write the answer. 2
Check. Four right angles make a whole turn, so 2 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Follow the minute hand clockwise, one quarter at a time.
- Find where the minute hand starts. It points to 12.start: 12
- Follow it round the clock. Each quarter turn moves it on to the next number.12 → 3
- Name the turn. 1 quarter turn is a quarter turn.quarter turn
Check. Going 1 quarter turns from 12 reaches 3. ✓
Watch out. Count the quarters of the clock, not the numbers. From 12 to 3 is a quarter turn.
Method. Look at each corner of the square.
- Look at the corners. A square has 4 corners.4 corners
- Check each corner. Each corner of a square is a right angle.a right angle at each corner
- Count the right angles. There is one right angle at each of the 4 corners.4
Check. Four quarter turns make a whole turn, and a square has 4 right angles. ✓
Watch out. A right angle is a square corner, like the corner of a book.
Watch out
Watch out: a half turn is not one quarter turn. A half turn is two quarter turns, so it makes 2 right angles.
Exam tip
Grown-up tip. Use a paper plate with a strip of paper stuck to the middle as the arrow. Turn it a quarter turn, then a half turn. Ask: “How many right angles have we turned?”
Exercise 9E Quarter, half and three-quarter turns
Non-calculatorMethod. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.1 quarter turn
- Count the right angles. Each quarter turn makes one right angle, so there are 1.1 right angles
- Write the answer. 1
Check. Four right angles make a whole turn, so 1 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.3 quarter turns
- Count the right angles. Each quarter turn makes one right angle, so there are 3.3 right angles
- Write the answer. 3
Check. Four right angles make a whole turn, so 3 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.4 quarter turns
- Count the right angles. Each quarter turn makes one right angle, so there are 4.4 right angles
- Write the answer. 4
Check. Four right angles make a whole turn, so 4 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Count the quarter turns along the orange arrow.
- Count the quarter turns. Each quarter turn is a quarter of the way round.2 quarter turns
- Name the turn. 2 quarter turns is a half turn.half turn
Check. Four quarter turns make a whole turn, so 2 quarter turns is less than a whole turn. ✓
Watch out. Two quarter turns make a half turn. Three make a three-quarter turn.
Method. Count the quarter turns along the orange arrow.
- Count the quarter turns. Each quarter turn is a quarter of the way round.3 quarter turns
- Name the turn. 3 quarter turns is a three-quarter turn.three-quarter turn
Check. Four quarter turns make a whole turn, so 3 quarter turns is less than a whole turn. ✓
Watch out. Two quarter turns make a half turn. Three make a three-quarter turn.
Method. Follow the minute hand clockwise, one quarter at a time.
- Find where the minute hand starts. It points to 12.start: 12
- Follow it round the clock. Each quarter turn moves it on to the next number.12 → 3 → 6
- Name the turn. 2 quarter turns is a half turn.half turn
Check. Going 2 quarter turns from 12 reaches 6. ✓
Watch out. Count the quarters of the clock, not the numbers. From 12 to 3 is a quarter turn.
Method. Follow the minute hand clockwise, one quarter at a time.
- Find where the minute hand starts. It points to 3.start: 3
- Follow it round the clock. Each quarter turn moves it on to the next number.3 → 6 → 9
- Name the turn. 2 quarter turns is a half turn.half turn
Check. Going 2 quarter turns from 3 reaches 9. ✓
Watch out. Count the quarters of the clock, not the numbers. From 12 to 3 is a quarter turn.
Method. Find the right angles in each turn, then add them together.
- Turn A: a half turn. A half turn makes 2 right angles.2 right angles
- Turn B: a quarter turn. A quarter turn makes 1 right angle.1 right angle
- Add them together. Two and one make three.2 + 1 = 3
- Write the answer. 3
Check. Count the quarter turns instead: a half turn is 2 quarter turns, plus 1 more, makes 3 right angles. ✓
Watch out. A half turn is two quarter turns, so it makes two right angles, not one.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.1 quarter turn
- Count the right angles. Each quarter turn makes one right angle, so there are 1.1 right angles
- Write the answer. 1
Check. Four right angles make a whole turn, so 1 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.2 quarter turns
- Count the right angles. Each quarter turn makes one right angle, so there are 2.2 right angles
- Write the answer. 2
Check. Four right angles make a whole turn, so 2 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.3 quarter turns
- Count the right angles. Each quarter turn makes one right angle, so there are 3.3 right angles
- Write the answer. 3
Check. Four right angles make a whole turn, so 3 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Count the quarter turns in the picture, then name the turn.
