You already know how to add the same number again and again. Multiplication is a faster way to do exactly that job. In this chapter you will learn the multiplication facts from 2 to 10, see how they connect to each other, and use three powerful properties that make a hard fact feel small.
1. What a multiplication fact tells you
A multiplication fact tells you how many in all when you have equal groups. If you see 3 rows with 4 dots in each row, you can add \(4 + 4 + 4 = 12\), or you can multiply \(3 \times 4 = 12\). A picture like this one, with rows and columns, is called an array.
In \(3 \times 4 = 12\), the numbers 3 and 4 are the factors. They tell you the number of groups and the size of each group. The answer, 12, is the product. The sign \(\times\) is read “times.”
You can also read \(3 \times 4\) as “3 groups of 4.” When you know the facts by heart, you can solve problems about teams, boxes, shelves, and tickets in seconds.
2. Facts of 2, 5 and 10
These are the friendliest facts, so start here.
- Facts of 2 are doubles. \(2 \times 7\) means 7 + 7 = 14. Every product of 2 is an even number.
- Facts of 5 are skip counting by 5: 5, 10, 15, 20, 25… Every product ends in 0 or 5.
- Facts of 10 are skip counting by 10. The product is the other factor with a 0 on the end: \(10 \times 6 = 60\).
| Factor | ×2 | ×5 | ×10 |
|---|---|---|---|
| 3 | 6 | 15 | 30 |
| 6 | 12 | 30 | 60 |
| 8 | 16 | 40 | 80 |
| 9 | 18 | 45 | 90 |
A pet store sells fish food in packs. Each shelf holds 5 packs and there are 8 shelves. How many packs are there in all?
\(8 \times 5 = 40\), because counting by 5 eight times gives 5, 10, 15, 20, 25, 30, 35, 40. There are 40 packs.
3. Facts of 3 and 4
You can build the facts of 3 and 4 from facts you already know.
- For \(3 \times n\), find the double of \(n\), then add one more group of \(n\).
- For \(4 \times n\), double \(n\), then double the answer again.
For example, \(3 \times 7\): the double of 7 is 14, and \(14 + 7 = 21\). For \(4 \times 7\): \(7 \to 14 \to 28\). So \(4 \times 7 = 28\).
Skip counting also works well. The picture below shows 6 hops of size 4 on a number line. The frog lands on 4, 8, 12, 16, 20 and then 24, so \(4 \times 6 = 24\).
A garage has 6 cars. Each car has 4 wheels on the road. How many wheels?
Double 6 is 12, and double 12 is 24. So \(4 \times 6 = 24\). There are 24 wheels.
4. Facts of 6, 7, 8 and 9
These facts feel harder, but you almost know them already. Most of them are the same fact in a different order, and the others can be built from the 5s and the 10s.
- 5s plus one more group. \(6 \times 7\) is \(5 \times 7 = 35\) plus one more 7, so \(35 + 7 = 42\).
- Doubling. \(8 \times n\) is the double of the double of the double of \(n\). For \(8 \times 6\): \(6 \to 12 \to 24 \to 48\).
- Tens minus one group (for 9s). \(9 \times 7\) is \(10 \times 7 = 70\) minus one 7, so \(70 - 7 = 63\).
On every product of 9, the two digits add up to 9: 18 (1 + 8), 27 (2 + 7), 36 (3 + 6)… And the tens digit is always one less than the other factor, so \(9 \times 8 = 72\) because 7 is one less than 8 and \(7 + 2 = 9\).
Many students mix up 54, 56 and 63. Remember the staircase 5, 6, 7, 8: \(56 = 7 \times 8\).
A choir stands in 7 rows with 9 singers in each row. How many singers?
\(7 \times 9 = 70 - 7 = 63\). Check with the digits: 6 + 3 = 9. There are 63 singers.
