Answers for parents. Each unit has a worked example, then the answers for every level.
Unit 1: Numbers to 10,000,000
Worked example. Find the value of the 6 in 4,602,718. 1. Read the columns from the left: 4 millions, 6 hundred thousands, 0 ten thousands. 2. The 0 in the ten thousands place is a placeholder: there are no ten thousands. 3. The 6 is in the hundred thousands place, so it is worth 6 hundred thousands: 600,000. So the 6 in 4,602,718 is worth 600,000.
Level 1
5285 hundreds is 500, 2 tens is 20 and 8 ones is 8. 500 + 20 + 8 = 528.
3,425Each part goes in its own column: 3 thousands, 4 hundreds, 2 tens, 5 ones.
200The 2 is in the hundreds place, so it is worth 2 × 100 = 200.
3From the right the places are ones, tens, hundreds, thousands. The fourth digit from the right is 3.
306,042There are no ten thousands and no hundreds, so a 0 goes in each of those places: 306,042.
900,000The 9 is in the hundred thousands place, so it is worth 9 × 100,000 = 900,000.
3,000,000The 3 is in the millions place, so it is worth 3 millions: 3,000,000.
6The millions digit is the first digit of a 7-digit number: 6.
800,000The 8 is in the hundred thousands place, so it is worth 8 × 100,000 = 800,000.
3,200,0003 millions is 3,000,000 and 2 hundred thousands is 200,000. Together: 3,200,000.
50,000The 0 is in the hundred thousands place, so the 5 is in the ten thousands place: 5 × 10,000 = 50,000.
0The places from the left are millions 2, hundred thousands 9, ten thousands 0.
Level 2
700,000The 7 is in the hundred thousands place, so it is worth 7 × 100,000 = 700,000.
2,040,805There are no hundred thousands, no thousands and no tens, so each of those places holds a 0: 2,040,805.
2,070,0092 millions, 0 hundred thousands, 7 ten thousands, then 0 thousands, 0 hundreds, 0 tens and 9 ones: 2,070,009.
NoThe 3 is in the hundred thousands place, so it is worth 300,000, not 30,000.
3,000The digit in the thousands place is 3, so the missing part is 3,000.
60,000The 0 is in the hundred thousands place, so the 6 is in the ten thousands place: 60,000.
3,490,215In 3,490,215 the 9 is in the ten thousands place. The 9 in 9,204,610 is worth 9,000,000 and in 5,309,764 it is worth 9,000.
7,050,000Adding one hundred thousand to 9 hundred thousands makes 10, so the millions digit goes up by 1: 7,050,000.
4The digits are 5, 0, 0, 8, 0, 3, 0. Four of them are zeros.
Level 3
900,0003,640,000 + 280,000 = 3,920,000. The 9 is in the hundred thousands place, so it is worth 900,000.
6,009,0006,000,000 + 9,000 = 6,009,000. The hundred thousands and ten thousands places hold 0, and so do the last three.
1,307,000Use the smallest millions digit, 1, then 3, then 0 in the ten thousands place, then 7, then 0s: 1,307,000.
3,430,6002,000,000 + 1,400,000 + 30,000 + 600 = 3,430,600. The 14 hundred thousands make 1 million and 4 hundred thousands.
NoThe 2 is in the hundreds place, so it is worth 200, not 20. The parts add to 3,500,028.
2; 3; 4The missing digit is in the hundred thousands place, so it is worth d × 100,000. Only 2, 3 and 4 give values from 200,000 to 400,000.
Expert
8,115,121The 5 is in the thousands place and the 2 in the tens place. The four other places (hundred thousands, ten thousands, hundreds, ones) hold 1: 8,115,121.
Model answer: No. The 5s are in different places, so they are worth different amounts: 5,000,000, 50,000, 500 and 5.Look for: the value of a digit depends on its place; the four values are 5,000,000, 50,000, 500 and 5; the 0s only hold places.
1,000,001; 1,000,010; 1,000,100; 1,001,000; 1,010,000; 1,100,000; 2,000,000The millions digit is 1 or 2. If it is 2, all other digits are 0. If it is 1, one more 1 goes in any of the other six places.
Model answer: Dev put the 6 in the hundred thousands place. Sixty thousand is 60,000, so the 6 belongs in the ten thousands place: 4,060,005.Look for: 60,000 has its 6 in the ten thousands place; a 0 is needed in the hundred thousands place; correct number 4,060,005.
Worked example. Order smallest to largest: 3,482,615; 874,320; 3,417,090; 3,508,200. 1. 874,320 has 6 digits and the others have 7, so 874,320 is the smallest. 2. Millions are all 3. Hundred thousands: 4 in two numbers, 5 in 3,508,200. 3. Ten thousands: 3,417,090 has 1 and 3,482,615 has 8, so 3,417,090 is smaller. So the order is 874,320; 3,417,090; 3,482,615; 3,508,200.
Level 1
3,420Both have 4 digits. The hundreds digits are 4 and 2, and 4 is more.
56,030Both have 5 digits. They match until the hundreds: 0 is less than 3.
70,850; 78,050Both have 5 digits. The thousands digits are 8 and 0, so 70,850 is the smaller number.
91; 910; 9,100More digits means a greater number: 2 digits, 3 digits, 4 digits.
345,210345,210 has 6 digits and 99,870 has 5, so 345,210 is greater.
52,640; 250,460; 250,64052,640 has 5 digits, so it is smallest. The other two match until the hundreds digit: 4 is less than 6, so 250,460 comes first.
3,400,000Both have 7 digits. The millions digits are 3 and 2, and 3 is more.
5,280,000The millions match. The hundred thousands digits are 2 and 8, so 5,280,000 is less.
7,430,000The hundred thousands digits are 3 and 4, so 7,430,000 is the greater number.
1,900,000; 2,100,000; 2,400,000Compare the millions, then the hundred thousands: 1 million, then 2.1, then 2.4.
4,251,000The first two digits match. The ten thousands digits are 1 and 5, so 4,251,000 is larger.
8,003,000The millions match. 8,003,000 has 0 in both the hundred thousands and ten thousands, so it is smallest.
Level 2
<The numbers match until the hundreds digit: 1 is less than 8, so 462,185 is less.
6,350,012635,120 has 6 digits, so it is out. Of the other two, the hundred thousands are 3 and 3, then ten thousands 0 and 5, so 6,350,012 is largest.
2,909,090The numbers match for two digits, then the ten thousands digits are 0 and 9. So 2,909,090 is smaller.
9,870,0009,870,000 has 7 digits and 10,000,000 has 8 digits, so 9,870,000 is smaller.
4,320,000Multiples of 10,000 end in 0,000. The only one between 4,315,000 and 4,325,000 is 4,320,000.
1,950,0001,950,000 has 7 digits and 995,000 has 6, so 1,950,000 is greater.
880,880; 8,008,008; 8,080,080; 8,800,880880,880 has 6 digits so it is first. For the rest compare the hundred thousands: 0, 0 and 8, then the ten thousands: 0 and 8.
=4,010,000 + 90,000 = 4,100,000, so the two sides are equal.
7,205,000 kmThe numbers match until the ten thousands digit: 0 is less than 5, so 7,205,000 km is shorter.
5,505,000; 5,050,000All have 7 digits. Hundred thousands: 5, 0 and 5, so 5,050,000 is smallest. Of the other two, the thousands digits are 5 and 0, so 5,505,000 is greatest.
Level 3
1,205,000Largest first: 1,520,000; 1,250,000; 1,205,000; 980,000. The third in that list is 1,205,000.
6,600,000; 6,700,000; 6,800,000Multiples of 100,000 between the two numbers: 6,600,000; 6,700,000 and 6,800,000. 6,500,000 is too small and 6,900,000 is too big.
5,013,689The number must start with 5 to be over 5,000,000 and as small as possible. Then use the rest smallest first: 0, 1, 3, 6, 8, 9.
4The numbers are 4,999,999; 5,000,000; 5,000,001 and 5,000,002. That is 4 numbers.
SecondThe first probe flew 3,850,000 + 100,000 = 3,950,000 km. That is less than 3,950,500 km, so the second probe flew further.
2,355,000Multiples of 5,000 go up in steps of 5,000: 2,350,000; 2,355,000; 2,360,000. The middle one is 2,355,000.
Expert
Model answer: No. 7,000,000 has 7 digits and 770,000 has only 6, so 770,000 is less. Kai compared 77 with 7 and ignored the place values.Look for: counting digits (7 against 6) or place values (7 millions against 7 hundred thousands), and the idea that 77 and 7 are in different columns.
6,800,000; 6,080,000; 6,008,000; 6,000,800; 6,000,080; 6,000,008The number must start with 6: a leading 8 is too big and a leading 0 makes fewer than 7 digits. The 8 can then go in any of the other six places.
7The thousands digit must be 0. The last three digits are each 0 or 9, which gives 8 ways, but 000 would make 9,000,000 itself. So 7.
Model answer: Sometimes. It is smaller when the millions digit is greater than the hundred thousands digit, such as 6,230,000 becoming 2,630,000. It is greater when it is less, and the same when the digits are equal.Look for: 'sometimes', the leading digit deciding the size, and an example each way or a mention of equal digits.
Worked example. Round 1,183,614 to the nearest 1,000, 100,000 and 1,000,000. 1. Look at the digit one place to the right: 5 or more rounds up, 4 or less rounds down. 2. Nearest 1,000: the hundreds digit is 6, so round up to 1,184,000. 3. Nearest 100,000: the ten thousands digit is 8, so round up to 1,200,000. 4. Nearest 1,000,000: the hundred thousands digit is 1, so round down to 1,000,000. So 1,183,614 rounds to 1,184,000, 1,200,000 and 1,000,000.
Level 1
50The ones digit is 7, so round up to 50.
700The tens digit is 8, so round up to 700.
4,000The hundreds digit is 2, so round down to 4,000.
70,000The thousands digit is 2, so round down to 70,000.
6,850The ones digit is 5, and 5 rounds up, so 6,850.
500,000The ten thousands digit is 6, so round up from 400,000 to 500,000.
8,000,0008,380,000 is 380,000 above 8,000,000 and 620,000 below 9,000,000.
7,000,000The hundred thousands digit is 2, so round down to 7,000,000.
4,700,000The ten thousands digit is 3, so round down to 4,700,000.
8,200,000The ten thousands digit is 6, so round up from 8,100,000 to 8,200,000.
7,000,000The hundred thousands digit is 5, and 5 rounds up, so 7,000,000.
9,000,000The ten thousands digit is 7, so round up from 8,900,000 to 9,000,000.
Level 2
49,000The hundreds digit is 6, so round up to 49,000.
7,284,000The hundreds digit is 0, so round down and keep 7,284,000.
3,980,000The thousands digit is 5, so round up from 3,970,000 to 3,980,000.
6,400,000The ten thousands digit is 8, so round up from 6,300,000 to 6,400,000.
TrueThe thousands digit is 9, so round up from 5,440,000 to 5,450,000.
750,000The hundreds digit is 5, so round up from 749,000 to 750,000.
1,300,000The thousands digit is 6, so round up from 1,290,000 to 1,300,000.
2,500,0002,500,000 is exactly halfway between 2,000,000 and 3,000,000, so it rounds up to 3,000,000.
10,000,000The ten thousands digit is 6, so round up from 9,900,000 to 10,000,000.
Level 3
6,000,0003,849,000 rounds up to 4,000,000 and 2,476,000 rounds down to 2,000,000. The total is 6,000,000.
5,500,000; 6,499,999Halfway numbers round up, so 5,500,000 is the smallest. One less than 6,500,000 is the greatest: 6,499,999.
6,500,000; 7,000,0006,500,000 is halfway and rounds up, and 7,000,000 stays. 7,500,000 rounds up to 8,000,000.
No4,449,800 has 4 ten thousands and rounds down to 4,400,000. 4,450,200 has 5 ten thousands and rounds up to 4,500,000.
1,500,000To the nearest 100,000 it is 1,450,000 to 1,549,999. To round to 2,000,000 it must be 1,500,000 or more, so the fewest is 1,500,000.
1,000,0008,962,000 rounds up to 9,000,000. Then 10,000,000 − 9,000,000 = 1,000,000.
