69 review solutions · 60 exam solutions · 181 examination points
Suggested partial credit is one point for each listed criterion. Equivalent correct mathematical methods receive credit. Human judgment is needed for diagrams and open explanations.
Question: Write 73,408 in words and expanded form. Explain what the 7 means.
Reasoning: Read the value of each digit, not just the digit itself. The 7 is in the ten-thousands place.
Question: Explain how ten groups of 100,000 compare with one million. Write 1,000,000 − 100,000.
Reasoning: Each place is ten times its neighbor on the right. Removing 100,000 from 1,000,000 leaves nine hundred-thousands.
Question: Continue 26,750; 26,500; 26,250; __; __. State the rule.
Reasoning: Consecutive differences are −250, so the next two values follow the same rule.
Question: Order 405,090; 450,009; 405,900; 449,999 from least to greatest. Explain the first differing place, then show their approximate positions on a number line from 400,000 to 500,000.
Reasoning: First compare hundred-thousands, then ten-thousands and lower places. On an evenly scaled line from 400,000 to 500,000, the first two points cluster around 405,000 while the other two straddle 450,000.
Question: Round 67,450 to the nearest hundred and to the nearest thousand. Why are the answers different?
Reasoning: The hundreds midpoint is 67,450, so standard school rounding goes to 67,500. The thousands midpoint is 67,500; 67,450 lies below it.
Question: Round 685,500 to the nearest ten thousand and nearest hundred thousand (separately).
Reasoning: For ten-thousands, compare with 685,000 midpoint. For hundred-thousands, 685,500 is above the 650,000 midpoint.
Question: Find 36 × 1,000 and 48,000 ÷ 100. Explain each place-value change.
Reasoning: Multiplying by 1,000 scales each place value by three positions; dividing by 100 scales by two positions in the other direction.
Question: Add 487,659 + 75,884. Show one regrouping and check by estimation.
Reasoning: Add aligned place values; the ones make 13, so exchange ten ones for one ten. 487,659 + 75,884 = 563,543; estimate about 490,000 + 80,000 = 570,000.
Question: Subtract 402,006 − 178,589. Explain how you regroup across zeros.
Reasoning: Rename one thousand as hundreds, tens and ones as necessary across the zero columns; verify 223,417 + 178,589 = 402,006.
Question: Use compensation to find 39,998 + 7,456 without long column addition.
Reasoning: Raise 39,998 by 2 to 40,000 and lower 7,456 by 2 to 7,454; the total remains unchanged.
Question: Find 50,002 − 29,998 by adding the same number to both quantities.
Reasoning: Add 2 to both quantities: 50,004 − 30,000 = 20,004. Equal adjustment preserves a difference.
Question: A library had 7,650 books. It received 875 and donated 1,240. How many books remain? Give a bar-model or equation explanation.
Reasoning: The total first increases then decreases: 7,650 + 875 − 1,240 = 7,285. A labeled before/after bar tracks the initial amount and both changes.
Question: List the first five positive multiples of 14 and explain why 0 is also a multiple of 14.
Reasoning: A multiple is 14 times a whole number; the first five positive multipliers are 1 through 5.
Question: Find the least positive common multiple of 8 and 12. Show two lists.
Reasoning: Multiples of 8 include 8, 16, 24; multiples of 12 include 12, 24. The first shared positive multiple is 24.
Question: List all positive factors of 36 by writing factor pairs.
Reasoning: Factor pairs: 1×36, 2×18, 3×12, 4×9, and 6×6; combine without double-counting 6.
Question: Classify 0, 1, 2, 21, and 29 as prime, composite, or neither. Explain each exception.
Reasoning: A prime has exactly two distinct positive factors, a composite has more than two. Zero has infinitely many positive divisors but is neither prime nor composite; one has only one positive factor; 21=3×7.
Question: Find the greatest common factor of 24 and 40 and explain how it relates to equal-size grouping.
Reasoning: Shared factors are 1, 2, 4 and 8, so the greatest is 8; both collections can be split into eight equal groups.
Question: Compute 25 × 36 mentally in two ways: using the distributive property and using doubling/halving. Explain why the two methods agree.
Reasoning: Distributivity separates products without changing their sum; doubling one factor while halving the other preserves the product.
Question: Use partial products to calculate 212 × 4 without regrouping.
