69 original questions · all 9 chapters · worked solutions revealed on request
Use this as a learning review, not a timed placement examination. Attempt a question first and then open its explanation. Drawing and explanation tasks may have multiple acceptable responses.
Write 73,408 in words and expanded form. Explain what the 7 means.
Read the value of each digit, not just the digit itself. The 7 is in the ten-thousands place.
Explain how ten groups of 100,000 compare with one million. Write 1,000,000 − 100,000.
Each place is ten times its neighbor on the right. Removing 100,000 from 1,000,000 leaves nine hundred-thousands.
Continue 26,750; 26,500; 26,250; __; __. State the rule.
Consecutive differences are −250, so the next two values follow the same rule.
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.
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.
Round 67,450 to the nearest hundred and to the nearest thousand. Why are the answers different?
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.
Round 685,500 to the nearest ten thousand and nearest hundred thousand (separately).
For ten-thousands, compare with 685,000 midpoint. For hundred-thousands, 685,500 is above the 650,000 midpoint.
Find 36 × 1,000 and 48,000 ÷ 100. Explain each place-value change.
Multiplying by 1,000 scales each place value by three positions; dividing by 100 scales by two positions in the other direction.
A stadium records 695,480 visitors. Round independently to the nearest 1,000 and 100,000, then explain why the two estimates differ.
Nearest thousand midpoint 695,500; 695,480 is below it. Nearest hundred-thousand midpoint 650,000; 695,480 is above it.
A child says “Adding one thousand always changes only the thousands digit.” Explain why this is false using 99,500.
Regrouping across 99 thousands to 100 thousands changes the hundred-thousands and ten-thousands digits.
Add 487,659 + 75,884. Show one regrouping and check by estimation.
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.
Subtract 402,006 − 178,589. Explain how you regroup across zeros.
Rename one thousand as hundreds, tens and ones as necessary across the zero columns; verify 223,417 + 178,589 = 402,006.
Use compensation to find 39,998 + 7,456 without long column addition.
Raise 39,998 by 2 to 40,000 and lower 7,456 by 2 to 7,454; the total remains unchanged.
Find 50,002 − 29,998 by adding the same number to both quantities.
Add 2 to both quantities: 50,004 − 30,000 = 20,004. Equal adjustment preserves a difference.
A library had 7,650 books. It received 875 and donated 1,240. How many books remain? Give a bar-model or equation explanation.
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.
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?
August−July = 14,390−9,865=4,525. Total = 12,750+9,865+14,390=37,005.
A student claims 70,000−39,999 = 30,001 because it is the same as 70,000−40,000+1. Is the claim valid?
70,000−40,000=30,000; subtracting one fewer means adding 1, giving 30,001.
List the first five positive multiples of 14 and explain why 0 is also a multiple of 14.
A multiple is 14 times a whole number; the first five positive multipliers are 1 through 5.
Find the least positive common multiple of 8 and 12. Show two lists.
Multiples of 8 include 8, 16, 24; multiples of 12 include 12, 24. The first shared positive multiple is 24.
List all positive factors of 36 by writing factor pairs.
Factor pairs: 1×36, 2×18, 3×12, 4×9, and 6×6; combine without double-counting 6.
Classify 0, 1, 2, 21, and 29 as prime, composite, or neither. Explain each exception.
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.
Find the greatest common factor of 24 and 40 and explain how it relates to equal-size grouping.
Shared factors are 1, 2, 4 and 8, so the greatest is 8; both collections can be split into eight equal groups.
Find the least positive common multiple of 6 and 10, then the greatest common factor of 6 and 10. Explain why they differ.
Common multiples are shared products; common factors are shared exact divisors.
A student says 1 is prime because it cannot be divided equally by any number except 1. What has been overlooked?
Prime numbers must have exactly two distinct positive factors, 1 and the number itself; 1 has just one.
Compute 25 × 36 mentally in two ways: using the distributive property and using doubling/halving. Explain why the two methods agree.
Distributivity separates products without changing their sum; doubling one factor while halving the other preserves the product.
Use partial products to calculate 212 × 4 without regrouping.
(200×4)+(10×4)+(2×4)=800+40+8=848; each partial product represents its correct place value.
Compute 487 × 6 with regrouping and check the approximate size.
6×7 = 42; record 2 ones and exchange 4 tens; continue by place value. Estimate 500×6≈3,000.
Calculate 47 × 30 by interpreting 30 as 3 tens.
47×3×10 = 141×10 = 1,410.
Use four partial products to calculate 34 × 26.
(30+4)(20+6) = 600+180+80+24 = 884.
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.
Both methods add the same four underlying partial products, giving 3,196. An estimate of 70×45≈3,150 confirms the magnitude.
A rectangular tile design has 24 rows of 35 tiles. Find the total using a split model.
24×(30+5)=720+120=840.
Explain why 46×20 is ten times 46×2, without saying “add a zero.”
Associativity and base-ten scaling justify multiplying by ten.
Divide 936 ÷ 6 using convenient partial quotients. Then calculate (936 × 2) ÷ (6 × 2) and explain why both quotients are equal.
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.
Estimate 2,848 ÷ 7 using compatible numbers. Then calculate the exact quotient.
7×400=2,800; 2,848−2,800=48; 48=7×6+6. Exact quotient 406 remainder 6.
Calculate 4,368 ÷ 7 and check by multiplication.
7×624=4,368. In the written algorithm, exchange between thousands, hundreds, tens and ones.
