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Semester 3A · Answer key · ORIGINAL PUBLIC PRACTICE

Complete worked solutions

Full explanations for every question on the cumulative review and practice examination.

Educator and learner solutions

Every answer has an explanation. The practice examination also includes a suggested per-point marking guide. Accept mathematically valid alternative methods, models, and responses. These solutions are public; do not treat this examination as secure or use its score alone for placement.

Review answers and worked reasoning

  1. 3A-R-001 · Write 6,407 in words.

    Answer: six thousand four hundred seven

    Read the 6 thousands, 4 hundreds, 0 tens, and 7 ones.

    1 points · C1 PV-01
  2. 3A-R-002 · In 8,353, what is the value of the second 3 (reading left to right)?

    Answer: 3 ones

    The second 3 is in the ones place, so its value is 3. The first 3 is 300.

    2 points · C1 PV-02
  3. 3A-R-003 · Write 5,090 as a sum of thousands, hundreds, tens, and ones.

    Answer: 5,000 + 90

    Five thousands, zero hundreds, nine tens, zero ones: 5,000 + 90.

    2 points · C1 PV-03
  4. 3A-R-004 · How many hundreds equal 3,200? Explain your grouping.

    Answer: 32 hundreds

    Each hundred is 100, and 3,200 ÷ 100 = 32; 3 thousands become 30 hundreds plus 2.

    2 points · C1 PV-04
  5. 3A-R-005 · Order these from least to greatest: 7,205; 7,025; 7,250; 7,052.

    Answer: 7,025; 7,052; 7,205; 7,250

    All have 7 thousands. Compare the hundreds next (0 before 2), then the tens.

    2 points · C1 PV-05
  6. 3A-R-006 · On a number line marked 2,000, 2,200, 2,400, 2,600 at equal distances, what value lies halfway between 2,200 and 2,400?

    Answer: 2,300

    Half of the 200-unit gap is 100; 2,200 + 100 = 2,300.

    2 points · C1 PV-06
  7. 3A-R-007 · Continue the pattern for two more terms: 5,875; 5,625; 5,375; __; __.

    Answer: 5,125; 4,875

    Each step subtracts 250, so continue twice.

    2 points · C1 PV-07
  8. 3A-R-008 · Round 6,521 to the nearest thousand.

    Answer: 7,000

    The hundreds digit is 5, and 6,521 is above the midpoint 6,500.

    1 points · C1 PV-08
  9. 3A-R-009 · Round 4,349 to the nearest hundred.

    Answer: 4,300

    4,349 is 49 away from 4,300 and 51 away from 4,400.

    1 points · C1 PV-09
  10. 3A-R-010 · Round 8,765 to the nearest ten.

    Answer: 8,770

    The ones digit is 5, so round up to the next multiple of 10.

    1 points · C1 PV-10
  11. 3A-R-011 · A student rounds 5,449 first to 5,400 (nearest hundred), then to 5,000 (nearest thousand). Is 5,000 also the correct result from rounding 5,449 directly to the nearest thousand? Explain.

    Answer: Yes, 5,000

    5,449 is below the 5,500 midpoint, so direct rounding gives 5,000. The two-step procedure agrees here, but two-step rounding is not always safe.

    3 points · C1 PV-11
  12. 3A-R-012 · Give a four-digit number that rounds to 3,500 to the nearest hundred and 4,000 to the nearest thousand; justify your example.

    Answer: Example: 3,500

    3,500 is already a multiple of 100 and sits at the midpoint between 3,000 and 4,000, so under the standard half-up rule it rounds to 4,000. Values 3,500–3,549 also work.

    3 points · C1 PV-11, C1 PV-08, C1 PV-09
  13. 3A-R-013 · Find 438 + 250 by splitting 250 into hundreds and tens.

    Answer: 688

    438 + 200 = 638; 638 + 50 = 688.

    2 points · C02-L01
  14. 3A-R-014 · Compute 598 + 167 using compensation; show why the adjusted sum must be corrected.

    Answer: 765

    (600 + 167) − 2 = 767 − 2 = 765. We added 2 extra to 598, so subtract it.

    2 points · C02-L02
  15. 3A-R-015 · Find 763 − 240 mentally; record useful jumps.

    Answer: 523

    763 − 200 = 563; then 563 − 40 = 523.

    2 points · C02-L03
  16. 3A-R-016 · Use equal adjustment to calculate 803 − 298. Show both adjusted numbers.

    Answer: 505

    Add 2 to both: 805 − 300 = 505. Equal shifts preserve the difference.

    2 points · C02-L04
  17. 3A-R-017 · What must be added to 674 to make 1,000?

    Answer: 326

    674 + 26 = 700; another 300 reaches 1,000, so 326.

    2 points · C02-L05
  18. 3A-R-018 · Without using column addition, calculate 397 + 205.

    Answer: 602

    Think 400 + 205 − 3 = 602.

    2 points · C02-L06
  19. 3A-R-019 · Two teams together earned 94 points. One scored 18 more than the other. Find both scores and describe a bar model.

    Answer: 38 and 56

    Remove the 18 extra points: 94 − 18 = 76. Two equal bars each represent 38; the larger is 38 + 18 = 56.

