Multiplication
When you combine equal groups of objects, multiplication gives you the total number of objects. Multiplication is equivalent to repeatedly adding a number (the multiplicand) a certain number of times. Both these figures are referred to as factors.
- For example, 4 multiplied by 3 (often written as \(4\times3\) and spoken as “4 times 3”) can be calculated by adding 3 four times.
\(4\times3\) = 3 + 3 + 3 + 3 = 12
The factors are 4 (the multiplier) and 3 (the multiplicand), and the product is 12.
What are models?
A model is a grid that helps in the multiplication and division of decimals and whole numbers. It’s a grid with 10 columns and 10 rows. So in total, we have 100 boxes, which means each box is \((\frac{1}{100})\)or 1 hundredth, and each row or column is a tenth.
Multiplication of decimals using models
If we have two decimal values to be inserted in the grid, we can shade the row with one decimal and the column with the other decimal.
The result of the multiplication of decimal by decimal is represented in the grid by the overlapping shaded region. The grid below represents the multiplication of 0.2 x 0.2 = 0.4. As we can see, 4 grids overlap. So, the result will be \(\frac{4}{10}\) = 0.4.
Example 1: Find the product of 0.5 x 0.8 using models.
Solution:
We can write 0.5 x 0.8 as \(\frac{5}{10}\) x \(\frac{8}{10}\), that is 5 tenths x 8 tenths.
Each column is a tenth, so we first shade 5 columns.
Each row is also a tenth. So, we have to shade 8 rows in the same grid.
Now count the number of overlapped boxes. There are 40 overlapped grids in the model shown above, which represents the product.
Overlapped grid = 40 x \(\frac{1}{100}\) = 0.40
So, the final result = 0.5 x 0.8 = 0.40.
Example 2: Find the product of 0.6 x 0.3 using models.
Solution:
We can write 0.6 x 0.3 as \(\frac{6}{10}\) x \(\frac{3}{10}\) or 6 tenths x 3 tenths.
Now, we have to shade 6 columns, as each column is a tenth.
We have to shade 3 rows in the same grid. Each row is also a tenth.
As there are 18 overlapped grids in the above model,
overlapped grid = 18 x \(\frac{1}{100}\) = 0.18.
So, the final result = 0.6 x 0.3 = 0.18.
Multiplication of whole numbers and decimals using models
The decimal is first represented either by a row or a column. Then the shading of the same number of grids is repeated as many times as the whole number. The product is the total number of boxes or grids shaded.
Example: Find the product of 2 x 0.3 using models.
Solution:
We can write 2 x 0.3 as 2 x \(\frac{3}{10}\) or 2 x 3 tenths.
Now, we have to shade 3 columns 2 times.
As there are 60 shaded grids in the above shown model,
shaded grid = 60 x \(\frac{1}{100}\) = 0.6.
So, the final result = 2 x 0.3 = 0.6.
Real-life examples
Example 1: The heart of a whale beats once every 0.06 seconds.
How long does it take for its heart to beat 9 times?
Solution:
We have to multiply the number of beats by 0.06 to get the time for 9 heartbeats.
Use the grid model and shade 0.06 or 6 tenths 9 times.
The total number of grids shaded is 54. Therefore, 54 x \(\frac{1}{100}\) = 0.54.
So, \(9\times 0.06=0.54\)
It takes 0.54 seconds for the whale’s heart to beat 9 times.
Example 2: The height of a plant is 0.8 metres. A tree is 1.9 times the height of that plant. How much taller is the tree than the plant?
Solution:
We have to multiply the height of the plant by 1.9 to get the height of the tree.
1.9 means that there are 1 ones and 9 tenths. Each grid or box is \(\frac{1}{100}\) to get a one we need to shade the entire model
100 x \(\frac{1}{100}\) = 1.
Draw a grid model of 20 x 10 to carry out the given multiplication.
So, we shade 1 entire model and 9 columns from the next 10-by-10 grid to represent 1.9.
0.8 is represented by shading 8 rows of the model.
There are 152 squares that are shaded in the above grid.
So, \(\frac{152}{100}\) = 1.52
Therefore, the height of the tree is 1.9 x 0.8 = 1.52 meters.
Subtract the height of the plant from the height of the tree = 1.52 – 0.8 = 0.72
So, the tree is 0.72 meters taller than the plant.
If you want to multiply a decimal by a whole number, simply leave out the decimal point and multiply as usual. Count the number of decimal places in the decimal. Then, for the answer, reintroduce the decimal point after those many decimal places, counting from right to left.
It is a simple and creative way to teach students about the multiplication and division of decimals and whole numbers.