Khan 4th Grade Math Diagnostic

Units 4–5 (12 Questions)
Suggested time: 60 minutes
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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

📖 Guide Saved

Use basic facts and place-value patterns to multiply.

a. 30 × 40 =

b. 70 × 80 =

c. 50 × 600 =

d. Fill in the missing number: 90 × = 5,400

e. Explain why 50 × 60 = 3,000 ends in three zeros instead of two zeros.

✓ 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

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Estimate products by rounding numbers to the nearest ten.

a. Estimate 48 × 32 by rounding each number to the nearest ten.

48 rounds to:

32 rounds to:

Estimated product:

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.

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.

✓ 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.

3. Area models for 2-digit multiplication

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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:

Box B:

Box C:

Box D:

b. Add all 4 partial areas to find the total product of 34 × 27.

✓ 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

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Multiply 53 × 46 by breaking apart both numbers: (50 + 3) × (40 + 6).

Find the 4 partial products:

Ones × Ones: 3 × 6 =

Tens × Ones: 50 × 6 =

Ones × Tens: 3 × 40 =

Tens × Tens: 50 × 40 =

e. Add all 4 partial products to find the final product.

✓ 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

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Multiply vertically using the standard algorithm: 64 × 38.

64 × 38 ────

a. Row 1: Multiply 64 by the ones digit (64 × 8):

b. Row 2: Multiply 64 by the tens digit (64 × 30, with placeholder 0 in the ones place):

c. Row 3: Add both rows together to find the final product:

✓ 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

6. Multi-step multiplication word problems

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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?

b. Step 2: What is the total cost for all students' museum tickets?

$

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?

$

✓ 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

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Find the quotient and remainder for each division problem.

a. 38 ÷ 5

Quotient: Remainder:

b. 53 ÷ 7

Quotient: Remainder:

c. Write the check equation (using multiplication and addition) that proves your answer to part b is correct.

d. Explain why the remainder can NEVER be 7 or greater when dividing by 7.

✓ 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.

8. Dividing multiples of 10, 100, and 1,000

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Use basic division facts and place value patterns to divide.

a. 280 ÷ 7 =

b. 3,600 ÷ 4 =

c. 42,000 ÷ 6 =

d. Fill in the missing dividend: ÷ 5 = 800

e. An art supplier packs 4,800 colored markers equally into 8 large storage crates. How many markers are in each crate?

✓ Solution & Answer Key

a. 40 (28 tens ÷ 7 = 4 tens)
b. 900 (36 hundreds ÷ 4 = 9 hundreds)
c. 7,000 (42 thousands ÷ 6 = 7 thousands)
d. 4,000 (5 × 800 = 4,000)
e. 4,800 ÷ 8 = 600 markers per crate.

9. Estimating quotients with compatible numbers

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Find friendly compatible numbers that divide easily to estimate each quotient.

a. Estimate 347 ÷ 5:

Friendly number close to 347:

Estimated quotient:

b. Estimate 2,745 ÷ 9:

Friendly number close to 2,745:

Estimated quotient:

c. Is 438 ÷ 6 between 60 and 70, or between 70 and 80? Explain using multiplication.

✓ 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.

10. Division area models & friendly chunks

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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):

b. Find the length of Part B (36 ÷ 6):

c. Add the lengths to find the total quotient of 156 ÷ 6:

d. Break apart 225 ÷ 5 into two friendly multiples of 5 and solve:

( ÷ 5 ) + ( ÷ 5 ) =

✓ 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

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Divide multi-digit numbers. Show your work or record partial quotients.

a. 546 ÷ 3 =

b. 789 ÷ 4

Quotient: Remainder:

c. Divide a 4-digit number: 3,294 ÷ 6 =

✓ Solution & Answer Key

a. 546 ÷ 3 = 182
• 3 × 100 = 300 (leaves 246)
• 3 × 80 = 240 (leaves 6)
• 3 × 2 = 6 (leaves 0)
• 100 + 80 + 2 = 182
b. 789 ÷ 4 = 197 R 1 (4 × 197 = 788; 789 - 788 = 1)
c. 3,294 ÷ 6 = 549
• 6 × 500 = 3,000 (leaves 294)
• 6 × 40 = 240 (leaves 54)
• 6 × 9 = 54 (leaves 0)
• 500 + 40 + 9 = 549

12. Division word problems: Interpreting the remainder

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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 = R

How many vans are needed so EVERY student has a seat? 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? 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: 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.