Category Archives: lessons

Hallelujah! Finding two-step function rules from tables

Maybe this method that I discovered from an exhaustive internet search is well-known to most, but maybe it’s not.  So, I decided to post it just in case.  That and, quite frankly, I am so freaking excited about it and my husband will have NO idea what I’m talking about if I decide to spew it all over him!

When I was in school a billion years ago, many things were taught as “guess-and-check.”  For instance, factoring quadratics. (I learned an awesome no-fail method for that last fall from my cooperating teacher during my internship- I’ll post that sometime soon.)  I don’t know about any of you, but I HATE guess-and-check.  Besides, how do you even TEACH guess-and-check?

 

What prompted my diehard search for an easier way of finding function rules?  My kiddos.  I was out of work last week due to an emergency surgery (yeah, 4th week of my first year teaching is a fabulous time to have your gall bladder removed!), so my sub covered the lesson involving finding function rules given a table.  She’s completely competent (she’s my ESE teacher for one of my classes and she’s certified in math), so I don’t think it was a lack of instruction on the concept.  She probably taught them the same exact way I would have: guess-and-check until you’re good enough to just “see” the rule.  Well, the majority of my poor kids don’t get it- at all!  So, today I was frustrated due to the gazillion questions during their end-of-unit test.  Clearly, if they are all asking questions about the same test item something needs to be done.

Cue in research.

I found this explanation in a math forum.  I’m so happy I did.  It’s not the prettiest or clearest of explanations, so I’ll try to make it a bit prettier here.

Take, for instance, the below table.

 

Input, x

Output, y

1

7

2

11

3

15

4

19

5

23

 

The first thing you do is take 2 sets of numbers (ordered pairs):

(1,7) and (4,19) ; any 2 sets will work

Next, find the difference in x-values and y-values:

x-values: 4-1 =3

y-values: 19-7 =12

Notice that the difference in y’s is 4 times the difference in x’s:

3 * 4 = 12

Therefore, 4 is your multiplier for the rule.

So, we know that 4x plus or minus something = y.

From here, it’s easy to tell what we have to add or subtract to get y.

4(1) = 4

4+3=7

Thus, y= 4x + 3!

 

Again, this may be somewhat common knowledge in the math community at this point.   However, I figured if I was clueless to this method then maybe (hopefully!) someone else is, too.  At least, that is what I am going to tell myself to fall asleep tonight!

I cannot wait to show this to my kids tomorrow!!!

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Toys in Math Class?!

I was student teaching in an Intensive Algebra I class and was trying to think of a hands-on activity to reinforce the concept of the unions and intersections of sets.  It dawned on me that ANYTHING could make up a set, so I wandered into my favorite place- the dollar store!  I found some cool little toys; little green army men, bouncy balls, googly eyes, red and black checkers.  I decided to add paper clips just to get another item thrown into the mix.  I separated all of the items and put them into plastic baggies.  I made enough baggies for each group of students.

After discussing what the words “union” and “intersection” mean in real life and asking for examples of these things (union of people in marriage, intersection of roads, and several others the students came up with), we applied those definitions to sets.  I then passed out the baggies and Venn diagrams to each group.

 I displayed different sets, asked different questions and had the groups use the toys manipulatives to figure out the unions and intersections.  I will say that using bouncy balls is not recommended!  I guess they are just TOO fun!  The kids loved being able to play with toys in math class and were actually eagerly awaiting the next question to be displayed on the SmartBoard.

I searched the internet for the pictures to put into the SmartBoard.

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If I had a million dollars…

During my final teaching internship, I taught one section of 8th grade Gifted Math/Algebra I.  These guys may have been my favorite class, even if they were the chattiest.  I also noticed they were VERY competitive.

I had to do an introductory exponential growth and decay lesson with them and remembered reading about a problem involving a guy walking into the classroom and offering either one million dollars or starting with one cent and doubling the money every day for 30 days to do a job.

*Cue in google.

I honestly do not remember where I found this problem, so I apologize for not giving credit.  However, this lesson (one that was very inquiry-based) worked out fantastic with this group of students.  (It’s based on the Conceptual Change Model, a model I learned about in my fantastic Math Methods course.)

I started with displaying the scenario on the SmartBoard:

You’re sitting in math class when in walks some rich and flashy guy and he has a job offer for you.  He doesn’t give too many details, hints about the possibility of danger.  He’s going to need you for 30 days, and you’ll have to miss school. (Won’t that just be awful?)  But do you ever sit up at the next thing he says.

You’ll have your choice of two payment options:

One cent that day, two cents on the next day, and double your salary every day thereafter for the thirty days; or

Exactly $1,000,000. (That’s one million dollars!)

I had the students write down which payment option they would prefer and why.  It’s important to only give them a minute to do this, otherwise many students will begin doing calculations.  I wanted them to commit to an answer prior to actually figuring it out mathematically.  They were then told to get into groups and share their thoughts.  One student from each group would then (anonymously) share the predictions.

Now is the time when I wanted them to figure out which payment option would be the most profitable.  I handed each group a partially filled out table and simply told them to figure it out in their groups. 😉

As they worked in their groups, I walked around to check on their progress, make sure they were on task, etc.  This class (as I mentioned earlier) was very competitive, so everyone was feverishly doubling the money in the right hand column.  When I heard complaints about it taking too long, I asked if there was an equation that would help them move things along faster.  When they heard this, all the groups started trying to figure out an equation.

I honestly do not recall how many groups figured out that it was 2^x, but I know it was well more than one group. Time was running away from us, so I didn’t get to have each group come up to explain their process in solving it (boo!). So, we moved on to discuss the results and what the graph of 2^x would look like, what the domain and range were, whether the graph of x^2 would increase faster or slower, etc. We then discussed what -2^x would look like and compared linear, quadratic and exponential functions.

All in all, it turned out to be a great lesson and students really seemed to have grasped the concept of exponential functions.

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