Chapter 1 — worked answers and marking guidance
All chapter review and practice examination items
Chapter review solutions
How many complete groups of 100 can be made from 58,143 objects, and how many remain?
58,143 = 581 × 100 + 43.
In 200,962, what is the value of the digit in the hundred-thousands place?
The digit 2 represents 2 groups of 100,000, or 200,000.
Extend this pattern: 934,120, 934,370, 934,620, ___.
Each term increases by 250; add that step once more.
Order these numbers from least to greatest: 873,669, 856,294, 948,515.
Compare digit lengths first, then compare the greatest differing place.
Round 48,003 to the nearest 100.
The bounding multiples are 48,000 and 48,100. The midpoint is 48,050, so the nearer benchmark is 48,000.
Round 507,649 to the nearest 1,000.
The bounding multiples are 507,000 and 508,000. The midpoint is 507,500, so the nearer benchmark is 508,000.
A teacher partitions 613,900 counters into equal groups of 100. How many counters are in each group?
Each group has 613,900 ÷ 100 = 6,139 counters.
In 18,820, what is the value of the digit in the ten-thousands place?
The digit 1 represents 1 groups of 10,000, or 10,000.
Find 893,434 − 892,434 and explain the place-value change.
Only the increment changes: 893,434 − 892,434 = 1,000.
A sequence begins at 428,216 and increases by 1,000 each time. What is its fifth term?
The fifth term is four equal steps after the first: 428,216 + 4 × 1,000.
Insert <, >, or =: 149,180 ___ 158,948.
Compare the leftmost place where the digits differ: 149,180 is less than 158,948.
Round 14,467 to the nearest 100.
The bounding multiples are 14,400 and 14,500. The midpoint is 14,450, so the nearer benchmark is 14,500.
What is the midpoint between 925,000 and 926,000?
Half of the interval of 1,000 is 500. Add it to the lower benchmark.
Calculate 7,321 × 100 using place value.
Every place-value unit becomes 100 times as large; 7,321 × 100 = 732,100.
Find 24,542 − 14,542 and explain the place-value change.
Only the increment changes: 24,542 − 14,542 = 10,000.
How many complete groups of 100 can be made from 181,798 objects, and how many remain?
181,798 = 1817 × 100 + 98.
Extend this pattern: 992,785, 993,035, 993,285, ___.
Each term increases by 250; add that step once more.
Insert <, >, or =: 376,090 ___ 366,451.
Compare the leftmost place where the digits differ: 376,090 is greater than 366,451.
What is the midpoint between 21,500 and 21,600?
Half of the interval of 100 is 50. Add it to the lower benchmark.
Round 601,061 to the nearest 1,000.
The bounding multiples are 601,000 and 602,000. The midpoint is 601,500, so the nearer benchmark is 601,000.
Calculate 292,000 ÷ 100 by grouping place-value units.
Dividing by 100 reduces each place-value unit by a factor of 100. The quotient is 2,920.
How many complete groups of 100 can be made from 67,606 objects, and how many remain?
67,606 = 676 × 100 + 6.
In 696,662, what is the value of the digit in the hundred-thousands place?
The digit 6 represents 6 groups of 100,000, or 600,000.
Extend this pattern: 421,870, 426,870, 431,870, ___.
Each term increases by 5,000; add that step once more.
Order these numbers from least to greatest: 251,265, 269,772, 634,634.
Compare digit lengths first, then compare the greatest differing place.
Round 68,623 to the nearest 100.
The bounding multiples are 68,600 and 68,700. The midpoint is 68,650, so the nearer benchmark is 68,600.
Round 765,211 to the nearest 1,000.
The bounding multiples are 765,000 and 766,000. The midpoint is 765,500, so the nearer benchmark is 765,000.
A student says the 2 in 24,105 means 2 ones. Explain its actual value.
The 2 occupies the ten-thousands position, representing 2 groups of 10,000.
A student removes the zeros from 304,006, saying zeros never affect place value. Explain why this is incorrect.
The zeros preserve each digit position: the number contains 3 hundred-thousands, 4 thousands, and 6 ones. Removing them changes the value.
A student predicts 12,526 after the pattern 12,126, 12,226, 12,326. Explain and give the correct next term.
Adjacent terms each differ by 100; add it once to 12,326.
Which is greater, 178,090 or 179,010? A student compares only the last digit. Explain the mistake.
Compare from the highest place where digits differ. 179,010 exceeds 178,090 by 920, regardless of the final digit.
A student rounds 67,450 to the nearest hundred as 67,400. Explain why this is wrong.
67,450 is halfway between 67,400 and 67,500; our rounding convention sends halfway values up.
A student rounds 685,500 down to 685,000 to the nearest thousand. Explain and correct it.
The midpoint between 685,000 and 686,000 is 685,500; halfway rounds upward.
A learner multiplies 7,200 by 100 and writes 72,000. Identify the scaling mistake and give the correct result.
Multiplying by 100 makes every place value one hundred times larger. The correct product is 720,000.
