Halves and quarters · applications

Applications: Halves and Quarters

10 question types · Primary 2 · each worked step by step with a figure that follows the steps

PSLE

01

A Chocolate Bar Broken into Parts That Are Not Equal

strategyCheck That the Parts Are Equal / Then Share into Equal Parts

A chocolate bar has 12 small squares in a row. Dad breaks it into 4 parts. The parts have 6, 2, 2 and 2 squares. Ben takes a part with 2 squares. (a) Is Ben's part 14 of the bar? (b) Dad breaks a new bar of 12 squares into 4 equal parts. How many squares are in each part?

Dad's bar6222
The bar has 12 squares. It is broken into 4 parts with 6, 2, 2 and 2 squares.
Draw the bar as 12 squares in a row. Mark the 4 parts. They have 6, 2, 2 and 2 squares.
step 1 of 5

A quarter is 1 of 4 equal parts. Count the parts, and then check that every part is the same size.

  1. Draw the bar as 12 squares in a row. Mark the 4 parts. They have 6, 2, 2 and 2 squares.
  2. Look at the sizes. One part has 6 squares and the other parts have 2 squares. The parts are not equal.
  3. (a) A quarter is 1 of 4 equal parts. These 4 parts are not equal, so Ben's part is not 14 of the bar.
  4. Now share the 12 squares of the new bar into 4 equal parts. Give 1 square to each part in turn until all the squares are used.
  5. (b) Each part gets 3 squares. Check: 3 + 3 + 3 + 3 = 12. Now each part is 14 of the bar.

answer(a) No, because the 4 parts are not equal; (b) 3 squares

techniqueEqual Parts

examsPSLE

Common pitfalls

  • Saying that Ben's part is 14 because the bar has 4 parts. Counting the parts is not enough. The parts must also be equal, and a part with 6 squares is not equal to a part with 2 squares.
  • Writing 4 squares for part (b) because there are 4 parts. Four parts of 4 squares make 4 + 4 + 4 + 4 = 16 squares, and the bar has only 12.
02

A Sandwich Cut in Half in Two Different Ways

strategyDraw on Square Paper / Count the Squares in Each Part

Mei and Tom each have a square sandwich of the same size. On square paper, each sandwich covers 4 small squares. Mei cuts her sandwich in half from side to side. Tom cuts his sandwich in half from corner to corner. Each child eats 1 part. (a) What fraction of her sandwich does Mei eat? (b) Who eats more, Mei or Tom?

MeiTom
Each sandwich is 4 small squares. Mei cuts from side to side. Tom cuts from corner to corner.
Draw each sandwich as 4 small squares. Draw Mei's cut from side to side. Draw Tom's cut from corner to corner.
step 1 of 5

Draw each sandwich on square paper as 4 small squares. Parts that cover the same number of squares are the same size, even when they have different shapes.

  1. Draw each sandwich as 4 small squares. Draw Mei's cut from side to side. Draw Tom's cut from corner to corner.
  2. (a) Mei's cut makes 2 equal parts. One of 2 equal parts is a half, so Mei eats 12 of her sandwich.
  3. Count the squares in Mei's part. It has 2 whole squares.
  4. Count the squares in Tom's part. It has 1 whole square and 2 half squares. Two half squares make 1 whole square, so his part has 1 + 1 = 2 squares.
  5. (b) Both parts have 2 squares, so they are the same size. Tom also eats 12 of his sandwich. Mei and Tom eat the same amount.

answer(a) 12; (b) They eat the same amount. Each part has 2 small squares

techniqueEqual Parts · Halves, Quarters, Thirds

examsPSLE

Common pitfalls

  • Saying that Tom eats more because his cut is longer. The length of the cut does not tell the size of the part. Count the squares: both parts have 2 squares.
  • Saying that Tom's part is not a half because it is a triangle. A half does not need a special shape. It is 1 of 2 equal parts of the whole.
03

A Pizza in Four Equal Slices with One Slice Eaten

strategyCount the Equal Slices / Slices Eaten and Slices Left

A pizza is cut into 4 equal slices. Sam eats 1 slice. (a) What fraction of the pizza does Sam eat? (b) What fraction of the pizza is left?

