BIG+ideas


 * BIG  Ideas Discussed In Class​ **; hoping that this word change will make a difference! Okay, so Big vs Main, not much difference. It's in the details you provide behind these big ideas.

__**Wednesday, 11/18/09**__ (by Cindy)

We continued to explore quadratic functions in class. We saw as "a" increased, the parabola got skinnier and as "a" decreased the parabola widened. We then began to explore how to create a parabola given 2 linear functions (one being y=x). We decided when given 2 linear functions, if you multiply their y-coordinates together, the product of those coordinates are the new y-coordinates for the parabola. We also decided their were 3 critical points to know when creating a parabola, one being the vertex. The vertex is the top or bottom most point of a parabola and it halfway between the other two critical points. This occurs because parabolas are symmetric.

__**Monday, 11/16/09**__ (by Cora)

We returned to the problem "when b<0, exponents have to be integers" and discovered that fractional exponents also work in this situation. Even roots do not work, but if you take the number to an even power first, such as (-3)^6/4, to make the number positive first, you can still take the even root of it. However, for some reason unknown, the calculator will not solve that type of problem all in one step. It gives an error.

Complex #'s were developed to be able to take an even root of a negative #. For example, the square root of -1 = i, i² = -1, i³ = -i, and so on. The system will repeat itself. Complex #'s cannot be graphed on the normal coordinate plane, so they made a different one. On this plane, the expression used is a+bi.

We began to explore quadratic functions. The formula for a quadratic function is **y= ax²+c**. The graph of a quadratic function produces a parabola. By looking at the equations: y=7x²+4, y=.5x²+10, y=x²-2, y=.002x²+6, and x²+2, we decided that: //*Opening up: a is positive; c is the y-intercept*// __**Wednesday, 10/21/2009 (Dan)**__

We discussed the article **__Why Should I Justify__** by John Lannin, David Barker and Brian Townsend. The article discussed four different possible ways children justify an answer. These included no justification/procedural Justification, Empirical Justification, Generic example, and deductive Justifications. We were split into groups and each group was given a sheet of paper describing two of the justification types to the Making Ballots problem.

Afterwards we created a list of words and terms we used to discribe each Justification. No justification/ proceedural justification meant that the person who provided little to no explination as to how they got the answer or they just stated step by step their process. For Empirical justification they create a generalization for the problem after one or two instances proving their generalization correct. Generic examples merely how each component of their method applies to the problem. Deductive justification provides a clear explanation as to why their rule applies to all cases.

We went over the hand out that we received last week: Requesting a Reward and Making Ballots For the Making Ballots problem the NOW - NEXT : NEXT = NOW x 2 with a starting value of 2 For each cut you will get twice as many more ballots. The explicit formulat b = 2 to the power of c Again, for every cut you make you will get double or twice as many cuts as before. Requestiong a Reward: We were looking at how to calculate the number of ruba on each of the squares of the chess board. r = 2 to the n-1 power We graphed that and it showed a curving line upward = this is an exponetial function
 * __Monday, 10/19/2009__** (by Melissa)

We also discussed how exponents work 5 to the 4th power = 5X5X5X5 5 to the 3rd power = 5X5X5 5 to the 2nd power = 5X5 5 to the 1st power = 5 5 to the 0 power = 1 5 to the neg 1 power = 1/5 5 to the neg 2 power = 1/5X5 5 to the neg 3 power = 1/5X5X5

We practiced using the CBRs and we were trying to match different graphs: to get curves (on the graph) **we have to change our rate of change** The steepness is the rate of change

Rate of change is the comparison between the change of the x and the change of y: they don't have to be the same change, but change at a consistant rate in order for it to be linear.

AHE Inv#2 1,4,7,8 Summarize a, b Inv#3 p. 298: read through problem and complete #5 Read article that was sent to our WMU account. -pg 226 #10 - writing equivalent expressions and simplifying expressions: important to find and recognize equivalent expressions and why they work (distributive property) -pg 227 # 12- writing equivalent expressions that deal with multiple variables (area of a trapezoid) base and height pg 236 summarize part a - using a scatter plot to help figure out if two variables were related by a linear function( if there is a correlation, cloud of points scatter along a "line" then its related. if there is no cloud or patter that follows a line, or just looks like a jumbled mess, then there is no correlation and the two variables are not related linearly -we used the motion detectors along with our graphing calculators to help us connect real life situations to graphs. We learned that it was easier to match the graph after we could assign some value to the tic marks on the graph. This helps us to realize that specifying the scale when we show a graph, and it also needs to be appropriate, is an important part of understanding the data.
 * __Monday, 10/12/09__** (by Ann)

__**Wednesday, 10/7/09**__ (by Rob) - # of solutions of linear functions, and graphing linear functions and inequalities (pg. 210 #28). **Looking at the # of solutions to linear functions like c=a+bx, we found that depending on any x value there will only be one value for c. It was challenging to work through this problem because instead of looking at it as having variables c, a, b, and x a and b are constants in the problem so for any x value there will be only one possible answer for c.** -comparing graphs of two functions and making inequalities (pg. 209 #26) -revisited Now, Next; remember __**START VALUES**__ What would be good here is to think about these Big Ideas (I'm going to change the top wording from Main Ideas; guess I don't remember writing that) and write out why they are "big ideas". What about the # of solutions to linear functions? Why was it challenging to work through that problem? Begin to think a bit more deeply about the mathematical ideas.

