Kiss those Math Headaches GOODBYE!

Archive for November, 2019

How to quickly find the y-intercept (b-value) of a line


Of course there’s a standard way to find the y-intercept of any line, and there’s nothing wrong with using that approach.

High-Octane Boost for Math
High-Octane Boost for Math Ed

But the method I’ll present here is a bit faster and therefore easer. And hey, if we can save time when doing math, it’s worth it … right?

So first let’s recall that the y-intercept of any function is the y-value of the function when the x-value = 0. That’s because the y-intercept is the y-value where the function crosses or touches the old, vertical y-axis, and of course all along the y-axis the x-value is always 0 (zero).

So the standard slope-intercept formula is y = mx + b. In a problem asking for the y-intercept, you’ll be given one point that the line passes through (that point’s coordinates will provide you with an x-value and a y-value), and you will also be told the slope of the line (the line’s m-value).
So then, to get the b-value, which is the value of the y-intercept, you just grab your y = mx + b equation (dust it off if you haven’t used it in a while), and plug in the three value you’ve been given: those for x, y and m. Then you solve the equation for the one variable that’s left: b, the value of the y-intercept.

Let’s look at an example: a line with a slope of 2 passes through the point (3, 10). What is this line’s y-intercept.

Now, according to the problem, the m-value = 2, the x-value = 3, and the y=value = 10. We just take these values and plug them into the equation:
y = mx + b, like this:

10 = (2)(3) + b

After doing these plug-ins, you just solve the equation for b, finding that
b = 4. That means that the y-intercept of the line = 4.

Now let’s see how you can do the same problem, but a little bit faster.
To do so, we first need to play around with the y = mx + b equation by subtracting the mx-term from both sides, like this:

y = mx + b [Standard equation.]
– mx = – mx [Subtracting mx from both sides.]
y – mx = b [Result after subtracting.]
b = y – mx [Result after flipping left & right sides
of the equation above.]

Aha! Look at that final, beautiful equation. This equation has b isolated on the left-hand side. So now if we want to solve for b, all we do is plug in the x, y and m values into the right-hand side of the equation and simplify the value, and the value we get will be the b-value.

For the problem we just solved, with x = 3, y = 10, m = 2, watch how easy it is to solve:

b = y – mx
b = 10 – (2)(3)
b = 10 – 6
b = 4

So notice that this technique, just like the first technique, reveals that the
y-intercept of the line is 4, or (0, 4). The techniques agree, they just get to the same end in slightly different ways.

Notice that with the second, quicker technique, you don’t need to add or subtract any terms. And that’s a key reason that this technique is faster and easier to use than the standard method. So try it out and stick with it if you like it.



Mental Math is Changing my Tutee’s Mindset


“Oh my god, I’m getting these!”

Sierra (not her real name) lay down her phone on the tutoring desk and shot me a quick glance, eyes wide.

“Two minutes ago I would’ve had no way to do these in my head,” she said.

Sierra had just completed a round of mental math addition problems through a phone app*. She was adding two two-digit numbers and had just gotten four problems correct in a row.

The app just showed Sierra how to do addition from left to right, not right to left, thus “chunking” the problem.

For example, take the problem 58 + 79. The app encouraged Sierra to add the tens digits first, helping her see that the 5 and 7, respectively, actually represent 50 and 70, which sum to 120.

The app instructed Sierra to hold that 120 in her short-term memory, then sum the digits in the ones place, 8 and 9, to get 17.

Finally, the app told Sierra to add the two subtotals to find the answer:
120 + 17 = 137.

Sierra said she had never realized that an addition problem could be tackled this way, even though she’s now 16 years old. She left the tutoring session saying she was excited to continue working with the app. The best feature of the app, she said, is that it provides tips on how to do each problem quickly and efficiently.

As a longtime tutor, I’ve long noticed that teaching mental math strategies often helps boost students’ “math self-esteem.” Today’s session with Sierra reinforced that point in a big way.

Before today’s session, Sierra had been voicing a range of negative feelings about her relationship to math. She said she can’t be “a good logical thinker” because she struggles with math. She said that she has “always” been bad at math. When I explored that statement, I found that its roots trace back to 4th grade when she had a teacher who forced students to do problems on the board, whether or not they were prepared. On a number of occasions Sierra was called to the board and confused. Rather than helping Sierra through her confusion, the teacher allowed her to wallow in it publicly. I know … terrible, right?!

But, back to the healing side of things … how does the development of mental math skills make such a big difference for many students? I think the answer relates largely to two concepts: labor/efficiency, and power.

If a student lacks good mental math skills, s/he will find the process of doing arithmetic laborious, i.e., like drudgery. If you doubt that, remind yourself of what it felt like to do a long division problem of the sort: 25,682 ÷ 479. By contrast, a student with strong mental math skills finds the process of arithmetic easy and somewhat fun. Knowing those shortcuts is actually kind of exhilarating, especially if you know A LOT of shortcuts.

Mental math skills also make a student feel powerful. For example, a student who has the rules of divisibility solidly locked down will have no problem when learning the skill of prime factorizing numbers; those divisibility rules will make the factoring process fast and fun. On the other hand, a student who doesn’t know divisibility rules will need to rely on multiplication facts (a rather backward way of working), or on a calculator (slow for this kind of task) — just to perform the division aspect of prime factorizing. As a result, this student will not enjoy prime factorizing numbers, and won’t even know how it might be possible to enjoy the process.

So it’s critically important that we help students nurture and maintain a solid toolbox of mental math skills The internet is loaded with resources for mental math. Just go to YouTube and search for this topic. This blog, also, has many articles and videos on the topic. Just search it for mental math, and see what pops up.


* the phone app Sierra was using was Magoosh’s Mental Math Flashcards app.