Expert
Model answer: Amira rounded twice. Rounding 3,444,999 to the nearest 1,000 gives 3,445,000, and that rounds up to 3,450,000. The thousands digit of the original number is 4, so it should round down to 3,440,000.Look for: rounding only once from the original number, using the digit 4 in the thousands place, and the correct answer 3,440,000.
6,148; 6,184; 6,418; 6,481The number must start with 6 and its hundreds digit must be 4 or less. That leaves 1 or 4 in the hundreds, with the other two digits in either order.
Model answer: Sometimes. 5,200,000 rounds to 5,000,000 and starts with 5. But 4,600,000 also rounds to 5,000,000 and starts with 4.Look for: numbers from 4,500,000 to 5,499,999 round to 5,000,000, so the first digit can be 4 or 5, with an example of each.
10,000To the nearest 10,000, the numbers run from 4,295,000 to 4,304,999, which is 10,000 numbers. All of them round to 4,000,000 at the nearest 1,000,000.
Worked example. The temperature rises from −8 °C to 6 °C. How many degrees is that? 1. From −8 °C up to 0 °C is 8 degrees. 2. From 0 °C on up to 6 °C is 6 degrees. 3. Add the two parts: 8 + 6 = 14. So the temperature rises by 14 °C.
Level 1
−2Count up 3 from −5: −4, −3, −2.
−4Any negative number is smaller than any positive number.
−3Count back 4 from 1: 0, −1, −2, −3.
−5The pattern goes down by 2 each time, so −3 − 2 = −5.
−9Count back 6 from −3. Each step goes further below 0: −9.
−11−11 is closer to 0 than −15, so −11 is larger.
4Count up from −4 to 0: −3, −2, −1, 0 is 4 steps.
7From −3 up to 0 is 3, then on to 4 is 4. 3 + 4 = 7.
FalseFrom −5 to 0 is 5 and from 0 to 3 is 3, so the interval is 8, not 2.
3Up 6 reaches 0, and 3 more steps reach 3.
9 °CFrom −7 °C to 0 °C is 7 degrees, then 2 more. 7 + 2 = 9.
−3Down 5 reaches 0, and 3 more steps reach −3.
Level 2
25From −18 up to 0 is 18, then on to 7 is 7. 18 + 7 = 25.
−9; −7; −2; 0; 4The further below 0, the smaller the number: −9 is smallest, then −7, −2, 0 and 4.
−15Among the negatives, −15 is furthest below 0, so it is the lowest.
11A is at −7 and B is at 4. From −7 to 0 is 7 and from 0 to 4 is 4, so 7 + 4 = 11.
7Up 23 reaches 0, and 7 more of the 30 reach 7.
−7 °CWarming by 11 °C moves up from −18 °C: −18 + 11 = −7. It stays below 0.
False−8 is further below 0 than −3, so −8 is smaller than −3.
15From 6 down to −9 is 6 steps to 0 and 9 more: 6 + 9 = 15.
−15 °CA is halfway between −20 and −10 on the scale, which is −15 °C.
−3The interval is 14 + 8 = 22. Half is 11, and 11 up from −14 is −3.
Level 3
−60 mRising 20 m gives −35 + 20 = −15. Sinking 45 m gives −15 − 45 = −60.
8 °CWork backwards: undo the fall of 14 by rising 14 from −6. −6 + 14 = 8.
325 mFrom −240 m up to 0 m is 240 m, then 85 m more. 240 + 85 = 325.
−6; −5; −4The number must be above −7. Adding 3 gives less than 0, so the number is below −3. So it is −6, −5 or −4.
−2The lowest floor is −4. Then −4 + 5 = 1, 1 − 2 = −1 and −1 − 1 = −2.
5 °CA is −15 °C and B is −5 °C, a rise of 10. Another rise of 10 from −5 °C gives 5 °C.
Expert
Model answer: No. −7 is further below 0 than −2, so −7 is smaller. The number nearer to 0 is greater, so −2 is greater than −7.Look for: Kai is not correct; negative numbers get smaller as the digit gets bigger; a number line or temperature (−7 °C is colder) as a reason.
−5; −4; −3; −2The larger number is the smaller plus 7 and must be 5 or less, so the smaller is at most −2. It is at least −5: −5, −4, −3, −2.
−6; 3Try pairs 9 apart: −6 and 3 are 9 apart (6 + 3) and −6 + 3 = −3. Both conditions fit.
Model answer: Always. The interval is the distance down to 0 plus the positive number. The distance down to 0 is more than 0, so the total is more than the positive number.Look for: always; the interval is the negative part plus the positive part; the negative part is always more than 0; an example such as −3 to 5 is 8, more than 5.
Worked example. Smallest number with 6 ten thousands that rounds to 4,700,000 (nearest 100,000). 1. Numbers that round to 4,700,000 go from 4,650,000 up to 4,749,999. 2. The smallest has 6 hundred thousands. Its ten thousands digit 6 is 5 or more. 3. For the smallest number, every lower digit is 0. That gives 4,660,000. So the smallest number is 4,660,000.
Level 1
3,460Adding 10 changes the tens digit from 5 to 6.
25,284 has 5 thousands, 2 hundreds, 8 tens and 4 ones.
60,000The 6 is in the ten thousands place, so it is worth 60,000.
4,400The tens digit is 8, which is 5 or more, so round up to 4,400.
52,030Both have 52 thousands. 52,030 has 0 hundreds and 52,300 has 3 hundreds.
540,000Take 1 from the hundred thousands digit: 6 becomes 5.
400,000The 4 is in the hundred thousands place, so it is worth 400,000.
3,340,000Add 1 to the millions digit: 2 becomes 3.
6,000,000The hundred thousands digit is 4, which is less than 5, so round down.
8,210,000Both have 8 millions. Compare hundred thousands: 2 is more than 1.
TrueThe 5 is in the hundred thousands place, so it is worth 500,000.
7,340,000Add the three parts: 7 millions, 3 hundred thousands and 4 ten thousands.
Level 2
90,000The 0 in the hundred thousands place is a placeholder. The 9 is in the ten thousands place, so it is worth 90,000.
3,708,764The 2 is in the hundred thousands place. Replacing it with 7 gives 3,708,764.
FalseBoth have 6 millions. The hundred thousands digits are 0 and 7, so 6,070,500 is smaller.
−12 °CCounting up 6 from −18 reaches −12. It is still below 0.
45,030; 450,300; 4,503,000Count the digits first: 5, 6 and 7 digits. More digits means a greater number.
8,000,000The ten thousands digit is 6, so round up. 7,900,000 up one step is 8,000,000.
3,058A number cannot start with 0, so the smallest start is 3. Then 0, 5 and 8 in order.
3,009,0003 millions is 3,000,000 and 9 thousands is 9,000. The places between hold 0.
3,850,000The ones digit is 5, so round up. The change carries through to 3,850,000.
−33 − 8 = −5 (below ground). Then −5 + 2 = −3.
Level 3
4,000,000 km2,468,000 rounds down to 2,000,000 and 5,731,000 rounds up to 6,000,000. 6,000,000 − 2,000,000 = 4,000,000.
2,093,700Order the visitors from most to least: 2,390,100; 2,309,500; 2,093,700; 965,800. The third is 2,093,700.
10,396The last digit must be 0 or 6. Ending in 6 allows 10,396, which is smaller than any number ending in 0.
2P: ten thousands digit 5, so it rounds up to 2,500,000. Q rounds to 2,100,000. R: digit 4, so it rounds down to 2,500,000. P and R match.
657; 675The six numbers are 567, 576, 657, 675, 756 and 765. Only 657 and 675 round to 700, since their tens digits are 5 and 7.
Expert
Model answer: No. Only the ten thousands digit decides. It is 4, which is less than 5, so 4,449,999 rounds down to 4,400,000.Look for: the deciding digit is the ten thousands digit (4); 4 is less than 5 so round down; the 9s further right do not matter; the answer is 4,400,000.
5,000; 5,005; 5,050; 5,055The number must start with 5. The hundreds digit must be 0 (a 5 would round up to 6,000). The tens and ones digits can each be 0 or 5, giving four numbers.
20,001; 20,010; 20,100; 21,000; 30,000The first digit is 2 or 3. If 3, the other digits are 0: 30,000. If 2, one other digit is 1 and the rest are 0: the 1 can be in four places.
Model answer: Sometimes. 3,200,000 rounds to 3,000,000, which has 7 digits. But 9,600,000 rounds up to 10,000,000, which has 8 digits.Look for: the answer 'sometimes'; an example that stays 7 digits; an example from 9,500,000 or more that rounds up to 10,000,000, which has 8 digits.
Worked example. Calculate 6,350,000 − 1,997,000. 1. 1,997,000 is 3,000 less than 2,000,000, so take away 2,000,000 first. 2. 6,350,000 − 2,000,000 = 4,350,000. 3. Too much was taken away: 3,000 too many, so add 3,000 back. So 6,350,000 − 1,997,000 = 4,353,000.
Level 1
190,00013 ten thousands and 6 ten thousands make 19 ten thousands.
4,0009 thousands take away 5 thousands leaves 4 thousands.
6,500Add the thousands: 2 + 4 = 6. The 500 stays.
55,00035 thousands + 20 thousands = 55 thousands.
41,00071 thousands − 30 thousands = 41 thousands.
280,000420,000 needs 80,000 to reach 500,000, then 200,000 more to reach 700,000.
8,000,0003 millions + 5 millions = 8 millions.
3,000,0009 millions − 6 millions = 3 millions.
3,700,000Add the hundred thousands: 2 + 5 = 7. The millions stay.
6,600,000Subtract the hundred thousands: 9 − 3 = 6.
600,000From 2 to 8 in the hundred thousands is 6, so 600,000.
1,550,0001,300,000 + 250,000: 300,000 + 250,000 = 550,000, so 1,550,000.
Level 2
5,400,0004,650,000 + 350,000 = 5,000,000, then 400,000 more is 5,400,000.
6,400,0007,200,000 − 200,000 = 7,000,000, then 600,000 less is 6,400,000.
2,444,000Add 100,000 to get 2,445,000, then take 1,000 off.
4,802,000Take away 2,000,000 to get 4,800,000, then add 2,000 back.
2,400,000; 3,750,000Try pairs. 2,400,000 + 3,750,000 = 6,150,000. No other pair makes 6,150,000.
Expert
Model answer: No. 3,999,999 is 1 less than 4,000,000, so subtract 4,000,000 to get 5,400,000, then add 1 back. The answer is 5,400,001.Look for: 3,999,999 is one less than 4,000,000; too much is taken away so 1 is added back, not taken off; correct answer 5,400,001.
500,000; 600,000The smaller number is the larger minus 400,000. 500,000 and 100,000 total 600,000. 600,000 and 200,000 total 800,000. 700,000 and 300,000 total 1,000,000, which is too big.
Model answer: Yes. 999,000 is 1,000 less than 1,000,000. Adding 1,000,000 adds 1,000 too much, so taking 1,000 away gives the right total every time.Look for: 999,000 = 1,000,000 − 1,000; adding too much then correcting by taking the extra off; it works for any number. A worked example is a bonus.
6,499,998Taking 1,999,999 and adding 2,000,001 changes the number by +2. Working backwards: 6,500,000 − 2 = 6,499,998.
Worked example. A tour sold 612,480 and 735,915 tickets. How many more to make 2,000,000? 1. Two steps: add the two amounts, then subtract the total from 2,000,000. 2. Columns: 612,480 + 735,915 = 1,348,395. 3. Columns, exchanging across the zeros: 2,000,000 − 1,348,395 = 651,605. So 651,605 more tickets are needed to reach 2,000,000.
Level 1
46,687Add in columns: ones 7, tens 8, hundreds 6, thousands 6, ten thousands 4. No exchange is needed.
Model answer: Yes. 1,000,000 is 1 more than 999,999, so the answer is 1 more: 999,999 − 456,789 = 543,210, and 543,210 + 1 = 543,211. No exchanging across zeros is needed.Look for: 1,000,000 is 1 more than 999,999; subtracting from 999,999 needs no exchange; add the 1 back at the end; answer 543,211.
570,000; 330,000Take away the difference: 900,000 − 240,000 = 660,000, which is two equal smaller numbers. Half of 660,000 is 330,000, so the larger is 330,000 + 240,000 = 570,000.
