Chapter 7Notes for InstructorsContentThe focus of this chapter is the Real Number System. Up to this point in the course theneed for richer number systems has been obvious because we desire to have closure underthe basic operations of arithmetic. For example, in moving from the Whole Number Systemto the Integers, we gain closure under subtraction. It may not be so obvious why we needto move from the Rational Number System to the Real Number System. Consequently, Ithink it is important to prove that the√2 is irrational. Beyond that, I did not have muchtime to spend on Chapter 7. Consequently, I spent very little time on the last two sectionsof this chapter; it was unfortunate, but necessary.ManipulativesWe have the base ten blocks which can be used to represent decimals. Note that it isimportant that you do not call the small cube the unit when working with decimals. Theunit will have to be the long, the flat, or the large block, depending on the magnitude of thenumbers you wish to represent. For example, if I needed to represent 134.5, then my unitwould be the long, and I would represent this with one large block, three flats, four longs(units) and five small cubes. On the other hand, if I wanted to represent 23.56, then myunit would be the flat, and I would represent this with two large block, three flats (units),five longs, and six small cubes.You can also use money to represent decimals. I haven’t seen any problems with thisapproach, but there does appear to be some discussion about this technique among educators.I’m not sure what their objections are, but you should be aware that a certain faction ofeducators is somewhat leery of this approach.The place value cards that we used to represent whole number arithmetic can be easilyadjusted to represent decimal addition and subtraction.Notes and Suggestions:Notes on Section 7.1: Decimals• I believe that it is important to prove that the√2 is irrational. This proof will providestudents with a wealth of irrational numbers. In order to understand this proof, stu-dents will need to be reminded about the Fundamental Theorem of Arithmetic. Youwill also need to familiarize them with the following idea based on the FTA: If a and pare integers, p is prime, and p is a factor of a2, then p is a factor of a. You should alsoconvince students that this is not true if we remove the restriction that p is prime.• I wanted my students to be able to do problems similar to those in Examples 7.3, 7.4,7.5, and 7.6 on pages 419–421 of the textbook.Notes on Section 7.2: Computations with Decimals1• When working with terminating decimals, students should see the connection betweenfraction arithmetic and decimal arithmetic.• In theory, these students should see scientific notation and significant digits in a scienceclass, so I tried not to spend too much time on this section.Notes on Sections 7.3 and 7.4:Ratio and Proportion and Percents• I spent very little time on these sections. I did try to ensure that they could do basiccalculations with percents similar to those in problems 1–11 of section 7.4. Any extratime you have to devote to these sections would certainly be worthwhile.WorksheetsI have included two worksheets with this
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