Posted by: jwhiff | July 18, 2018

MN3: Largest Countries (Population)

If you don’t follow up the largest countries (area) post with this one on population, you are missing out on an interesting opportunity to delve deep into issues concerning population density.  More of the full explanation will follow! If this is not a follow up lesson, it is still interesting as a stand alone.

If this is your first time trying Mystery Numbers, have a read here for the necessary background info.

Here is this week’s list:











Unit = people

Expert Source = CIA World Fact Book

Country List:














United Kingdom

I would present students with a population density map to help them narrow down which regions appear to be most populated. Have them make their initial lists before presenting them with the country list. Then it’s back to the maps!



Posted by: jwhiff | July 16, 2018

MN2: Largest Countries (Area)

This is a completely different list than last time!  It is a great one to do when you are about to cover area and perimeter. Make sure you have access to some atlases, even if they are old and out of date.  Longest coastlines will follow. Ooh…then populations and population densities!

By the way, I’m not going to give a big, long explanation every time I come up with new list.  If you want to know how to use Mystery Numbers in your class see my first posts to guide you:

Mystery Numbers Explained!

Mystery Numbers Sample #1: Basketball

Here’s the new list:











Here is the web source for your expert: World Altlas

Unit = square km (I recommend that you spend some time bringing this to life, too.  How big is this exactly? How big is our school property compared to a square km?)

Here are the countries (some correct, some put in there to throw them off.  Provide atlases to search and create their own lists ahead of time. Also a good time to introduce the idea of map projections!)


Greenland (Denmark)







United States


Democratic Republic of Congo


South Africa



I’m really hoping that the Big Reveal in this case is controversial!  It should be…map projections really fool us!  Plus, there is a good opportunity for kids to write about what surprised them here.  Greenland looks so huge!  I thought for sure it would be in the top 10.  Crazy that Australia is smaller than Brazil!  Russia is THAT much bigger than Canada?  Really?

If you have students finished way ahead of the others, get them to start looking for the very smallest countries and generate their own lists.  Use maps or atlases, not technology in this case.  Why?  Technology gives too much away and maps are so tactile and satisfying to explore.

Posted by: jwhiff | July 13, 2018

MN1: Basketball

I’m trying to keep my Mystery Number titles plain and numbered to help you keep track.  I’ve also added categories under Mystery Numbers so that you have an idea about what you are getting.

Now! I’ve started with a list that interests my youngest son right now.  My goal in this endeavour is to have as many lists as possible, covering a wide range of topics and possible interests.  I’m hoping that you (teachers) will find lists that will match topics that you are covering in class.  I would link this one up with gym class, for sure.

Here’s the first list:











Remember that the kids should demonstrate some understanding of the numbers FIRST.  Get them to order the numbers, for sure.  You could also ask them to provide word forms, expand them and represent them in some way (build them, draw them, compare them). You can also break down place value (important when you have very large numbers or very small numbers with decimals).  What you decide to do should match your teaching focus.   By the way, copy and paste this list into a plain word document for them to see or use.  Don’t give students a link to this site.  Too much will be given away!

Next, the kids need to come up with the title of the list.  Here is a link to the site I used to find those numbers in the first place: Expert Source 1

To help them, they need to have an expert. I recommend that you select a student expert (or two).  It would be great if you can choose students who are keen on the topic.  Experts should be willing to research and understand the topic well enough that they can lead 20 questions.  They can also make a powerpoint for the BIG REVEAL.  It’s nice to have pictures and a visual count down.  Here is a sample:

Basketball Points

For now, just play 20 questions and come up with the title.  You can generate a basic title, or really specific. Now either provide them with the unit (in this case points) or have this emerge from a quick discussion about what might make sense in this case.

Once that they know the basic details, they need to brainstorm what they already might know about the topic.  They can do this in groups.  Hopefully you have been sneakily exposing them to some of this information in gym class.  This will give them something to remember come brainstorming time.

Play it out a little and see how the kids do.  Stuck?  Provide them with this randomized list of people, some correct and some way off:

Kobe Bryant

Sydney Crosby

Dirk Nowitzki

Elvin Hayes

Sergio Aguero

LeBron James

Brock Boeser

Moses Malone

Kareem Abdul-Jabbar

Michael Jordan

Shaquille O’Neal

Wayne Gretzky

Karl Malone

Wilt Chamberlain

John Horgan

See if they can whittle this down to the ones that make sense and then match the ones that make sense up with their numbers.

