Thousands, angles and time · applications

Applications: Thousands, Angles and Time

10 question types · Primary 3 · each worked step by step with a figure that follows the steps

PSLE · GCSE Higher

01

Place-Value Blocks, Then One Flat More and One Cube Fewer

strategyWrite the Value of Each Kind of Block / Change One Digit at a Time

Mei has place-value blocks. She has 2 thousand cubes, 5 hundred flats, 3 ten rods and 8 ones. (a) What number do her blocks show? (b) Mei puts in 1 more hundred flat and takes away 1 thousand cube. What number do her blocks show now?

ThHTOBlocksNumber2538
2 thousands are 2000, 5 hundreds are 500, 3 tens are 30 and 8 ones are 8.
Write the value of each kind of block. 2 thousands are 2000. 5 hundreds are 500. 3 tens are 30. 8 ones are 8.
step 1 of 5

Each kind of block has its own value. Write the number in a place-value chart. A hundred flat changes only the hundreds digit, and a thousand cube changes only the thousands digit.

  1. Write the value of each kind of block. 2 thousands are 2000. 5 hundreds are 500. 3 tens are 30. 8 ones are 8.
  2. (a) Add the values: 2000 + 500 + 30 + 8 = 2538. The blocks show 2538.
  3. One more hundred flat is 100 more. Only the hundreds digit changes, from 5 to 6: 2538 + 100 = 2638.
  4. One thousand cube fewer is 1000 less. Only the thousands digit changes, from 2 to 1: 2638 − 1000 = 1638.
  5. (b) The blocks now show 1638. Check: there are now 1 thousand, 6 hundreds, 3 tens and 8 ones.

answer(a) 2538; (b) 1638

techniqueThe Thousand Cube · Numbers to Ten Million

examsPSLE · GCSE Higher

Common pitfalls

  • Adding the numbers of blocks, 2 + 5 + 3 + 8 = 18. Each kind of block has its own value: a thousand cube is worth 1000, not 1.
  • Changing the wrong digit for 100 more, for example writing 2548. That is only 10 more. 100 more changes the hundreds digit, which is the 5 in 2538.
02

The Same Digit in Two Different Places

strategyPlace-Value Chart / Find the Value of the Digit in Each Number, Then Subtract

A toy shop sold 2568 toy cars last year and 5310 toy cars this year. (a) What is the value of the digit 5 in 2568, and what is its value in 5310? (b) What is the difference between the two values?

ThHTOLast year25685 hundreds = 500
In 2568 the digit 5 is in the hundreds column, so its value is 500.
Write 2568 in a place-value chart. The digit 5 is in the hundreds column, so its value is 500.
step 1 of 4

Write both numbers in a place-value chart. The column a digit stands in gives its value.

  1. Write 2568 in a place-value chart. The digit 5 is in the hundreds column, so its value is 500.
  2. Write 5310 in the chart. The digit 5 is in the thousands column, so its value is 5000.
  3. (a) The digit 5 stands for 500 in 2568 and for 5000 in 5310.
  4. (b) Subtract the smaller value from the larger value: 5000 − 500 = 4500. Check: 4500 + 500 = 5000.

answer(a) 500 and 5000; (b) 4500

techniqueNumbers to Ten Million · The Thousand Cube

examsPSLE · GCSE Higher

Common pitfalls

  • Saying that the digit 5 is worth 5 in both numbers. The value of a digit depends on its column: 5 hundreds are 500 and 5 thousands are 5000.
  • Subtracting the two whole numbers, 5310 − 2568. The question asks for the difference between the two values of the digit 5, which are 5000 and 500.
03

The Greatest and the Smallest Number from Four Digit Cards

strategyOrder the Digits / The Greatest Digit Goes First, and Zero Cannot Go First

Ravi has four digit cards: 4, 0, 7 and 2. He uses all four cards to make a 4-digit number. (a) What is the greatest number he can make, and what is the smallest number he can make? (b) What is the difference between these two numbers?

