## Central Tendencies Puzzles

Central Tendency Puzzle templates for you to check out. I’d love to hear some feedback on these.

Data management is becoming an increasingly important topic as our students try to make sense of news, social media posts, advertisements… Especially as more and more of these sources aim to try to convince you to believe something (intentionally or not).

Part of our job as math teachers needs to include helping our students THINK as they are collecting / organizing / analyzing data. For example, when looking at data we want our students to:

• Notice the writer’s choice of scale(s)
• Notice the decisions made for categories
• Notice which data is NOT included
• Notice the shape of the data and spatial / proportional connections (twice as much/many)
• Notice the choice of type of graph chosen
• Notice irregularities in the data
• Notice similarities among or between data
• Consider ways to describe the data as a whole (i.e., central tendency) or the story it is telling over time (i.e., trends)

While each of these points are important, I’d like to offer a way we can help our students explore the last piece from above – central tendencies.

## Central Tendency Puzzle Templates

To complete each puzzle, you will need to make decisions about where to start, which numbers are most likely and then adjust based on what makes sense or not. I’d love to have some feedback on the puzzles.

Linked here are the Central Tendencies Puzzles.

## Questions to Reflect on:

• How will your students be learning about central tendencies before doing these puzzles? What kinds of experience might lead up to these puzzles? (See A Few Simple Beliefs)
• How might puzzles like these offer your students practice for the skills they have been learning? (See purposeful practice)
• How might puzzles like this relate to playing Skyscraper puzzles?
• What is the current balance of questions / problems in your class? Are your students spending more time calculating, or deciding on which calculations are important? What balance would you like?
• How might these puzzles help you meet the varied needs within a mixed ability classroom?
• If students start to understand how to solve one of these, would you consider asking your students to make up their own puzzles? (Ideas for making your own problems here).
• Would you want students to work alone, in pairs, in groups? Why?
• Would you prefer all of your students doing the same puzzle / game / problem, or have many puzzles / games / problems to choose from? How might this change class conversations afterward?
• How will you consolidate the learning afterward? (See Never Skip the Closing of the Lesson)
• As the teacher, what will you be doing when students are playing? How might listening to student thinking help you learn more about your students? (See: Noticing and Wondering: A powerful tool for assessment)

## Skyscraper Puzzles – printable package

An area of mathematics I wish more students had opportunities to explore is spatial/visualization. There are many studies that show just how important spatial/visual reasoning is for mathematical success (I discuss in more depth here), but often, we as teachers aren’t sure where to turn to help our students develop spatial reasoning, or now to make the mathematics our students are learning more spatial.

One such activity I’ve suggested before is Skyscraper Puzzles. I’ve shared these puzzles before (Skyscraper Puzzles and Skyscraper Templates – for relational rods). With the help of my own children, I decided to make new templates. The package includes a page dedicated to explain how to solve the puzzles, as well as instructions on each page.

## New Puzzles can be accessed here

*The above files are open to view / print. If you experience difficulties accessing, you might need to use a non-educational account as your school board might be restricting your access.

You’ll notice in the package above that some of the puzzles are missing information like the puzzle below:

Puzzles like these might include information within the puzzle. In the puzzle above, the 1 in the middle of the block refers to the height of that tower (a tower with a height of 1 goes where the 1 is placed).

You might also be interested in watching a few students discussing how to play:

### A few thoughts about how you might use these:

As always, I’d love to hear from you. Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).

## One-Hole Punch Puzzle Templates

Recently Lowrie et al. published an article in the Journal of Experimental Education where they looked at the effects of a spatial intervention program for grade 8 students. This study followed the 876 grade 8 students across 9 schools as they received 20 hours of spatial interventions (as well as other grade 8 students in schools that conducted “business as usual” mathematics classes). Their findings were quite interesting. They found students who received spatial intervention programs achieved:

• significantly better on spatial tasks (13% higher)
• significantly better on Geometry – Measurement problems that included material not discussed in either class
• significantly better on Number – Algebra problems
• equally well on Statistic – Probability problems

While this study supports many studies showing a link between spatial abilities and mathematics performance (Mix and Cheng, 2012), and others showing how students’ spatial abilities are malleable at any age (can be learned with the right experiences) (Uttal et al., 2013), there seems to be a definite need for teachers to have at their fingertips good examples of experiences that will help our students develop spatial reasoning. At the end of this post are possible examples for us to try.