    4 points · C02-L07
  20. 3A-R-020 · A museum counted 1,248 visitors over the weekend. If 375 came on Saturday, how many visited on Sunday?

    Answer: 873

    The weekend total has two parts: Saturday and Sunday. Sunday = 1,248 − 375 = 873.

    2 points · C02-L08
  21. 3A-R-021 · Maya has 235 beads. She has 47 fewer than Niko. How many beads does Niko have?

    Answer: 282

    Niko has more, so add: 235 + 47 = 282.

    2 points · C02-L09
  22. 3A-R-022 · A library had 1,420 books. It received 235 more, then lent out 189. How many remain?

    Answer: 1,466

    Start 1,420 + 235 = 1,655. Then 1,655 − 189 = 1,466.

    3 points · C02-L10
  23. 3A-R-023 · Two students calculate 702 − 299. One writes 703 − 300. The other writes 702 − 300 and adds 1. Are both valid? Find the answer.

    Answer: Both are valid; 403

    Equal adjustment: 703 − 300 = 403. Compensation: 702 − 300 + 1 = 403. Both preserve the original difference.

    3 points · C02-L02, C02-L04
  24. 3A-R-024 · Ava has 36 more stickers than Ben. Ava gives Ben 8. Who has more afterward, and by how many?

    Answer: Ava by 20

    Giving 8 decreases Ava by 8 and increases Ben by 8, so the gap shrinks by 16: 36 − 16 = 20.

    3 points · C02-L09
  25. 3A-R-025 · Calculate 2,768 + 1,457. Show at least one place-value exchange.

    Answer: 4,225

    8 ones + 7 ones = 15 ones: write 5 ones, exchange 10 ones for 1 ten. Continue to get 4,225.

    3 points · C03-L01
  26. 3A-R-026 · Calculate 4,503 − 1,278. Show why borrowing is required.

    Answer: 3,225

    Regroup across the empty tens; 4,503 − 1,278 = 3,225. Add 1,278 back to check.

    3 points · C03-L02
  27. 3A-R-027 · Solve 6,000 − 2,487 and explain how you regroup across zeros.

    Answer: 3,513

    Rename 6,000 as 5 thousands, 9 hundreds, 9 tens and 10 ones, then subtract columns: 3,513.

    3 points · C03-L03
  28. 3A-R-028 · Estimate 3,462 + 2,153 by rounding each addend to the nearest hundred.

    Answer: 5,700

    3,462 → 3,500 and 2,153 → 2,200; sum 5,700.

    2 points · C03-L04
  29. 3A-R-029 · Estimate 6,187 − 2,741 by rounding to the nearest thousand, then calculate the exact difference.

    Answer: Estimate 3,000; exact 3,446

    6,187 → 6,000; 2,741 → 3,000; estimate = 3,000. Exact: 6,187 − 2,741 = 3,446.

    3 points · C03-L05
  30. 3A-R-030 · A school collected 2,385 cans in September and 1,756 in October, then recycled 2,094. Estimate the number left by rounding each quantity to the nearest hundred, then calculate exactly.

    Answer: Estimate 2,100; exact 2,047

    Estimate 2,400 + 1,800 − 2,100 = 2,100. Exact: 2,385 + 1,756 = 4,141; 4,141 − 2,094 = 2,047.

    3 points · C03-L06
  31. 3A-R-031 · A learner says 4,002 − 1,978 = 3,976 because they subtracted the smaller digit from the larger in each column. Explain the error and find the correct answer.

    Answer: 2,024

    Column subtraction must exchange place-value units, not always subtract smaller digit from larger. Correct: 4,002 − 1,978 = 2,024.

    3 points · C03-L03
  32. 3A-R-032 · Which is more reasonable for 2,908 + 3,096: 5,000 or 6,000? Justify before finding the exact sum.

    Answer: 6,000; exact 6,004

    Both numbers are near 3,000, so their sum is near 6,000. The exact sum is 6,004.

    3 points · C03-L04, C03-L05
  33. 3A-R-033 · Draw or describe an array showing 7 rows of 4 dots. How many dots are there?

    Answer: 28 dots

    Each of 7 rows has 4 dots, so 7 × 4 = 28. Any 7-by-4 array has 28 dots.

    2 points · C04-L01
  34. 3A-R-034 · Use the distributive property to find 9 × 7 by splitting 7 into 5 and 2.

    Answer: 63

    9 × (5 + 2) = 45 + 18 = 63.

    2 points · C04-L02
  35. 3A-R-035 · Thirty-six pencils are shared equally among 6 boxes. How many pencils go in each box?

    Answer: 6 pencils

    36 ÷ 6 = 6. Check 6 × 6 = 36.

    2 points · C04-L03
  36. 3A-R-036 · Explain why 0 ÷ 7 = 0 but 7 ÷ 0 is not defined.

    Answer: 0 ÷ 7 = 0; 7 ÷ 0 is undefined

    Zero objects can be shared among 7 groups as 0 each. No number multiplied by 0 yields 7, so dividing 7 by 0 is undefined.

    3 points · C04-L04
  37. 3A-R-037 · Forty-seven counters are packed in bags of 6. How many full bags and how many counters left over?