After you have played this out, time for the big reveal!  Remember that it would be nice to use a student-generated powerpoint to help with this. Then, follow up as student interest dictates!  There are a lot more interesting stats on the topic, plus all kinds of opportunities for reading and research.


Posted by: jwhiff | July 10, 2018

Shape Shifters: Reflection and Rotation Symmetry

I worked away to update this one.  I added some rotation animations and a couple of guided practice pages to help out.  I hope they help!


Practice Pages:

Reflection Symmetry Worksheet

Rotational Symmetry Worksheet


Posted by: jwhiff | July 8, 2018

Mystery Numbers!

This is an excellent intermediate (grade 5-8) routine! This will take at least 2-4 math periods to complete at a time. It involves developing and practicing number sense across whole numbers, fractions and percents. Devices are handy (especially for accessing and recording information), but not totally necessary. It can all be done on paper, too. Have base ten materials ready for representing and building numbers, if you can. The kids will need to read, write, hypothesize, discuss, offer and listen to opinions, make connections and hopefully be blown away.

Step 1: The Experts

An expert’s job is to read ahead and absorb information about the chosen mystery numbers topic. They will help their classmates uncover the mystery by being there to take questions. Experts can also create power points or some sort of presentation for the big reveal (Sample: Basketball Points)

The first time around, this should be your job. You are modelling it for your future student experts. By the way, I will post up the necessary links!

Step 2: Introducing the Mystery Numbers

Show students the raw numbers randomized in a list. Their first job is to simply organize and break these number down a little. They should be able to read them, build or visualize them, expand them, explain place values and place them in some sort of order. They can do this on paper or on their devices. Handy tip: after students have built the numbers, they should take a photo with their devices and link it to their work.

Step 3: Game Time

Once the kids feel comfortable with the numbers, they’ll need to determine the topic. This is where your experts come in. Play a little 20 questions as a class. The experts can only answer yes or no to these questions. Keep going until the topic is uncovered! They range from light and fun to really serious.

Step 4: Story Details

Now that the topic is uncovered, the kids need to revisit the numbers. Any thoughts on what these numbers are describing? Write down hypothesis/guesses. Compare these through discussions (whole class, small group, partners or a bit of everything!). You may want to generate a list of units that make sense. Dollars? Grams? Might some numbers actually represent percentages? At this point, you can reveal the list with the units attached. Does this make the story clearer? Time to give the kids a second chance to revise their thinking. They should do some research during this phase, too.

Final Step: The Big Reveal

Time to compare results! Once the real list is revealed and kids have had a chance to see how close they came, they can dig into the story a little further. For example, once the kids know the GDP of various nations, they should be introduced to wealth sharing percentages. It creates a fuller, more complex picture. They should have a chance to ask and wrap their heads around possible answers. Why are there two completely different lists for the top 10 box office earning movies of all time? Why are civilian casualty numbers for World War 2 rounded to the nearest thousand? How did they determine world populations ten thousand years ago? They say that numbers don’t lie, but how can we trust them for sure?

These are all interesting and important questions. Stoke your students’ enthusiasm the best you can during this phase. Pay close attention to conversations, mining them for insights and questions. Be curious yourself. This is my favourite part and will be your’s too.

Stay tuned for the next posts…I’ll be laying out topics and number lists next.

Posted by: jwhiff | July 4, 2018

What Students Bring: Observing During Math Play

You know when you’ve planned this really great lesson and you place those colourful, hands-on materials in the middle of each group and the kids just dig in and start building a bunch of stuff that has nothing to do with your lesson?

We all anticipate this, right? A lot of wise teachers actually make sure they allow for some of this free play at the start of lessons. Why fight it?

You can actually build a really useful math routine around this natural urge to play.

Ever heard of math talks? You know- you place up an image of something in the real world like a couple of full egg cartons and you lead a discussion on the math that the kids notice? It’s a great routine. This just builds on this routine.

All you need to do is snap pictures of the things the kids are building while exclaiming “Woah! It’s so weird, but I’m noticing a lot of interesting math in the things you are building! Did anybody else notice this?”

Then you load up the images and show them one by one on the screen. Check it out! Leah! Did you realize your were making math this whole time? Crazy! Can anyone tell what kind of math Leah was making?

And they are in! The kids want to astound everyone with their math creations. You just need to pay close attention and order up some serious investigations when something interesting comes up.

This is a perfect intro to math at the start of the year. And it’s a gift that keeps on giving.