A recent meta-analysis of 217 studies, representing more than two decades of research on spatial training, found that a variety of activities improve spatial reasoning across all age groups. Not only did the authors find that spatial training led to improvements on spatial tasks closely related to the training task, but improvements were also seen on other types of tasks that were not part of the training.

Taking Shape, 2016

A focus on spatial reasoning, from my experience, has helped the students in my schools make sense of connections between concepts, it has been the underpinning of new learning, and has been the vehicle for so much of OUR learning as educators.

## One-Hole Punch puzzles

You might be familiar with various cognitive tests that ask students to think through mental rotations, 2D/3D visualization, paper folding or other tasks where students’ abilities to visualize are measured. Spatial tasks like these are very predictive of a students’ math success, however, few resources are aimed to specifically help our students develop their spatial reasoning (see Taking Shape as an ideal K-2 example).

Below is an example of one of these cognitive test questions. As you can see, students here are asked to mentally fold a piece of paper and then punch 1 hole through the folds, then imagine what the paper would look like once opened.

## Templates

Instead of trying to measure our students’ abilities by giving sample problems like this, it is far more productive to offer experiences where our students can learn to think spatially. This is why I have created a few sample experiences. Below are sample templates that you can use with your students. They will need squares of paper (linked below), as well as the puzzles they would be aiming to solve. Take a look:

## How to Solve a One-Hole Punch puzzle:

Directions for solving the puzzles are only written on the first page:

“These are One-Hole Punch puzzles.  To complete a puzzle, take a square of paper, fold it using as many folds as needed so that if you punch one hole and unfold it, you will match one of the puzzles.  Complete the following puzzles in any order.  Be prepared to discuss your strategies with others.”

Students will need several blank squares of paper and a single hole puncher. Student will need to visualize how to fold their paper and where to punch their one hole to match the puzzle they are working on. These puzzles are tricky because only 1 hole can be punched to create several holes.

Pro Tip: Make sure you have a lot of squares of paper for every student.

## A few thoughts about using these:

• How will you introduce these puzzles to your students?  How much information about strategies and tips will you provide?  Will this allow for productive struggle, or will you attempt to remove as much of the struggle as possible?
• Would you use these as an activity you give all students, or something you provide to just some.  Why? (Something for students who finish early or something for everyone to try?)
• How would giving a page of puzzles to a pair of students be different than if you gave it to individuals?  Which were you assuming to do here?  What if you tried the other option?
• How will you orchestrate a conversation for your class to help consolidate the learning here? How might this be helpful before you give the next set of puzzles?
• What will you do if students give up quickly?  What questions / prompts will you provide?
• Who is successful at solving these puzzles? Sometimes, those who are often left out of mathematical discussions can become more involved when given spatial tasks. How might you capitalize on increasing your students’ interest in mathematics? Do you see tasks like this helping improve your students’ agency, identity and authority in mathematics?
• How might you see intentional decisions to focus on spatial reasoning as helping your students with mathematics in other areas? This post might be helpful.

A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity.  Hopefully, this post has helped you consider your own decision making processes!

As always, I’d love to hear your thoughts. Feel free to write a response here or send me a message on Twitter ( @markchubb3 ).

## The Types of Questions we Ask: which categories of questions should we focus on?

I think we can all agree that there are many different ways for our students to show what they know or understand, and that some problems ask for deeper understanding than others. In fact, many standardized math assessments, like PISA, aim to ask students questions at varying difficult levels (PISA uses 6 difficulty levels) to assess the same concept/skill. If we can learn one thing from assessments like these hopefully it is how to expect more of our students by going deeper… and in math class, this means asking better questions.

Robert Kaplinsky is a great example of an educator who has helped us better understand how to ask better questions. His work on Depth of Knowledge (DOK) has helped many teachers reflect on the questions they ask and has offered teachers examples of what higher DOK questions/problems look like.