5. The commutative and distributive properties
You can swap the factors and the product stays the same: \(a \times b = b \times a\). For example, \(4 \times 9 = 9 \times 4 = 36\). This cuts the number of facts you must learn almost in half.
The distributive property lets you break one factor into two easier parts. Look at the array below. It has 6 rows and 7 columns. Cut the 7 columns into 5 blue columns and 2 orange columns.
Multiplying a number by a sum gives the same result as multiplying by each part and adding: \(a \times (b + c) = a \times b + a \times c\).
So \(6 \times 7 = 6 \times (5 + 2) = 6 \times 5 + 6 \times 2 = 30 + 12 = 42\). You can break either factor, and you should split into the facts you know best, such as 5 or 2.
Find \(8 \times 7\) by breaking 7 into 5 + 2.
\(8 \times 7 = 8 \times 5 + 8 \times 2 = 40 + 16 = 56\).
6. The associative property, zero and one
When you multiply three numbers, you can group them in any way and the product stays the same: \((a \times b) \times c = a \times (b \times c)\).
Parentheses tell you which multiplication to do first. Try \(3 \times 2 \times 5\). Grouping as \((3 \times 2) \times 5 = 6 \times 5 = 30\), or as \(3 \times (2 \times 5) = 3 \times 10 = 30\). Both give 30, but the second is faster because it uses a ten. Look for pairs that make 10.
Two special facts are worth keeping in mind. Any number times 1 is that same number: \(8 \times 1 = 8\). Any number times 0 is 0: \(8 \times 0 = 0\), because zero groups, or groups with nothing in them, hold nothing.
Find \(4 \times 5 \times 2\) in the easiest way.
Group \(5 \times 2 = 10\) first. Then \(4 \times 10 = 40\). The product is 40.
7. Multiplying by multiples of 10
A multiple of 10 is a number like 20, 30, 40 or 90. To multiply by a multiple of 10, think in tens. The number 30 is 3 tens, so \(4 \times 30\) is 4 groups of 3 tens, which is 12 tens, or 120.
- Cover the 0 and say the basic fact: \(4 \times 3 = 12\).
- Put the 0 back: 12 tens is 120.
You are using the associative property: \(4 \times 30 = 4 \times (3 \times 10) = (4 \times 3) \times 10 = 12 \times 10 = 120\).
\(6 \times 50\) is 30 tens, which is 300, not 30. When the basic fact ends in 0, such as \(6 \times 5 = 30\), you get two zeros in the answer.
8. Patterns in the multiplication table
The table below shows every product from \(1 \times 1\) to \(10 \times 10\). Use it to spot patterns; patterns are a great way to check your answers.
- The table is a mirror along the diagonal, which is the commutative property at work. \(3 \times 8\) and \(8 \times 3\) both give 24.
- The blue diagonal shows the products of a number times itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Products in the row or column of 2 are always even. Products of 5 end in 0 or 5. Products of 10 end in 0.
- An even number times any whole number is even. An odd number times an odd number is odd.
- In the orange 9s, the tens digit goes up by 1 and the ones digit goes down by 1.
Is 47 a product of 5? Is 45?
Products of 5 end in 0 or 5. Since 47 ends in 7, it is not. Since 45 ends in 5, it is: \(5 \times 9 = 45\).
Key takeaways
- A multiplication fact tells how many in all when there are equal groups; the numbers multiplied are factors and the answer is the product.
- Facts of 2 are doubles, facts of 5 end in 0 or 5, and facts of 10 end in 0.
- Build facts of 3 and 4 from doubles, and facts of 6, 7, 8 and 9 from the 5s, doubling, or ten groups minus one.
- Commutative: \(a \times b = b \times a\). Associative: group three factors any way you like. Distributive: split one factor into easier parts.
- To multiply by a multiple of 10, do the basic fact first, then write the tens: \(4 \times 30 = 120\).
- Use patterns in the table, such as even products and the digits of the 9s, to check your answers.
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