Reasoning: (200×4)+(10×4)+(2×4)=800+40+8=848; each partial product represents its correct place value.
Question: Compute 487 × 6 with regrouping and check the approximate size.
Reasoning: 6×7 = 42; record 2 ones and exchange 4 tens; continue by place value. Estimate 500×6≈3,000.
Question: Calculate 47 × 30 by interpreting 30 as 3 tens.
Reasoning: 47×3×10 = 141×10 = 1,410.
Question: Use four partial products to calculate 34 × 26.
Reasoning: (30+4)(20+6) = 600+180+80+24 = 884.
Question: Calculate 68 × 47 using a four-part area model and then a written partial-product algorithm. Explain how the two match; check with an estimate.
Reasoning: Both methods add the same four underlying partial products, giving 3,196. An estimate of 70×45≈3,150 confirms the magnitude.
Question: Divide 936 ÷ 6 using convenient partial quotients. Then calculate (936 × 2) ÷ (6 × 2) and explain why both quotients are equal.
Reasoning: 936=600+300+36 gives 100+50+6=156. Multiplying both dividend and divisor by 2 gives 1,872÷12=156 because the quotient is unchanged.
Question: Estimate 2,848 ÷ 7 using compatible numbers. Then calculate the exact quotient.
Reasoning: 7×400=2,800; 2,848−2,800=48; 48=7×6+6. Exact quotient 406 remainder 6.
Question: Calculate 4,368 ÷ 7 and check by multiplication.
Reasoning: 7×624=4,368. In the written algorithm, exchange between thousands, hundreds, tens and ones.
Question: There are 1,258 crayons packed in boxes holding 9 each. How many full boxes can be filled and how many crayons remain?
Reasoning: 9×139=1,251, and 1,258−1,251=7. The remainder is less than 9.
Question: A school has 1,245 tickets to share equally among 6 classes. Each class receives the same whole number; what remains? Then the school receives 23 additional tickets. How many tickets are not yet shared?
Reasoning: 1,245÷6 = 207 remainder 3. The 23 donated tickets add to the remainder: 3+23 = 26. State both stages clearly.
Question: Show why 3/4 and 9/12 represent equal parts of the same whole.
Reasoning: Dividing each fourth into three equal smaller parts creates 12 equal parts; 3 fourths become 9 twelfths.
Question: Order 2/3, 5/6, and 3/4 from smallest to largest. Mark their approximate positions on a number line from 0 to 1.
Reasoning: Rename using twelfths, divide the same whole line into twelve equal steps, and mark the points at steps 8, 9, and 10.
Question: Express 13/5 as a mixed number and represent its full wholes and leftover fifths.
Reasoning: Ten fifths make two wholes, and three fifths remain; 13=2×5+3.
Question: Convert 29/8 to simplest mixed-number form.
Reasoning: 29÷8 gives quotient 3 and remainder 5; five eighths is already in simplest form.
Question: Convert 4 3/7 to an improper fraction and explain the numerator.
Reasoning: Four wholes contain 28 sevenths; adding 3 sevenths gives 31 sevenths.
Question: Three pizzas are shared equally by four students. What fraction of a pizza does each receive?
Reasoning: Each pizza is divided into four equal parts; each student gets three quarters in all. This is 3÷4.
Question: Compute 7/10 + 8/10. Simplify the result and explain the unit being counted.
Reasoning: Count tenths: 7+8=15 tenths = 15/10 = 3/2 = 1 1/2.
Question: Calculate 2/3 + 1/6 using sixths.
Reasoning: Rename 2/3 as 4/6, then add 1/6 to obtain 5/6.
Question: Compute 2 3/4 + 1/2. Rename the fractional sum if necessary.
Reasoning: 1/2=2/4; 3/4+2/4=5/4=1 1/4; with 2 wholes this is 3 1/4.
Question: Compute 1 5/6 + 2 3/4; show a common denominator.
Reasoning: 5/6=10/12 and 3/4=9/12; fractional sum 19/12=1 7/12, plus three wholes gives 4 7/12.
Question: Compute 5 1/4 − 3/4 by regrouping one whole.
Reasoning: Rename 5 1/4 as 4 5/4. Then subtract 3/4 to get 4 2/4=4 1/2.
