Answers for parents. Each unit has a worked example, then the answers for every level.
Unit 1: Thousands, hundreds, tens and ones
Worked example. Write the value of each digit in 6,047, from left to right. 1. Thousands: the 6 is in the thousands place, so it is worth 6,000. 2. Hundreds: the 0 is a placeholder. It shows no hundreds, so it is worth 0. 3. Tens: the 4 is worth 40. Ones: the 7 is worth 7. So the values are 6,000, 0, 40 and 7.
Level 1
40The 4 is in the tens place, so it is worth 4 tens, which is 40.
500The 5 is in the hundreds place, so it is worth 5 hundreds, which is 500.
3623 hundreds, 6 tens and 2 ones make 362.
50451 has 4 hundreds, 5 tens and 1 one, so the missing part is 5 tens, which is 50.
1617 has 6 hundreds, 1 ten and 7 ones, so the digit in the tens place is 1.
FalseThe 8 is in the hundreds place, so it is worth 800, not 80.
6,000The 6 is in the thousands place, so it is worth 6 thousands, which is 6,000.
800The 8 is in the hundreds place, so it is worth 8 hundreds, which is 800.
5,3245 thousands, 3 hundreds, 2 tens and 4 ones make 5,324.
49,461 has 9 thousands, 4 hundreds, 6 tens and 1 one, so the hundreds digit is 4.
500The 5 is in the hundreds place, so it is worth 500. The 0 in the tens place is a placeholder.
8,076There are 8 thousands, no hundreds, 7 tens and 6 ones, so a 0 goes in the hundreds place: 8,076.
Level 2
300The 3 is in the hundreds place, so it is worth 300. The 0 in the tens place is a placeholder.
90The 9 is in the tens place, so it is worth 9 tens, which is 90. The 0 holds the hundreds place.
5,038The chart shows 5 thousands, 0 hundreds, 3 tens and 8 ones. The number is 5,038.
3,6143,614 has 3 thousands, 6 hundreds, 1 ten and 4 ones. In the others the 6 is in the tens place or the thousands place.
08,052 has 8 thousands, 0 hundreds, 5 tens and 2 ones. The 0 is a placeholder that shows there are no hundreds.
7,025There are no hundreds, so a 0 goes in the hundreds place. The number is 7,025.
6,000The 6 is in the thousands place, so it is worth 6,000.
FalseThe 9 is in the hundreds place, so it is worth 900. The 3 is the digit in the thousands place.
9,000; 300; 0; 59 thousands is 9,000. 3 hundreds is 300. The 0 in the tens place is worth 0. The 5 is worth 5.
10The 1 is in the tens place, so it is worth 1 ten, which is 10. The 0 in the hundreds place is a placeholder.
Level 3
8,0208 thousands, 0 hundreds, 2 tens and 0 ones make 8,020.
1,3901 is in the thousands place and 9 in the tens place. The 0 is in the ones place, so the 3 must be in the hundreds place: 1,390.
1,011; 1,101; 1,110Three digits are 1 and one is 0. The 0 cannot be in the thousands place, so it can be in the hundreds, tens or ones place: 1,011; 1,101; 1,110.
560The 6 is worth 600 and the 4 is worth 40. 600 − 40 = 560.
4,638A digit worth 600 is 6 in the hundreds place. The chart then shows 4 thousands, 6 hundreds, 3 tens and 8 ones: 4,638.
9,052The 5 must be in the tens place and the 9 in the thousands place. Only 9,052 fits.
Expert
Model answer: In 5,070 the 7 is in the tens place, so it is worth 70. In 5,700 the 7 is in the hundreds place, so it is worth 700. The same digit is worth different amounts in different places.Look for: the 7 is in a different column in each number; the values are 70 and 700; the place decides the value.
4,45614 hundreds is 1,400. 3,000 + 1,400 + 50 + 6 = 4,456. Ten hundreds make one more thousand.
Model answer: Always. The thousands digit is at least 1, so it is worth at least 1,000. The hundreds digit is at most 9, so it is worth at most 900. 1,000 is more than 900.Look for: the thousands digit cannot be 0 in a four-digit number; its smallest value is 1,000; the largest hundreds value is 900.
3,005; 4,105; 5,205; 6,305; 7,405; 8,505; 9,605The hundreds digit can be 0 to 6, so that the thousands digit is 3 to 9. Each gives one number, ending in 05.
37; 475,740 take away 40 is 5,700. One thousand leaves 47 hundreds, two thousands leave 37, three leave 27 which is not more than 30.
Model answer: The 7 is in the tens column, so it is 7 tens (70), not 7 hundreds. 5,070 = 5 thousands + 7 tens, and there are 0 hundreds.Look for: the 0 holds the hundreds place; the 7 is worth 70; 5 thousands + 7 hundreds is 5,700, not 5,070.
3,700; 5,1402 thousands, 14 hundreds, 30 tens gives 3,700. 2 thousands, 30 hundreds, 14 tens gives 5,140. Any order with 14 or 30 thousands is over 10,000.
5; 4; 21,000 + 4,500 + 30 + 12 = 5,542. After the 5 thousands come 5 hundreds, 4 tens and 2 ones.
Worked example. Fill the gap: 650, 675, ___, 725. 1. Each step is 25 more: 650 to 675 is 25 more. 2. The last two digits repeat: 00, 25, 50, 75, then 00 again. 3. After 675 the last two digits are 00, so the hundreds go up: 700. 4. Check: 700 + 25 = 725. So the missing number is 700.
Level 1
700Each number is 100 more. 600 + 100 = 700.
500Each number is 50 more. 450 + 50 = 500.
16Each number is 4 more. 12 + 4 = 16.
40The next number in the count in 8s is 8 more. 32 + 8 = 40.
250Counting back, each number is 50 less. 300 − 50 = 250.
56Counting back, each number is 8 less. 64 - 8 = 56.
5,000Each number is 1,000 more. 4,000 + 1,000 = 5,000.
75Each number is 25 more. 50 + 25 = 75.
6,000Counting back, each number is 1,000 less. 7,000 - 1,000 = 6,000.
300Each number is 25 more. 275 + 25 = 300.
100The missing number is 25 more than 75. 75 + 25 = 100.
FalseThe count in 1,000s goes 5,000 then 6,000, so 5,500 is not in it.
Level 2
825The missing number is 25 more than 800. 800 + 25 = 825.
25; 0Each number is 25 less: 50 − 25 = 25, then 25 − 25 = 0.
3,000; 4,000Each number is 1,000 more: 3,000, then 4,000, then 5,000.
875Numbers in the count in 25s end in 00, 25, 50 or 75. Only 875 ends in 75.
525; 500Each number is 25 less: 550 − 25 = 525, then 525 − 25 = 500, then 475.
175Count on from 0 in 25s: 25, 50, 75, 100, 125, 150, 175. The 7th number is 175.
TrueFour steps of 25 make 100, so ten hundreds come to 1,000 exactly. The count in 25s reaches 1,000.
225 mThe 9th marker is the 9th number in the count in 25s: 25, 50, 75, 100, 125, 150, 175, 200, 225.
10,000The next number is 1,000 more. 9,000 + 1,000 = 10,000.
41,000 - 900 = 100, and 4 steps of 25 make 100.
Level 3
410,000 − 6,000 = 4,000 m left. Each stage is 1,000 m, so 4 more stages.
825; 850; 875; 900Numbers in the count in 25s end in 00, 25, 50 or 75. Between 810 and 910 these are 825, 850, 875 and 900.
125Going back to 0, the numbers are 125, 100, 75, 50, 25, 0. There are 5 steps of 25 from the 1st number to 0.
200 mFrom the 5th cone to the 13th cone is 13 − 5 = 8 gaps. Eight gaps of 25 m is 200 m.
275; 725A number in the count in 25s ends in 25 or 75 here, since there is no 0. That gives 275 and 725.
125The count goes ... 175, 150, 125. 150 is not below 150, so Chloe stops at 125.
Expert
175; 625; 850Last two digits are 00, 25, 50 or 75. 25 needs hundreds digit 6 (625), 50 needs 8 (850), 75 needs 1 (175). 00 would need 13.
50; 75; 100The middle number is the average: 225 shared into three is 75. The numbers are 25 less and 25 more: 50, 75, 100.
Model answer: Every 4 numbers add another 100. 36 is 9 lots of 4, so the 36th number is 9 hundreds, which is 900.Look for: 4 steps of 25 make 100; 36 ÷ 4 = 9 hundreds; so 900. Also accept 36 × 25 = 900.
200; 400; 600; 800; 1,000The first number in both counts is 200, since 8 × 25 = 200. Then each further 200: 400, 600, 800, 1,000.
Worked example. Find 1,000 more, 1,000 less and 100 more than 3,960. 1. 1,000 more: the thousands digit 3 becomes 4, so 4,960. 2. 1,000 less: the thousands digit 3 becomes 2, so 2,960. 3. 100 more: hundreds 9 + 1 = 10, so write 0 and the thousands digit 3 becomes 4: 4,060. So the answers are 4,960; 2,960; 4,060.
Level 1
44Adding 10 adds 1 to the tens digit: 34 becomes 44.
48Taking away 10 takes 1 from the tens digit: 58 becomes 48.
350Adding 100 adds 1 to the hundreds digit: 250 becomes 350.
TrueTaking away 100 takes 1 from the hundreds digit: 640 becomes 540.
4059 tens + 1 ten = 10 tens, so the hundreds digit goes up: 405.
590600 is 60 tens, so 10 less is 59 tens: 590.
3,345Adding 1,000 adds 1 to the thousands digit: 2,345 becomes 3,345.
5,712Taking away 1,000 takes 1 from the thousands digit: 6,712 becomes 5,712.
5,508Only the thousands digit changes: 4 becomes 5.
7,263Only the thousands digit changes: 8 becomes 7.
False1,000 more than 5,164 is 6,164. The number 5,264 is only 100 more.
3,906Add 1,000: the thousands digit goes from 2 to 3.
Level 2
3,570Only the hundreds digit changes: 4 becomes 5.
7,195The tens digit is 0, so exchange: 7,205 is 720 tens and 5 ones, and 719 tens and 5 ones is 7,195.
5,950The hundreds digit is 0, so the thousands digit drops: 6,050 becomes 5,950.
3,030Only the thousands digit changes, and the zeros stay: 4 becomes 3.
6,640Go back: 1,000 less than 7,640 is 6,640.
True3,999 + 1 = 4,000, then 9 more makes 4,009.
4,942Take 1 from the hundreds digit, which is 0, so the thousands digit drops: 4,942.
5,0509 hundreds + 1 hundred = 10 hundreds, so the thousands digit goes up: 5,050.
8,000 m99 tens + 1 ten = 100 tens, which is 8,000 m.
3,215 mWork backwards. Undo the drop: 4,205 + 10 = 4,215. Undo the rise: 4,215 - 1,000 = 3,215.
7,903Nia took 1,000 instead of 100. 100 less than 8,003 is 7,903.
2,000The other number must be at least 1,000, and 1,000 + 1,000 = 2,000.
Box BBox A holds 3,950 + 100 = 4,050. Box B holds 5,100 - 1,000 = 4,100. Box B holds more.
