Answers for parents. Each unit has a worked example, then the answers for every level.
Unit 1: Roman numerals to 1,000
Worked example. Read MCMXCII as a number. 1. Split the numeral into parts: M, CM, XC and II. 2. M is 1,000. CM is 1,000 − 100 = 900. 3. XC is 100 − 10 = 90. II is 2. 4. Add the parts: 1,000 + 900 + 90 + 2 = 1,992. So MCMXCII is 1,992.
Level 1
8V is 5 and each I is 1, so 5 + 1 + 1 + 1 = 8.
14X is 10. IV is 5 − 1 = 4, so 10 + 4 = 14.
25X + X + V = 10 + 10 + 5 = 25.
34XXX is 30. IV is 4, so 30 + 4 = 34.
46XL is 50 − 10 = 40. VI is 6, so 40 + 6 = 46.
98XC is 100 − 10 = 90. VIII is 8, so 90 + 8 = 98.
200C is 100, so CC is 100 + 100 = 200.
165C + L + X + V = 100 + 50 + 10 + 5 = 165.
340CCC is 300. XL is 50 − 10 = 40, so 300 + 40 = 340.
112C is 100, X is 10 and II is 2, so 100 + 10 + 2 = 112.
TrueCD is D − C = 500 − 100 = 400. So the statement is true.
CDXXCDXX is 400 + 20 = 420 and CCCLXX is 300 + 70 = 370, so CDXX is greater.
Level 2
49; 64; 94XLIX is 40 + 9 = 49. LXIV is 50 + 10 + 4 = 64. XCIV is 90 + 4 = 94.
447CD is 400, XL is 40 and VII is 7, so 400 + 40 + 7 = 447.
774D is 500, CC is 200, LXX is 70 and IV is 4, so 500 + 200 + 70 + 4 = 774.
1,660M is 1,000, D is 500, C is 100, L is 50 and X is 10. The total is 1,660.
1,845M is 1,000, DCCC is 800, XL is 40 and V is 5, so 1,000 + 800 + 40 + 5 = 1,845.
FalseM is 1,000, CD is 400 and XL is 40, so MCDXL is 1,440, not 1,640. The statement is false.
MCMXLIX1,949 is 1,000 + 900 + 40 + 9, which is M + CM + XL + IX. MCMXLIX is the numeral.
2,016MM is 2,000, X is 10, V is 5 and I is 1, so 2,000 + 10 + 5 + 1 = 2,016.
MICMXCIX is 900 + 90 + 9 = 999. MI is 1,000 + 1 = 1,001. So MI is greater.
Model answer: MCMLXXXVIILook for: 1,987 is 1,000 + 900 + 50 + 30 + 7, which is M + CM + L + XXX + VII.
Level 3
755DXL is 500 + 40 = 540 and CCXV is 200 + 15 = 215. Then 540 + 215 = 755.
34MCM is 1,000 + 900 = 1,900. Then 1,934 − 1,900 = 34, which is XXXIV.
14; 16XIV is 10 + 4 = 14 and XVI is 10 + 5 + 1 = 16. No other order of the three letters is a correct numeral.
13XL is 40 and XXVII is 27. Then 40 − 27 = 13 books are left.
XIV is 4. Opposite numbers on a clock are 6 apart, and 4 + 6 = 10, which is X.
1,937MMXII is 2,012. Then 2,012 − 75 = 1,937.
Expert
4; 9; 40; 90; 400; 900Only I, X and C can go first. They give IV 4, IX 9, XL 40, XC 90, CD 400 and CM 900. Pairs such as IL or VX are not correct.
Model answer: No. I can only go before V or X, so IL is not allowed. 49 is 40 + 9, which is XL then IX, so the numeral is XLIX.Look for: I is only taken away from V and X; 49 is split into 40 and 9; the correct numeral is XLIX.
Model answer: No. D is one letter and stands for 500, but XVIII has five letters and stands for only 18.Look for one counter-example, such as D against XVIII or CM against MI, with both values given.
1,666Writing the letters from largest to smallest, MDCLXVI, adds every value: 1,000 + 500 + 100 + 50 + 10 + 5 + 1 = 1,666. Any subtraction makes the total smaller.
Worked example. Find the value of the 8 in 80,413. 1. Say it: eighty thousand, four hundred and thirteen. 2. Columns: 8 ten thousands, 0 thousands (a placeholder), 4 hundreds. 3. The 8 is in the ten thousands column, so it is worth 80,000. So the 8 in 80,413 is worth 80,000.
Level 1
7Read the columns of 8,725: 8 thousands, 7 hundreds, 2 tens, 5 ones. The hundreds digit is 7.
6,000The 6 is in the thousands column, so it is worth 6,000.
4,2514 thousands, 2 hundreds, 5 tens and 1 one make 4,251.
200The 2 is in the hundreds column, so it stands for 200.
6,040There are no hundreds, so write 0 in the hundreds column: 6,040.
80The 8 is in the tens column, so it is worth 80.
34,000Put 3 in the ten thousands column and 4 in the thousands column, then zeros: 34,000.
40,000The 4 is in the ten thousands column, so it is worth 40,000.
74,3007 ten thousands, 4 thousands and 3 hundreds, then 0 tens and 0 ones: 74,300.
3,000The 3 is in the thousands column, so it is worth 3,000.
TrueThe 5 is in the ten thousands column, so it is worth 50,000. The statement is true.
40,0094 ten thousands and 9 ones, with 0 in the thousands, hundreds and tens: 40,009.
Level 2
90,4069 ten thousands, 4 hundreds and 6 ones, with 0 in the thousands and tens: 90,406.
5,000The 5 is in the thousands column, so it is worth 5,000.
200The 2 is in the hundreds column, so the missing part is 200. The tens digit is 0.
FalseThe 9 is in the thousands column, so it is worth 9,000, not 90,000.
3,000The 3 is in the thousands column, so it is worth 3,000.
50,309There are no thousands and no tens, so write 0 in both columns: 50,309.
61,205Only 61,205 has its 6 in the ten thousands column. The others have 6 thousands and 6 hundreds.
0Read the columns: 2 ten thousands, 0 thousands, 4 hundreds. The digit is 0.
80; 60,000The 8 is in the tens column: 80. The 6 is in the ten thousands column: 60,000.
41,006Forty-one thousand and six has 0 hundreds and 0 tens, so it is 41,006. Femi put the 6 in the hundreds.
Level 3
40,000The ten thousands digit went from 3 (30,000) to 7 (70,000). Then 70,000 - 30,000 = 40,000.
50,070The 5 is worth 50,000 and the 7 is worth 70. Then 50,000 + 70 = 50,070.
6363,000 is 6 ten thousands and 3 thousands. 6 ten thousands is 60 thousands, so 60 + 3 = 63.
10,358A 5-digit number cannot start with 0, so start with 1, then put 0 next: 10,358.
29,0602 ten thousands, 9 thousands and 6 tens. The two zeros go in the hundreds and ones: 29,060.
36,000; 37,000; 38,000; 39,000The thousands digit must be worth more than 5,000, so it is 6, 7, 8 or 9. That gives four numbers.
Expert
Model answer: No. The 5 in 50,040 is in the ten thousands column, so it is worth 50,000. The 5 in 5,040 is in the thousands column, so it is worth 5,000.Look for: the value depends on the column, not on being the first digit; 50,000 against 5,000.
10,009; 10,090; 10,900; 19,000; 90,001; 90,010; 90,100; 91,000The first digit is 1 or 9, never 0. For each, the other non-zero digit goes in one of four places: 4 + 4 = 8 numbers.
21,050; 42,050; 63,050; 84,050The pairs (ten thousands, thousands) are 2 and 1, 4 and 2, 6 and 3, 8 and 4. Thousands 0 would need ten thousands 0, which is not allowed.
3; 30; 300; 3,000; 30,000The 3 can be in any of the five columns, so it is worth 3 × 1, 10, 100, 1,000 or 10,000.
Worked example. Write 507,246 in words and give the value of the 7. 1. 507 thousand is five hundred and seven thousand, then two hundred and forty-six. 2. The first three digits are hundred thousands (5), ten thousands (0), thousands (7). 3. The 7 is in the thousands column, so it is worth 7 × 1,000. So 507,246 is five hundred and seven thousand, two hundred and forty-six; the 7 is worth 7,000.
Level 1
66,482 has 6 thousands, 4 hundreds, 8 tens and 2 ones.
40The 4 is in the tens place, so it is worth 4 tens, which is 40.
57,159 has 7 thousands, 1 hundred, 5 tens and 9 ones.
9,000The 9 is in the thousands place, so it is worth 9 thousands, which is 9,000.
6,038There are no hundreds, so a 0 holds the hundreds place: 6,038.
75,070 has 5 thousands, 0 hundreds, 7 tens and 0 ones.
3The first digit of a 6-digit number is in the hundred thousands place: 3.
50,000The 5 is in the ten thousands place, so it is worth 5 ten thousands: 50,000.
300The 3 is in the hundreds place, so it is worth 3 hundreds: 300.
406,0004 hundred thousands is 400,000 and 6 thousands is 6,000. Together: 406,000.
5705,300 has 7 hundred thousands, 0 ten thousands, 5 thousands and 3 hundreds.
8,000The 8 is in the thousands place, so it is worth 8 thousands: 8,000.
Level 2
5481,576 has 4 hundred thousands, 8 ten thousands, 1 thousand, 5 hundreds.
60,000The 6 is in the ten thousands place, so it is worth 6 ten thousands: 60,000.
806,040No ten thousands, hundreds or ones, so zeros hold those places: 806,040.
135,206Look at the ten thousands place. In 135,206 the 3 is there. The other 3s are worth 3,000 or 300,000.
FalseThe 2 is in the ten thousands place. The thousands digit of 520,013 is 0.
40,000The 4 is in the ten thousands place, so it is worth 40,000.
774,000 is 7 ten thousands and 4 thousands.
0608,150 has 6 hundred thousands, then 0 ten thousands, then 8 thousands.
640640,000 is 640 thousands, because the three zeros hold the hundreds, tens and ones.
604,020604 thousands, then 0 hundreds, 2 tens and 0 ones: 604,020.
Level 3
540,319The 4 is in the hundred thousands place and the 5 in the ten thousands place. After the swap: 5, 4, 0, 3, 1, 9.
703,008Put each digit in its place: 7 hundred thousands, 3 thousands and 8 ones. Zeros fill the other places.
170,000The first digit cannot be 0, so it is 1. Then 7 in the ten thousands place and zeros after it: 170,000.
100,001; 100,010; 100,100; 101,000; 110,000The first digit must be 1. The second 1 can be in any of the other five places, giving five numbers.
The 3The 9 is in the thousands place: 9,000. The 3 is in the ten thousands place: 30,000.
400,000Adding 1 makes the ones, tens, hundreds, thousands and ten thousands each carry, giving 400,000.
Expert
Model answer: Yes. The first digit is at least 1 hundred thousand, which is 100,000. Any other digit is worth at most 9 ten thousands, which is 90,000.Look for: yes; the left digit is at least 100,000; every other digit is at most 90,000 (a 9 in the ten thousands place); a clear comparison.
635,2416, 5 and 4 fix the hundred thousands, thousands and tens places. The ten thousands, hundreds and ones places are left: 3 goes in the biggest place, then 2, then 1.
501,200; 502,400; 503,600; 504,800The thousands digit can be 1, 2, 3 or 4, because twice 5 is 10, which is too big for one digit. The hundreds digit is then 2, 4, 6 or 8.
Model answer: Jun wrote 406 for the thousands, then only 50, so the hundreds place has no digit. Fifty is 0 hundreds, 5 tens and 0 ones, so a 0 must hold the hundreds place. The correct number is 406,050.Look for: 40,650 has only 5 digits; the 0 in the hundreds place is missing; so the 4 has slipped to the ten thousands place; the correct number is 406,050.
Worked example. Partition 910,260 by place value, then in one other way. 1. Read each digit with its place: 9 hundred thousands, 1 ten thousand, 2 hundreds, 6 tens. 2. The 0 in the thousands place and the 0 in the ones place need no parts of their own. 3. By place value: 910,260 = 900,000 + 10,000 + 200 + 60. 4. Another way: join the first two parts to make 910,000, and keep 260. So 910,260 = 900,000 + 10,000 + 200 + 60, or 910,000 + 260.
Level 1
4,625Add the parts: 4,000 + 600 = 4,600, then + 20 + 5 makes 4,625.
86,718 has 8 ones, so the missing part is 8.
900The 9 is in the hundreds place, so it is worth 9 hundreds, which is 900.
2008,205 has 2 hundreds and no tens, so the missing part is 200.
60The 6 is in the tens place, so it is worth 60. The zeros need no parts.
7,0637,000 + 60 + 3 has no hundreds, so the hundreds digit is 0: 7,063.
23,456Write each part in its place: 2, 3, 4, 5, 6 gives 23,456.
800The 8 in 52,814 is in the hundreds place, so the missing part is 800.