    Answer: 7 full bags; 5 left over

    6 × 7 = 42 and 47 − 42 = 5. The remainder 5 is smaller than 6.

    2 points · C04-L05
  38. 3A-R-038 · Without adding individually, decide whether the sum of five odd whole numbers is odd or even. Explain.

    Answer: Odd

    Pairing two odd numbers gives an even sum. Two pairs give even + even, and one unpaired odd makes the total odd.

    3 points · C04-L06
  39. 3A-R-039 · There are 8 tables with 6 chairs at each. How many chairs are there?

    Answer: 48 chairs

    8 equal groups of 6: 8 × 6 = 48.

    2 points · C04-L07
  40. 3A-R-040 · Jules has four times as many marbles as Kai. Kai has 9. Describe the equal-unit bar model and find Jules’s number.

    Answer: 36 marbles

    Kai has one 9-unit bar. Jules has four bars each the same length, totaling 4 × 9 = 36.

    3 points · C04-L08
  41. 3A-R-041 · A baker makes 9 trays with 8 rolls each. She packs the rolls into bags of 5. How many full bags and loose rolls?

    Answer: 14 full bags; 2 loose rolls

    9 × 8 = 72 rolls. 72 ÷ 5 = 14 remainder 2.

    3 points · C04-L09
  42. 3A-R-042 · Thirty-eight children need seats in vans that each hold 8. How many vans are needed? Explain why the remainder matters.

    Answer: 5 vans

    38 ÷ 8 = 4 remainder 6. Four vans cannot seat the remaining six children, so a fifth van is needed.

    3 points · C04-L05
  43. 3A-R-043 · Find 7 × 300 using hundreds, not repeated counting.

    Answer: 2,100

    7 groups of 3 hundreds = 21 hundreds = 2,100.

    2 points · C05-L01
  44. 3A-R-044 · Calculate 23 × 3 using (20 + 3) × 3.

    Answer: 69

    20 × 3 = 60; 3 × 3 = 9; total 69.

    2 points · C05-L02
  45. 3A-R-045 · Calculate 42 × 6; show what happens when 4 tens × 6 = 24 tens.

    Answer: 252

    6 × 2 ones = 12 ones: write 2 ones and carry 1 ten. Then 6 × 4 tens plus 1 ten = 25 tens, giving 252.

    3 points · C05-L03
  46. 3A-R-046 · Calculate 58 × 4 with place-value exchanges.

    Answer: 232

    8 × 4 = 32 ones = 3 tens + 2 ones. 5 tens × 4 plus 3 tens = 23 tens = 232.

    3 points · C05-L04
  47. 3A-R-047 · Calculate 67 × 8 and identify both exchanges.

    Answer: 536

    8 × 7 ones = 56 ones, exchange 5 tens; 8 × 6 tens plus 5 tens = 53 tens, exchange 5 hundreds, yielding 536.

    3 points · C05-L05
  48. 3A-R-048 · A factory packs 4 boxes with 213 buttons each. How many buttons altogether?

    Answer: 852

    213 × 4 = (200 × 4) + (10 × 4) + (3 × 4) = 800 + 40 + 12 = 852.

    3 points · C05-L06
  49. 3A-R-049 · Calculate 376 × 5; estimate first to check whether the result is reasonable.

    Answer: 1,880

    376 × 5 = (300 + 70 + 6) × 5 = 1,500 + 350 + 30 = 1,880; roughly 400 × 5 = 2,000.

    3 points · C05-L07
  50. 3A-R-050 · A theater has 6 sections, each with 128 seats. If 49 seats are empty, how many people are seated?

    Answer: 719 people

    6 × 128 = 768 seats; 768 − 49 = 719 occupied seats.

    3 points · C05-L06, C05-L07
  51. 3A-R-051 · Someone claims 46 × 7 = 282 by writing 6 × 7 = 42 as just 2 ones, then doing 4 × 7 = 28 tens. What exchange was omitted? Find the correct product.

    Answer: 322

    The 4 tens carried from 42 ones must be added to 28 tens: 28 + 4 = 32 tens, so 46 × 7 = 322.

    3 points · C05-L05
  52. 3A-R-052 · Calculate 840 ÷ 4 by interpreting the hundreds and tens.

    Answer: 210

    840 is 84 tens. 84 tens ÷ 4 = 21 tens = 210.

    2 points · C06-L01
  53. 3A-R-053 · Calculate 64 ÷ 2 using equal tens and ones.

    Answer: 32

    Split 6 tens into 3 tens for each group, and 4 ones into 2 ones each: 32.

    2 points · C06-L02
  54. 3A-R-054 · Calculate 74 ÷ 2. Explain how regrouping a leftover ten works.

    Answer: 37

    Give each group 3 tens, leaving 1 ten; exchange that for 10 ones and combine with 4 ones. 14 ones ÷ 2 = 7 each, so 37.

    3 points · C06-L03
  55. 3A-R-055 · Divide 96 apples equally into 4 crates. How many apples per crate?

    Answer: 24 apples

    96 ÷ 4 = 24; check 4 × 24 = 96.

    2 points · C06-L04
  56. 3A-R-056 · Calculate 438 ÷ 2. Explain how many hundreds each group receives.

