Important Formulae And Points To Remember Expressions Using Letter Numbers - Class 7 Mathematics (Ganita Prakash) | Complete Learning Material PDF

Table of Contents
1. Concept of Letter-Numbers
2. Writing Expressions from Situations
3. Common Arithmetic Expression Properties
4. Evaluating Expressions
5. Writing Your Own Expressions
6. Omitting Multiplication Sign
7. Sequences and nth Term Formula
8. Perimeter of a Rectangle
9. Like and Unlike Terms
10. Distributive Property in Use
11. Non-equivalent Expressions
12. Common Mistakes in Simplification
13. Number Machines (Input-Output Rules)
14. Patterns in a Calendar (2 × 2 Squares)
15. Matchstick Patterns
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1. Concept of Letter-Numbers

• Letter-numbers are letters used to represent unknown or variable values (e.g., a, n, x).

• Algebraic expressions are combinations of numbers and letter-numbers, joined by operations (+, -, ×, ÷).

2. Writing Expressions from Situations

3. Common Arithmetic Expression Properties

• Swapping (Commutative Property): Order of addition doesn't affect the result:
  a + b = b + a

• Grouping (Associative Property): Regrouping terms doesn't change the sum:
  (a + b) + c = a + (b + c)

• Distributive Property:
  a × (b + c) = ab + ac

4. Evaluating Expressions

• Replace letter-numbers with given values.

• Follow BODMAS (Brackets, Orders, Division/Multiplication, Addition, Subtraction).

Example:
If a = 23, then a + 3 = 23 + 3 = 26.

5. Writing Your Own Expressions

6. Omitting Multiplication Sign

• 4 × n is written as 4n
• Always write the number before the letter:

7. Sequences and nth Term Formula

What is a Sequence?

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term.

Example Sequence:

4, 8, 12, 16, ...

This sequence increases by 4 each time, so it's called an arithmetic sequence (a sequence where each term increases or decreases by the same fixed number).

For a sequence like: 4, 8, 12, 16... (multiples of 4) 

n^th term = 4n

8. Perimeter of a Rectangle

Expression:
p = l + b + l + b 

p = 2l + 2b 
(where $ll$l = length, $bb$b = breadth)

9. Like and Unlike Terms

• Like Terms: Same letter-numbers (e.g., 5c, 3c)

• Unlike Terms: Different letter-numbers (e.g.,  5c, 11d)

Rule: Only like terms can be added or subtracted.

10. Distributive Property in Use

Example: 4 (x+y) -y 
= 4x + 4y - y 
= 4x+3y

11. Non-equivalent Expressions

Sometimes, two expressions look similar but are not equal because they follow different mathematical rules or structures.

To check if two expressions are equivalent, we can:

• Simplify both expressions, or

• Substitute a value into each and compare the results

Exmaple: Check whether 5u and 5 + u are equal or not 

These two expressions look alike, but they're not the same.

Let's substitute u = 2:

• 5u = 5 × 2 = 10

• 5 + u = 5 + 2 = 7

Since 10 ≠ 7,
5u and 5 + u are not equivalent.​

12. Common Mistakes in Simplification

13. Number Machines (Input-Output Rules)

• A formula or expression represents the operation performed on two inputs.

• Example: If the machine uses inputs a and b, and performs 2a - b, then:
  Output=2a-b

• Practice finding formulas for different number machines.

14. Patterns in a Calendar (2 × 2 Squares)

In any 2 × 2 block on a calendar:

$Top-left number\mathrm{<br><br>}\backslash text\{Top-left\; number\}\; =\; a$Top-left number = a

Other positions:

• Top-right: $a\; +\; 1$a + 1

• Bottom-left: $a\; +\; 7$a + 7

• Bottom-right: a + 8

• Diagonal Sums:
    a + (a + 8) = (a + 1) + (a + 7) = 2a + 8

15. Matchstick Patterns

Pattern of Triangles using Matchsticks:

• Step 1:  3 matchsticks

• Each new triangle shares 1 side with the previous triangle, so you only add 2 new matchsticks each time.

• To find the number of matchsticks needed for any step y, use the formula:

  Matchsticks at Step y = 2y + 1