1/41/41/41/4
The pizza has 4 equal slices. Each slice is 14 of the pizza.
Draw the pizza as a circle cut into 4 equal slices. Each slice is 14 of the pizza.
step 1 of 4

Draw the pizza and its 4 equal slices. Count the slices that are eaten and the slices that are left. Each count goes on top of the 4.

  1. Draw the pizza as a circle cut into 4 equal slices. Each slice is 14 of the pizza.
  2. (a) Sam eats 1 of the 4 equal slices. He eats 14 of the pizza.
  3. Count the slices that are left: 4 − 1 = 3 slices.
  4. (b) 3 of the 4 equal slices are left, so 34 of the pizza is left. Check: 1 + 3 = 4 slices, which is the whole pizza.

answer(a) 14; (b) 34

techniqueTop and Bottom · Halves, Quarters, Thirds

examsPSLE

Common pitfalls

  • Writing 13 for part (a) because 1 slice is eaten and 3 slices are left. The bottom number counts all the equal slices in the whole pizza, and there are 4 of them.
  • Writing 3 for part (b). The 3 is the number of slices left. The question asks for a fraction, so say 3 out of 4 equal slices, which is 34.
04

Half of a Group of Marbles Given Away

strategyShare into 2 Equal Groups / One Group Is a Half

Ali has 14 marbles. He gives half of them to Ben. (a) How many marbles does Ali give to Ben? (b) Ben had 3 marbles before. How many marbles does Ben have now?

14 marbles to share
Half means 1 of 2 equal groups. There is one plate for each group.
Half means 1 of 2 equal groups. Draw 2 plates, one for each group.
step 1 of 4

Half of a group is 1 of 2 equal groups. Share the marbles into 2 equal groups and count one group.

  1. Half means 1 of 2 equal groups. Draw 2 plates, one for each group.
  2. Share the 14 marbles. Put 1 marble on each plate in turn until all the marbles are used.
  3. (a) Each plate has 7 marbles. Check: 7 + 7 = 14. Half of 14 is 7, so Ali gives 7 marbles to Ben.
  4. (b) Ben had 3 marbles and he gets 7 more: 3 + 7 = 10. Ben has 10 marbles now.

answer(a) 7 marbles; (b) 10 marbles

techniqueA Fraction of a Group

examsPSLE · GCSE Higher

Common pitfalls

  • Taking away 2 and writing 14 − 2 = 12. Half does not mean take away 2. It means share into 2 equal groups and take 1 group.
  • Making two groups that are not equal, such as 8 and 6. Then neither group is a half. Both groups must have the same number, and 7 + 7 = 14.
05

A Quarter and Three Quarters of a Group of Stickers

strategyShare into 4 Equal Groups / Count One Group, Then Three Groups

Lina has 20 stickers. She gives 14 of them to her brother and keeps 34 of them. (a) How many stickers does her brother get? (b) How many stickers does Lina keep?

The bottom number is 4, so the 20 stickers are shared onto 4 plates.
The bottom number is 4, so draw 4 plates. Share the 20 stickers onto them, 1 sticker to each plate in turn.
step 1 of 4

The bottom number 4 says to share the stickers into 4 equal groups. The top number says how many of the groups to count.

  1. The bottom number is 4, so draw 4 plates. Share the 20 stickers onto them, 1 sticker to each plate in turn.
  2. Each plate has 5 stickers. Check: 5 + 5 + 5 + 5 = 20.
  3. (a) 14 is 1 of the 4 groups. Her brother gets 5 stickers.
  4. (b) 34 is 3 of the 4 groups: 5 + 5 + 5 = 15. Lina keeps 15 stickers. Check: 5 + 15 = 20.

answer(a) 5 stickers; (b) 15 stickers

techniqueA Fraction of a Group · Top and Bottom

examsPSLE · GCSE Higher

Common pitfalls

  • Taking away 4 and writing 20 − 4 = 16. The 4 in 14 is not a number of stickers. It says to share the stickers into 4 equal groups.
  • Saying that 34 of the stickers is 3 stickers. The 3 counts groups, not stickers. Each group has 5 stickers, so 3 groups have 15.
06

A Third of a Tin of Biscuits on Three Plates

strategyShare into 3 Equal Groups / One Group Is a Third

There are 18 biscuits in a tin. Mum puts all of them onto 3 plates. Each plate has the same number of biscuits. (a) How many biscuits are 13 of the 18 biscuits? (b) The children eat all the biscuits on 1 plate. How many biscuits are left?

