Official Grade: 51.5 / 57 (90.4% — Grade A) • Total Time: 1h 14m 21s • Completed October 5, 2026 at 7:58 PM.
Test Submitted & Graded
Official Grade: 51.5 / 57 (90.4% — Grade A) • Autograded: 47 / 57
Total Time Taken: 1h 14m 21s
All answers have been evaluated. See correct answers and step-by-step solutions below.
4th Grade Math Diagnostic — Units 4–5
Suggested time: 60 minutes. Answer every question on your own. No calculator, Khan Academy, hints, or answer checking. Show your work for multi-digit multiplication and division.
Unit 4 — Multiply by 2-Digit Numbers
1. Multiplying multiples of 10 and 100
Completed
Use basic facts and place-value patterns to multiply.
a. 30 × 40 =
Correct
b. 70 × 80 =
Correct
c. 50 × 600 =
Correct
d. Fill in the missing number: 90 × Correct = 5,400
e. Explain why 50 × 60 = 3,000 ends in three zeros instead of two zeros.
Correct
Solution & Answer Key
a.1,200 (3 × 4 = 12, plus two zeros) b.5,600 (7 × 8 = 56, plus two zeros) c.30,000 (5 × 6 = 30, plus three zeros: one from 50 and two from 600) d.60 (54 ÷ 9 = 6; 5,400 has two zeros and 90 has one zero) e.5 × 6 = 30, which already has a zero! When you attach the zero from 50 and the zero from 60, you get three zeros total (3,000).
2. Estimating products with 2-digit numbers
Completed
Estimate products by rounding numbers to the nearest ten.
a. Estimate 48 × 32 by rounding each number to the nearest ten.
48 rounds to: Correct
32 rounds to: Correct
Estimated product: Correct
b. A farmer packs 68 apples in each crate. She has 21 crates. Estimate the total number of apples by rounding both numbers to the nearest ten.
Accepted (Full Credit)
c. Is your estimate of 70 × 20 = 1,400 an overestimate or an underestimate of the exact number of apples (68 × 21 = 1,428)? Explain why.
Incorrect (0 / 1 pt)
Solution & Answer Key
a. 48 rounds to 50, 32 rounds to 30. Estimated product = 50 × 30 = 1,500. b. 68 rounds to 70; 21 rounds to 20. 70 × 20 = 1,400 apples. c. It is an underestimate because 1,400 is less than the exact product of 1,428.
Concept & Review Notes (Parts b & c)
Part b (Operation Sign): You calculated the correct estimated product (1,400)! Be careful to write the multiplication sign: 70 × 20 = 1,400 instead of +, since 70 + 20 would be 90.
Part c (Overestimate vs. Underestimate): To decide if an estimate is higher or lower, compare the estimate directly to the exact answer: because 1,400 < 1,428, it is an underestimate. Looking only at 68 turning into 70 misses that 21 rounded down to 20; both factors must be compared together!
3. Area models for 2-digit multiplication
Completed
The area model represents 34 × 27 = (30 + 4) × (20 + 7).
×
30
4
20
Box A
(20 × 30)
Box B
(20 × 4)
7
Box C
(7 × 30)
Box D
(7 × 4)
a. Find each partial area.
Box A: Correct
Box B: Correct
Box C: Correct
Box D: Correct
b. Add all 4 partial areas to find the total product of 34 × 27.
Correct
Solution & Answer Key
a. Partial areas:
Box A = 600 (20 × 30)
Box B = 80 (20 × 4)
Box C = 210 (7 × 30)
Box D = 28 (7 × 4) b. Total product: 600 + 80 + 210 + 28 = 918
4. Partial products & the break-apart rule
Completed
Multiply 53 × 46 by breaking apart both numbers: (50 + 3) × (40 + 6).
Find the 4 partial products:
Ones × Ones: 3 × 6 = Correct
Tens × Ones: 50 × 6 = Correct
Ones × Tens: 3 × 40 = Correct
Tens × Tens: 50 × 40 = Correct
e. Add all 4 partial products to find the final product.