    Answer: 219

    4 hundreds ÷ 2 = 2 hundreds per group, then 3 tens leave 1 ten to exchange; result 219.

    3 points · C06-L05
  57. 3A-R-057 · Calculate 684 ÷ 4, showing the place values of each quotient digit.

    Answer: 171

    6 hundreds ÷ 4 gives 1 hundred and 2 hundreds left; exchange to get 28 tens, giving 7 tens, then 4 ones ÷ 4 gives 1 one. Quotient 171.

    3 points · C06-L06
  58. 3A-R-058 · Calculate 336 ÷ 4. Why does the quotient have two digits rather than three?

    Answer: 84

    3 hundreds cannot make 4 full groups of 1 hundred, so exchange them into tens. 33 tens ÷ 4 gives 8 tens and 1 ten left, then 16 ones ÷ 4 = 4 ones. Quotient 84.

    3 points · C06-L07
  59. 3A-R-059 · A school has 285 pencils. It places 5 pencils in each packet. How many packets can be made?

    Answer: 57 packets

    285 ÷ 5 = 57; check 57 × 5 = 285.

    3 points · C06-L04, C06-L07
  60. 3A-R-060 · The bar graph shows books borrowed by four classes. Which class borrowed the most?

    Answer: Cedar

    Cedar has a bar reaching 18, taller than Oak 10, Pine 14, and Maple 12.

    2 points · C07-L01
  61. 3A-R-061 · Using the same graph, how many more books did Cedar borrow than Oak?

    Answer: 8 books

    Cedar borrowed 18 and Oak borrowed 10: 18 − 10 = 8.

    2 points · C07-L01
  62. 3A-R-062 · The table shows recycled cans by day. How many were recycled Monday through Wednesday altogether?

    Answer: 57 cans

    18 + 24 + 15 = 57.

    2 points · C07-L02
  63. 3A-R-063 · From the same table, which day was exactly 6 cans higher than Wednesday?

    Answer: Thursday

    Wednesday 15; Monday 18 (3 higher), Tuesday 24 (9 higher), Thursday 21 (6 higher). Correct day is Thursday.

    2 points · C07-L02
  64. 3A-R-064 · A picture graph uses one star to represent 5 votes. One team has 4 stars. How many votes did it earn?

    Answer: 20 votes

    4 symbols × 5 votes per symbol = 20. A picture graph key is part of the data.

    2 points · C07-L01
  65. 3A-R-065 · Two bar graphs display the same survey data. One vertical axis begins at 0; the other begins at 10. Could the pictures make the same difference look different? Explain.

    Answer: Yes

    Truncating the axis at 10 can exaggerate the visual difference; check axis labels and numerical scale before comparing bars.

    3 points · C07-L02
  66. 3A-R-066 · A class has 2,675 paper squares and another class brings 1,459. Estimate the total to the nearest hundred and then calculate it exactly.

    Answer: Estimate 4,200; exact 4,134

    Round 2,675 → 2,700 and 1,459 → 1,500 for estimate 4,200. Exact: 2,675 + 1,459 = 4,134.

    3 points · C03-L04, C1 PV-09
  67. 3A-R-067 · Five packages each contain 124 cards. The cards are then shared equally among four classes. How many does each class get?

    Answer: 155 cards

    5 × 124 = 620 cards, and 620 ÷ 4 = 155 cards per class.

    4 points · C06-L06, C05-L06

Practice examination answers and worked reasoning

  1. 3A-E-001 · Calculate 796 + 208 using compensation and show the corrected adjustment.

    Answer: 1,004

    800 + 208 − 4 = 1,004.

    2 points · C02-L01, C02-L02

    Suggested partial-credit guide:

    • 1 point — Uses 800 as a friendly substitute and shows the +208 step
    • 1 point — Correctly subtracts the extra 4 and obtains 1,004
  2. 3A-E-002 · Write “seven thousand thirty-six” in digits and expanded form.

    Answer: 7,036 = 7,000 + 30 + 6

    7 thousands, zero hundreds, 3 tens, 6 ones: 7,036 and 7,000 + 30 + 6.

    2 points · C1 PV-01, C1 PV-03

    Suggested partial-credit guide:

    • 1 point — Writes 7,036 in digits
    • 1 point — Gives an equivalent expanded form 7,000 + 30 + 6
  3. 3A-E-003 · Find 8 × 400 using place-value groups.

    Answer: 3,200

    8 × 4 hundreds = 32 hundreds = 3,200.

    2 points · C05-L01

    Suggested partial-credit guide:

    • 1 point — Identifies eight groups of four hundreds or equivalent multiplication
    • 1 point — Gives 3,200
  4. 3A-E-004 · Describe two different equal-group stories for 6 × 5. Give the product.

    Answer: 30

    Six bags of five objects each or five rows of six each both make 30, using the commutative property.

    3 points · C04-L01

    Suggested partial-credit guide:

    • 1 point — Gives one valid equal-group situation for 6×5
    • 1 point — Gives a distinct correct grouping or array orientation
    • 1 point — States the product 30
  5. 3A-E-005 · Calculate 3,485 + 2,769, showing the regrouping from ones to tens.