6,000The 6 is in the thousands place, so it is worth 6 thousands, which is 6,000.
FalseThe 1 is in the thousands place, so it is worth 1,000, not 10.
300The 3 is in the hundreds place, so the missing part is 300. The 0 in the tens place needs no part.
80,509There are no thousands and no tens, so write 0 in those places: 80,509.
Level 2
10The tens digit of 528,914 is 1, so the missing part is 10.
12,800Take 600,000 from 612,800. What is left is 12,800.
403,080There are no ten thousands, no hundreds and no ones, so those places hold 0: 403,080.
FalseThe parts make 79,107, which is a 5-digit number. 709,107 has an extra 0 in the ten thousands.
8,450238,450 − 230,000 leaves 8,450.
172,500The books upstairs are the other part: 372,500 − 200,000 = 172,500.
4,000Take 780,000 and 90 away: 784,090 has 4 thousands left, so 4,000.
800The hundreds digit is 8, so the missing part is 800. The 0s need no parts.
30,000480,000 − 450,000 = 30,000, because 48 ten thousands less 45 ten thousands is 3 ten thousands.
714,00014 thousands is 14,000. Add it to 700,000 to make 714,000.
Level 3
176,000Lake B has 235,000 + 41,000 = 276,000 birds. Then 276,000 − 100,000 = 176,000.
318,000The other load is the whole minus the known part: 468,000 − 150,000 = 318,000.
No8 ten thousands is 80,000 and 4 hundreds is 400, which make 80,400. Priya needed 4 thousands, not 4 hundreds.
615,08015 thousands is 15,000. Then 600,000 + 15,000 + 80 = 615,080.
300; 6,00096,300 − 90,000 leaves 6,300. That is 6 thousands and 3 hundreds: 6,000 and 300.
10,000; 20,000; 30,000; 40,000; 50,000110,000 is 11 ten thousands. The smaller part can be 1, 2, 3, 4 or 5 ten thousands: 10,000 to 50,000.
Expert
Model answer: No. 30 thousands + 5 thousands is 35 thousands, which is 35,000. 305,000 is 305 thousands.Look for: 30 + 5 = 35 thousands, so 35,000; the answer says 305,000 is 305 thousands, or 300 thousands + 5 thousands.
30,000; 40,000; 50,000; 60,000The box is more than 29,700 and less than 69,700. The multiples of 10,000 between are 30,000, 40,000, 50,000 and 60,000. 20,000 and 70,000 are too small and too large.
521,000; 542,000; 563,000; 584,000The thousands digit t can be 1, 2, 3 or 4, since twice t must be a single digit. That gives 521,000, 542,000, 563,000 and 584,000.
Model answer: Both are correct. 300,000 + 40,000 = 340,000 and 330,000 + 10,000 = 340,000. A number can be partitioned in many ways, not just by place value.Look for: both pairs add to 340,000, so both are correct. Hugo's parts do not follow place value, but a partition only needs the parts to add up.
Worked example. Count back in 10,000s four times from 214,500. Find the number reached. 1. 214,500 − 10,000 = 204,500. The ten thousands digit changes from 1 to 0. 2. 204,500 − 10,000 = 194,500. Ten thousands 0 to 9, so hundred thousands 2 to 1. 3. 194,500 − 10,000 = 184,500, then 184,500 − 10,000 = 174,500. So the number reached is 174,500.
Level 1
3,350Add 1 to the thousands digit: 2 becomes 3. The other digits stay the same.
5,408Take 1 from the thousands digit: 6 becomes 5. The other digits stay the same.
3,200The sequence goes up by 1,000 each time, so the missing number is 3,200.
6,015Take 1 from the thousands digit: 7 becomes 6. The 0 hundreds, 1 ten and 5 ones stay.
10,4509 thousands plus 1 thousand makes 10 thousands, so the answer is 10,450.
4,800The sequence goes down by 1,000 each time, so the missing number is 4,800.
33,400Add 1 to the ten thousands digit: 2 becomes 3. The other digits stay the same.
5,540Take 1 from the hundreds digit: 6 becomes 5. The other digits stay the same.
43,000The sequence goes up by 1,000 each time, so the next number is 43,000.
2,540Add 1 to the tens digit: 3 becomes 4. The other digits stay the same.
48,000The sequence goes down by 10,000 each time, so the next number is 48,000.
False4,095 + 10 = 4,105, not 4,005. The hundreds digit changes from 0 to 1.
Level 2
56,200Add 1 to the ten thousands digit: 4 becomes 5.
72,950The hundreds digit is 0, so take 1 thousand and add 9 hundreds: 72,950.
105,400The sequence goes up by 10,000 each time. 95,400 + 10,000 = 105,400, which crosses 100,000.
299,000300,000 is 300 thousands. One thousand less is 299 thousands, so 299,000.
FalseAdding 100,000 changes the hundred thousands digit: 4 becomes 5, giving 550,230. 460,230 is 10,000 more.
530,000The marks go up in 10,000s. A is one mark after 520,000, so it is 530,000.
108,30098,300 + 10,000: the ten thousands digit 9 becomes 10, so the number gains a digit: 108,300.
296,420Take 1 from the ten thousands digit: 0 becomes 9, so 3 becomes 2 in the hundred thousands: 296,420.
100,050Work back by adding 100 to 99,950. The hundreds digit 9 becomes 10, giving 100,050.
498,500; 500,500499,500 − 1,000 = 498,500. 499,500 + 1,000 = 500,500, which bridges the hundred thousands.
Level 3
12After 11 months the club has $49,500, which is not more than $50,000. After 12 months it has $50,500.
460,000A is 430,000. Counting on three 10,000s gives 440,000, 450,000 and 460,000.
403,400Work backwards: add 1,000 to get 503,400, then take away 100,000 to get 403,400.
240,000; 270,000A number is said if it is a whole number of 10,000s from 220,000. 240,000 and 270,000 are; 238,000 and 250,500 are not.
7,250Only the ten thousands and above change, so the last 4 digits stay 7,250. The count ends at 7,250; one more step would go below 0.
109,50096,500 + 10,000 = 106,500. Then add three 1,000s: 109,500.
Expert
10,000500,000 − 399,000 = 101,000, so the first two steps were 100,000 and 1,000. The step left is 10,000.
130,000; 139,000; 148,000Two 1,000s give 130,000. A 1,000 and a 10,000 give 139,000 in either order. Two 10,000s give 148,000.
190,000The hundred thousands digit changes only when the ten thousands digit is 9. The smallest such number is 190,000, since 190,000 + 10,000 = 200,000.
Model answer: Yes. Taking away 10,000 only changes the ten thousands digit, and sometimes the hundred thousands digit. The thousands digit never changes, so it stays 5.Look for: 10,000 is a whole number of ten thousands, so digits in the thousands, hundreds, tens and ones columns are untouched.
Worked example. Order 725,408, 752,048, 725,084 and 725,840, smallest first. 1. Ten thousands digits: 2, 5, 2 and 2, so 752,048 is the greatest. 2. The other three match in the thousands (5), so the hundreds digits decide: 4, 0 and 8. 3. Hundreds: 0 < 4 < 8, so 725,084 < 725,408 < 725,840. So the order is 725,084; 725,408; 725,840; 752,048.
Level 1
2,486Both have 2 thousands and 4 hundreds. The tens decide: 8 is more than 6.
9,031Both have 9 thousands. The hundreds decide: 0 is less than 3.
TrueThe thousands and hundreds match. The tens decide: 4 is more than 0.
4,021; 4,102; 4,210All start with 4 thousands. Compare the hundreds: 0, 1 and 2.
4,0014,001 has 4 thousands. The others have only 3 thousands.
<Both have 8 thousands and 2 hundreds. Tens: 0 is less than 5, so 8,205 is smaller.
45,200The ten thousands and thousands match. Hundreds: 2 is more than 0.
61,089Both start 61. The hundreds decide: 0 is less than 8.
31,025; 31,205; 31,250All start 31. Hundreds: 2, 2 and 0, so 31,025 is smallest. Tens then decide the other two.
80,421Compare the ten thousands digits: 1, 8 and 1. The 8 is greatest.
60,00060,000 has 6 ten thousands. 59,999 has only 5 ten thousands.
35,064All start 35. Hundreds: 6, 0 and 4. The 0 is smallest.
Level 2
<The first three digits match. Hundreds: 2 is less than 6, so 405,260 is smaller.
342,105All start 342. Hundreds: 0, 1 and 0, so 342,105 is greatest.
False86,430 has 5 digits and 864,300 has 6 digits, so 864,300 is greater.
83,520Both start 83. Hundreds: 5 is more than 2, so Sunday was busier, with 83,520 fans.
98,450; 128,045; 128,45098,450 has only 5 digits, so it is smallest. Then 128,045 is less than 128,450 (tens of thousands match, hundreds 0 against 4).
64,01964,019 has 5 digits and 640,190 has 6 digits, so 64,019 is smaller.
56,780506,780 has 6 digits. Of the two 5-digit numbers, 56,780 has fewer ten thousands than 65,780.
450,300450,300 is 1 more than 450,299 and 1 less than 450,301.
910,050; 910,005; 901,500; 91,50091,500 has 5 digits so it is last. Then 901,500 is below the two that start 910. Tens: 5 is more than 0.
520,030500,000 + 20,030 = 520,030. 520,003 has 0 tens and 520,030 has 3 tens, so 520,030 is greater.
Level 3
56,302Order them: 56,230; 56,302; 56,320. The middle one is 56,302.
99,998; 99,999; 100,000; 100,001Count on from 99,997 and stop before 100,002: 99,998, 99,999, 100,000, 100,001.
248,000Library A now has 238,000 + 10,000 = 248,000. That is more than 247,500.
4A is 140,000 and B is 180,000. Four of the numbers are between them: 145,000, 155,000, 165,000 and 175,000.
20,479Smallest digits first, but 0 cannot start a 5-digit number. Use 2, then 0, 4, 7, 9.
Expert
Model answer: Zara compared 305 and 35 as if the digits lined up. Both numbers have 3 hundred thousands, but the ten thousands are 0 and 5. As 5 is more than 0, 350,000 is greater.Look for: comparing digits by place value from the left, not by the size of the leading digits as a number. Both have 3 hundred thousands; the ten thousands (0 and 5) decide.
12Thousands digit 5 gives 6 numbers (all above 5,000). Thousands digit 7 gives 6 more. Thousands digit 3 is too small, and 0 cannot lead. So 12.
149From 78,951 to 78,999 is 49 numbers. From 79,000 to 79,099 is 100 numbers. 49 + 100 = 149.
Model answer: Sometimes. It is greater when the hundreds digit is more than the thousands digit, for example 523,410 becomes 524,310. It is smaller when the hundreds digit is less (524,310 becomes 523,410) and the same when the digits match.Look for: the answer is sometimes, a clear example that gets greater and one that gets smaller, and the idea that the digit moving into the higher (thousands) place decides.
Worked example. Round 462,987 to the nearest 10, 100 and 1,000. 1. Nearest 10: the ones digit is 7. 5 or more: round up to 462,990. 2. Nearest 100: the tens digit is 8. 5 or more: round up. 462,900 + 100 = 463,000. 3. Nearest 1,000: the hundreds digit is 9. 5 or more: round up to 463,000. So 462,987 is 462,990 to the nearest 10, and 463,000 to the nearest 100 and 1,000.
Level 1
50The ones digit is 7, which is 5 or more, so round up to 50.
7068 is 2 away from 70 and 8 away from 60, so 68 is nearer to 70.
700The tens digit is 2, which is less than 5, so round down to 700.
7,000The hundreds digit is 7, which is 5 or more, so round up to 7,000.
640The ones digit is 5, which is 5 or more, so round up to 640.
5,100The tens digit is 8, so round up. The hundreds digit 0 becomes 1, giving 5,100.
12,000The hundreds digit is 4, which is less than 5, so 12,436 is nearer to 12,000.
23,760The ones digit is 4, which is less than 5, so round down to 23,760.
31,300The tens digit is 8, which is 5 or more, so round up to 31,300.
TrueThe hundreds digit is 7, which is 5 or more, so 26,741 rounds up to 27,000. The statement is true.
30,210The ones digit is 6, so round up. The tens digit 0 becomes 1, giving 30,210.
17,500The tens digit is 5, so 17,450 is exactly halfway. 5 or more rounds up, giving 17,500.
Level 2
681,000The hundreds digit is 3, which is less than 5, so round down to 681,000.
90,410The ones digit is 7, so round up. The tens digit 0 becomes 1, giving 90,410.
FalseThe hundreds digit is 5, exactly halfway, so round up to 355,000. The statement is false.
49,000The tens digit is 7, so round up: 48,900 + 100 = 49,000.
483,000The hundreds digit is 6, which is 5 or more, so round up to 483,000.
52,49952,499 has hundreds digit 4, so it rounds down to 52,000. The others round to 51,000, 51,000 and 53,000.