1; 10; 1009,350 + 1,000 = 10,350, which is too big. 9,350 + 100 = 9,450, 9,350 + 10 = 9,360 and 9,350 + 1 = 9,351 all fit.
Expert
Model answer: No. Adding 100 to 7,950 gives 8,050, so the thousands digit changes too, because 9 hundreds + 1 hundred makes 10 hundreds.Look for: a correct example where the hundreds digit is 9, and the idea that 10 hundreds makes 1 thousand.
3,900; 4,100; 5,900; 6,100The first move gives 6,000 or 4,000. Each then goes up or down 100: 6,100, 5,900, 4,100 and 3,900.
6,960100 more reaches 7 thousands, so the hundreds digit is 9. Then 6 + 9 + tens + 0 = 21 gives tens digit 6: 6,960. Check: 6,960 + 100 = 7,060 and 6,960 - 10 = 6,950.
Model answer: No. Every order finishes at 5,186, because adding and taking away the same amounts gives the same total whichever order is used.Look for: the finishing number 5,186 found for at least one order, and the idea that the moves add up to the same total, +1,090, in any order.
Worked example. Read the number that arrow A points to on the line. 1. 10 steps cover 1,000, so each step is 100. 2. Arrow A is 6 steps after 2,000: 6 hundreds more. 3. 2 thousands and 6 hundreds make the number. So arrow A points to 2,600.
Level 1
30The numbers go up by 10 each time, so the gap is 20 + 10 = 30.
150Each mark is 10 more than the one before: 140 + 10 = 150.
800Each mark is 100 more than the one before: 700 + 100 = 800.
50The mark before 60 is one step of 10 back: 60 - 10 = 50.
30Halfway between 20 and 40 is 30, because 20 and 10 more is 30 and 30 and 10 more is 40.
390One step of 10 back from 400 crosses into the next hundred down: 390.
2,000Each mark is 1,000 more than the one before: 1,000 + 1,000 = 2,000.
8,000Each mark is 1,000 more than the one before: 7,000 + 1,000 = 8,000.
4,000One step back of 1,000: 5,000 - 1,000 = 4,000.
3,000Each step is 1,000 and arrow A is 3 steps from 0, so it points to 3,000.
4,000Halfway between 0 and 8,000 is 4,000, because 4,000 + 4,000 = 8,000.
4Count in 1,000s: 1,000; 2,000; 3,000; 4,000. That is 4 steps.
Level 2
2,300One step of 100 back from 2,400 is 2,300.
5,100One step of 10 on from 5,090 reaches the next hundred: 5,100.
3,470The gap of 100 has 10 steps, so each step is 10. Arrow A is 7 steps after 3,400, so 3,470.
5,010Each step is 10. 3 steps after 4,980: 4,990, 5,000, 5,010.
7,500Each mark is 100 more than the one before: 7,400 + 100 = 7,500.
8,250Marks in 10s end in 0. Only 8,250 ends in 0.
7,000 mThe next marker is 10 m on from 6,990 m, which is 7,000 m.
6,300Between 6,200 and 6,400 there is one mark, 6,300, and it is halfway.
1,0631,000 + 60 + 3 has no hundreds, so 0 goes in the hundreds place: 1,063.
9,2005 steps of 100 is 5 hundreds more. 8,700 + 5 hundreds = 9,200.
Level 3
130Arrow A points to 5,640 and arrow B to 5,770. The gap is 5,770 − 5,640 = 130.
4,400; 4,500The marks in 100s after 4,300 are 4,400, 4,500 and 4,600. Only 4,400 and 4,500 are between the two numbers.
2,700 mEach marker is 100 m further on, so 27 markers is 27 hundreds, which is 2,700 m.
1,420 mArrow R is 7 steps of 10 above 1,300, so 1,370 m. Then 1,370 + 50 = 1,420 m, above the top of the scale.
59,020 m − 8,970 m = 50 m, and 50 m is five gaps of 10 m.
3,200 mWork backwards: take 5 hundreds from 3,700. 3,700 m − 500 m = 3,200 m.
Expert
Model answer: Each step is 10, not 1, because the gap of 100 has 10 steps. 7 steps of 10 is 70, so arrow A points to 9,570.Look for: each step is 10 (100 split into 10 equal steps); 7 steps is 70, not 7; correct number 9,570.
4,690; 4,960A mark in 10s ends in 0, so 0 is the ones digit. Less than 5,500 means 4 is the thousands digit, leaving 6 and 9 to swap: 4,690 and 4,960.
Model answer: Yes. Each step of 100 is the same as 10 steps of 10, so the line in 10s has a mark at every number the line in 100s has.Look for: 100 is 10 tens, so every multiple of 100 is a multiple of 10. The reverse is not true: 10 is a mark on the second line but not the first.
7,710Rosa ends 15 − 8 = 7 marks left of the start, which is 70 less. So the start is 7,640 + 70 = 7,710.
Worked example. Which is the best estimate for arrow A: 800, 2,400 or 3,100? 1. The line runs from 0 to 3,200, so halfway is 1,600. 2. Arrow A is right of halfway, so the number is more than 1,600. 3. 800 is less than 1,600, so it is too small. 4. Arrow A is not near the end, so 3,100 is too big. 2,400 is best. So arrow A shows about 2,400.
Level 1
03 is 3 away from 0 and 7 away from 10, so 0 is nearer.
5Half of 10 is 5.
50Half of 100 is 50.
True15 is 5 away from 20 but 15 away from 0, so the statement is true.
1,000900 is 100 away from 1,000 and 900 away from 0, so 1,000 is nearer.
250The gap is 100 and half of 100 is 50, so halfway is 200 + 50 = 250.
1,000Half of 2,000 is 1,000.
5,0004,000 is 1,000 from 5,000 and 4,000 from 0, so 5,000 is nearer.
3,000Half of 6,000 is 3,000, because half of 6 is 3.
8,0007,300 is more than 7,000 and less than 8,000, so the next thousand is 8,000.
False1,000 is only 1,000 from 0 but 8,000 from 9,000, so the statement is false.
3,000The gap is 4,000 and half of it is 2,000, so halfway is 1,000 + 2,000 = 3,000.
Level 2
1,500The gap is 600 and half of it is 300, so halfway is 1,200 + 300 = 1,500.
6,0005,140 is more than 5,000 and less than 6,000, so the next thousand is 6,000.
5,800Halfway from 0 to 10,000 is 5,000. 2,500 is below halfway and 9,500 is far above it, so 5,800 is a little more than halfway.
TrueHalfway from 0 to 5,000 is 2,500. 2,350 is less than 2,500, so the statement is true.
1,700 mHalf of 3,400 m is 1,700 m, because half of 34 hundreds is 17 hundreds.
7,4006,300 is 1,100 from 7,400 and 3,400 from 2,900, so 7,400 is nearer.
3,140Halfway between 2,700 and 3,300 is 3,000. Only 3,140 is more than 3,000.
6,700From 6,100 to 6,400 is 300, so go another 300 past 6,400 to reach 6,700.
5,000 mMarkers are at 5,000 m and 6,000 m. Yara has passed 5,000 m but not yet 6,000 m.
4,1505,000 − 4,150 = 850 and 5,900 − 5,000 = 900. 850 is smaller, so 4,150 is closer.
Level 3
2,250 mA is at 1,000 m and B is at 3,500 m. The gap is 2,500 m, half is 1,250 m, and 1,000 + 1,250 = 2,250 m.
4,000; 5,000; 6,000The markers are the multiples of 1,000. Those after 3,450 and before 6,900 are 4,000, 5,000 and 6,000.
2,700 cmThe flag is halfway, so the whole depth is double: 1,350 + 1,350 = 2,700 cm.
CEach mark is 250. 1,250 is the fifth mark. Arrow C is one mark away, arrow B is three marks away and arrow A is six marks away.
3,999Halfway between 3,600 and 4,400 is 4,000. The number must be less than 4,000, so the largest whole number is 3,999.
2,550 mHalfway is 5,600 ÷ 2 = 2,800 m. Luca stops 250 m before that: 2,800 − 250 = 2,550 m.
Expert
Model answer: No. The gap is 3,700 − 2,300 = 1,400 and half of that is 700, so halfway is 2,300 + 700 = 3,000, not 3,150.Look for: Ines is not correct, and the child finds the gap (1,400), halves it (700) and adds it to 2,300 to get 3,000. Adding the two numbers and halving also works.
4,000; 4,500; 5,000Halfway between 2,800 and 4,200 is 3,500, which is not allowed because the number must be more than 3,500. The multiples of 500 up to 5,400 are 4,000, 4,500 and 5,000.
Model answer: Sometimes. Halfway between 5,300 and 5,700 is 5,500, a multiple of 100. Halfway between 5,300 and 5,800 is 5,550, which is not.Look for: the answer sometimes, one example that works and one that does not. It works when the gap is a multiple of 200, so half the gap is a multiple of 100.
5,050; 5,650Half of 600 is 300... the numbers sit 300 either side of halfway: 5,350 − 300 = 5,050 and 5,350 + 300 = 5,650.
Worked example. Write the greater number: 6,573 or 6,537. 1. Thousands: 6 and 6 are the same, so look at the hundreds. 2. Hundreds: 5 and 5 are the same, so look at the tens. 3. Tens: 7 is greater than 3, so 6,573 is the greater number. So 6,573 is greater than 6,537.
Level 1
354The hundreds are the same. Tens: 5 is more than 4, so 354 is greater.
618The hundreds are the same. Tens: 1 is less than 8, so 618 is less.
FalseTens: 3 is less than 6, so 436 is less than 463. The statement is false.
705Hundreds: 7 is more than 5, so 705 is greater.
>The hundreds are the same. Tens: 3 is more than 0, so 530 > 503.
999999 has three digits and 1,000 has four digits, so 999 is less.
6,305Thousands: 6 is more than 2, so 6,305 is greater.
5,280The thousands are the same. Hundreds: 2 is less than 8, so 5,280 is less.
3,4683,468 is one more than 3,467 and one less than 3,469, so it is between them.
4,510 mThe thousands are the same. Hundreds: 5 is more than 1, so 4,510 m is longer.
6,005Thousands, hundreds same. Tens: 0 is less than 5, so 6,005 is smaller.
8,000Thousands: 8 is more than 7, so 8,000 is greater, even with its zeros.
Level 2
4,327Thousands and hundreds match. Tens: 2 is less than 7, so 4,327 is smaller.
8,105The thousands match. Hundreds: 1 is more than 0, so 8,105 is greater.
FalseThousands and hundreds match. Tens: 9 is more than 3, so 5,693 is greater. The statement is false.
9871,002 has four digits and 987 has three, so 987 is smaller.
8The hundreds digit must be more than 7. A 7 would make the numbers equal, so the smallest digit is 8.
5,3605,000 + 300 + 6 is 5,306. Tens: 6 is more than 0, so 5,360 is greater.
3,206 mTens: 0 is less than 6, so 3,206 is less than 3,260. The shorter walk is 3,206 m.
TrueThousands match. Hundreds: 8 and 8 match. Tens: 6 is more than 0, so the statement is true.
=900 + 2,000 is 2,900, written in a different order. The numbers are equal.
4,980Thousands and hundreds match. Tens: 8 is more than 0, so 4,980 is larger.