Model answer: Sometimes. 600,000 + 700,000 = 1,300,000 has 7 digits, but 123,456 + 234,567 = 358,023 has only 6 digits.Look for one example that makes 7 digits and one that stays at 6 digits; both are needed to say sometimes.
Worked example. Find the common factors of 44 and 66. 1. Factors of 44: 1, 2, 4, 11, 22, 44. 2. Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66. 3. The numbers in both lists are 1, 2, 11 and 22. The greatest is 22. So the common factors are 1, 2, 11 and 22, and the greatest is 22.
Level 1
1; 2; 5; 1010 = 1 × 10 and 2 × 5, so the factors are 1, 2, 5 and 10.
412 ÷ 3 = 4, so the third pair is 3 × 4.
1; 2; 3; 6; 9; 1818 = 1 × 18, 2 × 9 and 3 × 6.
True5 × 7 = 35, so 5 is a factor of 35.
1; 3Factors of 6: 1, 2, 3, 6. Factors of 9: 1, 3, 9. Both lists have 1 and 3.
1; 2; 4Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 16: 1, 2, 4, 8, 16. Both lists have 1, 2 and 4.
4590 = 2 × 45 and 135 = 3 × 45. 2 and 3 share no factor, so 45 is the greatest.
Level 3
5The longest piece is the greatest common factor, 28 cm. 56 ÷ 28 = 2 and 84 ÷ 28 = 3, so 2 + 3 = 5 pieces.
3; 6; 12; 24; 48Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 32: 1, 2, 4, 8, 16, 32. Remove the shared ones.
5; 8; 10; 16Rows must be a factor of 80 and at least 5. Chairs per row, 80 ÷ rows, must also be at least 5, so rows are 16 or fewer: 5, 8, 10, 16.
6The greatest common factor of 48 and 72 is 24. For 90 and 120 it is 30. 30 − 24 = 6.
3The number of sets is the greatest common factor of 60, 90 and 150, which is 30. Grey tiles in each set: 90 ÷ 30 = 3.
56; 112; 140; 196The number is a multiple of 28: 56, 84, 112, 140, 168 or 196. 84 and 168 give a greatest common factor of 84, so they are out.
Expert
Model answer: Sometimes. For 8 and 12 it is 4, smaller than both. For 6 and 12 it is 6, which is not smaller than 6.Look for one example that works and one counter-example, such as a smaller number that is a factor of the larger.
15Fewest tiles means the longest side, the greatest common factor: 84 cm. 252 ÷ 84 = 3 and 420 ÷ 84 = 5, so 3 × 5 = 15 tiles.
4Both numbers are multiples of 6. Testing 6, 12, 18, 24, 30, 36 and 42 as the smaller number, only 6 and 84, 12 and 78, 24 and 66, 42 and 48 have a greatest common factor of 6.
Model answer: 8 is not a common factor, because 36 ÷ 8 is not a whole number. The correct list is 1, 2, 3, 4, 6 and 12, so 12 is still the greatest.Look for: 8 divides 24 but not 36; the corrected list; and that 12 is still greatest because the mistake was a smaller number.
Worked example. Find the first two common multiples of 6 and 8. 1. List the multiples of 8, the larger number: 8, 16, 24, 32, 40, 48. 2. Test each for 6: 24 = 6 × 4 and 48 = 6 × 8. The others are not in the 6 times table. 3. The smallest of the common multiples is 24. So the first two common multiples of 6 and 8 are 24 and 48, and the smallest is 24.
Level 1
5; 10; 15; 20Count on in fives four times: 5, 10, 15, 20.
27The 9th multiple of 3 is 3 × 9 = 27.
48The pattern goes up in twelves, so the missing number is 36 + 12 = 48.
42The multiples of 6 go 36, 42, so the next one after 40 is 42.
21; 28; 35The multiples of 7 are 7, 14, 21, 28, 35, 42. Three lie between 20 and 40.
13212 × 11 = 132.
1212 = 3 × 4 and 12 = 4 × 3, so 12 is in both times tables. No other number under 20 is.
10Multiples of 5: 5, 10. Only 10 is also a multiple of 2.
3636 = 4 × 9 and 36 = 6 × 6. No other number from 31 to 39 is in both tables.
True20 = 4 × 5 and 20 = 5 × 4, so 20 is in both times tables.
12; 24Multiples of 6: 6, 12, 18, 24. The multiples of 4 among them are 12 and 24.
18Multiples of 9: 9, 18. 18 = 6 × 3, so it is the first one in both tables.
Level 2
30Multiples of 10: 10, 20, 30. 30 = 6 × 5, so 30 is the first in both tables.
12; 24; 36; 48The common multiples go up in twelves: 12, 24, 36, 48.
True60 = 12 × 5 and 60 = 15 × 4, so both divide 60.
2424 is already in the 8 times table (8 × 3), so the smallest common multiple is 24 itself.
6045 and 75 are not multiples of 6. 60 = 6 × 10 = 15 × 4.
40 minutesThe smallest common multiple of 8 and 10 is 40, so both leave again in 40 minutes.
24Multiples of 12: 12, 24. 24 = 8 × 3, so 24 plates and 24 cups.
36Multiples of 9: 9, 18, 27, 36. 36 = 4 × 9 = 6 × 6, so 36 is the first in all three tables.
42Multiples of 21: 21, 42. 42 = 14 × 3, so 42 is the first in both tables.
108; 144; 180The smallest common multiple of 12 and 18 is 36. Multiples of 36 from 100 to 200: 108, 144, 180.
Level 3
60Multiples of 15: 15, 30, 45, 60. 60 = 12 × 5, so the drums next beat together at 60.
4The smallest common multiple is 30. Both are back at the start at 30, 60, 90 and 120 seconds, and 150 is too late.
5; 10; 15The number must divide 30 but not be a multiple of 6: 5, 10 and 15 work. 1, 2, 3 and 6 give a smaller common multiple.
75; 150The smallest common multiple of 15 and 25 is 75. The multiples of 75 below 160 are 75 and 150.
432The smallest common multiple of 18 and 24 is 72. 72 × 6 = 432, and 72 × 7 = 504 is too many.
72 minutesThe smallest common multiple of 6 and 8 is 24. Multiples of 24: 24, 48, 72. Only 72 is a multiple of 9.
Expert
Model answer: No. 6 × 9 = 54 is a common multiple of 6 and 9, but 18 is also a common multiple and is smaller. So the smallest common multiple is 18, not 54.Look for a counter-example: 18 is in the 6 and 9 times tables and is smaller than 54. The product is only sometimes the smallest.
336; 456Common multiples of 8 and 12 go up in 24s: 312, 336, 360 and so on. Only 336 and 456 end in 6.
61The number is one more than a common multiple of 3, 4 and 5. The smallest common multiple is 60, so the number is 61.
945Common multiples of 7 and 9 are multiples of 63. Those ending in 5 are 315 and 945, and only 945 has digits adding to 18.
Worked example. Find the smallest prime factor of 91. 1. Test 2: 91 is odd, so 2 is not a factor. 2. Test 3: the digits 9 + 1 = 10, not a multiple of 3. Test 5: 91 ends in 1. 3. Test 7: 7 × 13 = 91, so 7 is a factor and 91 is not prime. So the smallest prime factor of 91 is 7.
Level 1
2; 3; 5; 7A prime number has exactly two factors. The first four are 2, 3, 5 and 7.
False9 = 3 × 3, so 9 has three factors: 1, 3 and 9. It is not prime.
13Only 13 lies between 12 and 14, and 13 has just the factors 1 and 13.
2Every even number above 2 has 2 as a factor, so it has more than two factors. 2 is the only even prime.
2The factors of 11 are 1 and 11. Every prime number has exactly two factors.
23; 29Test 21 (3 × 7), 22, 24, 25, 26, 27, 28: none is prime. 23 and 29 have only two factors.
73 × 7 = 21, so 21 has factors other than 1 and 21.
3133 = 3 × 11 and 35 = 5 × 7. Only 31 has just the factors 1 and 31.
5The factors of 25 are 1, 5 and 25. Only 5 is prime.
4744, 45 and 46 each have a factor of 2 or 5. 47 has only the factors 1 and 47.
41; 43; 47Test each odd number: 45 is 5 × 9 and 49 is 7 × 7. 41, 43 and 47 are prime.
133 × 13 = 39, so 39 has the extra factors 3 and 13.
Level 2
5351 = 3 × 17, 57 = 3 × 19 and 63 = 7 × 9. Only 53 has no factor except 1 and 53.
False7 × 7 = 49, so 7 is a factor. 49 is not prime.
53; 59Test the odd numbers: 51 = 3 × 17, 55 = 5 × 11, 57 = 3 × 19. Only 53 and 59 are prime.
2; 3; 5The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. The prime ones are 2, 3 and 5.
1166 = 2 × 3 × 11. The prime factors are 2, 3 and 11, and the largest is 11.
2Even numbers and 85 are out. 81 = 9 × 9 and 87 = 3 × 29. The primes are 83 and 89, so two cards.
777 = 7 × 11. The prime factors are 7 and 11, and the smallest is 7.
9791 = 7 × 13, 93 = 3 × 31 and 95 = 5 × 19. Even numbers are out. 97 is the first prime.
2113, 19 and 23 are prime. 21 = 3 × 7 is odd but not prime, so it is a counter-example.
14471 and 73 are prime; 72 and 74 are even and 75 is out of range. Then 71 + 73 = 144.
Level 3
2; 3; 7The row length must be a factor of 42 and prime. The factors are 1, 2, 3, 6, 7, 14, 21 and 42, and the primes are 2, 3 and 7.
198Below 100 the nearest prime is 97. Above 100 the nearest is 101, which is prime. Then 97 + 101 = 198.
43The primes between 40 and 50 are 41, 43 and 47. Their digit sums are 5, 7 and 11, so the number is 43.
7; 23The pairs of primes with sum 30 are 7 and 23, 11 and 19, and 13 and 17. Their differences are 16, 8 and 4, so the pair is 7 and 23.
199Start at the top. 200 is even. 199 is odd and not in the 3, 5, 7, 11 or 13 times tables, so it is prime.
51This needs a number that is not prime. 41, 37 and 53 are prime. 51 = 3 × 17, so 3 rows of 17 chairs work.
Expert
Model answer: Sometimes. Two odd primes such as 3 and 5 add to 8, which is even. But 2 + 3 = 5 is odd, because 2 is the only even prime.Look for: the answer sometimes; an example with an even sum (two odd primes); an example with an odd sum that uses 2 as the only even prime.
11; 13; 17; 31; 37The primes are 11, 13, 17, 19, 23, 29, 31 and 37. Swapped they make 11, 31, 71, 91, 32, 92, 13 and 73. 91 = 7 × 13 and 32 and 92 are even, so 19, 23 and 29 are out.
13; 17Test primes in order: 2, 3, 5, 7 and 11 do not divide 221. 221 ÷ 13 = 17, and 17 is prime, so the numbers are 13 and 17.
Model answer: Dev is wrong. 21 ends in 1 but 21 = 3 × 7, so it has the factors 3 and 7 as well as 1 and 21 and is not prime. (27, 33 and 49 also work.)Look for: a counter-example ending in 1, 3, 7 or 9 (21, 27, 33, 49, 51 and so on) and a pair of factors that multiply to it, showing it is not prime.
Model answer: Tariq forgot the placeholder 0. The tens digit 3 means 30, so the row is 214 × 30 = 6,420, not 642.Look for: the tens row multiplies by 30, not 3; a placeholder 0 shifts the row; the correct row is 6,420.
28; 29; 30; 31; 32; 33; 3429 × 28 = 812 is above 800, but 29 × 27 = 783 is below. 29 × 34 = 986 is below 1,000, but 29 × 35 = 1,015 is above.
22,412The best choice is 431 × 52 = 22,412. Other close choices such as 521 × 43 = 22,403 are smaller.
100,870 km3,845 × 26 = 99,970 km. Then 99,970 + 900 = 100,870 km.
78,694 is close to 380 × 23, so try A = 7: 378 × 20 = 7,560 and 378 × 3 = 1,134. Add: 8,694, so A = 7.