207,390; 207,400; 207,000The ones digit 6 rounds up to 207,390. The tens digit 8 rounds up to 207,400. The hundreds digit 3 rounds down to 207,000.
$64,400The tens digit is 5, exactly halfway, so round up to $64,400.
300,100The ones digit is 6, so round up from 300,090 to 300,100. The 9 tens carry into the hundreds.
1,000,000The hundreds digit is 6, so round up. 999,000 + 1,000 = 1,000,000.
Level 3
82,000; 83,000; 84,00083,520 rounds up to 84,000. 82,480 rounds down to 82,000. 83,499 rounds down to 83,000. In order: 82,000; 83,000; 84,000.
39,000The tens digit of 38,450 is 5, so 38,450 rounds up to 38,500. The hundreds digit of 38,540 is 5, so 38,540 rounds up to 39,000. The greater answer is 39,000.
100To the nearest 10, $45,780 stays the same. To the nearest 100, $45,780 becomes $45,800. To the nearest 1,000, $45,780 becomes $46,000. So the answer is 100.
TrueNearest 100: the tens digit is 5, so round up to 55,000. Nearest 1,000: the hundreds digit is 9, so round up to 55,000. Both are 55,000, so the statement is true.
245,349 kmEvery number from 245,250 to 245,349 rounds to 245,300. 245,350 rounds up to 245,400, so the greatest is 245,349 km.
36,445; 36,446; 36,447; 36,448; 36,449A number rounds up to 36,450 when its ones digit is 5 or more. The whole numbers below 36,450 that do this are 36,445 to 36,449.
Expert
Model answer: No. 62,449 is closer to 62,400 than to 62,500, because the tens digit is 4. Ravi rounded twice: the first step gave 62,450, which then rounded up to 62,500.Key points: the correct answer is 62,400; the tens digit 4 means round down; the first rounding changed 62,449 to 62,450, so the second rounding went up wrongly.
354; 363; 372; 381; 390; 408; 417; 426; 435; 444Numbers that round to 400 run from 350 to 449. Those with digits adding to 12 are 354, 363, 372, 381 and 390, then 408, 417, 426, 435 and 444.
42,450; 42,49942,000 to the nearest 1,000 allows 41,500 to 42,499. 42,500 to the nearest 100 allows 42,450 to 42,549. Both fit only from 42,450 to 42,499.
1,000The numbers run from 8,500 to 9,499. Count them: 9,499 − 8,500 + 1 = 1,000.
Worked example. Round 954,600 to the nearest 10,000 and to the nearest 100,000. 1. Nearest 10,000: the thousands digit is 4, so 4 or less: round down to 950,000. 2. Nearest 100,000: the ten-thousands digit is 5, so 5 or more: round up. 3. Rounding up 900,000 by one hundred thousand gives 1,000,000. So 954,600 rounds to 950,000 and to 1,000,000.
Level 1
70The ones digit is 4, which is less than 5, so round down to 70.
600The tens digit is 4, so round down to 600.
3,000The hundreds digit is 4, so 3,420 is nearer to 3,000.
3,000The hundreds digit is 8, so round up to 3,000.
6,000The hundreds digit is 6, so round up to 6,000.
4,990The ones digit is 5, so round up to 4,990.
40,000The thousands digit is 7, so round up to 40,000.
40,000The thousands digit is 1, so round down to 40,000.
70,000The thousands digit is 8, so round up to 70,000.
70,000The thousands digit is 3, so round down to 70,000. The 9 hundreds do not matter.
90,000The thousands digit is 5, so round up to 90,000.
100,000The thousands digit is 6, so round up from 90,000 to 100,000.
Level 2
480,000The thousands digit is 2, so round down to 480,000.
400,000The ten-thousands digit is 7, so round up to 400,000.
FalseThe thousands digit is 5, so round up to 750,000, not 740,000.
600,000550,000 is exactly halfway. The ten-thousands digit is 5, so round up to 600,000.
600,000The thousands digit is 3, so round down. The zeros stay as placeholders: 600,000.
60,000The thousands digit is 9, so round up. 50,000 becomes 60,000.
254,000254,000 has ten-thousands digit 5, so it rounds up to 300,000. 249,500 rounds to 200,000 and 351,200 to 400,000.
100,000The ten-thousands digit is 4, so round down to 100,000.
810,000; 800,000Thousands digit 7, so round up to 810,000. Ten-thousands digit 0, so round down to 800,000.
530,000A is 7 marks after 520,000, so it is 527,000. Round up to 530,000.
Level 3
10,00044,900 rounds to 40,000 and 45,100 rounds to 50,000. The difference is 50,000 − 40,000 = 10,000.
650,000Numbers from 650,000 up round to 700,000. The smallest is exactly halfway, 650,000.
150,000; 249,999150,000 is halfway and rounds up. 249,999 has ten-thousands digit 4, so it rounds down. 149,999 gives 100,000 and 250,000 gives 300,000.
900,000A is five marks after 800,000, so it is 850,000, exactly halfway. Round up to 900,000.
449,999The crowd must be below 450,000 or it would round to 500,000. The largest whole number is 449,999, which still rounds to 450,000 and 400,000.
True128,400 rounds up to 130,000 (thousands digit 8). 131,900 rounds down to 130,000 (thousands digit 1).
Expert
Model answer: No. 649,500 rounds to 650,000, and 650,000 rounds to 700,000. Rounded straight to the nearest 100,000, the ten-thousands digit is 4, so the answer is 600,000.Look for: rounding twice gives 700,000; rounding once gives 600,000; so the answers differ. The reason is that the first rounding changes the digit that decides the second.
10,000; 1,000For the nearest 10,000, the numbers run from 65,000 to 74,999, which is 10,000 numbers. For the nearest 1,000, they run from 69,500 to 70,499, which is 1,000 numbers.
7,529; 7,592; 7,925; 7,952The number must start with 7 and have a hundreds digit of 5 or more. So the hundreds digit is 5 or 9, which gives four numbers.
Model answer: Sometimes. 502,000 rounds to 500,000 both times. But 520,000 rounds to 500,000 to the nearest 100,000 and to 520,000 to the nearest 10,000.Look for: one example where it is true and one where it is false, so the answer is sometimes. Only numbers from 495,000 to 504,999 round to 500,000 both ways.
Worked example. The temperature is 7 °C and falls by 15 degrees. Find the new temperature. 1. Fall 7 degrees to reach 0 °C. 2. 15 − 7 = 8 degrees are left to fall, so go 8 below 0. 3. 8 below 0 is written with a minus sign: −8. So the new temperature is −8 °C.
Level 1
−1Counting back one more from 0 goes below zero, to −1.
−3Counting on in 1s, each number is one more: after −4 comes −3.
−5 °CA temperature below zero is colder than one above zero, so −5 °C is colder.
−2Take 2 from 0 to get −2.
−3Count back 2 to reach 0, then 3 more below zero: −3.
−3; 0; 2The number below zero is smallest, then 0, then 2.
−5Take 5 from 0 to get −5.
−10Counting on in 10s from −20 gives −10, then 0.
−6Count back 4 to reach 0, then 6 more below zero: −6.
−3 °C−3 is closer to zero than −8, so −3 °C is warmer.
True−15 is further below zero than −9, so it is smaller.
−3 °CFall 5 degrees to reach 0 °C, then 3 more: −3 °C.
Level 2
−5The sequence counts back in 10s. 5 take away 10 is −5, then −15.
−11 take away 2 is −1, and −1 take away 2 is −3. The odd numbers pass through 0 without landing on it.
False−250 is further below zero than −205, so −250 is smaller. The statement is false.
−14; −9; −3; 0; 6The negative number furthest from zero is smallest: −14, then −9, then −3, then 0, then 6.
−12Of the negatives, −12 is furthest below zero, so it is the coldest.
−6 °CThe marks go up in 2s. A is three marks below 0, so it shows −6 °C.
−100200, 100, 0, and 100 less than 0 is −100.
10From −4 up to 0 is 4 degrees, then 6 more to reach 6: 4 + 6 = 10.
−11 °CWarming moves up towards zero: −18 + 7 = −11, still below zero.
5 °CUndo the fall by going up 8 from −3: 3 up to 0, then 5 more, giving 5 °C.
Level 3
−4 °C−7 + 12 = 5, then 5 − 9 = −4. The temperature crosses 0 twice.
12A is 4 °C and B is −8 °C. From B up to 0 is 8 degrees and from 0 up to A is 4 more: 12.
5Work backwards: go up 9 floors from −4. Four floors reach floor 0, then 5 more reach floor 5.
−4; −3; −2; −1; 0; 1The whole numbers between −5 and 2, not including either, are −4, −3, −2, −1, 0 and 1.
430 mFrom the cave up to sea level is 280 m, then 150 m more up to the hut: 280 + 150 = 430 m.
−25The numbers are 35, 25, 15, 5, −5, −15, −25. The first one below −20 is −25.
Expert
Model answer: No. −8 is further below zero than −3, so −8 is less than −3. For example, −8 °C is colder than −3 °C.Look for: the minus sign reverses the order; the number further below zero is smaller; a temperature or number line example.
8; 2; −4From 14 to −10 is 24 in four equal steps, so each step is 6. The numbers are 14, 8, 2, −4, −10.
−18; −14; −12; −7; −6Multiples of 6 below 20 are 6, 12 and 18. Multiples of 7 below 20 are 7 and 14. Put a minus sign on each.
Model answer: Dev says 0. Counting back in 5s gives 10, 5, 0. Counting back in 4s gives 10, 6, 2, −2, so Ines skips over 0.Look for: Dev only; 10 is a multiple of 5 but not of 4; a list of both sequences or the idea that Ines passes 0 landing on 2 then −2.
Worked example. A number is 400,000 + 7,000 + 20. Round it to the nearest 10,000. 1. Add the parts: 400,000 + 7,000 + 20 = 407,020. 2. The ten thousands digit is 0. Look at the thousands digit: 7. 3. 5 or more: round up, so 400,000 becomes 410,000. So 407,020 rounds to 410,000 to the nearest 10,000.
Level 1
4,450Add 1 to the thousands digit: 3,450 becomes 4,450.
6,100Take 1 from the hundreds digit: 6,200 becomes 6,100.
50The ones digit is 7, which is 5 or more, so round up to 50.
3008,300 is 8 thousands and 3 hundreds, so the missing part is 300.
3,500The tens digit is 6, which is 5 or more, so round up to 3,500.
4,098; 4,809; 4,908All have 4 thousands. Compare the hundreds: 0, 8 and 9.
44,500Add 1 to the ten thousands digit: 34,500 becomes 44,500.
46,000The hundreds digit is 3, which is less than 5, so round down to 46,000.
−2 °CCount back 4 to reach 0, then 2 more below 0, which is −2 °C.
28,640Both start 28 thousands. Compare the hundreds: 6 is more than 4, so 28,640 is greater.
80,000The pattern goes down by 1,000 each time, so the next number is 80,000.
19,95020,050 − 100 crosses the thousand: 20,050 − 50 = 20,000, then 50 more gives 19,950.
Level 2
680,000The thousands digit is 3, which is less than 5, so round down to 680,000.
50Reading from the right, the digits are 0 ones, 5 tens. So the 5 is worth 50.
295,200Take 1 from the ten thousands digit. 0 ten thousands means exchanging, so 305,200 becomes 295,200.
False1,000 less than 300,040 is 299,040. The statement has a 9 in the hundreds place, so it is false.
450,000The hundreds digit is 9, so round up. 449,000 + 1,000 = 450,000.
5,248; 52,048; 52,480; 524,800Fewer digits means smaller. Of the two 5-digit numbers starting 52, compare the thousands: 52,048 is less than 52,480.
440,200The numbers go down by 100,000 each time, so the next is 440,200.
38,500The tens digit is 5, so round up (5 or more) to 38,500.
−8Going down 5 floors from −3 gives −8.
10 °CFrom −4 up to 0 is 4 degrees, then 6 more to reach 6 °C. 4 + 6 = 10 °C.
Level 3
30,00034,600 rounds to 30,000 and 56,300 rounds to 60,000. 60,000 − 30,000 = 30,000.
65,000Exactly halfway rounds up, so 65,000 rounds to 70,000. 64,999 rounds down to 60,000.
110,500Work backwards: add 5 × 10,000 = 50,000 to 60,500. This crosses 100,000 and gives 110,500.
−4 °CA is at −7 °C. Rising 12 °C gives 5 °C, then falling 9 °C gives −4 °C.
4,995; 4,996; 4,997; 4,998; 4,999; 5,000; 5,001; 5,002; 5,003; 5,0044,995 is halfway and rounds up. Below 5,000 the ones digit must be 5 or more. Above 5,000 it must be 4 or less.
8,530An even number ends in 0 or 8. Ending in 0 leaves 8, 5, 3 in order: 8,530. Ending in 8 gives only 5,308.
Expert
269; 296List all six numbers. 269 and 296 have 2 hundreds and 6 or more tens, so both round up to 300. The others round to 600, 700, 900 or 1,000.
