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WELCOME TO THE REAL WORLD! LINKING ALGEBRA TO REAL WORLD SITUATIONS

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STRAP STRESS Teaching Guidelines Subject: Physical Science, Mathematics Topics: Forces (Stress and Strain), Algebra (Interpreting x-y graphs) Grades: 7-10 Concepts: Stress Strain Knowledge and Skills: Can relate aspects of a graphical model to the real world situation which is being modeled

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Procedure: Prepare for presentation the Futures Channel movie, “Designing Backpacks.” Tell students that as they watch the movie, you want them to think about this question (which should be posted): What qualities or characteristics would you look for in material that is used to make a backpack? At the end of the movie, ask students to work in teams of two or three to list answers to this question. Generate a list of all answers, and be sure that it includes the characteristics of “strength” and “weight.” Through further discussion, guide the class to the agreement that the ideal backpack material would be both light and strong.

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Ask students now to think about how they would measure the strength of a piece of material being considered for use in making backpacks. Discuss some answers, then distribute the handout to the teams. Ask the teams to study it and talk it over for a few minutes. As a class discussion, ask students to summarize the problem, so that it is clear to everyone what they are being asked to do.

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Ask questions to ensure that they understand how the graphs each show a relationship between the variables “stress” and “strain.” Student descriptions should reflect their understanding that the top graph shows a material which stretches very little at first as more stress is applied, then suddenly tears, so that a little more stress causes a lot more stretching. The material in the bottom graph, on the other hand, stretches continuously as more force is applied. Since a backback strap should NOT stretch when the backpack is loaded, the material shown in the top graph would be preferred for backpack use.

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Strap Stress When force is applied to a material, it will usually cause the material to change shape, even if just a little bit. In such a situation, the force that is applied is called “stress,” and the change of shape of the material is called “strain.” For example, when you sit on a wooden chair, the weight of your body exerts stress on the chair. The wooden frame of the chair will bend, very slightly, and with the right instruments, you could measure exactly how much it bends--the strain.

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The two graphs below represent the relationship between stress and strain for two different types of material, which are being considered for use as backpack straps. The graphs were created by applying a stretching-type stress to the material, and measuring how far it stretched (the strain). 1)Describe what each graph shows, in as much detail as you can. 2) Which material would you choose for the backpack strap, and why?

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GENERAL GUIDELINES FOR TEACHING ALGEBRA Curriculum – Often times, the curriculum adopted by school districts moves too quickly and students may not fully understand the current concept before they are forced to move on to the next concept.It is important that teachers provide students with ample time to learn the concept and a sufficient number of opportunities to practice the concept. Language of mathematics – It is important that students be able to define and use algebra terminology. A list of important vocabulary terms and strategies for teaching them are included in this case study. Prerequisite skills – Students must master prerequisite skills prior to learning algebra. These skills include but are not limited to basic facts, problem solving skills, and probability skills. It may be necessary to review these skills prior to working with algebra concepts.

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Modeling by teacher – Teachers must model strategies prior to allowing students to complete work on their own. During modeling, teachers talk aloud as they demonstrate how to solve a problem. This should be continued until enough problems have been modeled so that students understand the concept and how to use the manipulatives provided. Real-life examples – It is imperative that algebra problems be related to real-life situations. Students often ask why algebra is necessary; relating it to real-life situations will encourage the connection. It is also important for students to have strategies for deciding how to set up the problems they need to solve. Effective instruction – Teachers must make certain they understand algebra well enough to teach it to their students. Effective teaching behaviors (e.g., specific praise, questioning) should also be included in all lessons. Error analysis – Error analysis is the process of looking closely at student errors to determine what they are doing incorrectly. Error analysis can be done by examining the problems or by interviewing students and asking them to demonstrate what they have done.

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Reviewing material – It is important that students receive ample opportunities to review what has previously been learned, in order for them to maintain the knowledge. Calculators – The use of calculators will assist students in completing complex math problems, including algebra.The use of calculators in algebra is complex, and students will need explicit instruction on how to use the calculators. Concrete materials – The use of concrete materials or manipulatives will assist students in understanding the abstract level of algebra. These manipulatives may include algebra tiles, algeblocks, or other items. Promoting a positive attitude toward math – Teachers must show enthusiasm when teaching algebra.

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Real-life examples It is imperative that algebra problems be related to real-life situations. NCTM: Connections Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. Recognize and apply mathematics in contexts outside of mathematics

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Students often ask why algebra is necessary; relating it to real- life situations will encourage the connection. NCTM: Problem Solving Build new mathematical knowledge Solve problems that arise in mathematics and other contexts Apply and adapt a variety of appropriate strategies to solve problems

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It is also important for students to have strategies for deciding how to set up the problems they need to solve. NCTM: Representation Create and use representations to organize, record and communicate mathematical ideas. Select, apply and translate among mathematical representations to solve problems. Use representations to model and interpret physical, social and mathematical phenomena.

