Picture
Source: http://www.zazzle.ca/satan_alphabet_in_math_tee_shirt-235801256845953337
At some point kids learned that algebra is hard, confusing, or some other description preceded usually by an expletive, and followed by a sigh.  As a lover of Math, I never understood that algebra was difficult, and it became stunningly obvious the first time I tried to teach basic equations that I was not "normal."  I stood in front of the board, and told my grade eights "Just do the opposite operation, that's it!  If you see adding just subtract."  It seemed so obvious to me.  No stress, just tell them.  We see it all the time. It just didn't work for my students, and it didn't work for me.  So I thought long and hard, and this is what I came up with.

Balancing Scales

With my mind on 3 acts so often, I wanted to boil down my problems to a simple question.  I began to ask my students "How many pennies are in that cup?"
If you ask students how many pennies were in that first cup (assuming I divided the pennies equally), they will have the answer before the video is finished.
Picture
The thing that arose as the difficiency of this method, is that the problems became so easy in this case, that students couldn't see the connection between the algebra and the cups.  I worked for a while on a basic set of work that would bridge the gap between the visual and the abstract.  My process for my students went as follows:
  1. Student views visual and solves.
  2. Student develops visual from words and solves.
  3. Student views equation, develops visual (if necessary), and solve.

To see a copy of what this looks like just click on the link below.
one_step_equations_-_addition.docx
File Size: 590 kb
File Type: docx
Download File

I hummed and hawed about how to bridge two and three, but I found by the time students had finished step 2, they could complete step three without any hesitation.  It was fantastic! As soon as students get a bit stuck on the equation, I tell them, "Let's draw it on this lovely whiteboard."

To really hammer home the connection between them we play lots and lots of Algebranary!

Downfalls

There are a number of downfalls with this method, that I am still trying to work through, and I would love any critique that you can give.
  • When kids can do it in their head, they feel no need to draw anything; when the numbers are too large drawing is cumbersome, and inefficient (the necessary breaking of visual models to get to the abstract I guess).
  • Visually subtraction is impossible (very hard?) to show in a static image, so in my slides you will see I resort to number lines (I save the visual of that for last), since students will have a hard time drawing on their own.
  • Though students are able to UNDERSTAND the steps of solving equations, this set up does not create the desire FOR algebraic knowledge like some of the awesome projects we often see on the blogosphere.

It is not perfect, but this has all been incredibly valuable for student understanding the basics of solving linear equations.

Moving Further

I have always imagined this would also work great for inequalities (unbalanced scales and ask "What is the minimum number of pennies to do this?"). Dig through the slides, and the pictures, and the videos, take what you like, tell me where you would go, and hopefully enjoy.

The Goods

penny_equations_-_student_work.zip
File Size: 3610 kb
File Type: zip
Download File

penny_equations_-_mess_of_all_my_media.zip
File Size: 104463 kb
File Type: zip
Download File

I started to organize the video files, then I got lazy.  If you want to deal with the raw files however, you can download this mess of files. Otherwise see how I organized them in my PowerPoint.
linear_equations_with_pennies.pptx
File Size: 108444 kb
File Type: pptx
Download File

 
 
How do we inspire a curiosity in number sense?  I think one way is by having awesome internet colleagues who make ridiculously great resources, but I have been noticing one spot on the Youtubes that continuously shows some outlandish, beautiful, and yet humorous representations of things that we normally wouldn't think about.  I want students to learn in my class that math is not so much about "using in the real world" as if we are to know that a particular concept is used in scenario A, but rather to use it with your own impetus. I think Buzzfeed (and possibly more specifically Ze Frank) strikes this magnificent harmony between beauty, humour, and number sense. Check out these videos, and share anything else that you have seen.

Also if you haven't seen it yet go to Math Munch NOW!  The spirit of their posts is very much in line with how I feel about these particular videos, but definitely at a more mathematically sophisticated level.  Enjoie.
 
 
In case you were wondering...
As you can see over the past two months I have been quite busy.  Some of you may have seen that I went in for an interview, and I am happy to say that I am now getting ready to be a grade seven teacher! It is going to be a HUGE growth opportunity, as this is not a strictly math position.  Heck it is not even my usual math, science, and French.  I am going to be responsible for PE, social studies, and even English (eek!).

I've had a huge amount of support from you guys, and I just wanted to thank you all, and I hope to see you on here a lot more (I know I need to share more of my stuff, and I'll try).  Right now I just have to get ready for this wild world of Elementary teaching.

