Exponential Functions

After our Linear Function unit (which you can find here and here), we jumped right into Exponential functions.  

Our beautiful tabs!

Our learning targets

We started by describing exponential functions algebraically, graphically, and numerically.  Here is the foldable we did.  It is very similar to the one we did with linear functions:

Foldable closed

Foldable opened

Then we did a card sort.  We did exponential vs. not exponential.

I asked them to write a reason for why each card was on the side it was on.  Also after our notes on the base and y-intercept, we went back and found the base and y-intercept for each of our exponential functions.

Next we worked on finding the base and the y-intercept.  Notes with a graphic organizer, which is very similar to what we did with linear functions:

I gave them a practice worksheet.  I thought I was really clever and designed it so that they could easily complete, tape in their notebooks, and view later...

...however I then proceeded to copy it upside-down.  :S

Then we graphed exponential functions.  This is where it because crucial to have studied negative exponents prior to this unit.

I had to do another practice worksheet like the one before so that I could prove that I was capable of copying correctly.

Practice WS outside

Practice WS inside

Our last learning target was on writing equations to model exponential situations.  I gave minimal notes that tied to our learning of linear functions (the y-intercept is the start) and the learning we had done with exponential functions.  We had been wondering and noticing that some exponential functions increase and some decrease, but today I FINALLY made it clear what causes that difference.

I did a stations activity with them.  I had a handout for students to use to facilitate the process and our discussion.  Basically I just had students go to nine stations and write the start and the change.  Then we had a class discussion over the equations.  They seemed to think it was pretty easy, which I didn't really expect...I guess I'm just an amazing teacher :)

Handout that they used to go from station to station

NOTE:  To view files click the appropriate link.  It will open in Google drive; it will not show correctly in drive.  Choose to download file.  It will download the word doc/ppt for you with all of the correct formatting.

That's all I have for now.  I don't feel much like reflecting on the unit right now, so I guess this post is finished :)  Sequences are coming up next!  (Well, actually we've started them.  Post to come after the 20th!)

-Kathryn

Units for Intervention Class

I have been teaching an additional intervention class all year.  This is part of the Tiered Algebra project that our local AEA has been working with schools in our area to implement.  We had been given some guidelines, but not much, for what to do in that time.

• Connect it to what is happening in the Algebra class
• Don't just reteach/review what you are doing in Algebra
• Don't just have "do your Algebra homework" time
• Do more hands-on stuff
• Do more explicit instruction
• Use appropriate scaffolding
• And others, but that's what I could think of off the top of my head

So here I am, a general Algebra teacher, with very little understanding of how to differentiate in my regular classroom, now seeing 24 of my students twice a day, and doing my very best to make it worthwhile.  But I've been mostly clueless.  I tried to do a lot of pre-intervention with them... (see Micheal Pershan's argument for this type of intervention here) which looked like covering the coordinate plane and plotting points before we graphed linear functions in Algebra, for example.  But I still felt like my students weren't able to use much of what we were doing in a way that truly benefited them in the regular classroom.  Not that I actually had any real way of measuring it...

Then the AEA shared with us about a presentation they heard from a school that has a similar model in 8th grade.  They decided that vocabulary was really important, because IF THEY DON'T KNOW THE VOCABULARY, THEY CANNOT ACCESS THE LEARNING IN THE GENERAL CLASSROOM.  This was an argument I had never heard before.  Now I always knew that vocabulary was important.  And I teach it...sort of, but I've never really emphasized it.  This has changed that.  Along with other things that came from this summary of the presentation, I have changed some things, and I think it is for the better.

I now have week-long units.  These units focus on a particular skill and the vocabulary associated with it.  For example we studied exponents prior to working with exponential functions.  We had vocabulary:  exponent, power, base, exponential, reciprocal, expanded form.  We studied the vocabulary each day in different ways:  matching; create your own example; which could be used for x, which would be used for y; etc.  We also had scaffolded lessons on simplifying exponents.  We started with whole number exponents with only positive numbers.  Then we discussed things such as -2^4 vs. (-2)^4.  We simplified expressions using the order of operations (with exponents).  And finally we saw negative exponents.

