Central Tendencies Puzzles

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).
• How do these puzzles help your students build their mathematical intuitions? (See ideas 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)

I’d love to continue the conversation about these puzzles.  Leave a comment here or on Twitter @MarkChubb3

Skyscraper Templates – for Relational Rods

Many math educators have come to realize how important it is for students to play in math class. Whether for finding patterns, building curiosity, experiencing math as a beautiful endeavour, or as a source of meaningful practice… games and puzzles are excellent ways for your students to experience mathematics.

Last year I published a number of templates to play a game/puzzle called Skyscrapers (see here for templates) that involved towers of connected cubes. This year, I decided to make an adjustment to this game by changing the manipulative to Relational Rods (Cuisenaire Rods) because I wanted to make sure that more students are becoming more familiar with them.

Skyscraper puzzles are a great way to help our students think about perspective while thinking strategically through each puzzle.  Plus, since they require us to consider a variety of vantage points of a small city block, the puzzles can be used to help our students develop their Spatial Reasoning!

How to play a 4 by 4 Skyscraper Puzzle:

• Build towers in each of the squares provided sized 1 through 4 tall
• Each row has skyscrapers of different heights (1 through 4), no duplicate sizes
• Each column has skyscrapers of different heights (1 through 4), no duplicate sizes
• The rules on the outside (in grey) tell you how many skyscrapers you can see from that direction
• The rules on the inside tell you which colour rod to use (W=White, R=Red, G=Green, P=Purple, Y=Yellow)
• Taller skyscrapers block your view of shorter ones

Below is an overhead shot of a completed 4 by 4 city block.  To help illustrate the different sizes. As you can see, since each relational rod is coloured based on its size, we can tell the sizes quite easily.  Notice that each row has exactly 1 of each size, and that each column has one of each size as well.

To understand how to complete each puzzle, take a look at each view so we can see how to arrange the rods:

If you are new to completing one of these puzzles, please take a look here for clearer instructions: Skyscraper Puzzles

Relational Rod Templates

Here are some templates for you to try these puzzles yourself and with your students:

4 x 4 Skyscraper Puzzles – for Relational Rods

5 x 5 Skyscraper Puzzles – for Relational Rods

A few thoughts about using these:

• How will you introduce these puzzles to your students?  Would you use the cube version I shared first? 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 puzzle 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 might using Relational rods be different than paper-and-pencil or electronic versions? How might it be different than the version with cubes?
• How will you orchestrate a conversation for your class to help consolidate the learning here?
• What will you do if students give up quickly?  What questions / prompts will you provide?

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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!

I’d love to hear how you and/or your students do!

Making Math Visual

A few days ago I had the privilege of presenting at MAC2 to a group of teachers in Orillia on the topic of “Making Math Visual”.   If interested, here are some of my talking points for you to reflect on:

To get us started I shared an image created by Christopher Danielson and asked the group what they noticed:

We noticed quite a lot in the image… and did a “how many” activity sharing various numbers we saw in the image.  After our discussions I explained that I had shared the same picture with a group of parents at a school’s parent night followed by the next picture.

I asked the group of teachers what mathematics they noticed here… then how they believed parents might have answered the question.  While we, as math teachers, saw patterns in the placements of utensils, shapes and angles around the room, quantities of countable items, multiplicative relationships between utensils and place settings, volume of wine glasses, differences in heights of chairs, perimeter around the table…..  the group correctly guessed that many parents do not typically notice the mathematics around them.

So, my suggestion for the teachers in the room was to help change this:

I then asked the group to do a simple task for us to learn from:

After a few minutes of thinking, I shared some research of the different ways we use fractions:

When we looked at the ways we typically use fractions, it’s easy to notice that WE, as teachers, might need to consider how a focus on representations might help us notice if we are providing our students with a robust (let’s call this a “relational“) view of the concepts our students are learning about.

Data taken from 1 school’s teachers:

We continued to talk about Liping Ma’s work where she asked teachers to answer and represent the following problem:

Followed by a quick story of when a student told me that the following statement is true (click here for the full story).

So, why should we focus on making math visual?

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We then explored a statement that Jo Boaler shared in her Norms document:

…and I asked the group to consider if there is something we learn in elementary school that can’t be represented visually?

If you have an idea to the previous question, I’d love to hear it, because none of us could think of a concept that can’t be represented visually.

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I then shared a quick problem that grade 7 students in one of my schools had done (see here for the description):

Along with a few different responses that students had completed:

Most of the students in the class had responded much like the image above.  Most students in the class had confused linear metric relationships (1 meter = 100 cm) with metric units of area (1 meter squared is NOT the same as 100cm2).

In fact, only two students had figured out the correct answer… which makes sense, since the students in the class didn’t learn about converting units of area through visuals.

We wrapped up with a few suggestions:

And finally some advice about what we DON’T mean when talking about making mathematics visual:

You might recognize the image above from Graham Fletcher’s post/video where he removed all of the fractional numbers off each face in an attempt to make sure that the tools were used to help students learn mathematics, instead of just using them to get answers.

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I want to leave you with a few reflective questions:

• Can all mathematics concepts in elementary school be represented visually?
• Why might a visual representation be helpful?
• Are some representations more helpful than others?
• How important is it that our students notice the mathematics around them?
• How might a focus on visual representations help both us and our students deepen our understanding of the mathematics we are teaching/learning?

I’d love to continue the conversation.  Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).

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If you are interested in all of the slides, you can take a look here