Here's to the drawing board!
 
 
Over the past months I have been slightly invisible.  This makes me sad, but I know that blogging, tweeting etc. is an ebb and flow process for me.  My mind is not constantly bursting with new ideas, and when I have new ideas these days, I have not been in the space to bring them to fruition. I am also right now at a difficult point in my career.  This year I have suffered for the first time the dreaded "cutback." The school that I have lovingly served for three years has gone down in enrollment, and as such they needed to bring the budget down by one full time teacher; I'm on the lowest rung, and therefore I had to say good bye.  

I am surprisingly at peace with this, because I do feel I am now given a chance to pursue my passions, especially in regards to teaching math as I truly want to devote myself fully to the discipline of math (many of you know that I have been teaching science, and French as well).  I am looking forward to seeing what is next in store for me, as I know something will show up; I just don't know what that will look like yet!

...But that's not really the point of this...

For awhile I have wanted to hop on the Virtual Filing Cabinet Bandwagon and as I see myself in a spot of not really knowing what to plan for in the next year I decided to plan for EVERYTHING. This is partially a way of practicing  for my future, but also to prepare for the possibility of having a position at the end of summer, and having no resources.  I also want to give a Canadian spin (because we all know I play that card ALL THE TIME) on the virtual filing cabinet process.  You can see the first efforts here.  I will be updating throughout the summer.

The Organization

I have separated the topics in subjects based on British Columbia's IRPs and curriculum.  I have lifted (plagiarized?) the standards as they are written in our government documents so that anyone (me mainly) may go directly to the assessment standard and see resources for that specific standard.  I am not sure how much they link to the other provinces, but hey, I'm from BC I can't make everyone in Canada (no matter how nice they are).

What You Can Do

You can read, enjoy, and hopeful put to use this collection, but what be even BETTER is to help me out.  At the end of each page I have a small form that you can fill out.  If you see a topic that is missing an awesome activity, or has NO activity (designated by the TBA), fill in the form and let me know.  We are a gift culture on here, and I am not ashamed to ask for a showering of gifts in the form of links!  So please, help me out if you can, and maybe we can get an awesome resource going for the Canucks!
 
 
Language is a funny thing.  I have talked about it before but the use of language and our ability to reason numerically is so interesting.  I had a conversation with a student today where he told me about his dogs. It was one of those off topic conversations.  He was describing the size of his dogs and he said "I have two hundred ten pound dogs."  Now I have intentionally left out any dashes, because I want to let you in on what I understood.  When he said that I thought of this massive group of hundreds of these ten-pound puppies.  He meant he had two dogs that were 110 pounds.

Which brings me to my thought.  In one sentence we can have three different meanings, the likes of which are such.

I have:
  1. I have 210 pound dogs
  2. I have 200 10 pound dogs
  3. I have 2 110 pound dogs*


What does this tell us about the nature of quantity?  They all sound the same but all produce different quantities.  In scenario 1 we have 210x pounds of dogs. We do not know how many dogs I have, but they are all around the same size.  In scenario 2 I have 2000 pounds of dogginess, and in scenario 3 I have 220 pounds of dogs.  In some weird linguistic sense, these seem like they should all be similar in some sense, but they all produce different images, and different quantities entirely.

I do not know why this particular quantity pun amuses me so much, but I feel there is something here.

* Ya, I realise that mathematically we should say two one-hundred-ten pound dogs, but conversationally we rarely say that.

**Another fun quantity pun to ask kids especially is would rather have one and a half million dollars or one million and a half dollars? Something seems eerily the same about those, but they are screamingly different.

 
 
Well Geoff and Michael have been bugging me, and I  was also really inspired by Stadel's recent post on exponents that I wanted to quickly share my little intro to exponent rules for my grade eights this year. They (and I) really enjoyed it, and I think it touches a bit on what Dan Meyer is trying to get at with Tiny Math games.  In Grade 8 students do not need to know exponents, but our school wants them to be familiar with them, before grade nine, so I concocted this little activity.

The Setup

Nothing too fancy here, but basically I printed out this sheet multiple times
exponent_explorations.docx
File Size: 15 kb
File Type: docx
Download File

exponent_explorations.pdf
File Size: 63 kb
File Type: pdf
Download File

I sliced each section, and put them in envelopes at the front of the board titled "Product Rule," "Quotient Rule," "Power Rule," and a^0=?  I told students a few simple directions
  1. Follow the directions on the first slip of paper.
  2. Complete all directions on your large whiteboard.
  3. When you think you have it figured have me come and take a picture.
  4. Practice that rule with some basic worksheets.
  5. Move to the next rule.

