Number & Operations in Base Ten

Multi-digit Arithmetic: Place Value, Multiplication, and Division

Grade 4

✓By the end of this lesson students will be able to identify the place value of digits in multi-digit whole numbers up to 1,000,000.
✓By the end of this lesson students will be able to read, write, and compare multi-digit whole numbers using standard form, word form, and expanded form.
✓By the end of this lesson students will be able to multiply a whole number of up to four digits by a one-digit whole number.
✓By the end of this lesson students will be able to multiply two two-digit numbers.
✓By the end of this lesson students will be able to find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors.

Key concepts

Place Value

Place value tells us the value of each digit in a number based on its position. In our base-ten system, each place to the left is ten times greater than the place to its right. We will focus on places up to the millions: Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions.

Standard Form

Standard form is the usual way we write numbers using digits. For example, 5,432,100 is in standard form.

Word Form

Word form is writing a number using words. For example, 5,432,100 in word form is 'five million, four hundred thirty-two thousand, one hundred'.

Expanded Form

Expanded form is writing a number as the sum of the values of each digit. For example, 5,432,100 in expanded form is 5,000,000 + 400,000 + 30,000 + 2,000 + 100.

Multi-digit Multiplication

Multi-digit multiplication is finding the product of two numbers where at least one has more than one digit. We use strategies based on place value and the properties of operations, often using the standard algorithm. When multiplying, we multiply each digit of one factor by each digit of the other factor, remembering to carry over and add partial products.

Multi-digit Division

Multi-digit division is finding how many times one number (the divisor) goes into another number (the dividend). We use strategies based on place value, often using the standard algorithm (long division). The result is a quotient, and sometimes a remainder if the division is not exact.

Key facts to remember

1Our number system is a base-ten system, meaning each place value is ten times greater than the place to its right.
2Place value helps us understand the value of each digit in a number.
3Numbers can be written in standard form (digits), word form (words), or expanded form (sum of place values).
4When comparing multi-digit numbers, start comparing digits from the greatest place value (leftmost).
5The standard algorithm for multiplication involves multiplying by each digit and adding partial products.
6The standard algorithm for division (long division) involves dividing, multiplying, subtracting, and bringing down digits.
7A remainder in division is the amount left over when a number cannot be divided exactly.

Worked examples

Example 1

Consider the number 847,305.\na) Write the number in word form.\nb) Write the number in expanded form.\nc) What is the value of the digit 4?\nd) Compare 847,305 with 847,350 using >, <, or =.

Ia) To write in word form, read the number from left to right, grouping by periods (thousands, ones). 847,305 is 'eight hundred forty-seven thousand, three hundred five'.

IIb) To write in expanded form, identify the value of each digit based on its place. 8 is in the hundred thousands place (800,000), 4 is in the ten thousands place (40,000), 7 is in the thousands place (7,000), 3 is in the hundreds place (300), 0 is in the tens place (0), and 5 is in the ones place (5). So, 800,000 + 40,000 + 7,000 + 300 + 5.

IIIc) The digit 4 is in the ten thousands place. Its value is 4 x 10,000 = 40,000.

IVd) To compare 847,305 and 847,350, start from the leftmost digit and compare digits in the same place value. Both numbers have 8 in the hundred thousands place, 4 in the ten thousands place, 7 in the thousands place, and 3 in the hundreds place. Next, compare the tens place: 0 in 847,305 and 5 in 847,350. Since 0 < 5, then 847,305 < 847,350.

Answer

a) Eight hundred forty-seven thousand, three hundred five\nb) 800,000 + 40,000 + 7,000 + 300 + 5\nc) 40,000\nd) 847,305 < 847,350

Remember to use commas in word form and standard form to separate periods (thousands, millions).

Example 2

Multiply 2,145 by 6. Then, multiply 37 by 24.

IFor 2,145 x 6:

II1. Multiply the ones digit: 6 x 5 ones = 30 ones. Write down 0 in the ones place and carry over 3 tens.