730Hana: 1,340 × 15 = 20,100. Lena: 1,490 × 13 = 19,370. The difference is 730.
209208 boxes hold 208 × 24 = 4,992 cakes, which is not enough. 209 boxes hold 5,016 cakes, which is more than 5,000.
Expert
Model answer: Noah forgot that 20 is 2 tens, so the answer must be 10 times bigger. 2,314 × 20 = 46,280.Look for: 20 is 2 × 10; the 2 in 20 stands for 2 tens; the answer 4,628 needs a 0 placed after it, giving 46,280.
13; 23; 33; 43; 53; 63; 73; 83; 93Only the ones digits matter: 7 × 3 = 21 ends in 1, and no other digit works. So every 2-digit number ending in 3.
Model answer: 99 is one less than 100, so 99 lots of 3,415 is 100 lots of 3,415 with one lot taken away.Look for: 99 = 100 − 1; 100 lots minus 1 lot leaves 99 lots; it works for any number, not just 3,415.
Worked example. Calculate 5,436 ÷ 12 using short division. 1. 12 into 5 does not go. 12 into 54 is 4 (12 × 4 = 48), remainder 6. Write 4, carry 6. 2. The carried 6 and the 3 make 63. 12 into 63 is 5 (12 × 5 = 60), remainder 3. Write 5. 3. The carried 3 and the 6 make 36. 12 into 36 is 3 exactly. Write 3. So 5,436 ÷ 12 = 453.
Level 1
127 into 8 is 1 remainder 1. Carry 1 to make 14, and 7 into 14 is 2.
1235 into 6 is 1 remainder 1. 5 into 11 is 2 remainder 1. 5 into 15 is 3.
3646 into 21 is 3 remainder 3. 6 into 38 is 6 remainder 2. 6 into 24 is 4.
181; 14 into 7 is 1 remainder 3. 4 into 32 is 8. 4 into 5 is 1 remainder 1. So 181 remainder 1.
2212 into 26 is 2 remainder 2. 12 into 24 is 2. So the answer is 22.
4511 into 49 is 4 remainder 5. 11 into 55 is 5. So the answer is 45.
6312 into 75 is 6 (72) remainder 3. 12 into 36 is 3. So the answer is 63.
48; 211 into 53 is 4 (44) remainder 9. 11 into 90 is 8 (88) remainder 2. So 48 remainder 2.
Level 2
13211 into 14 is 1 remainder 3. 11 into 35 is 3 remainder 2. 11 into 22 is 2.
27412 into 32 is 2 remainder 8. 12 into 88 is 7 remainder 4. 12 into 48 is 4.
29425 into 73 is 2 remainder 23. 25 into 235 is 9 remainder 10. 25 into 100 is 4.
False15 into 24 is 1 remainder 9. 15 into 91 is 6 remainder 1. 15 into 15 is 1. The answer is 161, not 151.
50712 into 60 is 5. 12 into 8 is 0 remainder 8, so write 0. 12 into 84 is 7.
187; 520 into 37 is 1 remainder 17. 20 into 174 is 8 remainder 14. 20 into 145 is 7 remainder 5.
1131,350 ÷ 12 is 112 remainder 6. The 6 left over need one more box, so round up to 113.
1704,260 ÷ 25 is 170 remainder 10. Only full packs count, so round down to 170.
Level 3
15214 + 9 = 223 people. 223 ÷ 15 is 14 remainder 13, so the 13 left over need another minibus: 15.
1,027Work backwards: 12 × 85 = 1,020, then add the remainder 7 to get 1,027.
$5883,675 ÷ 25 = 147 boxes. 147 × 4 = 588, so the boxes sell for $588.
303; 328; 353; 378The multiples of 25 in this range are 300, 325, 350 and 375. Add 3 to each: 303, 328, 353, 378.
9,9909,999 ÷ 15 is 666 remainder 9. Take away the remainder: 9,999 − 9 = 9,990.
Expert
Model answer: No. The remainder must be less than 12. A remainder of 12 would make one more full group of 12, so the largest remainder is 11.Look for: remainder is always smaller than the divisor; 12 left over could be shared out again; possible remainders are 0 to 11.
275; 550; 825A number divisible by 11 and 25 is a multiple of 275. The 3-digit multiples are 275, 550 and 825.
930; 944The smallest has remainder 0: 15 × 62 = 930. The largest has remainder 14: 930 + 14 = 944.
Worked example. Calculate 594 ÷ 18 using long division. 1. Multiples of 18: 18, 36, 54, 72, 90. 2. 18 does not go into 5, so divide 59: 3 × 18 = 54, and 59 − 54 = 5. 3. Bring down the 4 to make 54. 54 ÷ 18 = 3, and 54 − 54 = 0. So 594 ÷ 18 = 33.
Level 1
249 ÷ 4 = 2 carry 1, then 16 ÷ 4 = 4. The answer is 24.
4225 ÷ 6 = 4 carry 1, then 12 ÷ 6 = 2. The answer is 42.
32112 ÷ 4 = 3, 8 ÷ 4 = 2, 4 ÷ 4 = 1. The answer is 321.
1,0081,000 ÷ 36 is 27 with some left over, so try 28 × 36 = 1,008. That is the first multiple of 36 above 999.
282,496 ÷ 32 = 78 rows in all. Then 78 − 50 = 28 rows are left.
39The tiles in one row are 1,326 ÷ 34. 132 ÷ 34 = 3 (102), left 30, then 306 ÷ 34 = 9. The answer is 39.
Expert
Model answer: No. 36 × 60 = 2,160, which is more than 1,872, so the answer must be less than 60. 62 is too big.Look for: Accept any estimate showing the answer is near 50 (36 × 50 = 1,800), or 36 × 60 = 2,160 being too big. Checking 36 × 62 is more than 1,872 also works. The true answer is 52.
2748 is double 24, so each share is half as big: 54 ÷ 2 = 27.
13; 14; 21; 26; 39; 42Test two-digit divisors of 546: 13 × 42, 14 × 39, 21 × 26. Each pair gives two answers, as 42, 39 and 26 can also be the divisor.
Worked example. Calculate 2,190 ÷ 24. Give the answer as a mixed number. 1. 219 ÷ 24 = 9 (216), remainder 3. Bring down 0: 30 ÷ 24 = 1, remainder 6. 2. The quotient is 91 and the remainder is 6, so 2,190 ÷ 24 = 91 r 6. 3. The remainder over the divisor is 6/24. Divide both by 6 to get 1/4. So 2,190 ÷ 24 = 91 1/4.
Level 1
25 × 3 = 15, and 17 − 15 = 2.
18; 34 × 18 = 72, and 75 − 72 = 3.
False5 × 7 = 35, and 38 − 35 = 3, so the remainder is 3, not 2.
6There is a remainder, so round up to the next whole number: 6.
7; 112 × 7 = 84, and 85 − 84 = 1.
1020 × 7 = 140, and 150 − 140 = 10.
26; 515 × 26 = 390, and 395 − 390 = 5.
True14 × 31 = 434, and 438 − 434 = 4.
Level 2
135; 1221 × 135 = 2,835, and 2,847 − 2,835 = 12.
40 5/1212 × 40 = 480, remainder 5. The fraction is 5/12, which cannot be simplified.
25 7/1224 × 25 = 600, remainder 14. 14/24 simplifies to 7/12 by halving.
12530 ÷ 48 = 11 r 2. The 2 pupils left over need another coach, so 12.
4724 × 47 = 1,128 and 24 × 48 = 1,152, which is too many. So 47 full boxes.
True15 × 244 = 3,660, remainder 12. 12/15 simplifies to 4/5, so the statement is true.
910Multiply back: 17 × 53 = 901, then add the remainder: 901 + 9 = 910.
2426 × 66 = 1,716, and 1,740 − 1,716 = 24.
Level 3
27385 + 14 = 399 people. 399 ÷ 15 = 26 r 9, so round up to 27 minibuses.
84 3/816 × 84 = 1,344, remainder 6. 6/16 simplifies to 3/8, so each tank holds 84 3/8 litres.
765Working backwards from a division: 52 × 14 = 728, then 728 + 37 = 765.
617; 63518 × 33 = 594, so 599 is the one before. Add 18 each time: 617, 635, then 653, which is too big.
93524 × 38 = 912. The remainder must be less than 24, so the largest is 23: 912 + 23 = 935.
Expert
Model answer: No. The fraction is the remainder over the divisor, so it is 5/15, which simplifies to 1/3. The answer is 6 1/3.Look for: the quotient 6 and remainder 5 are right; the denominator should be the divisor 15, not 6; 5/15 simplifies to 1/3.
6; 18The fraction is the remainder over 24. 6/24 = 1/4 and 18/24 = 3/4. No other remainder under 24 simplifies to quarters.
51; 111; 171Numbers leaving 1 when divided by 5 end in 1 or 6. List 3, 15, 27, 39, 51 and so on, and keep those ending in 1 or 6: 51, 111, 171.
12Dev: 4 × 5 = 20, then 3 + 20 = 23. Amira: 7 × 5 = 35. The difference is 35 − 23 = 12.
$6Books: 3 × 12 = 36. Pens: 2 × 4 = 8. Total 44, and 50 − 44 = 6.
15Work backwards: 76 + 14 = 90. Then 90 ÷ 6 = 15.
5; 6; 7Try each: 3 × 5 − 4 = 11, 3 × 6 − 4 = 14, 3 × 7 − 4 = 17, but 3 × 8 − 4 = 20 is too big and 3 × 4 − 4 = 8 is too small.
1313 × 7 + 5 = 96, which is less than 100. 14 × 7 + 5 = 103, which is too big.
Expert
Model answer: No. 18 − 6 + 3 is worked left to right: 12 + 3 = 15. With brackets, 6 + 3 = 9 is done first, so 18 − 9 = 9. The answers are different.Look for: left to right without brackets (15), brackets first (9), and a clear statement that Bilal is not correct.
2; 7Second box 5: 5 × 5 = 25, so first box 7. Second box 6: 6 × 5 = 30, so first box 2. Other second boxes give a first box that is too big or too small.
Model answer: Jun did 4 × 3 = 12 first, then 24 ÷ 12 = 2. Multiplying and dividing are worked left to right, so 24 ÷ 4 = 6 first, then 6 × 3 = 18.Look for: Jun multiplied first, left to right is the rule for × and ÷, and the correct answer 18.
Worked example. Calculate 70,000 × 8 mentally. 1. Use the table fact: 7 × 8 = 56. 2. 70,000 is 7 × 10,000, so make 56 ten thousand times larger: 560,000. 3. Check with the inverse: 560,000 ÷ 8 = 70,000. So 70,000 × 8 = 560,000.
Level 1
42This is a times table fact: 6 × 7 = 42.
78 × 7 = 56, so 56 ÷ 8 = 7.
350Multiplying by 10 moves each digit one place left: 350.
90Dividing by 10 moves each digit one place right: 90.
25,000Multiplying by 1,000 moves each digit three places left: 25,000.
50045 ÷ 9 = 5, and 4,500 is 45 hundreds, so the answer is 5 hundreds: 500.
39,2006 × 4,200 = 25,200 and 4 × 3,500 = 14,000. Then 25,200 + 14,000 = 39,200.
2,000,000Work backwards with the inverse: 250,000 × 8 = 2,000,000.
1920 × 50,000 = 1,000,000, which is not below 1,000,000. So the greatest is 19.
181180 boxes hold 9,000 screws, which is not enough. 181 boxes hold 9,050 screws.
Expert
72015 × 48 = 30 × 24 = 60 × 12 = 720.
6; 7; 8; 9; 10; 115 × 22,000 = 110,000 is too small, and 6 × 22,000 = 132,000 fits. 11 × 22,000 = 242,000 fits, but 12 × 22,000 = 264,000 is too big.
Model answer: Yes. Multiplying by 5 is the same as multiplying by 10 and halving. A multiple of 10 ends in 0. Halving it gives an answer ending in 0 or 5.Key points: Hugo is correct; link ×5 to ×10 then ÷2 (or the 5 times table pattern 5, 10, 15, 20); the tens-and-ones ending alternates 5 and 0.