- Count the quarter turns. The arc shows 3 quarter turns.3 quarter turns
- Name the turn. 3 quarter turns is a three-quarter turn.three-quarter turn
- Decide. The claim says whole turn.No
Check. A whole turn is not the same as a three-quarter turn. ✓
Watch out. A quarter turn moves one quarter of the way round. Count the quarters, then name the turn.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. The turn has 2 quarter turns.2 quarter turns
- Count the right angles. That makes 2 right angles.2 right angles
- Decide. The claim says 2.Yes
Check. Each quarter turn is one right angle, so the count is 2. ✓
Watch out. Count the right angles, one for each quarter turn.
Clockwise and anticlockwise turns
Clockwise is the way the hands of a clock move. Clockwise goes up, right, down, then left.
Anticlockwise is the other way round. It goes up, left, down, then right.
The orange arrow in each picture shows the turn. Follow it to see which way it goes.
Key ideas
- Clockwise goes up, right, down, left.
- Anticlockwise goes up, left, down, right.
- Clockwise and anticlockwise go opposite ways round.
- Check the orange arrow before you turn.
Method. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces up.start: up
- Turn it clockwise, a quarter turn. Each quarter turn moves it one step round.up → right
- Write the way it faces now. right
Check. Turning it back the other way by the same amount makes it face up again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Follow the orange arrow round from the car.
- Find where the arrow starts. It starts at the way the car faces, right.start: right
- Follow the arrow round. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.right → up
- Say which way it turns. anticlockwise
Check. Going the other way from right would give a different order. ✓
Watch out. Clockwise is the way the hands of a clock move. Check the way the orange arrow points.
Method. Turn the car anticlockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces right.start: right
- Turn it anticlockwise, a half turn. Each quarter turn moves it one step round.right → up → left
- Write the way it faces now. left
Check. Turning it back the other way by the same amount makes it face right again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Watch out
Watch out: check the way the orange arrow goes before you turn. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Exam tip
Grown-up tip. Hold up a real clock or a plate with a hand on it. Turn the hand clockwise, then anticlockwise. Ask: “Which way did it go? Which way does the car face now?”
Exercise 9F Clockwise and anticlockwise turns
Non-calculatorMethod. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces up.start: up
- Turn it clockwise, a quarter turn. Each quarter turn moves it one step round.up → right
- Write the way it faces now. right
Check. Turning it back the other way by the same amount makes it face up again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Turn the car anticlockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces up.start: up
- Turn it anticlockwise, a quarter turn. Each quarter turn moves it one step round.up → left
- Write the way it faces now. left
Check. Turning it back the other way by the same amount makes it face up again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces right.start: right
- Turn it clockwise, a half turn. Each quarter turn moves it one step round.right → down → left
- Write the way it faces now. left
Check. Turning it back the other way by the same amount makes it face right again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces left.start: left
- Turn it clockwise, a three-quarter turn. Each quarter turn moves it one step round.left → up → right → down
- Write the way it faces now. down
Check. Turning it back the other way by the same amount makes it face left again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Turn the car anticlockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces down.start: down
- Turn it anticlockwise, a quarter turn. Each quarter turn moves it one step round.down → right
- Write the way it faces now. right
Check. Turning it back the other way by the same amount makes it face down again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Follow the orange arrow round from the car.
- Find where the arrow starts. It starts at the way the car faces, up.start: up
- Follow the arrow round. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.up → right
- Say which way it turns. clockwise
Check. Going the other way from up would give a different order. ✓
Watch out. Clockwise is the way the hands of a clock move. Check the way the orange arrow points.
