Math = Love

Sunday, March 26, 2017

Electron Magnets

Today I have a super quick post for you.  It's been sitting in my drafts folder for MONTHS.  I decided it was time to either post it or delete it!  

My sister gave me a set of colored, circular magnets (affiliate link) for my classroom.  I wasn't exactly sure what I was going to use them for at the time, but they came in SUPER-handy when we were making Bohr models of atoms in physical science.  

This happened to coincide with my almost month long experience of having no projector and therefore no SMARTboard in my classroom.  Let's just say that magnets make awesome electrons!    

Maybe this is one of those ideas that every other science teacher has already figured out.  But, I was pretty proud of myself for coming up with this idea!  

Saturday, March 25, 2017

Can You Level the Towers?

Recently, I've been trying to find a way to organize many of the amazing resources I've found online so I can use them more easily in the future.  Don Steward's blog is an absolute treasure trove of activities and interesting problems for any mathematics classroom.  A few weeks ago, I went through every single one of his blog posts and took note of all of the activities that aligned with my Algebra 1 standards.  There were A LOT of them!  

One of the activities that caught my eye was called "Equal Sharing."  Students are given a set of towers and asked to make all of the towers have the same height.  You must maintain the same number of towers.  The only thing you can change is the arrangement of the blocks.  

I took Don's activity sheet that featured 10 problems, stuck it in a publisher file, enlarged it to fit on 11" x 17" paper, and added a title with a cute font.  I printed the puzzle on some 11" x 17" cardstock (affiliate link) that is amazing for creating activities that can easily be worked on by multiple students at the same time.  It also makes pretty awesome posters!     

Then, I stuck each activity into one of my new 11" x 17" dry erase pockets (affiliate link) to make the activity reusable.

I allowed students to work on this activity in groups of 2 or 3.  A few of my students chose to work alone.  The majority of students were in pairs.

I didn't give them many details before I let them loose.  I instructed them that their challenge was to rearrange the blocks so that each tower was the same height.  They had to end with the same number of towers that they started with.

Even though I told students before we started that they had to end up with the same number of towers that they started with, I still found myself answering that question over and over and over.

It was interesting to watch students try to convince their partners that they had a superior strategy as I circulated.

A few minutes into the activity, I saw students changing up their strategy.  At the beginning of the problems, the students were marking out boxes and moving them one at a time.  This worked, but students soon found it tedious.  Instead, the students started counting all of the boxes in all of the towers.

I stopped by a group just after they reached the conclusion that there were 28 blocks in the towers.  With a perplexed look on my face, I asked them why that information was useful.  The group informed me that because they knew how many towers they needed to end up with that they could take the number of blocks and just divide by the number of towers.

I do have to admit that not all of my groups made this realization.  Some were plenty happy to keep using the same method that worked over and over and over.


When my students got to question 6, they began to become confused.  Really confused.

My students who were rearranging the blocks one at a time found themselves left with 4 extra blocks and 8 towers.  How does that work?

Some groups decided to cut the four blocks in half and distribute one half block to each tower.  The groups that were adding up the blocks and dividing by the towers seemed to have less trouble with this problem.  Many groups, however, got frustrated and gave up when the problem didn't work out evenly.

After the groups had time to attempt MOST of the problems, I began a class discussion.  I asked the groups to share their strategies.  One student in each class would bring up the fact that they had found a shortcut.  They then proceeded to describe how they had added up all the blocks and divided by the number of towers.

After they finished explaining, I asked them what mathematical concept matched up with what they had just done.  It was so exciting to see their eyes light up as they realized they had just found the mean!

I hope that my students walked away from this lesson with a better understanding of what the mean is instead of just knowing the steps to arrive at the mean.

Thanks Don Steward for this awesome activity!

I have uploaded my 11 x 17 sized file of Don's activity here.

Step-by-Step Directions for Factoring Polynomials Using the Box Method

When I posted my interactive notebook pages for our Algebra 1 unit on Polynomials, I said that I was going to post step-by-step photographs of how to use the box method to factor polynomials.  Here's that post.  

Begin by drawing a box.  Quadratic trinomials require a 2 x 2 box for factoring.  This box will also work for difference of squares factoring.  

ALWAYS check to see if you can factor out a GCF from the polynomial first.  If you can, this goes in front of the parentheses in your answer.  This year, I added a space between the equal sign and the first parenthesis to give students a place to write the GCF.

There isn't a number that divides evenly into 2, 11, and 5, so we can skip this step!

The x squared term is written in the top left hand box.  The constant term is written in the bottom right hand box.  Students should be used to this as a result of lots of multiplying polynomial problems using the box method!

This takes care of the 2x^2 and the 5 from 2x^2 + 11x + 5.  The 11x term tells me that my two missing boxes add to 11x.  I have my students draw an arrow and write this fact next to the arrow.

We know from examining patterns with multiplying polynomials that are two missing boxes multiply to the exact same value as the two boxes we have filled in.

This means we are looking for two terms that add to 11x and multiply to 10x^2.

Those two terms are 10x and 1x.  Write those terms in the missing boxes.

Now, we're almost done.  We need to figure out what to put on the outside of our boxes to give us these multiplication results.  I have my students begin by picking two boxes and finding the GCF of those two boxes.

For example, the GCF of 2x^2 and 10x is 2x.  So, I can write 2x on the outside of those two boxes.

