Substitutions in expressions

Any number in an expression can be replaced by the same number but written in a different form. Take the following expression, which has already been calculated, as an example:

15 + 3 = 18

Let's replace the number 15 with itself, but written in a different form:

(10 + 5) + 3 = 18

You can see that we replaced the number 15 with the expression in parentheses (10 + 5). But the main expression 15 + 3 = 18 has not changed because 15 and (10 + 5) are the same. After all, 10 + 5 = 15.

Let's replace the number 18 with itself, but written in a different form:

(10 + 5) + 3 = 3 × 6

Now replace the last 6 with the 6 itself, but written in a different form again:

(10 + 5) + 3 = 3 × 2 × 3

Now let's compare two expressions: the first one we had and the new one we modified:

15 + 3 = 18

(10 + 5) + 3 = 3 × 2 × 3

At first glance, it would seem that these are two different expressions. And that is what anyone who sees these two expressions for the first time will think. But we know that they are the same expression. The only difference is that we have modified some of its parameters.

You can change the expression indefinitely. The main thing is not to break equality. The equal sign (=) must justify its position. Remember the second lesson? An equal sign is placed between numbers or expressions only when they are equal to each other.

Such operations, where one number or expression is replaced by itself, but written in a different form, are called transformations or representations.

Representation as a sum

Any number or expression can be represented as a sum. For example, number 10 can be represented as a sum of 5+5 or 7+3 or 8+2. As long as there is equality between the number and the sum presented. This might look like this:

10 = 5 + 5

10 = 7 + 3

10 = 8 + 2

10 = 6 + 4

In books you can find the following tasks: represent as a sum, and then there are numbers or expressions that should be represented as a sum. This is exactly the case when you need to use your imagination and decide which numbers (or expressions) to use in order to complete the task.

Representation as a subtraction

We know from past lessons that a subtraction is the result of subtracting one number from another. But it is also an expression that is connected by a subtraction sign (-). For example, the following expressions are differences:

15 – 5

10 – 6

20 – 10

Any number can be represented as a subtraction. For example, the number 50 can be represented as a difference of 90-40 or 80-30 or 60-10. As long as there is equality between the number 50 and the presented difference. This might look like this:

50 = 90 − 40

50 = 80 − 30

50 = 60 − 10

Representation as a product

We know from past lessons that a product is the result of multiplying one number by another. But a product is also an expression that is joined by a multiplication sign (×). For example, the following expressions are products:

3 × 2

15 × 2

12 × 3

Any number can be represented as a product. For example, the number 30 can be represented as a product of 5×6 or 10×3 or 15×2. As long as there is equality between the number 30 and the product represented. This may look as follows:

30 = 5 × 6

30 = 10 × 3

30 = 15 × 2

Presentation as a division

We know from past lessons that a division is the result of dividing one number by another. But a division is also an expression that is joined by a division sign (÷). For example, the following expressions are divisions:

15 ÷ 5

30 ÷ 6

12 ÷ 4

Any number can be represented as a division. For example, number 5 can be represented as a division of 15÷3 or 25÷5 or 30÷6. As long as there is equality between the number 5 and the presented division. This could look like this:

5 = 15 ÷ 3

5 = 25 ÷ 5

5 = 30 ÷ 6

To practice with the material, try the following exercises:

Task 1. Express the following numbers as a sum: 20, 30, 45, 50. You may represent any number. For example, you could represent the first number 20 as 15 + 5.

Task 2. Express the following numbers as a difference: 10, 15, 12, 5 Can be represented by any numbers. For example, you could represent the first number as 15 - 5.

Task 3. Express the following numbers as a product: 30, 40, 72.

Task 4. Express the following numbers as a division: 7, 5, 9, 3

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