I was actually able to see my students apply what we were learning in class (no actual way of measuring other than my observations).  I depended on them to lead the other students when we saw negative exponents.  This felt amazing.  I am excited to continue using this model, although I am very frustrated that it took until November for me to find something this useful for interventions.  Hopefully as I continue to apply it I will see more improvement from these students.

But I am also frustrated by the fact that it is still group all of my intervention students together.  What if student a needs this and student b needs that?  How do I make that work?  How do I know what they need?

Just for an FYI here is an outline of the exponent unit:

-Kathryn

Math Modeling Contest

On of the math teachers at my school (the one who head's up math club), has been doing a 36-hour Mathematical Contest in Modeling for the past few years.   I am only a helper, but when sharing about the contest, it seemed as though many other were interested in hearing about it, so I decided to share what I know.

The contest is put on by COMAP.  Information about the contest can be found here:  http://www.comap.com/highschool/contests/himcm/index.html.  It is a yearly competition that occurs in November.  It is for groups of  up to 4 students, and costs about $75 per group.  The competition includes a maximum of 36 hours to work on 1 of 2 given real-world problems.  Each group gets to select which problem they would prefer to work on and they work toward a solution.  Then they must write a paper outlining their research, solution, and weakness of it.  Sometimes there are other things they must include, such as a letter or a memo.

I don't think I can share any of the previous problems on here, but they are good modeling problems.  There is not one solution that is obviously the best.  There are many different ways to approach each problem.  Also the problems are low-entry, so even Algebra 1 students can work on them.  Upper-level students can do more sophisticated mathematics with them, but all students can do something to tackle the problems.

Some things we do at Shenandoah:

• Students register before hand and pay $10-15 to help cover the group fee of $75.  The rest of the cost comes from the math club account, filled by fundraisers.
• Students may sign up as a group, or individually.  Most students sign up individually, try to convince all their friends to do it too, and then work out groups with the sponsor once our registration is closed.  Teachers have the final say in groups.  We keep most groups all girls or all boys.
• Each team is assigned a classroom as home base.  Teachers give consent for their classrooms to be used ahead of time, and the students love having control of the room for the weekend.  They can move things around, but by the time they leave it must be back in the shape it was when they arrived.
• In addition to all necessary classrooms, we use the FCS room for food and the library and the gym as hang out places.
• We start at 8am on Saturday so that we can be finished by 8pm Sunday.  Of the weekends the contest is available, we try to choose one where few events are happening.  (Also we like it if it is the weekend before Thanksgiving because then we only have to survive a short week afterwards.)
• Parents sign up to bring meals.  Several sign up together to help feed the kids for each meal that is needed:  lunch and supper on Saturday and breakfast and lunch on Sunday.  Most of the teams are done by supper on Sunday, and for those who aren't there are plenty of leftovers.
• We encourage teams to make good progress on their projects before lunch on Saturday, but do allow for some brain breaks.  We have the gym and the library open for team co-mingling.
• Saturday night around 9 we play a game in the gym as a whole group (like line tag) and then watch a movie.  Around 1 or 2 am we have lights out and kids are expected to stay in their sleeping rooms until morning.  Most teams sleep in their classrooms, which is why we usually have all girls or all boys.  If there is a mixed group, then they sleep elsewhere.  Teachers "sleep" in the halls.  We choose locations where kids would have to walk over us to get anywhere if they decided to sneak around.
• That's all I can think of now.  If you have other questions please, please let me know I'd be happy to help!

-Kathryn

Negative Exponents by Patterns

Right in between our unit on linear functions and our unit on exponential functions, I did a little "review" of zero and negative exponents.  I tried to teach it strictly by patterns.  Last year I taught the rules and then explained via the patterns why the rules were there.

