Rate of Change

Many of the 7th grade standards in South Carolina have me cover topics very briefly. If I follow the standard exactly I have to literally fight to stay away from such obvious progressions of the math that are supposed to be taught in Algebra 1 and beyond. I find this extremely frustrating, because I feel like I'm trying to fool my students into thinking that this really is all there is to whatever topic we're covering.

Most recently this has come up in a 2-week unit I just finished with my 7th grade advanced students on Rate of Change. The standards say:

7-3.2 Analyze tables and graphs to determine the rate of change between and among quantities.
7-3.3 Understand slope of a constant rate of change.

I began this unit with filling in a function table with a given x-value and rule to find y. We then wrote a sentence describing the rate of change such as:

        As x increases by 3, y decreases by 1.

The students were then to determine if the 'Rate of Change' is constant or not constant, based on the ability to write of a sentence structured like the one above. This went pretty well. Most students were able to fill in the function table with no problem, and determining the Rate of Change was easy! Yay!

The next day we began studying graphs. Each graph had a line with drawn points. Students were to study the graph, translate the points to a table, and write a Rate of Change sentence in the same format as the day before. This was a bit more of a struggle since we haven't done an direct instruction on reading a graph, or plotting points. (Every holiday I have them graph a holiday picture for a day of 'fun math' that is still relevant and standards-based. The students seem to enjoy it and they get the much needed practice.)

The next day was supposed to be focused on slope. I put several graphs on my Promethean board. We wrote a Rate of Change sentence for each. We colored in the graph. I then urged the students to look at each picture carefully. I asked, "What do you notice about all the graphs?", "What do they all have in common?" It took some time, but we eventually got to the desired answer: "They are all straight lines!". Yay!

Now this is really where the standard ends. All I had to do was tell them that if the Rate of Change is constant then the line is straight. If the line is straight then it has as "Slope". And finally, Slope is a fancy math word for "Rate of Change".

But I couldn't stand it. I hate to leave my kids hanging. There is such an obvious and easy step right in front of them. I don't have to teach it. In fact, I try to only teach the standards in my classroom because standardized test acheivement is so 'important'. Rarely do I go above and beyond. I hate it, but it's true. This time I couldn't take it. We had to talk about slope. We defined slope, measured/calculated slope using Rise/Run and then made connections between tables, graphs, ROC sentences, slope and the function rule.

To sum it all up we completed a card sort where students were given 6 graphs, 6 rules, 6 tables, 6 slopes, and 6 sentences. Students worked in groups of 4-5 to sort the cards into groups were the rule matched the table, matched the graph, matched the slope, and finally matched the sentence. There were 4 sets of these cards that were traded between groups. Here are a few pics:

This card sort really helped my students make the connections between graphs, tables, Rate of Change, and slope. It was struggle for them to 'sort' it all out, but once they did, I knew they knew it!

In the end I am so glad I took the extra step. The kids got it. I probably saved the Algebra 1 teacher a day or two  of instruction for next year (if the kids remember what they learned), and the students understand how this all works together. It was all I could do to not keep going into equations of a line, y-intercepts, etc! Oh well...back to the standards it is...