This pdf includes the following topics:- Simplifying Equations What is algebra Equation Operator Associative Property Commutative property Distributive property Multiplication Division
1. Maths Refresher Simplifying Equations
2. Simplifying Equations Learning intentions …. • Algebra • Order of operations • Commutative property • Associative Property • Distributive property • Simplify with grouping symbols
3. What is algebra • Algebra has many definitions; some adults have grown up believing that algebra is somewhat scary and difficult. • Algebra involves finding and communicating number patterns and relationships • As number patterns become more complex they are more difficult to communicate verbally… • Hence, notation is used to simplify the task. • Algebra is actually a very useful and simple concept, the complex part is familiarising yourself with the language. • The simplest definition for algebra is: – A mathematical method for finding an unknown number.
4. • 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.
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6. Glossary example
7. 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, 𝑥𝑥 + 0 = 𝑥𝑥 • 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
11. 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. 𝟏𝟏 • Multiplicative Inverse: 𝒙𝒙 × = 𝟏𝟏 𝒙𝒙 • Any number multiplied by its reciprocal equals one. • Symmetric Property: 𝒙𝒙 = 𝒚𝒚 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒚𝒚 = 𝒙𝒙 • Perfect harmony! • Transitive Property: 𝑰𝑰𝑰𝑰 𝒙𝒙 = 𝒚𝒚 𝒂𝒂𝒂𝒂𝒂𝒂 𝒚𝒚 = 𝒛𝒛 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝒛𝒛 • For example, if apples cost $2 and oranges cost $2 then apples and oranges are the same price.
12. Understanding an algebraic Let’s investigate this pattern • The first thing we do is number each element, if you do not number the element you will not be able to see the relationship between the ‘element number’ (which is the ‘term’) and the ‘total number of sticks’ in each element. If you cannot see the relationship, how the pattern changes, you will not be able to work out a formula.
13. Consider: Number of Total number elements or of sticks a)What is changing in this term T1 4 b) What is the repeating part? c) What stays the same? T2 7 • Each time it grows by 3 sticks. T3 10 T4 13 • The first stick remains the Tn y same.
14. • 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: – nx3+1=y – 3n + 1 = y • (The number of elements (the term) times three plus one = the total number of matchsticks)
15. Commutative property • Think of the term ‘commutative’ in relation to being able to move things around – to commute. • Hence, Commutative Property is the property where we can move things around – The Commutative Law of Addition: 𝒙𝒙 + 𝒚𝒚 = 𝒚𝒚 + 𝒙𝒙 For example, 2+3 = 3+2 – The Commutative Law of Multiplication: 𝒙𝒙 × 𝒚𝒚 = 𝒚𝒚 × 𝒙𝒙 For example, 2 × 3 = 3 × 2
16. Associative Property • The Associative Law of Addition: 𝒙𝒙 + 𝒚𝒚 + 𝒛𝒛 = 𝒙𝒙 + (𝒚𝒚 + 𝒛𝒛) 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. =
17. Distributive property • The Distributive Law: multiplication distributes over addition or subtraction through the brackets (parentheses) 𝒙𝒙 𝒚𝒚 + 𝒛𝒛 = 𝒙𝒙𝒙𝒙 + 𝒙𝒙𝒙𝒙 For example, 2 3 + 4 = 2 × 3 + 2 × 4 2 7 =6+8 14 = 14
18. Your turn …
19. • Rewrite 3 × 2 × 𝑥𝑥 by using the ‘commutative property’ 𝟑𝟑 × 𝟐𝟐𝟐𝟐 𝒐𝒐𝒐𝒐 𝟐𝟐 × 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 (simplified). • Rearrange 2(4𝑥𝑥) in using the ‘associative property’ 8 × 𝒙𝒙 𝟖𝟖𝟖𝟖 (simplified) • Rewrite 8(2 + 𝑥𝑥) using the ‘distributive property’ 𝟖𝟖 × 𝟐𝟐 + (𝟖𝟖𝟖𝟖) 𝟏𝟏𝟏𝟏 + 𝟖𝟖𝟖𝟖 (simplified)
20. 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-3
21. Like terms Often real life algebra problems look like the following: • 7𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6𝑥𝑥 + 2 = 14 • It is difficult to even try to start solving a problem so large. What we need to do is simplify the problem into a smaller problem. We do this by collecting like terms. • A like term is a term which has the same variable to the same power only the coefficient is different. • Looking at the example: 7𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6𝑥𝑥 all coefficients have the same variable. • We can treat them as a simple equation: 7 + 2 + 3 − 6 which equals 6, so: 7𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6𝑥𝑥 = 6𝑥𝑥 ∴ 6𝑥𝑥 + 2 = 14
22. Like terms Watch this short Khan Academy video for further explanation: “Combining like terms and the distributive property” https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-variables-expressions/cc-7th-manipulating- expressions/v/combining-like-terms-and-the-distributive-property