The Inspirations

If you can't tell or don't recognize the questions, I was inspired by Exeter.  Reading their problems I realise they have a knack for assessing, and teaching within the context of a clear and concise question.  The question itself pushes students to work through the problem solving in a natural way (even if it is pure math not applied). I worked on these questions for awhile, to make sure that they put the students through the algebraic clarity but they also had a chance to play and punch numbers in their calculator.

(aside) This in my mind is also a less rigorous version of David Cox's fantastic lesson.

The Context

Students had a concept of repeated multiplication.  They knew how to evaluate exponents within whole numbers and could hit the "equals" button on their calculator a repeated number of times.  They were not instructed how to find the exponential form of a number (for example they can find 2^3=8 but they had never tried 8=2^3).

The Payoffs

Since this only needed to be an overview, and we just needed to touch on these concepts I felt that I didn't need to make sure that everyone learned every rule.  Most students were very proficient with the product rule and quotient rule, and figured that the power rule would somehow involve multiplication of the exponents.

What students LOVED is that they could work through it "at their own pace" (KA buzzwords I KNOW, but still!).  Students struggled through it, but it was set up in a way that was just out of their intuitive reach!  The students who arrived at a^0=1 were very easily convinced of that fact, and because of that, I knew I had a winner.

The Drawbacks

Sure they know the rules, and that is fine and dandy, but do they know the reasons behind the rules?  I do not teach associative property in grade eight, and therefore reasoning through WHY these concepts work (at this stage, rather than because my teacher said so, it is kind of because my calculator said so), but I think this is only a small drawback, because students are convinced of these rules, and they feel they are natural because they found them.  The rigour can come later.

Conclusion

I will do this again with students, especially as review.  I think it puts them in a state of problem solving that is not to laborious, but also not rote procedural notes.  Tell me what y'all think!
Picture
The MONEY shot!
 
 
It has been far too long since I have posted here, and for that I am sorry.  To tell you the truth I have been busy, but more than I busy I can admit that I have been a bit down on myself.  I still have so much to do to become the nguyeningest teacher that I want to be, and I have been a bit under a rock by recognizing that I am not there.

We can talk a big game on these blogs and share all these awesome lessons that work (when they work), but even those great ideas can fall or be used improperly, or rushed to the point that no meaning is taken from them.  I am a the point right now where I find myself trying to catch up to all the curriculum that needs to be done, and I don't like that, but I have a professional responsibility to make sure that kids are prepared, so I have to work better and harder than I do at this point.  Have I led students through inquiry? Are they engaging with their world?  Why can't I be more like (insert awesome blogger/teacher/awesome person name here)?  I feel as though I am not measuring up, and that's a hard place to be.

So I am introducing volume and I asked them to notice and wonder, and this is what I got...
This board of notice and wonder seemed a little too good to be true, so I asked the class (a class who I trust to be honest) this:*  
Raise your hand to vote.  Here are your two options: Did you notice or wonder because you actually want to know, or did you notice and wonder this because you think it is what I want you to notice and wonder?
The majority of the students put their hand up for option A.  They looked at pop-cans and wanted to know legitimately awesome things.  They were curious, and they were taking it seriously.  They had cool discussions on why the companies would make these decisions.  Now granted, we have talked about pop companies before when we did surface area, and I walked them through that, BUT I can see that they have caught a bit of this math bug.  They have caught a bit of the curiosity that I had when driving home and watching road lines whiz by.

I am not a perfect teacher by any means.  I lose control of my class. I take forever to assess.  I give them notes, and I even *dun dun dun* lecture.  Not every class is inquiry based.  I am not always fully prepared.  I am very disorganized. I have more faults than I care to name, but somewhere, hidden under the heap of my self-deprecation is something that has attracted kids toward curiosity. We often say to ourselves "If I but do this ONE thing, it will have all been worth it."  Today this board represented my one thing.  Today I saw that I am not in the wrong profession; I am not the worst person in the world; I am not a terrible teacher. Today I cast off my self-deprecation and embrace encouragement.


*Keep in mind this is ONE of my classes.  My morning class was not as indepth.
 