In Ontario though we actually have an achievement chart that is aimed to help us think more about the types of questions/problems we expect our students be able to do. Basically, it is a rubric showing 4 levels of achievement across 4 categories. In Ontario it is expected that every teacher evaluate their students based on each the these categories. Many teachers, however, struggle to see the differences between these categories. Marian Small recently was the keynote speaker at OAME where she helped us think more about the categories by showing us how to delineate between the different categories of questions/problems:

• Knowledge
• Understanding
• Application
• Thinking

## Knowledge vs. Understanding

Below are a few of Marian Small’s examples of questions that are designed to help us see the difference between questions aimed at knowledge and questions aimed at understanding:

As you can see from the above examples, each of the knowledge questions ask students to provide a correct answer. However, each of the understanding questions require students to both get a correct answer AND be able to show that they understand some of the key relationships involved. Marian’s point in showing us these comparisons was to tell us that we need to spend much more time and attention making sure our students understand the math they are learning.

Each of the questions that asks students to show their understanding also help us see what knowledge our students have, but the other way around is not true!

Hopefully you can see the potential benefits of striving for understanding, but I do believe these shifts need to be deliberate. My recommendation to help us aim for understanding is to ask more questions that ask students to:

• Draw a visual representation to show why something works
• Provide an example that fits given criteria
• Explain when examples will or won’t work
• Make choices (i.e., which numbers, visual representations… will be best to show proof)
• show their understanding of key “Big Ideas” and relationships

## Application vs. Thinking

Below are a few examples that can help us delineate the differences between application and thinking:

These examples might be particularly important for us to think about. To begin with, application questions often use some or all of the following:

• use a context
• require students to use things they already should know
• provide a picture(s) or example(s) for students to see
• provide almost all of the information and ask the student to find what is missing

Thinking questions, on the other hand, are the basis for what Stein et. al called “Doing Mathematics“. In Marian’s presentation, she discussed with us that these types of questions are why those who enjoy mathematics like doing mathematics. Thinking and reasoning are at the heart of what mathematics is all about! Thinking questions typically require the student to:

• use non-algorithmic thinking
• make sense of the problem
• use relevant knowledge
• notice important features of the problem
• choose a possible solution path and possibly adjust if needed
• persevere to monitor their own progress

Let’s take a minute to compare questions aimed at application and questions aimed at thinking. Application questions, while quite helpful in learning mathematics concepts (contexts should be used AS students learn), they typically offer less depth than thinking questions. In each of the above application questions, a student could easily ignore the context and fall back on learned procedures. On the other hand, each of the thinking questions might require the student to make and test conjectures, using the same procedures repeatedly to find a possible solution.

Ideally, we need to spend more time where our students are thinking… more time discussing thinking questions… and focus more on the important relationships/connections that will arise through working on these problems.

## Final Thoughts

Somehow we need to find the right balance between using the 4 types of questions above, however, we need to recognize that most textbooks, most teacher-made assessments, and most online resources focus heavily (if not exclusively) on knowledge and occasionally application. The balance is way off!

Focusing on being able to monitor our own types of questions isn’t enough though. We need to recognize that relationships/connections between concepts/representations are at the heart of expecting more from our students. We need to know that thinking and reasoning are HOW our students should be learning. We need to confront practices that stand in the way of us moving toward understanding and thinking, and set aside resources that focus mainly on knowledge or application. If we want to make strides forward, we need to find resources that will help US understand the material deeper and provide us with good examples.

## Questions to Reflect on:

• What did your last quiz or test or exit card look like? What is your current balance of question types?
• What resources do you use? What balance do they have?
• Where do you go to find better Understanding or Thinking questions?
• What was the last problem you did that made you interested in solving it? What was it about that problem that made you interested? Likely it was a Thinking question. What was it about that problem that made it interesting?
• Much of the work related to filling gaps, intervention, assessment driving learning… points teachers toward students’ missing knowledge. How can we focus our attention more toward understanding and thinking given this reality?
• How can we better define “mastery” given the 4 categories above? Mastery must be seen as more than getting a bunch of simple knowledge questions correct!

If you haven’t already, please take a look at Marian Small’s entire presentation where she labels understanding and thinking as the “fundamentals of mathematics”

Open problems are probably my most used strategy to help meet each student where they are.  Problems that offer a low floor and high ceiling are great because all students engage in the learning, then can participate and learn from each other.  However, some teachers also like to offer Parallel Tasks as a way to differentiate instruction.  The idea here is that students can be given a choice of a task/problem, some being more difficult than others, yet all of the tasks/problems deal with the same standard (curriculum expectation).  Let’s take a look at an example of how a quality parallel task can work:

Take a look at the problem below.  What is it asking us to do?

