Combining integers … does any early algebraic skill cause more problems?

If so, I can’t think of one.

Fortunately, though, using the double-slash notation that I’ve been talking about this week helps students make sense of this tricky topic.

Even a problem as simple as the following can be made easier with the double-slash:

– 2 + 5 – 3 + 7 – 9

How? Use a double-slash to separate negatives from positives. To mirror the left-right orientation on the number line, have students put negative numbers on the left; positives on the right, like this:

– 2 + 5 – 3 + 7 – 9

= – 2 – 3 – 9 // + 5 + 7

Students proceed:

– (2 + 3 + 9) // + (5 + 7)

= – 14 // + 12

= – (14 – 12)

= – 2

Notice how the use of the double-slash encourages many important thought-steps:

**1) ** It gives students a way to organize the numbers into two groups: a positive group and a negative group.

**2)** It helps students see symmetry in the same-sign rule step, for the expressions:

– (2 + 3 + 9) and + (5 + 7) look similar; only the leading signs differ.

**3)** It continues to keep the numbers organized up through the step before the final step.

**4)** Doing the problem this way keeps students from having to “jump across zero” as they would need to do if they worked the problem sequentially:

– 2 + 5 = + 3 [jumping across zero left to right, going from – 2 to + 3]

+ 3 – 7 = – 4 [jumping across zero right to left], going from + 3 to – 4, etc.

**5)** Using the double-slash, students see the “big picture.” They see that this problem boils down to, first, combining a bunch of negative numbers and a bunch of positive numbers, and secondly combining the sum of the positives with the sum of the negatives by boiling them down to a final answer.

Once again the double-slash helps us pursue the “art” of teaching algebra well. Through teaching well we help students conceptualize big patterns and grasp a higher level of organization: skills that help them in more than just math class.

Saw your post on Facebook inviting questions – I have a son who breezed through Saxon 7/6 last year, and has done some hodgepodge curricula this year without problems. He sees how to do the math easily. He is working on Matmatical Reasoning right now through the summer. Questions – Is Pre-Algebra necessary? Is 8th Grade Algebra necessary? (hes 13). Is there an 8th grade curriculum that you’d recommend? I did purchase your book – Amazon said it is mailing it today – even though it is Sunday. 😉

Hello,

Thanks for your questions. I do think that PreAlgebra is necessary and very helpful for anyone who wants to do a solid year of Algebra 1.

And I do think that Algebra 1 (8th grade algebra) is absolutely necessary! Algebra is the gateway to all higher math, and it is the one course that, more that any other, determines whether or not a student will go on to college. For an 8th grade curriculum for Algebra, you could do: first, the Algebra Survival Guide and Workbook, to give someone a general understanding of algebra, and then after that, a standard curriculum, like any of the McDougal Littell Algebra 1 curriculuae. I like the McDougal Littel textbooks because they are well organized and well edited. Here, for example, is a page with three options by this company. I would not recommend the Saxon Algebra 1 curriculum, as Saxon has not kept up with modern math standards and so their Algebra 1 book is outdated. It makes virtually no attempt to relate algebra to real word events, and so it reinforces the notion that math is not related to the world, when in fact it is very much related. Also, the Saxon books jump from topic to topic so quickly that students end up not getting a thorough sense of any given topic. I hope that helps.

I want the algebraic wtuhoit the data types. And am still pondering how to do that; something to do with algebraic structures being overlain on things, while types underlie them, I think. I’m also thinking it might be related to some sort of conceptual flaw in the programming notion of symbol.