Level 3
3,508Omar's number is 3,085. Hundreds: 0 is less than 5, so 3,508 is greater.
4,381; 4,382; 4,383The numbers between 4,380 and 4,384 are 4,381, 4,382 and 4,383. Neither end is included.
5,399Below 6,000 the thousands digit is at most 5. Keep 3 in the hundreds and make the tens and ones 9: 5,399.
2,608The thousands match. Hundreds: 6 is the least, so 2,608 is the fewest.
6The hundreds digit must be more than 5 and less than 7, so it is 6. Check: 4,578 < 4,678 < 4,778.
4,706Arjun scored 4,706. The thousands match. Hundreds: 7 is the most, so 4,706 is the top score.
Expert
Model answer: No. Compare from the left: the thousands and hundreds match, then the tens are 0 and 9. So 4,790 is greater. Sami compared the ones digits first.Look for: compare from the greatest place value; tens 0 and 9 decide, so 4,790 is greater; the ones digits do not matter here.
0; 1; 2; 3The right side is 9,468. The hundreds digit must be less than 4; a 4 makes the numbers equal. A 0 is allowed: 9,068 < 9,468.
6,951; 6,971; 6,981The number starts 69 and ends in 1, so it is 69_1 with the tens digit 5 or more. Tens 6 and 9 repeat a digit, so the tens digit is 5, 7 or 8.
Model answer: Always. The smallest 4-digit number is 1,000 and the largest 3-digit number is 999, so more digits means a greater number. This works for any number of digits.Look for: the smallest number with more digits beats the largest with fewer digits; an example such as 1,000 and 999; digits do not start with 0.
Worked example. Write smallest first: 5,260, 764, 6,015, 5,206. 1. 764 has three digits, so it is the smallest. 2. Thousands digits of the others: 5, 6 and 5. So 6,015 is the largest. 3. 5,260 and 5,206 have the same thousands and hundreds digits. 4. Tens digits: 0 and 6. Since 0 is less than 6, 5,206 comes before 5,260. So the order is 764, 5,206, 5,260, 6,015.
Level 1
345Both have 3 hundreds. Compare the tens: 4 tens is less than 5 tens, so 345 is smaller.
270All have 2 hundreds. The tens are 0, 7 and 1, and 7 is the most, so 270 is largest.
350; 503; 530The hundreds digits are 5, 3 and 5, so 350 is smallest. For 503 and 530, compare the tens: 0 is less than 3.
681; 618; 168168 has 1 hundred, so it is smallest. For 618 and 681, compare the tens: 8 is more than 1, so 681 is largest.
FalseBoth have 9 hundreds. The tens are 0 and 5, so 905 is smaller than 950.
790; 797; 799; 800800 has 8 hundreds, so it is largest. The others have 7 hundreds and 9 tens, so compare the ones: 0, 7 and 9.
6,381Compare the thousands first: 6 thousands is more than 3 thousands.
4,250Both have 4 thousands. Compare the hundreds: 2 hundreds is less than 5 hundreds.
3,570; 5,370; 7,350The thousands digits are 5, 3 and 7. Put them in order: 3, 5, 7.
2,841; 2,814; 2,481All have 2 thousands. The hundreds are 8, 8 and 4, so 2,481 is smallest. For the others, compare the tens: 4 is more than 1.
TrueThe thousands and hundreds match. The tens are 0 and 8, so 7,208 is less than 7,280.
8,590; 8,905; 9,0589,058 has 9 thousands, so it is largest. For 8,590 and 8,905, compare the hundreds: 5 is less than 9.
Level 2
5,640; 6,450; 6,5405,640 has 5 thousands, so it is smallest. For 6,450 and 6,540, compare the hundreds: 4 is less than 5.
8,170817 and 871 have only 3 digits, so they are smaller. For 8,017 and 8,170, compare the hundreds: 1 is more than 0.
True906 is less than 960 (tens 0 and 6). Both have 3 digits, so 6,009 is larger than both.
5,2995,190 is less than 5,280 and 5,320 is more than 5,310. Only 5,299 fits.
MayBoth have 7 thousands. Compare the hundreds: 2 is more than 1, so May has more.
1,098; 1,089; 986; 908The 4-digit numbers are larger. Compare 1,098 and 1,089 by the tens: 9 is more than 8. Then 986 is more than 908.
4,037All have 4 thousands. 4,037 and 4,073 have 0 hundreds. Compare the tens: 3 is less than 7.
2,305In order the numbers are 2,053, 2,305, 2,350 and 2,530. The second smallest is 2,305.
5,9906,010 is not below 6,000. Of the rest, 5,990 has the most hundreds (9) and then the most tens.
999; 9,099; 9,999; 10,000999 has 3 digits and 10,000 has 5 digits. For 9,099 and 9,999, compare the hundreds: 0 is less than 9.
Level 3
2,805; 2,580; 2,085; 2,058All have 2 thousands. Compare hundreds: 8, 0, 5, 0. Then break the tie between 2,058 and 2,085 with the tens.
3,045; 3,405; 3,450Write each as a number: 3,045, 3,405 and 3,450. Then compare the hundreds: 0, 4 and 4, and the tens: 0 and 5.
3,280Largest first the numbers are 3,820, 3,280, 3,208 and 3,082. The second is 3,280.
4,397; 4,398; 4,399; 4,400Count on from 4,396 and stop before 4,401. The numbers are 4,397, 4,398, 4,399 and 4,400.
7,853The number must be 7,850 to 7,860, so the thousands and hundreds are 7 and 8. With 5 tens and 3 ones it is 7,853.
6,701Every number from 6,601 to 6,699 has two 6s, and 6,700 has two 0s. 6,701 uses the digits 6, 7, 0 and 1, all different.
Expert
6,010; 6,020; 6,021The number is 6,000 to 6,029, so the tens digit is 0, 1 or 2. Tens 0 gives none, tens 1 gives 6,010, and tens 2 gives 6,020 and 6,021.
Model answer: Both have 4 thousands, so compare the hundreds: 0 is less than 1. So 4,099 is smaller. How many 9s there are does not matter.Look for: compare digits from the left, one place at a time. The thousands match, and 0 hundreds is less than 1 hundred, so 4,099 is smaller.
3,048; 3,084; 3,408; 3,480; 3,804; 3,840Choose the hundreds digit (0, 4 or 8), then swap the other two digits. That gives 3 choices, then 2 choices, so six numbers, listed in order of the hundreds digit.
Model answer: Always. The smallest 4-digit number is 1,000 and the largest 3-digit number is 999, so every 4-digit number is larger.Look for: the smallest 4-digit number (1,000) compared with the largest 3-digit number (999), or the idea that more digits means a larger number when there is no leading zero.
Worked example. XL is 50 − 10, V is 5 and I is 1. What number is XLVII? 1. Split the numeral into XL, V and II. 2. X is before L, so take away: XL = 50 − 10 = 40. 3. V = 5 and II = 2, so add them on: 40 + 5 + 2. So XLVII is the number 47.
Level 1
3III is three ones: 1 + 1 + 1 = 3.
8V is 5 and III is 3, so 5 + 3 = 8.
TrueX is the Roman numeral for 10, so the statement is true.
7V is 5 and II is 2, so 5 + 2 = 7.
9I is before X, so take away: 10 − 1 = 9.
12X is 10 and II is 2, so 10 + 2 = 12.
15X is 10 and V is 5, so 10 + 5 = 15.
14X is 10 and IV is 5 − 1 = 4, so 10 + 4 = 14.
FalseXX is 20 and IV is 4, so XXIV is 24, not 26.
19X is 10 and IX is 9, so 10 + 9 = 19.
50L is the Roman numeral for 50.
40X is before L, so take away: 50 − 10 = 40.
Level 2
28XX is 20 and VIII is 8, so 20 + 8 = 28.
42XL is 50 − 10 = 40 and II is 2, so 40 + 2 = 42.
XIX19 is 10 + 9, and 9 is IX, so 19 is XIX. XIV is 14 and XXI is 21.
FalseLX is 60 and IV is 4, so LXIV is 64, not 66.
90X is before C, so take away: 100 − 10 = 90.
77L is 50, XX is 20 and VII is 7, so 50 + 20 + 7 = 77.
94XC is 90 and IV is 4, so 90 + 4 = 94.
XLIX49 is 40 + 9, so XL then IX. I can only go before V or X, so IL is not allowed.
35XXXIV is 30 + 4 = 34, so the next chapter is 35.
99XC is 90 and IX is 9, so 90 + 9 = 99.
Level 3
100XLIII is 40 + 3 = 43 and LVII is 50 + 7 = 57. Then 43 + 57 = 100.
XXXIXXL is 40, so the door before it is 39 = 30 + 9, which is XXXIX. XLI is 41 and XXIX is 29.
1; 2; 3; 9; 10; 11; 12; 13; 19; 20I, II, III, IX, X, XI, XII, XIII, XIX and XX use no V or L. Numbers like IV, V, XIV and XV use a V.
4The single letters are I, V, X and L, which are the numbers 1, 5, 10 and 50. That makes 4 chapters.
8XLV is 45 and LII is 52. Counting 45 to 52 including both gives 8 tiles.
9599, 98, 97 and 96 need four or more letters (XCIX, XCVIII, XCVII, XCVI). 95 is XCV, which has three.
Expert
Model answer: No. I can only go before V or X, so IC is not allowed. 99 is 90 + 9, which is XC then IX: XCIX.Look for: I only before V or X; 99 split as 90 and 9; the correct numeral XCIX.
14; 16XIV is 10 + 4 = 14 and XVI is 10 + 6 = 16. Orders such as VIX, VXI, IVX and IXV break the rules.
Model answer: X always means 10 wherever it sits, so XX is 10 + 10. In 20 the 0 holds the ones place so the 2 means tens. Roman numerals have no 0 and no place value.Look for: each letter has a fixed value; XX is 10 + 10; our 0 is a placeholder that gives the 2 its place value; Roman numerals have neither.
38VIII has four letters, so XXXVIII has seven (XXX, V, III). No other number to 40 has seven: 28 has six, 37 has six.
Worked example. Round 7,825 to the nearest 10. 1. 7,825 lies between 7,820 and 7,830. 2. Look at the ones digit: 5. 3. A ones digit of 5 or more rounds up, so the answer is 7,830. So 7,825 rounded to the nearest 10 is 7,830.
Level 1
20; 3023 is 2 tens and 3 ones, so it sits between 20 and 30.
60; 7068 is 6 tens and 8 ones, so it sits between 60 and 70.
70Counting in tens: 60, 70, 80. The one just below 76 is 70.
True50 is less than 52 and 60 is more than 52, so the statement is true.
120Counting in tens: 100, 110, 120. The one just above 119 is 120.
290; 300295 is 29 tens and 5 ones, so it sits between 290 and 300.
3034 is 4 away from 30 and 6 away from 40, so it is nearer to 30.
60The ones digit is 2, which is less than 5, so round down to 60.
90The ones digit is 7, which is 5 or more, so round up to 90.
270273 is 3 away from 270 and 7 away from 280, so it is nearer to 270.
100The ones digit is 8, so round up. 98 is 2 away from 100.
260The ones digit is 5, and 5 rounds up, so 255 becomes 260.
