Finding the Regression Line without Calculus

Last month, my latest professional article, Deriving the Regression Line with Algebra, was published in the April 2017 issue of Mathematics Teacher (Vol. 110, Issue 8, pages 594-598). Although linear regression is commonly taught in high school algebra, the usual derivation of the regression line requires multidimensional calculus. Accordingly, algebra students are typically taught the keystrokes for finding the line of best fit on a graphing calculator with little conceptual understanding of how the line can be found.

In my article, I present an alternative way that talented Algebra II students (or, in principle, Algebra I students) can derive the line of best fit for themselves using only techniques that they already know (in particular, without calculus).

For copyright reasons, I’m not allowed to provide the full text of my article here, though subscribers to Mathematics Teacher should be able to read the article by clicking the above link. (I imagine that my article can also be obtained via inter-library loan from a local library.) That said, I am allowed to share a macro-enabled Microsoft Excel spreadsheet that I wrote that allows students to experimentally discover the line of best fit:

http://www.math.unt.edu/~johnq/ExploringTheLineofBestFit.xlsm

I created this spreadsheet so that students can explore (which is, after all, the first E of the 5-E model) the properties of the line of best fit. In this spreadsheet, students can enter a data set with up to 10 points and then experiment with different slopes and $y$ -intercepts. As they experiment, the spreadsheet keeps track of the current sum of the squares of the residuals as well as the best guess attempted so far. After some experimentation, the spreadsheet can also provide the correct answer so that students can see how close they got to the right answer.

Difference of Two Squares (Part 1)

In Algebra I, we drill into student’s heads the formula for the difference of two squares:

$x^2 - y^2 = (x-y)(x+y)$

While this formula can be confirmed by just multiplying out the right-hand side, innovative teachers can try to get students to do some exploration to guess the formula for themselves. For example, teachers can use some cleverly chosen multiplication problems:

$9 \times 11 = 99$

$19 \times 21 = 399$

$29 \times 31 = 899$

$39 \times 41 = 1599$

Students should be able to recognize the pattern (perhaps with a little prompting):

$9 \times 11 = 99 = 100 - 1$

$19 \times 21 = 399 = 400 - 1$

$29 \times 31 = 899 = 900 - 1$

$39 \times 41 = 1599 = 1600 - 1$

Students should hopefully recognize the perfect squares:

$9 \times 11 = 99 = 10^2 - 1$

$19 \times 21 = 399 = 20^2 - 1$

$29 \times 31 = 899 = 30^2 - 1$

$39 \times 41 = 1599 = 40^2 - 1$ ,

so that they can guess the answer to something like $59 \times 61$ without pulling out their calculators.

Continuing the exploration, students can use a calculator to find

$8 \times 12 = 96$

$18 \times 22 = 396$

$28 \times 32 = 896$

$38 \times 42 = 1596$

Students should be able to recognize the pattern:

$8 \times 12 = 10^2 - 4$

$18 \times 22 = 20^2 - 4$

$28 \times 32 = 30^2 - 4$

$38 \times 42 = 40^2 -4$ ,

and perhaps they can even see the next step:

$8 \times 12 = 10^2 - 2^2$

$18 \times 22 = 20^2 - 2^2$

$28 \times 32 = 30^2 - 2^2$

$38 \times 42 = 40^2 -2^2$ .

From this point, it’s a straightforward jump to

$(10-2) \times (10+2) = 10^2 - 2^2$

$(20-2) \times (20+2) = 20^2 - 2^2$

$(30-2) \times (30+2) = 30^2 - 2^2$

$(40-2) \times (40+2) = 40^2 -2^2$ ,

leading students to guess that $(x-y)(x+y) = x^2 -y^2$ .

Divisibility tricks

Based on personal experience, about half of our senior math majors never saw the basic divisibility rules (like adding the digits to check if a number is a multiple of 3 or 9) when they were children. I guess it’s also possible that some of them just forgot the rules, but I find that hard to believe since they’re so simple and math majors are likely to remember these kinds of tricks from grade school. Some of my math majors actually got visibly upset when I taught these rules in my Math 4050 class; they had been part of gifted and talented programs as children and would have really enjoyed learning these tricks when they were younger.

Of course, it’s not my students’ fault that they weren’t taught these tricks, and a major purpose of Math 4050 is addressing deficiencies in my students’ backgrounds so that they will be better prepared to become secondary math teachers in the future.

My guess that the divisibility rules aren’t widely taught any more because of the rise of calculators. When pre-algebra students are taught to factor large integers, it’s no longer necessary for them to pre-check if 3 is a factor to avoid unnecessary long division since the calculator makes it easy to do the division directly. Still, I think that grade-school students are missing out if they never learn these simple mathematical tricks… if for no other reason than to use these tricks to make factoring less dull and more engaging.