Question: Compute 6 1/3 − 2 5/6 with a clearly shown exchange.
Reasoning: 6 1/3=6 2/6=5 8/6; subtract 2 5/6 to obtain 3 3/6=3 1/2.
Question: Find 1/7 of 63 counters and explain why division represents one equal share.
Reasoning: Divide the 63 counters into seven equal groups: 63÷7=9.
Question: Calculate 4 × 3/8 as repeated addition and simplify.
Reasoning: 3/8+3/8+3/8+3/8=12/8=3/2=1 1/2.
Question: Compute 6 × 5/9. Explain how to simplify the improper result.
Reasoning: 6×5/9=30/9=10/3=3 1/3.
Question: A jar has 40 beads. Three-fifths are green. How many beads are green?
Reasoning: Split 40 into five groups of 8; choose three groups: 3×8=24.
Question: Find 24 × 3/4 as three-quarters of a whole-number quantity.
Reasoning: First 24÷4=6, then multiply 6×3=18.
Question: Compute 36 × 5/6 two ways and show why the results agree.
Reasoning: Partition first: 36÷6=6 then 6×5=30. Multiply first: 36×5÷6=180÷6=30.
Question: A farm harvests 84 apples and sells 5/7 of them. How many apples are sold? Show a unit bar.
Reasoning: Divide 84 into seven equal units of 12, then select five units: 12×5=60.
Question: A baker has 120 rolls. She gives 1/4 away, then sells 2/3 of what remains. How many rolls are left?
Reasoning: After giving away 30, 90 remain. Sell 2/3 of 90=60, so 30 remain. The second fraction is of 90, not 120.
Question: The graph shows library visitors on four days. How many more visited on Thursday than Monday?
Reasoning: Read Monday as 24 and Thursday as 42; 42−24=18.
Question: Draw a line graph of plant height: Week 1 = 8 cm, Week 2 = 12 cm, Week 3 = 11 cm, Week 4 = 15 cm. Label both axes and choose a sensible scale.
Reasoning: A scale of 0, 5, 10, 15, 20 cm or steps of 2 cm is sensible. The line rises, dips, then rises again.
Question: Draw a line plot for seven reading times (minutes): 10, 15, 15, 20, 20, 20, 25. Label the units, then tell how many students read at least 20 minutes.
Reasoning: Place each of the seven readings as an X above its numerical position. There are three Xs at 20 and one X at 25, giving four readings of at least 20 minutes.
Question: A stadium records 695,480 visitors. Round independently to the nearest 1,000 and 100,000, then explain why the two estimates differ.
Reasoning: Nearest thousand midpoint 695,500; 695,480 is below it. Nearest hundred-thousand midpoint 650,000; 695,480 is above it.
Question: A child says “Adding one thousand always changes only the thousands digit.” Explain why this is false using 99,500.
Reasoning: Regrouping across 99 thousands to 100 thousands changes the hundred-thousands and ten-thousands digits.
Question: A store sold 12,750 items in June, 9,865 in July, and 14,390 in August. How many more did it sell in August than July, and what was its three-month total?
Reasoning: August−July = 14,390−9,865=4,525. Total = 12,750+9,865+14,390=37,005.
Question: A student claims 70,000−39,999 = 30,001 because it is the same as 70,000−40,000+1. Is the claim valid?
Reasoning: 70,000−40,000=30,000; subtracting one fewer means adding 1, giving 30,001.
Question: Find the least positive common multiple of 6 and 10, then the greatest common factor of 6 and 10. Explain why they differ.
Reasoning: Common multiples are shared products; common factors are shared exact divisors.
Question: A student says 1 is prime because it cannot be divided equally by any number except 1. What has been overlooked?
Reasoning: Prime numbers must have exactly two distinct positive factors, 1 and the number itself; 1 has just one.
Question: A rectangular tile design has 24 rows of 35 tiles. Find the total using a split model.
Reasoning: 24×(30+5)=720+120=840.
Question: Explain why 46×20 is ten times 46×2, without saying “add a zero.”
Reasoning: Associativity and base-ten scaling justify multiplying by ten.
Question: A school has 3,205 pencils to distribute in bundles of 8. How many full bundles and what remainder?
Reasoning: 3,205=8×400+5. The remainder is below eight.