Level 2
540The ones digit is 8, so round up from 530 to 540.
4,670The ones digit is 2, so the tens stay the same: 4,670.
6,410The ones digit is 0, so the number is already a multiple of 10 and stays 6,410.
FalseThe ones digit is 8, so round up to 5,100, not 5,090.
806806 has ones digit 6, so it rounds up to 810. 794 gives 790, 803 gives 800 and 818 gives 820.
2,370The ones digit is 5, and 5 rounds up, so 2,365 becomes 2,370.
10,000The ones digit is 6, so round up. The tens, hundreds and thousands all change, giving 10,000.
4,350 mThe ones digit is 3, so round down to 4,350 m.
2,990; 3,0102,994 has ones digit 4, so it rounds down to 2,990. 3,006 has ones digit 6, so it rounds up to 3,010.
3,603Numbers that round to 3,600 run from 3,595 to 3,604. Only 3,603 ends in 3.
Level 3
2,6501,284 rounds down to 1,280 and 1,367 rounds up to 1,370. Then 1,280 + 1,370 = 2,650.
735; 744735 rounds up to 740 because its ones digit is 5. 745 would round up to 750, so the largest is 744.
Yes7,455 rounds up to 7,460 because its ones digit is 5. 7,461 has ones digit 1, so it rounds down to 7,460. They match.
4,996; 5,0034,996 rounds up to 5,000 and 5,003 rounds down to 5,000. 5,008 rounds up to 5,010.
20Hugo gets 2,670 and Maya gets 2,690. Then 2,690 − 2,670 = 20.
5,9905,995 rounds up to 6,000. Then 6,000 − 10 = 5,990.
Expert
1; 2; 3; 4Rounding down happens for ones digits 1, 2, 3 and 4. A ones digit of 0 gives no change and 5 to 9 round up.
696; 698; 700; 702; 704The numbers that round to 700 run from 695 to 704. The ones that end in 0, 2, 4, 6 or 8 are 696, 698, 700, 702 and 704.
Model answer: No. The ones digit is 6, so 3,496 rounds up. 3,500 is 4 away and 3,490 is 6 away, so the answer is 3,500.Look for: the ones digit (6) decides, not the tens digit; 6 is 5 or more so round up; the answer is 3,500; 3,500 is nearer than 3,490.
Model answer: Always. Numbers that round to the same multiple of 10 sit in one block of 10 numbers, such as 65 to 74. The furthest apart two can be is 9.Look for: each multiple of 10 owns a block of 10 numbers (5 below to 4 above); a block has gaps of at most 9; an example such as 65 and 74.
Worked example. Round 8,653 to the nearest 100. 1. 8,653 lies between 8,600 and 8,700. 2. Look at the tens digit: 5. 3. 5 or more rounds up, so the number goes up to 8,700. So 8,653 rounded to the nearest 100 is 8,700.
Level 1
30The ones digit is 4, which is less than 5, so 34 rounds down to 30.
70The ones digit is 7, which is 5 or more, so 67 rounds up to 70.
9083 is more than 80 and less than 90, so it lies between 80 and 90.
TrueThe ones digit is 1, so 81 rounds down to 80. The statement is true.
370The ones digit is 2, so 372 rounds down to 370.
250The ones digit is 5, and 5 rounds up, so 245 rounds up to 250.
400330 is more than 300 and less than 400, so it lies between 300 and 400.
400The tens digit is 2, which is less than 5, so 420 rounds down to 400.
700The tens digit is 8, which is 5 or more, so 680 rounds up to 700.
200The tens digit is 3, which is less than 5, so 236 rounds down to 200.
800The tens digit is 5, and 5 rounds up, so 750 rounds up to 800.
900The tens digit is 6, so 862 rounds up to 900.
Level 2
4,600The tens digit is 3, which is less than 5, so 4,632 rounds down to 4,600.
7,500The tens digit is 8, so 7,481 rounds up to 7,500.
2,400The tens digit is 5 and the ones digit is 0. A 5 in the tens place rounds up, so 2,350 rounds to 2,400.
FalseThe tens digit is 4, so 8,249 rounds down to 8,200. The statement is false.
5,900The tens digit is 0, so 5,908 rounds down to 5,900. The 8 ones do not matter.
6,400The tens digit is 1, so 6,415 rounds down to 6,400.
7,000The tens digit is 6, so the hundreds go up from 9 to 10. That makes 7,000.
3,7923,792 has tens digit 9, so it rounds up to 3,800. 3,741 gives 3,700, and 3,850 and 3,911 give 3,900.
1,300 mThe tens digit is 9, so the hundreds go up from 2 to 3. 1,296 m rounds to 1,300 m.
10,000The tens digit is 5, so 9,950 rounds up. The hundreds go from 9 to 10, giving 10,000.
Level 3
7,0004,372 rounds to 4,400 and 2,640 rounds to 2,600. Then 4,400 + 2,600 = 7,000.
5,250The tens digit must be 5 or more to round up to 5,300, so the smallest is 5,250. 5,249 would round down to 5,200.
6,400The tens digit is 4, so 6,449 rounds down to 6,400. Kai looked at the ones digit 9 and rounded up by mistake.
450; 504; 540The possible numbers are 405, 450, 504 and 540, because 0 cannot be first. 405 rounds to 400. The other three round to 500.
1004,350 rounds up to 4,400 and 4,349 rounds down to 4,300. Then 4,400 − 4,300 = 100.
4,700 m4,262 + 400 = 4,662. The tens digit is 6, so 4,662 rounds up to 4,700 m.
Expert
Model answer: No. 3,449 has a tens digit of 4, so it rounds down to 3,400. Amira should look at the tens digit of 3,449 once, not round twice.Look for: the correct answer 3,400; the tens digit 4 decides; rounding twice gives a wrong answer because the first rounding creates a 5.
100The numbers run from 3,950 up to 4,049. That is 50 numbers from 3,950 to 3,999, plus 50 from 4,000 to 4,049, which makes 100.
6,850; 6,875; 6,900; 6,925Numbers from 6,850 to 6,949 round to 6,900. The multiples of 25 in that range are 6,850, 6,875, 6,900 and 6,925. 6,950 rounds up to 7,000.
Model answer: Sometimes. It changes when the hundreds digit is 9 and the tens digit is 5 or more, for example 2,975 rounds to 3,000. It stays the same for 2,425, which rounds to 2,400.Look for: the answer sometimes; hundreds digit 9 with tens digit 5 or more causes the change; one example that changes and one that does not.
Worked example. Round 1,462 to the nearest 10, 100 and 1,000. 1. Nearest 10: between 1,460 and 1,470. The ones digit is 2, so round down to 1,460. 2. Nearest 100: between 1,400 and 1,500. The tens digit is 6, so round up to 1,500. 3. Nearest 1,000: between 1,000 and 2,000. The hundreds digit is 4, so round down. So 1,462 rounds to 1,460, 1,500 and 1,000 to the nearest 10, 100 and 1,000.
Level 1
340The ones digit is 2, which is under 5, so round down to 340.
720The ones digit is 8, which is 5 or more, so round up to 720.
200The tens digit is 3, which is under 5, so round down to 200.
700The tens digit is 8, which is 5 or more, so round up to 700.
300The ones digit is 6, so round up. 296 is between 290 and 300, so the answer is 300.
800The tens digit is 5, and 5 rounds up, so the answer is 800.
4,000The hundreds digit is 2, which is under 5, so round down to 4,000.
8,000The hundreds digit is 8, which is 5 or more, so round up to 8,000.
2,000The hundreds digit is 3, which is under 5, so round down to 2,000.
5,0005,100 is 100 above 5,000 and 900 below 6,000, so it is nearer to 5,000.
6,000The hundreds digit is 6, which is 5 or more, so round up to 6,000.
5,000The hundreds digit is 9, so round up. 4,920 is between 4,000 and 5,000, so the answer is 5,000.
Level 2
5,000The hundreds digit is 2, which is under 5, so round down to 5,000.
8,0007,500 is exactly halfway between 7,000 and 8,000. A halfway number rounds up, so the answer is 8,000.
10,000The hundreds digit is 6, so round up. 9 thousands and 1 more thousand make 10,000.
0480 is between 0 and 1,000 and is under halfway (500), so round down to 0.
FalseThe hundreds digit is 4, which is under 5, so 2,499 rounds down to 2,000. The statement is false.
7,5017,501 has hundreds digit 5, so it rounds up to 8,000. 6,890 gives 7,000; 8,501 and 9,200 give 9,000.
4,000The hundreds digit is 8, so round up to 4,000.
6,380; 6,400; 6,000Ones digit 2: down to 6,380. Tens digit 8: up to 6,400. Hundreds digit 3: down to 6,000.
3,000The hundreds digit is 5, which is 5 or more, so round up to 3,000.
8,000; 9,0008,499 has hundreds digit 4, so it rounds down to 8,000. 8,500 has hundreds digit 5, so it rounds up to 9,000.
Level 3
9,0003,640 rounds up to 4,000 and 5,380 rounds down to 5,000. Then 4,000 + 5,000 = 9,000.
1,0002,486 rounds down to 2,000 and 2,513 rounds up to 3,000. The difference is 3,000 − 2,000 = 1,000.
Lucia onlyFemi is wrong: the tens digit is 6, so 9,962 rounds up to 10,000 to the nearest 100. Lucia is right: the hundreds digit is 9, so it rounds up to 10,000.
500; 1,000500 rounds up to 1,000 and 1,000 stays 1,000. 1,500 rounds up to 2,000 and 0 stays 0, so neither counts.
7,499Numbers from 6,500 to 7,499 round to 7,000. A hundreds digit of 4 leaves 7,400 to 7,499, so the largest is 7,499.
4,444The number must be between 3,500 and 4,499. The only number there with four equal digits is 4,444.
Expert
Model answer: No. 3,499 does round to 3,500 to the nearest 100, but for the nearest 1,000 only the hundreds digit matters. It is 4, so 3,499 rounds down to 3,000.Look for: rounding in two stages is wrong; round the original number once, using the hundreds digit 4; answer 3,000.
8,509; 8,590; 8,905; 8,950; 9,058; 9,085Numbers from 8,500 to 9,499 round to 9,000. Start with 8 and a hundreds digit of 5 or 9, or start with 9 and a hundreds digit of 0. That gives six numbers.
4,650; 4,651; 4,652; 4,653; 4,654To the nearest 10, the numbers are 4,645 to 4,654. To the nearest 100 they must be 4,650 or more, so 4,645 to 4,649 are out.
Model answer: Sometimes. 2,580 rounds up to 3,000 and up to 2,600. But 2,520 rounds up to 3,000 and down to 2,500.Look for: rounding up to 1,000 depends on the hundreds digit; rounding to 100 depends on the tens digit; one example that works and one that does not.
Worked example. Count back in 2s: 5, 3, 1, ___, ___. 1. Count back 2 from 1: 1, then 0, then −1. The next number is −1. 2. Count back 2 from −1: −1, then −2, then −3. The next number is −3. 3. Numbers below 0 are written with a − sign and get smaller as the count goes on. So the missing numbers are −1 and −3.
Level 1
4Each number is 4 less than the one before: 8 − 4 = 4.
