8.2. Multiplication of Algebraic Expressions: Introduction

Look at the following patterns of dots.

Pattern of dots Total number of dots 4 × 9 5 × 7

To find the number of dots we have to multiply the expression for the number of rows by the m × n expression for the number of columns. Here the number of rows is increased by 2, i.e., m + 2 and number of columns increased by 3, i.e., n + 3.

This chunk introduces how multiplication can be visualized using patterns, like dots. For instance, if you have a rectangle made of dots, the total number of dots can be found by multiplying the number of rows (m) by the number of columns (n). Here, if we increase the number of rows by 2 and the columns by 3, we update the expressions to (m + 2) and (n + 3) respectively to represent this change.

Imagine you are arranging chairs in a hall for an event. If there are 4 rows of chairs with 9 chairs in each row, you simply multiply 4 by 9 to find a total of 36 chairs. Now, if two more rows are added and three more chairs in each existing row, you can see it’s like expanding your seating arrangement.

Can you now think of similar other situations in which two algebraic expressions have to be multiplied? Ameena gets up. She says, “We can think of area of a rectangle.” The area of a rectangle is l × b, where l is the length, and b is breadth. If the length of the rectangle is increased by 5 units, i.e., (l + 5) and breadth is decreased by 3 units, i.e., (b – 3), then the area of the new rectangle will be (l + 5) × (b – 3).

Here, the concept of multiplying algebraic expressions is connected to a concrete example: the area of a rectangle. The area is calculated by multiplying length (l) by breadth (b). If these dimensions change, such as if the length increases and breadth decreases, we adapt our multiplication expressions to account for these new dimensions, exemplified by the algebraic expression (l + 5) × (b – 3).

Consider a garden where you originally plan to plant flowers in a space measuring 4 meters long and 3 meters wide. If you decide to extend the length by 5 meters and decrease the width by 3 meters, the area calculation now needs to involve (4 + 5) and (3 - 3), allowing you to see how the layout changes with dimensions.

Can you think about volume? (The volume of a rectangular box is given by the product of its length, breadth and height). Sarita points out that when we buy things, we have to carry out multiplication. For example, if price of bananas per dozen = p and for the school picnic bananas needed =z dozens, then we have to pay = p × z. Suppose, the price per dozen was less by 2 and the bananas needed were less by 4 dozens. Then, price of bananas per dozen = (p – 2) and bananas needed =(z – 4) dozens. Therefore, we would have to pay = ` (p – 2) × (z – 4).

This section emphasizes multiplication in everyday life scenarios such as calculating the volume of a box or the total cost of items purchased. When calculating volume, the product of length, breadth, and height gives you the total space within a box. Likewise, for purchases, if you know the price per dozen bananas and how many dozens you want, you can use multiplication to find your total cost. Adjusting for changes in price and quantity also translates into a multiplication format of algebraic expressions.

Imagine you're hosting a picnic and need to buy bananas. If each dozen costs 3 dollars and you plan to buy 6 dozens, you quickly calculate the total cost as 3 × 6 = 18 dollars. If the price drops by 1 dollar and you decide on buying only 4 dozens, you would set up your new cost calculation as (3 - 1) × (6 - 2). This shows how changes affect your multiplication and pricing situation.

In all the above examples, we had to carry out multiplication of two or more quantities. If the quantities are given by algebraic expressions, we need to find their product. This means that we should know how to obtain this product. Let us do this systematically. To begin with we shall look at the multiplication of two monomials.

This final chunk summarizes that many common situations require multiplication of quantities, often expressed as algebraic expressions. The objective is to understand how to find the product step by step. The content indicates a shift toward learning about specific methods of multiplication, starting with monomials, which are algebraic expressions that contain a single term.

Think of baking where you often need to multiply quantities of ingredients. If a recipe calls for 2 cups of sugar and you want to double it, you mathematically set up 2 × 2 which naturally leads to needing 4 cups. This same logic applies to variables and expressions in math, forming the basis of algebraic multiplication.

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