Cards4072ThHTOGreatest7420
The greatest digit goes in the thousands place, then the next greatest each time: 7420.
For the greatest number, put the greatest digit in the thousands place. Then put the next greatest digit in each place after it: 7, 4, 2, 0. The greatest number is 7420.
step 1 of 4

The thousands place has the greatest value, so the digit put there matters most. Fill the places from the left.

  1. For the greatest number, put the greatest digit in the thousands place. Then put the next greatest digit in each place after it: 7, 4, 2, 0. The greatest number is 7420.
  2. For the smallest number, the smallest digit should go first. But 0 cannot go first, because 0247 is only the 3-digit number 247.
  3. So put 2 first, then 0, then 4, then 7. (a) The greatest number is 7420 and the smallest number is 2047.
  4. (b) Subtract the smallest number from the greatest number: 7420 − 2047 = 5373. Check: 2047 + 5373 = 7420.

answer(a) 7420 and 2047; (b) 5373

techniqueNumbers to Ten Million

examsPSLE · GCSE Higher

Common pitfalls

  • Writing 0247 as the smallest number. A 4-digit number cannot start with 0. The smallest digit that is not 0 goes first, and 0 goes second.
  • Writing the smallest number as 2470, with 0 at the end. A small digit should go in a place with a great value, so 0 goes in the hundreds place: 2047.
04

A Number Pattern That Crosses a Thousand

strategyFind the Step Between Neighbors / Count On Through the Next Thousand

Five lockers in a row are numbered in a pattern. The numbers are 3780, 3880, 3980, a number that has rubbed off, and 4180. (a) By how much does the number go up from one locker to the next? (b) What is the number that has rubbed off?

378038803980?4180+100+100
3880 − 3780 = 100 and 3980 − 3880 = 100.
Subtract the neighbors that you can see: 3880 − 3780 = 100 and 3980 − 3880 = 100.
step 1 of 4

Subtract two neighbors to find the step of the pattern. Then add the step to the number just before the gap.

  1. Subtract the neighbors that you can see: 3880 − 3780 = 100 and 3980 − 3880 = 100.
  2. (a) The number goes up by 100 from one locker to the next.
  3. Add 100 to 3980. 9 hundreds and 1 more hundred make 10 hundreds, which is 1 more thousand: 3980 + 100 = 4080.
  4. (b) The number that has rubbed off is 4080. Check: 4080 + 100 = 4180, which is the last locker.

answer(a) 100; (b) 4080

techniqueThe Thousand Cube · Numbers to Ten Million

examsPSLE · GCSE Higher

Common pitfalls

  • Writing 3990 for the missing number. That adds only 10, but the numbers go up by 100 each time.
  • Writing 3080 when the hundreds digit passes 9. Ten hundreds are exchanged for one thousand, so the thousands digit goes up from 3 to 4 and the hundreds digit becomes 0.
05

The Minute Hand Turns Through Right Angles

strategyEvery 15 Minutes Is One Quarter Turn / Count the Right Angles

The minute hand of a clock points to 12. Ben reads a book for 45 minutes, and the minute hand turns all that time. (a) How many right angles does the minute hand turn through? (b) How many degrees is that?

12369
A full turn takes 60 minutes and is 4 right angles, so one right angle takes 60 ÷ 4 = 15 minutes.
The minute hand goes once round the clock in 60 minutes. A full turn is 4 right angles, so one right angle takes 60 ÷ 4 = 15 minutes.
step 1 of 4

The minute hand makes a full turn in 60 minutes, and a full turn is 4 right angles. So each 15 minutes is one right angle.

  1. The minute hand goes once round the clock in 60 minutes. A full turn is 4 right angles, so one right angle takes 60 ÷ 4 = 15 minutes.
  2. In 15 minutes the hand turns from 12 to 3. That is one quarter turn, which is one right angle.
  3. Count in fifteens up to 45: 15, 30, 45. That is 3 lots of 15 minutes. (a) The minute hand turns through 3 right angles, from 12 round to 9.
  4. (b) One right angle is 90°. 3 right angles are 90° + 90° + 90° = 270°.