200Each number is 50 less than the one before: 250 − 50 = 200.
16Each number is 8 less than the one before: 24 − 8 = 16.
700Each number is 100 less than the one before: 800 − 100 = 700.
48 − 4 = 4, and 4 − 4 = 0, so the gap is 4.
08 − 8 = 0, so the count reaches 0.
01 less than 1 is 0.
−1The number after 0 when counting back is −1, one below 0.
−2Counting back 2 from 0 gives −2.
−10Counting back 10 from 0 gives −10.
−5Counting back 5 from 0 gives −5, and 5 more back gives −10.
−1 °COne degree below 0 °C is −1 °C.
Level 2
1; −12 − 1 = 1, and 1 less than 0 is −1. Then −2 follows.
10; 0; −1020, 10, 0, −10: each is 10 less than the one before.
−55 − 10 = −5, and 10 less than −5 is −15.
−2; −60 − 2 gives −2. Then −4, and 2 further back is −6.
−2Count back 4 floors from 2: 1, 0, −1, −2. Floor −2 is below ground.
−17, 5, 3, 1, −1. Odd numbers skip 0 and land on −1.
−5 °CCount back in 5s: 10 °C, 5 °C, 0 °C, −5 °C.
Level 3
−30 m6 counts back in 5s from 0: −5, −10, −15, −20, −25, −30.
−5 °CCount back in 5s four times from 15 °C: 10, 5, 0, −5.
6Three jumps of 2 reach 0, so the start is 6: 6, 4, 2, 0.
−50The count goes 50, 25, 0, −25, −50. The first number below 0 is −25, so the second is −50.
−25Each jump is 15 back (20 to 5). From −10, a jump of 15 back lands on −25.
−10Kai says −5, −10, −15, ... and Zara says −10, −20, ... The first number both say is −10.
Expert
Model answer: Yes. The count goes 75, 50, 25, 0, −25, −50, so it lands on −50.Look for: writing out the count past 0 to −50, or saying 75 + 50 = 125 is a multiple of 25.
10; 20; 30; 40; 50In 10s the start must be 10, 20, 30, 40 or 50. Each of these also lands on −10 in 5s.
Model answer: No. After 1, counting back 2 gives −1, not 0. The count is 9, 7, 5, 3, 1, −1, −3.Look for: odd numbers never reach 0, so the next number after 1 is −1, and then −3.
8The numbers below 0 are −5, −10, −15, −20, −25, −30, −35 and −40. That is 8.
Worked example. Use cards 0, 4, 6 and 9 once each. Make the largest and smallest number. 1. Largest: the digits go in order from biggest, 9, 6, 4, 0, giving 9,640. 2. Smallest: 0 cannot be the thousands digit, so the thousands digit is 4. 3. The other digits go in order from smallest, 0, 6, 9, giving 4,069. So the largest number is 9,640 and the smallest number is 4,069.
Level 1
8In 583 the digits are 5 hundreds, 8 tens and 3 ones, so the tens digit is 8.
40The 4 is in the tens place, so it is worth 4 tens, which is 40.
2592 hundreds, 5 tens and 9 ones make 259.
350Taking 100 away changes the hundreds digit from 4 to 3, so the answer is 350.
705There are no tens, so 0 goes in the tens place: 705.
853Put the biggest digit in the hundreds place, then the next biggest: 8, 5, 3.
600The 6 is in the hundreds place, so it is worth 6 hundreds, which is 600.
7,2597 thousands, 2 hundreds, 5 tens and 9 ones make 7,259.
6,280Only the thousands digit changes, from 5 to 6, so the answer is 6,280.
TrueThe hundreds digit of 7,305 is 3, so the statement is true.
6,040There are no hundreds and no ones, so 0 goes in both places: 6,040.
4,208The thousands match, so compare the hundreds: 2 is more than 0, so 4,208 is greater.
Level 2
8,052The 0 holds the hundreds place, so the number is 8,052.
90The 9 is in the tens place, so it is worth 90. The 0s are placeholders.
5,9956,005 is 6,000 and 5 ones. Taking 10 away needs an exchange, giving 5,995.
0; −5Take away 5 each time: 5 − 5 = 0, then 0 − 5 = −5.
7,000The tens digit is 6, so round up. 6,900 becomes 7,000.
>4,000 + 500 + 6 is 4,506. Its hundreds digit 5 is more than 0, so it is greater than 4,056.
9,050The hundreds digit 9 goes up by 1, which makes 10 hundreds, so 8,950 becomes 9,050.
5,720A 7 worth 700 sits in the hundreds place. Only 5,720 has 7 hundreds.
539; 5,039; 5,309539 has 3 digits so it is smallest. Both others have 5 thousands, so compare hundreds: 0 then 3.
3,0580 cannot be the thousands digit, so start with 3. Then the rest go smallest first: 0, 5, 8.
Level 3
6,2701,000 more makes 6,370. Then 100 less makes 6,270.
4,050 mWork backwards. Undo the fall: 4,950 + 100 = 5,050. Undo the rise: 5,050 − 1,000 = 4,050.
6,8308 cannot be the thousands digit, or the number is 8,000 or more. Use 6, then the rest largest first: 8, 3, 0.
4,500; 4,600; 4,700; 4,800; 4,900; 5,000; 5,100; 5,200; 5,300; 5,400The numbers run from 4,500 (rounds up) to 5,499. The multiples of 100 in that range are listed. 5,500 rounds up to 6,000.
−25Count back in 25s: 100, 75, 50, 25, 0, then −25.
1,006; 3,004; 5,002; 7,000Only the thousands and ones digits are not 0, and they add to 7. The ones digit must be even: 0, 2, 4 or 6.
Expert
Model answer: The place of each digit matters. In 3,040 the 3 is worth 3,000, but in 340 it is worth 300. The 0 in 3,040 holds the hundreds place, so the numbers are not equal.Look for: digit value depends on place; 3 is 3,000 against 300; 0 is a placeholder; 3,040 is bigger.
4,123; 4,213; 4,231; 4,321The ones digit must be 1 or 3. Ones 1 gives 4,231 and 4,321. Ones 3 gives 4,123 and 4,213.
7,450; 7,470; 7,490To round to 7,500 and 7,000, the number is from 7,450 to 7,499. The ones digit is 0 and the tens digit is odd (5, 7 or 9).
Model answer: Sometimes. It gets bigger when the hundreds digit is more than the thousands digit, as 2,500 becomes 5,200. It gets smaller when the hundreds digit is less, as 6,100 becomes 1,600.Look for: sometimes. Bigger if the hundreds digit is more than the thousands digit (2,500 becomes 5,200). Smaller if less (6,100 becomes 1,600). Same if equal.
Worked example. Calculate 5,214 + 372 in columns. 1. Line up the ones. Ones: 4 + 2 = 6. 2. Tens: 1 + 7 = 8. Hundreds: 2 + 3 = 5. 3. Thousands: only 5 is left, so 5 is written in the thousands column. So 5,214 + 372 = 5,586.
4,733Find the number that adds to 3,126 to make 7,859, column by column: ones 6 + 3 = 9, tens 2 + 3 = 5, hundreds 1 + 7 = 8, thousands 3 + 4 = 7.
7,483Each column of the two numbers can total at most 9: ones 6 + 3, tens 1 + 8, hundreds 5 + 4, thousands 2 + 7. The answer is 7,483.
4,376Saturday is 1,121 + 2,134 = 3,255. Both days: 1,121 + 3,255 = 4,376.
2,415; 4,274Try pairs. Only 2,415 + 4,274 = 6,689. The other pairs give different totals.
Expert
Model answer: The 2 of 235 stands for 2 hundreds, but under the 7 it would be added as 2 thousands. Digits must be lined up by place value: ones under ones, tens under tens.Look for: the digits are in the wrong columns; 235 has 2 hundreds, 3 tens and 5 ones; each digit must sit in the column of its own place value.
11; 22; 33; 44The ones digit must be at most 4, since 5 + 5 makes an exchange. Tens digit 4 + 4 = 8 is fine. So 11, 22, 33 and 44 work, but 55 does not.
15With no exchanges, the digits of the answer total the same as the digits of both numbers together. So 29 − 14 = 15.
$3,154Count on from 3,846: 54 to reach 3,900, then 3,100 to reach 7,000. 54 + 3,100 = 3,154.
1,836Put the biggest digits in the hundreds: 9 and 8. Next 7 and 6 in the tens, 4 and 2 in the ones. For example 974 + 862 = 1,836.
100; 200; 300; 400; 500; 600; 7006,375 + 700 = 7,075, which is under 7,100. 6,375 + 800 = 7,175, which is too big, so 800 and 900 are out.
9,347Ones: 8 + 9 = 17. Tens: 6 + 7 + 1 = 14. Hundreds: 4 + 8 + 1 = 13. Thousands: 3 + 5 + 1 = 9. Ben left out the last carried 1.
Expert
6,543No carry means each pair of digits totals 9 or less: ones 6 + 3, tens 5 + 4, hundreds 4 + 5, thousands 3 + 6. The largest digits that fit give 6,543.
53; 71The ones digits must total 10 or more: 3 and 8 (13 + 58 = 71) or 5 and 8 (15 + 38 = 53). 1 with 3, 5 or 8 never reaches 10, and 3 with 5 makes only 8.
Model answer: Always true. The biggest digits are 9 and 9, and with a carried 1 they make 19, so at most 1 ten, hundred or thousand is carried.Look for: biggest possible column total is 9 + 9 + 1 = 19; 19 is less than 20, so the carry is never 2 or more.
Worked example. Calculate 5,687 − 342. 1. Line up the digits by place: 342 has no thousands digit. 2. Ones: 7 − 2 = 5. Tens: 8 − 4 = 4. 3. Hundreds: 6 − 3 = 3. 4. Thousands: nothing is taken away, so the 5 stays. So 5,687 − 342 = 5,345.
319,764 − 3,521 = 6,243 and 8,695 − 2,483 = 6,212. Then 6,243 − 6,212 = 31.
1; 2; 3; 4; 5; 6Only the ones column matters. The ones digit is 6, so any digit from 1 to 6 can be taken without an exchange.
Expert
Model answer: Yusuf lined 324 up from the left. The ones digit 4 must go under the ones digit 6, so the answer is 7,262.Look for: digits must be lined up by place value from the right, ones under ones. Correct sum: 6 − 4 = 2, 8 − 2 = 6, 5 − 3 = 2, 7 stays, so 7,262.
8,000; 8,665Each digit taken must be no more than the digit above it. Largest to take: 765, giving 8,000. Smallest 3-digit number to take: 100, giving 8,665.
103; 112; 121; 202; 211; 220; 301; 310Hundreds digit 1 to 3, tens digit 0 to 2, ones digit 0 to 5. List by hundreds digit: 103, 112, 121; 202, 211, 220; 301, 310.
Worked example. Calculate 6,352 − 2,481. 1. Ones: 2 − 1 = 1. Tens: 5 is less than 8, so exchange 1 hundred for 10 tens. 2. Tens: 15 − 8 = 7. Hundreds: 2 is left, and 2 is less than 4, so exchange 1 thousand. 3. Hundreds: 12 − 4 = 8. Thousands: 5 are left, and 5 − 2 = 3. So 6,352 − 2,481 = 3,871.