 
This lesson is not a feat of great technology.  This is not a three act post. This is however something that I am proud of because it was simple, clear, and really really fun.  I am in my third year of teaching, and every year I notice misconceptions that I want to get out of students heads.  I also want to introduce them to natural understandings of mathematics that transitions students easily to higher level thinking, and higher level mathematics.  I love this lesson because I feel as though it did that.

When (plural noun) understand they (adverb) learn!

My struggle has been to get kids to understand that a variable is a number.  They see x and they think that it is a letter, but I want them to know that it is a number.  However, I want them to know that it is any number.  Depending on the number we put in, we get a certain outcome.  Rather than tell them this, I start with a favourite past time of mine, mad libs.

Students in my class have a vague concept of variable but I am really driving home that a variable can be anything, so I begin with variables in English.  I prepared this spread sheet and mail merge document so that they can see clearly how variables work.
mad_lib.docx
File Size: 16 kb
File Type: docx
Download File

mad_lib.xlsx
File Size: 9 kb
File Type: xlsx
Download File

The paragraph first looks like this:

  • Once upon a/an «Noun_1», there were three little pigs. The first pig was very «Adjective_1», and he built a house for himself out of«Pl_Noun». The second little pig was «Adjective_2», and he built a house out of «Pl_Noun_2».
And wonderful grade eights turn it into this beauty:
  • Once upon a fire breathing llama, there were three little pigs. The first pig was very colourful, and he built a house for himself out of cats. The second little pig was scary, and he built a house out of apes
I talk a bit about programming here and say, "This is what computers do, they take information and insert it where it is supposed to be.  Here wherever I have 'Noun 1' what information is stored there gets displayed." I then talk about we get a certain outcome based on the variables we put in.  I ask about the first variable 'noun 1', "What if I put in this variable the word 'time'? How would that change the outcome of the sentence?"

"It wouldn't be funny!" Most students say. At which point we move on to our next session...

Wow, that _______________________ smells really good!

I tell students here that we are going to use ________________________ as a variable.  I give them a chart, and I want them to fill in the chart with variables that produce a factual sentence, a nonsensical sentence, and a funny sentence.  Here are a few examples....

Factual

  • Apple Pie
  • Food
  • Soap
  • Parfume

Nonsensical

  • Dream
  • Colour
  • Funny
  • Light

Funny

  • Farts
  • Cannibalistic Llamas
  • bucket of degrading cat carcasses
At this point you can't help but have wonder and awe at the buy in factor! It goes with out saying that you have to remind students to be appropriate (the cat carcass one is weird, but it made me laugh!), but kids love thinking of these and sharing them.

It is important that you wear a/an X before you jump into a pool!

At this point I tell students that they have used variables for ever in math.  They have all done 3+__ = 5. The __ is a variable, you can put any number in there (although only one number makes that statement true!).  So in this example we do the same thing, but instead of _____________ we now say X.  We discuss again what is factual what is not.  We can even begin to talk about one to one correspondence (is "speedo" factual or funny? Hint: the answer is yes).

But wait, are we not in math class?

The jump then becomes ridiculously easy. I have students do the same thing with numbers.  There tables now read even and odd (instead of funny and factual), and they insert values of x that produce even and odd. They do three functions: x+3, 3x, and 2x+1.

Now all you wiley math teachers out there are saying 
Becausewith 2x+1 you get some interesting results.  Kids start to X out the even section.  Here's where the messing with your kids in a totally awesome way part comes in. Ask them, "you can't find a value that produces an even number?" They say "no it is impossible." You say, "Yes it is." I let the class know that there is a number that works, they just have to 'open their mind' (but not in a hippy way). They rack their brains, and finally asking, "can we use decimals?" to which you respond "is it a number?" because remember a variable means any number.

Going even further!

Picture
That's right, there's more!
Why stop at odd and even?  This I feel is the homerun to this lesson.  I gave them first 4x+1>5 and then 2x+11=7. They fill in a true and false table for each of these, and what they notice is fantastic!  In the first example students ask, "What about when x=1?" To which I reply, "Is it greater than 5?" "No" "Then is it true?"  I had students writing down 1- (-infinity) in the false section.  This wasn't 1 minus negative infinity, this was 1 TO negative infinity!  In one lesson I had students figuring out inequalities, variables functions, and equations.
At that point all we needed to do was show a little notation, and they were totally on board for inequalities and equations.  Awesome! I am going to keep giving them these right up until I formally teach them equations (inequalities aren't even touched on in grade 8, but hey, get them ahead!).  Because once they get bored of filling these in, we can say, "How can we speed this process up?"