I’d love some actual responses here.  Build it using actual manipulatives (ideally) or using virtual manipulatives (This Illuminations Applet might help).

Notice that each choice allows students to do the same expectation related to proportional thinking, however, students are given choice about what numbers they want to think about.

Think about what the answers would look like?  When we discuss designs afterward, we should be able to discuss the solutions to each problem and compare the similarities and differences.

Here are some designs students made.  Can you tell which option each student chose?

Actually try to match the designs to each of the tasks/problems.  Take a moment to think this through.  What do you notice about the 4 images above?

This task was designed very cleverly to help make a point… to help us bring ALL of our students together to have a conversation  (Even when we ask our students to do different things from each other, we still need to make sure we come together and have shared experiences).

Did you notice anything about the 3 options?  Did you try decimals or fractions to solve any of the students’ designs?  If you did, you would notice that all 3 options used the same proportions.

A great parallel task helps us to learn things together…  It helps us see others’ thinking…  It allows every student to start to think where they are comfortable, yet be able to learn and grow from the ideas of others.

If you were to offer a parallel problem/task for your students would you:

• Choose which students get each choice, or allow students to pick themselves? (Does this matter?)
• Expect all students to create the design using blocks or digitally?  (Does this matter?)
• Ask students to work independently or in a small group?  (Does this matter?)
• Offer calculators or not?  (Does this matter?)
• Engage in 3 different conversations – 1 per group – or 1 conversation all together?  (Does this matter?)

The small decisions we make tell a lot about what we value!    Personally, IF I want to offer Parallel tasks/problems, I want to make sure that all of my students feel successful, that everyone realize their ideas are valued, that there isn’t a hierarchy of ability in the room… and of course, that the mathematics we are engaged is important.

## Seeking Challenges in Math

I was working with a grade 7 teacher and his students a while back.  The teacher came to me with an interesting problem, his students were doing quite well in math (in general) but only wanted to do work out of textbooks, only wanted to work independently, and were very mark-driven. The teacher wanted his students to start being able to solve non-routine problems, not just be able to follow the directions from the textbook, and he wanted his students to see the value in working collaboratively and to listen to each other’s thoughts.

Our conversations quickly moved to the topic of mindsets. It sounded like many of his students had fixed mindsets, and didn’t want to take any risks.

For those of you who are not familiar with growth and fixed mindsets, students with fixed mindsets believe that their ability (in math for example) is an inborn trait.  They believe how smart they are in math is either a gift or a curse they are born with.  Those with growth mindsets, however, believe that their ability improves over time with the right experiences, attitude and effort.

When confronted with challenges, those with growth mindsets are willing to struggle, willing to make mistakes, knowing that they will continue to learn and grow throughout the learning process.  On the other hand, those that have fixed mindsets tend to avoid challenges.  They believe that struggle, making mistakes, and being challenged are signs of weakness.  Psychologically, they will avoid the feeling of discomfort in not knowing, as this threatens their belief about how smart they are.

Knowing this, we devised a plan to see whether or not his students were able to take on challenges.  We started the class by giving each student their own unique 24 card (see below).

We explained that each card had 4 numbers that could be manipulated to equal 24.  For instance, the card above could be solved by doing 5 x 4 x 1 + 4 = 24.

We then explained that we would give them time to solve their own card (which had a front and a back), and that we would give them additional cards if they completed both problems.  We also explained the little white dots in the center of the card, 1 dot being an easy card, 2 dots being more complicated, and 3 dots being the most difficult.

As students continued to work, we noticed some students eagerly trying to solve the cards, and others starting to become frustrated by others’ successes.  After a few minutes, the first few students had completed both problems and asked for their next card.  We asked, “Would you like another easy card, or would you like to challenge yourself?” to which the vast majority asked for another easy card.  In fact, some students completed many cards, front and back, all at the easy level, never accepting a more challenging card (even bragging to others about how many they had completed).  Others, after giving up pretty quickly, asked if they could work with a classmate to make a pair.  While we were happy at first with this, none of the pairs had students working cooperatively together for most of the time.

Take a look at some of the challenging cards.  What do you do when confronted with something challenging?  Do you skip it and move on, or do you keep trying?

