Multiplication: Tables, Written Method and Strategies

Multiplication finds the total in equal groups and also describes scaling a quantity by a factor. In brief: 6 × 4 = 24 can describe six groups of four objects. Knowing multiplication facts helps you calculate larger products without repeatedly adding every group.

This guide connects equal groups, reference facts, mental strategies and written multiplication. The examples focus on whole numbers, where repeated addition is a useful starting model. You will see why place value controls partial products and how a quick estimate can reveal a missing zero or an incorrectly shifted row.

What does multiplication mean?

The numbers being multiplied are factors, and the result is the product. In 7 × 5 = 35, 7 and 5 are the factors and 35 is the product. An equal-groups story might describe seven bags containing five counters each, giving 35 counters altogether.

An array displays the same calculation in rows and columns. Seven rows of five objects contain the same total as five rows of seven objects. Turning the array helps explain why 7 × 5 = 5 × 7, even though the two stories assign different roles to the group count and group size.

Multiplication also scales quantities. Doubling a length of 9 centimetres gives 2 × 9 = 18 centimetres. For whole-number factors, repeated addition supports understanding, but multiplying fractions later needs a broader scaling interpretation. A factor does not universally make a quantity larger; even among whole numbers, multiplying by zero gives zero.

Multiplication reference table for factors 1 to 10

The complete reference below covers every pair of factors from 1 through 10. Each row fixes one factor and gives its products with all ten possible second factors. The products are grouped into two cells so that the table stays within a narrow format while retaining every value.

Read a row in order. For factor 7, the first cell lists 7 × 1 through 7 × 5, and the second lists 7 × 6 through 7 × 10. The difference between consecutive products is always 7 because each new product includes one more group of seven.

Learn facts through relationships as well as recall. The five-times facts are half the ten-times facts, and a nine-times fact can be found from ten groups minus one group. For example, 9 × 8 = 80 − 8 = 72. Reversed facts share a product, reducing the number of independent results you must memorize.

How do you multiply a large number by one digit?

Use place value and work from right to left. For 247 × 6, the ones calculation is 7 × 6 = 42. Record 2 ones and carry 4 tens. The carry represents the forty already produced by the ones calculation, so it must be included exactly once in the next column.

The tens calculation is 4 × 6 = 24 tens, plus the carried 4 tens, giving 28 tens. Record 8 tens and carry 2 hundreds. The hundreds calculation is 2 × 6 = 12 hundreds, plus 2 carried hundreds, giving 14 hundreds. The full product is 1482.

An expanded-form check gives the same answer: 200 × 6 = 1200, 40 × 6 = 240 and 7 × 6 = 42. Adding 1200 + 240 + 42 yields 1482. The written algorithm compresses these separate place-value products into columns, making the process shorter without changing the reasoning.

1. Multiply ones: 7 × 6 = 42; record 2 and carry 4.
2. Multiply tens: 4 × 6 + 4 = 28; record 8 and carry 2.
3. Multiply hundreds: 2 × 6 + 2 = 14.
4. Read the completed product as 1482.

How does written multiplication by two digits work?

For 324 × 26, split 26 into 20 + 6. Calculate the ones partial product, 324 × 6 = 1944. Then calculate the tens partial product, 324 × 20 = 6480. Add the two partial products to obtain 8424.

The second partial product represents twenty groups, not two groups. This is why the row must reflect a tens shift or an explicit final zero when using the traditional layout. Writing 648 rather than 6480 would calculate 324 × 2 and therefore solve a different problem.

Estimate before or after calculating: 324 × 26 is near 300 × 30 = 9000, while the exact product 8424 is plausible. A more direct check uses 324 × (25 + 1) = 8100 + 324 = 8424. Using a different partition can help detect an error in either partial product.

Which mental multiplication strategies help?

Partition a factor into friendly parts. For 18 × 7, calculate 10 × 7 + 8 × 7 = 70 + 56 = 126. For a value near a round number, compensate: 49 × 6 equals 50 × 6 − 6 = 294. The correction must account for every group added by the temporary adjustment.

Doubling and halving can simplify a product when one factor is even. For 16 × 25, halve 16 and double 25 to obtain 8 × 50 = 400. Repeating the change gives 4 × 100 = 400. The product remains unchanged because one factor is divided by 2 while the other is multiplied by 2.

Multiplying a whole number by 10 shifts each digit’s contribution one place to the left. Thus 73 × 10 = 730 and 73 × 100 = 7300. Thinking in place values is safer than the phrase ‘add a zero’, which can become misleading when you later work with decimals or other representations.

What multiplication properties explain these methods?

The commutative property allows factors to exchange order: a × b = b × a. The associative property allows a product to be grouped differently: (a × b) × c = a × (b × c). For 4 × 17 × 25, grouping 4 and 25 produces 100 × 17 = 1700.

The distributive property connects multiplication and addition: a × (b + c) = a × b + a × c. Written multiplication uses this property when a two-digit factor is split into tens and ones. It also supports subtraction-based compensation, such as 8 × (30 − 1) = 240 − 8 = 232.

One is the multiplicative identity, so n × 1 = n. Zero gives n × 0 = 0, which differs from adding zero. Keep these identities separate: a group count of zero gives no objects, while adding an empty group to an existing collection leaves the collection unchanged. Both statements follow the quantity each operation represents.

Where is multiplication used and which mistakes should you avoid?

Multiplication is useful whenever equal quantities repeat, including packing objects in boxes, finding repeated journey distances and calculating rectangular arrays. Written methods grew naturally from place-value notation because the number can be partitioned into consistent units. Different layouts, including area models, show the same partial products with different visual organization.

When applying multiplication, keep the units meaningful. Seven boxes with 12 pencils per box contain 84 pencils, while a rectangle 7 metres by 12 metres has area 84 square metres. The identical numerical product does not mean the quantities share the same units or interpretation.

Common errors include dropping a carry, multiplying a carry again and forgetting that a tens partial product represents tens. Record each intermediate result, check the table fact involved and estimate the product’s size. For a positive whole-number multiplier greater than 1, the product should exceed the original positive number, providing one useful reasonableness check.

Frequently asked questions

What are factors and a product?

Factors are the numbers multiplied, and the product is their result. In 8 × 9 = 72, the factors are 8 and 9 and the product is 72.

Why is the second partial product shifted left?

It is shifted because it represents multiplication by tens. Multiplying by 20 creates a result ten times the product obtained from multiplying by 2.

Does multiplication always make numbers larger?

Multiplication does not always make numbers larger. Multiplying by 1 leaves a value unchanged, and multiplying by 0 gives zero.

How can I work out a forgotten nine-times fact?

Use ten times the other factor and subtract one group. For example, 9 × 7 = 70 − 7 = 63.

Can division check a multiplication?

Division can check a product when the checking factor is nonzero. For example, 1482 ÷ 6 = 247 checks 247 × 6 = 1482.

Practice exercises

1. Calculate 8 × 7.
2. Complete 9 × ___ = 63.
3. Find the correct tens partial product in 215 × 34.
4. Calculate 236 × 7.
5. There are 24 cartons with 36 bottles each. How many bottles are there?

Answers to the exercises

1. 56. The product is the seventh entry in the eight-times row.
2. 7. Seven groups of nine give 63.
3. 6450. The tens part is 30, so calculate 215 × 30 rather than 215 × 3.
4. 1652. The place-value products are 1400, 210 and 42, which sum to 1652.
5. 864 bottles. Calculate 24 × (30 + 6) = 720 + 144 = 864.