III2. Multiply the tens digit: 6 x 4 tens = 24 tens. Add the carried over 3 tens: 24 + 3 = 27 tens. Write down 7 in the tens place and carry over 2 hundreds.

IV3. Multiply the hundreds digit: 6 x 1 hundred = 6 hundreds. Add the carried over 2 hundreds: 6 + 2 = 8 hundreds. Write down 8 in the hundreds place.

V4. Multiply the thousands digit: 6 x 2 thousands = 12 thousands. Write down 12 (1 in the ten thousands place, 2 in the thousands place).

VIFor 37 x 24:

VII1. Multiply 37 by the ones digit of 24 (which is 4):

VIII 4 x 7 ones = 28 ones. Write down 8, carry over 2 tens.

9 4 x 3 tens = 12 tens. Add carried over 2 tens: 12 + 2 = 14 tens. Write down 14.

10 This gives the first partial product: 148.

112. Multiply 37 by the tens digit of 24 (which is 2 tens, or 20):

12 Write a 0 in the ones place as a placeholder because we are multiplying by tens.

13 2 x 7 ones = 14 ones. (This is actually 20 x 7 = 140). Write down 4, carry over 1 hundred.

14 2 x 3 tens = 6 tens. (This is actually 20 x 30 = 600). Add carried over 1 hundred: 6 + 1 = 7 hundreds. Write down 7.

15 This gives the second partial product: 740.

163. Add the partial products: 148 + 740.

Answer

2,145 x 6 = 12,870\n37 x 24 = 888

Always remember to add a zero placeholder when multiplying by the tens digit (or any higher place value) in multi-digit multiplication.

Example 3

Divide 5,847 by 4. Express any remainder.

I1. Divide the thousands digit: How many 4s are in 5? One 4. Write 1 above the 5.

II Multiply: 1 x 4 = 4.

III Subtract: 5 - 4 = 1.

IV Bring down the next digit (8) to make 18.

V2. Divide the hundreds: How many 4s are in 18? Four 4s. Write 4 above the 8.

VI Multiply: 4 x 4 = 16.

VII Subtract: 18 - 16 = 2.

VIII Bring down the next digit (4) to make 24.

93. Divide the tens: How many 4s are in 24? Six 4s. Write 6 above the 4.

10 Multiply: 6 x 4 = 24.

11 Subtract: 24 - 24 = 0.

12 Bring down the next digit (7) to make 7.

134. Divide the ones: How many 4s are in 7? One 4. Write 1 above the 7.

14 Multiply: 1 x 4 = 4.

15 Subtract: 7 - 4 = 3.

16 Since there are no more digits to bring down, 3 is the remainder.

Answer

5,847 ÷ 4 = 1,461 R 3

To check your answer, multiply the quotient by the divisor and add the remainder: (1,461 x 4) + 3 = 5,844 + 3 = 5,847.

Common mistakes

✗Misaligning digits when adding partial products in multiplication or when subtracting in long division.
✗Forgetting to carry over digits in multiplication or to borrow in subtraction steps within division.
✗Incorrectly handling zeros, especially when a zero is in the middle of a number or when multiplying by a multiple of ten.
✗Not understanding the meaning of a remainder or how to express it correctly.
✗Confusing place value names (e.g., 'ten thousands' with 'hundred thousands').

Exam tips

★Always show your work step-by-step for multiplication and division problems. This helps you catch errors and allows for partial credit.
★Use estimation to check if your answer is reasonable. For example, 2,145 x 6 is roughly 2,000 x 6 = 12,000, so 12,870 is a reasonable answer.
★Practice your basic multiplication facts (times tables) regularly. This will make multi-digit multiplication and division much faster and more accurate.
★When comparing numbers, carefully look at each digit from left to right. Even a small difference in a higher place value makes a big difference in the number's overall value.

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