Question: Explain why 4,080 ÷ 8 cannot be 51 using an inverse check.
Reasoning: 8×51=408, but 8×510=4,080; the place-value zero matters.
Question: A ribbon is 3 1/4 meters long. Write the length as an improper fraction and in fourths.
Reasoning: Three meters is twelve fourths; adding one fourth gives thirteen fourths.
Question: A student says 3/5 > 3/4 because 5 > 4. Explain the error using equal-sized wholes.
Reasoning: If the whole is divided into five parts, each part is smaller than a fourth; three fifths is less than three fourths.
Question: A recipe uses 1 3/4 cups of oats and 2 5/8 cups of flour. How much is that altogether?
Reasoning: 1 6/8 + 2 5/8 = 3 11/8 = 4 3/8.
Question: A student adds 1/2 and 1/3 and writes 2/5. Explain what has gone wrong and calculate the correct sum.
Reasoning: The pieces are different sizes; rename both in sixths: 3/6+2/6=5/6.
Question: A class has 120 pencils. Two-fifths are blue. Of the rest, one-third are green. How many are neither blue nor green?
Reasoning: Blue = 120×2/5=48; remaining 72. Green = 72÷3=24. Neither=72−24=48.
Question: A child claims that 3/4 of 32 is 32×4÷3. Explain and correct the error.
Reasoning: Divide the whole into four equal parts, then take three: 32÷4×3=24.
Question: Read the line graph showing results over four equal days. What is the overall change from the first to last day? Does it increase every day?
Reasoning: The endpoint change is 35−20=15, but the values decline from 30 on the second day to 25 on the third day.
Question: A student draws a line graph with equal time intervals spaced unevenly on the horizontal axis. What should be changed?
Reasoning: Distances on the axes communicate scale; unequal spacing falsely changes the represented rate of change.
Question: Write 56,091 in words and expanded form. State the value of the 6. Show your work and briefly explain or check your reasoning.
Reasoning: Decompose into 5 ten-thousands, 6 thousands, 0 hundreds, 9 tens, 1 one; the zero acts as a placeholder.
Question: Calculate 596,785 + 48,679, showing two place-value exchanges and a reasonable estimate. Show your work and briefly explain or check your reasoning.
Reasoning: Add by place value, carrying when a column totals ten or more: 596,785 + 48,679 = 645,464. Estimate about 600,000 + 50,000 = 650,000.
Question: Write four positive multiples of 18 greater than 50, and decide whether 0 is a multiple of 18. Show your work and briefly explain or check your reasoning.
Reasoning: Since 18×3 = 54, continue by adding 18. Zero is the product with multiplier zero.
Question: Calculate 18 × 45 mentally using a distributive decomposition and a second doubling/halving method. Explain why both methods preserve the product.
Reasoning: A difference of products represents the same 18 groups of 45. Halving an even factor and doubling the other factor gives the same product.
Question: Compute 864 ÷ 8 using friendly multiples. Verify that (864 × 2) ÷ (8 × 2) has the same quotient, and explain why.
Reasoning: 864=800+64 gives 100+8=108. The scaled quotient 1,728÷16=108; multiplying both dividend and divisor by the same nonzero factor preserves a quotient.
Question: Write two fractions equivalent to 5/8, and justify one with a partition model. Show your work and briefly explain or check your reasoning.
Reasoning: Multiply numerator and denominator by 2 or by 3; the quantity is unchanged because each original part is subdivided equally.
Question: Compute 11/12 − 5/12 and express the result in simplest form. Show your work and briefly explain or check your reasoning.
Reasoning: Both fractions use twelfths: 11−5=6 twelfths, or 1/2.
Question: Find 1/8 of 96 stickers. Describe the eight equal groups. Show your work and briefly explain or check your reasoning.
Reasoning: Each of eight equal groups has 96÷8=12 stickers.
Question: Read the line graph. What is the total increase in water collected from Tuesday to Friday? Show your work and briefly explain or check your reasoning.
Reasoning: Tuesday is 15 L and Friday is 27 L; difference 27−15=12 L.
Question: Write 806,050 in expanded form. How many hundred-thousands are represented in 1,000,000? Show your work and briefly explain or check your reasoning.
Reasoning: Zeros occupy the ten-thousands, hundreds and ones places in 806,050. One million can be exchanged for ten hundred-thousands.