The 18 biscuits are shared onto 3 plates, 1 biscuit to each plate in turn.
Draw 3 plates. Share the 18 biscuits onto them, 1 biscuit to each plate in turn.
step 1 of 4

A third is 1 of 3 equal groups. The 3 plates are the 3 equal groups, so one plate holds a third of the biscuits.

  1. Draw 3 plates. Share the 18 biscuits onto them, 1 biscuit to each plate in turn.
  2. Each plate has 6 biscuits. Check: 6 + 6 + 6 = 18.
  3. (a) 13 is 1 of the 3 equal groups. 13 of 18 is 6 biscuits.
  4. (b) The children eat 1 plate of 6 biscuits. Two plates are left: 6 + 6 = 12 biscuits. Check: 18 − 6 = 12.

answer(a) 6 biscuits; (b) 12 biscuits

techniqueA Fraction of a Group · Halves, Quarters, Thirds

examsPSLE · GCSE Higher

Common pitfalls

  • Sharing the biscuits into 2 groups because a third sounds like a half. Look at the bottom number. It is 3, so there are 3 equal groups.
  • Writing 18 − 3 = 15 for part (b). The children eat 1 plate, and that is 6 biscuits, not 3. The 3 is the number of plates.
07

A Window with Three of Its Four Panes Painted

strategyBottom Number Counts All the Equal Parts / Top Number Counts the Parts Asked About

A window has 4 equal panes of glass. Dad paints 3 of the panes yellow. (a) What fraction of the window is yellow? (b) How many panes are not yellow?

The window has 4 equal panes. 3 panes are painted yellow.
Draw the window with 4 equal panes. Color 3 panes yellow.
step 1 of 5

A fraction has two numbers. The bottom number counts all the equal parts in the whole. The top number counts the parts that are asked about.

  1. Draw the window with 4 equal panes. Color 3 panes yellow.
  2. Count all the equal panes. There are 4. This is the bottom number of the fraction.
  3. Count the yellow panes. There are 3. This is the top number of the fraction.
  4. (a) 3 of the 4 equal panes are yellow, so 34 of the window is yellow.
  5. (b) Count the panes that are not yellow: 4 − 3 = 1. One pane is not yellow, and it is 14 of the window.

answer(a) 34; (b) 1 pane

techniqueTop and Bottom

examsPSLE

Common pitfalls

  • Writing 43 with the two numbers the wrong way round. The number of all the equal panes, 4, goes at the bottom. The number of yellow panes, 3, goes on top.
  • Writing 31 by putting the yellow panes over the panes that are not yellow. The bottom number counts all the panes in the window, yellow or not, so it is 4.
08

Pieces of a Play Mat Put Together to Make the Whole

strategyCount On in Quarters and in Halves / Fill the Whole

A play mat is a square made of 4 equal pieces. Each piece is 14 of the mat. Jun has put down 1 piece. (a) How many more pieces does Jun need to make the whole mat? (b) Mei's mat is the same size. Each of her pieces is 12 of the mat. How many of her pieces make the whole mat?

Jun's mat1/4
Jun's mat has 4 equal parts. The 1 piece he has put down is shaded.
Draw Jun's mat as a square with 4 equal parts. Shade the 1 piece he has put down.
step 1 of 5

Put the pieces down one at a time and count them until the square is full. The bottom number of the fraction tells how many pieces fill the whole.