8 °CWork backwards with opposite moves: −3, add 6 gives 3, take 4 gives −1, add 9 gives 8.
Model answer: No. 6,349 is closer to 6,300 than to 6,400, so it rounds to 6,300. Hana rounded twice, and the first rounding changed the digit that decides.Look for: rounding once from the original number; the tens digit 4 means round down to 6,300; rounding twice can move a number past halfway.
Model answer: Sometimes. 50,000 rounds to 50,000 both ways. But 48,000 rounds to 50,000 to the nearest 10,000, and to 48,000 to the nearest 1,000.Look for: the answer sometimes; one example that works and one that does not; numbers from 45,000 to 54,999 all round to 50,000 to the nearest 10,000.
Worked example. Calculate 413,250 + 5,999 in your head. 1. 5,999 is 1 less than 6,000, so add 6,000 first. 2. 413,250 + 6,000 = 419,250. 3. Only 5,999 was meant, so take away the extra 1: 419,250 − 1 = 419,249. So 413,250 + 5,999 = 419,249.
Level 1
248Only the ones digit changes: 3 + 5 = 8.
372Only the ones digit changes: 6 − 4 = 2.
548Add 3 tens: the tens digit goes from 1 to 4.
642Take away 5 tens: the tens digit goes from 9 to 4.
554Add 2 hundreds: the hundreds digit goes from 3 to 5.
570640 − 40 = 600, then 600 − 30 = 570.
4,80045 hundreds + 3 hundreds = 48 hundreds.
6,20067 hundreds − 5 hundreds = 62 hundreds.
5,420Add 2 thousands: the thousands digit goes from 3 to 5.
22,600Take 3 thousands from 25 thousands to leave 22 thousands.
56,00047 thousands + 9 thousands = 56 thousands.
41,300Take 2 ten thousands from 6 ten thousands: 41,300.
False52,000 − 9,000 = 43,000, then add 1 back: 43,001, not 43,011.
325,600Add 8 ten thousands: 24 + 8 = 32 ten thousands, so 325,600.
650A is at 7,350. Count on to 7,400 (50), then to 8,000 (600): 650.
66,799Add 20,000 to get 66,800, then take away 1.
6,300Count on from 63,700 to 64,000 (300), then to 70,000 (6,000).
68,449 kmAdd 30,000 to 38,450 to get 68,450, then take away 1.
98,001100,000 − 2,000 = 98,000, then add 1 back.
355,001345,000 + 11,000 = 356,000. Then 356,000 − 1,000 = 355,000, and add 1 back.
Level 3
136,801Sunday: 72,400 − 8,000 = 64,400, plus 1 is 64,401. Total: 72,400 + 64,401 = 136,801.
82,001Work backwards: undo the return (take away 3,999) and undo the borrowing (add 25,000). 61,000 + 25,000 = 86,000, then 86,000 − 3,999 = 82,001.
12,499500,000 − 487,500 = 12,500 would give exactly 500,000, so the largest number that stays below it is 12,499.
51,000 kmA is 43,000 and B is 47,000, so one trip is 4,000 km. The bus finishes at 47,000 + 4,000.
39,000; 51,000Take away the 12,000 and halve the rest: half of 78,000 is 39,000. The other number is 39,000 + 12,000 = 51,000.
60,999; 61,999; 62,999Totals end in 999. Test multiples 56,000, 57,000 and 58,000. 55,000 gives 59,999 (too small); 59,000 gives 63,999 (too big).
Expert
Model answer: Yes. Adding 9,999 is adding 10,000 and taking 1 away. Taking away 10,001 takes away 10,000 and 1 more. The 10,000s cancel, so 2 is taken away in all.Look for: 9,999 is 1 less than 10,000; 10,001 is 1 more; the 10,000s cancel; 1 + 1 = 2 less. Any example that works is a bonus.
164,001Work backwards: add 4,000 to 200,000 to get 204,000, then take away 39,999: 204,000 − 40,000 = 164,000, plus 1.
112,500; 117,500The multiples are 110,000; 112,500; 115,000; 117,500; 120,000. Adding 7,500 gives a multiple of 5,000 only for 112,500 (120,000) and 117,500 (125,000).
Model answer: Taking away 7,000 takes away 1 too many, so 1 must be added back, not taken off. The answer is 293,001.Look for: 6,999 is less than 7,000, so 7,000 removed too much; adjust by adding 1; correct answer 293,001.
103,122Ones 12, tens 7 + 4 + 1 = 12, hundreds 8 + 2 + 1 = 11, thousands 9 + 3 + 1 = 13, ten thousands 9 + 1 = 10. The total has six digits.
Level 3
380,005Add the first two: 225,185. Then add 154,820 to get 380,005. Check each exchange column by column.
530,460 kmYear two is 245,860 + 38,740 = 284,600 km. Both years: 245,860 + 284,600 = 530,460 km.
48,975 mDay two is 16,325 m and day three is 18,325 m. Add the three days: 14,325 + 16,325 + 18,325 = 48,975 m.
7The ones column must make 6 after carrying: 9 + 7 = 16, so write 6 and carry 1. The other number ends in 7 (it is 32,517).
110,100Add each pair: 64,435; 78,268; 82,589; 105,779; 110,100; 123,933. Only 110,100 and 123,933 are over 110,000; the smaller is 110,100.
Expert
Model answer: Kemi forgot to carry 1 hundred from the tens column: 7 + 4 + 1 = 12. The hundreds should be 6 + 8 + 1 = 15, so the total is 86,521.Look for: the correct total 86,521; the error is in the hundreds digit (4 instead of 5); a carry was missed from the tens column.
66,666; 77,777; 88,888; 99,999Test 55,555 + 38,457 = 94,012, which is too small. 66,666 + 38,457 = 105,123 works, and so do the larger ones.
Model answer: Sometimes. 60,000 + 70,000 = 130,000 has six digits, but 11,000 + 22,000 = 33,000 has five.Look for: the answer sometimes, with one example of each kind, or the idea that the ten thousands digits must make 10 or more.
Worked example. Calculate 842,007 − 58,609. 1. Ones: 7 is less than 9. Tens and hundreds are 0, so exchange from the thousands. 2. Thousands 2 becomes 1, hundreds 9, tens 9, ones 17. Ones: 17 − 9 = 8. Tens: 9 − 0 = 9. 3. Hundreds: 9 − 6 = 3. Thousands: 1 is less than 8, so exchange: 11 − 8 = 3. 4. Ten thousands: 3 is less than 5, so exchange: 13 − 5 = 8. Hundred thousands: 7. So 842,007 − 58,609 = 783,398.
Level 1
3,253Subtract column by column from the ones: 4 − 1 = 3, 8 − 3 = 5, 6 − 4 = 2, 5 − 2 = 3. No exchange is needed.
3,343Tens: 2 is less than 8, so exchange one hundred: 12 − 8 = 4. Then hundreds 6 − 3 = 3, thousands 4 − 1 = 3, ones 5 − 2 = 3.
256,004Working backwards, add the bricks used to the bricks left: 207,329 + 48,675 = 256,004. Check: 256,004 − 48,675 = 207,329.
500,000Add the two numbers to undo the subtraction: 351,743 + 148,257 = 500,000. Check: 500,000 − 148,257 = 351,743.
187,456The smallest answer allowed is 100,000, so the smallest number is 100,000 + 87,456 = 187,456. Check: 187,456 − 87,456 = 100,000.
Expert
Model answer: No. 70,000 − 27,586 = 42,414. Check by adding: 27,586 + 43,414 = 71,000, not 70,000, so 43,414 is 1,000 too big.Look for: the correct answer 42,414 and a reason, such as adding back (27,586 + 42,414 = 70,000) or noticing the answer is 1,000 too big because of a missed exchange across the zeros.
46,000; 47,000; 48,000; 49,00060,000 − 45,000 = 15,000, so the multiple must be more than 45,000. The multiples of 1,000 between 45,000 and 50,000 are 46,000, 47,000, 48,000 and 49,000.
Model answer: Sometimes. 120,000 − 30,000 = 90,000 has 5 digits, but 700,000 − 20,000 = 680,000 has 6 digits.Look for: the word sometimes, and two examples, one with a 5-digit answer and one with a 6-digit answer. Never or always means a single example was used.
Worked example. Check 54,318 + 26,845 = 81,163 by rounding to the nearest 10,000. 1. 54,318: the thousands digit is 4, less than 5, so round down to 50,000. 2. 26,845: the thousands digit is 6, 5 or more, so round up to 30,000. 3. Add the rounded numbers: 50,000 + 30,000 = 80,000. So the estimate is 80,000. This is close to 81,163, so the answer is reasonable.
Level 1
8048 rounds to 50 and 31 rounds to 30. 50 + 30 = 80.
4072 rounds to 70 and 29 rounds to 30. 70 − 30 = 40.
600396 rounds to 400 and 204 rounds to 200. 400 + 200 = 600.
500780 rounds to 800 and 310 rounds to 300. 800 − 300 = 500.
4,0005,820 rounds to 6,000 and 2,190 rounds to 2,000. 6,000 − 2,000 = 4,000.
7,0004,507 has hundreds digit 5, so it rounds up to 5,000. 2,350 rounds down to 2,000. 5,000 + 2,000 = 7,000.
74,00042,300 rounds to 42,000 and 31,800 rounds to 32,000. 42,000 + 32,000 = 74,000.
40,00017,400 rounds to 20,000 and 21,600 rounds to 20,000. 20,000 + 20,000 = 40,000.
46,00068,900 rounds to 69,000 and 23,100 rounds to 23,000. 69,000 − 23,000 = 46,000.
24,000The hundreds digit is 8, so round up from 23,000 to 24,000.
False7,500 rounds up to 8,000 and 1,400 rounds down to 1,000. The estimate is 9,000, not 8,000.
12,0009,600 rounds up to 10,000 and 2,300 rounds down to 2,000. 10,000 + 2,000 = 12,000.
Level 2
59,00038,460 rounds to 38,000 and 21,290 rounds to 21,000. 38,000 + 21,000 = 59,000.
40,00074,820 rounds to 70,000 and 29,640 rounds up to 30,000. 70,000 − 30,000 = 40,000.
68,00048,300 rounds to 48,000 and 19,800 rounds to 20,000. 48,000 + 20,000 = 68,000.
100,00056,700 rounds up to 60,000 and 43,200 rounds down to 40,000. 60,000 + 40,000 = 100,000.
Not reasonable48,615 rounds to 50,000 and 32,874 to 30,000, so the estimate is 80,000. 91,489 is more than 10,000 away, so it is not reasonable.
300,000612,000 rounds to 600,000 and 287,000 rounds up to 300,000. 600,000 − 300,000 = 300,000.
1,000Rounding 38,640 to the nearest 1,000 gives 39,000. The nearest 100 gives 38,600 and the nearest 10,000 gives 40,000.
81,000The numbers round to 25,000, 38,000 and 18,000. 25,000 + 38,000 + 18,000 = 81,000.
70,00045,000 is exactly halfway, so it rounds up to 50,000. 24,500 rounds down to 20,000. 50,000 + 20,000 = 70,000.
13,00052,400 rounds to 52,000 and 38,650 rounds up to 39,000. 52,000 − 39,000 = 13,000.
Level 3
$83,000$38,400 rounds to $38,000 and $44,900 rounds up to $45,000. $38,000 + $45,000 = $83,000.
114,999Numbers from 105,000 to 114,999 round to 110,000. 115,000 would round up to 120,000, so the largest is 114,999.
92,015The estimate is 90,000. Only 92,015 is close to it, and 63,845 + 28,170 = 92,015.
23,00061,500 − 38,240 = 23,260. The hundreds digit is 2, so round down to 23,000.
345; 346; 347; 348; 349Rounding to 350 gives 345 to 354. Rounding to 300 needs less than 350, so only 345 to 349 work.
11,480Numbers from 11,500 to 12,499 round to 12,000. 11,480 is below 11,500, so it rounds to 11,000.
Expert
Model answer: No. Both numbers round to 500,000, so the estimate is 0, but the real difference is about 10,000. Rounding to the nearest 1,000 gives 482,000 − 472,000 = 10,000.Look for: both numbers round to 500,000; the estimate 0 is far from the true difference of 10,400; a smaller place is needed when numbers are close.
41,000; 42,000; 43,000; 44,000The first number is from 35,000 to 44,000. The second must be at most 24,000, so the first is at least 41,000. The pairs run from 41,000 + 24,000 to 44,000 + 21,000.
4,5004,500 rounds up to 5,000, which then rounds up to 10,000. Straight to the nearest 10,000, 4,500 rounds down to 0. No smaller number differs.
Model answer: More. Both numbers round up, 36,700 to 40,000 and 27,900 to 30,000, so each rounded number is bigger than the real one and the estimate is too big.Look for: both numbers round up; each rounded number is larger than the original; so the total of the rounded numbers is larger.
76,097Add to undo the subtraction: 51,729 + 24,368 = 76,097.
False45,860 + 36,740 = 82,600, not 82,500. The correct answer to the subtraction is 45,760.
