Understanding Expressions and Equations
Explore the world of expressions and equations through constructing, expanding brackets, factorizing, and substituting into expressions. Learn about unknowns, like terms, and formulas. Gain insights into simplifying expressions and dealing with variables with powers. Dive into examples and get ready to master the language of mathematics.
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Presentation Transcript
EXPRESSIONS Chapter 2
OBJECTIVES: Learn about constructing expressions. Learn about expanding brackets. Learn about factorizing expressions. Learn about substituting into expressions.
EXPRESSIONS, FORMULAE AND EQUATIONS: Things to know: 1. An unknown is the part of the equation that you don t know the value of. It is usually represented by a variable such as ? . 2. Expressions have no equal sign such as 3? + 4 3. When you simplify expressions, you collect like terms. 4. Expressions consist of terms (in the expressions 2? + 4, 2? is a term where 2 is the coefficient of 2? and 4 is a term which is known as the constant term.) ? = ? ? is a formula. The letters are variables, and ? is the subject of the formula. It cannot be solved, it tells you the relationship between variables. ? = 2? + 3 is a formula and a linear function with two variables. If you know ? , you can find out the value of ? . 5. 6.
CONSTRUCTING EXPRESSIONS 1. Constructing expressions means expressing a word problem as an expression using variables. 2. Variables might be provided to you in the questions, otherwise you can use a variable of your choice. Example: if one pen costs $3 and I bought 4 pens, then the cost of 4 pens would be 3x4= $12. Similarly, if I have ? pens costing $3 each, then the total cost would be 3 x ? = 3?
P. 27 Ex. 2A: Q1 Q2 Q3 (c,d) Q4 Q5 Q6 Q8 Q9 Q10 Q11 Q12 (a) Q14
THINGS TO REMEMBER WHEN SIMPLIFYING EXPRESSIONS: 1. 2. 3. You can t add/subtract different terms (Ex: ? + ? stays the same) You can t add/subtract letters and numbers (Ex: ? + 3 stays the same) You can multiply letters and numbers (Ex: 3 ? = 3?) because it s the same as ? + ? + ? you can multiply different letters (Ex: ? ? = ??). Remember commutative law: ? x ? = ? x ?) You can multiply same letters (Ex: ? ? ? = ?3). When you Simplify ? ? x ? to ??, this is using index notation. 8. 9. a 10. Letters of the same power are grouped together (Ex: ?2+ ?2= 2?2and 3?3 ?3= 2?3). Answers should be written in order of power (highest power first). Https://www.youtube.com/watch?v=yigRz8b9eL0
FOR VARIABLES WITH POWERS ?3 ?2 ? Pretend that each power is a different color, you cannot mix colors together, only same color cards can be collected together.
EXAMPLES: 4?2 ?2 ?2 ?2 ?2 ?3+ 2?2+ 3? ?2 ?2 ?3 ? ? ? ?3 ?3 ?3 ?3 ?3 ?3 ?2?2?2 6?3+ 3?2+ 2? ? ?
