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Probing Questions to Address Algebraic Misconceptions

Probing Questions to Address Algebraic Misconceptions. Slides by Teresa Robinson teresaarob@yahoo.co.uk. Algebra Misconceptions. How do the following expressions differ? 2 n and n + 2 3(c + 5) and 3c + 5 n ² and 2n 2n ² and (2n) ².

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Probing Questions to Address Algebraic Misconceptions

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  1. Probing Questions to Address Algebraic Misconceptions Slides by Teresa Robinson teresaarob@yahoo.co.uk

  2. Algebra Misconceptions • How do the following expressions differ? 2n and n + 2 3(c + 5) and 3c + 5 n² and 2n 2n² and (2n)²

  3. AlgebraProbing Questions • Can you write an expression that would simplify to 6m – 3n? Repeat for 8(3x + 6). Are there others? • Can you give me an expression that is equivalent to 4p + 3q – 2? Are there others?

  4. Algebra Probing Questions • What do you look for when you have an expression to simplify? • What are the important stages? • What hints and tips would you give to someone about simplifying expressions?

  5. AlgebraProbing Questions • Multiply out the following brackets:- • 3(x + 1) • 2(x – 3) • 4(2x + 5) • -2(x + 2) • -3(2x – 4) • x(x + 1) • What hints and tips would you give to someone about removing a bracket from an expression?

  6. Algebra Probing Questions • Are these expressions correct? 4(b + 2) = 4b + 2 3(p – 4) = 3p – 7 –2 (5 – b) = –10 – 2b 12 – (n – 3) = 9 – n • If not, what is the error and how can they be corrected?

  7. AlgebraProbing Questions • What do you get when you substitute x = -1 into the formula y = 5x – 2? Make up some more formulae that also give y = –7 when x = –1? • What do you get when you substitute a = 2 and b = 7 into the formula t = ab + 2a? Make up some more formulae that also give t = 18 when a = 2 and b = 7?

  8. Algebra Probing Questions p = 4a² and p = (4a)² Are these two formulae the same or different? When are they the same? Why? Write down two formula which are written differently, but are in fact the same.

  9. AlgebraTrial and Improvement • Use a systematic trial and improvement method to find approximate solutions to the equation x³ + x = 20.

  10. Trial and Improvement - x³ + x = 20 Probing Questions • How do you go about choosing a value (of x) to start? • How do you use the previous outcomes to decide what to try next? • How do you know when to stop? • How would you improve the accuracy of your solution? • Is your solution exact? • Can this equation be solved using any other methods? Justify your answer?

  11. AlgebraLinear Equations 6 = 2p – 8 How many solutions does this equation have? Give other equations with the same solution? Why do they have the same solution? How do you know?

  12. AlgebraLinear Equations • Which of these linear equations are easy to solve? Which are difficult and why? What strategies are important with the difficult ones? 3c – 7 = –13 1.7m² = 10.625 4(z + 5) = 8 4(b – 1) – 5(b + 1) = 0 12 = 21 (x + 1) (x + 4)

  13. AlgebraGraph work • If I wanted to plot the graph y = 2x how should I start? • Why is the point (3,6) not on the line y = x + 2? • Write down the equations of some graphs that pass through (0,1)? • Write down the equations of some graphs that pass through (0,0)?

  14. AlgebraGraph work • Use graphing software or graphic calculators to plot the following graphs. What do you notice? y = x y = x y = -x y = x + 1 y = 2x y = x - 1 y = x + 2 y = 3x y = x² y = x + 3 y = 4x y = x + 4... y = 5x...

  15. AlgebraGraph work • Write down some equations of graphs:- • parallel to the x-axis • with a gradient of 2 • with a gradient of 1/3 • with a gradient of -1 • with a gradient of -1/2 • Write down the equation of a graph parallel to the graph of y = 4x + 1. Repeat for 2y = x + 1 • Write down the equation of a graph perpendicular to the graph of y = ½ x – 3. Repeat for 3y + x = 1

  16. Are these lines parallel or not?

  17. AlgebraGraph work • Describe the graphs • y = 2x + 1 • y = 4x – 2 • 2y + x = 5 • How can you tell if two lines are parallel? • How can you tell if two lines are perpendicular?

  18. Graph work – Drawing graphsProbing Questions • How do you go about finding a set of coordinates for the straight line graph y = 2x + 4? • How do you decide on the range of numbers to put on the x and y axes? • How do you decide on the scale you are going to use?

  19. Graph WorkProbing Questions • If you increase/decrease the value of m, what effect does this have on the graph? What about changes to c? • What have you noticed about the graphs of functions of the form y = mx + c? What are the similarities and differences?

  20. Graph WorkProbing Questions • How do you go about finding the gradient for a straight-line graph: • that has been drawn on a set of axes? • from the equation given in the form y = mx + c? • from a table of coordinates? • What happens when m changes? (Increases, decreases, is negative?) • What happens as c changes?

  21. AlgebraGraph work • How would you go about identifying the graph of y = 3x – 5 on a set of axes? • How can you draw the graph of the equation y = ½ x + 3? What different methods can you use? • Given a straight line graph, how can you find it’s equation? What different methods can you use?

  22. Graph WorkProbing Questions • Without drawing the graphs, compare and contrast features of graphs such as: y = 3x y = 3x + 4 y = x + 4 y = x – 2 y = 3x – 2 y = –3x + 4 x + y = 6

  23. Explain why (n+1)(n+20) is an even number • If n is an even number • n+1 is • n+20 is • (n+1)(n+20) is odd x even = even • If n is an odd number • n+1 is • n+20 is • (n+1)(n+20) is even x odd = even • n can only be odd or even and BOTH CASES GIVES AN EVEN ANSWER.

  24. Prove that the difference between the squares of any two consecutive numbers is odd • Let the consecutive numbers be n and n+1 • (n)² = n² • (n+1)² = (n+1)(n+1) = n²+2n+1 • Difference between the squares = n²+2n+1-n² = 2n+1 2n+1 is odd for all n 

  25. Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8 • Let the odd numbers be 2n+1 and 2n+3 • (2n+1)²= (2n+1)(2n+1) = 4n²+4n+1 • (2n+3)² = (2n+3)(2n+3) = 4n²+12n+9 • Difference between the squares = 4n²+12n+9-(4n²+4n+1) = 4n²+12n+9-4n²-4n-1 = 8n+8 = 8(n+1) Careful with the negative signs – use brackets!! Always end up with an expression to factorise OOH – I do like those 8’s!!! 

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