Background. A Transformation from Traditional College Algebra to Freudenthal College Algebra. Background. Traditional

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1 A Transformation from Traditional College Algebra to Freudenthal College Algebra Monica Geist, Ph.D. Front Range Community College Background Started teaching College Algebra in 1997 Very traditional/formal algebra Logarithms were a disaster November of 005 Attended a 4 hour Realistic Mathematics Education workshop at CU Henk Van der Kooij presented on logarithms Background Spring 006 asked Henk for his help with logarithms Summer 006 Henkwrote a replacement chapter for Exponential and Logarithms Fall 006 Pilot tested this chapter Spring 007 Changed entire course modeling after the Freudenthal philosophy Traditional No education courses required State mandated curriculum Book dependent As books changed, so did we Followed the order the topics were presented in the book No learning trajectories built into these book Freudenthal Traditional Must know your pre requisites don t have time to re teach these.1 Lines and Slope. Distance and Midpoint Formulas; Circles.3 Basic of Functions.4 Graphs of Functions.5 Transformations of Functions.6 Combinations of Functions; Composite Functions.7 Inverse Functions

2 Traditional Day One Point-Slope Form of the Equation of a Line The point-slope equation of a nonvertical line of slope m that passes through the point (x 1, y 1 ) is y y 1 = m(x x 1 ). Traditional Formal Algebra Equations of Lines Point-slope form: y y 1 = m(x x 1 ) Slope-intercept form: y = m x + b Horizontal line: y = b Vertical line: x = a General form: Ax + By + C = 0 Freudenthal Day One LINEAR GROWTH Suppose gasoline costs $.50 per gallon. Draw a graph that models putting g gallons of gas into a car. Let the horizontal axis be number of gallons and the vertical axis be the total cost, in dollars. Let C be the total cost for the gasoline. Let g be the number of gallons. Create an equation that models the scenario described above. Traditional Polynomials The Standard Form of a Quadratic Function The quadratic function f (x) a(x h) k, a 0 is in standard form. The graph of f is a parabola whose vertex is the point (h, k). The parabola is symmetric to the line x h. If a 0, the parabola opens upward; if a 0, the parabola opens downward. Freudenthal Polynomials Calculator Exploration What do you get if you multiply a line times a line? What do you get if you multiply three lines? What about four lines? Freudenthal Polynomials Linear function times linear function results in: Quadratic function (graphically) Line times line results in parabola Linear function times linear function times linear function results in: Cubic function (graphically) Line times line times line results in a cubic graph

3 Freudenthal Polynomials Linear Factorization Theorem All polynomials, f(x) = a n x n + a n 1 x n a 1 x+a 0, can be written as a product of linear factors f(x) = a n (x c 1 )(x c ) (x c n ) Traditional Exponential Functions Definition of the Exponential Function The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than 1 ( b 0 and b 1) and x is any real number Here are some examples of exponential functions. f (x) = x g(x) = 10 x h(x) = 3 x+1 Base is. Base is 10. Base is 3. On the same day two friends both buy a foal. Both foals have a weight of 50 kilogram. After one month they compare the weights. Fred says, "My foal grew 10 kg." Andy answers, "My foal grew 0 %. After another month they meet again and compare the weights. Fred: "another 10 kg!"; Andy: "another 0%." 1. What are the weights of the foals two months after they were bought?. Compare the weight of Andy's foal after two months to the weight on the day it was bought. What is the percentage increment (percent increase)? # months after purchase Weight of Fred s foal = x 50 = = x 60 = Weight of Andy s foal # months after purchase Weight of Fred s foal = x 50 = = x 60 = = x 7 = 86.4 Weight of Andy s foal = x 86.4 = = x = = x = = x = = x = The growth of Andy's foal deserves a closer look! Looking at the weights in the table, you can find the following relationship: weight this month 1. weigth last month

4 This relationship can also be written as: weight this month = 1. weight last month We call the consecutive weights of Andy's foal: W A (t) with t = 0, 1,, 3,... With this function we can write the statement as: W A (1) = 1. W A (0) W A () = 1. W A (1) = W A (0) = 1. W A (0) W A (3) = 1. W A () = W A (0) = 1. 3 W A (0) W A (4) = 1. W A (3) = W A (0) = 1. 4 W A (0) Bacteria: Escherichia coli (E. coli), is one of the main species of bacteria that live in the lower intestines of mammals including us. They are necessary for the proper digestion of food and are part of the intestinal flora. The human body has the same number of human cells and E.coli cells: In real life, E.coli duplicate every two hours. Notice what E. coli looks like magnified to 10,000x. E. coli magnified to 10,000x. Suppose a culture of E.coli is grown in a lab. We put 18 of them in an ideal environment and they duplicate every two hours. How many are there after one day? And how many after one week? The number of E.coli is E(t) and t indicates the time (in units of hours) after the start. What formula describes the number of E.coli as a function of time t? How many are there after one day? 5488 And how many after one week?.48 x 10 7 The number of E.coli is E(t) and t indicates the time (in units of hours) after the start. What formula describes the number of E.coli as a function of time t? E(t) = 18*() t A growth process that can be described with a function like G(t) = c(g) t is called an exponential growth process. Linear growth is characterized by: every fixed time step results in a fixed addition Exponential growth is characterized by: every fixed time step results in a fixed multiplication.