Posted by: jwhiff | July 4, 2018

Investigating What Students Bring

I don’t even want to call this assessment, although really this is what investigating means.  Formative investigation.  Investigation for learning.

I don’t want to call it assessment, because beginning of the year assessments look an awful lot like tests in my opinion.  I understand that these tests are simple to administer, but they don’t give kids much of a chance to explain to you what they actually bring with them to your class.  And what do they actually show you?  That Jimmy doesn’t know how to subtract 1.447 from 2? Maybe Jimmy does, but the test doesn’t give him room to use a strategy he learned last year.  Or maybe he knows he has a good strategy, but the stress of a beginning of the year test has made it hard for him to recall it.  Maybe he doesn’t even get to share his greatest strength in math because there are no questions that allow him to show this to you.

How can you find out what students bring with them?  One way is to  ask them, one at a time.

When on earth can you find the time for this?  This may seem a bit simplistic, but I ask them at silent reading.  I spend the first couple of weeks at school setting a rock-solid silent reading routine.  It is a very serious routine in my class.  All kids must read.  I set up reading expectations and kids document with a reading log.  Anyone who can’t focus during reading time gets extra reading coaching from me before being allowed out for lunch.  I help anyone who can’t find a good book find one.  They figure expectations out pretty quickly: they must be silent and they must read.  Then I introduce the “one math question a day” routine.

If I want to investigate their subtraction skills, I get give them a few questions to choose from.  I show them in different ways.  For example, one question will show numbers lined up, one over the other.  Another will show numbers side by side.  I even have one with words (If I spent $14.58 on books, and I started with $20, how much do I have left over?).  They choose one and try it out in their planner.  I then visit and ask questions:

I see you have come up with a pretty reasonable answer here.  Can you show me your strategy?

I see you crossed out your work.  Why?  Ah.  You can’t quite remember. Can I show you a few ways to try this?  You tell me which way looks familiar to you.

Are all of these too easy?  Wow!  How about this one? 

I can actually visit quite a number of students during one silent reading period, especially with no serious distractions.  I then repeat the process with slightly different questions the next day.  I keep this up for the entire year.  Yep.  The whole year.

It works incredibly well for me.  I find out what they are interested in learning, what they know and what they need more practice with.  It is a daily, one-on-one tutoring session for students who need it.  These students range from the mathematically motivated kids who really want to learn something new to the kids that need extra time with multiplication.  It is a perfect time to set goals and to check in to see how those goals are coming along.

I can’t imagine not doing this in my class.  I find it simple and incredibly valuable.  No students complain about this extra one question at planner/silent reading time.

There are other ways to find out what students bring, too.  I’ll share another way next post.

Posted by: jwhiff | July 3, 2018

What Students Bring

I’ve been giving some thought to math practice over the last…well…decade or two.

I’m not one who relishes spending math class completing worksheet after worksheet of adding or dividing or whatever.  I feel terrible giving kids practice when they already know how to do it and find it too easy.  I feel awful giving practice to kids when they struggle and can’t get through it because they need you to sit with them through every tough spot.  I can’t stand it when I don’t give enough of the right practice when kids need it.  I have developed some pretty interesting routines to try to address this problem and to give us time as a class to explore some more meaningful math activities.

However, when kids, parents or next year’s teacher let me know that they were assessed on some pretty tough basics right off the bat, I feel demoralized and worried.  Was my attempt to make math meaningful during their time with me wasted time?  Should I have stuck to drilling the basics since that was what they were judged on right away?  Did I give them enough practice?

To simplify this confessional, let’s just say that I find this bit of teaching math tough.

To top it all off, I know with 100% certainty that if you want to do well at anything, it takes time and practice.  So what to do?

Well, I feel like I need to honour the need to practice and practice well.  Plus, I don’t want to give up my meaningful activities.  Plus, I want to get better at both.  And, oh ya…I still have to teach language arts and science and social studies  and gym and art and, and, and….!

All of these things take time and practice!!!  We need to prioritize, of course.  When parents bemoan the fact that their kids aren’t good at cursive writing, I am not afraid to tell them what I know…that to get good at anything, you need lots of practice time.  And then to ask them, “How much time in the day do you actually want me to dedicate to cursive writing so that the kids these days know how to do it well…and at the expense of what?”   When I put it that way, they reasonably conclude not much.  When they come to me asking about math practice, I feel less sure of what to say.

