About a week ago I listened to a podcast called "Can we help kids learn to love math?" which featured math educators Christopher Danielson (@trianglemancsd) and Sara VanDerWerf (@saravdwerf). One of the key takeaways from this is that math is NOT about calculating fast. In fact, we destroy kids views of themselves as mathematicians when we tell them that being 5 seconds slower at a calculation is bad. The truth is math is about exploring and playing - things kids enjoy. Math is about noticing, describing and generalizing. These things can happen naturally when students are given time to play with math.
So what does playing with math have to do with Wally's math books? If you look at my grandfather's college math textbooks from the 1920's, you'd think that math is all about solving things in one way, using theorems and performing calculations. However, I did find glimpses of what mathematicians actually do in this book - the making observations, looking for patterns and drawing conclusions part of mathematics. Here is one such question in the chapter called "General Theory of Equations":
I graphed these three functions on Desmos. It was easy to see that the functions were just vertical transformations of each other. However, I was curious about what happens with the x-intercepts. So, I created a table which included these functions and a few others
Wow! The sum of the x-intercepts is 3. This leads to a whole list of questions.
- Why is the sum of the roots 3?
- Is it related to coefficient of the quadratic term, which is -3?
- What would happen if I changed that coefficient?
- If I changed the coefficient to -4, would the sum of the x-intercepts be 4?
- What if I added a linear term to the function? How would the sum change?
- What about the product of the x-intecepts? How is it related to the coefficients of the cubic polyonomial?
- How is the sum and product of the roots impacted if the polynomial is an even degree?