Question: Calculate 600,004 − 258,677. Check using addition. Show your work and briefly explain or check your reasoning.
Reasoning: Exchange from the hundred-thousands place through intermediate zero places as needed. Verify 341,327 + 258,677 = 600,004.
Question: Find the least positive common multiple of 9 and 15. Explain your method. Show your work and briefly explain or check your reasoning.
Reasoning: Multiples of 9: 9, 18, 27, 36, 45. Multiples of 15: 15, 30, 45. First common positive multiple = 45.
Question: Write the place-value partial products for 312 × 3 and combine them. Show your work and briefly explain or check your reasoning.
Reasoning: 300×3 = 900; 10×3 = 30; 2×3 = 6. Add to get 936.
Question: A student estimates 3,237 ÷ 8 as about 40. Explain why that is unreasonable, then give a better estimate and calculate the quotient with remainder. Show your work and briefly explain or check your reasoning.
Reasoning: 40×8 would be only 320, far from 3,237. 8×400=3,200; remaining 37 = 8×4+5, so 404 remainder 5.
Question: A student says 5/8 is greatest because 8 is the largest denominator. Explain the error, order 5/8, 3/4, and 2/3 from greatest to least, and locate them on a number line from 0 to 1.
Reasoning: For the same whole, more denominator parts means smaller individual pieces. Use twenty-fourths: 18/24 > 16/24 > 15/24; mark the three positions on one equally scaled number line.
Question: A student calculates 5/8 − 1/4 = 4/4 by subtracting numerators and denominators. Explain the error and calculate correctly. Show your work and briefly explain or check your reasoning.
Reasoning: Denominators identify equal-size pieces and cannot be subtracted directly. First 1/4=2/8, then 5/8−2/8=3/8.
Question: Calculate 5 × 2/7. Explain its repeated-addition meaning. Show your work and briefly explain or check your reasoning.
Reasoning: Five copies of two sevenths make 10/7 = 1 3/7.
Question: Draw a line graph for temperatures at 9 a.m., 10 a.m., 11 a.m., noon: 16°C, 18°C, 22°C, 20°C. Give a title, labeled axes and a consistent vertical scale.
Reasoning: A vertical scale 0–24 °C in increments of 4 °C is valid. Make successive horizontal intervals equal; do not put completed points in the question grid.
Question: Continue 84,600; 85,200; 85,800; __; __. Explain the change at each step. Show your work and briefly explain or check your reasoning.
Reasoning: Calculate 85,200 − 84,600 = 600 and 85,800 − 85,200 = 600, then extend by two equal increases.
Question: Find 68,997 + 5,608 using an adjusted addend; explain why your adjustment works. Show your work and briefly explain or check your reasoning.
Reasoning: Move 3 from 5,608 to 68,997: 69,000 + 5,605 = 74,605, preserving the sum.
Question: Write all positive factors of 42 and organize them into factor pairs. Show your work and briefly explain or check your reasoning.
Reasoning: Pairs are 1×42, 2×21, 3×14 and 6×7.
Question: Compute 698 × 7 and explain why the result is near 4,900. Show your work and briefly explain or check your reasoning.
Reasoning: 700×7 = 4,900, minus 2×7=14 yields 4,886; same as regrouped algorithm.
Question: Compute 5,742 ÷ 6 and verify using multiplication. Show your work and briefly explain or check your reasoning.
Reasoning: 6×957=5,742; use successive hundreds, tens and ones groups in the algorithm.
Question: Convert 17/6 to a mixed number and explain what the remainder tells you. Show your work and briefly explain or check your reasoning.
Reasoning: 17=2×6+5; the quotient is the number of wholes and the remainder counts remaining sixths.
Question: Calculate 4 5/6 + 1/3 in simplest form. Show your work and briefly explain or check your reasoning.
Reasoning: 1/3=2/6; 5/6+2/6=7/6=1 1/6; plus four wholes is 5 1/6.
Question: Calculate 8 × 3/10 and simplify the answer. Show your work and briefly explain or check your reasoning.
Reasoning: 8×3/10=24/10=12/5=2 2/5.
Question: Construct a labeled line plot for pages read by seven students: 4, 6, 6, 6, 8, 8, 10. State the total number of observations and the most common value.