As soon as we were finished, we showed the class this video:

Watch the 3 minute video above as it ties in perfectly with the 24 problem from above.  We had a quick discussion about the video and why some of the students wanted to choose the easier puzzles.  The class quickly saw the parallels between the problems we had just done and the video.

While we had a great discussion about fixed and growth mindsets, it took most of the year to be able to get this group to see the value in collaboration, to focus on their learning instead of their marks, to be able to take on challenges and not get frustrated when they didn’t have immediate success.

Changing our mindset takes time and the right experiences!

I am really interested in why students who believe themselves to be “smart” at math would opt out of challenging themsleves.

Do any of your students exhibit any of the same signs as these students:

• Not comfortable with tasks that require thinking
• Eager for formulas and procedures
• Competitive with others to show they are “smart”
• Preference to work alone
• Preference to work out of textbooks/ worksheets instead of on rich problems/tasks
• See math as about being fast / right, not about thinking / creativity
• Eager to do easy work that is repetative

So I leave you with some reflective questions:

What previous experiences must these students have had to create such fixed mindsets?

What would you do if your students avoided challenges?

What would you do if your students groaned each time you asked them to work with a partner?

How are you helping your students gain a growth mindset in math?

Can you recognize those in your class that have fixed mindsets?  Are you noticing those from different achievement levels, or just those who are struggling?

If our students find everything we do “easy” what will happen to them when they get to a math course that actually does offer them some challenge???

P.S.  Did you solve any of the 24 cards above?  Did you skip over them?  What do you typically do when confronted with challenges?

## An Unsolved Problem your Students Should Attempt

There are several great unsolved math problems that are perfect for elementary students to explore.  One of my favourites is the palindrome sums problem.
In case you aren’t familiar, a palindrome is a word, phrase, sentence or number that reads the same forward and backward.

The problem itself comes out of this conjecture:

If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome.

Let’s take a look at the numbers 12, 46 and 95:

As you can see with the above examples, some numbers can become palindromes with 1 simple addition, like the number 12.  12 + 21 = 33.  Can you think of others that should take 1 step?  How did you know?

Other numbers when added to their inverse will not immediately become a palindrome, but by continuing the process, will eventually, like the numbers 46 and 95.  Which numbers do you think will take more than 1 step?

After going through a few examples with students about the process of creating palindromes, ask your students to attempt to see if the conjecture is true (If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome).  Have them find out if each number from 0-99 will eventually become a palindrome. Ask students if they need to find the answer for each number?  Encourage them to make their own conjectures so they don’t need to do all of the calculations for each number.

Mathematically proficient students look closely to discern a pattern or structure. They notice if calculations are repeated, and look both for general methods and for shortcuts. (SMP 7)

Some students will notice:

• Some numbers will already be a palindrome
• If they have figured out 12 + 21… they will know 21 + 12.  So they don’t have to do all of the calculations.  Nearly half of the work will have already been completed.
• A pattern emerging in their answers … 44, 55, 66, 77, 88, 99, 121… and see this pattern regularly (well, almost)
• A pattern about when numbers will have 99 as a palindrome, or 121… 18+81 = 27+72 = 36+63…

• What is the goal of this lesson?  (Practice with addition?  Looking for patterns?  Perseverance?  Making conjectures? …)
• Do we share some of the conjectures students are making during the time when students are working, or later in the closing of the lesson?
• How will my students record their work?  Keep track of their answers?  Do I provide a 0-99 chart or ask them to keep track somehow?
• Will students work independently / in pairs / in small groups?  Why?
• Do I allow calculators?  Why or why not?  (think back to your goal)
• How will I share the conjectures or patterns noticed with the class?
• Are my students gaining practice DOING (calculations) or THINKING (noticing patterns and making conjectures)?  Which do you value?

The smallest of decisions can make the biggest of impacts for our students!

So, at the beginning of this post I shared with you that this problem is currently unsolved.  While it is true that the vast majority of numbers have been proven to easily become palindromes, there are some numbers that require many steps (89 and 98 require 24 steps), and others that have never been proven to either work or not work (198 is the smallest number never proven either way).