    Answer: 6,254

    5 + 9 = 14 ones, carry 1 ten; continue adding aligned place values for 6,254.

    3 points · C03-L01

    Suggested partial-credit guide:

    • 1 point — Explains the 14 ones regroup as 1 ten and 4 ones
    • 1 point — Correctly aligns/combines tens and hundreds including further exchange
    • 1 point — Obtains 6,254
  6. 3A-E-006 · Calculate 960 ÷ 3 using place value.

    Answer: 320

    96 tens ÷ 3 = 32 tens = 320.

    2 points · C06-L01

    Suggested partial-credit guide:

    • 1 point — Explains that 960 is 96 tens or 9 hundreds and 6 tens
    • 1 point — Calculates 320
  7. 3A-E-007 · A bar graph shows club members. How many fewer members does Willow have than Cypress?

    Answer: 6 members

    Willow 12, Cypress 18, so 18 − 12 = 6.

    2 points · C07-L01

    Suggested partial-credit guide:

    • 1 point — Reads Willow as 12 and Cypress as 18 from the graph
    • 1 point — Finds the difference of 6 members
  8. 3A-E-008 · Find 905 − 398 using two methods: (1) subtract 300 and then 98; (2) add 2 to both numbers. Explain why both work.

    Answer: 507

    Method 1: 905 − 300 = 605, then 605 − 98 = 507. Method 2: 907 − 400 = 507; adding the same amount to both numbers preserves the difference.

    3 points · C02-L03, C02-L04

    Suggested partial-credit guide:

    • 1 point — Subtracts 300 and then 98, obtaining 507
    • 1 point — Uses equal adjustment to form 907−400, obtaining 507
    • 1 point — Explains the equal-adjustment method preserves the difference
  9. 3A-E-009 · In 9,404, what is the value of the first 4? How many tens equal that value?

    Answer: 400; 40 tens

    The first 4 is in the hundreds position: 400. Forty tens make 400.

    3 points · C1 PV-02, C1 PV-04

    Suggested partial-credit guide:

    • 1 point — Identifies first 4 as hundreds
    • 1 point — States the place-value amount 400
    • 1 point — States 40 tens
  10. 3A-E-010 · Find 31 × 3 by breaking apart tens and ones.

    Answer: 93

    (30 × 3) + (1 × 3) = 90 + 3 = 93.

    2 points · C05-L02

    Suggested partial-credit guide:

    • 1 point — Decomposes 31 as 30+1 and multiplies both parts
    • 1 point — Combines the parts to obtain 93
  11. 3A-E-011 · Use 8 × (10 − 3) to find 8 × 7.

    Answer: 56

    8 × 10 − 8 × 3 = 80 − 24 = 56.

    2 points · C04-L02

    Suggested partial-credit guide:

    • 1 point — Shows 8×10 − 8×3 or equivalent distributive reasoning
    • 1 point — Obtains 56
  12. 3A-E-012 · Calculate 5,206 − 1,897 and check by addition.

    Answer: 3,309

    5,206 − 1,897 = 3,309; check 3,309 + 1,897 = 5,206.

    3 points · C03-L02

    Suggested partial-credit guide:

    • 1 point — Performs valid subtraction with any required regrouping
    • 1 point — Obtains 3,309
    • 1 point — Checks by adding 1,897 to 3,309
  13. 3A-E-013 · Calculate 84 ÷ 2 and 94 ÷ 2. Explain why only the second calculation requires exchanging a ten.

    Answer: 42; 47

    For 84 ÷ 2, 8 tens split evenly into 4 tens each and 4 ones into 2 ones each: 42. For 94 ÷ 2, 9 tens give 4 tens each with 1 ten left; exchange it for 10 ones. Then 14 ones ÷ 2 = 7 ones each, giving 47.

    3 points · C06-L02, C06-L03

    Suggested partial-credit guide:

    • 1 point — Computes 84÷2=42
    • 1 point — Computes 94÷2=47
    • 1 point — Explains that 94 requires exchanging one remaining ten for ten ones while 84 does not
  14. 3A-E-014 · Using the same graph, which club has 15 members?

    Answer: Birch

    Birch bar reaches 15 on the graph.

    2 points · C07-L01

    Suggested partial-credit guide:

    • 1 point — Identifies Birch
    • 1 point — Supports the choice by reading its bar value as 15
  15. 3A-E-015 · Fill the missing addend: 1,000 = 587 + __.

    Answer: 413

    587 + 13 = 600; 600 + 400 = 1,000, so 413.

    2 points · C02-L05

    Suggested partial-credit guide:

    • 1 point — Recognizes the missing addend is 1,000−587
    • 1 point — Obtains 413
  16. 3A-E-016 · Arrange in descending order: 6,018; 6,180; 6,081; 6,108.

    Answer: 6,180; 6,108; 6,081; 6,018

    All have 6 thousands; compare hundreds, then tens, then ones.

    2 points · C1 PV-05

    Suggested partial-credit guide:

    • 1 point — Places 6,180 and 6,108 before the other two in correct order
    • 1 point — Places 6,081 and 6,018 afterward in correct order
  17. 3A-E-017 · Calculate 47 × 6 with written regrouping.