answer(a) 3 right angles; (b) 270°

techniqueAngles · Telling the Time

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Saying the hand turns through 9 right angles because it stops at the 9. The number on the clock face is not the number of right angles. Count the quarter turns: 12 to 3, 3 to 6, and 6 to 9.
  • Thinking about the hour hand. In 45 minutes the hour hand moves less than the space between two numbers. The question asks about the minute hand, which turns through one right angle every 15 minutes.
06

The Angles of a Shape Sorted Against a Right Angle

strategyCompare Each Angle with 90° / Sort into Three Groups, Then Count

A kite maker cuts a shape with five corners, A, B, C, D and E. She measures the angle at each corner. Angle A is 90°, angle B is 120°, angle C is 140°, angle D is 60° and angle E is 130°. (a) How many of the angles are greater than a right angle? (b) How many of the angles are smaller than a right angle?

A90°B120°C140°D60°E130°
A right angle is 90°. Compare each angle with 90°.
A right angle is 90°. Compare each of the five angles with 90°.
step 1 of 5

A right angle is 90°. Compare each angle with 90° and put it in one of three groups: smaller than a right angle, equal to a right angle, or greater than a right angle.

  1. A right angle is 90°. Compare each of the five angles with 90°.
  2. 120°, 140° and 130° are all more than 90°. So angles B, C and E are greater than a right angle.
  3. (a) 3 of the angles are greater than a right angle.
  4. 60° is less than 90°, so angle D is smaller than a right angle. Angle A is exactly 90°, so it is a right angle and it belongs to neither group.
  5. (b) 1 angle is smaller than a right angle. Check: 3 + 1 + 1 = 5, so every corner has been sorted.

answer(a) 3 angles; (b) 1 angle

techniqueAngles · Measuring and Drawing Angles

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Counting angle A as greater than a right angle. Angle A is exactly 90°. It is equal to a right angle, not greater than one.
  • Judging an angle by the length of its sides. Angle D has the longest sides, but it is the smallest angle. The size of an angle is how far one side has turned from the other.
07

A Turn Made in Two Parts

strategyA Quarter Turn Is 90° and a Half Turn Is 180° / Subtract the Part Already Turned

A toy robot turns on the spot. (a) The robot has to make a quarter turn, which is a right angle. It turns 25° and stops. How many more degrees must it turn? (b) Next the robot has to make a half turn, so that it faces the opposite way along a straight line. It turns 130° and stops. How many more degrees must it turn?

startstopquarter turn25°?A quarter turn is 90°.
A quarter turn is a right angle, and a right angle is 90°.
A quarter turn is a right angle, and a right angle is 90°.
step 1 of 4

Write the size of the whole turn in degrees first. The part already turned and the part still to turn add up to the whole turn, so subtract.

  1. A quarter turn is a right angle, and a right angle is 90°.
  2. (a) The robot has turned 25° of the 90°. It must turn 90° − 25° = 65° more. Check: 25° + 65° = 90°.
  3. A half turn makes a straight line, and the angle on a straight line is 180°.
  4. (b) The robot has turned 130° of the 180°. It must turn 180° − 130° = 50° more. Check: 130° + 50° = 180°.

answer(a) 65°; (b) 50°

techniqueMeasuring and Drawing Angles · Angles

examsPSLE · O-Level

Common pitfalls

  • Subtracting from 100° instead of 90° in part (a), which gives 75°. A right angle is 90°, not 100°.
  • Using 90° again in part (b). A half turn is two right angles, which is 180°, so the 130° is taken away from 180°.
08

A Clock Read to the Minute, and the Minutes to the Next Hour

strategyCount in Fives to the Number, Then Count the Small Marks / Subtract from 60

Lina looks at the clock before school. The hour hand is between 7 and 8. The minute hand points 2 small marks past the 4. (a) What time does the clock show? (b) How many minutes is it until 8 o'clock?

12346789Hour hand: past 7, not yet 8
The hour hand has passed 7 and has not reached 8, so the hour is 7.
The hour hand is between 7 and 8. It has passed 7 and has not reached 8, so the hour is 7.
step 1 of 4

The hour hand gives the hour it has passed. For the minute hand, each number on the clock is 5 minutes and each small mark is 1 minute.