Correct
Solution & Answer Key
a. 3 × 6 = 18 b. 50 × 6 = 300 c. 3 × 40 = 120 d. 50 × 40 = 2,000 e. Total: 18 + 300 + 120 + 2,000 = 2,438
5. Standard multiplication algorithm
Completed
Multiply vertically using the standard algorithm: 64 × 38.
64
× 38
────
a. Row 1: Multiply 64 by the ones digit (64 × 8):
Correct
b. Row 2: Multiply 64 by the tens digit (64 × 30, with placeholder 0 in the ones place):
Half Credit (0.5 / 1 pt)
c. Row 3: Add both rows together to find the final product:
Correct
Solution & Answer Key
a. Row 1: 64 × 8 = 512 b. Row 2: 64 × 30 = 1,920 (with placeholder 0) c. Final Product: 512 + 1,920 = 2,432
Concept & Review Note (Part b)
Row 2 in Standard Vertical Multiplication: In 64 × 38, Row 2 means multiplying 64 by 30 (not estimating). Place the 0 in the ones place first, then multiply the exact digits: 4 × 3 = 12 (write 2, carry 1), and 6 × 3 + 1 = 19 to get 1,920. Do not round 64 down to 60 during standard algorithm vertical steps.
6. Multi-step multiplication word problems
Completed
The 4th-grade class is planning a field trip to the science museum. There are 4 classrooms, and each classroom has 24 students. Each museum ticket costs $15.
a. Step 1: How many students are going on the trip in total?
Correct
b. Step 2: What is the total cost for all students' museum tickets?
$ Correct
c. Step 3: The school set aside a total trip budget of $1,600. After paying for all tickets, how much money is left over for bus transportation?
$ Correct
Solution & Answer Key
a. Total students: 4 classrooms × 24 students = 96 students. b. Total ticket cost: 96 students × $15 = $1,440. c. Leftover money: $1,600 - $1,440 = $160 left for buses.
Unit 5 — Division
7. Division with remainders and checking work
Completed
Find the quotient and remainder for each division problem.
a. 38 ÷ 5
Quotient: Correct
Remainder: Correct
b. 53 ÷ 7
Quotient: Correct
Remainder: Correct
c. Write the check equation (using multiplication and addition) that proves your answer to part b is correct.
Accepted (Full Credit)
d. Explain why the remainder can NEVER be 7 or greater when dividing by 7.
Half Credit (0.5 / 1 pt)
Solution & Answer Key
a. 38 ÷ 5 = 7 R 3 (5 × 7 = 35; 38 - 35 = 3) b. 53 ÷ 7 = 7 R 4 (7 × 7 = 49; 53 - 49 = 4) c. Check equation:(7 × 7) + 4 = 53 (or 49 + 4 = 53) d. If the remainder was 7 or greater, you could make another whole group of 7! The quotient would increase by 1.
Concept & Review Notes (Parts c & d)
Part c (Check Equation Format): Your math is correct: 7 × 7 = 49, and 49 + 4 = 53. To write clean mathematical equations, write it as one complete statement: (7 × 7) + 4 = 53 without chained equals signs like 49=+4.
Part d (Divisor & Remainder Rule): Division means sorting into equal groups of 7. If 7 or more were left over, you haven't finished dividing yet—you can make another whole group of 7! That is why the remainder must always be strictly less than the divisor.
8. Dividing multiples of 10, 100, and 1,000
Completed
Use basic division facts and place value patterns to divide.
a. 280 ÷ 7 =
Correct
b. 3,600 ÷ 4 =
Correct
c. 42,000 ÷ 6 =
Correct
d. Fill in the missing dividend: Incorrect (0 / 1 pt) ÷ 5 = 800
e. An art supplier packs 4,800 colored markers equally into 8 large storage crates. How many markers are in each crate?
Finding the Missing Dividend: In [ ] ÷ 5 = 800, the missing number is the dividend (the starting total). To find it, use multiplication: 800 × 5 = 4,000. Dividing 80 ÷ 5 = 16 makes the starting number tiny (16 ÷ 5 is only 3 with remainder 1, not 800!).
