Topic: Algebra

Today’s lesson: variables

By the end of this lesson, you will be able to:

  • Define pronumerals and variables
  • Use algebraic conventions correctly
  • Simplify algebraic expressions using the four operations

Pronumerals

Copy this definition:

A pronumeral is a letter used to represent a number.

Examples:

  • In the expression \(5a\), the letter \(a\) is a pronumeral
  • In the formula \(P = 2l + 2w\), the letters \(P\), \(l\), and \(w\) are pronumerals

Variables

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A variable is a type of pronumeral where the letter stands for a number that can change.

Example:

  • In \(y = 3x + 2\), both \(x\) and \(y\) are variables
  • As \(x\) changes, \(y\) also changes

Algebraic Conventions

Copy these rules:

  1. Multiplication: Write \(4p\) instead of \(4 \times p\)
  2. Coefficient of 1: Write \(x\) instead of \(1x\)
  3. Order: Write the number before the letter: \(7m\) not \(m7\)
  4. Division: Write \(\frac{a}{3}\) instead of \(a \div 3\)

Algebra writing conventions

Copy these examples:

Instead of writing: We write:
\(3 \times n\) \(3n\)
\(1 \times y\) \(y\)
\(p \times 5\) \(5p\)
\(a \div 4\) \(\frac{a}{4}\)
\(2 \times x \times 3\) \(6x\)

Worked Example 1

Attempt first, then we’ll review:

Rewrite using algebraic conventions:

  1. \(6 \times m\)

  2. \(1 \times k\)

  1. \(t \times 9\)

  2. \(b \div 7\)

Worked Example 2

Attempt first, then we’ll review:

Rewrite using algebraic conventions:

  1. \(4 \times p \times 2\)
  2. \(5 \times 1 \times n\)
  3. \(w \times 3 \times x\)
  4. \(2 \times a \div 5\)

Your turn

Attempt these, then we’ll review:

Rewrite using correct algebraic conventions:

  1. \(8 \times q\)
  2. \(1 \times z\)
  3. \(r \times 12\)
  4. \(c \div 6\)
  5. \(3 \times d \times 4\)

Simplifying Expressions

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When we simplify an algebraic expression, we make it as simple as possible by:

  • Combining numbers
  • Following the order of operations
  • Using algebraic conventions

For example, \(3b+2b = 5b\). \(5b\) is the simplified form.

Simplifying: Addition

Attempt first, then we’ll review:

Simplify:

  1. \(3a + 5a\)
  2. \(7p + 2p\)
  3. \(m + 6m\)
  4. \(4x + x + 2x\)

Simplifying: Subtraction

Attempt first, then we’ll review:

Simplify:

  1. \(8b - 3b\)
  2. \(10n - n\)
  3. \(5y - 2y\)
  4. \(9k - 4k - k\)

Simplifying: Multiplication

Attempt first, then we’ll review:

Simplify:

  1. \(3 \times 4a\)
  2. \(5 \times 2m\)
  3. \(6p \times 3\)
  4. \(2n \times 4 \times 3\)

Simplifying: Division

Attempt first, then we’ll review:

Simplify:

  1. \(\dfrac{12x}{3}\)
  2. \(\dfrac{20a}{4}\)
  3. \(15m \div 5\)
  4. \(8p \div 2\)

Complete Exercise 3.01

  • all questions
  • odd numbered subparts

Note: Some questions ask you to find the expanded form. This is the opposite of simplifying.

Example:

  1. \(5bc = 5 \times b \times c\)

  2. \(-4kr^2 = -4 \times k \times r \times r\)

Mixed Practice

Attempt these, then we’ll review:

Simplify:

  1. \(4x + 3x\)
  2. \(7a - 2a\)
  3. \(5 \times 3b\)
  4. \(\frac{18n}{6}\)
  5. \(2m \times 4\)

Challenge Questions

Attempt these, then we’ll review:

Simplify:

  1. \(6p + 4p - 2p\)
  2. \(3 \times 2x \times 5\)
  3. \(\frac{24a}{8} + 2a\)
  4. \(5n - n + 3n\)

Summary

What we learned today:

  • Pronumerals are letters representing numbers
  • Variables are pronumerals that can change
  • Algebraic conventions make expressions simpler
  • We can simplify expressions using the four operations

Next Lesson

Coming up:

  • Collecting like terms
  • More complex algebraic expressions