Home | Chapter 24 |

This chapter addresses numeration topics that are typically developed in the middle grades yet have not been addressed in earlier chapters. These include exponents, integers, absolute value, and an investigation of rational and irrational numbers as they make up the real numbers.

Exponents may be important simply because they have traditionally been given such instrumental treatment in algebra: "When multiplying numbers with a like base add the exponents," and so on and so on. With graphing calculators, there are some nice things that can be done with these topics.

The integers also provide a context for the broader discussion of the use of models in learning mathematics - a review of ideas addressed in Chapter 3. As noted in Chapter 3, the two suggested models, vectors and sets, are superficially quite different. However, onto each of these models can be imposed the concepts of opposite and quantity. Some teachers tend to find one model or the other easier to understand. (I have found this to be a function of how I have introduced and worked with the models.) Why work with both or require students to understand the use of both models. My own response is that if students (your teachers or children in school) cannot "see" the same concepts in both models then perhaps they do not really have a firm understanding of the ideas involved.

Indicated Division. It has always been a bit of a mystery to me that somehow children in about the seventh grade are to begin to believe that 3/5 can mean 3 whole things divided into 5 parts when for all of their prior school years they have been told that it means 3 of 5 equal parts making up one whole. A review of popular textbook series offers limited assistance here. I have offered some ideas for you to consider. Chapter 16 on fraction meanings begins with a development of fractional parts based on sharing problems or what are essentially partitive division problems. Based on this development, the denominator of a unit fraction can be seen as a divisor of one whole. Then the numerator is a multiplier - indicating the number of thee parts are shared with each sharer. Activity 24.3 on page 504, suggests a partitive division context with various symbolic solutions, all of which are correct but which look very different. The discussion around a problem like this may be useful. Perhaps the whole issue of indicated division is simply one that bothers me more than it does you. I think it is important to at least discuss it.

Density of the Rationals. It has always seemed to me that the density of the rational numbers was a fairly mind-boggling idea. Imagine being able to have an infinite number of fractions exist between any two that I pick, regardless of how close they may be to begin with. And then to think that there are even more numbers in between all of those. Wow! But I must admit that you do not run into many middle school teachers who are as blown away by that idea as I am. Further, I am not convinced that I can excite many seventh or eighth grade students about these ideas. So, there is ample argument that perhaps these topics are not worth discussing. To argue on the other side, it seems really interesting that 0.999999… is exactly the same as one and the idea of an infinite decimal may be important. Note that 1/9 = 0.1111111…, 2/9 = 0.22222…, 3/9 = 0,33333…, etc. So, 9/9 = 0.99999… . The existence of the rationals and the density of these things seems to be important if we want students to believe that even though you plot only a few discrete points from a continuous function, that it makes sense to connect those points in a continuous curve. There really are a lot of connected ideas in mathematics.

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How much time and effort you give to this probably depends on how well you think your students understand these topics from their methods classes and perhaps how profound or significant you believe these ideas are for middle-grade students.