Reasoning: Draw equally spaced tick marks 4, 6, 8, 10 on a number line labeled Pages Read, stacking X marks 1, 3, 2, 1. The mode is 6 pages with three Xs.
Question: Order 702,400; 720,040; 702,040; 719,999 from greatest to least. Show their approximate locations on an evenly scaled number line from 700,000 to 730,000, explaining the first differing place.
Reasoning: Compare the ten-thousands and hundreds digits to order. The 30,000-unit interval must have equal spacing; place 702,040 then 702,400 close to 700,000, and 719,999 then 720,040 close to 720,000.
Question: Use counting up or equal adjustment to calculate 73,004 − 39,997. Show your work and briefly explain or check your reasoning.
Reasoning: Add 3 to both: 73,007 − 40,000 = 33,007. This is equivalent to counting up from 39,997 to 73,004.
Question: Is 51 prime? Decide, showing a nontrivial factor pair. Classify 0 and 1 as prime, composite, or neither, explaining why.
Reasoning: 51 has additional positive factors. Zero is excluded from prime/composite classifications; 1 has one positive factor instead of the required two.
Question: Calculate 68 × 40. Explain the factor of ten. Show your work and briefly explain or check your reasoning.
Reasoning: 68×4=272, then multiplying by 10 gives 2,720.
Question: A museum has 1,174 brochures and places 8 in each packet. How many complete packets can it make, and how many brochures remain? Show your work and briefly explain or check your reasoning.
Reasoning: 8×146=1,168, leaving 6. A partial packet does not count as a complete packet.
Question: Convert 38/12 to a mixed number in simplest form. Show your work and briefly explain or check your reasoning.
Reasoning: 38÷12=3 remainder 2. Rename 2/12 as 1/6.
Question: Calculate 3 2/5 + 1 7/10 with all intermediate conversions. Show your work and briefly explain or check your reasoning.
Reasoning: 2/5=4/10; 4/10+7/10=11/10=1 1/10; add whole parts 3+1+1=5.
Question: There are 72 books. Five-eighths are novels. How many are novels? Show your work and briefly explain or check your reasoning.
Reasoning: 72÷8=9 books per group, and five groups give 5×9=45.
Question: Read the line plot. Find the range of the observations and the number of observations.
Reasoning: The smallest plotted value is 2, the largest is 8, and seven X marks appear. Range=8−2=6.
Question: Round 43,650 independently to the nearest ten and the nearest thousand (start with the original number each time). Show your work and briefly explain or check your reasoning.
Reasoning: Already a multiple of ten. For the nearest thousand, 43,650 is past the 43,500 midpoint, so it rounds to 44,000.
Question: A charity raised $18,450, spent $7,895 on supplies and received another $2,650. How much money does it have now? Show your work and briefly explain or check your reasoning.
Reasoning: Start with 18,450; after spending 7,895 there are 10,555; adding 2,650 gives 13,205. Track the order of events.
Question: Find the greatest common factor of 30 and 45. Use it to explain how many identical packs can be made from 30 pencils and 45 erasers without leftovers. Show your work and briefly explain or check your reasoning.
Reasoning: The greatest common factor is 15; each of 15 equal packs contains 30÷15=2 pencils and 45÷15=3 erasers.
Question: Use a 2×2 area model to calculate 47 × 32. Label the four products. Show your work and briefly explain or check your reasoning.
Reasoning: 40×30=1,200; 7×30=210; 40×2=80; 7×2=14. Sum = 1,504.
Question: A charity has 2,350 notebooks. It packs 7 in each complete bag and sets aside leftovers. Afterward it receives 15 more notebooks. How many complete bags were made originally and how many notebooks are outside the bags afterward? Show your work and briefly explain or check your reasoning.
Reasoning: 7×335=2,345, leaving 5. Adding 15 makes 20 outside the 335 original bags.
Question: Convert 5 2/9 to an improper fraction and justify the result. Show your work and briefly explain or check your reasoning.
Reasoning: Five wholes give 45 ninths; add two ninths for 47 ninths.
Question: Calculate 7 1/6 − 5/6 in simplest form. Show your work and briefly explain or check your reasoning.
Reasoning: Rename 7 1/6 as 6 7/6; subtract 5/6 to get 6 2/6=6 1/3.