Model answer: Yusuf forgot that 30 is 3 tens. 60,000 × 30 = 6 × 3 × 10,000 × 10 = 18 × 100,000 = 1,800,000, so 180,000 is ten times too small.Key points: 60,000 is 6 ten thousands and 30 is 3 tens; 18 must be made 100,000 times larger; 1,800,000.
Worked example. Estimate 6,284 × 38 by rounding to the nearest 1,000 and 10. 1. 6,284 rounds to 6,000: the hundreds digit is 2, so round down. 2. 38 rounds to 40: the ones digit is 8, so round up. 3. Estimate 6,000 × 40: 6 × 4 = 24, then four zeros give 240,000. So 6,284 × 38 is about 240,000.
Level 1
50The ones digit is 7, so round up to 50.
400The tens digit is 8, so round up to 400.
7,000The hundreds digit is 5, so round up to 7,000.
30,000The thousands digit is 8, so round up to 30,000.
9038 rounds to 40 and 53 rounds to 50. 40 + 50 = 90.
200412 rounds to 400 and 189 rounds to 200. 400 − 200 = 200.
9,0006,480 rounds to 6,000 and 2,730 rounds to 3,000. 6,000 + 3,000 = 9,000.
3,0006,100 rounds to 6,000 and 2,900 rounds to 3,000. 6,000 − 3,000 = 3,000.
16038 is close to 40. 40 × 4 = 160.
100590 is close to 600. 600 ÷ 6 = 100.
False1,940 rounds to 2,000 and 3,060 rounds to 3,000. The estimate is 5,000, not 7,000.
1,50048 rounds to 50. 50 × 30 = 1,500.
Level 2
8,0002,914 rounds to 3,000 and 5,087 rounds to 5,000. 3,000 + 5,000 = 8,000.
5,3008,640 rounds to 8,600 and 3,270 rounds to 3,300. 8,600 − 3,300 = 5,300.
2,40061 is about 60 and 39 is about 40. 60 × 40 = 2,400.
2008,000 ÷ 40 = 800 ÷ 4 = 200.
False48,300 rounds to 50,000 and 29,800 rounds to 30,000. The estimate is 80,000, not 90,000.
1,00052 rounds to 50 and 18 rounds to 20. 50 × 20 = 1,000.
12,000350 is halfway, so it rounds up to 400. 28 rounds to 30. 400 × 30 = 12,000.
No5,000 × 20 = 100,000, so the answer should be near 100,000. 10,206 is ten times too small.
$3,000$6,180 rounds to $6,000 and $2,840 rounds to $3,000. $6,000 − $3,000 = $3,000.
6,0004,960 rounds to 5,000, 3,070 to 3,000 and 1,980 to 2,000. 5,000 + 3,000 − 2,000 = 6,000.
Level 3
800,000 km38,900 rounds to 40,000 and 21 rounds to 20. 40,000 × 20 = 800,000 km.
1,15036,000 ÷ 30 = 1,200, so the rounded number is 1,200. The smallest number that rounds up to 1,200 is 1,150.
35; 36; 37; 38; 39; 40; 41; 42; 43; 44800 ÷ 20 = 40, so the rounded number is 40. Numbers from 35 to 44 round to 40.
$120,000The tickets round to 3,000 and 3,000, making 6,000. The price rounds to $20. 6,000 × $20 = $120,000.
5,870 × 316,000 × 30 = 180,000 and 4,000 × 40 = 160,000. The first product is greater.
4,000,0001,960,000 rounds to 2,000,000 and 2,070,000 rounds to 2,000,000. The total is about 4,000,000.
Expert
Model answer: More. Rounding 51 down to 50 takes away a whole 4,950, but rounding 4,950 up to 5,000 adds only about 50 × 50 = 2,500. The estimate loses more than it gains, so the exact answer is more.Look for: the effect of each rounding is compared (51 to 50 loses 4,950; 4,950 to 5,000 gains about 2,500), and the conclusion is more than 250,000. Exact: 252,450.
Model answer: Sometimes. 16 + 17 = 33 but 20 + 20 = 40, which is greater. 14 + 13 = 27 but 10 + 10 = 20, which is less.Look for: the answer sometimes, one example where both numbers round up (estimate greater), and one where both round down (estimate less).
350; 450The estimate is greater only when the number rounds up. 350 rounds up to 400 and 450 rounds up to 500. 300, 400 and 500 do not change.
3,500,000; 4,499,9997,000,000 − 3,000,000 = 4,000,000, so the number rounds to 4,000,000. The smallest is 3,500,000 (halfway rounds up) and the largest is 4,499,999.
Worked example. Find the change from $9,500 for 24 tickets at $385 each. 1. Operation 1: multiply, 24 × 385, because 24 equal prices are added. 2. 385 × 4 = 1,540 and 385 × 20 = 7,700, so 24 × 385 = 9,240. 3. Operation 2: subtract the cost from the money paid: 9,500 − 9,240 = 260. So the change is $260.
Level 1
8345 + 38: 45 + 30 = 75, then 75 + 8 = 83.
10810 × 9 = 90 and 2 × 9 = 18, so 90 + 18 = 108.
98 × 9 = 72, so 72 ÷ 8 = 9.
850Take 4 hundreds from 12 hundreds and 50: 850.
1,6501,750 + 1,650 = 3,400, so the answer is 1,650.
1058 × 100 = 800, and 40 ÷ 8 = 5, so 105.
43236 × 10 = 360 and 36 × 2 = 72, so 360 + 72 = 432.
1,5006 × 250 = 1,500 (6 × 25 = 150, times 10).
Level 2
10,0324,236 + 8,795 = 13,031. Then subtract 3,000 and add 1: 10,032.
673,270 − 870 = 2,400. Then 2,400 ÷ 36 = 66 remainder 24, so round up to 67 packs.
Expert
Model answer: No. 7,800 is a hundred times 78, so the answer is a hundred times 6, which is 600. Check: 13 × 600 = 7,800.Look for: 7,800 is 78 hundreds; the quotient scales by 100 to 600; a check by multiplying.
12; 24; 36; 48Test each number: 12 = 4 × 3, 24 = 4 × 6, 36 = 4 × 9, 48 = 4 × 12. The ones digit must be double the tens digit.
0; 5; 10Notebooks cost a multiple of $8, and the rest must be a multiple of $5. 0 notebooks: 16 pens; 5: 8 pens; 10: 0 pens.
Worked example. Simplify 40/56. 1. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56. 2. The common factors are 1, 2, 4 and 8, so the greatest common factor is 8. 3. Divide both by 8: 40 ÷ 8 = 5 and 56 ÷ 8 = 7. 4. Check: 5 and 7 share no factor except 1, so 5/7 is in simplest form. So 40/56 in simplest form is 5/7.
Level 1
24 is 2 × 2, so multiply the top by 2 as well: 1 × 2 = 2.
103 became 6, so multiply by 2. The bottom becomes 5 × 2 = 10.
83 × 4 = 12, so multiply the top by 4: 2 × 4 = 8.
125 × 2 = 10, so the bottom is 6 × 2 = 12.
212 ÷ 4 = 3, so divide the top by 4 as well: 8 ÷ 4 = 2.
True4/6 simplifies to 2/3 and 6/9 simplifies to 2/3. They are equal.
1/2Divide both numbers by 4: 4 ÷ 4 = 1 and 8 ÷ 4 = 2.
2/3Divide both numbers by 3: 6 ÷ 3 = 2 and 9 ÷ 3 = 3.
5/6Divide both numbers by 2: 10 ÷ 2 = 5 and 12 ÷ 2 = 6.
2/58 ÷ 4 = 2 and 20 ÷ 4 = 5.
False7 and 21 are both in the 7 times table, so 7/21 simplifies to 1/3.
3/4Divide both numbers by 2: 6 ÷ 2 = 3 and 8 ÷ 2 = 4.
Level 2
2/3The greatest common factor of 14 and 21 is 7. 14 ÷ 7 = 2 and 21 ÷ 7 = 3.
5/8The greatest common factor of 20 and 32 is 4. 20 ÷ 4 = 5 and 32 ÷ 4 = 8.
3/49 of 12 parts are shaded. Divide both by 3 to get 3/4.
FalseThe greatest common factor of 28 and 42 is 14, so 28/42 is 2/3, not 3/4.
335 ÷ 7 = 5, so divide the top by 7 as well: 21 ÷ 7 = 3.
12/3514/35 and 35 share 7, and 15/35 shares 5. 12 and 35 share only 1.
3/5The greatest common factor of 24 and 40 is 8. 24 ÷ 8 = 3 and 40 ÷ 8 = 5.
4/5The greatest common factor is 12: 48 ÷ 12 = 4 and 60 ÷ 12 = 5. 8/10 and 12/15 equal 4/5 but can be simplified again.
5/6The greatest common factor of 30 and 36 is 6. 30 ÷ 6 = 5 and 36 ÷ 6 = 6.
3/4The greatest common factor of 45 and 60 is 15. 45 ÷ 15 = 3 and 60 ÷ 15 = 4.
Level 3
3/848 − 18 − 12 = 18 cakes are left, so 18/48. The greatest common factor is 6: 18 ÷ 6 = 3 and 48 ÷ 6 = 8.
7After dividing by 2 the fraction is 21/28. The greatest common factor of 21 and 28 is 7, giving 3/4.
The same12/20 simplifies to 3/5 and 21/35 simplifies to 3/5. The fractions are equal.
6384 ÷ 4 = 21, so both numbers were multiplied by 21 in the original. 3 × 21 = 63.
14; 352 + 5 = 7 and 49 ÷ 7 = 7, so multiply 2 and 5 by 7. The fraction is 14/35.
1; 3; 7; 9; 11; 13; 17; 1920 = 2 × 2 × 5, so n cannot be even or a multiple of 5. The numbers left are 1, 3, 7, 9, 11, 13, 17 and 19.
Expert
Model answer: No. 12 and 18 still share the factors 2, 3 and 6, so 12/18 can be simplified again to 2/3. Kai did not use the greatest common factor, which is 12.Look for: 12 and 18 still have a common factor above 1; simplest form is 2/3; dividing by the greatest common factor 12 does it in one step.
28; 2412 is 3 × 4, so the denominator is 7 × 4 = 28. 56 is 7 × 8, so the numerator is 3 × 8 = 24.
15; 20; 30; 60The simplified fraction is 1/5, 2/5, 3/5 or 4/5. Then 12/60 = 1/5, 12/30 = 2/5, 12/20 = 3/5 and 12/15 = 4/5.
Model answer: Always. Any common factor of the two numbers would also divide their difference, which is 1. So the only common factor is 1. For example, 7/8 and 9/10 cannot be simplified.Look for: the numbers differ by 1; a common factor must divide the difference; so no factor above 1. Examples such as 7/8 or 9/10 support the reason.
Worked example. Write 3/10 and 5/8 with their smallest common denominator. Give the numerators. 1. Multiples of 10: 10, 20, 30, 40. The number 8 goes into 40, so the denominator is 40. 2. 3/10: 40 ÷ 10 = 4, so multiply top and bottom by 4: 3/10 = 12/40. 3. 5/8: 40 ÷ 8 = 5, so multiply top and bottom by 5: 5/8 = 25/40. So the numerators are 12 and 25, because 3/10 = 12/40 and 5/8 = 25/40.
Level 1
24 is 2 times 2, so multiply the top by 2 as well: 1 × 2 = 2.
26 is 2 times 3, so the top is 1 × 2 = 2.
8The top went from 3 to 6, which is times 2, so the bottom is 4 × 2 = 8.
410 is 2 times 5, so the top is 2 × 2 = 4.
812 is 4 times 3, so the top is 2 × 4 = 8.
312 divided by 3 is 4, so divide the top by 3 as well: 9 ÷ 3 = 3.
2Eighths are a common denominator, since 4 goes into 8. 1/4 = 2/8.
8The top went from 1 to 4, which is times 4, so the bottom is 2 × 4 = 8.
36 is 3 times 2, so the top is 1 × 3 = 3.
312 is 3 times 4, so the top is 1 × 3 = 3.
True2 and 5 both go into 10, so 1/2 = 5/10 and 1/5 = 2/10.
12Multiples of 6: 6, 12. The number 4 goes into 12, so the answer is 12.
Level 2
2024 is 4 times 6, so the top is 5 × 4 = 20.