Method. Follow the orange arrow round from the car.
- Find where the arrow starts. It starts at the way the car faces, right.start: right
- Follow the arrow round. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.right → up
- Say which way it turns. anticlockwise
Check. Going the other way from right would give a different order. ✓
Watch out. Clockwise is the way the hands of a clock move. Check the way the orange arrow points.
Method. Count the quarter turns along the orange arrow.
- Count the quarter turns. Each quarter turn is a quarter of the way round.2 quarter turns
- Name the turn. 2 quarter turns is a half turn.half turn
Check. Four quarter turns make a whole turn, so 2 quarter turns is less than a whole turn. ✓
Watch out. Two quarter turns make a half turn. Three make a three-quarter turn.
Method. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces up.start: up
- Turn it clockwise, a quarter turn. Each quarter turn moves it one step round.up → right
- Write the way it faces now. right
Check. Turning it back the other way by the same amount makes it face up again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Turn the car anticlockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces up.start: up
- Turn it anticlockwise, a quarter turn. Each quarter turn moves it one step round.up → left
- Write the way it faces now. left
Check. Turning it back the other way by the same amount makes it face up again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces up.start: up
- Turn it clockwise, a half turn. Each quarter turn moves it one step round.up → right → down
- Write the way it faces now. down
Check. Turning it back the other way by the same amount makes it face up again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Follow each turn from the same start, one quarter at a time.
- Start both turns from the same way. Both start with the car facing up.start: up
- Turn clockwise. A clockwise turn of 1 quarter turn ends right.up → right
- Turn anticlockwise. An anticlockwise turn ends left.up → left
- Compare the two ends. They end different.No
Check. Clockwise and anticlockwise go opposite ways round, so they end on different sides. ✓
Watch out. Clockwise and anticlockwise go opposite ways round. Check both ends.
Method. Do each turn in order. Each turn starts from the way the car faces after the turn before.
- Find which way it faces at the start. The car faces up.start: up
- Turn it clockwise, 2 quarter turns. It was facing up.up → right → down
- Turn it anticlockwise, 1 quarter turn. It was facing down.down → right
- Write the way it faces now. right
Check. Check each turn one at a time, starting from up. ✓
Watch out. Do the turns in the order they are given. Each turn starts from where the car is now.
Method. Do each turn in order. Each turn starts from the way the car faces after the turn before.
- Find which way it faces at the start. The car faces left.start: left
- Turn it clockwise, 2 quarter turns. It was facing left.left → up → right
- Turn it anticlockwise, 2 quarter turns. It was facing right.right → up → left
- Write the way it faces now. left
Check. Check each turn one at a time, starting from left. ✓
Watch out. Do the turns in the order they are given. Each turn starts from where the car is now.
Chapter summary
| Skill | Method |
|---|---|
| Patterns | A pattern repeats. Find the part that repeats, then carry it on. In a growing pattern each step adds the same amount. |
| Arrangements | An arrangement is a different order. List every order, then count each one once. |
| Combinations | A combination picks one thing from each group. List every pair, then count each pair once. |
| Grid position | Row 1 is the top row. Column 1 is the left column. Count rows down and squares across. |
| Movement | Moving right or left stays in a row. Moving up or down stays in a column. A turn starts a new line. |
| Quarter, half, three-quarter | A quarter turn makes 1 right angle. A half turn makes 2. A three-quarter turn makes 3. A whole turn makes 4. |
| Clockwise and anticlockwise | Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right. |
Can you…?
- I can say what comes next in a pattern
- I can list the different ways to arrange or pair objects
- I can find an object on a grid, using rows and columns
- I can say where an object is after a move in a straight line
- I can name a quarter, half or three-quarter turn
- I can count the right angles in a turn
- I can say whether a turn is clockwise or anticlockwise
Key terms
- pattern
- objects that repeat in the same order again and again.
- sequence
- a list of things in a set order.
- arrangement
- a different order of the same objects.
- combination
- a choice of one thing from each group.
- row
- a line of squares going across a grid. Row 1 is the top row.
- column
- a line of squares going up and down a grid. Column 1 is the left column.