There are 1,258 crayons packed in boxes holding 9 each. How many full boxes can be filled and how many crayons remain?
9×139=1,251, and 1,258−1,251=7. The remainder is less than 9.
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?
1,245÷6 = 207 remainder 3. The 23 donated tickets add to the remainder: 3+23 = 26. State both stages clearly.
A school has 3,205 pencils to distribute in bundles of 8. How many full bundles and what remainder?
3,205=8×400+5. The remainder is below eight.
Explain why 4,080 ÷ 8 cannot be 51 using an inverse check.
8×51=408, but 8×510=4,080; the place-value zero matters.
Show why 3/4 and 9/12 represent equal parts of the same whole.
Dividing each fourth into three equal smaller parts creates 12 equal parts; 3 fourths become 9 twelfths.
Order 2/3, 5/6, and 3/4 from smallest to largest. Mark their approximate positions on a number line from 0 to 1.
Rename using twelfths, divide the same whole line into twelve equal steps, and mark the points at steps 8, 9, and 10.
Express 13/5 as a mixed number and represent its full wholes and leftover fifths.
Ten fifths make two wholes, and three fifths remain; 13=2×5+3.
Convert 29/8 to simplest mixed-number form.
29÷8 gives quotient 3 and remainder 5; five eighths is already in simplest form.
Convert 4 3/7 to an improper fraction and explain the numerator.
Four wholes contain 28 sevenths; adding 3 sevenths gives 31 sevenths.
Three pizzas are shared equally by four students. What fraction of a pizza does each receive?
Each pizza is divided into four equal parts; each student gets three quarters in all. This is 3÷4.
A ribbon is 3 1/4 meters long. Write the length as an improper fraction and in fourths.
Three meters is twelve fourths; adding one fourth gives thirteen fourths.
A student says 3/5 > 3/4 because 5 > 4. Explain the error using equal-sized wholes.
If the whole is divided into five parts, each part is smaller than a fourth; three fifths is less than three fourths.
Compute 7/10 + 8/10. Simplify the result and explain the unit being counted.
Count tenths: 7+8=15 tenths = 15/10 = 3/2 = 1 1/2.
Calculate 2/3 + 1/6 using sixths.
Rename 2/3 as 4/6, then add 1/6 to obtain 5/6.
Compute 2 3/4 + 1/2. Rename the fractional sum if necessary.
1/2=2/4; 3/4+2/4=5/4=1 1/4; with 2 wholes this is 3 1/4.
Compute 1 5/6 + 2 3/4; show a common denominator.
5/6=10/12 and 3/4=9/12; fractional sum 19/12=1 7/12, plus three wholes gives 4 7/12.
Compute 5 1/4 − 3/4 by regrouping one whole.
Rename 5 1/4 as 4 5/4. Then subtract 3/4 to get 4 2/4=4 1/2.
Compute 6 1/3 − 2 5/6 with a clearly shown exchange.
6 1/3=6 2/6=5 8/6; subtract 2 5/6 to obtain 3 3/6=3 1/2.
A recipe uses 1 3/4 cups of oats and 2 5/8 cups of flour. How much is that altogether?
1 6/8 + 2 5/8 = 3 11/8 = 4 3/8.
A student adds 1/2 and 1/3 and writes 2/5. Explain what has gone wrong and calculate the correct sum.
The pieces are different sizes; rename both in sixths: 3/6+2/6=5/6.
Find 1/7 of 63 counters and explain why division represents one equal share.
Divide the 63 counters into seven equal groups: 63÷7=9.
Calculate 4 × 3/8 as repeated addition and simplify.
3/8+3/8+3/8+3/8=12/8=3/2=1 1/2.
Compute 6 × 5/9. Explain how to simplify the improper result.
6×5/9=30/9=10/3=3 1/3.
A jar has 40 beads. Three-fifths are green. How many beads are green?
Split 40 into five groups of 8; choose three groups: 3×8=24.
Find 24 × 3/4 as three-quarters of a whole-number quantity.
First 24÷4=6, then multiply 6×3=18.
Compute 36 × 5/6 two ways and show why the results agree.
Partition first: 36÷6=6 then 6×5=30. Multiply first: 36×5÷6=180÷6=30.
A farm harvests 84 apples and sells 5/7 of them. How many apples are sold? Show a unit bar.
Divide 84 into seven equal units of 12, then select five units: 12×5=60.
A baker has 120 rolls. She gives 1/4 away, then sells 2/3 of what remains. How many rolls are left?
After giving away 30, 90 remain. Sell 2/3 of 90=60, so 30 remain. The second fraction is of 90, not 120.
A class has 120 pencils. Two-fifths are blue. Of the rest, one-third are green. How many are neither blue nor green?
Blue = 120×2/5=48; remaining 72. Green = 72÷3=24. Neither=72−24=48.
A child claims that 3/4 of 32 is 32×4÷3. Explain and correct the error.
Divide the whole into four equal parts, then take three: 32÷4×3=24.
The graph shows library visitors on four days. How many more visited on Thursday than Monday?
Read Monday as 24 and Thursday as 42; 42−24=18.
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.
A scale of 0, 5, 10, 15, 20 cm or steps of 2 cm is sensible. The line rises, dips, then rises again.
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.
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.
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?
The endpoint change is 35−20=15, but the values decline from 30 on the second day to 25 on the third day.
A student draws a line graph with equal time intervals spaced unevenly on the horizontal axis. What should be changed?
Distances on the axes communicate scale; unequal spacing falsely changes the represented rate of change.