Maths Refresher Simplifying Equations, Slides of Mathematics

Download Maths Refresher Simplifying Equations and more Slides Mathematics in PDF only on Docsity!

Maths Refresher

Simplifying Equations

Learning intentions ….

• Algebra
• Order of operations
• Commutative property
• Associative Property
• Distributive property
• Simplify with grouping symbols

Simplifying Equations

Glossary

• Equation : Is a mathematical sentence. It contains an equal sign meaning that both sides are equivalent.
• Expression : An algebraic expression involves numbers, operation signs, brackets/parenthesis and pronumerals that substitute numbers.
• Operator : The operation (+ , − ,× ,÷) which separates the terms.
• Term : Parts of an expression separated by operators.
• Pronumeral: A symbol that stands for a particular value.
• Variable : A letter which represents an unknown number. Most common is 𝑥𝑥, but it can be any symbol.
• Constant : Terms that contain only numbers that always have the same value.
• Coefficient : Is a number that is partnered with a variable. Between the coefficient and the variable is a multiplication. Coefficients of 1 are not shown.

Glossary

Some algebra rules …

Expressions with zeros and ones

• Zeros and ones can be eliminated, why:
• When we add zero it does not change the number,

• If we multiply by one, then the number stays the

same, for example: 𝑥𝑥 × 1 = 𝑥𝑥

• What we do to one side we do to the other
• …and the BODMAS rule

Order of Operations

Revision: Example 1

Revision: Example 2

Revision: Example 3

50 − 3 × 2 × 5 =

50 − 6 × 5 =

10 + 2 − 3 × 4 =

32 ÷ 2 − 2 × 3 =

16 − 2 × 3 =

Answers

Some algebra rules

• Multiplicative Property: 𝟏𝟏 × 𝒙𝒙 = 𝒙𝒙
  • Multiplying any number by one makes no difference.
• Additive Inverse: 𝒙𝒙 + −𝒙𝒙 = 𝟎𝟎
  • Any number added to its negative equals zero.
• Multiplicative Inverse: 𝒙𝒙 × 𝟏𝟏𝒙𝒙 = 𝟏𝟏
  • Any number multiplied by its reciprocal equals one.
• Symmetric Property: 𝒙𝒙 = 𝒚𝒚 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒚𝒚 = 𝒙𝒙
  • Perfect harmony!
• Transitive Property: 𝑰𝑰𝑰𝑰 𝒙𝒙 = 𝒚𝒚 𝒂𝒂𝒕𝒕𝒂𝒂 𝒚𝒚 = 𝒛𝒛 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝒛𝒛
  • For example, if apples cost $2 and oranges cost $ then apples and oranges are the same price.

…continued

Consider:

a)What is changing in this

pattern

b) What is the repeating part?

c) What stays the same?

• Each time it grows by 3 sticks.
• The first stick remains the

same.

Number of elements or term

Total number of sticks

T 1 4

T 2 7

T 3 10

T 4 13

Tn y

…continued

• Each new term grows by three, so for each term
  • step in the pattern – another ‘group’ of three is added.
• However, there is always one matchstick that stays the same – ‘the constant’
• Therefore, the generalisation (general rule) or ‘algebraic equation’ for the matchstick pattern would be: - n x 3 + 1 = y - 3 n + 1 = y
• (The number of elements (the term) times three plus one = the total number of matchsticks)

Associative Property

• The Associative Law of Addition:

The order you add numbers does not matter. The

difference is that we ‘regroup’ the numbers

• The Associative Law of Multiplication:

𝒙𝒙 × 𝒚𝒚 × 𝒛𝒛 = 𝒙𝒙 × (𝒚𝒚 × 𝒛𝒛)

The order you multiply numbers does not matter.

The difference is that we ‘regroup’ the numbers,

whereas in commutative property the numbers are

moved around – not regrouped.

=

Distributive property

• The Distributive Law: multiplication distributes over addition or subtraction through the brackets (parentheses) 𝒙𝒙 𝒚𝒚 + 𝒛𝒛 = 𝒙𝒙𝒚𝒚 + 𝒙𝒙𝒛𝒛

For example, 2 3 + 4 = 2 × 3 + 2 × 4 2 7 = 6 + 8

14 = 14

Answers

• Rewrite 3 × 2 × 𝑥𝑥 by using the ‘commutative property’

𝟑𝟑 × 𝟐𝟐𝒙𝒙 𝒐𝒐𝒐𝒐 𝟐𝟐 × 𝟑𝟑𝒙𝒙 𝟔𝟔𝒙𝒙 (simplified).

• Rearrange 2(4𝑥𝑥) in using the ‘associative property’

8 × 𝒙𝒙 𝟖𝟖𝒙𝒙 (simplified)

• Rewrite 8(2 + 𝑥𝑥) using the ‘distributive property’

𝟖𝟖 × 𝟐𝟐 + (𝟖𝟖𝒙𝒙) 𝟏𝟏𝟔𝟔 + 𝟖𝟖𝒙𝒙 (simplified)

Collecting ‘like’ terms

Watch this short Khan Academy video for further explanation: “Combining like terms, but more complicated” https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-variables-expressions/cc-7th-manipulating-expressions/v/combining-like-terms-