Trick them into learning... yes...

variables_and_expressions.docx
File Size: 22 kb
File Type: docx
Download File

This is the worksheet I whipped up.  Take it and do WHATEVER you want with it!
 
 
I assume I am not the first person to think of this, but just in case I am, this is the game that I played today in class (if you have seen this before tell me in the comments)!

The Rules

It is pictionary, but with integers!  Basically it runs like this.
  • The teacher has a list of x number of worked out equations involving integers (ex. (-3)+(+2)=(-1)) written on a piece of paper.
  • Students in pairs/groups send one person who is the "drawer" to see the equation.
  • Students return to their group, where on one mini white board they draw the equation (using a number line or army men [what lesser people call "integer chips"]).
  • Other students in the group use their own mini whiteboards to write down the equation exactly.
  • When students correctly answer on their whiteboard they send the next student up to the front to receive an equation.
  • The process continues until all teams are finished (if you are into rewards give the top team some candy or something).

Why I like it

I really want students to understand the models that they are applying to these problems.  I feel that drawing it out is way harder for students than "doing it" as they seem to think memorizing mnemonics is true math.  It gets students thinking laterally about mathematics, and translating something concrete into something abstract.

Other Uses

I can easily see using this with algebra tiles, fractions, etc.  I really do not think it is for integers alone. What do you think?
 
 
In my last post I proposed a possible introduction to order of operations.  It was by no means a complete planned lesson, and neither shall this post be. What happened after, I posted on Twitter "how do we teach order of operations as a concept, rather than a rule to be memorised?"  My goal was to teach students a reason why multiplication precedes addition, rather than simply saying...
Or is that all we can do? It is a convention right?

Conventions can still be conceptual

The first thing I thought was that the order of operations are not simply a random guess in the dark convention.  I do not imagine the world's greatest mathematicians developing this necessary convention by the toss of a coin.  Rather, I expected that there is some reason the great minds of the past did in fact decide that there was an optimal way to follow the procedure of calculation now infamously known as BEDMAS or PEMDAS depending on the mnemonic device you have learned.

For me the order of operations has always made sense in terms of algebra, and that is exactly what Dr. Math (referenced in paper) claims. Well, great, I want to teach students order of operations conceptually, but they do not know algebra yet, what can be made of this?

Terms are the cornerstone

My algbraic understanding moves me to discuss terms.  The standard convention makes sense in my mind when we discuss terms: 6x+6y.  I know that whatever x is, I have 6 of them, same with y, but since I do not know the values of x and y, therefore I cannot complete and simplify this term any further, because I am not given enough information. 6x+6y is necessarily simplified at this moment.

This is a great way to think of the operations, if you know the basics of algebra, but when we teach order of operations, students do not have a strong (if any) concept of a variable and especially like terms.  My struggle over the last few days has been this: "How do I reconcile my algebraic understanding of the order of operations within arithmetic?"

Models, models, models

I think I just heard @fnoschese cheer, but it really comes down to understanding our models of addition and multiplication. When I think of my conceptual understanding of addition, this comes to mind...
That is, addition is largely a combining operation.  I have this many things here, I have this many things there, I bring them together and TADA!  However when I think of multiplication, I think of this...
When I think of multiplication I often think in groupings.  I do not take a quantity and combine it (like addition), but I create a quantity (seemingly from nowhere) via grouping.

The point that expresses this for me the most is when we look at the number line.  With addition we start at a particular number, and then move onward.  With multiplication we start at zero! Multiplication is an ex nihilo process, we get something from nothing. With addition we combine two quantities that are there, whereas with multiplication we simply start making groups. I think this is the "multiplication is a stronger" operation argument, but I just had to put it in my terms.

Putting it all together, and bringing it back

This is not an airtight argument for multiplication to necessarily precede addition, but since order of operations is a convention it need not be.  However I think this is my first fruits of grasping why this is a well defined convention, not a coin flip.  If I am uncertain of which order to use when given the problem 3x4+2x5, it makes sense to me that I cannot do any addition until I am guaranteed to have separated values that I can combine, multiplication doesn't have that requirement, so logically I can start with that multiplication.

I'd like to hear all of your thoughts on this though. Is there a point to all of this or should we just say, "Bedmas everyone! Look it's bedmas!"?  Do you want to get mad at me for my lame models or start the war on multiplication is not repeated addition?  Let's talk this through so we can make the order of operations a less daunting, more natural task for students...