Level 1
45Ones: 3 is less than 8, so exchange 1 ten: 13 − 8 = 5. Tens: 6 − 2 = 4.
348Ones: 15 − 7 = 8. Tens: 2 is less than 8, so 12 − 8 = 4. Hundreds: 5 − 2 = 3.
TrueOnes: 12 − 5 = 7. Tens: 0 is less than 7, so 10 − 7 = 3. Hundreds: 8 − 4 = 4. So 437 is correct.
2,367Tens: 5 is less than 9, so exchange 1 hundred: 15 − 9 = 6. Ones: 8 − 1 = 7. Hundreds: 6 − 3 = 3. Thousands: 4 − 2 = 2.
3,236Ones: 4 is less than 8, so 14 − 8 = 6. Tens: 5 − 2 = 3. Hundreds: 3 − 1 = 2. Thousands: 5 − 2 = 3.
6,473Tens: 4 is less than 7, so 14 − 7 = 7. Hundreds: 7 − 3 = 4. The 6 thousands stay.
3,722Hundreds: 2 is less than 5, so exchange 1 thousand: 12 − 5 = 7. Thousands: 6 − 3 = 3.
Level 2
4,364Tens: 4 is less than 8, so 14 − 8 = 6. Hundreds: 6 − 3 = 3. Ones 4 and thousands 4 follow.
5,183Ones: 11 − 8 = 3. Tens: 5 is less than 7, so 15 − 7 = 8. Hundreds: 3 − 2 = 1. Thousands: 8 − 3 = 5.
FalseIn columns: 12 − 4 = 8, 8 − 7 = 1, 16 − 8 = 8, 4 − 1 = 3. The answer is 3,818, not 3,828.
5,455 kmSubtract 1,897 from 7,352 in columns. Every column needs an exchange: 12 − 7 = 5, 14 − 9 = 5, 12 − 8 = 4, 6 − 1 = 5.
4,576The missing number is the whole minus the part: 9,134 − 4,558. Columns: 14 − 8 = 6, 12 − 5 = 7, 10 − 5 = 5, 8 − 4 = 4.
6,657Ones: 13 − 6 = 7. Tens: 3 is less than 8, so 13 − 8 = 5. Hundreds: 1 is less than 5, so 11 − 5 = 6. Thousands: 6 stay.
2,659Find the difference: 6,418 − 3,759. Ones: 18 − 9 = 9. Tens: 0 is less than 5, so 10 − 5 = 5. Hundreds: 13 − 7 = 6. Thousands: 5 − 3 = 2.
3,847Ones: 15 − 8 = 7. Tens: 0 is less than 6, so 10 − 6 = 4. Hundreds: 11 − 3 = 8. Thousands: 7 − 4 = 3.
Level 3
3,025Two steps: 9,468 − 2,759 = 6,709, then 6,709 − 3,684 = 3,025. Each step needs exchanges.
7,444 gWorking backwards, add what was used to what is left: 2,865 + 4,579 = 7,444.
7,263The largest number is 9,742 and the smallest is 2,479. Then 9,742 − 2,479 = 7,263, with two exchanges.
4,007Match C had 5,478 − 1,359 = 4,119 fans. Then 8,126 − 4,119 = 4,007.
1,367Hana took the smaller digit from the larger digit in each column. With exchanges: 13 − 6 = 7, 13 − 7 = 6, 11 − 8 = 3, 4 − 3 = 1.
Expert
Model answer: A digit cannot be taken from a smaller digit above it, so 1 must be exchanged from the next column. The correct answer is 4,566, not 5,434.Look for: top number minus bottom number, never the other way round; exchanging 1 from the column to the left; the correct answer 4,566.
6,315; 6,335; 6,375Every ones digit is 5, less than 8, and every hundreds digit is 3, less than 4, so those always need an exchange. The tens need one when the tens digit less 1 is below 7, so for 1, 3 and 7 but not 8 or 9.
3,087The first two answers are both 3,087. Each number goes up by 1,111, so the difference stays the same.
3,454Subtract 7,040 − 3,586 in columns, exchanging across the zero in the hundreds place. Check: 3,454 + 3,586 = 7,040.
5,625 gSubtract 8,000 − 2,375 in columns, exchanging across three zeros. Check: 5,625 + 2,375 = 8,000.
3,1577,002 − 3,845 = 3,157. The answer 4,157 comes from forgetting the exchange from the thousands.
Level 3
3,153Subtract 7,000 − 3,847 in columns, exchanging across three zeros. Check: 3,153 + 3,847 = 7,000.
5,644Subtract 10,000 − 4,356 in columns, exchanging across four zeros. Check: 5,644 + 4,356 = 10,000.
5,503Working backwards: the start is the tiles sold plus the tiles left, 2,468 + 3,035 = 5,503. Check: 5,503 − 2,468 = 3,035.
3,008 kmAdd the two stages: 2,485 + 3,507 = 5,992. Then subtract 9,000 − 5,992 = 3,008 across the zeros.
5,661The largest is 9,740. The smallest is 4,079, because 0 cannot be the thousands digit. Then 9,740 − 4,079 = 5,661.
Expert
Model answer: No. Hana took the smaller digit from the larger in each column. The ones need an exchange because 2 is less than 7. The correct answer is 3,655.Look for: Hana is not correct; the ones digit 2 is smaller than 7, so an exchange is needed across the zeros; 5,002 − 1,347 = 3,655; adding 4,345 + 1,347 does not give 5,002.
667The answer is between 3,001 and 3,900, so it must be 3,333. Then 4,000 − 3,333 = 667.
701The answers are 1, 101 and 201: each time the number taken away drops by 100, the answer rises by 100. Five steps from 1,799 to 1,299 gives 201 + 500 = 701. Check: 701 + 1,299 = 2,000.
Worked example. Estimate 3,642 + 1,258 by rounding each to the nearest 100. 1. Round 3,642: the tens digit is 4, so round down to 3,600. 2. Round 1,258: the tens digit is 5, so 5 or more rounds up to 1,300. 3. Add the rounded numbers: 3,600 + 1,300 = 4,900. So the estimate of 3,642 + 1,258 is 4,900.
Level 1
6041 rounds to 40 and 19 rounds to 20. 40 + 20 = 60.
3072 rounds to 70 and 38 rounds to 40. 70 − 40 = 30.
8052 rounds to 50 and 28 rounds to 30. 50 + 30 = 80.
6096 rounds up to 100 and 41 rounds to 40. 100 − 40 = 60.
200148 rounds to 150 and 52 rounds to 50. 150 + 50 = 200.
170384 rounds to 380 and 213 rounds to 210. 380 − 210 = 170.
1,400920 rounds to 900 and 480 rounds to 500. 900 + 500 = 1,400.
500760 rounds to 800 and 270 rounds to 300. 800 − 300 = 500.
900630 rounds to 600 and 280 rounds to 300. 600 + 300 = 900.
7,0002,900 rounds to 3,000 and 4,100 rounds to 4,000. 3,000 + 4,000 = 7,000.
True2,600 rounds to 3,000 and 2,400 rounds to 2,000. 3,000 + 2,000 = 5,000, so it is true.
3,0004,720 rounds up to 5,000 and 1,560 rounds up to 2,000. 5,000 − 2,000 = 3,000.
Level 2
5,0003,470 rounds to 3,000 and 2,230 rounds to 2,000. 3,000 + 2,000 = 5,000.
4,6006,840 rounds to 6,800 and 2,190 rounds to 2,200. 6,800 − 2,200 = 4,600.
7,0004,320 rounds to 4,000 and 2,690 rounds to 3,000. 4,000 + 3,000 = 7,000.
8,0003,480 rounds down to 3,000 and 4,620 rounds up to 5,000. 3,000 + 5,000 = 8,000.
490365 ends in 5, so it rounds up to 370. 124 rounds to 120. 370 + 120 = 490.
3,8002,450 has 5 tens, so it rounds up to 2,500. 1,320 rounds to 1,300. 2,500 + 1,300 = 3,800.
No6,812 + 2,975 is about 7,000 + 3,000 = 10,000. Ben's answer of 7,787 is much too small.
5,6009,951 rounds up to 10,000 and 4,449 rounds down to 4,400. 10,000 − 4,400 = 5,600.
8,0002,385 rounds to 2,000, 4,712 to 5,000 and 1,296 to 1,000. 2,000 + 5,000 + 1,000 = 8,000.
4,0001,850 has hundreds digit 8, so it rounds up to 2,000. 2,350 has hundreds digit 3, so it rounds down to 2,000. 2,000 + 2,000 = 4,000.
Level 3
1,520 + 2,4601,520 + 2,460 rounds to 2,000 + 2,000 = 4,000. The first sum gives 1,000 + 2,000 = 3,000 and the third gives 3,000 + 3,000 = 6,000.
200Luca: 4,800 + 2,400 = 7,200. Omar: 5,000 + 2,000 = 7,000. The difference is 7,200 − 7,000 = 200.
5,000Round each number to the nearest 1,000: 10,000, 3,000 and 2,000. Then 10,000 − 3,000 − 2,000 = 5,000.
$700$3,260 rounds to $3,300 and $1,540 rounds to $1,500, so about $4,800 is raised. $5,500 − $4,800 = $700.
55; 56; 57; 58; 59; 60; 61; 62; 63; 6437 rounds to 40, so the other number rounds to 60. Numbers from 55 (5 rounds up) to 64 (4 rounds down) round to 60.
1,0503,620 rounds to 3,600, so the other rounds to 1,100. The smallest number that rounds up to 1,100 is 1,050.
Expert
Model answer: Hugo looked at the ones digit. The digit that decides is the hundreds digit, 4, so 3,496 rounds down to 3,000.Look for: the deciding digit for the nearest 1,000 is the hundreds digit (4, which is less than 5); the answer is 3,000.
50Rounding to 2,500 needs 2,450 to 2,549, and rounding to 3,000 needs 2,500 or more. So 2,500 to 2,549 works: 50 numbers.
5; 6; 7; 8; 95,350 rounds to 5,000, so the other number must round to 5,000. That needs a hundreds digit of 5 or more.
Model answer: Always. Rounding moves each number by at most 500, so the two rounded numbers together move the sum by at most 1,000.Look for: each number changes by 500 or less when rounded to the nearest 1,000, so the sum changes by at most 500 + 500 = 1,000.
7,847Work backwards with the inverse: 5,379 + 2,468 = 7,847. Check: 7,847 − 2,468 = 5,379.
2,6549,000 − 6,347 = 2,653, which gives a total of exactly 9,000. One more, 2,654, gives 9,001, which is more than 9,000.
3,856Check: 3,956 + 2,359 = 6,315, not 6,215. The correct answer is 6,215 − 2,359 = 3,856.
5,036; 5,2088,000 − 5,208 = 2,792 and 8,000 − 5,036 = 2,964, both below 3,000. The other two give 3,085 and 3,208.
Expert
Model answer: Yes. 5,743 − 2,129 = 3,614 and 5,743 − 3,614 = 2,129. Adding can be done in either order, so taking away either part leaves the other part.Look for: both subtractions tried and shown to work; a reason such as addition being the same in either order, so either part can be removed.