    Answer: 282

    7 × 6 = 42 ones → write 2, carry 4 tens; 4 × 6 + 4 = 28 tens → 282.

    3 points · C05-L03, C05-L04

    Suggested partial-credit guide:

    • 1 point — Shows 7×6=42 with exchange of four tens
    • 1 point — Correctly computes the tens contribution 4×6+4=28 tens
    • 1 point — Obtains 282
  18. 3A-E-018 · 54 stickers are divided equally among 9 students. How many does each receive?

    Answer: 6 stickers

    54 ÷ 9 = 6. Check 6 × 9 = 54.

    2 points · C04-L03, C04-L07

    Suggested partial-credit guide:

    • 1 point — Uses equal sharing or a valid division equation 54÷9
    • 1 point — Obtains 6 stickers per student
  19. 3A-E-019 · Explain the exchange needed to calculate 7,000 − 3,658, then find the difference.

    Answer: 3,342

    Regroup 7,000 into 6 thousands, 9 hundreds, 9 tens, 10 ones; subtract to get 3,342.

    3 points · C03-L03

    Suggested partial-credit guide:

    • 1 point — Explains exchanging a thousand/hundred/ten correctly across zeros
    • 1 point — Maintains correct place values in the subtraction
    • 1 point — Obtains 3,342
  20. 3A-E-020 · Find 75 ÷ 5 and verify by multiplication.

    Answer: 15

    75 ÷ 5 = 15; 15 × 5 = 75.

    2 points · C06-L04

    Suggested partial-credit guide:

    • 1 point — Obtains quotient 15
    • 1 point — Verifies 15×5=75
  21. 3A-E-021 · The table shows trays delivered by Teams A–D. How many trays did Teams A and C deliver altogether? Then draw a bar graph representing all four teams with an appropriate scale and labeled axes.

    Answer: 25 trays; graph bars A=11, B=17, C=14, D=17

    Team A 11 + Team C 14 = 25. Draw four bars with heights 11, 17, 14 and 17. Use equally spaced categories and a consistent vertical scale that includes zero and at least 17.

    4 points · C07-L02

    Suggested partial-credit guide:

    • 1 point — Reads 11 for Team A and 14 for Team C and adds to 25 trays
    • 1 point — Uses a sensible graph scale (including zero) with labeled axes/categories
    • 1 point — Plots A=11 and C=14 at correct heights
    • 1 point — Plots B=17 and D=17 at correct, equal heights
  22. 3A-E-022 · Calculate 499 + 304 mentally; explain the adjustment.

    Answer: 803

    499 + 304 = 500 + 304 − 1 = 803.

    2 points · C02-L06

    Suggested partial-credit guide:

    • 1 point — Replaces 499 with 500 and explains subtracting the extra 1
    • 1 point — Obtains 803
  23. 3A-E-023 · A number line has equally spaced labels 3,600; 3,750; 3,900; __. What number is next, and what is the interval?

    Answer: 4,050; interval 150

    Each step rises by 150, so 3,900 + 150 = 4,050.

    2 points · C1 PV-06, C1 PV-07

    Suggested partial-credit guide:

    • 1 point — Identifies the equal interval of 150
    • 1 point — Obtains 4,050
  24. 3A-E-024 · Calculate 86 × 7; identify both exchanges.

    Answer: 602

    6 × 7 = 42 ones → carry 4 tens. 8 × 7 + 4 = 60 tens → 6 hundreds, 0 tens; 602.

    3 points · C05-L05

    Suggested partial-credit guide:

    • 1 point — Explains 6×7=42 and exchanges 4 tens
    • 1 point — Explains 8×7 tens plus 4 gives 60 tens or 6 hundreds
    • 1 point — Obtains 602
  25. 3A-E-025 · Fill the blanks and explain: 38 × 1 = __; 38 × 0 = __; 0 ÷ 5 = __.

    Answer: 38; 0; 0

    Multiplying by 1 preserves the number; by 0 yields 0; zero objects divided into 5 groups gives 0 each.

    3 points · C04-L04

    Suggested partial-credit guide:

    • 1 point — Correctly states 38×1=38 with identity reasoning
    • 1 point — Correctly states 38×0=0 with zero-product reasoning
    • 1 point — Correctly states 0÷5=0 with grouping reasoning
  26. 3A-E-026 · Estimate 4,381 + 2,678 by rounding each addend to the nearest hundred.

    Answer: 7,100

    4,381 → 4,400 and 2,678 → 2,700; add 7,100.

    2 points · C03-L04

    Suggested partial-credit guide:

    • 1 point — Rounds 4,381 to 4,400 and 2,678 to 2,700
    • 1 point — Adds to 7,100
  27. 3A-E-027 · Calculate 526 ÷ 2. Explain the middle digit.

    Answer: 263

    5 hundreds ÷ 2 = 2 hundreds remainder 1 hundred → 12 tens ÷ 2 = 6 tens; 6 ones ÷ 2 = 3, giving 263.

    3 points · C06-L05

    Suggested partial-credit guide:

    • 1 point — Handles 5 hundreds÷2 with one hundred carried into tens
    • 1 point — Explains why the tens digit is 6
    • 1 point — Obtains 263
  28. 3A-E-028 · Teams B and D each delivered 17 trays. If the pictograph key is one symbol = 2 trays, can we draw their values using only whole symbols? Explain.

