# Traditional Algorithms – the better way??

I have, on several occasions, had a student walk into my grade 9 class on the first day and state “I’m going to fail this class!”.  When asked why, I get responses such as “I suck at math”, “I’ve never been good at math” “I can’t do fractions”….etc.

“I can’t do long division.”

This one surprises me most.  Really?  Long division?  You think that is what is going to prevent you from passing math?  Long division is certainly not high on my list of key ideas in mathematics!

Last week Jessica and I had the opportunity to work with a great group of elementary teachers and to observe a grade 5/6 class.  It was a great learning experience for us as most of our teaching experiences have been in secondary math with some time in grades 7/8.  The students had just finished working with multiplication and exploring the array model for multiplication.  We decided to see how they might tackle a “division” problem and chose one from Van de Walle.

” Aidan has a bag with 783 jelly beans.  Aidan and his four friends want to share them equally. How many jelly beans will Aidan and each of his friends get?”

Having spent very little time in junior classes I wasn’t sure what to expect in terms of strategies.  The teacher mentioned that most of the grade 5’s would NOT have seen the standard algorithm for long division but the grade 6’s would have explored it last year.   We also wanted to see if students could make a reasonable estimate before they began as the teachers wanted students to work on their ability to judge the reasonableness of their answers.

We saw students approach the problem using a wide variety of strategies.  Some tackled it by using a strategic guess and check approach using either repeated additon or multiplication.  Others approached it by “distributing” the jelly beans to the five students (i.e. let’s give everyone 100, now how many more do we have….now lets give everyone 10, etc) until they ran out.

I have been thinking  a lot about this lesson since that day – specifically about the benefits of student generated algorithms vs. standard algorithms.

There were a few students that used the standard algorithm for long division successfully.   What was interesting is that these students had a hard time thinking about the problem in any other way or explaining how the algorithm worked.  They were very successful in executing it (right down to extending it to decimal values) but there seemed to be little understanding as to why it worked.  In fact, one of the groups that took this approach had an estimate that was very low (5 – 20 jellybeans).

There were a few groups that tried the traditional algorithm, likely because they remembered it vaguely from last year and thought they should use it.  I found looking at their misunderstanding of the algorithm interesting.

What was nice was that these students recognized that their answers using this approach didn’t make sense and decided to try a different strategy that actually made sense to them.  They are in a classroom where alternate approaches are encouraged and celebrated and the ability to “make sense” of what they are doing is what is most important.

One pair of students that I watched took an approach that I find fascinating and although it looks a bit chaotic on paper, it was quite strategic.

It is somewhat difficult to follow on paper but essentially these students broke up 783 into more “friendly” numbers and then divided each of those.  The teacher helped the other students follow what these students were doing by writing it this way.

When asked how they knew that 3/5 was .6, they explained that they first tried 0.5, but when they counted up by .5’s they ended up at 2.5.  They then realized that since they were 0.5 away from their “goal of 3” that they just needed to add .1 to .5 and it would work.

I have had many conversations, mainly with colleagues who have children in elementary school, about why they aren’t teaching math “the way we were taught”.  “Why aren’t they teaching them how to do long division properly – they have this new way that students make up themselves – don’t they have to eventually learn it the right way…”.

So I wonder, which of these groups of students that we watched have a better understanding of the problem they were working on and number sense in general?  The students that did it the “right way” – using long division – or this group that broke 783 down into more friendly numbers?

Some people would cry “but the standard algorithm is more efficient!”  I’m not sure the students that developed their own method took any longer….

# Orangey Orange Juice!

We were in a grade 8 classroom yesterday and presented them with this problem which Liisa originally saw presented by Nathalie Sinclair at an OMCA conference.  She mentioned it was from the Connected Math Project but we have adjusted the numbers slightly for a variety of reasons.

Liisa spent some time discussing the problem with the grade 8’s.  She asked them if they knew what orange concentrate was.  Then she asked them to guess (there’s power in guessing! Dan Meyer talks about how guessing can make tasks more engaging in this post).  Out of the twenty kids in the class, 18 chose Mix A!!!!  That surprised me (it didn’t surprise Liisa… she’s done this task before and found the same results).  I wondered why they picked it.  Did they truly have a gut sense of the proportions?  Was it because it was the first one?  Was it because everyone else was doing it?  The two students who didn’t pick A guessed Mix C and D.