4,000; 5,500Take the extra 1,500 away: 9,500 − 1,500 = 8,000, then share equally: 4,000 each. The larger is 4,000 + 1,500 = 5,500. Check: 4,000 + 5,500 = 9,500.
Model answer: No. The check should give 7,000 but gives 7,100, so the subtraction is wrong. The correct answer is 4,282.Look for: the check total 7,100 does not match 7,000, so the answer is wrong. The error is in the hundreds; 7,000 − 2,718 = 4,282, and 4,282 + 2,718 = 7,000.
Worked example. 3,486 tickets are sold, then 2,759 more. How many more are needed for 7,000? 1. Add to find the total sold: 3,486 + 2,759 = 6,245, exchanging in three columns. 2. Subtract from the target to find the gap: 7,000 − 6,245. 3. Exchange across the zeros: ones 10 − 5 = 5, tens 9 − 4 = 5, hundreds 9 − 2 = 7. 4. Thousands: 6 − 6 = 0, so the gap is 755. So 755 more tickets are needed to reach 7,000.
1,670Add the three nights: 2,318 + 1,927 + 3,085 = 7,330. Then 9,000 − 7,330 = 1,670.
3,940Work backwards: undo getting 940 by subtracting, undo giving away 1,265 by adding. 3,615 − 940 = 2,675, then 2,675 + 1,265 = 3,940.
5,6234,376 + 5,624 = 10,000, which is not less than 10,000. One less gives 5,623, and 4,376 + 5,623 = 9,999.
335 mBilal walked 3,470 + 2,765 = 6,235 m. Then 6,235 − 5,900 = 335 m.
6,343Ones 13, tens 6 + 7 + 1 = 14, hundreds 8 + 4 + 1 = 13, thousands 3 + 2 + 1 = 6. Rosa forgot to carry 1 into the hundreds.
Expert
5,250; 2,750Take the extra 2,500 off the total: 8,000 − 2,500 = 5,500. Share it equally: 2,750 each for the smaller. The larger is 2,750 + 2,500 = 5,250.
5,200; 5,750Check all six pairs. Totals: 3,950, 5,200, 3,050, 6,650, 4,500 and 5,750. Only 5,200 and 5,750 are between 5,100 and 5,800.
Model answer: Yes. 2,998 is 2 less than 3,000, so taking away 3,000 removes 2 too many. Adding 2 back gives 2,004 + 2 = 2,006, which is the correct answer.Look for: 2,998 is 2 less than 3,000; subtracting 3,000 takes away too much; adding 2 corrects it; result 2,006.
Worked example. Find the area of the shape by counting the squares in each row. 1. The area is the number of squares the shape covers. 2. Bottom two rows: 6 squares in each row. 3. Top two rows: 4 squares in each row. 4. Add the rows: 6 + 6 + 4 + 4 = 20. So the area of the shape is 20 squares.
Level 1
102 rows of 5 is 5 + 5 = 10.
124 rows of 3 is 3 + 3 + 3 + 3 = 12.
186 threes make 18.
246 rows of 4 is 6 × 4 = 24.
9Think 3 × 9 = 27, so the missing number is 9.
735 ÷ 5 = 7 because 5 × 7 = 35.
6Add the two rows: 4 + 2 = 6 squares.
15The area is the number of squares covered: 3 × 5 = 15.
BThe shape that covers more squares has the larger area. 9 is more than 7, so B.
14Adding 3 squares makes the area 11 + 3 = 14 squares.
13Add the squares covered by the two shapes: 6 + 7 = 13.
91 row of 9 squares covers 9 squares.
Level 2
284 rows of 7 squares: 4 × 7 = 28.
11Top row 5, then 2 in each of the three rows below: 5 + 2 + 2 + 2 = 11.
4Share 24 squares into 6 equal rows: 24 ÷ 6 = 4.
True2 rows of 8 is 16 squares and 4 rows of 4 is 16 squares, so they match.
233 rows of 6 is 18 tiles. Then 18 + 5 = 23.
193 rows of 4 is 12 squares. Add the row of 7: 12 + 7 = 19.
RCompare 18, 21 and 16. The smallest is 16, which is shape R.
2617 + 9 = 26 squares.
4X covers 24 squares and Y covers 20, so X has 24 − 20 = 4 more.
21The hole takes away 4 squares: 25 − 4 = 21.
Level 3
395 rows of 9 is 45 tiles. Taking away 6 leaves 45 − 6 = 39.
4Count the squares by rows: 6 + 5 + 3 + 2 = 16. 3 packs give 15, which is not enough, so 4 packs.
9Before the 4 extra tiles there were 58 − 4 = 54 tiles. 54 ÷ 6 = 9 in each row.
1; 2; 3; 4; 6; 12Rows of 1, 2, 3, 4, 6 or 12 squares all fill exactly 12 squares: 12 × 1, 6 × 2, 4 × 3 and so on.
17The grid has 4 rows of 7, which is 28 squares. 11 are shaded, so 28 − 11 = 17 are not.
CB covers 3 × 5 = 15 squares and C covers 15 − 2 = 13. 13 is less than 14 and 15, so C.
Expert
179 × 9 = 81 and 8 × 8 = 64, so 81 − 64 = 17. Each new square adds the next odd number.
Model answer: No. A rectangle of 2 rows of 6 squares also covers 12 squares, and a shape with corners cut out can too. Area only counts the squares covered, not the shape.Look for: area is the number of squares covered; different shapes can have the same area; an example such as 2 rows of 6 or an L-shape of 12 squares.
36; 44Areas are multiples of 4 between 30 and 45: 32, 36, 40, 44. A row of 8 or 10 squares is even, so keep 36 (rows of 9) and 44 (rows of 11).
55Step n has n squares, so add 1 + 2 + 3 + ... + 10. Pair them: 1 + 10, 2 + 9, 3 + 8, 4 + 7 and 5 + 6 make five 11s, so 55.
12Each number is 3 more than the one before: 9 + 3 = 12.
927 ÷ 3 = 9, so 3 × 9 = 27.
1212 threes make 36, so 36 ÷ 3 = 12.
12Two sixes are 6 + 6 = 12.
30Five sixes are 30 (half of 6 × 10 = 60).
24Each number is 6 more than the one before: 18 + 6 = 24.
44 sixes make 24, so 24 ÷ 6 = 4.
366 × 6 is double 6 × 3, and 18 + 18 = 36.
488 boxes of 6 eggs is 8 × 6 = 48.
Level 2
488 × 3 = 24, and double 24 is 48.
99 × 6 = 54, so 54 ÷ 6 = 9.
1272 ÷ 6 = 12, so 12 × 6 = 72.
30; 36Add 6 each time: 24 + 6 = 30, then 30 + 6 = 36.
False6 × 9 = 54, not 56.
549 packs of 6 pencils is 9 × 6 = 54.
6010 × 6 = 60. The others are not multiples of 6.
27 sixes is 5 sixes plus 2 sixes.
1166 ÷ 6 = 11, because 11 × 6 = 66.
266 × 6 = 36 cards, then 36 − 10 = 26.
Level 3
84 × 12 = 48 eggs, then 48 ÷ 6 = 8 days.
8Work backwards: 53 − 5 = 48, then 48 ÷ 6 = 8.
5436 is marked. Count on in 6s: 42, 48, 54.
6312 × 6 = 72 paints, then 72 − 9 = 63.
42; 48; 54; 60; 66The 6 table goes 36, 42, 48, 54, 60, 66, 72. Keep those between 40 and 70.
6Isla has 9 × 6 = 54 and Dev has 6 × 8 = 48. The difference is 6.
Expert
18; 36; 54; 72List 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72 and add the digits of each. 18, 36, 54 and 72 give 9.
Model answer: Yes. 3 × 8 = 24 and double 24 is 48, which is 6 × 8. This works for any number because 6 is double 3.Look for: Kai is correct; 3 × 8 = 24, doubled is 48 = 6 × 8; reason that 6 is double 3, so it always works.
12; 24; 36; 48; 60; 72List the 6 table: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72. Keep those also in the 4 table: 12, 24, 36, 48, 60, 72.
Model answer: No. Counting in 6s goes 96, then 102, so 100 is skipped. 100 is not in the 6 times table.Look for: not correct; 16 × 6 = 96 and 17 × 6 = 102 (or counting on from 90: 96, 102), so 100 is missed.
Worked example. Use 10 × 6 to calculate 9 × 6. 1. Use the 10 table: 10 × 6 = 60. 2. 9 × 6 is one 6 less than 10 × 6. 3. Take away one 6: 60 − 6 = 54. So 9 × 6 = 54, and the matching division fact is 54 ÷ 9 = 6.
Level 1
505 tens make 50.
808 tens make 80.
12Count in 3s four times: 3, 6, 9, 12.
9Dividing by 10 moves each digit one place right: 90 ÷ 10 = 9.
83 × 8 = 24, so 24 ÷ 3 = 8.
True3 × 9 = 27, so the statement is true.
18Double 9: 9 + 9 = 18.
455 × 9 is half of 10 × 9 = 90, so 45.
36Double 9 twice: 18, then 36.
39, 18, 27 is three nines, so 27 ÷ 9 = 3.
77 × 9 = 63, so the missing number is 7.
False9 × 8 = 72, not 82, so the statement is false.
Level 2
637 × 10 = 70, take away 7 to get 63.
44 × 9 = 36, so 36 ÷ 9 = 4.
810 × 9 = 90, 90 − 18 = 72, so 8 nines.
546 × 9 = 54. The digits of 54 add to 9; 47 and 65 do not.
99 × 9 = 81, so 81 ÷ 9 = 9 rows.
99 × 9 is one 9 less than 10 × 9, so take away 9 from 90 to get 81.
False10 × 12 = 120, take away 12 to get 108. So 106 is wrong.
108The missing number is 12 × 9 = 108.
9911 × 9 = 10 × 9 + 9 = 90 + 9 = 99.
11912 × 9 = 108, then 108 + 11 = 119.
Level 3
9312 × 9 = 108. Then 108 − 15 = 93.
11Work backwards: 99 ÷ 9 = 11.
10811 × 9 = 99 is too small. 12 × 9 = 108 is the first over 100.
27; 63The 9s up to 90: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90. Only 27 and 63 end in 7 or 3.
$2111 × 9 = 99. Then $120 − $99 = $21.
109 × 6 = 54 and 9 × 4 = 36. Then 54 + 36 = 90, and 90 = 9 × 10.
Expert
Model answer: Yes. Each time 9 is added, the tens digit goes up by 1 and the ones digit goes down by 1, so the digits always add up to 9.Look for: a yes; a check of some or all (18, 27, 36 and so on); a reason such as adding 10 and taking away 1.
3; 6; 9; 126 × 3 = 18, 6 × 6 = 36, 6 × 9 = 54 and 6 × 12 = 72 are all in the 9 table. The other numbers are not.
27; 72The digits must be 2 and 7, giving 27 or 72. Both are in the 9 table: 3 × 9 and 8 × 9.
Model answer: Femi took away 9, but 9 × 7 is one 7 less than 10 × 7. The correct answer is 70 − 7 = 63.Look for: one seven is taken away, not a nine; the correct answer 63.
