Graphing non-linear inequalities

Learning target: students can graph non-linear inequalities.

This is a multiple day lesson, where students first start with a game of graphing pictionary.  I hand students this packet and tell them that the person holding it must


Students have to complete the 4 problems

I've also learned that these types of activities work great but they have to be accompanied by practice almost immediately after in order to cement the ideas down so students can actually apply the concepts.


The second day I open with a warm up that requires students to write the equations for a system of inequalities based off of a graph.  Then I hand them a sheet, tell them to figure out what the equations are.  the most important part is that they are going to double check themselves on their own calculator.  I've learned by now that with my 11th grade students activities always run smoother if the lesson is student centered.  They must be able to check their own work.

This turns out to be a great review of function transformations as well

Linear Programming high challenge

Learning target: students can solve linear programming problems unassisted.



students get to pick from one of the three problems and make a poster about it.  I show a poster of previous student work of Joe's coffee shop, which is a problem all of them have already solved.:

(most of the students choose the juice problem)



One thing that could make this activity better is if these same students had to complete one problem of the high support, where there is no context.

Anticipate students defining their variables half haphazardly, which can lead to confusion latter, especially for the juice problem where students have to convert their unit to be in all quarts or all gallons.

Linear programming

Learning Target: students can maximize the profit of a system of inequalities.


The lesson starts off based on the previous walk-though of feasible region and Fred's coffee shop.  then, if students understand they move forward, and if not we go back to basics.

Determining factor: can the students do this problem while using Fred's coffee shop as a guide






High challenge:
Students pick on of three problems and make a poster that shows the solution using the template we have been using.

Anticipate: students not defining x and y well enough.


High support:
 Here are the three questions.  Students had to pick one depending on where they think they are at.

Question 1 student work

:

Question 2 student work

Question 3 student work.

Scavenger Hunt-feasible region and linear programming introduction

Learning Target: students can identify points inside and outside of the feasible region.

Intro: I hand out the problem to each student as they come in the door and most of them can begin without any prompt.

the idea is that we are going to set up a scavenger hunt and need to set up the boundaries using a system of inequalities. (thank you Google Maps)

Students graph these and label 5 points that would serve as suitable stations for the scavenger hunt.  Inevitably, one student will pick some point where x=4, which gives us a good talking point when we review it.

there is one question about the northern-most point, which students will eyeball from there sometimes haphazard graphs, which gives us an avenue to review solving systems.

Plugging the point back in and finding the y-location is not always easy to remember.

Here is a PDF of the lesson, and below is the worksheet.

Scale of the Universe

Lesson objective: students can plot data of varying magnitudes on a semi-log scale.

This is a lab day, and students log on to find this website: http://htwins.net/scale2/lang.html

They then pick 12 objects of various sizes and write down their names and sizes in meters.  The interactive website displays data between the two extremes shown below, and naturally student choose some from each side of the spectrum, which makes it hard to graph using a base 10 scale

As students enter I hand them this worksheet that will guide them through the activity.  After they pick their 12 objects they plot them on semi-log scale graph paper.

Logarithmic Lions

Lions: 1,2,...many, logarithmic thinking.

This lesson was designed after a national geographic article I read some years back, and I wish I had a copy or knew the name of the article.

Here's the lesson:



Show this flip (its in Smart Notebook, sorry ActiveInspire).  Each student has to write down the number of predators they see, but I only show the slide for a fraction of a second.





Which leads to the idea that the human brain was wired to think logrithmically.  Evolution favored those who could discern the difference between 1 lion, 2 lions and 3 lions, but no favor is granted to those who can tell the difference between 9 and 11, as they are dead meat anyways.

This intro is a fun activity and is a non-threatening way to show semi-log scales.

Formal introduction to logarithms

Learning target: students can apply logarithms

DAY 1

1st I throw this puzzle at them, and they try to solve for the unknown with guess and check.  I give them about 5 minutes, which allows them to get close on about 3 or 4 of them.  One student may even figure out the relationship between 3=10^x and 30=10^x.

Common questions:
"Can we write on this?"
"How close do we have to be?"
"Is there an easier way to do this?"

then I introduce the common log as a way to undo a base, while first reminding them that they just learned how to undo a power (raise to the reciprocal power).

I solve 1=10^x by taking the log of both sides, and show how things reduce.  Then students find the rest of the values by using the log function. 

Then we talk about how this is all in base 10 and the 10 is not written but is implied.  Someone, almost on cue, asks "how do you solve if the base isn't 10?"  which leads us into the notes on uncommon logs.  I hand out a slip (calculator instructions) that students glue into their notebook, and go over this process.  then we do one example, base 3.