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ALGEBRA TIC-TAC-TIMES FROM “THE MATHEMATICS TEACHER” (NCTM)

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"Algebra tic-tac-times" combines mathematical skills with a competitive strategy. It is a highly motivational skill-review exercise that involves the problem-solving strategy of working backward.

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Directions for Play Objective The object of the game is similar to that of tic-tac-toe; the winner is the first of two players to place four tokens in a row, either vertically, horizontally, or diagonally. Materials The materials necessary to play algebra tic-tac-times include a factor board and a game board (fig. 1) and forty translucent tokens of two different colors. One token of each color is used as the factor marker, and the remainder are used as game tokens. The game board should be laminated so that it can be saved from year to year. The tokens should be stored in a bag so that they don't get lost. Method of play Player 1 begins the game by placing a factor marker and one of player 2's factor markers on any factors on the factor board. The product of these factors determines the placement of player 1's game token. In figure 1, player 1 placed a factor marker on x + 1 and player 2's marker on x -1. Player 1 then placed a game token on x^2 -1 because (x -1)(x + 1) = x^2 -1.

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x²-7x+12x²-3x+2x²-16x²+8x+16x²-x x²-5x+4x²-4xx²+2x-3x²+xx²-1 x²-8x+12x²-5x+6x²-4x+4x²+7x+12x²-2x-8 X-4X-3X-2X-1xX+1X+2X+3X+4

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x²-7x+12x²-3x+2x²-16x²+8x+16x²-x x²-5x+4x²-4xx²+2x-3x²+xx²-1 x²-8x+12x²-5x+6x²-4x+4x²+7x+12x²-2x-8 X-4X-3X-2X-1xX+1X+2X+3X+4

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QUADRATIC FORMULA PROBLEM

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Hitting a Home Run With Algebra

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Quadratic Formula

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BAGEL ALGEBRA NCTM ILLUMINATIONS

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Objectives : Decipher series of equations and interpret results Symbol manipulation Consider proportional relationships and interpret results

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COMPARE ROTE ALGORITHM PRACTICE CONNECTING TO REAL PROBLEMS

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EXETER MATHEMATICAL INSTITUTE PRESENTED TO CMSD TEACHERS OVER THE PAST 3 SUMMERS AT JOHN CARROLL UNIVERSITY

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ALGEBRA I Tying it into the OGT preparation time

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PAGE #6 -- PERIMETER For what values of x will the square and the rectangle shown at right have the same perimeter? X + 5X + 3 X + 7

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PAGE #9 – VOLUME AND SURFACE AREA The diagram at the right shows the wire framework for a rectangular box. The length of this box is 8 cm greater than the width and the height is half the length. A total of 108 cm of wire was used to make this framework. a.What are the dimensions of the box? b. The faces of the box will be panes of glass. What is the total area of the glass needed for the six panes? c.What is the volume of the box?

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PAGE #9 – SYSTEM OF EQUATIONS When asked to solve the system of equations 5x + 2y = 8 x – 3y = 22 Kelly said “Oh that’s easy—you just set them equal to each other.” Looking puzzled, Wes replied, ”Well, I know the method of linear combinations and I know the method of substitution, but I do not know what method you are talking about.” First, explain each of the methods to which Wes is referring, and show how they can be used to solve the system. Second explain why Wes did not find sense in Kelly’s comment. Third, check that your answer agrees with the diagram.

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GEOMETRY Tying it into the OGT preparation time

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PAGE 1 -- #4 – DISTANCE FORMULA Instead of walking along two sides of a rectangular field, Fran took a shortcut along the diagonal, thus saving distance equal to half the length of the longer side. Find the length of the long side of the field, given that the length of the short side is 156 meters.

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PAGE 8 -- #8 -- TRANSLATIONS A triangle has vertices A = (1,2), B = (3,-5), and C = (6,1). Triangle A’B’C’ is obtained by sliding triangle ABC 5 units to the right (in the positive x-direction, in other words) and 3 units up (in the positive y- direction). It is also customary to say that the vector [5,3] has been used to translate triangle ABC. What are the coordinates of A’, B’, and C’? By the way, “A prime” is the usual way of reading A’.

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PAGE 8 -- #9 -- TRANSLATIONS (Continuation) When vector [h,k] is used to translate triangle ABC, it is found that the image of vertex A is (-3,7). What are the images of vertices B and C ?

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PAGE 8 -- #7 – DISTANCE-RATE FORMULA Leaving home on a recent business trip, Kyle drove 10 miles south to reach the airport, then boarded a plane that flew a straight course – 6 miles east and 3 miles north each minute. What was the airspeed of the plane? After two minutes of flight, Kyle was directly above the town of Greenup. How far is Greenup from Kyle’s home? A little later, the plane flew over Kyle’s birthplace, which is 50 miles from home. When did this occur?

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IMPORTANT NUMBERS Session # Topics# 14, 21, 41 Professional Development Goal# 200 Session Outcome 1 Strategic Focal Point# 1 Hours 6.0

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