5 Traditional Logarithmic Functions Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log b x is the logarithmic function with base b. Location of Base and Exponent in Exponential and Logarithmic Forms Exponent Exponent Logarithmic form: y = log b x Exponential Form: b y = x. Base Base Traditional Logarithmic Functions Properties for Expanding Logarithmic Expressions For M > 0 and N > 0: 1. log b (MN) log b M log b N M. log b log N b M log b N 3. log b M p plog b M Let s return to exponential growth. Duckweed (or lemna minor) is a type of plant that multiplies fast. The number doubles every day. A pond that has this plant can get completely covered in a short time. The surface then looks like green grass. Because the area that is covered is important we will look at the area (in m ) covered by the plant instead of thinking about numbers of individual plants. Beginning with 1 m of plants on one day (starting time t = 0), after 1 day (t = 1) there are m of it. The formula that describes this process is a = t The graph shows this process for the first four days. Use this graph to answer the following questions. 1. At what moment will there be 3 m of duckweed in the pond? And 6 m? And 1 m? Now without the help of the graph: at what moment are there 4 m of duckweed?. The same question for 5 m, 10 m and 0 m area (in m ) time (in days) 3. Complete the empty cells in the following table using the information from problem 1 and. 3. Complete the empty cells in the following table using the information from problem 1 and. Area (m ) Time (days) 0 1 Area (m ) Time (days) 1.6 Area (m ) Time (days).3 Area (m ) Time (days) 0 Area (m ) Time (days) Area (m ) Time (days) Area (m ) Time (days) Area (m ) Time (days)

6 a t a t a t a t t() t(3) t(6) t(4) t(6) t(4) log log 3 log 6 log A log B log ( AB) a t a t a t a t t(0) t(5) t(4) t(15) t(3) t(5) log 0 log log log B log A log 4 A B a t a t a t a t t() t() t() 3* t() t(8) 1 log log log A B B log A

7 Traditional Systems of Equations Solving Linear Systems by Addition If necessary, rewrite both equations in the form Ax + By = C. If necessary, multiply either equation or both equations by appropriate nonzero numbers so that the sum of the x-coefficients or the sum of the y- coefficients is 0. Add the equations in step. The sum is an equation in one variable. Solve the equation from step 3. Back-substitute the value obtained in step 4 into either of the given equations and solve for the other variable. Check the solution in both of the original equations. Freudenthal Systems of Equations How many bananas do you need to balance the third scale? Explain your reasoning. Traditional End of Term Students felt rushed through the course No depth of knowledge Even A students complained of not understanding what logarithm were Rates of A, B, C s ranged from 5 40% Rates of D, F, W s ranged from 60 75% Freudenthal Students come alive and are more engaged Students ask deeper questions Students can explain what they are doing and have a deeper understanding of concepts 95% of students can explain the meaning of a logarithm Rates of A, B, C s ranged from 55 65% Rates of D, F, W s ranged from 35 45% Comments from Students I understood the main idea. It wasn t just performing problems. It was understanding why behind the problem. It used to be if you put a problem in front of me, I couldn t do it unless you told me what you wanted me to do here I understand everything and can just do it. Comments from Students My big math moment was when I realized the relationship between logarithms and exponential equations. My worst math experience was one day, while I was driving home, I realized I was trying to model the car in front of me with a graph.

8 Teaching Algebra: An Autoethnographic Poem algebra what is it good for when will we ever use this algebra skills do not transfer to life algebra skills do not transfer to critical thinking am I wasting my life by teaching algebra disillusionment disenchantment disappointment I must change my career do something useful contribute to society in a better way While returning to school with new career plans in the works Realistic Mathematics Education entered my life sure, I ll try it I ll try anything that might help teaching algebra is just temporary I still need this job for now different focus different style slower pace deeper learning more engagement increased participation class is turned upside down high achieving students are frustrated can t memorize their way out of this unit low achieving students finally wake up no memorizing needed just thinking thinking about the problem what it means student responses it was surprisingly enjoyable I understood it very well we talked a lot about exponential/logarithms in many different settings seeing it over and hearing it over helps a lot I came out of this with some understanding the visual charts and graphs were very helpful I did feel a different type of learning going on inside my head I enjoyed the graph discussions can you teach like this every day I had a good day I finally understood math today

9 teacher response I had a good day, too I finally taught math today