24,288The books added are the difference: 72,903 − 48,615 = 24,288. Check: 48,615 + 24,288 = 72,903.
56,491Add: 40,619 + 15,872 = 56,491. Subtracting instead gives 24,747, and 55,491 forgets a carry.
$86,050Last year was more, so add: $58,400 + $27,650 = $86,050.
273,625Subtract the part left from 400,000, exchanging across the zeros: 400,000 − 126,375 = 273,625.
19,425 kmSubtract the first trip from the total: 31,905 − 12,480 = 19,425 km.
Level 3
$18,825Add the weeks: $29,750 + $31,425 = $61,175. Then $80,000 − $61,175 = $18,825.
51,623The largest total below 100,000 is 99,999. Subtract: 99,999 − 48,376 = 51,623.
52,72752,827 + 38,675 = 91,502, not 91,402, so Hugo is wrong. The correct answer is 91,402 − 38,675 = 52,727.
55,825Work backwards. Before buying: 61,275 − 9,800 = 51,475. Before giving away: 51,475 + 4,350 = 55,825.
18,450; 41,550For each number, find what is needed to reach 60,000. 60,000 − 18,450 = 41,550, which is in the list.
Expert
Model answer: No. The check should give back 36,945, but it gives 35,845, so the total is wrong. The correct total is 64,331.Look for: the subtraction should return the other number, 36,945; it does not; so there is a mistake (a missed carry of 1,000). Bonus: the correct total is 64,331.
55,000; 35,000Take 20,000 off the total to leave two equal numbers: half of 70,000 is 35,000. The larger is 35,000 + 20,000 = 55,000. Check: 55,000 + 35,000 = 90,000.
Model answer: Sometimes. 70,000 + 80,000 = 150,000 has 6 digits, but 12,000 + 23,000 = 35,000 has only 5 digits.Look for: the answer 'sometimes', one example with a 6-digit total (large numbers) and one with a 5-digit total (small numbers).
Unit 16: Multi-step addition and subtraction problems
Worked example. A club has $34,250, then raises $28,675. How much more to reach $75,000? 1. Step 1, add to find the total so far: 34,250 + 28,675 = 62,925. 2. Step 2, subtract the total from the target: 75,000 − 62,925. 3. Exchange across the zeros, then subtract column by column: 12,075. So the club needs $12,075 more.
Level 1
590Add the tens: 40 + 50 = 90, and the hundreds: 300 + 200 = 500. The total is 590.
550900 − 300 = 600, then 600 − 50 = 550.
700Subtract to find the missing number: 1,900 − 1,200 = 700.
True46 hundreds − 21 hundreds = 25 hundreds, which is 2,500. The statement is true.
16,800123 hundreds + 45 hundreds = 168 hundreds, which is 16,800.
53,00058 thousands − 5 thousands = 53 thousands, which is 53,000.
24,700In all means add: 21,500 + 3,200 = 24,700.
SubtractMoney spent is taken away from the money the club has, so subtract: 9,500 − 4,100 = 5,400.
34,19980,000 − 45,800 = 34,200, but that makes exactly 80,000. The greatest whole number below that is 34,199.
64,795Saturday and Sunday: 57,915 + 49,260 = 107,175. Then 107,175 − 42,380 = 64,795.
$79,600Work backwards: add what was spent and what is left. 36,450 + 27,800 + 15,350 = 79,600.
Expert
Model answer: 48,700 is less than 50,000 and 5,360 is less than 6,000, so the total is less than 56,000. 102,300 is far too big.Look for a rounding or estimate: the total must be a little over 54,000, well below 102,300. Zara probably misaligned the columns or added a digit twice.
33,000; 57,000Take 24,000 away from 90,000 to leave 66,000, which is two equal smaller numbers: 33,000 each. The larger is 33,000 + 24,000 = 57,000.
Worked example. Is 195 a multiple of 15? If so, find the next one. 1. Ten 15s make 150, so count on from 150. 2. 195 − 150 = 45, and 45 = 15 × 3. 3. So 195 = 15 × 13, which makes 195 a multiple of 15. 4. The next multiple adds one more 15: 195 + 15 = 210. So 195 is a multiple of 15, and the next multiple of 15 is 210.
Level 1
153 groups of 5 make 15.
244 × 6 = 24.
94 × 9 = 36, so 36 ÷ 4 = 9.
567 × 8 = 56.
79 × 7 = 63, so the missing number is 7.
13212 × 10 = 120 and 12 × 1 = 12, so 120 + 12 = 132.
3; 6; 9; 12Count in 3s: 3, 6, 9, 12.
True4 × 5 = 20, so 20 is in the 4 times table.
80Count on one more 10 from 70 to get 80.
24The 6th multiple of 4 is 6 × 4 = 24.
217 × 3 = 21. Neither 15 nor 30 is in the 7 times table.
486 rows of 8 chairs is 6 × 8 = 48.
Level 2
8; 16; 24; 32; 40Count in 8s five times: 8, 16, 24, 32, 40.
True7 × 12 = 84, so 84 is a multiple of 7.
546 × 8 = 48 is too small and 6 × 9 = 54 is more than 50.
75; 125Add 25 each time: 50 + 25 = 75 and 100 + 25 = 125.
8045 = 9 × 5, 63 = 9 × 7 and 81 = 9 × 9. 80 is between 72 and 81.
False12 × 14 = 168 and 12 × 15 = 180. 175 is between them, so it is not a multiple of 12.
96The 8th multiple of 12 is 8 × 12 = 96.
48Only a multiple of 6 is possible: 6 × 8 = 48.
550; 600; 650Multiples of 50 end in 00 or 50. Those between 520 and 680 are 550, 600 and 650.
105; 120Count on in 15s from 90: 90 + 15 = 105, then 105 + 15 = 120.
Level 3
412 × 16 = 192 is below 200 and 12 × 17 = 204 is the next multiple. 204 − 200 = 4.
87 × 8 = 56, so the number is 8.
45; 63Multiples of 9 in the range are 36, 45, 54 and 63. Only 45 and 63 are odd.
275 mDistances with lamps are multiples of 25. 25 × 11 = 275, while 190 and 380 are not.
51015 × 33 = 495 is under 500. The next multiple is 495 + 15 = 510.
404 multiples of 25 make 100, and 1,000 is ten hundreds, so 10 × 4 = 40.
Expert
Model answer: Always. 6 is 3 × 2, so every multiple of 6 is a number of 3s doubled, which is still a multiple of 3. For example 18 = 3 × 6.Look for: 6 = 3 × 2, so each multiple of 6 can be split into groups of 3; an example such as 12 or 18 is not enough alone.
450; 900Multiples of 50 end in 00 or 50. Ending in 00: 900 (9 + 0 + 0). Ending in 50: 450 (4 + 5 + 0).
96The 2-digit multiples of 12 are 12, 24, 36, 48, 60, 72, 84 and 96. Only 96 has digits that differ by 3 with 9 first.
2612 + 6 + 1 = 9, so 261 follows the pattern. The others add to 8, 12 and 13. Check: 9 × 29 = 261.
Worked example. Find all the factors of 56 by finding factor pairs. 1. Test 1, 2, 3 and so on. 1 × 56 = 56, so 1 and 56 are a pair. 2. 2 × 28 = 56 gives 2 and 28. 3 does not divide 56. 4 × 14 = 56 gives 4 and 14. 3. 5 and 6 do not divide 56. 7 × 8 = 56 gives 7 and 8. 4. The next number to test is 8, which is already found, so the pairs repeat: stop. So the factors of 56 are 1, 2, 4, 7, 8, 14, 28 and 56.
Level 1
9Multiplying in either order gives the same answer, so the missing number is 9.
408 × 5 = 40.
64 × 6 = 24, so 24 ÷ 4 = 6.
721 ÷ 3 = 7, so 3 × 7 = 21.
TrueThe order of multiplying does not change the answer: both are 60.
7Division undoes multiplication, so 84 ÷ 12 = 7.
6Every number is a factor of itself, so the list ends with 6.
True4 × 5 = 20, so 4 divides 20 exactly.
1; 2; 5; 101 × 10 = 10 and 2 × 5 = 10, so the factors are 1, 2, 5 and 10.
43 × 4 = 12, so 3 pairs with 4.
4The pairs are 1 and 14, and 2 and 7, so the factors are 1, 2, 7 and 14: four.
152 × 15 = 30 and no larger number below 30 divides 30, so the answer is 15.
Level 2
1; 2; 3; 6; 9; 18The pairs are 1 and 18, 2 and 9, and 3 and 6.
True6 × 13 = 78, so 6 divides 78 exactly.
4The pairs are 1 and 24, 2 and 12, 3 and 8, and 4 and 6: four pairs.
342 ÷ 14 = 3, so 3 pairs with 14.
77 × 9 = 63, so 7 rows of 9 works. 63 is not in the 4 or 8 times tables.
1; 2; 3; 4; 6; 9; 12; 18; 36The pairs are 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6. List 6 only once.
FalseA factor divides the number exactly, so it cannot be larger than the number. 15 is a multiple of 5.
2; 3; 6; 954 = 2 × 27 = 3 × 18 = 6 × 9, so 2, 3, 6 and 9 rows work; 4, 5, 7 and 8 do not divide 54.
42; 48Count on in sixes: 36, 42, 48, 54. Only 42 and 48 are between 40 and 50.
12The pairs are 1 and 72, 2 and 36, 3 and 24, 4 and 18, 6 and 12, 8 and 9: six pairs, twelve factors.
Level 3
3Count the tiles: 4 rows of 9 is 36 tiles. 36 ÷ 12 = 3, so 12 columns need 3 rows.
12The factors of 20 are 1, 2, 4, 5, 10 and 20. Those below 10 are 1, 2, 4 and 5, and 1 + 2 + 4 + 5 = 12.
633 × 7 = 21, so the number is in the 21 times table: 21, 42, 63. Only 63 is between 60 and 70.
3Test 4 to 9: 4, 6 and 8 divide 48 exactly (4 × 12, 6 × 8, 8 × 6). So three row lengths work.
5The pairs are 1 and 80, 2 and 40, 4 and 20, 5 and 16, and 8 and 10. Each pair gives one rectangle, so there are five.
12Check numbers in order. 12 has factors 1, 2, 3, 4, 6 and 12. No smaller number has six factors.
Expert
4; 9; 25Three factors means one pair plus a repeated factor, so the number is a number multiplied by itself: 2 × 2, 3 × 3, 5 × 5. 6 × 6 is too big, and 4 × 4 = 16 has five factors.
Model answer: No. 16 has five factors: 1, 2, 4, 8 and 16. The pair 4 and 4 is only one factor, so the count is odd.Look for: a counter-example (16 has five factors); the factors listed; the reason that factors usually come in pairs but 4 pairs with itself.
5List the pairs of 60: 1 and 60, 2 and 30, 3 and 20, 4 and 15, 5 and 12, 6 and 10. Only 5 and 12 differ by 7, so 5 rows.
Model answer: No. There are 24 tiles. 5 is not a factor of 24, because 5 × 4 = 20 and 5 × 5 = 25, so 5 equal rows leave tiles over.Look for: the tile count (4 × 6 = 24); the test that 5 must be a factor of 24; the evidence that it is not (24 is not in the 5 times table).
Worked example. Find the common factors of 56 and 84. 1. Factors of 56, from pairs: 1, 2, 4, 7, 8, 14, 28, 56. 2. Factors of 84, from pairs: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. 3. Factors in both lists: 1, 2, 4, 7, 14, 28. So the common factors of 56 and 84 are 1, 2, 4, 7, 14 and 28.
Level 1
324 × 8 = 32.
74 × 7 = 28, so 28 ÷ 4 = 7.
427 × 6 = 42.
6012 × 5 = 60 (10 × 5 = 50 and 2 × 5 = 10).
98 × 9 = 72, so the missing number is 9.
812 × 8 = 96, so 96 ÷ 12 = 8.
1; 2; 3; 61 × 6 = 6 and 2 × 3 = 6, so the factors are 1, 2, 3 and 6.
True3 × 5 = 15, so 3 is a factor of 15.
1; 3; 91 × 9 = 9 and 3 × 3 = 9, so the factors are 1, 3 and 9.
10Every number is a factor of itself, so 10 is the missing factor.
1; 2Factors of 4: 1, 2, 4. Factors of 6: 1, 2, 3, 6. Both lists have 1 and 2.
3Factors of 6: 1, 2, 3, 6. Factors of 9: 1, 3, 9. The only shared one besides 1 is 3.
Level 2
1; 2; 4Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. Both have 1, 2 and 4.
5Factors of 10: 1, 2, 5, 10. Factors of 15: 1, 3, 5, 15. The other shared factor is 5.
False4 × 4 = 16, but 30 is not in the 4 times table, so 4 is not a factor of 30.
99 × 2 = 18 and 9 × 3 = 27. The others do not divide both numbers.
44 is a factor of 20, but 30 ÷ 4 is not a whole number. 2, 5 and 10 divide both.