After using this problem with many different students I have noticed that many start to see that mathematics can be a much more intellectually interesting subject than they had previously experienced.  This problem asks students to notice things, make conjectures, try to prove their conjectures and be able to communicate their conjectures with others…  The problem provides students with the opportunity to both think and do.  It offers students from various ability levels access to the problem (low floor), and many different avenues to challenge those ready for it (high ceiling).  It tells students that math is still a living, growing subject… that all of the problems have not yet been figured out!  And probably the most important for me, it sends students messages about what it means to really do mathematics!

## Is That Even A Problem???

Ask others what problem solving means with regard to mathematics.  Many will explain that a problem is when we put a real-world context to the mathematics being learned in class… others might explain a process of how we solve a problem (look at what you know and determine what you want to know, or some other set of strategies or a creative acronym that we have likely seen in school).  Sadly, much of what most others would point to as a problem is not really even a problem at all.

I think we all need to consider the real notion of what it means to problem solve…

George Polya shared this:

…Thus, to have a problem means: to search consciously for some action appropriate to attain a clearly conceived, but not immediately attainable, aim. To solve a problem means to find such action. … Some degree of difficulty belongs to the very notion of a problem: where there is no difficulty, there is no problem.

What Polya is suggesting here is that if we show students how to do something, and then ask the students to practice that same thing in a context IT ISN’T A PROBLEM!!!  A problem in mathematics is like the DOING MATHEMATICS Tasks listed below.  Take a look at all 4 sections for a minute:

My thoughts are simple… If a student can relatively quickly determine a course of action about how to get an answer, it isn’t a problem!   Even if the calculations are difficult or take a while.  The vast majority of what we call problems are actually just contextual practice of things we already knew.  Doing word problems IS NOT the same as problem solving!

On the other hand, if a student has to use REASONING skills, they are thinking, actively trying to figure something out, then and only then are they problem solving!!!

Marian Small has written a short article on her thoughts about problem solving:  Marian Small – Problem Solving

What are her main messages here?  Does this or Polya’s quote change your definition of a problem?

I’ve already written about What does Day 1 Look Like where I shared the importance of starting with problems.  So, why should we start with problem solving?  If we don’t start there, we aren’t likely ever doing any problem solving at all!

I also think that many hearing this might assume that this means we just hand students problems that they wouldn’t be successful with…  Ask everyone to attempt something that they wouldn’t know how to do.

Let’s look at an example:

Take a look at these two grade 8 expectations from Ontario curriculum:

• determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies;
• solve problems involving right triangles geometrically

Many teachers look at these expectations, think to themselves, Pythagorean Theorem… I can teach that… followed by explicit teaching on a Smartboard (showing a video, modeling how the pythagorean theorem works, followed by some examples for the class to work on together…), then finally some problems that students have to answer in a textbook like this:

This progression neither shows how we know students learn, nor does it even get students to be able to do what is asked!

Let’s take a step back and start to notice what the curriculum is saying more clearly:

• determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies;
• solve problems involving right triangles geometrically, using the Pythagorean relationship;

When we pull apart the verbs and the tools/strategies from the content, we start to notice what the curriculum is telling us our students should actually be doing that day!  Remember… these expectations are what OUR STUDENTS should be doing… NOT US!!!

Above I have colored the verbs blue (These are the actions our students should be doing that day) and tools/strategies orange (specifically HOW our students should be accomplishing the verb).

Now let’s take a quick look at how this might actually play out in the classroom if we are starting with problems like our curriculum states:

Let’s start with the first expectation.  Our curriculum often includes the statement “determine through investigation,” yet it is overlooked far too often!  Students need to determine this themselves!  We need to assess students’ ability to “determine the Pythagorean relationship through investigation.”

This doesn’t mean we tell students what the theorem is, nor does it mean that we expect everyone to reinvent the theorem… so what does it mean???

Well, it could mean lots of things.  Here is one possible suggestion…

Show the figure on the left.  Ask students the area of the blue square in the middle.  Possibly give a geoboard or paper and scissors for this task.  How might students come up with the area?  How many different ways might students accomplish this?

Share different approaches as a group.  (By the way, some calculate the whole shape and subtract the 4 corner triangles.  Others calculate the 4 black rectangles and divide by 2 then add 1 for the middle… others rearrange the shapes to make it make sense).

Now explore how the two pictures are similar / different.  What do you notice between the two pictures?