I start with what students know.  I ask students to evaluate 3^1, 3^2, 3^3, and 3^4.  They can do this and they can even show why it works.  We recorded that part in this table (sorry all I have pictured is the final, but imagine half of it is blank :)

We notice that the the pattern from 3 to 9 to 27 to 81 is times 3 (duh!), but still important to state.  Then I ask students to think about the pattern backwards... Divided by 3!  So we continue that pattern to see 3 divided by 3 is 1 :)  And we have to continue the pattern on the top showing that 3^0 is 1. So then we talk about how multiplying 0 threes (or 0 anythings is 1).  It's still hard for them, but they can see.  I try to draw the connection to 0 in addition  (the identity) and 1 in multiplication (the identity).  But they are freshmen, so that is advanced.  If a few students make that connection from our quick discussion I'm happy.

Then we continue dividing.  1 divided by 3 is 1/3 (yes I have to force them to use fractions--I just say that they will recognize the pattern better if they use fractions).  Then by 1/9 there is a bunch of "oh"s and almost every students (or at least every students who is still paying attention) can say that the next in the pattern is 1/27.

We use the word reciprocal (instead of opposite as they tend to) and come to the rule that to evaluate negative exponents, we evaluate the positive first.  At this point I gave them several problems to practice in their notebooks.

That about sums it up.  If you have other questions about this, let me know :)  Or other ideas to make it better I'd be happy to hear it!!!!

-Kathryn

Linear Functions (Part 2)

See my first post on linear functions here.  This is what the rest of the unit looked like:

Above was supposed to be a matching game, but the students didn't enjoy it very well.  I was frustrated, but maybe we should have played bingo like @mpershan (see his post it's trig bingo).  Anyway here is my template.  (To view file open in google docs and download original in word.)  I had each student graph 6 equations before starting, and then pair up with someone who had six different equations and play memory.

Then we wrote linear equations.  Kids practiced a few and then came up with their own.  It is hard to figure out how to think about something that is happening over and over again, so it was a struggle.  Also I feel this is something that is very difficult to differentiate for and provide interventions for....

That was the end of our unit on linear functions.  We moved onto exponential functions!

-Kathryn

Linear Functions

We jumped into linear functions this week.  Here are some of the things we did (and by some I mean pretty much my entire week of lessons).

What does it mean to be linear?
I really wanted to emphasize the various representations of linearity, so that students could see how they all work together.  So I started with this foldable to define linear.  I gave them the algebraic and graphic definitions, but we used those to come up with what it means to be linear in a table.

I did this by giving them each a graph of a linear function, with four points marked on it.  (Here are the graphs I used.)  Then I had them in groups of two create a table on the board for their graph.  I was hoping this would help students with the concept that a line is made up of points as we have been struggling with that idea.  Once we were done with that we made lots and lots of observations about the tables.  This took a LONG time.  I let them notice anything, so it took a while to get away from "they all have a 0/1/-2" type of statements into noticing the various patterns that were there.

But eventually we came up with "x and y both change by constant amounts".  I was very happy because in every class students recognized that they could choose any of the tables on the board as an example.  And students were willing to try to think of non-examples.  So each student got to make that part their own.  I know that is the purpose of things like this, but we don't get there often enough.  It was nice to get that far with this.

Here is the inside.  (And here is a link to the foldable.)

I created a card sort as a way to help them practice telling if something is linear.  I really wanted to get at the fact that linear is a word that describes a group of functions.  So something must be a function before it can be linear.  So we sorted between "Not a Function", "Function-Linear", and "Function-Nonlinear".  It was a lot harder for them than I thought it would be, but eventually we began to get the hang of it.  They still struggle with messing up whether VLT is for function or linear...but one step at a time.

As you can see, instead of gluing the cards in, we recorded the answers in our notebooks.  I hope that students will try to sort the cards on their own as a way to review.