At this point I don't have time to prove the rules so I show them were they come from and we write them down in our exponent booklet. for example 2^3*2^5=2^(2+3)=2^8 leads to the log property log(a*b)=log(a)+log(b)

Exit slip: expand this logarithm

Day 2: We talk about how the scale they used in the Moore's law activity is a logarithmic scale, increasing by exponents. then we look back on the logarithmic lions, and talk about how the scale is adjusted.  then its back to logarithms rules in their booklet.  I will on the side also introduce natural logarithms but we wont work much with them.  Homework time.  This introduction on day 2 was too boring, and needs to be beefed up.  Also, I should have done a few more example on the homework packet before letting students loose.

Cowboy Bob vs. Big-City Slick

Learnign target: students can model multiple years at a time with compound interest.

Inverse functions and Sneeches

Learning target: students can write equations for inverse functions

First we start out reviewing composition of functions f(g(x)).  I've never taught inverses this way before so we'll see.

This lesson is partially a tribute to a personal hero of mine.

after they've been working for a while we take a break by watching this video, its about 12 minutes and not about math, but students are due for something novel in class, they've been grinding the wheel for a while, so we watch this:

Here is the worksheet that I've been using:

Dead Puppy Theorem

I have to say I stole this from a college of mine, Chris Lusto wrote a post about it:
http://linesoftangency.wordpress.com/2012/01/20/the-dead-puppy-theorem/

Dead puppy theorem: every time a student distributes an exponent to a binomial a puppy dies.  That is...

I introduced the idea during a warm up, and the students really liked it.  It always helps to throw up a picture like this:

would you kill this puppy?

Every so often a student kills a puppy:

Exponential Alice

Learning target: students can model exponential situations.

This was a good lesson for sorting out repeated multiplication issues.

Dispite the fact that I;ve been writting equations with ratios like this for some time students still want to multiply the 4 by 30% and add it back on to 4.  Aside from the fact that such steps make it hard to see what the pattern is (are we multiplying or adding? Both?) the only problem I see with this is an economy of calculations, if a student makes a mistake they feel like they have to start over because they were balancing so many things in their head at the time.

On the worksheet students are asked to calculate how tall she would be if she ate 2 oz of cake.  When students rush to calculate 2 oz without first calculating 1 oz they usually do 4(1.00+.60) because 30%+30%=60%.  

When modeling this with an equation a common mistake is for students to write y=x(1.3), mixing up the recursive and explicit method of describing this pattern.

Then, every so often I can tell that students want to know this stuff.  I find evidence of a student putting in EXTRA effort to make sense of a problem.  #2 show the meaning of each part of this equation.  Beautiful.

Here is an available version of the worksheet:

Moore's Law

Learning target: students can make predictions based on exponential growth

Right off the bat I throw this packet at them and ask them to read it.  Its the MIT photo essay of various computer chips and transistors throughout computer history.


I ask students what their questions are and I get a range of answers including:

"What is this?"
"How does this relate to what we are doing in class?"
"Why are we reading this?"

So I tell them that the first picture is the birth of the digital age, and give them a brief overview of what a transistor is and how it related to computing power.

From their questions it was clear I was not going to get a question that would involve making a prediction so I gave them the task:

1. how many transistors are their in a computer chip made in 2013/2014? 

(This we can look afterwards and see what reasonable answer would be)

2. When will the number of transistors pass the trillion transistor mark?

Step 1: make a guess.  I ask students to make a guess, based on what their gut says and the data points. I can steer them a little "what number do you know it has to be bigger than?"  Studies show student engagement increases if they make a guess 1st.  I had students write down their guesses and I ut them into an excel file.

Step 2: attack!  It took a while for students to even begin to start this problem.  Then eventually someone figured out they had better make a table just to get something.

Then they found the common differences and realized that wasn't useful, so to set students up for the next part I told them a graph was another strategy they could try.  After some time I got the question "Mr. Olson, what should the scale be?  How am I going to fit 1 and 371 million on the same graph?"  -this will be a great lead in to logarithms.

While trying to make an exponential line of best fit students needed to know how to find the average ratio from year to year.  This proved difficult since the data isn't pretty and the years aren't uniform.

Exponent rules 4-in-a-row

Learning target: students can apply exponent rules.

This activity comes from http://busynessgirl.com/exponent-block-and-factor-pair-block/

It was a very engaging way to practice exponent rules, which can be a very dry topic.

Two students playing one another.

 A veiw of some sample calculations, scratch paper is a must for this activity.

What I liked about this is none of my students picked up their calculator. 

Exponent rules

Learning target: students can apply exponent rules


to try to shake up the way we take notes on this we make a little booklet of all the exponent rules.  I saves the last page for Logarithms and the second to last for examples of solving.

questions to address: if 2^5/2^3=2^2 than what does 2^3/2^5 equal?  If all the twos on top cancel out whats left on top?  Zero?

how do you expand (3x^2)^3 with the power to a power rule?  We need to talk more about making 3=3^1

after which I gave out this A-B sheet stolen from f(t)

Row-game exponent rules