SIMPLIFY THE EXAMPLES BELOW: * Ex 5: ?2 + 2? + 2?2 + 3? * Ex 1: ? + ? 2? + 2? = 3?2+5? = ? + 3? * Ex 2: 2? + ? + 3 3? + 4 = 2? 2? + 7 * Ex 6: 2?3+ 2?2 3?3 + 9? 6?2 = ?3 4?2+ 9? * Ex 3: 12? 10? + 3? + 5? 2? = 7? 9? * Ex 7: 2?2? 6?? 3?2? + 6?2? + 5?? + 5?2? 6?2 * Ex 4: = 7?2? + 3?2? 6?2 ?? 2?? + 3 + 4? + 6?? 7? 3 = 3? + 4?? Algebra Basics: Simplifying Polynomials - Math Antics - YouTube
EXPANDING BRACKETS: * Expanding brackets uses the distributive law. * Ex1: Expand the brackets: ? (? + ?) = ? x ? + ? x ? = ?? + ?? *Ex2: simplify: 3(? 2?) + ? (4 2?) = 3? 6? + 4? 2?? = 7? 6? 2?? * Ex3: simplify: 2?(3 2?) ? (4? 2) = 6? 4?? 4?? + 2? = 8? 8?? https://www.youtube.com/watch?v=v-6MShC82ow&t=405s
*Ex4: simplify: ?(3? + 2?) ? (3?2 4?) = 3?? + 2?2 3?3 + 4?? = 3?3 + 2?2 + 7?? *Ex5: simplify: 3?2(? 5?) (6?2? 4?3) = 3?3 15?2? 6?2? + 4?3 = 7?3 21?2? Remember: ? ? = ?2 ? ?2 = ?3 ? ?3 = ?4 ?2 ?2 = ?4 ?3 ?2 = ?5
P. 29 Ex. 2B: Q1 (b, d, e, g, I, j) Q2 (b,d) Q3 Q4 (a) Q5 (a) Q6 (a, c,d) Q7 Q8 Q9 Q10 (b, c) Q11 (a,b) Q12 (a,b,d,f,g,h) Q13 (a) *
FACTORIZING EXPRESSIONS *The opposite of expanding brackets is to put an expressions into brackets. This is called factorizing. *You do this by putting the highest common factor of all of the terms outside the brackets. *Examples: *In the expressions 3? + 6, the highest common factor is 3. *In the expressions 8? + 16?, the highest common factor is 8. *In the expressions 5? + 10??, the highest common factor is 5?. *In the expressions ?2+ 10?, the highest common factor is ?. 1) https://youtu.be/ctqviXu-mTE 2) Factoring Expressions Algebra - YouTube
*When you find the common factor, you out it outside and work out what s left by division. *Let s see what happens to our previous examples when we do that. *Examples: *In the expressions 3? + 6, the highest common factor is 3. After factorizing: ?(? + ?) *In the expressions 8? + 16?, the highest common factor is 8. After factorizing: ?(? + ??) *In the expressions 5? + 10??, the highest common factor is 5?. After factorizing: ??(? + ??) *In the expressions ?2+ 10?, the highest common factor is ?. After factorizing: ?(? + ??)
ACTIVITY TIME Match each expression from the first column, with its highest common factor from the second column and the factorized expression from the third column. First Column Expression Second column Highest common factor Third column Factorized Expression 5?2(3 + 2? ?2 3? + 6? 9? 2? 12? + 18? 2? 3?(3? + 4?) 3 9? 3?? 2?(2?? 1) 2?? 6?? + 8?? ? 3(? + 2? 3?) 4??(2?2+ 4?? + 7???) ?? + 2?? 4?? 4?? 6 10?? 12?? 16?? 2?(? 3? + 4?) 9?? + 12?2 5?2 6(2? + 3?) 15?2+ 10?3 5?4 3? 2?(5? + 6? + 8?) 4?2? 2? 3? ?(? + 2? 4?) -2? 8???2+ 16???2+ 28???2 3?(3 ?)
P. 31 Ex. 2C: Q1 (a,c,e,f,h) Q2 (b,c,d,f,g,h) Q3 Q4 Q6 (a,c,d,g,h,j,L) Q8 (a,c,d,f,g,h) Q10 (a,c,e,f) Q12 (a,d,f,g,h,i,j,L) Q13* Q14* (a,b,d,f,h) Q15 (b,d)
SUBSTITUTING INTO EXPRESSIONS *The word substitute means replace. In expressions, variables are substituted with given integers. *Remember, you should always solves using BIDMAS. *Pay attention to signs. *Examples: *If a= 3, find the value of 2? + 5. Answer: 2 ? + 5 = 6+ 5 =11 *Examples: *If ?= 2, find the value of 5?2. Answer: 5 ( ?)?= ? ? ? = ?? Algebra Substitution - GCSE Maths - YouTube
P. 33 Ex. 2D: Q1 (b,c,d,f,g) Q2 a,b,d,e,f,i,k,L) Q3 (d,e,f) Q4 (a,c,d,f,h) Q5 (a,d,e) Q6 (b,c,d) Q7 Q9 (a) Q10 **