- straight line
- a path that keeps going the same way and does not turn.
- turn
- going round, part of the way or all the way.
- quarter turn
- a quarter of the way round. It makes one right angle.
- half turn
- half of the way round. It makes two right angles.
- three-quarter turn
- three quarters of the way round. It makes three right angles.
- whole turn
- all the way round, back to the start. It makes four right angles.
- right angle
- a square corner, like the corner of a book.
- clockwise
- the way the hands of a clock move: up, right, down, left.
- anticlockwise
- the other way round from the hands of a clock: up, left, down, right.
Exercise 9Z Check-up
Non-calculatorMethod. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: bird, cat.bird cat ...
- Count the objects you can see. There are 6 objects, so the next one is number 7.next: number 7
- Carry the repeat on. Number 7 is the 1st object in the repeat.bird
Check. The last 2 objects match the repeat, so the next object is a bird. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Find the row first, then find the column. Read the object in the square.
- Count down to the row. Row 3 is 3 rows down from the top.row 3
- Count across to the column. Column 4 is 4 squares across from the left.column 4
- Read the object. It is in the square where the row and the column meet.boat
Check. Count 3 rows down and 4 squares across to check. ✓
Watch out. Count the rows from the top. Count the columns from the left.
Method. Start on the cake and move one square at a time, as the words say.
- Find the square you start on. It is in row 5, column 2.start: row 5, column 2
- Move 2 squares up. Move one square at a time, in a straight line.now at row 3, column 2
- Read the row number of the square you land on. 3
Check. Walk back the same way from row 3, column 2 and you reach row 5, column 2. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces left.start: left
- Turn it clockwise, a quarter turn. Each quarter turn moves it one step round.left → up
- Write the way it faces now. up
Check. Turning it back the other way by the same amount makes it face left again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Follow the minute hand clockwise, one quarter at a time.
- Find where the minute hand starts. It points to 6.start: 6
- Follow it round the clock. Each quarter turn moves it on to the next number.6 → 9
- Name the turn. 1 quarter turn is a quarter turn.quarter turn
Check. Going 1 quarter turns from 6 reaches 9. ✓
Watch out. Count the quarters of the clock, not the numbers. From 12 to 3 is a quarter turn.
Method. Each quarter turn makes one right angle.
- Count the quarter turns. Each quarter turn is a quarter of the way round.4 quarter turns
- Count the right angles. Each quarter turn makes one right angle, so there are 4.4 right angles
- Write the answer. 4
Check. Four right angles make a whole turn, so 4 is less than four. ✓
Watch out. A quarter turn makes one right angle. A right angle is the corner of a square.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 4 coats.hat 1 with coat 1; hat 1 with coat 2; hat 1 with coat 3; hat 1 with coat 4
- Take each of the other hats. Pair each one with every coat too.hat 2 with coat 1; hat 2 with coat 2; hat 2 with coat 3; hat 2 with coat 4; hat 3 with coat 1; hat 3 with coat 2; hat 3 with coat 3; hat 3 with coat 4
- Count every outfit once. There are 3 hats and 4 coats.12 outfits
- Write the answer. 12
Check. Count the outfits again, hat by hat: 12. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Count the balls in the first 6 objects, one at a time.
- Write the first objects in order. Count from the first object.1 ball, 2 ball, 3 star, 4 ball, 5 ball, 6 star
- Find the balls. The balls are in places 1, 2, 4, 5.places 1, 2, 4, 5
- Count them. There are 4 balls.4
Check. The repeat is ball, ball, star, so each part has 2 ball(s). ✓
Watch out. Count the places you listed. Do not count the other objects.
Method. Follow the orange arrow round from the car.
- Find where the arrow starts. It starts at the way the car faces, down.start: down
- Follow the arrow round. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.down → left
- Say which way it turns. clockwise
Check. Going the other way from down would give a different order. ✓
Watch out. Clockwise is the way the hands of a clock move. Check the way the orange arrow points.
Method. Follow the turn one quarter at a time, then check the claim.