The students set to work with big paper and markers in groups of two and the fun began!  Liisa led the lesson but there were 5 other teachers in the room that picked a group and observed without saying anything.  The two girls I was sitting with talked about the problem in an interesting way at first:

S1:  What do we do?
S2:  Well we don’t have to do math or anything.  We just, you know, think about it and decide which one is orangeiest.

They then began to write out the reasons that they felt Mixes B, C and D weren’t as orangey:  “It wouldn’t be Mix C because there wouldn’t be enough orange concentrate.  It wouldn’t be mix B because there’s too much water so it would make it watery and not orangey enough.  It wouldn’t be Mix D because there’s still too much water but not as watery as B and C.”  (I found that last sentence really interesting… they knew that!  They just didn’t have the words to explain how they knew!)

Liisa came by and asked the girls to explain their reasoning. They couldn’t, so she asked them to draw some pictures or find a way to justify their thinking.  This is what they came up with:

What a great way to represent the whole!!  Even after they had come up with this representation, they weren’t entirely sure how that helped them explain.

Liisa started a congress and had several groups present. There was some great thinking going on!  Some groups took the same strategy the girls did and drew a representation of the parts to whole.  Others used the strategy of reducing the ratio to a unit rate.

Some used percentages:

One of the students shared his interesting method:  He divided Mix A into a unit rate (1:1.5) and then multiplied it by 3 so that he could compare it with the ratio for Mix D (3:5).  That way he was able to see which mixture had more water for 3 cups.  Cool!

In sum, the lesson brought out some great proportional reasoning and thinking.  The students were engaged, asked good questions, and did some great problem solving.  These were grade 8’s… but the task could really be used in any grade that is trying to teach proportional reasoning.

A P.S. from Liisa….

Although I have tried this problem several times with teachers and students and am not surprised when many of them guess which solution is the “orangiest” – I am still surprised at the variety of strategies that people use to prove this and how rich the discussion is.   In this particular class the common theme in the strategies that the students used was the need to make something the “same” in order to compare.   Some adjusted their ratios to find the amount of water needed for 1 cup of orange juice, others turned everything into percents so that they “were all out of 100” and it would be easier to compare, some drew circle graphs so that their “whole” looked the same….What was nice is that many students recognized this similarity in their solutions even though they looked quite different.

# Some Websites…

Okay… just one more post on other good web resources. We won’t do this for every post, we promise! There is just some really great stuff online that everyone needs to know about! Sites that don’t necessarily show up when you google “Help! I need a lesson idea for ___________”. (Wouldn’t it be nice if it was that easy… maybe someone should figure out a way to create a search engine that would come up with amazing problems for a search like that!) Here’s the list… check them out and then comment with your favourites!

1)  101 Questions – A site Dan Meyer set up to “perplex” you students!  See perplexing photos and videos and ask the first question that comes to your mind.  If you log in you can see all the questions that other people have asked also.  I can see this as a nice “minds on” for a class… possibly turning into some great problems to solve!

2)  Visual Patterns – Ever trying to come up with creative patterns?  This site has a bunch of ideas ready for your using!! You can submit your own (or your students’!) patterns to the site also to add to the collection!  This would be a great for the patterning to algebra unit in grade 9!  They aren’t all linear either so use them for some great thinking in every grade!!

3)  Graphing Stories – I love this site!!!!  You will love it too… because we all love analyzing life through graphs!  This site has a collection of videos that your students can watch and graph!!! A downloadable handout is on the site with four blank __________ vs. time graphs.  Show your students a video, get them to graph what is happening and talk about it!  I think your students will love it as much as you do!

4)  Math Mistakes – I was reading a great resource yesterday (Malcolm Swan Standards Unit – it’s great! Check it out too!) that talked about the importance of exposing and discussing common misconceptions.  This site may be able to help you with that.  Teachers submit interesting math mistakes that students have made.  The mistakes are interesting because it shows a conceptual error in student thinking.  It’s good professional development as a teacher… but I think it could be powerful for students also!  And it’s real!!!