23. Like terms Collect the like terms and simplify: 5𝑥𝑥 + 3𝑥𝑥𝑥𝑥 + 2𝑦𝑦 − 2𝑦𝑦𝑦𝑦 + 3𝑦𝑦 2 Step 1: Recognise the like terms: (𝑥𝑥𝑥𝑥 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑦𝑦𝑦𝑦 − 𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙) 5𝑥𝑥 + 3𝒙𝒙𝒙𝒙 + 2𝑦𝑦 − 2𝒚𝒚𝒚𝒚 + 3𝑦𝑦 2 Step 2: Arrange the expression so that the like terms are together (Remember to take the operator with the term) 5𝑥𝑥 + 2𝑦𝑦 + 3𝒙𝒙𝒙𝒙 − 2𝒚𝒚𝒚𝒚 + 3𝑦𝑦 2 Step 3: Simplify 5𝑥𝑥 + 2𝑦𝑦 + 𝒙𝒙𝒙𝒙 + 3𝑦𝑦 2
26. Simplify with grouping symbols Watch this short Khan Academy video for further explanation: “Factoring and the distributive property 2” https://www.khanacademy.org/math/algebra-basics/quadratics-polynomials-topic/Factoring-simple-expressions-core- algebra/v/factoring-and-the-distributive-property-2
27. Simplify with nested grouping When there are two sets of brackets – one is nested inside the other – operations in the inner set must be worked first. Watch this short Khan Academy video for further explanation: “Expression terms, factors and coefficients” https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-writing-expressions/v/expression- terms-factors-and-coefficients
31. Simplify with grouping symbols • Now let’s work with an algebraic expression with brackets and simplify by removing the brackets and including powers. • Recap: • So if we multiply two numbers together, the order in which we multiply is irrelevant – commutative property • Simplify 4 3𝑥𝑥 • This could be written as 4 × 3 × 𝑥𝑥 • And then as 4 × 3 × 𝑥𝑥 • Therefore, we can simplify to 12𝑥𝑥
32. Simplify with grouping symbols and exponents Another example incorporating exponents: • Simplify (3𝑥𝑥)(6𝑥𝑥) • 3 × 𝑥𝑥 × 6 × 𝑥𝑥 • We can change the order (3 × 6) × (𝑥𝑥 × 𝑥𝑥) • Therefore, we can simplify to 18𝑥𝑥 2
33. Simplify with grouping symbols and exponents • Remember we need to follow the order of operations rule BODMAS • …and now we also apply the ‘INDEX LAWS’ from the previous session
34. Simplify with grouping symbols and exponents Is 2𝑥𝑥 2 the same as (2𝑥𝑥)2 ? • 2𝑥𝑥 2 is 2 × 𝑥𝑥 × 𝑥𝑥 – 𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑤𝑤𝑤𝑤𝑤𝑤 5 – 2 × 5 × 5 = 50 • (2𝑥𝑥)2 is 2 × 𝑥𝑥 × 2 × 𝑥𝑥 – 𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑤𝑤𝑤𝑤𝑤𝑤 5 – 10 × 10 = 100
35. Simplify with grouping symbols and exponents This example explains how any order property can be used: 6𝑥𝑥 × 2𝑦𝑦 × 3𝑥𝑥𝑥𝑥 = 6 × 𝑥𝑥 × 2 × 𝑦𝑦 × 3 × 𝑥𝑥 × 𝑦𝑦 = 6 × 2 × 3 × 𝑥𝑥 × 𝑥𝑥 × 𝑦𝑦 × 𝑦𝑦 = 36𝑥𝑥 2 𝑦𝑦 2
36. Simplify with grouping symbols and exponents Simplify 5𝑥𝑥 2 × 6𝑥𝑥 5 • So we can say that we have – (5 × 6) × (𝑥𝑥 × 𝑥𝑥) × (𝑥𝑥 × 𝑥𝑥 × 𝑥𝑥 × 𝑥𝑥 × 𝑥𝑥) – Which is 30 × 𝑥𝑥 2+5 – Therefore, 30𝑥𝑥 7 (Remember the first index law from last week.)
37. Simplify with grouping symbols and exponents • Simplify – What is the difference – 6𝑥𝑥 5𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 6𝑥𝑥 + 5𝑥𝑥 30𝑥𝑥 2 11𝑥𝑥 (Here we add like terms) • One more, are these the same: – (−5𝑎𝑎2 ) −2𝑎𝑎 – 10𝑎𝑎3 • −5 × −2 = 10 • 𝑎𝑎2 × 𝑎𝑎1 = 𝑎𝑎3 (index law one) • Therefore, (−5𝑎𝑎2 ) −2𝑎𝑎 is the same as 10𝑎𝑎3
40. Simplify systematically Steps to follow which may assist your reasoning 1. Simplify expressions that have grouping symbols first and work from the innermost to the outer. As you do this apply the BODMAS rule too. 2. Simplify powers 3. Multiply in order from left to right 4. Add and subtract in order from left to right. 5. Then work backwards to check
41. Your turn …
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43. Your turn … Simplifying expression with grouping symbols and exponents …and a challenge: What is the missing number? 4( +3)2 2 =16 5(14−3 )
44. Challenge answer 4( +3)2 • =16 5(14−9) 4( +3)2 • =16 5(5) 4( +3)2 • =16 so what divided by 25 = 16, 25 × 16 = 400 25 • Now we look at what divided by 4 = 400 which is 100 • What is the square root of 100? 10 • So now we can say that the missing number is 7 • Work backwards to see if this is correct 4( 7 +3)2 • =16 5(14−9)
45. Simplifying Equations Reflect on the learning intentions …. • Algebra • Order of operations • Commutative property • Associative Property • Distributive property • Simplify with grouping symbols
46. Australian Mathematical Sciences Institute. (2011). Algebraic expressions. Retrieved from http://www.amsi.org.au/teacher_modules/pdfs/Algebraic_ expressions.pdf Purple Math. (2013). Associative, commutative and distributive properties. Retrieved from http://www.purplemath.com/modules/numbprop2.htm Muschla, J. A., Muschla, G. R., Muschla, E. (2011). The algebra teacher’s guide to reteaching essential concepts and skills. San Francisco: Jossey-Bass
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