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Functions black box

Directions:
1. Load one of the following problems with a Geogebra-enabled Ipad.
2. Then, play around with the input in order to determine what the function is.
3. Keep track of your inputs and outputs, creating a table and graph with your data.
4. When putting inputs in don't press enter to engage the machine, instead put your desired input into the Geogebra "input bar" button and press enter.

Parent Functions:
Function Machine Problem 1
Function Machine Problem 2
Function Machine Problem 3
Function Machine Problem 4
Function Machine Problem 5
Function Machine Problem 6
Function Machine Problem 7





Transformations:

Permutations and combinations

I like this warm up because Students will take a variety of methods to represent it.  There are only two shapes, and its a quick problem that gets down to the idea of choosing.  

After this I ask them to extend their thinking with this ordered horse race problem:

common student  responses include 3*8, 8^3, 8*7*6

and most student responses are "its the same" 

Venn Diagrams, Probability and Roulette

This is one of my favorite days during the probability unit.  It isn't often I get to gamble with my students and say its educational.

I begin this day with a quick warm up involving placing cards in the right parts of the Venn diagram.  Then, I ask them a few questions about it, just as review of the notation.

The last one I stop myself and ask the students how to even say that and inevitably someone says not A or B, to which I have to make a point.

After everyone is okay with the basic of the Venn diagram I preface the next activity with "Don't tell you parents we're playing this but does anyone recognize this game?

Then I start taking bets.  I draw two student names at random, inform them what they could bet on and place their bets.  Before I spin we have to model the situation with a Venn diagram and talk about intersections, unions, and compliments.

Here is the graphic organizer

Imaginary numbers

Day 1: take notes on what imaginary numbers are.  Boring, but it is a topic that always gets the students riled up.  Their feather get ruffled once they know that their 7th grade teacher was mistaken when they were told "you can't take the square root of an negative number."  in one such instance I got to pull out my Star Wars  quotes and tell a student to "let go of their hatred."   Most of the students thought imaginary numbers were kind of cool, some thought I was making it all up and playing a trick on them, and unfortunately some through it was the dumbest thing ever and they should be learning something "useful."

On day two i show most of this video, and put it in the context of a robotic arm rotating to pick things up at various points.

Then we get to the good stuff.  I give students a little packet and have them do the top problem, they double check with me and if they did it right and get a quick "yep" or "not yet," they get to crumple up their paper and shoot it from the free throw line if they're correct, otherwise they have to keep working on the problem.  There is only get the one shot per correct answer and if they miss then they have to go fetch their shot and put it in by hand.   For all but one of my hours this was a great alternative to a worksheet, and I really saw them asking each other how to get the answer.  Who knew throwing the piece of paper would be such a great motivator?!

The questions increase in difficulty, making for great differentiation.  Students can go at their own pace and can know that they are right before continuing.  The pace is fast, and I was relatively surprised at how hard students tried on something that they had just the previous day been questioning the relevance of.

Here is the template, just photocopy it in order and put the staples in and you'll be ready to go after its cut up, no need to organize the booklets.

Quadratic patterns

For this lesson I hand students a slip of paper as they enter in the door, one that has a quadratic pattern printed on it.  Their 1st task is to make a guess, no calculations, how many squares/circles/shaded squares with the 50th term have?

All the patterns used for this lesson were taken from http://www.visualpatterns.org/

Then I will take a few answers and put them up on the board to incite some competition and buy-in from the students.

At this point I may show them how to do it, or remind them if they already know.

They have some time to work, most of the students can accurately find the equation using the table and the process I trained them with to find the equation from the table, but fewer of them can look at the pictures and write an equation relating x to side lengths and so forth.

Here are the patterns that we have been using:

Here is some student work that shows the process of finding the equations:

some students can look as the figures and relate the parts to x, but it is more common that they use the table approach as it is more formulaic.

Drawing the next picture is crucial to getting enough data for using a table.  It allows students to get the 2nd differences and work backwards to find the y-intercept.

Most of the questions that I received, or the students that I helped that were stuck had miscounted the squares, thus finding no pattern.

After they have done the work to generate the equation they use it to find the value of the 5th term, and at the end of the hour we look to see whose guess was the closest.

Students then go home with this practice Hw.  The first pattern is actually linear, and although it confuses students because that's not the unit we're in it is a good connection to make.