**__Monday, 10/5/09__** (by Bettina) Reminder: Biggie Quiz next Wednesday, 10/14! __AHE for this Wed.__ p. 217 #4-8 (old a.h.e.) p. 219 Inv. 2, pick one from #1 and explain in #2, check out #3 too p. 220 Read! p. 221 Do as much as you need to be comfortable __Coming up next:__ p. 224 2, 4-8, 10, 12, 18-19, 25 - 27, p. 234 #3-7 & summarize

-Reviewed p.198 #1 about internet cafes. Writing Now-Next rules and rules with T1 and Tn-1. __*Remember to give a starting value when writing a now-next rule!*__ ie. Next = Now + .05 starting value is 3.95 For Tn = Tn-1 +.05 we set Tsub zero = 3.95 and said n starts at 1

Solving part d - To find which cafe was cheaper for what amounts of minutes Can make a graph of both equations on calculator, then see which cafe costs less Can look at a table, either made by hand or on calc., and see which is less (2nd TBLSET and 2nd TABLE are handy) Can solve two inequalities, one for each cafe
 * Either way, __be sure to justify your answer with plenty of specific details/examples!__

-Reviewed p. 206 #17 More now-next and T rules! Important to define starting value! Part c iv made us put our answer into the context of the problem. Don't just give a number - be able to explain its meaning.

-Reviewed man-fit lines on calculator and linear regression on calculator using p. 207 - 207 #18 __Be able to:__ -determine independent (x) & dependent (y) variables -create a scatter plot -manually fit a line to the plot and get the equation of the line -use the linear regression function to get a fit line and equation -predict other points (not in the original data set) using the fit line along with TRACE function and/or by using the table and typing in a new point

[|abonanni] Hey guys, sorry for being late. The big ideas from Wednesday, september 30 are:

-recursive thinking, using now-next if next equals Yn then now is Yn-1. You add your rate of change, the slope, to the now value to get your next value. Can you provide an example of this so we see what we mean by "adding the rate of change"?

-height of soccer ball kicked in the air: the slope represents velocity, (why is that true that the slope is velocity?) at the highest point the ball stops going upward, and the velocity hits zero, because the ball stops moving upward. Then the curve of the graph is coming back down to the x-axis as the ball is coming back down to the ground. The slope increases, because the velocity is increasing, but the slope is negative and is negatively increasing. Posted 12 minutes ago**** hat the sI'm thinking that for these big ideas to really be helpful you need to put some more thought into them and provide more details about how and why vs a listing of "here are the problems we did or here are the key strokes". Those points do help but why did we do those things? If we can't write why they are important, they we need to talk about that in class. So kimberfaith, I won't go to the end of your notes to add these similar comments; you get the idea.

If you read any of these summaries now and they don't make sense to you without referring back to your own notes or the text to say "oh ya", then some details need to be added. **


 * [|kimberfaith] ** Big Ideas 9-30-2009
 * Equation? Table? Graph?

Table, Set Dependent ASK Automatic (teaching tool) Independent ASK You can put in any X value and get Y

List 2nd Stat Change List(Y Will give you the change in X and Y

Y header 2nd text “done EquationLX Y list is linked to X list Linking is done by “

How to read inequalities Is less than or equal to important?

Different ways to solve equations algebraically -balancing approach -“un-doer” approach

Website: nlvm.usu.edu Balancing equations

Looking at the way students solve equations algebraically How to form a question by looking @ an equation or inequality

Velocity + going up - going down Has a direction attached to it With Speed you don’t care about direction

Flip the inequality if you multiple or divide by a negative If two equations have the same slope they should run parallel

(9/23) -//pg. 175 #14, Is the circumference of a circle a linear function of the radius? Is the Area?// After much discussion, we decided that Circumference, C = 2//[pi//]r, is in the form of y = mx + b, because 2(//pi//) is the coefficient of the variable, which is r, and the + b is +0 for this problem. Therefore, circumference is a linear function of the radius. Area, A = (//pi//)r², is not in the form of y = mx + b, so therefore is not a linear function of the radius. We also graphed both equations, and saw from those graphs which one was linear and which was not, but were cautioned not to use a graph as proof of of an equation being linear or not, because of the possibility to zoom so much on a non linear equation to make it look linear.

-We also focused more on Now-Next rules and y=, and the difference between them. We made a table of values for a problem, and decided that the Now-Next rules focused only on the y values and how they changed from one to the next. We also decided that the y= equations focused on how x and y changed in relation to each other (eg. as x goes up by one, what does y do). It cleared confusion on the two types of equations.

