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Mastering Algebra and Sequences: Fundamentals and Applications

Learn to simplify algebraic expressions, substitute values, create models, solve equations, and understand sequences. Master concepts like algebraic terms and substitution to solve real-world problems efficiently.

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Mastering Algebra and Sequences: Fundamentals and Applications

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  1. Year 9 Mathematics http://www.mathsisfun.com/fractions.html Algebra and Sequences

  2. Learning Intentions • Pupils should be able to: • Simplify algebraic expressions • Substitute values into algebraic expressions • Create models to describe real world scenarios and solve equations • Find the general rule to generate simple sequences

  3. Algebraic Expressions • A mathematical expression consists of numbers, letters and mathematical operations • Sometimes the multiplication symbol is left out • Examples of algebraic expressions include: • 3x • 3x + 2y • 3x + 2y – 5z2 • 3(a + 2b)

  4. Algebraic Terms • Sometimes similar terms occur. When this happens we need to simplify the terms • For example • 3x + 2x becomes 5x • 5y – 2y becomes 3y • 3x2 + 4x2 becomes 7x2 • 5y2 + 2y – 2y2 becomes 3y2 + 2y • 4x x 3 becomes 12x • 3y x 4y becomes 12y2

  5. Substitution • Sometimes we need to find the actual value of an algebraic expression. • We do this by replacing (substituting) numbers for the letters. • For example, the perimeter of a square is 4l, where l is the length of one side. • To find the perimeter of a square with a side length of 10 cm, we replace the l with10. • Thus the perimeter of the square is 4 x l = 4 x 10 = 40 cm • Remember we need to use BIDMAS!

  6. Modelling • When we use algebra to investigate the real world, we describe this as modelling. • We can create an equation and then solve the equation. • For example, if we have 36m of fencing and wish to create a square pen we can use an equation to help. • Remember the equation for the perimeter of a square is P = 4l. • If the perimeter is 36m, then 36 = 4l. • This gives a length of 9m for each side.

  7. Equations and Expressions • Remember an algebraic expression consists of numbers, letters and mathematical operations. • An equation consists of algebraic expressions and an equals sign.

  8. Example • Ian is three times as old as his son Edward. Edward is r years old. A. Write an expression for Ian’s age B. Write an expression for their ages in 10 years time C. In 10 years time, Ian will be twice as old as Edward. How old are Ian and Edward now?

  9. Sequences • A mathematical sequence is a list of numbers that are connected together in some way. • For example: • the numbers 2, 4, 8, 16, 32, 64, .. form a sequence • Each number is double the value of the previous number

  10. Virus Infection

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  15. Linear Sequences • In a linear sequence the next number is found by adding a fixed value (common difference) to the previous number • For example: • The numbers 2, 6, 10, 14, 18, 22 form a linear sequence • The next number is found by adding 4 each time

  16. Linear Sequences • Sequence values can be written in a table indicating the position in the sequence (n) and the value • For example, the previous sequence can be put into a table as shown below

  17. Finding the Formula • To find the formula for a linear sequence we need to: • Find the common difference • Write the expression • Modify the rule to find the first term • Write down the rule • Check your answer!

  18. Matchsticks • Here is a pattern of triangles that is built using matchsticks.

  19. Matchsticks • Here is a pattern of triangles that is built using matchsticks.

  20. Matchsticks • Here is a pattern of triangles that is built using matchsticks.

  21. Differences • We can calculate the difference between two patterns • Lets look at the first pattern again +2 +2 +2 +2 +2 +2

  22. Common Difference • We can see that the difference between successive values is 2. This can help to create a formula. • We will multiply the sequence number by 2. • This is nearly right. Can you explain how to get the correct sequence?

  23. Simplifying our Wording • The rule is: • The number of matches is found by multiplying the sequence number by 2 and adding 1 • If we use n for the sequence number, the formula becomes: • Number of Matchsticks = 2n + 1

  24. The Final Sequence • This is still too long • We will use n to represent the position in the sequence • Number of Matchsticks = 2n + 1 • So for the third sequence the number of matches is • Number of Matchsticks = 2 x 3 + 1 = 7 • Is it correct?

  25. Second Pattern • Can you find the formula for this sequence?

  26. Matchsticks 2

  27. Matchsticks 2

  28. Matchsticks 2

  29. Matchsticks 2

  30. Matchsticks 2

  31. Matchsticks 2

  32. Matchsticks 2

  33. Matchsticks 2 Number of Matchsticks = 8n + 4

  34. Finding the Formula • Find the formula for the sequence • 1, 4, 7, 10, … • Common difference = 3 • Expression = 3n • Modify the rule to find the first term when n = 1, 3n = 3 • we need to subtract 2 • Write down the rule 3n – 2 • When n = 3, 3n – 2 = 3 x 3 – 2 = 7 

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