1; 2; 3; 6A pile size must be a factor of 24 and of 30. The common factors are 1, 2, 3 and 6.
4Row lengths are common factors of 16 and 24: 1, 2, 4 and 8. That is four lengths.
30The other number is a multiple of 6 but not of 12 or 4: 24 shares 12, 30 shares exactly 6. So it is 30.
15; 21; 33; 39The number is in the 3 times table but is odd and not in the 9 times table (27 shares 9): 15, 21, 33 and 39.
15Shared with 20: 8 has 1, 2, 4; 15 has 1, 5; 9 has 1; 12 has 1, 2, 4. Only 15 has two.
18The number of pots is a common factor of 36 and 54: 1, 2, 3, 6, 9 or 18. The most is 18.
6Factors of 20: 1, 2, 4, 5, 10, 20. Factors of 45: 1, 3, 5, 9, 15, 45. Shared: 1 and 5, which add to 6.
Expert
Model answer: No. 9 and 15 are both odd, but 3 is a common factor of them as well as 1.Look for one counter-example, such as 9 and 15 (common factors 1 and 3) or 21 and 35 (1 and 7). A single example is enough.
3Both numbers are multiples of 8 under 30: 8, 16 and 24. Each pair of them shares exactly 1, 2, 4 and 8: 8 and 16, 8 and 24, 16 and 24.
1; 7; 11; 13; 17; 19; 23; 2930 has factors 2, 3 and 5, so any number sharing one of them is out. Left: 1, 7, 11, 13, 17, 19, 23 and 29.
48Six common factors means the shared part is 12 or 18. Multiples of 12 and 18 above 36 are 48, 54 and so on, and 48 is the smallest (36 and 48 share 12).
Worked example. Is 38 prime or composite? List its prime factors. 1. Factor pairs of 38: 1 × 38 and 2 × 19. 2. 38 has more than two factors, so 38 is composite. (1 is neither prime nor composite.) 3. The factors 2 and 19 are prime numbers, so they are the prime factors. So 38 is composite. Its prime factors are 2 and 19.
Level 1
123 groups of 4 make 12.
945 ÷ 5 = 9, because 9 × 5 = 45.
55 × 8 = 40, so 40 ÷ 8 = 5.
7Multiplying in either order gives the same answer, so the missing number is 7.
936 ÷ 4 = 9, because 4 × 9 = 36.
12132 ÷ 11 = 12, because 11 × 12 = 132.
2A prime number has exactly two factors: 1 and itself.
1; 3; 5; 151 × 15 = 15 and 3 × 5 = 15, so the factors are 1, 3, 5 and 15.
False9 has three factors: 1, 3 and 9. A prime number has only two.
2Only 1 × 13 = 13 works, so the factors are 1 and 13. That makes two.
11; 13; 17; 1912, 14, 15, 16, 18 are composite. 11, 13, 17 and 19 have only two factors.
2; 5The factors of 10 are 1, 2, 5 and 10. The prime ones are 2 and 5.
Level 2
Composite21 = 3 × 7, so 21 has more than two factors and is composite.
TrueNo whole number other than 1 and 29 divides 29 exactly, so 29 is prime.
2; 3The factors of 18 are 1, 2, 3, 6, 9 and 18. Only 2 and 3 are prime.
3; 5There are 3 rows of 5, so 15 tiles. 15 = 3 × 5, and 3 and 5 are both prime.
1; 23The rows must divide 23 exactly. 23 is prime, so only 1 row of 23 or 23 rows of 1 work.
2727 = 3 × 9, so it has more than two factors. The others are prime.
41; 43; 47Test each odd number. 45 = 5 × 9 and 49 = 7 × 7 are composite. 41, 43 and 47 are prime.
True1 has only one factor, so it is not prime (two factors) and not composite (more than two).
2; 3; 7The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. The prime ones are 2, 3 and 7.
Level 3
3727 = 3 × 9, 33 = 3 × 11 and 39 = 3 × 13 all make rectangles. 37 is prime, so it only makes one row.
52The primes are 23 and 29. 23 + 29 = 52.
19; 37Digits adding to 10 gives 19, 28, 37 and 46. 28 and 46 are even, so only 19 and 37 are prime.
47Only a prime number of tiles gives one row. The largest prime below 50 is 47, because 49 = 7 × 7.
NoThe picture has 12 tiles, so 11 are left. 11 is prime, so only one row of 11 can be made.
5; 7The factor pairs of 35 are 1 × 35 and 5 × 7. Only 5 and 7 are both prime.
Expert
3; 5; 11; 17; 29; 41The pairs are 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43. Note 5 starts one pair and ends another.
Model answer: No. 9 is odd, but 9 = 3 × 3 has three factors (1, 3 and 9), so it is composite.Look for one odd number that is not prime, such as 9, 15, 21 or 27, with a reason (it has a factor other than 1 and itself).
33; 36; 39; 45; 48The multiples of 3 are 33, 36, 39, 42, 45, 48. 42 has three prime factors (2, 3, 7). The others each have two.
Model answer: Every even number can be divided by 2. An even number greater than 2 has the factors 1, 2 and itself, so it has at least three factors and is not prime.Look for: even numbers are multiples of 2; so 2 is a factor as well as 1 and the number; more than two factors means composite; 2 itself is the only exception.
Worked example. Find the smallest factor of 77 that is greater than 1. 1. Test 2: 77 is odd, so 2 is not a factor. 2. Test 3: 7 + 7 = 14, which is not in the 3 times table, so 3 is not a factor. 3. Test 5: 77 does not end in 0 or 5, so 5 is not a factor. 4. Test 7: 7 × 11 = 77, so 7 is a factor. So the smallest factor of 77 above 1 is 7, and 77 is not a prime number.
Level 1
426 × 7 = 42.
78 × 7 = 56, so 56 ÷ 8 = 7.
729 × 8 = 72.
954 ÷ 6 = 9.
1212 × 12 = 144, so 144 ÷ 12 = 12.
13211 × 12 = 132 (10 × 12 = 120, plus 12).
76 is 2 × 3, so it is not prime. 7 has only the factors 1 and 7.
1; 2; 3; 61 × 6 = 6 and 2 × 3 = 6, so the factors are 1, 2, 3 and 6.
True13 has only the factors 1 and 13, so it is prime.
179 = 3 × 3 and 15 = 3 × 5. 17 has only the factors 1 and 17.
2The only factors of 11 are 1 and 11, so it has two factors.
2321 = 3 × 7, 22 = 2 × 11 and 23 has only the factors 1 and 23.
Level 2
2921 = 3 × 7, 27 = 3 × 9 and 33 = 3 × 11. Only 29 has just two factors.
2; 3; 5; 7; 11; 13; 17; 19Test each number from 2 to 19 for a factor other than 1 and itself. The primes are 2, 3, 5, 7, 11, 13, 17 and 19.
31; 37Even numbers and multiples of 5 are out. 33 = 3 × 11, 35 = 5 × 7 and 39 = 3 × 13, so 31 and 37 are left.
2Every other even number has 2 as a factor as well as 1 and itself, so it is not prime.
False5 + 1 = 6, so 51 is in the 3 times table: 3 × 17 = 51. It has more than two factors.
137 × 13 = 91, so 7 is a factor of 91 other than 1 and 91.
Yes5 + 7 = 12, so 57 is in the 3 times table: 3 × 19 = 57. Three rows of 19 tiles work.
8381 = 9 × 9, 82 and 84 are even. 83 has no factor other than 1 and 83.
3The odd numbers are 41, 43, 45, 47 and 49. 45 = 5 × 9 and 49 = 7 × 7, so 41, 43 and 47 are prime.
5349 = 7 × 7, 65 = 5 × 13 and 87 = 3 × 29. Only 53 has just two factors.
Level 3
369 is in the 3 times table (6 + 9 = 15): 3 × 23 = 69. Two rows is not possible as 69 is odd, so 3 rows is the fewest.
150The smallest is 61 and the largest is 89. 61 + 89 = 150.
71The primes between 60 and 80 are 61, 67, 71, 73 and 79. Only 71 has digits that add up to 8.
6Ravi's number is 47 and Sofia's is 53. 53 − 47 = 6.
13; 19; 31The numbers are 13, 19, 31, 39, 91 and 93. 39 = 3 × 13, 91 = 7 × 13 and 93 = 3 × 31, so 13, 19 and 31 are prime.
97Only a prime number of tiles cannot make such a rectangle. 99, 98, 96 and 95 have factors, and 97 has none other than 1 and 97.
Expert
Model answer: No. 9 is odd but 9 = 3 × 3, so it has the factor 3 as well as 1 and 9. Other examples are 15 and 21.Look for a counter-example: an odd number with a factor other than 1 and itself, such as 9, 15, 21 or 51. Also accept the point that 2 is even and prime.
19; 37; 73Numbers with digit sum 10 are 19, 28, 37, 46, 55, 64, 73, 82 and 91. Even numbers, 55 and 91 (7 × 13) are not prime.
13; 17; 31; 37The primes are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. Swapped, 13, 17, 31 and 37 give 31, 71, 13 and 73, all prime. 11 gives 11, which is not different.
Model answer: A factor of 89 bigger than 10 would need a partner of 10 or less, because 11 × 11 is already more than 89. A partner of 4, 6, 8, 9 or 10 would mean 2, 3 or 5 is a factor, so testing 2, 3, 5 and 7 covers every possibility.Look for factor pairs, one of each pair being 10 or less, and for 4, 6, 8, 9 and 10 being covered by 2, 3 and 5.
Worked example. Find 9², then say whether 80 is a square number. 1. 9² means 9 × 9. 2. 9 × 9 = 81, so 9² = 81. 3. The squares either side of 80 are 8² = 64 and 9² = 81. 4. 80 is neither 64 nor 81, so 80 is not a square number. So 9² = 81, and 80 is not a square number.
Level 1
123 groups of 4 make 12.
357 × 5 = 35 from the 5 times table.
87 × 8 = 56, so 56 ÷ 7 = 8.
954 ÷ 6 = 9, so 9 × 6 = 54.
1089 × 10 = 90 and 9 × 2 = 18, so 90 + 18 = 108.
1212 × 12 = 144, so 144 ÷ 12 = 12.
93² means 3 × 3, which is 9.
255² = 5 × 5 = 25.
164² = 4 × 4 = 16.
True6 × 6 = 36, so 36 is a square number.
10010² = 10 × 10 = 100.
648 × 8 = 64, so 64 is a square number. 24 is not.
Level 2
507² = 7 × 7 = 49. 49 + 1 = 50.
12111 × 11 = 110 + 11 = 121.
False5² = 5 × 5 = 25, but 5 × 2 = 10. Squaring is not doubling.
819 × 9 = 81. The other two are between squares.
36The square has 6 rows of 6 tiles. 6 × 6 = 36.
25; 36; 495 × 5 = 25, 6 × 6 = 36 and 7 × 7 = 49. 4² = 16 and 8² = 64 are outside.
497 rows of 7 seeds is 7 × 7 = 49.
303 × 3 = 9, so 30 × 30 = 900. The missing number is 30.
179² = 81 and 8² = 64. 81 − 64 = 17.
166² = 36 beads. 36 − 20 = 16.
Level 3
37520 × 20 = 400 tiles. 400 − 25 = 375.
9A square has equal rows, so find the number that times itself is 81. 9 × 9 = 81.
36; 64Squares between 30 and 90: 36, 49, 64 and 81. The even ones are 36 and 64.
20A 6 by 6 square has 36 tiles. 4 × 4 = 16 are shaded, so 36 − 16 = 20 more.
No6² = 36 and 4 × 10 = 40. 36 is less than 40.
36; 64Try 100 minus each square. 100 − 36 = 64, and 64 is a square. No other square works.
Expert
36; 81The two-digit squares are 16, 25, 36, 49, 64 and 81. Digit sums: 7, 7, 9, 13, 10, 9. Only 36 and 81 give 9.
Model answer: Never. The ones digit of a square comes from the ones digit of the number times itself. These give 0, 1, 4, 5, 6 or 9 only, so no square ends in 2, 3, 7 or 8.Look for: squares of 1 to 9 end in 1, 4, 9, 6, 5, 6, 9, 4, 1. The answer is never, with a reason that covers every ones digit.
121; 144The difference is the two numbers added: 11 + 12 = 23. So the squares are 11² = 121 and 12² = 144.
Model answer: 20² means 20 × 20, not 20 × 2. Femi doubled instead of multiplying 20 by itself. 2 × 2 = 4, so 20 × 20 = 400.Look for: ² means multiply the number by itself, not by 2. The correct value is 400.
Worked example. Find 6³. 1. 6³ means 6 × 6 × 6, with three sixes multiplied together. 2. Square first: 6 × 6 = 36. 3. Multiply by 6 again: 36 × 6 = 216. So 6³ = 216.
Level 1
246 × 4 is a known table fact: 24.
567 × 8 = 56.
3610 × 3 = 30 and 2 × 3 = 6, so 36.