-//pg. 176 #20, Which of these situations involve linear functions and which do not? Explain.// //a) If a race car averages 150 miles per hour, the distance// d //covered is a function of driving time// t//.// This is a linear equation, that is put in the form of y = mx + b. d is a function of t, where y is usually a function of x, so d goes where y usually is, giving us d =. The distance is equal to 150 miles per hour (given in the problem), so the equation is d = 150t +0, or d = 150t, or the distance is equal to 150 miles multiplied by how many hours. //b) If the length of a race is 150 miles, time// t //to complete the race is a function of average speed// s//.// This is non linear, because it can't be put into the form of y = mx + b ( and also doesn't have a constant rate of change). t is a function of s, so it becomes t =. The time, t, is equal to the 150 miles divided by the speed s, so the equation is t = 150/s. //c) If the length of the race is 150 miles, average speed// s //for the race is a function of race time// t//.// This is non linear, because it can't be put into the form of y = mx + b ( and also doesn't have a constant rate of change). s is a function of t, so it becomes s =. The speed, s, is equal to the 150 miles divided by the race time t, so the equation is s = 150/t.

//-pg. 177 #23, We talked about the domain and range of certain values. a) V = 40 - 32T, where v is velocity of the ball, and t is time in seconds.// The range of T is 0 to 2.5 seconds, because that is the amount of time it took the ball to go up and then fall back down to the ground. The range of V is 40 to -40, because it starts with an upward velocity of 40, which slows to 0 when the ball reaches the top of the arch, and falls to -40 when the ball reaches back to the ground. //b) R = 500 - 133T, where R is the resale value of a video game, and T is time in years.// The range of R is 0 to 500, because the value of the game starts at $500, and can only go to $0, not be worth a negative amount. The range of T is 0 to 3.76 years, because it starts at 0 years, and only takes 3.76 years to get to a value of $0.

//-pg. 178 #25, Will certain points intersect? b) will y = 3x + 7, and y = 2 + 3x intersect//? These will not intersect, because they are parallel, having the same slope, and starting at different points. //c) will y = 3x +7, and y = 2 - 3x intersect?// Yes, these will intersect on the left side of the y axis, because t he have the same slope and one goes up and the other goes down.

-//We also discussed the pool problem (s + 2)² - s².// We decided that this is correct and appropriate because the problem gets the area of the pool including the sidewalk, and the corners, and then removes the area of the pool itself, leaving you with the sidewalk, including the corners only. It becomes (s +2) (s+2) - s². which becomes s² +2s +2s +4 -s², which becomes 4s +4, which is the original equation that we came up with for the pool problem.

(9/21) -//How can we display the Pool Problem recursively?//** Next= Now + 4 This formula is explicit (closed form) -//Can exponents represent a linear function?// No! They don't fit the slope-intercept formula. In class we discussed x squared (x^2), when we enter y= x^2 we are actually squaring the slope, which gives us a quadradic equation and a big U on the graph. (this is not a straight line so it can't be a linear function) -//Using 2 data points we can write rules for linear functions:// Pg 171 #9 d (-6,4) and (3,-8) Start by taking the change in y over the change in x __(-8) -4__ 3- (-6) Which equals -12/9 or simplified -4/3
 * We can display any problem recursively by using the Next Now formula

This -4/3 is our slope, we can use this to find the y-intercept by plugging in the slope and one of the data points y= -4/3 (x) + b 4= -4/3 (-6) + b I used (-6,4) 4= 8 + b 4 -8= 8 -8 + b -4 = b Therefore, -4 is where our line hits the y axis or it is our y-intercept

**y= -4/3x - 4
 * The last thing we need to do it put it all together!

- Using Man-Fit: -Put raw data into lists -Graph (paying close attention to window) -2nd STAT -left to go right (CALC is highlighted) -3 Manual-Fit - find first point, press enter, find second point, press enter - expression at the top is our slope-intercept formula -we can tilt our line by using arrow keys
 * //Hint: We can check our answer by pugging in the other point that was given//

Okay to keep track of key strokes but hopefully these aren't big mathematical ideas!! **

What happens if I pushed enter and my expression disapeared? - 2nd VARS - 3 Statistics... - right 2 spots to EQ, then press 1 for RegEQ, press enter
 * - Go to the home screen

-Using Linear Regressions: -Go to home screen - 2nd STAT - left to go right (CALC is highlighted) - 5 LinReg (ax+b), enter -2nd STAT L1** COMMA **2nd STAT for L2** COMMA - 2 Y-Vars, press enter - Choose which y= -press enter //You can see your slope-intercept equation by pressing the red button Y=//
 * - 2nd VARS

(9/16) -//What is zero? Zero is a whole number. -How can we convince someone the range is neg. infinitiy to pos. infinity? -Subscriptive notation: A recursive way to write an equation. Such as Tn=Tn-1+4 // I need to get more done!!! Not very many big ideas!

(9/14) -//Recursive equations: Recursive=Thinking about the same process over and over again. This idea can be expressed as an equation with a //**//Next=Now **equation. (or a subscript notation) -Expression: An expression is 4x+6. It is **not** set equal to anything. -Equation:// An equation is y=4x+6. It is set equal to something.