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Mental Math Starters

Mental Math Starters. Work it out!. 4 + 3 ×. 0.6. 43. –7. 8. 5. = 133. = –17. = 5.8. = 28. = 19. Work it out!. 7 ×. 0.4. 22. –3. 6. 9. 2. = –10.5. = 31.5. = 1.4. = 21. = 77. Work it out!. 0.2. 12. –4. 3. 9. 2 + 6. = 6.04. = 150. = 15. = 22. = 87.

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Mental Math Starters

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  1. Mental Math Starters

  2. Work it out! 4 + 3 × 0.6 43 –7 8 5 = 133 = –17 = 5.8 = 28 = 19

  3. Work it out! 7 × 0.4 22 –3 6 9 2 = –10.5 = 31.5 = 1.4 = 21 = 77

  4. Work it out! 0.2 12 –4 3 9 2 + 6 = 6.04 = 150 = 15 = 22 = 87

  5. Work it out! 2( + 8) –13 3.6 18 69 7 = 23.2 = 154 = –10 = 30 = 52

  6. Key Words and Introduction to Expressions

  7. Writing expressions 6 n Here are some examples of algebraic expressions: n + 7 a number n plus 7 5 – n 5 minus a number n 2n 2 lots of the number n or 2 ×n 6 divided by a number n 4n + 5 4 lots of a number n plus 5 a number n multiplied by itself twice or n× n × n n3 3 × (n + 4) or 3(n + 4) a number n plus 4 and then times 3.

  8. Writing expressions Miss Green is holding n number of cubes in her hand: Write an expression for the number of cubes in her hand if: She takes 3 cubes away. n – 3 She doubles the number of cubes she is holding. or 2 ×n 2n

  9. Equivalent expression match

  10. Simplifying Algebraic Expressions

  11. Like terms An algebraic expression is made up of terms and operators such as +, –, ×, ÷ and ( ). A term is made up of numbers and letter symbols but not operators. For example, 3a + 4b – a + 5 is an expression. 3a, 4b, a and 5 are terms in the expression. 3a and a are called like terms because they both contain a number and the letter symbol a.

  12. Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, 5 + 5 + 5 + 5 = 4 × 5 In algebra, a + a+a+a= 4a The a’s are like terms. We collect together like terms to simplify the expression.

  13. Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, (7 ×4) + (3 ×4)= 10 × 4 In algebra, 7 ×b + 3 ×b= 10 ×b or 7b + 3b= 10b 7b, 3b and 10b are like terms. They all contain a number and the letter b.

  14. Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, 2 + (6 ×2) – (3 ×2)= 4 × 2 In algebra, x + 6x – 3x = 4x x, 6x, 3x and 4x are like terms. They all contain a number and the letter x.

  15. Collecting together like terms When we add or subtract like terms in an expression we say we are simplifying an expression by collecting together like terms. An expression can contain different like terms. For example, 3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b = 7a + 8b This expression cannot be simplified any further.

  16. Collecting together like terms Simplify these expressions by collecting together like terms. 1) a + a + a + a + a = 5a 2) 5b – 4b = b 3) 4c + 3d + 3 – 2c + 6 – d = 4c – 2c + 3d – d + 3 + 6 = 2c + 2d + 9 4) 4n + n2 – 3n = 4n – 3n + n2= n + n2 Cannot be simplified 5) 4r + 6s – t

  17. Algebraic perimeters 2a 3b 5x 4y x 5x Remember, to find the perimeter of a shape we add together the lengths of each of its sides. Write algebraic expressions for the perimeters of the following shapes: Perimeter = 2a + 3b + 2a + 3b = 4a + 6b Perimeter = 4y + 5x + x + 5x = 4y + 11x

  18. Algebraic pyramids

  19. Algebraic magic square

  20. Using Index Notation

  21. Using index notation Simplify: x + x + x + x + x = 5x x to the power of 5 Simplify: x×x×x×x×x = x5 This is called index notation. Similarly, x×x = x2 x×x×x = x3 x×x×x×x = x4

