Chapter 3Notes for InstructorsContentThe third chapter of Long and DeTemple discusses place value and algorithms for wholenumber arithmetic. I tested students on the material from the first four sections. I spentone class discussing material from 3.5. I think this is a fun topic. This discussion was prettyrelaxed so I only included one question from 3.5 on the exam. I did not have time to coverthe material in section 3.6. Moreover, it is difficult to cover the calculator sections becausewe do not require that the students use a specific type of calculator.Notes and SuggestionsNotes on Section 3.1:Numeration Systems Past and Present• In section 3.1, Long and DeTemple introduce some of the numeration systems thathave been used throughout history. At first glance, it may appear that this section isextraneous and unnecessary, but I have found that this section really does provide afoundation for the positional system. It also introduces the concept of borrowing insubtraction and making exchanges with addition.• I tested students over the ideas from section 3.1, but I did not require that theymemorize the notation from section 3.1 because I do not think the systems themselvesare the point of section 3.1.• Manipulatives: Several types of manipulatives are discussed at the end of section 3.1.My personal favorites are the the classroom abacus and the base ten blocks. We donot have a classroom abacus, though you can improvise with some colored block and asheet of paper. We do have a set of base ten blocks. The also mention UnifixTMcubes.I do not believe that we have any of these, but we do have multilink cubes which couldbe used in a similar mannar.• An Activity Note: I have provided you with a worksheet that I had my students doin class. I did not lecture on section 3.1. I found that it was better to have them workthrough the ideas of this section in small groups. Of course, I was willing to answerquestions, but they should be able to figure out a lot of the ideas from the reading. Ibrought the Unifix block to class for the last quesiton on the worksheet. I doubt thedesigner of these blocks intended them for this purpose, but they did help the students.Notes on Section 3.2:Nondecimal Positional Systems• In section 3.2, students study the concept of place value in nondecimal system. Nor-mally students find the nondecimal positional systems frustrating. It is not uncommonto hear comments along the lines of “I will never teach this stuff to my students”and “Who would ever use this stuff?” With regards to the latter comment, you can1tell them that computer scientists use the base two and base sixteen systems. Withregards to the former comment, you can tell them that you are trying to help themunderstand how their students will feel when they begin to learn about the decimalsystem. Remember, that the decimal system may be familiar to us, but it will notbe familiar to their students. Moreover, we want these future teachers to know moreabout place value than their students.• An Activity Note with Manipulatives: They do not introduce place value cardshere, but I like these because they can be easily adjusted for problems in different basesand they are easy to make. All you need is a piece of paper and some poker chips. Longand DeTemple discuss place value cards in sections 3.3 and 3.4, but it does not hurt tomention them earlier. Place value cards can be very helpful when converting betweendifferent bases. For example, suppose you want to find the base six representation forfifty-four. Start with a pile of fifty-four poker chips. Place these in the ones place of abase six place value card. When we are finished, no place should have more than fivepoker chips. Remove a group of thirty-six chips from the ones place and exchange itfor one chip in the thirty-sixes place. This will leave eighteen chips in the ones placeand one chip in the thirty-sixes place. Now remove a group of six chips from the onesplace and replace it with one chip in the sixes place. Do this two more times. You willthen have a place value card with one chip in the thirty-sixes place and three chips inthe sixes place. Therefore the base six representation of fifty-four is 130six.Notes on Sections 3.3 and 3.4:Algorithms for Adding and Subtracting Whole Numbersand Algorithms for Multiplication and Division of Whole Numbers• I spent several days on the material from sections 3.3 and 3.4. Students should be ableto use manipulatives to justify algorithms for addition, subtraction, multiplication, anddivision of whole numbers. Moreover, they need to understand that the traditionalalgorithms for computations with whole number are not the only valid algorithms.• I am particularly fond of problem 14 in section 3.3. I was surprised at the number offuture teachers who would have told the hypothetical student in this problem that he orshe had incorrect reasoning. Many of the future teachers believed that the hypotheticalstudent had only happened on the correct answer by luck.• An Activity Note with Manipulatives: As I mentioned before, I like to use placevalue cards because they are easy to make and I find that students understand them.Certainly you can use other manipulatives such as base ten blocks (assuming that youare, in fact, working in base ten) in a similar manner. I am providing you with fourexamples of arithmetic problems which I have solved with place value cards to showyou how you can use these cards with