One way of thinking of multiplication is as repeated addition. Multiplicative situations arise when finding a total of a number of collections or measurements of equal size. Arrays are a good way to illustrate this. Some division problems arise when we try to break up a quantity into groups of equal size and when we try to undo multiplications.
Step-by-step explanation:
If we had 35 biscuits and wanted to share them equally amongst the family of 7, we would use sharing to distribute the biscuits equally into 7 groups.
We can write down statements showing these situations:
A farmer is filling baskets of apples. The farmer has 24 apples and 4 baskets. If she divides them equally, how many apples will she put in each basket?
twenty-four divided by four equals six
When you divide to find the number of groups, the division is called measuring or repeated subtraction. It is easy to see that you can keep subtracting 4 from 24 until you reach zero. Each 4 you subtract is a group or basket.
A farmer has 24 apples. She wants to sell them at 4 apples for $1.00. How many baskets of 4 can she fill?
twenty-four divided by four equals six
Manipulatives and visual aids are important when teaching multiplication and division. Students have used arrays to illustrate the multiplication process. Arrays can also illustrate division.
Three circled rows of four dots: twelve divided by four equals 3Four circled columns of three dots: twelve divided by three equals four
Since division is the inverse, or opposite, of multiplication, you can use arrays to help students understand how multiplication and division are related. If in multiplication we find the product of two factors, in division we find the missing factor if the other factor and the product are known.
In the multiplication model below, you multiply to find the number of counters in all. In the division model you divide to find the number of counters in each group. The same three numbers are used. The model shows that division “undoes” multiplication and multiplication “undoes” division. So when multiplying or dividing, students can use a fact from the inverse operation. For example, since you know that 4 x 5 = 20, you also know the related division fact 20 ÷ 4 = 5 or 20 ÷ 5 = 4. Students can also check their work by using the inverse operation.
inverse operation
Notice that the numbers in multiplication and division sentences have special names. In multiplication the numbers you multiply are called factors; the answer is called the product. In division the number being divided is the dividend, the number that divides it is the divisor, and the answer is the quotient. Discuss the relationship of these numbers as you explain how multiplication and division are related.
There are other models your students can use to explore the relationship between multiplication and division. Expose your students to the different models and let each student choose which model is most helpful to him or her. Here is an example using counters to multiply and divide.
Four circled groups of three dots
factor
4
number of
groups x factor
3
counters in
each group = product
12
total number of
counters
Three circled groups of four dots
dividend
12
total number
of counters ÷ divisor
4
number of
groups = quotient
3
counters in
each group
Here is an example using a number line.
number line
factor
4 x factor
5 = product
20
number line
dividend
20 ÷ divisor
5 = quotient
4
Another strategy your students may find helpful is using a related multiplication fact to divide. The lesson Relating Multiplication and Division focuses on this strategy. Here is an example.
18 ÷ 6 = ?
Think: 6 x ? = 18 Six times what number is 18?
6 x 3 = 18,
so 18 ÷ 6 = 3.
When students understand the concept of division, they can proceed to explore the rules for dividing with 0 and 1. Lead students to discover the rules themselves by having them use counters to model the division. A few examples follow.
Divide 4 counters into 4 groups.
four circled dots
4 ÷ 4 = 1
Divide 2 counters into 2 groups.
two circled dots
2 ÷ 2 = 1
When any number except 0 is divided by itself, the quotient is 1.
Put 3 counters in 1 group. Put 5 counters in 1 group.
Three dots in a circle five dots in a circle
3 ÷ 1 = 3 5 ÷ 1 = 5
When any number is divided by 1, the quotient is that number.
Divide 0 counters into 2 groups.
two circles
0 ÷ 2 = 0
Divide 0 counters into 4 groups.
four circles
0 ÷ 4 = 0
When 0 is divided by any number except 0, the quotient is 0.
Divide 6 counters into 0 groups. Divide 1 counter into 0 groups.
You cannot divide a number by 0.
Encourage students to think about the relationship between multiplication and division when they solve problems. For example, they can use a related multiplication fact to find the unit cost of an item—for example the cost of one baseball cap priced at 3 for $18.
Answers & Comments
Answer:
One way of thinking of multiplication is as repeated addition. Multiplicative situations arise when finding a total of a number of collections or measurements of equal size. Arrays are a good way to illustrate this. Some division problems arise when we try to break up a quantity into groups of equal size and when we try to undo multiplications.