- Find the start. The car faces down.start: down
- Turn it clockwise, a half turn. Each quarter turn moves it on one step round.down → left → up
- Check the claim. The claim says it faces up.Yes
Check. The car faces up after the turn, so the claim is right. ✓
Watch out. Follow the turn one quarter at a time. Do not guess the end from the start alone.
Total 13 marks. Allow about 13 minutes.
Exercise 9X End-of-chapter quiz
Non-calculatorMethod. Find the part that repeats, then carry the pattern on.
- Find the part that repeats. The pattern repeats: duck, cat.duck cat ...
- Count the objects you can see. There are 4 objects, so the next one is number 5.next: number 5
- Carry the repeat on. Number 5 is the 1st object in the repeat.duck
Check. The last 2 objects match the repeat, so the next object is a duck. ✓
Watch out. Find the whole part that repeats before you say what comes next.
Method. Find the row first, then find the column. Read the object in the square.
- Count down to the row. Row 1 is 1 rows down from the top.row 1
- Count across to the column. Column 5 is 5 squares across from the left.column 5
- Read the object. It is in the square where the row and the column meet.cat
Check. Count 1 rows down and 5 squares across to check. ✓
Watch out. Count the rows from the top. Count the columns from the left.
Method. Start on the car and move one square at a time, as the words say.
- Find the square you start on. It is in row 3, column 1.start: row 3, column 1
- Move 2 squares right. Move one square at a time, in a straight line.now at row 3, column 3
- Read the object on the square you land on. bee
Check. Walk back the same way from row 3, column 3 and you reach row 3, column 1. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the flag and move one square at a time, as the words say.
- Find the square you start on. It is in row 5, column 5.start: row 5, column 5
- Move 1 square up. Move one square at a time, in a straight line.now at row 4, column 5
- Read the row number of the square you land on. 4
Check. Walk back the same way from row 4, column 5 and you reach row 5, column 5. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Start on the star and move one square at a time, as the words say.
- Find the square you start on. It is in row 1, column 4.start: row 1, column 4
- Move 3 squares down. Move one square at a time, in a straight line.now at row 4, column 4
- Read the object on the square you land on. car
Check. Walk back the same way from row 4, column 4 and you reach row 1, column 4. ✓
Watch out. Count the squares you move onto. Do not count the square you start on.
Method. Turn the car clockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces up.start: up
- Turn it clockwise, a half turn. Each quarter turn moves it one step round.up → right → down
- Write the way it faces now. down
Check. Turning it back the other way by the same amount makes it face up again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Count the quarter turns along the orange arrow.
- Count the quarter turns. Each quarter turn is a quarter of the way round.3 quarter turns
- Name the turn. 3 quarter turns is a three-quarter turn.three-quarter turn
Check. Four quarter turns make a whole turn, so 3 quarter turns is less than a whole turn. ✓
Watch out. Two quarter turns make a half turn. Three make a three-quarter turn.
Method. Turn the car anticlockwise, one quarter turn at a time.
- Find which way it faces at the start. The car faces right.start: right
- Turn it anticlockwise, a quarter turn. Each quarter turn moves it one step round.right → up
- Write the way it faces now. up
Check. Turning it back the other way by the same amount makes it face right again. ✓
Watch out. Clockwise goes up, right, down, left. Anticlockwise goes up, left, down, right.
Method. Count along the pattern, one object at a time.
- Number the objects in order. Each object gets one number as you count.1 star, 2 ball, 3 ball, 4 star, 5 ball, 6 ball, 7 star, 8 ball
- Find number 8. Number 8 is the 2nd object in the repeat.8: ball
- Write the answer. ball
Check. The repeat is star, ball, ball, so number 8 is a ball. ✓
Watch out. Count from the first object. The repeat starts again after each full set.
Method. Follow the turn one quarter at a time, then check the claim.
- Find the start. The car faces left.start: left
- Turn it clockwise, a half turn. Each quarter turn moves it on one step round.left → up → right
- Check the claim. The claim says it faces right.Yes
Check. The car faces right after the turn, so the claim is right. ✓
Watch out. Follow the turn one quarter at a time. Do not guess the end from the start alone.