    Answer: No

    17 ÷ 2 = 8½ symbols. Whole symbols alone would represent an even total, so a half-symbol or different key is needed.

    3 points · C07-L01, C07-L02

    Suggested partial-credit guide:

    • 1 point — Explains whole symbols of two can produce only even totals
    • 1 point — Computes 17÷2=8½ symbols
    • 1 point — Suggests half-symbols or another valid key
  29. 3A-E-029 · Two children have 128 cards in total. The first has 22 more than the second. Find both amounts and label equal bars.

    Answer: 53 and 75

    Subtract the extra 22 from total: 128 − 22 = 106. Halve 106 to get 53; larger 53 + 22 = 75.

    4 points · C02-L07

    Suggested partial-credit guide:

    • 1 point — Represents equal bars and the 22-card extra correctly
    • 1 point — Subtracts 22 from the 128 total
    • 1 point — Divides 106 by 2 to obtain 53
    • 1 point — Finds larger quantity 75 and identifies which is which
  30. 3A-E-030 · Round the original number 5,486 separately to the nearest 10, the nearest 100, and the nearest 1,000. Do not round a previous answer.

    Answer: 5,490; 5,500; 5,000

    To ten: ones 6 rounds up. To hundred: tens 8 rounds up. To thousand: 5,486 is below midpoint 5,500 so down.

    3 points · C1 PV-08, C1 PV-09, C1 PV-10

    Suggested partial-credit guide:

    • 1 point — Rounds original 5,486 to nearest ten as 5,490
    • 1 point — Rounds original 5,486 to nearest hundred as 5,500
    • 1 point — Rounds original 5,486 to nearest thousand as 5,000
  31. 3A-E-031 · A gardener plants 4 rows of 206 bulbs. How many bulbs?

    Answer: 824 bulbs

    206 × 4 = 800 + 0 + 24 = 824.

    3 points · C05-L06

    Suggested partial-credit guide:

    • 1 point — Calculates four groups of 200 as 800
    • 1 point — Calculates four groups of 6 as 24
    • 1 point — Combines to 824 bulbs
  32. 3A-E-032 · 65 beads go into bags of 8. Find the number of full bags and leftover beads.

    Answer: 8 full bags; 1 bead left

    8 × 8 = 64; 65 − 64 = 1, less than divisor 8.

    2 points · C04-L05

    Suggested partial-credit guide:

    • 1 point — Finds 8 complete bags of 8
    • 1 point — Finds 1 bead remaining
  33. 3A-E-033 · Estimate 8,206 − 3,712 to the nearest thousand and then compute exactly.

    Answer: About 4,000; exact 4,494

    8,206 → 8,000; 3,712 → 4,000; estimate 4,000. Exact 8,206 − 3,712 = 4,494.

    3 points · C03-L05

    Suggested partial-credit guide:

    • 1 point — Gets estimate 8,000−4,000=4,000
    • 1 point — Computes the exact subtraction correctly
    • 1 point — Reports 4,494 and distinguishes it from the estimate
  34. 3A-E-034 · Calculate 648 ÷ 4 and check your quotient.

    Answer: 162

    6 hundreds ÷ 4 = 1 hundred, remainder 2 hundreds → 24 tens ÷ 4 = 6 tens, 8 ones ÷ 4 = 2. Check 162 × 4 = 648.

    3 points · C06-L06

    Suggested partial-credit guide:

    • 1 point — Shows a valid division procedure with exchanges
    • 1 point — Obtains 162
    • 1 point — Checks 162×4=648
  35. 3A-E-035 · A pond has 456 fish. A nearby pond has 89 fewer fish. How many fish are in the nearby pond?

    Answer: 367 fish

    456 − 89 = 367.

    2 points · C02-L09

    Suggested partial-credit guide:

    • 1 point — Recognizes the nearby pond has fewer fish, so subtracts 89
    • 1 point — Obtains 367 fish
  36. 3A-E-036 · Is it always correct to round a number to the nearest hundred first and then to the nearest thousand? Give a counterexample or justify your answer.

    Answer: No; for example 3,450

    For 3,450, rounding to the nearest hundred gives 3,500, which rounds to 4,000. But rounding 3,450 directly to the nearest thousand gives 3,000, because 3,450 is below 3,500. The two processes disagree.

    4 points · C1 PV-11

    Suggested partial-credit guide:

    • 1 point — Answers no with a valid counterexample such as 3,450
    • 1 point — Correctly shows 3,450→3,500 to the nearest hundred
    • 1 point — Correctly shows 3,500→4,000 to the nearest thousand
    • 1 point — Shows direct 3,450→3,000 and explains disagreement
  37. 3A-E-037 · An auditorium has 7 rows of 138 seats. How many seats is that, and is 1,000 a reasonable estimate?

    Answer: 966 seats; yes, 1,000 is a reasonable estimate

    138 × 7 = 966. Rounding 138 to the nearest ten gives 140, and 140 × 7 = 980, which is close to 1,000.