I know it takes time. They know it, too.  But math reality is more complicated than cursive writing reality.  There is a lot we are supposed to do in math, and to do any of these things, they all need time and practice to do them well.  If we try to spend a little time on everything, kids aren’t getting enough time to do any of those things well.  So what do we focus on?

I recently spoke to a student completing her PhD in math education and she argued that we shouldn’t bother practicing basic algorithms at all.  Her opinion was that students should be engaged in authentic problem solving, period.  At the risk of sounding like a luddite, I have to confess that I was uncomfortable with her statement.  It is true that students would become better intuitive problem solvers if all math time was dedicated to this practice and I love thinking about this possibility on a strictly philosophical level.  However, I could never do it in isolation.  I’d have to have pretty darn strong convictions to throw it down and go this way knowing the judgement and uncertainty my students would face the following year.  Private tutors would get a lot of work, that’s for certain.  If parents can afford them (which brings up another issue: two-tiered math education! Good topic for a future post.)

And that’s the thing!  No matter what I choose to do, people everywhere have very strong, often divergent opinions about math. And when people disagree, kids are caught in the middle.  The middle of a battle ground is a very uncomfortable place for anyone to be, let alone a child!  I won’t choose a strong, controversial position and engage in a philosophical battle with parents or with other teachers.  I can’t just say they are ignorant or wrong.  Especially when I know these people to be mostly reasonable, intelligent and concerned for kids.

So!  This is what I have done:

I have developed a practice that attempts to honour and accommodate what students bring with them to my class.  My early assessments are very gentle. I want them to teach me what they know about numbers and how they interact.  And if they can’t remember, I want to have a few possibilities ready for them to see if any look familiar so that I might help refresh memories and ease anxiety.  I then use my knowledge of what they bring with them to build the right kind of practice that helps them move forward on the foundations they have.  No deleting.  No huffing and puffing about last year’s methods or about what they did or did not learn.  I’m not worried about that.  I am worried about the child in front of me.  Kids feel so good when they know that the precious things they have are valued and will not be discarded by me.

I’ll go into detail about how I managed this next post.  This was a bit of a long essay, but I needed to write it.



Posted by: jwhiff | June 30, 2018

Outdoor Club: Birder’s Wing

This wing of the outdoor club was wonderful. It was so simple and attracted an amazing diversity of children.

I was worried at first that they would lose interest.  Bird watching takes patience and you don’t always have a lot of sightings.  I shouldn’t have worried, however.  The kids were thrilled by the possibility and loved to be out on an adventure.  I think they also simply loved belonging to this gentle group of bird-lovers.

Sometimes the adventure was circling the playground in a new, slightly beyond the fence way.  Sometimes we dipped into the forest and walked a peaceful trail.

We also created a bird wall were the whole school could record their sightings and test their knowledge.  I was surprised at the level of interest and often heard groups of kids chatting about birds, whether or not they were in the Outdoor Club: Birder’s Wing.

I loved my birders and was so glad to have created the opportunity.  If you like the idea, but are unsure, I encourage you to give it a shot.  My only regret, in typical fashion, is that I did not try this earlier.

Oh…and don’t you think this is wonderful data collection?  It worked really well.  Great opportunity to integrate with math.

Posted by: jwhiff | June 30, 2018

Builders of the Outdoor Club

I had so many kids sign up for the outdoor club, that I felt like I needed to break them up into interest groups.  I had gardeners, bird watchers, designers and builders.  Great idea, but my problem was that I hadn’t really figured out how to support them properly.  Builders?  I had some eager grade 5’s build the garden boxes and pull the future sand pit apart, but after that, what on earth were they supposed to build?

I initially thought they might test out a construction spot in the outdoor classroom.  I imagined stumps and nails and saws and sticks, but worried about the number of kids who wanted to build in that setting.  I then wondered if I should order a bunch of bird house kits and give them an official building project, but this is not really in the spirit of my outdoor class (too complicated, too expensive, not creative enough).

Finally, I realized the answer was right in front of me:  the forest.  I just needed a couple of staff members or parents to come out at lunch to help me supervise the kids in the best flexible construction space ever!  What would we build?  Forts of course!

It was simple and fun.  The kids (from grade 1 to 5) harvested bark off of deadfalls, gathered sticks, and built to their hearts content.  It was beautiful to witness and became a regular routine.  Staff members who help supervise were astounded at the richness of the experience.

Despite the fact that I will be leaving for a new job next year, I think the fort building will continue!

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