302 × 5 = 10, then 10 × 3 = 30.
453 × 3 = 9, then 9 × 5 = 45.
846 × 2 = 12, then 12 × 7 = 84.
82 × 2 = 4, then 4 × 2 = 8.
273 × 3 × 3: 3 × 3 = 9, then 9 × 3 = 27.
1Multiplying 1 by 1 any number of times gives 1.
644 × 4 = 16, then 16 × 4 = 64.
True3 × 3 × 3 = 27, so 27 is a cube number.
1255 × 5 = 25, then 25 × 5 = 125.
Level 2
352³ = 8 and 3³ = 27, so 8 + 27 = 35.
False2³ + 2³ = 8 + 8 = 16, but 4³ = 64.
644 × 4 × 4 = 64, which is 4³.
3³4² = 16 and 3³ = 27, so 3³ is greater.
87³ = 343 is too small. 8³ = 8 × 8 × 8 = 512.
1005³ = 125 and 5² = 25, so 125 − 25 = 100.
3437 × 7 = 49, then 49 × 7 = 343.
False5³ = 125 and 6³ = 216, so 150 is not a cube number.
1,00010 × 10 = 100, then 100 × 10 = 1,000.
7299 × 9 = 81, then 81 × 9 = 729.
Level 3
48The cube has 4 × 4 × 4 = 64 small cubes. Taking away 16 leaves 48.
4Try 3: 27 is too small. 4 × 4 × 4 = 64, so 4 along each edge.
8; 27; 642³ = 8, 3³ = 27 and 4³ = 64. 1³ = 1 is too small and 5³ = 125 is too big.
64The cubes there are 27 and 64. 64 = 8², but 27 is not a square.
615³ = 125 and 4³ = 64. The difference is 125 − 64 = 61.
5³ is greater5³ = 5 × 5 × 5 = 125 and 11² = 11 × 11 = 121, so 5³ is greater.
Expert
Model answer: Kai multiplied 4 by 3. 4³ means 4 × 4 × 4, which is 16 × 4 = 64.Look for: ³ means three fours multiplied, not 4 times 3; correct value 64.
8; 64Cubes below 72 are 1, 8, 27 and 64. Only 8 + 64 = 72 works.
8Add 1 to each cube: 2, 9, 28, 65 and 126. Only 9 is a square (3²), so the cube is 8.
Model answer: No. 1³ = 1 and 1² = 1, so they are equal. For 2 or more, the cube is greater, such as 2³ = 8 and 2² = 4.Look for: a counterexample using 1 (both 1); possibly that bigger numbers do give a greater cube.
Worked example. Multiply 748 by 1,000. 1. Multiplying by 1,000 moves every digit three places to the left. 2. 7 hundreds become 7 hundred thousands; 4 tens become 4 ten thousands. 3. 8 ones become 8 thousands. Zeros fill the hundreds, tens and ones places. So 748 × 1,000 = 748,000.
Level 1
360Multiplying by 10 moves each digit one place to the left: 36 becomes 360.
47Dividing by 10 moves each digit one place to the right: 470 becomes 47.
5,300Multiplying by 100 moves each digit two places left: 53 becomes 5,300.
82Dividing by 100 moves each digit two places right: 8,200 becomes 82.
6,140Each digit moves one place left and a zero fills the ones place: 6,140.
730Work backwards: 7,300 ÷ 10 = 730.
7,000Multiplying by 1,000 moves the digit three places left: 7 becomes 7,000.
6Dividing by 1,000 moves the digit three places right: 6,000 becomes 6.
82,000Each digit moves three places left: 82 becomes 82,000.
45Each digit moves three places right: 45,000 becomes 45.
False24,000 ÷ 100: the digits move two places right, giving 240, not 24, so it is false.
7,040Each digit moves one place right: 70,400 becomes 7,040.
Level 2
380,600Each digit moves two places left and two zeros fill the gaps: 380,600.
420Dividing by 1,000 moves each digit three places right: 420,000 becomes 420.
590Work backwards: 59,000 ÷ 100 = 590.
84,200Work backwards: 8,420 × 10 = 84,200.
False5,290 × 100 = 529,000, which is ten times larger than 52,900. So it is false.
$24,50010 × $2,450 moves each digit one place left: $24,500.
607,000Times 100 moves each digit two places left: 6,070 becomes 607,000.
380Share equally by dividing by 100: 38,000 ÷ 100 = 380.
370,00037 × 10 = 370. The missing number divided by 1,000 gives 370, so it is 370 × 1,000 = 370,000.
Expert
Model answer: The answer is 64. Dividing by 100 moves each digit two places right, but Hana moved them only one place, as if dividing by 10. Check: 64 × 100 = 6,400.Look for: correct answer 64; Hana divided by 10, not 100; digits move two places right; a check by multiplying back.
1,000; 2,000; 3,000; 4,000; 5,000; 6,000; 7,000; 8,000; 9,000100 times a 2-digit number ends in two zeros. To also be a multiple of 1,000, the tens digit of that number must be 0 as well.
3,000Times 100 then ÷ 10 makes the number 10 times as big. So 9 times the number is 27,000, and 27,000 ÷ 9 = 3,000.
Model answer: Yes. 365 × 10 = 3,650 and 3,650 × 100 = 365,000, which is 365 × 1,000. The digits move one place and then two, which is three places in all.Look for: Sami is correct; the two steps shown with 365; the idea that 1 + 2 = 3 places, so the same as × 1,000.
3; 4; 53 × 7,000 = 21,000, 4 × 7,000 = 28,000 and 5 × 7,000 = 35,000 fit. 2 gives 14,000 and 6 gives 42,000.
9054,000 ÷ 600 is 540 ÷ 6 = 90.
500 × 8040 × 900 = 36,000 and 500 × 80 = 40,000, so 500 × 80 is greater.
4030 × 80 = 2,400 seeds. Then 2,400 ÷ 60 = 40.
Expert
Model answer: No. 5 × 6 = 30 is right, but 50 and 60 each have a zero, so two zeros are added. 50 × 60 = 3,000.Look for: Amira is not correct; both numbers are tens so two zeros are needed; the answer is 3,000.
13036 = 4 × 9, so 40 × 90 = 3,600. 60 × 60 = 3,600 too, but the numbers are the same. So 40 + 90 = 130.
Model answer: Always. A multiple of 100 ends in two zeros and a multiple of 10 ends in one zero, so the product ends in at least three zeros, for example 300 × 20 = 6,000.Look for: the answer is always; the zeros from each number are added together; a correct example such as 300 × 20 = 6,000.
1; 2; 3; 4; 6; 936,000 is 36 thousands, so the number must divide 36. 1, 2, 3, 4, 6 and 9 do. 5, 7 and 8 do not.
Unit 26: Problems with factors, multiples, squares and cubes
Worked example. Find the 2-digit number that is a multiple of 9, a factor of 72 and a square. 1. 2-digit multiples of 9: 18, 27, 36, 45, 54, 63, 72, 81, 90, 99. 2. Factors of 72 among these: 18, 36 and 72, since 72 ÷ 18 = 4 and 72 ÷ 36 = 2. 3. Squares among 18, 36 and 72: only 36, because 6 × 6 = 36. So the number is 36.
Level 1
173 × 4 = 12, then 12 + 5 = 17.
386 × 5 = 30, then 30 + 8 = 38.
364 times as big means 4 × 9 = 36.
657 × 8 = 56, then 56 + 9 = 65.
605 times as big means 5 × 12 = 60.
483 × 8 = 24 and 4 × 6 = 24, so 24 + 24 = 48.
True7 × 5 = 35, so 35 is a multiple of 7.
1; 2; 5; 10The factor pairs are 1 × 10 and 2 × 5, giving 1, 2, 5 and 10.
255² means 5 × 5 = 25.
82³ means 2 × 2 × 2 = 8.
Prime13 has only two factors, 1 and 13, so it is prime.
49A square bed of 7 rows of 7 is 7² = 49 plants.
Level 2
1212 × 12 = 144, so the missing number is 12.
9003 × 3 = 9, and 30 × 30 is 100 times bigger, so 900.
1048 × 12 = 96 is too small, so the next multiple is 8 × 13 = 104.
5951 = 3 × 17, 57 = 3 × 19 and 63 = 7 × 9, but 59 has no factors other than 1 and 59.
True5 × 5 × 5 = 125, so 125 is 5³.
918² = 64 and 3³ = 27, so 64 + 27 = 91.
30012 × 25 = 10 × 25 + 2 × 25 = 250 + 50 = 300.
121The mosaic is 11 rows of 11 tiles: 11 × 11 = 121.
14The only multiple of 7 between 10 and 20 is 14, and 56 ÷ 14 = 4.
Level 3
100The square bed has 9 × 9 = 81 plants. Then 81 + 19 = 100.
37The bigger cube uses 4 × 4 × 4 = 64 blocks and the smaller 3 × 3 × 3 = 27. Then 64 − 27 = 37.
6Work backwards: 43 − 7 = 36, and 6 × 6 = 36, so the number is 6.
4; 6; 8; 12Factor pairs of 48 are 4 × 12, 6 × 8, 8 × 6 and 12 × 4. Each factor in both places is more than 3, giving 4, 6, 8 and 12 rows.
2; 4; 8The number of pots must be a factor of both 40 and 56. The common factors are 1, 2, 4 and 8, and 1 pot is not allowed, so 2, 4 and 8.
441A square multiple of 7 comes from 7, 14 or 21 squared: 49, 196, 441. The next, 28², is over 500.
Expert
1; 64Cubes up to 90 are 1, 8, 27 and 64. Only 1 (1 × 1) and 64 (8 × 8) are also squares.
Model answer: No. 2 is prime because its only factors are 1 and 2, but 2 is even.Key points: Zara is not correct; give 2 as the counter-example; 2 has exactly two factors and is even.
24; 602-digit factors of 120 that are multiples of 6 are 12, 24, 30 and 60. Their digit sums are 3, 6, 3 and 6, so 24 and 60.
Model answer: Always. Of two consecutive numbers one is odd and one is even, so one square is odd and one is even. Odd + even is odd.Key points: always odd; one number of each pair is odd and the other even; odd × odd is odd, even × even is even; odd + even is odd. Examples such as 1 + 4 = 5 or 9 + 16 = 25 support but do not prove it.
Worked example. Write 4/5 as ___/10 and as ___/100. 1. Fifths to tenths: 5 × 2 = 10, so multiply the numerator by 2 too. 2. Multiply the numerator and the denominator by 2: 4 × 2 = 8, so 4/5 = 8/10. 3. For hundredths, multiply 8/10 by 10 on the top and the bottom: 80/100. So 4/5 = 8/10 = 80/100.
Level 1
41 × 2 = 2, so multiply the denominator by 2 as well: 2 × 2 = 4.
52 × 5 = 10, so multiply the numerator by 5 as well: 1 × 5 = 5.
TrueMultiply the top and the bottom of 1/3 by 2 to get 2/6.
2Divide the top and the bottom of 2/4 by 2: 2 ÷ 2 = 1 and 4 ÷ 2 = 2.
64 × 2 = 8, so the numerator is 3 × 2 = 6.
83 × 4 = 12, so the numerator is 2 × 4 = 8.
45 × 2 = 10, so the numerator is 2 × 2 = 4.
4Divide the top and the bottom by 2: 6 ÷ 2 = 3 and 8 ÷ 2 = 4.
32 × 3 = 6, so the numerator is 1 × 3 = 3.
123 × 3 = 9, so the denominator is 4 × 3 = 12.
2Divide the top and the bottom by 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3.
31/4 = ___/12. Since 4 × 3 = 12, the numerator is 1 × 3 = 3.
Level 2
125 × 4 = 20, so the numerator is 3 × 4 = 12.
30One tenth is 10 hundredths, so 3 tenths is 3 × 10 = 30 hundredths.
245 × 3 = 15, so the denominator is 8 × 3 = 24.
2Divide the top and the bottom by 2: 4 ÷ 2 = 2 and 6 ÷ 2 = 3.
TrueMultiply the top and the bottom of 5/6 by 3: 5 × 3 = 15 and 6 × 3 = 18.
6No whole number takes 6 to 9, so use thirds: 4/6 = 2/3, and 2/3 = 6/9.
2825 × 4 = 100, so the numerator is 7 × 4 = 28.
93/4 = ___/12. Since 4 × 3 = 12, the numerator is 3 × 3 = 9.
3Divide the top and the bottom by 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4.
6; 9; 12Multiply 3/4 by 2, 3 and 4 on the top and the bottom: 6/8, 9/12 and 12/16.
Level 3
141/3 = 8/24 and 1/4 = 6/24, so roses take 8 sections and herbs take 6. 8 + 6 = 14.
2/3A is 8 marks along, so it is at 8/12. Dividing the top and the bottom by 4 gives 2/3.
245 × 4 = 20, so multiply the denominator by 4 too: 6 × 4 = 24.
9; 8; 15Each step multiplies 2/3 by 2, 3, 4 and 5. So 6/9, 8/12 and 10/15.
