{"id":27540,"date":"2017-01-15T03:01:28","date_gmt":"2017-01-14T17:01:28","guid":{"rendered":"http:\/\/www.rjmprogramming.com.au\/ITblog\/?p=27540"},"modified":"2017-01-15T08:05:16","modified_gmt":"2017-01-14T22:05:16","slug":"mathematics-equation-backtracking-game-primer-tutorial","status":"publish","type":"post","link":"https:\/\/www.rjmprogramming.com.au\/ITblog\/mathematics-equation-backtracking-game-primer-tutorial\/","title":{"rendered":"Mathematics Equation Backtracking Game Primer Tutorial"},"content":{"rendered":"
\"Mathematics<\/a>

Mathematics Equation Backtracking Game Primer Tutorial<\/p><\/div>\n

Thanks to New Century Maths Stages 5.2\/5.3<\/i> (page 432) we have Mathematics based subject matter for today’s “Backtracking Equations” game. When we were at school we were given extra marks when we showed how we worked towards a solution to a Mathematics problem. That is the go today, here, too, with our game.<\/p>\n

To quote our source book above …<\/p>\n

\nBacktracking is a process used to solve an equation by undoing what has been done to the variable in the equation.\n<\/p><\/blockquote>\n

So, for an equation like …<\/p>\n

3(n<\/i> - 2) \/ 6 = 15<\/code><\/p>\n

… using Backtracking techniques we solve this equation for n<\/i> in the four steps (starting with the “shorter” side (in our case the RHS)) …<\/p>\n

    \n
  1. Start with 15<\/i> … x 6 (because originally, we last (in the equation), \/ 6) … 15 x 6<\/i> … then …<\/li>\n
  2. Continue with 15 x 6<\/i> … \/ 3 (because the simplest LHS bit left that can be Backtracked is “3 x” (ie. “3()”)) … (15 x 6) \/ 3<\/i> … then …<\/li>\n
  3. Continue with (15 x 6) \/ 3<\/i> … + 2 (because we Backtrack the last and simplest LHS bit left, which is the “- 2” in “(n – 2)”) … ((15 x 6) \/ 3) + 2)<\/i><\/li>\n
  4. So with n<\/i> now isolated on the LHS, what is ((15 x 6) \/ 3) + 2)<\/i>? … n<\/i> = 32<\/i><\/li>\n<\/ol>\n

    So the user can jump straight to that last step as they wish, and score less, but score nonetheless, or discipline themselves to solve the equation in a logical method.<\/p>\n

    That’s all fine, and we mix it up a bit in the game, but we also add a Help<\/i> link for a clue using the Google Chart Line Chart functionality we talked about when we presented PHP\/Javascript\/HTML Google Charts Parabola Line Graph Tutorial<\/a>. The means by which we help is to present the user with a two dimensional Line Graph of Y<\/i> versus the letter you are solving for<\/i>. How can a Line Graph be useful, when, at school, you may recall that if our letter you are solving for<\/i> is X<\/i> the general form of a Linear Equation to suit a Linear Graph is …<\/p>\n

    Y<\/i> = (GradientOfLine) x X<\/i> + (Y-AxisCut)<\/code><\/p>\n

    … but let’s just take the case where Y<\/i> = 1 in our new equation …<\/p>\n

    3(n<\/i> - 2) \/ 6 = 15Y<\/i><\/code><\/p>\n

    … and we set out to isolate Y<\/i> (which equals 1) … the steps are …<\/p>\n

      \n
    1. Start with 3(n<\/i> – 2) \/ 6<\/i> … \/ 15 (to LHS) … (3(n<\/i> – 2) \/ 6) \/ 15<\/i> … then …<\/li>\n
    2. Continue with (3(n<\/i> – 2) \/ 6) \/ 15)<\/i> … -1 (that is Y) leads to … Y = ((3(n<\/i> – 2) \/ 6) \/ 15) – 1<\/i> = ((3(n<\/i> – 2) – 90) \/ 90)<\/i> = (n<\/i> – 32) \/ 30 … which …<\/i><\/li>\n
    3. Cuts the n<\/i>-Axis at 32 … where Y = 0 … (the solution for n<\/i> … so we would draw Y = 0.0333333333n<\/i> – 1.0666666666666<\/i> (as our “helpful” Line Graph equation, for where this line cuts the n<\/i>-Axis))<\/li>\n<\/ol>\n

      We’d like to direct you to a live run<\/a> link with underlying HTML and Javascript (DOM) code you could call backtracking.html<\/a> for your perusal.<\/p>\n

      But one last thing we need to clarify is the way we used “Parabola Line Graph” thoughts with “Line Graph” data above. However, what is the formula for a parabola …<\/p>\n

      Y = aX2<\/sup> + bX + c<\/code><\/p>\n

      … when a<\/i> above is zero … it’s a straight line (suitable for a Line Graph). However we hadn’t catered in the code for this rewording on the Google Chart so parabola_lgraph.php<\/a> is code you can look at here, changed in this way<\/a>. And here is the Line Graph talked about above …<\/p>\n