Worked example. Find 7 × 9 by splitting 7 into 5 and 2. 1. Split 7 rows of 9 into 5 rows of 9 and 2 rows of 9. 2. 5 × 9 = 45 and 2 × 9 = 18. 3. Add the two parts: 45 + 18 = 63. So 7 × 9 = 63.
Level 1
20Count in 5s four times: 5, 10, 15, 20.
12Double 6 is 12.
66 × 5 = 30, so 30 ÷ 5 = 6.
7Half of 14 is 7.
99 × 5 = 45, so the missing number is 9.
TrueDouble 9 is 18, so the statement is true.
14Double 7 is 14.
21Add 7 to 14 to get 21. Then 21 + 7 = 28.
35Count in 5s seven times, or use 5 × 7 = 35.
70Ten sevens make 70.
33 × 7 = 21, so 21 ÷ 7 = 3.
28Four weeks of 7 days is 4 × 7 = 28.
Level 2
425 × 6 = 30 and 2 × 6 = 12, so 7 × 6 = 42.
44 × 7 = 28, so 28 ÷ 7 = 4.
117 × 11 = 77, so the missing number is 11.
False7 × 8 = 56, not 54, so the statement is false.
568 rows of 7 is 8 × 7 = 56.
84The shaded 5 rows hold 5 × 12 = 60 and the other 2 rows hold 24. Together 60 + 24 = 84.
56; 63; 70; 77Count in 7s: 49, 56, 63, 70, 77, 84. Only 56, 63, 70 and 77 are between 50 and 80.
65 strings give 35, which is too few. 6 strings give 42, which is enough.
21Uma has 12 × 7 = 84 and Kai has 9 × 7 = 63, so 84 − 63 = 21.
407 − 2 = 5 rows are not shaded, and 5 × 8 = 40.
Expert
Model answer: Yes. 5 × 6 = 30 and 2 × 6 = 12, and 30 + 12 = 42, which is 7 × 6. Splitting 7 into 5 and 2 splits the 7 rows into two groups.Look for: Femi is correct; both sides equal 42; 7 sixes can be split into 5 sixes and 2 sixes.
35; 70Multiples of 7 ending in 0 or 5: 35 and 70. The next, 105, is over 80.
126The number is in both tables. Multiples of 7 between 100 and 144 are 105, 112, 119, 126, 133 and 140. Only 126 is also 6 × 21.
Model answer: Sometimes. 7, 21 and 35 are odd, but 14, 28 and 42 are even. An odd number of sevens is odd and an even number of sevens is even.Look for: sometimes; one odd example and one even example; odd × 7 is odd, even × 7 is even.
Worked example. Calculate 12 × 11. 1. Split 11 into 10 and 1: 12 × 10 = 120. 2. 12 × 1 = 12, and 120 + 12 = 132. 3. Check with the division fact: 132 ÷ 11 = 12. 4. 12 is two digits, so the answer does not repeat a digit. So 12 × 11 = 132.
Level 1
40Multiplying by 10 moves each digit one place left: 4 becomes 40.
707 tens is 70.
660 is 6 tens, so 60 ÷ 10 = 6.
909 tens is 90.
550 is 5 tens, so the missing number is 5.
11110 is 11 tens, so 110 ÷ 10 = 11.
2211 + 11 = 22.
555 × 10 = 50 and 5 × 1 = 5, so 55.
333 × 11 = 33, the digit 3 twice.
44 × 11 = 44, so 44 ÷ 11 = 4.
False7 × 11 = 77, not 78, so the statement is false.
999 × 10 = 90 and 9 × 1 = 9, so 99.
Level 2
888 × 10 = 80 and 8 × 1 = 8, so 88.
77 × 11 = 77, so the missing number is 7.
1111 × 11 = 121, so 121 ÷ 11 = 11.
False11 × 10 = 110 and 11 × 1 = 11, so 11 × 11 = 121, not 111.
999 teams of 11 players: 9 × 11 = 99.
711 is 10 plus 1, so 7 × 11 is 7 × 10 plus 7 × 1, which is 7.
666 × 11 = 66. Neither 54 nor 75 is a multiple of 11.
11011 × 10 = 110. The repeated-digit pattern stops here.
888 ÷ 11 = 8, because 8 × 11 = 88.
1329 × 11 = 99 and 3 × 11 = 33. Then 99 + 33 = 132.
Level 3
1049 × 11 = 99 seats in the rows. Then 99 + 5 = 104.
8Take away the 4 extra trees: 92 − 4 = 88. Then 88 ÷ 11 = 8.
66Multiples of 11 up to 130 end at 121. Of those, only 66 is also in the 6 times table (6 × 11 = 66).
687 × 11 = 77 players. Then 77 − 9 = 68.
44; 55; 66; 774 × 11 = 44, 5 × 11 = 55, 6 × 11 = 66 and 7 × 11 = 77. 33 is too small and 88 is too big.
66Work backwards: 10 − 4 = 6, then 6 × 11 = 66.
Expert
Model answer: Up to 9 × 11 the answer is that many tens and that many ones, like 4 tens and 4 ones is 44. 10 × 11 is 10 tens and 10 ones. The 10 ones make another ten, so it is 11 tens, which is 110, not one digit written twice.Look for: 11 is 10 + 1; up to 9 the same digit fills the tens and the ones; 10 ones cannot fit in the ones place, so they make another ten.
33; 132List 11, 22, 33 and so on up to 132 and add the digits of each. Only 33 (3 + 3) and 132 (1 + 3 + 2) give 6.
99Multiples of 11 between 50 and 105 are 55, 66, 77, 88 and 99. Only 99 is also in the 9 table (9 × 11 = 99).
4The pairs are 11 + 88, 22 + 77, 33 + 66 and 44 + 55. That is 4 pairs.
Model answer: Sofia added only one 7. 12 is 10 and 2, so 12 × 7 = 70 + 14 = 84, because there are two sevens to add as well as ten.Look for: only 7 × 10 plus one 7 was used; 7 × 2 = 14 is needed; correct answer 84.
Model answer: Yes. 12 is double 6, so every number in the 12 times table is double a number in the 6 times table, and that is still in the 6 times table. For example, 12 × 5 = 60 = 6 × 10.Look for: 12 = 2 × 6 (or 12 is double 6), so each 12 table number is double a 6 table number; an example.
Worked example. Calculate 0 × 58, 1 × 614 and 73 ÷ 1. 1. 0 × 58: no groups of 58 is nothing, so any number × 0 is 0. 2. 1 × 614: one group of 614 is 614, so any number × 1 is the number itself. 3. 73 ÷ 1: sharing 73 into 1 group leaves 73, so any number ÷ 1 is the number itself. So 0 × 58 = 0, 1 × 614 = 614 and 73 ÷ 1 = 73.
Level 1
183 groups of 6 is 6 + 6 + 6 = 18.
284 sevens: double 7 is 14, double again is 28.
38 × 3 = 24, so 24 ÷ 8 = 3.
84 × 8 = 32, so the missing number is 8.
False8 × 6 = 48, not 46, so the statement is false.
Model answer: Sometimes. It is true for any number above 0, such as 0 × 5 = 0, which is less than 1 × 5 = 5. It is false for 0, because 0 × 0 and 0 × 1 are both 0.Look for: sometimes; a true example above 0; the counter-example 0, where both products are 0.
0; 6; 8; 48Any pair with 0 gives 0. Then 1 × 6 = 6, 1 × 8 = 8 and 6 × 8 = 48.
1,001; 1,010; 1,100; 2,000Dividing by 1 keeps the number, so find 4-digit numbers whose digits add to 2: 2,000 or two 1s with the rest 0.
Model answer: Mateo has mixed up × 0 and × 1. Multiplying by 1 leaves a number the same. Multiplying by 0 gives 0, so 4,172 × 0 = 0.Look for: the rule for × 0 is that the answer is 0; the rule that leaves a number the same is × 1; the corrected answer 0.
Worked example. Work out 5 × 12 × 2. 1. Look for a pair that makes 10: 5 × 2 = 10. 2. Multiply that pair first: 5 × 2 = 10. 3. Then multiply by the third number: 10 × 12 = 120. 4. Another order agrees: 12 × 2 = 24, then 24 × 5 = 120. So 5 × 12 × 2 = 120.
Level 1
3Multiplying can be done in any order, so 3 × 5 = 5 × 3.
12; 122 × 6 = 12 and 6 × 2 = 12. The order does not change the answer.
33 × 6 is the same as 6 × 3, so the missing number is 3.
False5 × 9 = 45 but 9 × 4 = 36. Swapping the order only works with the same two numbers.
12Swap the order: 12 × 10 = 10 × 12.
549 × 6 is 6 × 9 in the other order, so the answer is also 54.
242 × 3 = 6, then 6 × 4 = 24.
405 × 2 = 10, then 10 × 4 = 40.
183 × 3 = 9, then 9 × 2 = 18.
363 × 2 = 6, then 6 × 6 = 36.
302 × 5 = 10 eggs on one tray, then 10 × 3 = 30.
TrueBoth sides use the same three numbers. 3 × 2 × 5 = 30 and 5 × 2 × 3 = 30.
Level 2
702 × 5 = 10 first, then 7 × 10 = 70.
724 × 3 = 12, then 12 × 6 = 72. Or 3 × 6 = 18, then 4 × 18 = 72.
633 × 3 = 9, then 9 × 7 = 63.
805 × 2 = 10, then 8 × 10 = 80.
92 × 5 = 10, so 10 × ___ = 90 and the missing number is 9.
603 × 4 = 12, then 12 × 5 = 60. Adding gives 12, which is wrong.
604 × 5 = 20 eggs in a crate, then 20 × 3 = 60.
False3 × 4 = 12, not 7. So 6 × 3 × 4 = 72 but 6 × 7 = 42.
968 × 3 = 24 flowers on a shelf, then 24 × 4 = 96. Or 3 × 4 = 12, then 12 × 8 = 96.
180Leave out the smallest card, 2. Then 4 × 5 × 9 = 180.
83 × 4 = 12 rows in all. Then 96 ÷ 12 = 8 cubes in each row.
4; 5; 63 × 2 = 6, so the answer is 6 times the missing number. 6 × 4 = 24, 6 × 5 = 30, 6 × 6 = 36 fit; 6 × 3 = 18 and 6 × 7 = 42 do not.
Expert
2The sets are 2, 2 and 6 (2 × 2 × 6 = 24) and 2, 3 and 4 (2 × 3 × 4 = 24). The smallest number must be 2, because 3 × 3 × 3 = 27 is too big. The other two then multiply to 12: 2 × 6 or 3 × 4.
2; 4; 8The numbers are a, 2a and 4a. Try 1, 2, 3...: 1 × 2 × 4 = 8 is too small, 2 × 4 × 8 = 64 works.
Model answer: No. Multiplication can be done in any order and the answer stays the same. For example, 2 × 3 × 5 = 30 and 5 × 3 × 2 = 30.Look for: Kai is wrong; the order does not matter; a worked example with both orders giving the same answer.
Model answer: Hugo added 5 and 3 to get 8, but they should be multiplied. 5 × 3 = 15 and 4 × 15 = 60, so the answer is 60.Look for: the mistake is adding 5 + 3 instead of multiplying; a correct route such as 4 × 5 = 20 then 20 × 3 = 60; the answer 60.