7/1028 of the 40 squares are shaded, so the sky is 28/40. Divide the top and the bottom by 4 to get 7/10.
8; 12; 16The denominator must be a multiple of 4 so that the numerator is whole: 8, 12 and 16 (6/8, 9/12, 12/16).
Expert
Model answer: No. Equivalent fractions come from multiplying or dividing the top and the bottom by the same number, not adding. 3/5 = 9/15 but 4/6 = 10/15, so 3/5 and 4/6 are not equal.Look for: no; equivalent fractions come from multiplying or dividing both numbers by the same number, not adding. A check such as 3/5 = 9/15 and 4/6 = 10/15 shows they differ.
16; 242/3 has 2 + 3 = 5 parts in all, so multiply by 40 ÷ 5 = 8. The fraction is 16/24, and 16 + 24 = 40.
3; 6; 9; 12The numerator must be divisible by 3 so that dividing the top and the bottom by 3 gives a whole number: 3/12, 6/12, 9/12 and 12/12 become 1/4, 2/4, 3/4 and 4/4.
Model answer: Always. A fraction equal to 1/2 has a denominator that is 2 times its numerator, and any number multiplied by 2 is even.Look for: always, because the denominator is double the numerator (1/2 multiplied by the same number top and bottom), and double any whole number is even.
Worked example. Write 17/5 as a mixed number. 1. 5 fifths make 1 whole bar, so count how many 5s fit into 17. 2. 3 × 5 = 15, so 3 whole bars fill, and 17 − 15 = 2 fifths are left over. 3. The 3 full bars and 2 parts of the next bar make 3 2/5. So 17/5 = 3 2/5.
Level 1
3/5The denominators match, so add the numerators: 2 + 1 = 3. The answer is 3/5.
3/8The denominators match, so subtract the numerators: 5 − 2 = 3. The answer is 3/8.
5/7Add the numerators: 3 + 2 = 5. The denominator stays 7.
5/9Subtract the numerators: 8 − 3 = 5. The denominator stays 9.
4/11Find what must be added to 3 elevenths to make 7 elevenths: 7 − 3 = 4, so 4/11.
15 + 7 = 12, so the sum is 12/12, which is 1 whole.
1 1/44 quarters make 1 whole, and 5 − 4 = 1 quarter is left, so 1 1/4.
12 wholes are 6 thirds. 7 − 6 = 1 third is left, so the missing number is 1.
True2 wholes are 4 halves, and 1 more half makes 5 halves, so the statement is true.
1 2/55 fifths make 1 whole, and 7 − 5 = 2 fifths are left, so 1 2/5.
1 1/56/5 is 5/5 (1 whole) and 1/5 more, so 1 1/5.
24 quarters make 1 whole, so 8 quarters make 2 wholes with nothing left over.
Level 2
2 3/42 × 4 = 8, so 2 wholes use 8 quarters. 11 − 8 = 3 quarters are left, so 2 3/4.
3 11/123 × 12 = 36 and 4 × 12 = 48 is too big, so the whole part is 3. 47 − 36 = 11, so 3 11/12.
223 − 7 = 16 eighths are whole ones. 16 ÷ 8 = 2, so the missing whole part is 2.
2 9/102 × 10 = 20 and 29 − 20 = 9, so the mixed number is 2 9/10.
False5 wholes are 20 quarters, and 25 − 20 = 5 quarters are left. So 25/4 is 6 1/4, not 5 1/4.
3 1/43 × 4 = 12, so 3 whole laps, and 13 − 12 = 1 quarter is left over.
1 5/87 + 6 = 13, so the sum is 13/8. 8/8 is 1 whole and 5/8 is left, so 1 5/8.
2 1/54 + 3 + 4 = 11, so the sum is 11/5. 2 × 5 = 10, so 2 wholes and 1/5, which is 2 1/5.
66 × 6 = 36 and 7 × 6 = 42 is too big, so the whole part is 6.
1 4/75 + 6 = 11, so the total is 11/7. 7/7 is 1 whole and 4/7 is left, so 1 4/7.
Level 3
3 2/7The bars show 7 + 7 + 3 = 17 sevenths. Adding 6 more makes 23/7. 3 × 7 = 21 and 23 − 21 = 2, so 3 2/7.
2 3/5The line is marked in fifths. A is 3 marks after 2, so A is 2 3/5.
2 3/85 + 14 = 19 slices, which is 19/8 of a pizza. 2 × 8 = 16 and 19 − 16 = 3, so 2 3/8.
6/7Work backwards: take 4/7 away from 10/7. 10 − 4 = 6, so the start was 6/7.
11/5; 12/5; 13/5; 14/52 wholes are 10 fifths, and 15 fifths would be 3 wholes. So the numerators are 11, 12, 13 and 14. 10/5 is a whole number, not a mixed number.
7 3/431 sprints are 31 quarters. 7 × 4 = 28 and 31 − 28 = 3, so 7 3/4 laps.
Expert
Model answer: The fraction part 7/4 is still more than 1 whole, so another whole can be taken out of it. 4 quarters make 1 whole, which leaves 3 quarters. The answer is 4 3/4.Look for: the fraction part of a mixed number must be less than 1 whole; 7/4 is improper; the correct answer is 4 3/4.
13/9; 22/9; 31/9The mixed numbers are 1 4/9, 2 4/9 and 3 4/9, since 4 4/9 is not less than 4. They are 13/9, 22/9 and 31/9.
Model answer: Sometimes. 9/4 is 2 1/4, a mixed number. But 8/4 is exactly 2, a whole number with no fraction part, so it is not a mixed number.Look for: sometimes, with an example that works (a numerator that is not a multiple of 4) and one that does not (a multiple of 4).
86 wholes are 24 quarters. Each step is 3 quarters, and 24 ÷ 3 = 8 steps.
Worked example. Write 3 3/8 as an improper fraction. 1. Each whole is 8 eighths, so 3 wholes are 3 × 8 = 24 eighths. 2. Add the 3 extra eighths: 24 + 3 = 27 eighths. 3. Write the total eighths over 8: 27/8. So 3 3/8 = 27/8.
Level 1
5Share 10 into 2 equal groups: 5 in each.
1/3One part out of 3 equal parts is 1/3.
1/3Sharing into 3 gives bigger parts than sharing into 6, so 1/3 is larger.
3Three quarters make 3/4, so the missing number is 3.
91/4 of 12 is 3, so 3/4 of 12 is 3 × 3 = 9.
7/9Add the numerators and keep the denominator: 4 + 3 = 7, so 7/9.
31 whole is 2 halves, and 1 more half makes 3 halves.
5/22 wholes are 4 halves, plus 1 half is 5 halves, so 5/2.
4The first bar has 3 thirds shaded and the second has 1, so 4 thirds in all.
71 whole is 4 quarters, and 3 more quarters make 7.
True2 wholes are 6 thirds, plus 1 third is 7 thirds, so the statement is true.
8/51 whole is 5 fifths, plus 3 fifths is 8 fifths, so 8/5.
Level 2
17/53 wholes are 15 fifths, plus 2 fifths is 17 fifths.
194 wholes are 16 quarters, plus 3 quarters is 19 quarters.
17/6Count the shaded sixths: 6 + 6 + 5 = 17, so 17/6.
False5 wholes are 15 thirds, plus 2 thirds is 17 thirds, not 15 thirds.
13/34 wholes are 12 thirds, plus 1 third is 13 thirds.
212 cakes are 16 eighths, plus 5 eighths is 21 eighths.
115 wholes are 10 halves, plus 1 half is 11 halves.
373 wholes are 30 tenths, plus 7 tenths is 37 tenths.
23/54 laps are 20 fifths, plus 3 fifths is 23 fifths.
83/126 wholes are 72 twelfths, plus 11 twelfths is 83 twelfths.
Level 3
134 5/6 cakes is 4 × 6 + 5 = 29 slices. 7 cakes is 42 slices, and 42 − 29 = 13.
5Take away the 3 extra eighths: 43 − 3 = 40 eighths. 40 ÷ 8 = 5 wholes.
14/53 is 15/5, so the largest fraction in fifths below it is 14/5.
38/9Arjun forgot the 2 extra ninths. 4 × 9 = 36, and 36 + 2 = 38, so 38/9.
30/122 3/6 is 15/6. Doubling the top and bottom gives 30/12.
Expert
Model answer: 6 wholes are 6 × 7 = 42 sevenths. The 2 extra sevenths make 44 sevenths, so 6 2/7 = 44/7.Look for: each whole is 7 sevenths; 6 × 7 = 42; add the 2 extra sevenths to get 44.
2 5/6; 2 3/7; 2 1/8The numerator is 2 × d + n, which must be 17, so n = 17 − 2d with n below d. Only d = 6, 7 and 8 work.
28/3The numerator goes up by 3 for each extra whole. The rule is whole × 3 + 1: 27 + 1 = 28.
1 3/6; 2 4/6; 4 1/6Numerators run 7 to 11, 13 to 17, 19 to 23 and 25 to 29. The squares are 9, 16 and 25.
Worked example. Order 2/3, 5/6 and 7/12, smallest first. 1. Denominators 3, 6 and 12 all go into 12, so write each fraction in twelfths. 2. 2/3 = 8/12 and 5/6 = 10/12. The fraction 7/12 stays as it is. 3. Compare the numerators: 7 is less than 8, and 8 is less than 10. So the order, smallest first, is 7/12, 2/3, 5/6.
Level 1
1/2Halves are bigger pieces than quarters, so 1/2 is larger.
7/10The denominators are the same, so the larger numerator is the larger fraction.
1/4A whole cut into 4 parts gives smaller parts than a whole cut into 3 parts, so 1/4 is smaller.
FalseBoth are ninths and 6 is more than 4, so 6/9 is greater than 4/9.
2/9; 5/9; 7/9All are ninths, so put the numerators in order: 2, 5, 7.
1/8With unit fractions, the larger the denominator, the smaller the fraction. The largest denominator is 8.
3/41/2 is the same as 2/4, and 3/4 is more than 2/4.
2Thirds and sixths: each third is two sixths, so 1/3 = 2/6.
5/83/4 is the same as 6/8, and 5/8 is less than 6/8.
3/61/3 is the same as 2/6, and 3/6 is more than 2/6.
1/2; 5/8; 3/4In eighths the fractions are 6/8, 4/8 and 5/8, so the order is 4/8, 5/8, 6/8.
Omar1/4 is the same as 2/8, and 3/8 is more than 2/8, so Omar ran further.
Level 2
7/92/3 is the same as 6/9. Then 7/9 is more than 6/9.
>1/3 is the same as 4/12. Then 5/12 is greater than 4/12.
3/5; 7/10; 4/5In tenths the fractions are 7/10, 6/10 and 8/10, so the order is 6/10, 7/10, 8/10.
103/4 is the same as 9/12. The numerator must be more than 9, so the smallest is 10.
TrueIn twelfths, 1 1/3 is 16/12, which is more than 13/12. So it is true.
=Multiply 3/8 top and bottom by 2 to get 6/16. The fractions are equal.
Kai5/8 is the same as 10/16, which is less than 11/16, so Kai ate more.
5/12In twelfths the fractions are 3/12, 5/12 and 4/12. The largest is 5/12.
1 1/2; 1 5/8; 7/4In eighths the numbers are 14/8, 12/8 and 13/8. The order is 12/8, 13/8, 14/8.
2 1/42 1/4 is 18/8. Then 17/8 is less than 18/8.
Level 3
5/8In eighths the fractions are 5/8, 7/8 and 4/8. The middle one is 5/8.
4; 5; 6; 7; 81/4 is 3/12 and 3/4 is 9/12, so n is more than 3 and less than 9.
11/8In eighths, 1 1/4 is 10/8 and 1 1/2 is 12/8. The only numerator between 10 and 12 is 11.
19/8In eighths the amounts are 18/8, 17/8 and 19/8. The greatest is 19/8.
BIn twelfths, bar A is 5/12, bar B is 6/12 and bar C is 4/12. Bar B is greatest.
2In tenths the fractions are 1, 4, 6, 9 and 5, and 1/2 is 5/10. Only 6/10 and 9/10 are greater.
Expert
Model answer: No. 3/4 is the same as 6/8, and 5/8 is less than 6/8. Comparing the tops and bottoms separately does not work.Look for: writing both in eighths (6/8 and 5/8), a clear no, and that the numerator and denominator cannot be compared separately.
3/8; 5/8; 7/8The numerator is more than 2 and less than 8: 3, 4, 5, 6, 7. The even ones (4/8 = 2/4 and 6/8 = 3/4) can be written in quarters, which leaves 3, 5 and 7.
Model answer: Always. The same number of parts is taken, and a larger denominator means smaller parts. For example, 3/8 is less than 3/4.Look for: always, the idea that more parts in the whole means smaller parts, and a worked example such as 3/8 and 3/4.
1 11/24In twenty-fourths the bounds are 1 10/24 and 1 12/24. The number halfway is 1 11/24.