Variation As Part Of Mastering Math.

 The final big idea in mastery math, is variation.  Variation refers to drawing closer attention to a key aspect of a mathematical concept by varying certain elements while keeping the rest constant.  This is similar to what a scientist
s does when they vary one thing and keep the rest the same.  

Variation can be done either in the conceptual region or the procedural element.  In conceptual variation, the variation happens in how it is represented and the representation is varied so students see it from more than one view point.

In the procedural variation, how the student goes through the learning sequence. The variation draws students attention to key features while scaffolding student knowledge so they are better able to see connections.

In addition, procedural variation provides repeated practice in one of three ways.  First, one can extend the numbers used by varying the number, the variable, or the context.  Variations might be 6 + 19, 16 + 19, 26 + 19........96 + 19 and then asking what do you notice?  In multiplication it might look like 7 x 4, 70 x 4, 700 x 4, etc so students have the opportunity to notice the pattern.

Next, is to vary the processes involved in solving a problem.  It might be 8(44) which could be solved that way or look at it as 8(40+4) which gives 8(40) + 8(4) = 320 + 32 = 352.  This is two different ways to solve the problem or the second one might be 8(50) - 8(6) = 400 - 48 = 352.  If you are looking at money, it might be something like Paul bought food for $23.76 and gave the person $30.00 to pay for it.  A student could subtract the numbers so $30.00 - $23.76 = $6.24 or a student can add coins to $23.76 and work their way to $30.00.  They would go with 4 pennies to $23.80, two dimes to $24.00 and $6.00 to $30.00 or $6.24.

Third, to vary the problems by applying the same process to it so students see how its the same and how its different.  For instance, if you use base 10's for multiplying 27 x 32, one can use the same process for (x + 1) (x -2) or even lattice multiplication works for both.  This helps students see the connections between the two. You could also ask what has changed and what has stayed the same.  This allows students to think about the answers and communicate them

This wraps up an in depth look at the five big ideas in mastery math.  Let me know what you think, I'd love to hear.  Have a great day.