False5 goes into 10, but 4 does not, so 10 is not a common denominator of 2/5 and 1/4.
12Both 4 and 6 must go into the number. Only 12 works: 4 × 3 and 6 × 2.
15; 83/4 = 15/20 because 3 × 5 = 15. 2/5 = 8/20 because 2 × 4 = 8.
30The top went from 4 to 24, which is times 6, so the bottom is 5 × 6 = 30.
2130 is 3 times 10, so the top is 7 × 3 = 21.
24Multiples of 8: 8, 16, 24. The number 6 goes into 24, and not into 8 or 16.
No3 and 4 do not go into 7. A common denominator is a common multiple, such as 12.
4; 9; 51/3 = 4/12, 3/4 = 9/12 and 5/12 stays the same.
25; 14The smallest common multiple of 6 and 15 is 30. 5/6 = 25/30 and 7/15 = 14/30.
Level 3
9; 10Bar A is 3/8 = 9/24 (times 3). Bar B is 5/12 = 10/24 (times 2).
5/936 is 4 times 9, so divide the top by 4: 20 ÷ 4 = 5. The fraction is 5/9.
2; 3; 4; 6; 12The box number must go into 12. The numbers from 2 to 12 that do are 2, 3, 4, 6 and 12.
96Common denominators are multiples of 24: 24, 48, 72, 96. The next one, 120, is over 100.
18Both 6 and 9 must go into the number of pieces. The smallest such number is 18.
27A = 8/12, B = 9/12 and C = 10/12. The numerators total 8 + 9 + 10 = 27.
Expert
Model answer: No. 6 goes into 30, but 4 does not (30 ÷ 4 is not a whole number). A common denominator must be a multiple of both 4 and 6.Look for: both denominators must divide the new denominator; 4 does not go into 30; 12 or 24 would work.
3; 6; 12; 24The box number must go into 24 and must make 24 the smallest multiple shared with 8. 3, 6, 12 and 24 work; 2, 4 and 8 give 8, and 9 gives 72.
7/9; 7/12The smallest common denominator is 36. So the fractions are 28/36 = 7/9 and 21/36 = 7/12.
Model answer: Take 1/4 and 1/6. The product is 4 × 6 = 24, but 4 and 6 both go into 12, so 12 is a smaller common denominator.Look for two denominators with a shared factor (such as 4 and 6, or 6 and 9) and a common multiple smaller than their product.
Worked example. Which is greater, 9/10 or 13/15? 1. 10 and 15 both divide into 30, so use thirtieths. 2. 9/10 = 27/30, because 10 × 3 = 30 and 9 × 3 = 27. 3. 13/15 = 26/30, because 15 × 2 = 30 and 13 × 2 = 26. 4. Compare the numerators: 27 is greater than 26. So 9/10 is greater than 13/15.
Level 1
7/8The denominators are the same, so the greater numerator is the greater fraction: 7 is more than 3.
1/41/2 is 2/4, which is more than 1/4. So 1/4 is smaller.
32 × 3 = 6, so multiply the top by 3 as well: 1 × 3 = 3.
123 × 3 = 9, so multiply the bottom by 3 as well: 4 × 3 = 12.
1/6; 1/3; 1/2Write them as sixths: 3/6, 1/6, 2/6. Smallest first: 1/6, 1/3, 1/2.
2/32/3 = 8/12, and 8/12 is greater than 7/12.
more1/2 = 4/8, and 5/8 is greater than 4/8.
1/3Thirds are bigger pieces than fifths, so 1/3 is greater.
3/42/3 = 8/12 and 3/4 = 9/12. 9 is more than 8, so 3/4 is greater.
45 × 2 = 10, so multiply the top by 2 as well: 2 × 2 = 4.
False3/4 = 9/12 and 5/6 = 10/12. 9 is less than 10, so the statement is false.
4/5Use thirtieths: 5/6 = 25/30 and 4/5 = 24/30. 24 is less than 25.
Level 2
7/10Use twentieths: 3/4 = 15/20 and 7/10 = 14/20. 14 is less than 15.
>Use fortieths: 5/8 = 25/40 and 3/5 = 24/40. 25 is more than 24.
lessHalf of 15 is 7 and a half. 7 is less than 7 and a half, so 7/15 is less than 1/2.
False6/9 = 2/3 and 4/6 = 2/3. They are equal, so 6/9 is not greater.
KaiUse twenty-fourths: 5/6 = 20/24 and 7/8 = 21/24. Kai ran further.
11/24Use twenty-fourths: 3/8 = 9/24, 5/12 = 10/24 and 11/24. 11 is the most.
7/12Use thirty-sixths: 5/9 = 20/36 and 7/12 = 21/36. 21 is more than 20.
5/121/3 = 4/12 and 1/2 = 6/12. The only twelfth between them is 5/12.
103/7 = 9/21 and 1/2 = 10 and a half over 21. The whole number between 9 and 10.5 is 10.
Level 3
NiaUse thirtieths: 7/30, 3/10 = 9/30 and 4/15 = 8/30. The greatest is 9/30, which is Nia.
5/12; 6/12; 7/12; 8/121/3 = 4/12 and 3/4 = 9/12. The numerators strictly between 4 and 9 are 5, 6, 7 and 8.
BRead 7/12, 5/8 and 6/10. Use one denominator: 70/120, 75/120 and 72/120. Bar B is the greatest.
5Use thirty-fifths: 4/5 = 28/35. 5/7 = 25/35 is less than 28/35, but 6/7 = 30/35 is not. So 5.
4/11; 4/9; 4/7; 4/5The numerators are the same, so more parts means smaller parts. Order the denominators from largest to smallest: 11, 9, 7, 5.
Class BWrite the fractions: 5/12 and 7/16. Use forty-eighths: 20/48 and 21/48. Class B is greater.
Expert
Model answer: No. Compare using sixty-thirds: 5/9 = 35/63 and 4/7 = 36/63, so 5/9 is less. Lena ignored the denominators.Look for: not correct; a common denominator (63); 35/63 compared with 36/63; the numerators alone cannot decide when the denominators differ.
3/7; 3/8; 3/9; 3/10; 3/113/6 = 1/2 and 3/12 = 1/4, so the denominator must be between 6 and 12: 7, 8, 9, 10 or 11.
4/5; 5/6; 6/7; 7/8; 8/9; 9/10; 10/11Each fraction is 1 minus 1/d. 7/9 is 1 minus 2/9 and 11/12 is 1 minus 1/12, so 1/d is between 1/12 and 2/9. Then d is 5 to 11.
Model answer: Never. With the same numerator, a greater denominator means smaller parts. For example 3/5 is greater than 3/8.Look for: never; the same number of parts taken, but a greater denominator makes each part smaller; a worked example such as 3/5 and 3/8.
Worked example. Write in order, smallest first: 3 1/4, 8/3 and 3 1/3. 1. Write 8/3 as a mixed number: 8 ÷ 3 = 2 remainder 2, so 2 2/3. 2. Compare the whole parts: 2 is less than 3, so 8/3 is the smallest. 3. Compare 1/4 and 1/3 as twelfths: 3/12 is less than 4/12, so 3 1/4 is less than 3 1/3. So the order is 8/3, 3 1/4, 3 1/3.
Level 1
1 3/47 quarters is 4 quarters (1 whole) and 3 quarters left, so 1 3/4.
7/32 wholes is 6 thirds, and 1 more third makes 7 thirds.
8Each whole is 4 quarters, so 2 wholes is 2 × 4 = 8 quarters.
4 1/29 halves is 4 wholes (8 halves) and 1 half left, so 4 1/2.
17/62 wholes is 12 sixths. 12 + 5 = 17, so 17/6.
35/84 wholes is 4 × 8 = 32 eighths. 32 + 3 = 35, so 35/8.
9/4The denominators are the same, so the larger numerator is the greater fraction.
<1 3/4 is less than 2 wholes, because 3/4 is less than 1 whole more.
2 1/2The whole parts are equal. A half is greater than a quarter.
211/5 is 2 1/5, which is more than 2, so 2 is smaller.
1 1/2; 5/2; 35/2 is 2 1/2. So the numbers are 2 1/2, 1 1/2 and 3. Smallest first: 1 1/2, 2 1/2, 3.
False9/4 is 2 1/4, and 1/4 is less than 1/2, so 9/4 is less than 2 1/2.
Level 2
3 2/517 ÷ 5 = 3 remainder 2, so 3 whole and 2/5.
39/839/8 is 4 7/8, and 7/8 is greater than 5/8, so 39/8 is greater.
2 1/313/6 is 2 1/6. The whole parts match, and 1/3 (2/6) is greater than 1/6.
54 wholes is 24 sixths. 29 − 24 = 5, so the missing number is 5.
2 1/3; 5/2; 11/4As mixed numbers: 2 1/2, 2 1/3 and 2 3/4. Compare 1/3, 1/2 and 3/4 as twelfths: 4, 6 and 9.
11/4A is 3 marks past 2. Each mark is 1/4, so A is 2 3/4. 2 × 4 + 3 = 11, so 11/4.
False11/3 is 3 2/3, so the two numbers are equal. Neither is greater.
2 1/27/3 is 2 1/3. 1/2 is greater than 1/3, so 2 1/2 km is the longer walk.
Same amount13/3 is 4 1/3 (12 thirds make 4 wholes, and 1 third is left), so the two swim the same.
5 1/321/4 is 5 1/4. The whole parts match, and 1/3 is greater than 1/4, so 5 1/3 is greatest.
Level 3
Bag C17/6 is 2 5/6, so Bags A and B hold the same. Compare 5/6 with 3/4 as twelfths: 10/12 and 9/12, so Bag C is least.
11/5; 12/5; 13/5; 14/52 is 10/5 and 3 is 15/5. The numerators strictly between 10 and 15 are 11, 12, 13 and 14.
11/4Each mark is 1/12. A is 4 marks past 2 (2 1/3) and B is 9 marks past 2 (2 3/4). B is greater: 2 × 4 + 3 = 11, so 11/4.
13/43 is 12/4 and 3 1/2 is 14/4. The only numerator strictly between 12 and 14 is 13, so 13/4.
10/3; 3 2/5; 31/9; 3 1/2As mixed numbers: 3 1/2, 3 1/3, 3 2/5 and 3 4/9. Compare the parts as 90ths: 45, 30, 36 and 40.
276 1/2 is 26/4, which is not greater than itself. So the smallest numerator is 27.
Expert
Model answer: No. 5/2 is 20/8, and 13/8 is less than 20/8. A greater numerator does not make a greater fraction when the denominators differ.Look for: the child says Amira is wrong; converts to a common denominator (20/8 against 13/8) or to mixed numbers (1 5/8 against 2 1/2); says the denominators must be compared too.
Model answer: Sometimes. 2 1/2 is greater than 7/4, but 1 1/4 is less than 9/4.Look for: the answer sometimes; one example where the mixed number is greater and one where it is less, each checked by converting to the same form.
3 11/20; 3 13/207/2 is 3 10/20 and 11/3 is just over 3 13/20 (3 13.3/20). The fraction part is 11/20, 12/20 or 13/20. 12/20 simplifies to 3/5, so only 11/20 and 13/20 remain.
3 5/8; 5 3/8; 8 3/5Each digit can be the whole part, and the other two make the fraction with the smaller digit on top: 3 5/8, 5 3/8 and 8 3/5. Order by the whole part.
Worked example. Calculate 2/3 + 1/4. 1. The smallest common multiple of 3 and 4 is 12, so use twelfths. 2. 2/3 = 8/12 and 1/4 = 3/12. 3. Add the numerators: 8 + 3 = 11, so the total is 11/12. So 2/3 + 1/4 = 11/12, which is already in its simplest form.
Level 1
3/5The denominators match, so add the numerators: 1 + 2 = 3 fifths.
5/9The denominators match, so subtract the numerators: 7 − 2 = 5 ninths.
7/83/4 = 6/8, so 6/8 + 1/8 = 7/8.
1/21/3 = 2/6, so 5/6 − 2/6 = 3/6, which is 1/2.
1 2/53 + 4 = 7 fifths, which is 5/5 + 2/5 = 1 2/5.
1/21/3 = 2/6, so the missing part is 5/6 − 2/6 = 3/6, which is 1/2.
