Chapter 8Notes for InstructorsContentAlgebra is the focus of this chapter. Students should be able to:1. recognize patterns2. write an algebraic expression for the nthterm in a sequence (which may be given as asequence of geometric diagrams)3. write an algebraic expressions for a situation posed in a word problem4. use the equality sign “=” appropriately5. solve simple equations6. understand the concept of a function as a formula, as a table, as a machine, and as agraph7. understand the concepts of domain and range8. explain the vertical line test9. identify points in the Cartesian Coordinate System10. apply the Pythagorean Theorem to find the distance between two points in the Carte-sian Coordinate System. (I am trying to help my students understand the DistanceFormula. Most people remember the Pythagorean Theorem. Few rember the DistanceFormula. I want my students to be able to solve distance problems even if they forgetthe Distance Formula.)11. understand that the slope of a line is invariant. (This could be used as the definitionof a line.)12. write the equation of a given line13. graph linear equations14. understand the Condition for ParallelismI spent about two weeks on this chapter, but I certainly could have spent much more time.I really think that I should not have spent more than one day reviewing MA 201 materialat the beginning of the course. I think it would have been better to review throughout thesemester. It was hard to teach this group of students because many had not had MA 201in its present form. Even so, I think it would have been best to jump right into the newmaterial and review as it became necessary.1ManipulativesThere are manipulatives that can be used for Algebra. I do not believe that we currentlyhave any of these manipulatives at UK. Nevertheless, there are some ideas which can be usedto make concepts in these chapters easier for students. I found that the traditional notion ofa function as a machine was quite useful (as you will see in the worksheet for this chapter).Notes and Suggestions:Notes on Section 8.1: Algebraic Expressions and Equations• A lot of the students in this course have difficulty with definitions. Consequently, theyhave difficulty writing what they mean. For example, when my class was studyingthe Geometry chapter, I saw the following sentences: “The sum of the angles is 360o”and, worse yet, “The angles are 360o.” My students meant to (but did not) say thatthe sum of the measures of the angles is 360o.” On another section in the Geometrychapter, my students had a lot of difficulty with a question regarding regular polygons.They did not read the section carefully. Consequently, they did not know that theregular polygons are a proper subset of the polygons. They just don’t fully understandthe precision involved with definitions. I tell you this a few chapters early because Ibelieve you need to stress the importance of of precise and acurate writing from thevery first day of this class. In particular, with respect to section 8.1, students shouldbe very careful with the “=” sign. It should not be used to represent “implies that.”They should only use it when quantities are indeed equal. For example, suppose astudent must square 3 and then add 1. It is not uncommon to see the following workon a student’s paper: “32= 9 + 1 = 10. Clearly, this is false and should be corrected.I think your only hope is to start early and to deduct points. I am not saying thatyou need to deduct a lot. Usually, a deduction of half a point will get their attentionwithout really affecting their grade.• I really like the rate problem in Example 8.5 on pages 484–485, but there are not anyrate problems in the exercises. I have included several rate problems on the Tradi-tionally, students have difficulty with these types of problems. I like these problemsbecause they involve fractions which is a good review. I also like these problems be-cause they provide a good writing exercise. Students should not be given credit forsimply solving an equation. They should explain how they arrived at the equation thatthey must solve.• I am fond of problem 13 in Section 8.1.Notes on Section 8.2: Funtions• I found that it was very useful to view functions as machines. I like to use the machinesto write compound functions. You can then work your way backwards through themachines to solve simple equations. For example, if we want to solve 2(x + 1)2= 8, wewrite the lefthand side as a sequence of machines. First x goes into the +1-machine.2From there it goes into the squaring machine. Finally, it goex into the ×2-machine.So, if we need to solve this equation, we must first divide both sides of the equation by2 (which undoes the ×2-machine), then we take the square root of both sides of theequation (which undoes the squaring machine), and finally we subtract 1 from bothsides of the equation (which undoes the +1-machine).• Students know about the vertical line test, but they should be able to explain thevertical line test.• I like Problems 3, 9–11, and 15–16 in Section 8.2.Notes on Section 8.3:Graphing Functions in the Cartesian Plane• I think it is a good idea to pass out the graphs of several lines to the class. Make sureto include a vertical line, a horizontal line, and a pair of parallel lines. Choose severalpoints on one of the lines. Calculate the slope of the line using different pairs of points.Students should come to the conclusion that the slope is invariant. Have them find theslopes of the parallel lines. Perhaps you could also include a pair of perpendicular lines.Have them find several points on the horizontal line and the vertical line. What is trueof the y values on the horizontal line? What is true of the x values on the vertical line?I have included a graph containing several lines with this documentation.• I found that my students had been graphing equations for years without truly under-stand what the graph of an graph is. It might not hurt to mention that a point is onthe graph if and only if it satisfies the equation.• I think it is really important that students in this class stop memorizing formulas withthe exception of some basic formulas. Students know the Pythagorean Theorem, butthey forget the Distance Formula (or they cram it before a test). I think it is importantthat they be able to use the Pythagorean Theorem to find the distance between twopoints. I had my students plot two specific points and draw an