If the curriculum tells us to “determine through investigation” that is exactly the experience our students need to conceptualize the concept.  It is also what we need to assess.  This can’t be put on a test easily though, it needs to be observed!

That second expectation is quite interesting to me too.  At the beginning of this post we started talking about what is and isn’t a problem.  If our students have now understood what the Pythagorean Theorem is, how can we now make things problematic?

Showing a bunch of diagrams with missing hypotenuses or legs isn’t really problematic!

However, something like Dan Meyer’s Taco Cart would be!  If you haven’t seen the lesson, take a look:

Oh… and by the way… problem solving isn’t always about answering a question… really, at it’s heart, problem solving is about making sense of things that we didn’t understand before… reasoning though things… noticing things we didn’t notice before… making conjectures and testing them out…  Problem solving is the process of LEARNING and DOING MATHEMATICS!

So I want to leave you with a problem for you to think about: what does this have to do with the Pythagorean Theorem?

## “Worst Problem Ever!”

A few years ago I was teaching a 6/7 split class.  We had been exploring surface area and volume of prisms (rectangular so far), but were quite early on in the learning cycle.  At the beginning of my math class I drew a picture of a rectangular prism and wrote the dimensions on the diagram like below:

I asked everyone to calculate the Surface Area.  Really this was just a check for me to see if we understood surface area… the problem for the day was coming up.

However, as I walked around, I realized something VERY unusual.  EVERYONE got the answer of 250… but not everyone did it correctly.  Let me show you:

Take a look at the 2 answers above.  Some students calculated the Surface Area correctly at 250 units squared… and others calculated 250 units cubed (they found the volume).

I apologized to the class for giving them the worst problem ever… THE ONLY POSSIBLE rectangular prism that has the same surface area as volume.  Then one student commented, “Is that the only possible prism?”

I didn’t know the answer, so we started seeing if it was the only possible prism.

Over the next 100 minutes, every student in the class, some in pairs, some on their own, started drawing prisms and calculating the surface area and volume.

Eventually, a student told me that one of the dimensions couldn’t possibly be a 1.  I asked him to prove it… After a few minutes he showed me several examples with numbers that were small or large.  He attempted to find the limits:

He explained to the class that if there is a 1 as any of the dimensions, the Surface Area would always be larger.   We explored why that happens.

Minutes later, another student told me that it was impossible to have a 2 as one of the dimensions.  I asked her to prove it… She showed me the work that she had completed and shared it with the class.

She explained that the numbers were getting close if she chose 2 of the numbers identical, but the end pieces of the shape would always add up to more than the volume would.

After lots of trial and error… guessing and refining ideas, several students started finding possible prisms that also had the same Surface Area and Volume…

The class started noticing a pattern between the dimensions and found limits between the smallest and largest possible prisms…

By the end of class, all students had calculated dozens of surface areas and volumes… all students were making conjectures or testing out the conjectures of others.

My original problem, which I intended as a quick warm-up was not a quality engaging problem.  However, I want you to think about what made this lesson better?

WHO posed the problem that day?  Did this have something to do with the shared responsibilities that happened later in the lesson?

What if I just moved on?  would the learning have been as rich?

Think about how the students picked the shapes they were testing?  Some students worked in teams to work strategically… others made their own conjectures and followed those patterns.

In this lesson, choice was key, but the choices didn’t come from me… Students were working together to reach an ultimate goal, not in competition with each other…  Conjectures were made and tested, not because I told everyone to, but because it served our purpose!

I think about Dan Meyer’s “Real World vs Real Work” a lot.  Why were my students so engaged here?  There was NO real world connection.  That wasn’t what was motivating my students at all!

I also think we need to reflect on the level of cognitive demand we ask our students to be engaged in:

“Doing Mathematics Tasks” seems like something that is hard to do, yet, every one of my students were engaged in this problem… everyone eagerly searched for patterns, many drew pictures, used snap cubes, visualized what was happening… all in the name of better understanding the relationships between the dimensions of a rectangular prism, and its surface area and volume!

By the end of the class, my students had found exactly 10 rectangular prisms that have the same value of its surface area as its volume (using only whole numbers), and could prove that these were the only 10 possible.

I’d love to hear your thoughts about our problem… or why the students were SO engaged… or about the conditions that must have been present in the class… or how problem solving can be used as a purposeful practice of procedures…..