  1. The hour hand is between 7 and 8. It has passed 7 and has not reached 8, so the hour is 7.
  2. For the minute hand, each number on the clock is 5 minutes. Count in fives to the 4: 4 × 5 = 20 minutes.
  3. Add the 2 small marks: 20 + 2 = 22 minutes. (a) The clock shows 7.22 a.m.
  4. (b) An hour has 60 minutes. From 22 minutes to 60 minutes is 60 − 22 = 38 minutes. Check: 22 + 38 = 60.

answer(a) 7.22 a.m.; (b) 38 minutes

techniqueTelling Time to the Minute · Telling the Time

examsPSLE

Common pitfalls

  • Reading the hour as 8 because the next number the hour hand will reach is 8. The hour hand has not reached 8 yet, so the hour is still 7.
  • Reading the minutes as 4 or as 6 because the minute hand is near the 4. For the minute hand the number 4 means 4 fives, which is 20 minutes, and the 2 small marks make it 22.
09

Minutes Changed to Hours and Minutes, and Back Again

strategyTake Out Groups of 60 Minutes / Put In 60 Minutes for Each Hour

A film lasts 165 minutes. After the film, Hana's bus ride home takes 1 h 20 min. (a) How long is the film in hours and minutes? (b) How long is the bus ride in minutes?

Film60 min60 min?165 minBus ride1 h20 min1 h 20 min
Two groups of 60 minutes fit in 165 minutes: 60 + 60 = 120. A third group would make 180.
1 hour is 60 minutes. Take groups of 60 minutes out of 165: 60 + 60 = 120. A third group would make 180, which is too many.
step 1 of 5

1 hour is 60 minutes. To change minutes to hours, take out groups of 60. To change hours to minutes, put in 60 minutes for each hour.

  1. 1 hour is 60 minutes. Take groups of 60 minutes out of 165: 60 + 60 = 120. A third group would make 180, which is too many.
  2. 2 hours use 120 minutes. The minutes left are 165 − 120 = 45.
  3. (a) The film lasts 2 h 45 min.
  4. For the bus ride, change the 1 hour to 60 minutes and add the other 20 minutes: 60 + 20 = 80.
  5. (b) The bus ride takes 80 minutes.

answer(a) 2 h 45 min; (b) 80 minutes

techniqueHours and Minutes

Common pitfalls

  • Writing 165 minutes as 1 h 65 min. The minutes part must be less than 60, because 60 of those minutes make another hour.
  • Treating 1 hour as 100 minutes and writing 1 h 20 min as 120 minutes. An hour is 60 minutes, so 1 h 20 min is 60 + 20 = 80 minutes.
10

Two Race Times, One in Minutes and Seconds

strategyChange Both Times to Seconds / The Smaller Time Is the Faster One

Ali and Ben each swim one lap of the pool. Ali takes 2 min 15 s. Ben takes 128 s. (a) What is Ali's time in seconds? (b) Who is faster, and by how many seconds?

Ali60 s60 s15 s2 min 15 sBen128 s
1 minute is 60 seconds, so 2 minutes are 60 + 60 = 120 seconds.
1 minute is 60 seconds, so 2 minutes are 60 + 60 = 120 seconds.
step 1 of 4

Two times can be compared only when they are in the same unit. 1 minute is 60 seconds, so change Ali's time to seconds first.

  1. 1 minute is 60 seconds, so 2 minutes are 60 + 60 = 120 seconds.
  2. (a) Add the 15 seconds: 120 + 15 = 135. Ali's time is 135 seconds.
  3. Now both times are in seconds. Ali took 135 s and Ben took 128 s. The faster swimmer takes less time, and 128 is less than 135.
  4. (b) Ben is faster, by 135 − 128 = 7 seconds.

answer(a) 135 seconds; (b) Ben, by 7 seconds

techniqueMinutes and Seconds

Common pitfalls

  • Writing 2 min 15 s as 215 seconds. The 2 stands for 2 minutes, and each minute is 60 seconds, so the time is 120 + 15 = 135 seconds.
  • Saying that Ali is faster because 135 is the bigger number. In a race the faster swimmer has the smaller time.
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