Question: Find 45 × 2/5 by partitioning first and scaling second. Show your work and briefly explain or check your reasoning.
Reasoning: 45÷5=9 per fifth, then 2×9=18.
Question: Round 749,950 independently to the nearest thousand and nearest hundred thousand. Show your work and briefly explain or check your reasoning.
Reasoning: The nearest-thousand midpoint is 749,500; round up. The nearest-hundred-thousand midpoint is 750,000; 749,950 is just below it.
Question: A warehouse held 40,005 cartons, shipped 18,999, then received 7,495 more. How many are there now? Show your work and briefly explain or check your reasoning.
Reasoning: 40,005−18,999=21,006; then add 7,495 = 28,501.
Question: What is the smallest positive number divisible by both 12 and 18? Explain using multiples or factors. Show your work and briefly explain or check your reasoning.
Reasoning: Multiples of 12: 12,24,36; multiples of 18:18,36; first common positive multiple is 36.
Question: A student writes 79 × 56 = 474 + 395, treating the 5 as five ones. Identify the mistake, show the correct tens partial product, and connect a four-part area model with a written multiplication calculation.
Reasoning: In 56, the 5 is in the tens place, so it represents 50. Both the area decomposition and written partial products sum to 4,424.
Question: There are 2,510 stickers packed in packets of 6. How many complete packets can be made, and how many stickers remain? Show your work and briefly explain or check your reasoning.
Reasoning: 6×418=2,508, leaving 2; check remainder below 6.
Question: Five identical sandwiches are shared equally among three children. Express each share as an improper fraction and a mixed number. Show your work and briefly explain or check your reasoning.
Reasoning: Each sandwich is split in thirds; five thirds per child equals one whole and two thirds.
Question: Calculate 8 1/4 − 3 7/8 and explain why a whole must be renamed. Show your work and briefly explain or check your reasoning.
Reasoning: 8 1/4=8 2/8=7 10/8; subtract 3 7/8: whole difference 4 and fraction difference 3/8.
Question: Compute 48 × 7/8 using two valid orders of operations. Show your work and briefly explain or check your reasoning.
Reasoning: 48÷8×7=6×7=42; equivalently 48×7÷8=336÷8=42.
Question: Calculate 72,000 ÷ 1,000 and 45 × 10,000. Explain why neither operation changes the order of the nonzero digits. Show your work and briefly explain or check your reasoning.
Reasoning: Use place-value groups: 72 thousands divided into single units is 72; 45 ten-thousands is 450,000.
Question: A garden has 37 rows of 28 seedlings. Find the total using two different decompositions. Show your work and briefly explain or check your reasoning.
Reasoning: 37×(20+8)=740+296=1,036; (40−3)×28=1,120−84=1,036.
Question: A student says 13/4 represents exactly 3 wholes. Explain and give the mixed number. Show your work and briefly explain or check your reasoning.
Reasoning: Twelve quarters make three wholes; one quarter remains.
Question: Calculate 4 1/6 + 2 3/4 and simplify. Show your work and briefly explain or check your reasoning.
Reasoning: 1/6=2/12, 3/4=9/12, so fractional sum=11/12 and wholes total 6.
Question: A club has 96 tickets. Three-eighths are sold in the morning. How many tickets are sold in the morning? Show your work and briefly explain or check your reasoning.
Reasoning: 96÷8=12 per unit; 3×12=36 sold.
Question: A town counted 598,499 people. Round independently to the nearest thousand and nearest ten thousand. Explain both. Show your work and briefly explain or check your reasoning.
Reasoning: At a thousand, 598,499 is below the 598,500 midpoint. At ten thousand, it exceeds 595,000.
Question: A gardener picks 144 flowers, gives 1/3 to neighbors, and sells 3/4 of the remainder. How many flowers are left? Explain why selling 3/4 of the original 144 would be a mistake. Show your work and briefly explain or check your reasoning.
Reasoning: Give away 48; remainder is 96. Sell 3/4 of 96=72; 96−72=24. The second fraction refers to 96 remaining flowers, not the original 144.
Question: A school orders 180 markers. Three-fifths are given to students. Of what remains, one-half is stored. How many are neither given away nor stored? Show your work and briefly explain or check your reasoning.
Reasoning: Given =108; remain=72; stored=36; neither=36.