9. Estimating quotients with compatible numbers
Completed
Find friendly compatible numbers that divide easily to estimate each quotient.
a. Estimate 347 ÷ 5:
Friendly number close to 347: Correct
Estimated quotient: Correct
b. Estimate 2,745 ÷ 9:
Friendly number close to 2,745: Half Credit (0.5 / 1 pt)
Estimated quotient: Accepted (Full Credit)
c. Is 438 ÷ 6 between 60 and 70, or between 70 and 80? Explain using multiplication.
Incorrect (0 / 1 pt)
Solution & Answer Key
a. Friendly number = 350 (since 5 × 7 = 35). Estimated quotient: 350 ÷ 5 = 70. b. Friendly number = 2,700 (since 9 × 3 = 27). Estimated quotient: 2,700 ÷ 9 = 300. c. It is between 70 and 80 because 6 × 70 = 420 and 6 × 80 = 480. 438 is between 420 and 480.
Concept & Review Notes (Parts b & c)
Part b (Friendly Number Place Value): You divided 27,000 ÷ 9 = 3,000 correctly! Just watch your place value digits: 2,745 has 4 digits (thousands), so the friendly number close to it is 2,700 (not 27,000). 2,700 ÷ 9 = 300.
Part c (Testing Quotient Ranges): Test the tens by multiplying by 6: 6 × 60 = 360, 6 × 70 = 420, and 6 × 80 = 480. Because 438 is between 420 and 480, 438 ÷ 6 must be between 70 and 80.
10. Division area models & friendly chunks
Completed
The rectangle represents 156 ÷ 6. It has a total area of 156 and height 6, split into two friendly chunks: 120 and 36.
Width:
120 ÷ 6 = [Part A]
36 ÷ 6 = [Part B]
Height: 6
Area = 120
Area = 36
a. Find the length of Part A (120 ÷ 6):
Correct
b. Find the length of Part B (36 ÷ 6):
Correct
c. Add the lengths to find the total quotient of 156 ÷ 6:
Correct
d. Break apart 225 ÷ 5 into two friendly multiples of 5 and solve:
( Correct ÷ 5 ) + ( Correct ÷ 5 ) = Correct
Solution & Answer Key
a. Part A: 120 ÷ 6 = 20 b. Part B: 36 ÷ 6 = 6 c. Total quotient: 20 + 6 = 26 d. (200 ÷ 5) + (25 ÷ 5) = 40 + 5 = 45
11. Multi-digit division: Partial quotients & standard division
Completed
Divide multi-digit numbers. Show your work or record partial quotients.
12. Division word problems: Interpreting the remainder
Completed
Solve each word problem and interpret what to do with the remainder based on the story.
a.Camping Trip (Round Up): There are 135 fourth-graders going on an outdoor trip. Each passenger van holds 8 students.
135 ÷ 8 = Correct R Correct
How many vans are needed so EVERY student has a seat? Incorrect (0 / 1 pt) vans
b.Bakery Boxes (Drop Remainder): A baker has 98 chocolate cookies. She puts 6 cookies in each gift box. She can only sell full boxes.
How many full boxes can she sell? Correct boxes
c.Leftover Cookies (Remainder is the Answer): After selling all the full boxes in part b, how many cookies are left over for the baker to take home?
Leftover cookies: Correct cookies
Solution & Answer Key
a. 135 ÷ 8 = 16 R 7. Since all 7 remaining students need a seat, you must round up: 17 vans. b. 98 ÷ 6 = 16 R 2. You drop the remainder because only full boxes can be sold: 16 boxes. c. The remainder is the answer for leftovers: 2 cookies left over.
Concept & Review Note (Part a)
Interpreting Remainders (Round Up): 135 ÷ 8 = 16 with remainder 7. 16 vans only hold 16 × 8 = 128 students. The 7 remaining students also need a seat to go on the trip! You must round up to 17 vans so no student is left behind.