Last year, when our school district finally decided to start using the Common Core standards for middle school, we created our own curriculum for Math 8CP. That’s when I decided to stop using “the slope equation” of “(y sub 2 minus y sub 1) divided by (x sub 2 minus x sub 1)” and concentrate on “change in y over change in x”. I also told the class about how we use the Greek letters to mean different things in math, (I gave the example of “we use pi to represent the ratio of any circle’s circumference divided by it’s diameter” and some kids started saying 3.14159… so I don’t think that example really hit the mark as they just were thinking about the number…sigh) and introduced the Greek letter delta, and we wrote it as “delta y over delta x” – using the triangles not the words.

We’d been graphing from tables, and had found the change in the y values and the x values and related it to the movement on the graph, so I decided it would be pretty easy to always write a little t-table for the 2 points and find the “change in y” and “change in x”, then write slope as m= delta y/delta x = whatever the numbers were. This year I also wrote delta y= and delta x = on the right and left side of the table before I wrote the numbers. (every time I’m writing delta in this post I’m really writing the triangle on my paper in class, and saying “change in y” or “delta y”)

It seems like my classes are understanding how to find the slope from 2 points without having a graph this year better than they have in the past. The problem I’m still seeing is sometimes the negative isn’t included. So (1,4) and (2,7) could have the same slope as (1,10) and (2,7) – they’re seeing the difference as 3 in both, not the first 3/1 and the second as -3/1. We’re working on that piece, but any suggestions you have are appreciated!

So we are piloting and just jumped into CMP3 with our Math 7 Accelerated classes. We are using the Modeling with Mathematics unit and it is going on the assumption that the students have had y=mx+b experience, so we had to do some teaching about what each letter means and what the slope means and how to find it from 2 points, rather quickly (I know we should have taken more time, but that’s another topic to blog on!).

(To interrupt myself, we had 3 lessons in Investigation 1 where found how many pennies it took to “break” bridges of different lengths and thicknesses of paper and graphed the results. For the summary day we decided to go to the computer lab and do Desmos’ Marble Slides – Lines with the kids. We started out together, then they went along on their own – some understanding what the m and b did, some just winging it. Bottom line – they LOVED it! The bell rang for the end of school and I heard, “Wait, I’m not done yet!” I told them they could use the code at home… don’t know if that actually happened (I guess I could check how far they got!) So I referred to the Desmos activity when we were talking about the m and b in the discussion of what the m and b meant in the y=mx+b equation.)

So we were working on looking at the line, and writing the 2 points in a little table and finding “m” as delta y over delta x. Then we would use one of the points for the “x” and “y” in the equation, and the “m” we just found to figure out the “b”. Then we could write the equation! The HW that night from the ACE problems included looking at some points and finding the slope. The next day one of the boys came in and said he used “y2 – y1/x2-x1 instead of making a table. I asked him why he did that – his brother told him how. So then I wrote it down so the class could see what we were talking about – and asked him what he thought was easier: doing the equation or finding the difference from the table. One other student said he thought the equation was easier – I said it was fine with me which ever method they used, just do it correctly! (just gotta love those older siblings, I swear the table makes more sense! – but maybe they do for me and not for them). Anyhow, it seems to be working well for my regular 8th graders, and my accelerated 7th graders…and they had their choice when we just reviewed it on Thursday.

I’m not thrilled at how we jumped into this (but it’s a pilot and I’ll stop justifying) and a lot of kids were stressing about not understanding (because basically they’ve just predominately had direct instruction with some group work – the last book we were piloting was Big Ideas Advanced so they aren’t really used to struggling) so we added a day to go over what we’ve learned so far. After going over the HW (in detail) I gave each student a little piece of paper and told them to write a point on it (-20 to 20, integers only, no (0,0)). Then I did one example using my point (-3,8) and one of theirs to find the slope, “b”, and wrote “my” equation. The kids found the slope with their neighbors, then the equation of “their” line. In my 6th period I had them tell me some of their lines, I should have done this with my 4th period, but I will next time. After finding their first line, the kids got to find a new partner and do it again. I think they liked it because they got to work with each other, walk around and find new people – in spite of having to figure out m and b! I was happy because I’d been able to do 2 different activities with 3 of my classes that let them do math, and get up and move around the class. And I think the activities helped cement some learning! Yea math!!

I like your commitment to delta and emphasizing change. I use cmp2 and also avoid the slope formula. I think it’s helpful to draw little arrows along the sides of a vertical t-table from one entry to another. In several years of this I’ve never seen a student draw arrows in opposite directions. Once the arrows are on the paper we label them showing how to “go from one number to the other”, That is we write the change that will be needed to “take us from one value to the other”, using negatives when the numbers decrease. This generally works well.

I have them write the arrows also. I think it helps – so they “see” which way they are going and don’t subtract “up” one time and then “down” the other! I need to stress that it isn’t the absolute value!

Thank you for your comment. 🙂

I agree with not starting with the slope formula. I started using tables and the delta y over delta x, about 25 years ago when I was also teaching a high school physics class. I always model saying “how are the ys changing” or “what is our vertical change” instead of the formula, as we look at slope. I do put up the formula, and say that they are free to use it, but I prefer to “reason it out in my head.” In think even with common core, there needs to be a balance with conceptual and procedural. All kids are different and prefer different methods. We need to get them to try something outside of their comfort zone, (like conceptual when they love procedural), but we shouldn’t mandate a method, or ban another, ever. I think the “spirit of common core” would have them critique one method vs another.

Thanks for your comment! I agree – I think I should have told them the formal equation. We are ending up systems so I think I can put it in when we do an overall review Thank you for the reminder!!