5)  Estimation 180 – Similar to the 101 questions site mentioned above, this site shows you a photo and asks you for a quantitative estimate about something in the photo.  You are asked to explain what cues you used to estimate and then you can see what other people have estimated and why.

# Apps for Math

We’ve been on the hunt this year for Ipad/Iphone apps that would be useful for math.  Here’s what we’ve found so far:

SketchExplorer – This is an app put out by Geometer’s Sketchpad.  It has some neat premade sketches that come with it and you can also upload your own sketches from GSP.

WolframAlpha – This app tells you everything about anything!  Type in any math word, non-math word, equation, anything and it will give you more information than you ever wanted to know!

Free GraCalc – A handy graphing calculator that includes a graphing function, tables of values, triangle solver and polynomial solvers.  Easy to use interface.

Nearpod – This app is kind of cool, although I haven’t quite decided how I would use it.  It allows the teacher to show a powerpoint-type presentation on each student’s iPad.  The teacher controls the pace of the slides but students are able to interact with each slide at their own pace.  You can add videos, polls, survey’s, questions, and other features.

Protractor 1st – a simple but useful protractor that shows measurements in degrees and radians

oScope Lite – An oscilloscope that measures decibles vs. time of sound.  Great for introducing periodic functions using instruments or voice.

Qrafter – A QR code generator and reader.  You can make scavenger hunt type activities with these, or just add them to a worksheet or assignment for hints or interesting facts.

Quick Graph – Another graphing calculator which allows you to graph functions in 2D or 3D.

Skitch – A neat app that lets you add details to photos, maps, or documents and save them.

Video Physics – Allows you to take a video and analyze the motion of an object with a distance time graph.  Some premade sketches are included.

MyScript Calculator – Write a numerical expression with your fingers and the app will calculate the answer.

DragonBox – A game that explores algebra concepts with kids hardly even knowing it!  It’s a bit pricey… but it’s fun!

A lot of reading I’ve been doing about ipads in the classroom emphasizes the fact that it’s not the effectiveness of the app that matters, but rather how the teacher uses it in the classroom.  We think the best apps will be ones that allow kids to use the iPad as a tool (Videophysics, oScope Lite, etc.) that allows them to see and explore the math.  Comment if there are any great apps that you have been using!

There are two expectations in our Grade 9 Academic curriculum that I felt I never did a great job of addressing and I know many other teachers feel the same. They state that students will:

• pose questions about geometric relationships, investigate them, and present their findings, using a variety of mathematical forms
• illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example, with or without the use of dynamic geometry software

Really, they seem like quite fantastic expectations.  They provide an opportunity for some great open exploration. Unfortunately, I think they are often left to the end of the course and aren’t given much attention because they don’t seem to be that necessary for future courses.  I have to agree that the actual “content” of these expectations may not be critical for future math courses.  However, I would have to argue that the “processes” in these expectations; the ability to pose questions, explore them and be able to prove or disprove ideas is indeed critical for future math courses and the very thing that many of us hope our students can do.

With that in mind, a grade 9 teacher let us “borrow” her students to see how they might deal with these expectations.

We started the lesson by talking about the word “conjecture”.  None of the students in the class were able to provide a definition so we gave them a few clues (posted a couple of tweets that had the word in it) and had them talk with a partner about what the word might mean.  After some time, a student suggest it means “a guess or a prediction”.

I made a conjecture that “all grade 9 students in this school have a cell phone” and asked whether they thought my conjecture was true.  A student quickly put his hand up and said “no”. I asked him how he knew and he smiled slyly and said “because I don’t have one…”  Which led to a discussion about the usefulness of a counterexample.

Students created a sketch of a parallelogram on GSP.  I told them a student had a conjecture that the angles that were diagonally opposite each other were equal.  We discussed what we would need to do to “prove” this conjecture and then the students proceeded to make the necessary measurements.  (Note:  I realize that testing many cases on GSP may not be considered a true “proof” in the academic sense, but for our purposes we decided it would be good enough for now).

The next task students drew the diagonals of the parallelogram and were asked to come up with their own conjectures.