Practice examination solutions and scoring
In 51,079, what is the value of the digit in the ten-thousands place?
The digit 5 represents 5 groups of 10,000, or 50,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find 12,412 − 11,412 and explain the place-value change.
Only the increment changes: 12,412 − 11,412 = 1,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
How many complete groups of 10 can be made from 81,005 objects, and how many remain?
81,005 = 8100 × 10 + 5.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
In 16,272, what is the value of the digit in the ten-thousands place?
The digit 1 represents 1 groups of 10,000, or 10,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find 47,378 − 46,378 and explain the place-value change.
Only the increment changes: 47,378 − 46,378 = 1,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
How many complete groups of 100 can be made from 81,744 objects, and how many remain?
81,744 = 817 × 100 + 44.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
In 78,807, what is the value of the digit in the ten-thousands place?
The digit 7 represents 7 groups of 10,000, or 70,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find 238,432 − 237,432 and explain the place-value change.
Only the increment changes: 238,432 − 237,432 = 1,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
How many complete groups of 100 can be made from 836,761 objects, and how many remain?
836,761 = 8367 × 100 + 61.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
In 353,980, what is the value of the digit in the hundred-thousands place?
The digit 3 represents 3 groups of 100,000, or 300,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find 456,669 − 455,669 and explain the place-value change.
Only the increment changes: 456,669 − 455,669 = 1,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
How many complete groups of 10 can be made from 302,721 objects, and how many remain?
302,721 = 30272 × 10 + 1.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
In 542,696, what is the value of the digit in the hundred-thousands place?
The digit 5 represents 5 groups of 100,000, or 500,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find 945,534 − 935,534 and explain the place-value change.
Only the increment changes: 945,534 − 935,534 = 10,000.
Scoring (2 points): Award one point for a sound method and one for the correct result.
A sequence begins at 676,500 and increases by 250 each time. What is its fifth term?
The fifth term is four equal steps after the first: 676,500 + 4 × 250.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Extend this pattern: 190,687, 190,787, 190,887, ___.
Each term increases by 100; add that step once more.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Extend this pattern: 700,335, 705,335, 710,335, ___.
Each term increases by 5,000; add that step once more.
Scoring (2 points): Award one point for a sound method and one for the correct result.
A sequence begins at 693,557 and increases by 50 each time. What is its fifth term?
The fifth term is four equal steps after the first: 693,557 + 4 × 50.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Extend this pattern: 530,049, 535,049, 540,049, ___.
Each term increases by 5,000; add that step once more.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Extend this pattern: 379,072, 384,072, 389,072, ___.
Each term increases by 5,000; add that step once more.
Scoring (2 points): Award one point for a sound method and one for the correct result.
A sequence begins at 600,942 and increases by 100 each time. What is its fifth term?
The fifth term is four equal steps after the first: 600,942 + 4 × 100.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Insert <, >, or =: 181,994 ___ 178,979.
Compare the leftmost place where the digits differ: 181,994 is greater than 178,979.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Insert <, >, or =: 184,403 ___ 199,590.
Compare the leftmost place where the digits differ: 184,403 is less than 199,590.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Order these numbers from least to greatest: 937,209, 942,835, 240,822.
Compare digit lengths first, then compare the greatest differing place.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Insert <, >, or =: 837,632 ___ 857,395.
Compare the leftmost place where the digits differ: 837,632 is less than 857,395.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Insert <, >, or =: 983,097 ___ 965,340.
Compare the leftmost place where the digits differ: 983,097 is greater than 965,340.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Order these numbers from least to greatest: 722,734, 716,040, 547,753.
Compare digit lengths first, then compare the greatest differing place.
Scoring (2 points): Award one point for a sound method and one for the correct result.
A student says the 8 in 84,105 means 8 ones. Explain its actual value.
The 8 occupies the ten-thousands position, representing 8 groups of 10,000.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
A student removes the zeros from 204,006, saying zeros never affect place value. Explain why this is incorrect.
The zeros preserve each digit position: the number contains 2 hundred-thousands, 4 thousands, and 6 ones. Removing them changes the value.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
A student predicts 17,455 after the pattern 16,455, 16,705, 16,955. Explain and give the correct next term.
Adjacent terms each differ by 250; add it once to 16,955.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
Which is greater, 128,090 or 129,010? A student compares only the last digit. Explain the mistake.
Compare from the highest place where digits differ. 129,010 exceeds 128,090 by 920, regardless of the final digit.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
A student rounds 40,450 to the nearest hundred as 40,400. Explain why this is wrong.
40,450 is halfway between 40,400 and 40,500; our rounding convention sends halfway values up.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
A student rounds 357,500 down to 357,000 to the nearest thousand. Explain and correct it.
The midpoint between 357,000 and 358,000 is 357,500; halfway rounds upward.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
A learner multiplies 4,100 by 100 and writes 41,000. Identify the scaling mistake and give the correct result.
Multiplying by 100 makes every place value one hundred times larger. The correct product is 410,000.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.