Thoughts:

• My card sort is not perfect.  Two of my tables are non-functions for the same reason, that was a typo...I want to fix that.
• Also didn't really mean to have the equation x=2 in there because we didn't discuss what makes an equation a function...but it did lead to a decent conversation.  I think I would prefer to replace the card with a linear function that is not in slope-intercept form.
• One class got into a really good discussion about whether or not y=(1/3)x was linear or not.  Even when b=0 was thrown out as an idea, one student was still adamant that it needed to be written down for it to be in slope-intercept form.  I kept the conversation going for a while and then moved on without giving up who was correct.  While I was talking a student looked to the student sitting next to him and asked, "so is it linear or not?".  Aren't I mean?  The next day I eventually showed them the graph and they all agreed that it was linear.  I also explained that mathematicians tend toward laziness and prefer not to write something if it is unnecessary.
• I went hard-core with colors in my notes, but students didn't in theirs.  That is something I need to work on being more intentional about.

Finding slope and y-intercept

To begin our discussion of slope and y-intercept, I had students write "what I know..." and "what I want to know..." on the board.  Here is one example:

We used this to jump start our graphic organizer for notes.  Starting with what they already knew, but still getting it into the notes was a win-win.  They got to feel smart for knowing it and I got them all to put it into their notes anyway, especially since not all students knew it.  Here is what we came up with.

I don't think they've seen the slope formula, and I really wanted to share that with them for the Numeric part, but they thought smarter than me and were ready with ideas by the time we got their.  They remembered finding the change in y and x when deciding if the table was linear or not, and figured that would work!  It is really quite brilliant, and I feel lame because I didn't think of it first, but by the time I made it to the end of the day I had abandoned the slope formula.  I want to work back to it for tables that don't have a constant change in x, but still might be linear...we'll see if I can manage that :)

Then we practiced finding slope and y-intercept with a boring worksheet.  But it was quick and worked as a check to see where students were at.  I used my name cards to call students for answers at the end of class.

Thoughts:

• I really wish I could have just had them create their own graphic organizer for this, but I'm not sure I could have been clear enough for them to understand the expectations
• Once again I went hard-core with color and they did not (but look how cute it is!)
• The worksheet was boring, but didn't take too much time.  I think that's OK, because I just needed enough for them to get back into things.  I would have done things slightly differently if this was the first time they had been introduced to slope-intercept form.

Proficiency Certificates

Because it is the end of the quarter, we had PD on Friday, part of which was some worktime to update grades.  All of the HS teachers at my school are supposed to have an updated "standards" document, that shows the level of proficiency each student has met for all standards or learning targets.  Mine is by learning targets (which are aligned to standards).  I had updated everything in PowerSchool, but needed to update my Google spreadsheet.  As I was doing this I noticed that there are several students who have achieved proficient (or above) on all learning targets this quarter.  I was really excited, so I decided to make little certificates and share it as a #Made4Math Monday :)

made4math.blogspot.com

I made it up quickly on PowerPoint (it really isn't anything spectacular).  I printed 2 slides per page because I thought little ones were cuter :) Also I printed on green paper because kids love colored paper.  Here is a picture:

Download File Here

What I'm most excited about though is to give them to students.  I have never done anything like this before, but I think this is something worth rewarding.  I didn't consider whether or not they reassessed, just where they were at when the quarter ended.  I really think this is a good way of encouraging students to do better.  I haven't done much all year to encourage grades, and I don't want to start doing that.  Some of my students who have an A or A- will not be receiving one of these certificates because they scored a 2.5 on one of the learning targets.  I'm excited that I feel as though this is encouraging them to learn more rather than encouraging them to get a good grade.  And I truly believe that this is an award all my students are capable of earning.  That is why I am most excited to present the certificates, because I want to tell that to my students:  YOU CAN DO THIS NEXT QUARTER!

- Kathryn