Multiplication (2)
In this topic you will
- Find connections between the $1\times$, $2\times$, $5\times$ and $10\times$ multiplication tables.
- Connect multiplying by $2$ with doubling.
- Recognise that number facts on either side of the $=$ sign have the same value.
Key Words
- product
Show Definitions
- product: The answer you get when you multiply two or more numbers together.
Multiplication tables are lists of facts about one number. They are very useful to know when working with equal groups in any situation.
You can use one multiplication fact to find another, doubling any fact about $5\times$ to find a $10\times$ fact.
EXERCISES
$1.$ Write the multiplication sentences.
The first one has been done for you.
Double: $5 \times 3 = 15 \;\rightarrow\; 10 \times 3 = 30$
Half of: $10 \times 4 = 40 \;\rightarrow\;$ _______________________
Double: $5 \times 8 = 40 \;\rightarrow\;$ _______________________
Half of: $10 \times 7 = 70 \;\rightarrow\;$ _______________________
👀 Show answer
Half of: $10 \times 4 = 40 \;\rightarrow\; 5 \times 4 = 20$
Double: $5 \times 8 = 40 \;\rightarrow\; 10 \times 8 = 80$
Half of: $10 \times 7 = 70 \;\rightarrow\; 5 \times 7 = 35$
$2.$ Marcus used $8$ hands to make two different multiplication facts from the multiplication table for $5$. What could those facts be?
Sofia also used $8$ hands, but she made two different multiplication facts for the multiplication table for $10$. What could those facts be?
👀 Show answer
Marcus (table for $5$): using $8$ hands means $8$ groups of $5$. Two facts could be $8 \times 5 = 40$ and $5 \times 8 = 40$.
Sofia (table for $10$): the hands are shown in pairs, so $8$ hands make $4$ pairs, and each pair is $10$. Two facts could be $4 \times 10 = 40$ and $10 \times 4 = 40$.
EXERCISES
$3.$ Use the connection between doubling and multiplying by $2$ to find the missing facts.
| Multiplying by $2$ | Doubling |
|---|---|
| __________ | $2 + 2 = 4$ |
| $5 \times 2 = 10$ | __________ |
| __________ | $1 + 1 = 2$ |
| $10 \times 2 = 20$ | __________ |
👀 Show answer
Missing fact: $2 \times 2 = 4$
Missing fact: $5 + 5 = 10$
Missing fact: $1 \times 2 = 2$
Missing fact: $10 + 10 = 20$
$4.$ The equal product machine makes equivalent multiplication calculations. What calculation might come out of the machine?
$5 \times 4 = 20$
$2 \times 3 = 6$
$10 \times 3 = 30$
Write your three equivalent facts.
__________ $\times$ __________ $=$ __________ $\times$ __________
__________ $\times$ __________ $=$ __________ $\times$ __________
__________ $\times$ __________ $=$ __________ $\times$ __________
👀 Show answer
$5 \times 4 = 20$ could become $10 \times 2 = 20$.
$2 \times 3 = 6$ could become $1 \times 6 = 6$.
$10 \times 3 = 30$ could become $5 \times 6 = 30$.
$5.$ The teacher pointed to this place on the counting stick. Which multiplication facts could this represent?
👀 Show answer
The arrow points to $8$, so it could represent $4 \times 2 = 8$.
It could also represent $2 \times 4 = 8$.
$6.$ Which pair of equivalent multiplication facts do these cubes represent?
👀 Show answer
The cubes show $4$ groups of $2$, so one fact is $4 \times 2 = 8$.
The equivalent fact is $2 \times 4 = 8$.
Think like a Mathematician
Let’s investigate
The products in some multiplication tables have the pattern odd, even, odd, even. Others have only even products. Why is that?
Method:
- Write the first $10$ products in the $3$ times table.
- Circle the odd products and underline the even products.
- Repeat steps $1$–$2$ for the $4$ times table.
- Repeat for one more table of your choice.
- Compare your patterns and look for a rule linking the parity (odd/even) of the number being multiplied to the parity of the product.
Follow-up Questions:
Show Answers
- $1:$ A table alternates odd, even, odd, even when the number you are multiplying by is odd (for example, the $3$ times table). As you multiply by $1,2,3,4,\dots$, the multiplier switches between odd and even, so the products switch too.
- $2:$ A table has only even products when the number you are multiplying by is even (for example, the $4$ times table). Every product includes an even factor, so the product must be even.
- $3:$ The key rules are: $\text{odd}\times\text{odd}=\text{odd}$, $\text{odd}\times\text{even}=\text{even}$, and $\text{even}\times\text{anything}=\text{even}$. So if the table number is odd, the product depends on whether the multiplier is odd or even (giving an alternating pattern). If the table number is even, every product is even.