3/41/2 = 2/4, so 2/4 + 1/4 = 3/4.
1/21/5 = 2/10, so 7/10 − 2/10 = 5/10, which is 1/2.
2/31/4 = 3/12, so 5/12 + 3/12 = 8/12, which is 2/3.
True3/4 = 6/8, and 6/8 − 1/8 = 5/8, so the statement is true.
5/6Use sixths: 3/6 + 2/6 = 5/6.
1/6Use sixths: 3/6 − 2/6 = 1/6.
Level 2
11/12Use twelfths: 9/12 + 2/12 = 11/12.
1/12Use twelfths: 10/12 − 9/12 = 1/12.
1 2/15Use fifteenths: 12/15 + 5/15 = 17/15, which is 1 2/15.
7/12Use twelfths: 10/12 − 3/12 = 7/12.
5/121/3 + 1/4 = 4/12 + 3/12 = 7/12 drunk, so 12/12 − 7/12 = 5/12 is left.
5/61/4 = 3/12, so 7/12 + 3/12 = 10/12. Divide both numbers by 2 to get 5/6.
FalseAdding numerators and denominators is wrong. 3/5 + 1/3 = 9/15 + 5/15 = 14/15.
11/15The bars show 1/3 and 2/5. Use fifteenths: 5/15 + 6/15 = 11/15.
17/24Use twenty-fourths: 9/24 + 8/24 = 17/24.
11/18Use eighteenths: 14/18 − 3/18 = 11/18.
Level 3
13/303/10 + 4/15 = 9/30 + 8/30 = 17/30 painted. So 30/30 − 17/30 = 13/30 is not painted.
1/33/4 = 9/12, so Sofia eats 9/12 − 5/12 = 4/12, which is 1/3.
1 7/12The largest are 3/4 and 5/6. 9/12 + 10/12 = 19/12, which is 1 7/12.
6; 71/3 = 4/12 and 3/4 = 9/12. The total 4 + n twelfths must be more than 9 and less than 12, so n is 6 or 7.
11/24Uma reads 3/8 + 1/6 = 9/24 + 4/24 = 13/24. So 24/24 − 13/24 = 11/24 is left.
1 19/20Use twentieths: 14/20 + 15/20 + 10/20 = 39/20, which is 1 19/20 km.
Expert
Model answer: Kai added the numerators and the denominators. The pieces are different sizes, so both fractions must be changed to twentieths first: 5/20 + 8/20 = 13/20.Look for: the denominators must be made the same before adding; 1/4 and 2/5 are not counted in the same size pieces; the correct total is 13/20.
3The pairs are 1/3 and 1/4, 1/4 and 1/6, and 1/6 and 1/12. Writing each as twelfths (6, 4, 3, 2, 1) makes the differences easy to see.
1/6In twelfths, 3a + 2b = 11. Either a = 1 and b = 4, or a = 3 and b = 1. Only a = 3 makes the quarter fraction larger, so the sixths fraction is 1/6.
Model answer: Sometimes. 1/4 + 1/4 = 1/2, which is less than 1. 3/4 + 1/2 = 1 1/4, which is more than 1.Look for the answer sometimes, with one example that totals less than 1 and one that totals more than 1.
Model answer: 1/3 is greater than 1/4, so 1/4 − 1/3 cannot be taken as 1/12. Exchange a whole: 4 5/4 is 4 15/12, so 4 15/12 − 2 4/12 = 2 11/12 is the answer.Look for: Hugo subtracted the smaller fraction from the larger; the first fraction is smaller, so a whole must be exchanged; correct answer 2 11/12.
7; 8; 9; 102 5/6 − 1 3/4 = 1 1/12. The box plus 1 1/12 must be between 8 and 12: 7 gives 8 1/12 and 10 gives 11 1/12, but 6 gives 7 1/12 and 11 gives 12 1/12.
3; 2Fractions: 1/2 + ?/3 must end in sixths: 3/6 + 4/6 = 7/6 = 1 1/6, so the second box is 2. Wholes: 6 − 2 − 1 = 3.
Model answer: Always. The fraction parts make 1/4 + 3/4 = 1 whole, and whole numbers added to whole numbers are whole, so the total is a whole number.Look for: 1/4 + 3/4 = 1; the wholes add to a whole number; a reason that holds for any whole parts, or an example such as 2 1/4 + 5 3/4 = 8.
Worked example. Calculate 3/4 × 2/9. Give the answer in simplest form. 1. Multiply the numerators: 3 × 2 = 6. 2. Multiply the denominators: 4 × 9 = 36. The product is 6 over 36. 3. The highest common factor of 6 and 36 is 6, so divide both by 6. So 3/4 × 2/9 = 1/6.
Level 1
3/53 lots of one fifth is three fifths.
4/94 lots of one ninth is four ninths.
False2 × 1/6 = 2/6. Only the numerator is multiplied; the denominator stays 6.
55 lots of one twelfth is five twelfths, so the missing numerator is 5.
6/113 × 2 = 6, and the denominator stays 11.
2/54 × 1/10 = 4/10. Divide the top and bottom by 2 to get 2/5.
1/6Multiply the denominators: 2 × 3 = 6. The numerators are 1 × 1 = 1.
1/10"Of" means multiply. 1/2 × 1/5 = 1/10.
1/8Multiply the denominators: 2 × 4 = 8. The answer is 1/8, not 1/6.
1/10Half of 1/5 is 1/2 × 1/5 = 1/10.
1/93 × 3 = 9 for the denominator, and 1 × 1 = 1 for the numerator.
5/85 × 3 = 15 and 6 × 4 = 24. Divide both by 3 to get 5/8.
3/104 × 3 = 12 and 5 × 8 = 40. Divide both by 4 to get 3/10.
2/56 of the 15 cells are shaded, so the product is 6/15. Divide both by 3 to get 2/5.
5/12"Of" means multiply. 1/2 × 5/6 = 5/12.
True5 × 3 = 15 and 6 × 10 = 60. Divide both by 15 to get 1/4, so the statement is true.
1/22/3 of 3/4 is 2/3 × 3/4 = 6/12, which simplifies to 1/2.
7/157 × 4 = 28 and 12 × 5 = 60. Divide both by 4 to get 7/15.
3/49 × 5 = 45 and 10 × 6 = 60. Divide both by 15 to get 3/4.
1/35/9 × 3/5 = 15/45. Divide both by 15 to get 1/3.
Level 3
2/3Left after the first part: 1/2. Then 1/3 × 1/2 = 1/6. Total walked: 1/2 + 1/6 = 3/6 + 1/6 = 4/6 = 2/3.
1/3The numerators are both 5, so the fraction is 1 over something. 6 × 3 = 18, so the fraction is 1/3.
1; 2; 35/6 × n/10 = 5n/60 = n/12. This is less than 1/3 = 4/12 when n is less than 4, so n is 1, 2 or 3.
3/8Luca keeps 1 − 2/5 = 3/5 of the 5/8. 3/5 × 5/8 = 15/40 = 3/8.
122/3 × 3/4 = 1/2, so the 6 girls are half of the club. The club has 12 pupils.
3/5The two largest cards give the largest product: 3/4 × 4/5 = 12/20 = 3/5.
Expert
Model answer: Always true. Multiplying by a proper fraction takes only part of the number, so the answer is smaller. For example 1/2 × 3/4 = 3/8, which is smaller than 1/2 and 3/4.Look for: always; multiplying by a fraction less than 1 makes a number smaller; a worked example with both fractions compared to the product.
2; 4; 5; 10The denominators multiply to 20: 2 × 10 and 4 × 5. So the possible denominators are 2, 4, 5 and 10.
1/8Each product is 1 over two consecutive numbers multiplied. 8 × 9 = 72, so 1/8 × 1/9 = 1/72 and the first fraction is 1/8.
Model answer: Sami added the denominators (5 + 4 = 9). The denominators should be multiplied: 5 × 4 = 20. So 2/5 × 3/4 = 6/20 = 3/10.Look for: numerators multiplied correctly (6); denominators added instead of multiplied; the correct answer 6/20, simplified to 3/10.
Worked example. Calculate 3/5 ÷ 2. 1. The numerator 3 does not divide by 2, so rename 3/5 with a bigger denominator. 2. 3/5 = 6/10, because both parts are multiplied by 2. 3. Divide the numerator: 6 ÷ 2 = 3, so 6/10 ÷ 2 = 3/10. 4. Check: 3/10 is 3 tenths, and two lots of 3 tenths make 6 tenths, which is 3/5. So 3/5 ÷ 2 = 3/10.
Level 1
24 is 2 × 2, so multiply the top by 2 as well: 1 × 2 = 2.
26 is 3 × 2, so multiply the top by 2: 1 × 2 = 2.
8The top went from 3 to 6, which is × 2, so the bottom is 4 × 2 = 8.
8/1212 is 3 × 4, so multiply the top by 4 as well: 2 × 4 = 8.
FalseMultiply the top and bottom of 3/4 by 3 to get 9/12, not 8/12. So the statement is false.
3The bottom went from 8 to 4, which is ÷ 2, so the top is 6 ÷ 2 = 3.
2/5Divide the numerator: 4 ÷ 2 = 2. The denominator stays 5.
1/8Divide the numerator: 3 ÷ 3 = 1. The denominator stays 8.
2/7Divide the numerator: 6 ÷ 3 = 2. The denominator stays 7.
2/9Sharing equally is dividing: 8/9 ÷ 4. Divide the numerator: 8 ÷ 4 = 2, so 2/9.
TrueDivide the numerator: 6 ÷ 3 = 2, and the denominator stays 10. So it is true.
1/610 ÷ 5 = 2, so the answer is 2/12. Divide the top and bottom by 2 to get 1/6.
Level 2
3/11Divide the numerator: 9 ÷ 3 = 3. The denominator stays 11.
1/12Dividing a unit fraction makes the denominator bigger: 3 × 4 = 12, so 1/12.
1/49 ÷ 3 = 3, so the answer is 3/12. Divide the top and bottom by 3 to get 1/4.
False6 ÷ 2 = 3, so 6/8 ÷ 2 = 3/8. Dividing the top and bottom by 2 gives 3/4, which is the same size as 6/8.
2/152 does not divide by 3. Multiply the denominator by 3 instead: 5 × 3 = 15, so 2/15.
3/83/4 ÷ 2: 3 does not divide by 2, so multiply the denominator by 2. Each gets 3/8.
1/62 ÷ 4 does not work, so 2/3 ÷ 4 = 2/12, which simplifies to 1/6.
55/6 ÷ 2 = 5/12, because 5 does not divide by 2 and 6 × 2 = 12.
1/83 does not divide by 6, so 4 × 6 = 24 gives 3/24. Divide the top and bottom by 3 to get 1/8.
2/214/7 ÷ 6 = 4/42, because 7 × 6 = 42. Divide the top and bottom by 2 to get 2/21.
Level 3
1/4The rest is 1 − 1/4 = 3/4. Then 3/4 ÷ 3 = 1/4, so each friend gets 1/4 of the cake.
5/6 ÷ 25/6 ÷ 2 = 5/12 and 3/4 ÷ 3 = 1/4 = 3/12. Since 5/12 is more than 3/12, 5/6 ÷ 2 is larger.
1/5The shaded part is 6/10. Then 6/10 ÷ 3 = 2/10, which is 1/5 of the whole bar.
2/3Work backwards: multiply by 5. 2/15 × 5 = 10/15, which simplifies to 2/3.
85/8 ÷ n = 5/(8n). Test values: n = 7 gives 5/56, which is more than 1/12. n = 8 gives 5/64, which is less.
4; 84/9 ÷ n = 4/(9n). The top must divide into the bottom. This works when n = 4 (1/9) and n = 8 (1/18).
Expert
Model answer: No. 4/5 ÷ 2 = 2/5, because only the numerator needs dividing. Hugo's 2/10 is equal to 1/5, which is half the correct answer.Look for: the correct answer 2/5; only the top is divided when it divides exactly; Hugo divided twice, so his answer is too small.
2; 3; 6; 93/4 ÷ n = 3/(4n). Test each n and simplify: 3/8, 1/4, 1/8 and 1/12 work for n = 2, 3, 6 and 9. The other values leave a denominator over 12.