Step-by-step explanation:
If we had 35 biscuits and wanted to share them equally amongst the family of 7, we would use sharing to distribute the biscuits equally into 7 groups.
We can write down statements showing these situations:
7 × 5 = 35 and 5 × 7 = 35
Also,
35 ÷ 5 = 7 and 35 ÷ 7 = 5
Answer:
For example:
A farmer is filling baskets of apples. The farmer has 24 apples and 4 baskets. If she divides them equally, how many apples will she put in each basket?
twenty-four divided by four equals six
When you divide to find the number of groups, the division is called measuring or repeated subtraction. It is easy to see that you can keep subtracting 4 from 24 until you reach zero. Each 4 you subtract is a group or basket.
A farmer has 24 apples. She wants to sell them at 4 apples for $1.00. How many baskets of 4 can she fill?
twenty-four divided by four equals six
Manipulatives and visual aids are important when teaching multiplication and division. Students have used arrays to illustrate the multiplication process. Arrays can also illustrate division.
Three circled rows of four dots: twelve divided by four equals 3Four circled columns of three dots: twelve divided by three equals four
Since division is the inverse, or opposite, of multiplication, you can use arrays to help students understand how multiplication and division are related. If in multiplication we find the product of two factors, in division we find the missing factor if the other factor and the product are known.
In the multiplication model below, you multiply to find the number of counters in all. In the division model you divide to find the number of counters in each group. The same three numbers are used. The model shows that division “undoes” multiplication and multiplication “undoes” division. So when multiplying or dividing, students can use a fact from the inverse operation. For example, since you know that 4 x 5 = 20, you also know the related division fact 20 ÷ 4 = 5 or 20 ÷ 5 = 4. Students can also check their work by using the inverse operation.
inverse operation
Notice that the numbers in multiplication and division sentences have special names. In multiplication the numbers you multiply are called factors; the answer is called the product. In division the number being divided is the dividend, the number that divides it is the divisor, and the answer is the quotient. Discuss the relationship of these numbers as you explain how multiplication and division are related.
There are other models your students can use to explore the relationship between multiplication and division. Expose your students to the different models and let each student choose which model is most helpful to him or her. Here is an example using counters to multiply and divide.
Four circled groups of three dots
factor
4
number of
groups x factor
3
counters in
each group = product
12
total number of
counters
Three circled groups of four dots
dividend
12
total number
of counters ÷ divisor
4
number of
groups = quotient
3
counters in
each group
Here is an example using a number line.
number line
factor
4 x factor
5 = product
20
number line
dividend
20 ÷ divisor
5 = quotient
4
Another strategy your students may find helpful is using a related multiplication fact to divide. The lesson Relating Multiplication and Division focuses on this strategy. Here is an example.
18 ÷ 6 = ?
Think: 6 x ? = 18 Six times what number is 18?
6 x 3 = 18,
so 18 ÷ 6 = 3.
When students understand the concept of division, they can proceed to explore the rules for dividing with 0 and 1. Lead students to discover the rules themselves by having them use counters to model the division. A few examples follow.
Divide 4 counters into 4 groups.
four circled dots
4 ÷ 4 = 1
Divide 2 counters into 2 groups.
two circled dots
2 ÷ 2 = 1
When any number except 0 is divided by itself, the quotient is 1.
Put 3 counters in 1 group. Put 5 counters in 1 group.
Three dots in a circle five dots in a circle
3 ÷ 1 = 3 5 ÷ 1 = 5
When any number is divided by 1, the quotient is that number.
Divide 0 counters into 2 groups.
two circles
0 ÷ 2 = 0
Divide 0 counters into 4 groups.
four circles
0 ÷ 4 = 0
When 0 is divided by any number except 0, the quotient is 0.
Divide 6 counters into 0 groups. Divide 1 counter into 0 groups.
You cannot divide a number by 0.
Encourage students to think about the relationship between multiplication and division when they solve problems. For example, they can use a related multiplication fact to find the unit cost of an item—for example the cost of one baseball cap priced at 3 for $18.
$18 ÷ 3 = $6
Think: 3 x ? = $18
3 x $6 = $18
So $18 ÷ 3 = ?
The cost is $6 for one baseball