This is how the conversation went:
S:   The opposite angles, where the diagonals intersect will be equal.
S:  The two diagonals meet in the middle of the parallelogram.
T:  What do you mean by middle?
S:  The center of the parallelogram.
T:  Taylor, you seem to have an idea…
S:  The sum of the angles where the diagonals intersect will be 360°
T:  Wes, you have an idea:
S:  The triangles that are opposite each other are congruent.
T:  What do you mean by the triangles opposite each other?
S:  The one on the left and the one on the right.
T:  And can you explain what you mean by congruent because some people may have forgotten that word.
S: Perfect mirror images of each other.

We recorded all of the conjectures that students proposed and then they set out trying to prove them.  We found that as students were trying to prove their conjectures they started to make new conjectures based on some of the measurements that they were making.

Students were then given some additional sketches to create and then make their own conjectures about. See the file here:  making conjectures

In general, students were quite engaged.  I think mainly because they had a stake in what they were doing.  It was “their” conjecture so they wanted to prove that they were right.   It reminded me that we need to give more ownership of the mathematical ideas to the students and allow them to be mathematicians as much as possible!

# I can never find good problems…..

This is something we all struggle with. We know there are good problems out there, and often we even come up with some ourselves….but I can never seem to come up with a good problem when I need one!

Here are some people that have been helping us through their own blogs about learning mathematics.

Dan Meyer

Fawn Nguyen

Matt Vaudrey

Chris Hunter

Nat Banting

Kate Nowak

Cathy Yenca

We’ve recently been talking with teachers a lot about the following problem from last year’s EQAO test:

The total cost of swimming at a community swimming pool is made up of a membership fee and a cost per swim. At this community centre, Jake pays a total of \$100 and swims 40 times. Paula pays a total of \$70 and swims 25 times. Which of the following statements is true?

a) The membership fee is \$20.

b) The membership fee is \$30.

c) The cost per swim is \$2.50.

d) The cost per swim is \$2.80.

How would you do it?  Or maybe more importantly, how would you expect your students to do it?

We’ve heard a variety of methods:  create a system of equations and solve, write as two points and find the slope and y-intercept, think about the context (i.e. I’m paying \$30 more for 15 more swims…), trial and error for each option, etc., etc.

Through talking with teachers we conjectured that students really don’t need to have any knowledge of linear equations at all to do the swimming pool problem.  They just need to have a sense of the context and operations.  We decided to test this conjecture with the Graduation Problem which we found on Dan Meyer’s website, and adapted to the class we were working with.

Students are given a class graduation roster (we got a list of the students currently in grade 9 and made up a “Class of 2016” roster) with the times that two students crossed the stage (see Dan’s activity).  Students know that graduation starts at 1:00, there are some speeches, and then students start crossing the stage one by one.  The overarching questions are:  a) how long are the speeches? and b) how long does it take each student to cross the stage.

Students were immediately engaged in the problem.  Many had experienced a Grade 8 graduation not that long ago so could relate to the context.  One student shouted out “I wonder what time I would cross the stage at?”  so we knew we had their interest.

Here’s some samples of student work:  (note that these students knew NOTHING about linear equations yet, they were just finishing their first unit of the course on measurement)

Most teams managed to solve the problem but the approaches varied.  Many students eventually chose to find a rate but they described the rate in different ways: 25 students/5 mins., 12 seconds/student, 5 students/1 minute.   We used the same numbers as Dan for the problem and we found this choice of numbers was helpful in allowing students to represent their rates in a way that they were comfortable with.  Students also used different strategies for determining the time of the initial speeches including just guessing initially but then returning to test their guess once they came up with a different approach.

After sharing solutions, Liisa told the class that the roster the students had been given actually was missing about 20 people from their grade (which was true!).  She asked them how they could figure out the length of the graduation if they were added to the list.

One girl raised her hand: “You could make an equation… it’d be like… um… 12 seconds times the number of people + 20 minutes for the speeches”.

This student essentially had developed an equation of a line given two points (with some minor issues with units….) and she did this without any formal lesson on how to find the equation of a line given two points, in fact, she hadn’t even heard about slopes and y-intercepts yet – she just made sense of the problem and solved it!

Would this be an effective way to introduce the equation of line given two points?  If you started with this problem, could students move to generalizing an approach that would work in any situation?