1/16Work back: 3/32 × 4 = 12/32 = 3/8 is the original. Then 3/8 ÷ 6 = 3/48, which simplifies to 1/16.
Model answer: Sometimes. 1/3 ÷ 2 = 1/6 has an even denominator, but 2/3 ÷ 2 = 1/3 has an odd one.Look for two examples, one that gives an even denominator and one that does not. A single example proves 'sometimes' only when both kinds are shown.
Worked example. Convert 3,045 g to kg. 1. 1,000 g make 1 kg, so divide by 1,000: each digit moves three places right. 2. 3,045 g becomes 3.045 kg: 3 ones, 0 tenths, 4 hundredths, 5 thousandths. 3. The 0 must stay as a placeholder: 3.45 kg would be 3,450 g. So 3,045 g = 3.045 kg.
Level 1
3,000 m1 km is 1,000 m, so multiply 3 by 1,000.
50 mm1 cm is 10 mm, so multiply 5 by 10.
7 km1,000 m is 1 km, so divide 7,000 by 1,000.
4 litres1,000 ml is 1 litre, so divide 4,000 by 1,000.
600 cm1 m is 100 cm, so multiply 6 by 100.
2,500 g1 kg is 1,000 g. 2 kg is 2,000 g and 0.5 kg is 500 g, so 2,500 g.
1,500 mlMultiply by 1,000: 1.5 × 1,000 = 1,500.
4.5 kgDivide by 1,000: 4,500 g is 4 kg and 500 g, which is 4.5 kg.
6.3 kmDivide by 1,000: 6,300 m is 6 km and 300 m, which is 6.3 km.
True1 cm is 10 mm, so 7.5 cm is 7.5 × 10 = 75 mm.
80 cm1 m is 100 cm, so 0.8 × 100 = 80.
1.8 litresDivide by 1,000: 1,800 ml is 1 litre and 800 ml, which is 1.8 litres.
Level 2
6.25 litresDivide by 1,000: 6,250 ml is 6.250 litres, written 6.25 litres.
450 mMultiply by 1,000: each digit moves three places left, so 0.45 km is 450 m.
5.008 kgDivide by 1,000. The zeros in 5,008 stay as placeholders, giving 5.008 kg.
3,060 ml3.06 × 1,000 = 3,060. The 0 in the tenths place stays.
False0.8 kg is 800 g, and 800 g is less than 850 g, so the statement is false.
3.4 m40 cm is 0.4 m, so 3 m 40 cm is 3.4 m.
1,850 ml2.75 litres is 2,750 ml. Then 2,750 − 900 = 1,850 ml.
1,400 gArrow A is four marks after 1, so 1.4 kg. Then 1.4 × 1,000 = 1,400 g.
2,450 mmIn cm the lengths are 240, 235 and 245, so 2,450 mm is the longest.
175 cm2,400 mm is 240 cm and 0.65 m is 65 cm. Then 240 − 65 = 175 cm.
Level 3
2.25 kg750 g is 0.75 kg. Then 1.5 + 0.75 = 2.25 kg.
1.5 kgWork backwards: add the butter used. 350 g is 0.35 kg, and 1.15 + 0.35 = 1.5 kg.
231; 232; 233; 2342.3 m is 230 cm and 2.35 m is 235 cm. The whole numbers of cm between them are 231 to 234.
2.2 kgEach mark is 0.25 kg, so B is 1.75 kg. Then 450 g is 0.45 kg, and 1.75 + 0.45 = 2.2 kg.
7.5 km2,850 m is 2.85 km. Then 3.4 + 2.85 + 1.25 = 7.5 km.
107.5 litres is 7,500 ml. Nine pots hold 7,200 ml, which is not enough, so Kai needs 10 pots.
Expert
Model answer: Dev lost the 0 in the tenths place. 4,030 g is 4 kg and 30 g, so it is 4.03 kg. 4.3 kg would be 4,300 g.Look for: the 0 placeholder dropped, the correct 4.03 kg, and a check that 4.3 kg is 4,300 g.
3,025; 3,075; 3,125; 3,175The multiples of 25 are 3,025 to 3,175. Those ending in 0 give fewer decimal places in kg, leaving 3,025; 3,075; 3,125 and 3,175.
575 m2.5 km is 2,500 m. Take away the extra 1,350 m to leave 1,150 m, then halve it: 575 m.
Model answer: No. 0.7 km is 700 m and 6,500 cm is 65 m, so 0.7 km is longer. Change both to the same unit before comparing.Look for: units must match before comparing, 0.7 km = 700 m, 6,500 cm = 65 m, so 0.7 km is longer.
Worked example. 5 miles is about 8 km. Convert 35 miles to km. 1. 35 ÷ 5 = 7, so 35 miles is seven lots of 5 miles. 2. Each 5 miles is about 8 km, so multiply: 7 × 8 = 56. 3. The answer is about 56 km, because 5 miles is only about 8 km. So 35 miles is about 56 km.
Level 1
24Four packs of 6 pens: 6 × 4 = 24.
$2Divide the cost by the number of stickers: $10 ÷ 5 = $2.
$121 bun costs $6 ÷ 3 = $2. Then 6 buns cost 6 × $2 = $12.
$408 books is double 4 books, so the cost doubles: $20 × 2 = $40.
60 km6 litres is double 3 litres, so the distance doubles: 30 km × 2 = 60 km.
16 km10 miles is two lots of 5 miles, so 2 × 8 km = 16 km.
24 km15 miles is three lots of 5 miles, so 3 × 8 km = 24 km.
1016 km is two lots of 8 km, so 2 × 5 miles = 10 miles.
32 km20 miles is four lots of 5 miles, so 4 × 8 km = 32 km.
2540 km is five lots of 8 km, so 5 × 5 miles = 25 miles.
48 km30 miles is six lots of 5 miles, so 6 × 8 km = 48 km.
Level 2
72 km45 ÷ 5 = 9 lots of 5 miles. Then 9 × 8 = 72 km.
6096 ÷ 8 = 12 lots of 8 km. Then 12 × 5 = 60 miles.
120 km75 ÷ 5 = 15 lots of 5 miles. Then 15 × 8 = 120 km.
3048 km is 3 times 16 km, so the miles are 3 times 10 miles: 30 miles.
150240 ÷ 8 = 30 lots of 8 km, and 30 × 5 = 150 miles.
4064 ÷ 8 = 8 lots of 8 km. Then 8 × 5 = 40 miles.
280 km175 ÷ 5 = 35 lots of 5 miles. Then 35 × 8 = 280 km.
85136 ÷ 8 = 17 lots of 8 km. Then 17 × 5 = 85 miles.
128 km55 + 25 = 80 miles, which is 16 lots of 5 miles. Then 16 × 8 = 128 km.
5048 + 32 = 80 km, which is 10 lots of 8 km. Then 10 × 5 = 50 miles.
Level 3
375600 ÷ 24 = 25, so the miles are 25 times 15 miles: 375 miles.
640 km215 + 185 = 400 miles, which is 80 lots of 5 miles. Then 80 × 8 = 640 km.
40; 45; 50; 55; 60; 65Miles m give m ÷ 5 × 8 km. The km must be between 60 and 110, which gives 40, 45, 50, 55, 60 and 65 miles (64 km to 104 km).
100320 ÷ 2 = 160 km each day. 160 ÷ 8 = 20 lots of 8 km, and 20 × 5 = 100 miles.
176 km25 + 35 + 50 = 110 miles, which is 22 lots of 5 miles. Then 22 × 8 = 176 km.
140 miles140 miles is 28 lots of 5 miles, which is 28 × 8 = 224 km. That is more than 220 km.
Expert
Model answer: Dev divided by 8 and multiplied by 5, which converts km to miles. To go from miles to km, divide by 5 and multiply by 8: 40 ÷ 5 × 8 = 64 km.Look for: Dev used the km to miles method on a miles value. Also, 25 km is less than 40 miles, which cannot be right because 1 mile is longer than 1 km. Correct answer 64 km.
Model answer: Always. A multiple of 5 miles is a whole number of lots of 5 miles, and each lot is about 8 km. So the km are that number of lots times 8, a multiple of 8.Look for: always, with the reason that each 5 miles gives 8 km, so n lots of 5 miles give n × 8 km. An example such as 15 miles giving 24 km supports but does not prove it.
115; 135400 km is 50 lots of 8 km, so 250 miles. Take off the extra 20 miles: 230, then halve to get 115. The longer leg is 115 + 20 = 135.
15; 30; 45; 60; 75; 90The km are 8 × n for n lots of 5 miles, and 8 has no factor 3, so n must be a multiple of 3. That gives 15, 30, 45, 60, 75 and 90 miles.
Worked example. Add 3.25 kg and 640 g, then find how many grams more make 5 kg. 1. Change kg to g first: 3.25 × 1,000 = 3,250 g. 2. Add the masses: 3,250 + 640 = 3,890 g. 3. Change 5 kg to 5,000 g, then subtract: 5,000 − 3,890 = 1,110 g. So 1,110 g more are needed to make 5 kg.
Level 1
180 minutesThere are 60 minutes in 1 hour, so 3 × 60 = 180.
240 secondsThere are 60 seconds in 1 minute, so 4 × 60 = 240.
48There are 24 hours in 1 day, so 2 × 24 = 48.
35There are 7 days in 1 week, so 5 × 7 = 35.
135 minutes2 hours is 120 minutes. Then 120 + 15 = 135.
9 hoursThere are 60 minutes in 1 hour, so 540 ÷ 60 = 9.
3,000 mThere are 1,000 m in 1 km, so 3 × 1,000 = 3,000.
2,500 gThere are 1,000 g in 1 kg, so 2.5 × 1,000 = 2,500.
4 litresThere are 1,000 ml in 1 litre, so 4,000 ÷ 1,000 = 4.
1 kg750 + 250 = 1,000 g. There are 1,000 g in 1 kg, so 1,000 g is 1 kg.
True1.5 × 1,000 = 1,500, so the statement is true.
600 g0.6 × 1,000 = 600 g.
Level 2
3,400 m3.4 × 1,000 = 3,400 m.
205 seconds3 minutes is 180 seconds. Then 180 + 25 = 205.
4,650 gIn grams the masses are 4,050; 4,005; 4,650 and 380. The greatest is 4,650 g.
6,350 m3.6 + 2.75 = 6.35 km. Then 6.35 × 1,000 = 6,350 m.
450 secondsOne song is 180 + 45 = 225 seconds. Two songs: 225 × 2 = 450 seconds.
Level 3
900 g2.4 kg is 2,400 g and 0.65 kg is 650 g. Then 2,400 − 850 − 650 = 900 g.
3.75 litres8,750 ml is 8.75 litres. Then 12.5 − 8.75 = 3.75 litres.
1.95 kg350 g is 0.35 kg and 400 g is 0.4 kg. Then 1.2 + 0.35 + 0.4 = 1.95 kg.
72.5 m is 250 cm. Then 250 ÷ 35 is 7 with some left over, so 7 whole pieces.
460; 470; 480; 4900.45 litres is 450 ml and 0.5 litres is 500 ml. The multiples of 10 between them are 460, 470, 480 and 490.
3,500 mOne hour is 60 minutes, which is 5 lots of 12 minutes. 0.7 km is 700 m, and 700 × 5 = 3,500 m.
Expert
Model answer: No. The 0.5 in 2.5 is half an hour, which is 30 minutes, so 2.5 hours is 2 hours 30 minutes. Amira wrote the 5 as 5 minutes.Look for: 0.5 hour means half of 60 minutes, so 30 minutes; a decimal part of an hour is not a number of minutes; correct time 2 hours 30 minutes.
200; 240; 300; 4001.2 litres is 1,200 ml. The sizes between 160 and 420 that divide 1,200 exactly are 200, 240, 300 and 400.
Model answer: Sometimes. 60 minutes is 1 whole hour, but 20 minutes is only a third of an hour. It is a whole number of hours only when it is also a multiple of 60.Look for: sometimes; an example that works (60 or 120) and one that does not (20 or 40); whole hours need a multiple of 60.
5 kmThe extra distance is 0 + 2.5 + 5 = 7.5 km. Then 22.5 − 7.5 = 15 km, and 15 ÷ 3 = 5 km on the first day.