  1. Draw Jun's mat as a square with 4 equal parts. Shade the 1 piece he has put down.
  2. Count on until the square is full: 1 quarter, 2 quarters, 3 quarters, 4 quarters. Four quarters make 1 whole.
  3. (a) Jun has 1 piece, so he needs 4 − 1 = 3 more pieces.
  4. Draw Mei's mat with 2 equal parts. Each part is 12 of the mat.
  5. (b) Count on until the square is full: 1 half, 2 halves. Two halves make 1 whole, so 2 of her pieces make the whole mat.

answer(a) 3 more pieces; (b) 2 pieces

techniqueHalves, Quarters, Thirds · Equal Parts

examsPSLE

Common pitfalls

  • Answering 4 for part (a). Four quarters make the whole mat, but Jun has already put down 1 of them, so he needs only 3 more.
  • Saying that Mei also needs 4 pieces because her mat is the same size. Her pieces are bigger. One half covers as much as 2 quarters, so only 2 halves fill the mat.
09

The Whole Group Found from a Half or a Quarter of It

strategyDraw One Group / Draw the Other Equal Groups and Count Them All

Half of the children in a class are girls. There are 9 girls. (a) How many children are in the class? (b) A quarter of the children in the art club are boys. There are 3 boys. How many children are in the art club?

9 girls?
Half means 1 of 2 equal groups. The 9 girls are one group.
Half means 1 of 2 equal groups. The 9 girls are one group. Draw 9 dots for them.
step 1 of 5

The number that is given is one of the equal groups. Draw that group, draw the other groups the same size, and count all the children.

  1. Half means 1 of 2 equal groups. The 9 girls are one group. Draw 9 dots for them.
  2. The other half is an equal group, so it also has 9 children. Draw 9 more dots.
  3. (a) Count both groups: 9 + 9 = 18. There are 18 children in the class.
  4. A quarter means 1 of 4 equal groups. The 3 boys are one group. Draw 4 groups with 3 dots in each.
  5. (b) Count all four groups: 3 + 3 + 3 + 3 = 12. There are 12 children in the art club.

answer(a) 18 children; (b) 12 children

techniqueA Fraction of a Group

examsPSLE · GCSE Higher

Common pitfalls

  • Answering 9 for part (a). The 9 girls are only half of the class. The whole class is two equal groups of 9, which is 18 children.
  • Adding the bottom number and writing 3 + 4 = 7 for part (b). The 4 is the number of equal groups, not a number of children. Four groups of 3 make 12.
10

A Quarter of an Hour and Half an Hour in Minutes

strategyDraw 1 Hour as 60 Minutes / Cut It into Equal Parts and Count in Fives

There are 60 minutes in 1 hour. Tim reads for a quarter of an hour. Then he plays for half an hour. (a) For how many minutes does Tim read? (b) For how many minutes does Tim play?

1 hour55555555555560 minutes
One hour is a bar of 12 blocks. Each block is 5 minutes, and the last block ends at 60 minutes.
Draw 1 hour as a bar of 12 blocks. Each block is 5 minutes. Count in fives along the bar to check: the last block ends at 60 minutes.
step 1 of 5

Draw 1 hour as a bar of 12 blocks. Each block is 5 minutes, like the space from one number to the next on a clock face. Cut the bar into equal parts and count in fives.

  1. Draw 1 hour as a bar of 12 blocks. Each block is 5 minutes. Count in fives along the bar to check: the last block ends at 60 minutes.
  2. A quarter is 1 of 4 equal parts. Cut the bar into 4 equal parts. Each part has 3 blocks.
  3. (a) Count in fives along 3 blocks: 5, 10, 15. A quarter of an hour is 15 minutes, so Tim reads for 15 minutes.
  4. Half is 1 of 2 equal parts. Cut the bar into 2 equal parts. Each part has 6 blocks.
  5. (b) Count in fives along 6 blocks: 5, 10, 15, 20, 25, 30. Half an hour is 30 minutes, so Tim plays for 30 minutes. Check: 30 + 30 = 60.

answer(a) 15 minutes; (b) 30 minutes

techniqueHalves, Quarters, Thirds · A Fraction of a Group

examsPSLE

Common pitfalls

  • Saying that a quarter of an hour is 25 minutes. That would be right if an hour had 100 minutes. An hour has 60 minutes, and 60 minutes cut into 4 equal parts gives 15 minutes in each part.
  • Writing 4 minutes for a quarter of an hour because of the 4 in 14. The 4 is the number of equal parts, not the number of minutes in a part.
Mr. Chalk Drill fraction arithmetic