  22. Using index notation We can use index notation to simplify expressions. For example, 3p× 2p = 3 ×p× 2 ×p = 6p2 q2×q3 = q×q×q×q×q = q5 3r×r2 = 3 ×r×r×r = 3r3 2t× 2t = (2t)2 or 4t2

  23. Expressions of the form (xm)n When a term is raised to a power and the result raised to another power, the powers are multiplied. For example, (a5)3 = a5 × a5 × a5 = a(5 + 5 + 5) = a15 = a(3 × 5) In general, (xm)n=xmn

  24. Expressions of the form (xm)n Rewrite the following without brackets. 1) (2a2)3 = 8a6 2) (m3n)4 = m12n4 3) (t–4)2 = t–8 4) (3g5)3 = 27g15 5) (ab–2)–2 = a–2b4 6) (p2q–5)–1 = p–2q5 1 7) (h½)2 = h 8) (7a4b–3)0 =

  25. Any number or term divided by itself is equal to 1. Look at the following division: y4 ÷ y4 = 1 The zero index But using the rule that xm÷xn=x(m – n) y4 ÷ y4 = y(4 – 4) = y0 That means that y0 = 1 In general, for all x 0, x0=1

  26. Fractional Indices

  27. x × x = x+ = 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 2 3 2 2 3 3 3 3 2 But, x × x = x x = x So, Similarly, x × x × x = x+ + = But, x × x × x = x 3 3 3 x = x So, 3 Indices can also be fractional. x1 = x Fractional indices The square root of x. x1 = x The cube root of x.

  28. 1 1 1 1 1 1 1 m m m m n n n n n n n n n n n Also, we can write x as x . ×m x × m= (x)m = (x)m n We can also write x as xm× . n x = (xm) = xm m× x = xm n or n x = (x)m In general, x = x n Fractional indices Using the rule that (xm)n=xmn,we can write In general,

  29. 1 xm × xn = x(m + n) x–1 = x 1 xm÷xn=x(m – n) x–n = xn 1 2 (xm)n=xmn x = x 1 m n n x = x n x1=x x0=1 (for x = 0) x = xm or (x)m n n Here is a summary of the index laws. Index laws

  30. Multiplying Algebraic Terms

  31. Multiplying terms together In algebra we usually leave out the multiplication sign ×. Any numbers must be written at the front and all letters should be written in alphabetical order. For example, 4 ×a = 4a We don’t need to write a 1 in front of the letter. 1 ×b = b b× 5 = 5b We don’t write b5. We write letters in alphabetical order. 3 ×d×c = 3cd 6 ×e×e = 6e2

  32. Brackets Look at this algebraic expression: 4(a + b) What do you think it means? Remember, in algebra we do not write the multiplication sign, ×. This expression actually means: 4 × (a + b) or (a + b) + (a + b) + (a + b) + (a + b) = a + b + a + b + a + b + a + b = 4a + 4b

  33. Using the grid methodtoexpandbrackets

  34. Expanding Algebraic Expressions

  35. Expanding brackets then simplifying Sometimes we need to multiply out brackets and then simplify. For example, 2(5 – x) We need to multiply the bracket by 2. – 2x 10

  36. Expanding brackets then simplifying Simplify 4 – (5n – 3) We need to multiply the bracket by –1 and collect together like terms. 4 + 3 – 5n = 4 + 3 – 5n = 7 – 5n

  37. Expanding brackets then simplifying Simplify 2(3n – 4) + 3(3n + 5) We need to multiply out both brackets and collect together like terms. – 8 + 15 6n + 9n = 6n + 9n – 8 + 15 = 15n + 7

  38. Expanding brackets then simplifying Simplify 5(3a + 2b) – 2(2a + 5b) We need to multiply out both brackets and collect together like terms. 15a + 10b – 4a –10b = 15a – 4a + 10b – 10b = 11a

  39. Algebraic multiplication square

  40. Pelmanism: Equivalent expressions

  41. Algebraic areas

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