Similarly, x is the GCF of 2x^2 and 1x.

5 is the GCF of 10x and 5.

And, 1 is the GCF of 1x and 5.

At this point, I instruct my students to always stop and double check that their multiplication is correct.  Every once in a while, there will be an issue with positives and negatives that needs to be cleared up.

Once everything is double-checked, it's time to write the answer.  Since we didn't have a GCF to factor out at the beginning, we don't need to worry about putting a value in front of the parentheses.  The terms from the side of the box (2x and +1) go in one set of parentheses.  The terms from the top of the box (x and +5) go in the other set of parentheses.  We have successfully factored a quadratic trinomial!

Let's try another one!

First, we need to check and see if we can factor out a GCF.  10, 80, and 70 are all divisible by 10.

When I divide all three terms by 10, I am left with x^2 + 8x + 7 to factor.

I'm going to go ahead and take the 10 I factored out and put it in front of the parentheses for my final answer.

My x^2 term goes in the top left square.  My constant of 7 goes in the bottom right square.

My two empty boxes must add up to 8x.

Lots of terms add up to 8x, so I need more information.  I know that my two missing terms multiply to the same value as the two given terms.  So, I'm looking for two terms that add to 8x and multiply to 7x^2.

Those two terms would have to be 7x and 1x.  So, I write those in the missing boxes.

Now, I start looking for the values that go on the outside of the boxes.  The GCF of x^2 and 1x is x.

Similarly, the GCF of x^2 and 7x is x.

The GCF of 1x and 7 is 1.

And, the GCF of 7x and 7 is 7.

Now, I need to stop and double check that all of my multiplication works out correctly.  If it does, I'm ready to write my final answer.  The 10 is already written outside my parentheses.  The x and +1 from the top of the box go in one set of parentheses.  The x and +7 from the side of the box go in the other set of parentheses.

We've finished another problem!  I have one more example I want to show you.  I don't teach factoring difference of squares as a separate topic.  I make my students use the exact same box method.  This keeps factoring the difference of squares from seeming like a "trick" to memorize.

x^2 - 16 can be rewritten as x^2 + 0x - 16.

There is no GCF to factor out.  So, I can write x^2 in the top left box and -16 in the bottom right box.

The middle 0x term tells me that my two missing boxes add to 0x.  At first, students ALWAYS think this is impossible.

The two boxes that are given multiply to the same value as the two boxes that are missing.  So, I know I am looking for two terms that add to 0x and multiply to -16x^2.

Those two terms would be -4x and +4x.  So, I need to write those in my empty boxes.

x^2 and -4x have a GCF of x.

x^2 and 4x also have a GCF of x.

The GCF of -4x and -16 is -4.

Meanwhile, the GCF of x^2 and 4x is 4.

So, I can write my answer using the x +4 from the side and the x -4 from the top.  x^2 - 16 is the same as (x+4)(x-4).

And, that's the beautiful box method.  I hope this step-by-step explanation has helped clear up any questions you might have had about how the box method works.  If you still have questions, please leave them in the comments!

Friday, March 24, 2017

How to Make Pockets for Storing Notes in your Interactive Notebooks

I've been asked a few questions lately about how I have my students make pockets for their interactive notebooks.  I thought for sure I had blogged about this already, but I couldn't find the blog post anywhere!  

I have my students make pockets for their interactive notebooks whenever I'm giving them a bunch of similar papers that don't need their own separate pages.  We take a LOT of notes, so many of my Algebra 1 students have had to start a new notebook part way through the year.  Making pockets helps prevent them from going through pages *too* fast.  Of course, I always have the students who refuse to write on the back side of their notebook pages.  These students FLY through notebooks no matter what!  

A few years ago, I used to have my students make pockets for their notebooks using the actual notebook pages.  This worked great, but I can't bring myself to waste a precious notebook page to make a pocket anymore! 

Here's what our pockets look like lately: 

Usually, I write a title on the pocket to describe the contents.  I did title this one "Measures of Central Tendency" after I took the photo!  

Begin with a sheet of colored paper. 

Fold the paper in half, "hamburger style."

Unfold the paper to reveal the crease.

Cut along the crease.  Each pocket only requires one sheet of paper.  Whenever we make pockets, I will have kids pair up and share a sheet of paper.  They always get into the cutest debates over what color of paper to use for their pockets!

Position the paper in front of you so the long edge is facing you.

Fold this edge away from you about 1-2 inches.

Fold the left edge 1-2 inches over to the right.

Find the papers that will be held in your pocket.  We will want to use them to make sure we get our pocket sized properly.  Often, kids will skip this step at the beginning of the year.  They won't realize that they have made their pocket ridiculously small until they go to put in their papers!

Slide the papers into the partial pocket.

Fold the right edge over.  Make sure you leave some wiggle room!

Now, we've made all of our creases.  We're almost done!

Unfold everything.

Place a bit of glue in the bottom right and left corners.

Fold them up.

Place a bit of glue on the new bottom right and left corners.

Fold them over.

Apply glue around the right, left, and bottom edges.

Stick the pocket in your interactive notebook.

Slide in your notes.  The first time you slide in your notes, it may be a bit tricky to get them to fit.  Sometimes some of the glue escapes and pins down part of the pocket.  This is usually easily fixed by sliding your hand into the pocket.