    4 points · C05-L07

    Suggested partial-credit guide:

    • 1 point — Computes 138×7=966
    • 1 point — Reports correct seat unit or total
    • 1 point — Gives a valid estimate near 980 or 1,000
    • 1 point — Explains why 1,000 is reasonable
  38. 3A-E-038 · Is the product of an odd number and an even number always even? Explain.

    Answer: Yes

    An even number has a factor of 2; multiplying by any whole number preserves that factor.

    3 points · C04-L06

    Suggested partial-credit guide:

    • 1 point — States the product is always even
    • 1 point — Explains the even factor has a factor of 2
    • 1 point — Shows multiplying by an integer preserves that factor
  39. 3A-E-039 · A fundraiser collected $3,275 on Friday and $2,468 on Saturday, then spent $1,895 on supplies. First estimate by rounding each amount to the nearest hundred; then calculate the exact amount remaining.

    Answer: Estimate $3,900; exact $3,848

    Estimate: $3,300 + $2,500 − $1,900 = $3,900. Exact: $3,275 + $2,468 = $5,743; $5,743 − $1,895 = $3,848.

    3 points · C03-L06

    Suggested partial-credit guide:

    • 1 point — Estimates 3,300+2,500−1,900=3,900 dollars
    • 1 point — Correctly combines and subtracts the exact dollar amounts
    • 1 point — Obtains exactly $3,848
  40. 3A-E-040 · Calculate 324 ÷ 4. Explain why the quotient is less than 100.

    Answer: 81

    3 hundreds cannot be shared into 4 groups of 1 hundred each; exchange into tens. 32 tens ÷ 4 = 8 tens, 4 ones ÷ 4 = 1, giving 81.

    3 points · C06-L07

    Suggested partial-credit guide:

    • 1 point — Explains 3 hundreds cannot provide a full hundred for each of 4 groups
    • 1 point — Repartitions to 32 tens and then ones
    • 1 point — Obtains 81 and explains it is below 100
  41. 3A-E-041 · A hall seats 680. There are 249 tickets sold in the morning and 175 more in the afternoon. How many seats remain unsold?

    Answer: 256 seats

    249 + 175 = 424 tickets sold; 680 − 424 = 256.

    3 points · C02-L08, C02-L10

    Suggested partial-credit guide:

    • 1 point — Computes tickets sold as 249+175=424
    • 1 point — Subtracts from the total capacity 680
    • 1 point — Obtains 256 seats
  42. 3A-E-042 · A school orders 3 cartons of 128 pencils each and splits them equally among 4 classrooms. How many pencils per room?

    Answer: 96 pencils

    3 × 128 = 384; 384 ÷ 4 = 96.

    4 points · C05-L07, C06-L06

    Suggested partial-credit guide:

    • 1 point — Calculates 3×128=384 pencils
    • 1 point — Divides 384 into four equal groups
    • 1 point — Obtains 96 pencils per room
    • 1 point — Gives a reasonable unit interpretation or verification
  43. 3A-E-043 · Sana owns 5 times as many stamps as Wei. Wei owns 7. Draw or describe bars and calculate Sana’s stamps.

    Answer: 35 stamps

    One bar of 7 for Wei, five equal bars for Sana: 5 × 7 = 35.

    3 points · C04-L08

    Suggested partial-credit guide:

    • 1 point — Represents Wei by one 7-unit bar
    • 1 point — Represents Sana by five equal such bars
    • 1 point — Computes 35 stamps
  44. 3A-E-044 · A shop sells 6 bundles of 149 tickets. Estimate by rounding 149 to the nearest ten, then find the exact number.

    Answer: Estimate 900 tickets; exact 894

    149 rounds to 150 at nearest ten, giving 6 × 150 = 900 as an estimate. The exact total is 149 × 6 = 894, which is close to 900.

    3 points · C05-L07

    Suggested partial-credit guide:

    • 1 point — Rounds 149 to 150
    • 1 point — Uses 6×150=900 for an estimate
    • 1 point — Finds exact 6×149=894
  45. 3A-E-045 · Lina and Omar have 90 buttons together. Lina has 14 more than Omar. Lina gives Omar 7 buttons. How many does each have after the transfer?

    Answer: 45 each

    Initially Omar = (90 − 14) ÷ 2 = 38, Lina = 52. After Lina gives 7, both have 45.

    4 points · C02-L07, C02-L10

    Suggested partial-credit guide:

    • 1 point — Finds original amounts 38 and 52 (or equivalent justified work)
    • 1 point — Applies the transfer of 7 in opposite directions
    • 1 point — Obtains 45 for each afterward
    • 1 point — Verifies the total remains 90 or both amounts equal
  46. 3A-E-046 · A group prepares 7 boxes of 9 muffins. They sell 25 muffins. How many remain, and how many complete groups of 5 can be made?

    Answer: 38 remain; 7 full groups of 5 (3 left)

    7 × 9 = 63; 63 − 25 = 38; 38 ÷ 5 = 7 remainder 3.

    4 points · C04-L09

    Suggested partial-credit guide:

    • 1 point — Computes original count 7×9=63
    • 1 point — Subtracts 25 to obtain 38 remaining
    • 1 point — Divides into 5s for seven full groups
    • 1 point — Gives 3 leftover and interprets full groups correctly