## 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)

## Whose problems? Whose game? Whose puzzle?

TRU Math (Teaching for Robust Understanding) a few years ago shared their thoughts about what makes for a “Powerful Classroom”. Here are their 5 dimensions:

Looking through the dimensions here, it is obvious that some of these dimensions are discussed in detail in professional development sessions and in teacher resources. However, the dimension of Agency, Authority and Identity is often overlooked – maybe because it is much more complicated to discuss. Take a look at what this includes:

This dimension helps us as teachers consider our students’ perspectives. How are they experiencing each day? We should be reflecting on:

• Who has a voice? Who doesn’t?
• How are ideas shared between and among students?
• Who feels like they have contributed? Who doesn’t?
• Who is actively contributing? Who isn’t?

Reflecting on our students’ experiences makes us better teachers! So, I’ve been wondering:

## Who created today’s problem / game / puzzle?

For most students, math class follows the same pattern:

This pattern of a lesson leaves many students disinterested, because they are not actively involved in the learning – which might lead to typical comments like, “When will we ever need this?”. This lesson format is TEACHER centered because it centers the teachers’ ideas (the teacher provides the problem, the teacher helps students, the teacher tells you if you are correct). In this example, students’ mathematical identities are not fostered. There is NO agency afforded to students. Authority solely belongs to the teachers. But there are ways to make identity / agency / authority a focus!

## STUDENT driven ideas

Today as a quick warm-up, I had students solve a little pentomino puzzle.  After they finished, I asked students to create their own puzzles that others will solve.  Here is one of the student created puzzles:

Here you can see a simple puzzle. The pieces are shown that you must use, and the board is included (with a hole in the middle). Now, as a class, we have a bank of puzzles we can attempt any day (as a warm-up or if work is finished).

You can read about WHY we would do puzzles like this in math class along with some examples (Spatial Reasoning).

What’s more important here is for us to reflect on how we are involving our own students in the creation of problems, games and puzzles in our class.  This is a low-risk way to allow everyone in class do more than just participate, they are taking ownership in their learning, and building a community of learners that value learning WITH and FROM each other!

## How to involve our students?

The example above shows us a simple way to engage our students, to expand what we consider mathematics and help our students form positive mathematical identities. However, there are lots of ways to do this:

• Play a math game for a day or 2, then ask students to alter one or a few of the rules.
• Have students submit questions you might want to consider for an assessment opportunity.
• Have students look through a bank or questions / problems and ask which one(s) would be the most important ones to do.
• Give students a sheet of many questions. Ask them to only do the 3 easiest, and the 3 hardest (then lead a discussion about what makes those ones the hardest).
• Lead 3-part math lessons where students start by noticing / wondering.
• Have students design their own SolveMe mobile puzzles, visual patterns, Which One Doesn’t Belong…

## Questions to Reflect on:

• Who is not contributing in your class, or doesn’t feel like they are a “math student”? Whose mathematical identities would you like to foster? How might something simple like this make a world of difference for those children?
• Does it make a difference WHO develops the thinking?
• Fostering student identities, paying attention to who has authority in your class and allowing students to take ownership is essential to build mathematicians. The feeling of belonging in this space is crucial. How are you paying attention to this? (See Matthew Effect)
• How might these ideas help you meet the varied needs within a mixed ability classroom?
• If you do have your students create their own puzzles, will you first offer a simplified version so your students get familiar with the pieces, or will you dive into having them make their own first?
• 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?
• 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 ).

## Spatial Puzzles: Cuisenaire Cover-ups

Foundational to almost every aspect of mathematics is the idea that things can be broken down into pieces or units in a variety of ways, and then be recomposed again. For example, the number 10 can be thought of as 2 groups of 5, or 5 groups of 2, or a 7 and a 3, or two-and-one-half and seven-and-one-half…

Earlier this year I shared a post discussing how we might decompose and recompose numbers to do an operations (subtraction). But, I would like us to consider why some students are more comfortable decomposing and recomposing, and how we might be aiming to help our students early with experiences that might promote the kinds of thinking needed.

Doug Clements and Julie Sarama have looked at the relationship between students’ work with space and shapes with students understanding of numbers.

“The ability to describe, use, and visualize the effects of putting together and taking apart shapes is important because the creating, composing, and decomposing units and higher-order units are fundamental mathematics. Further, there is transfer: Composition of shapes supports children’s ability to compose and decompose numbers”

Contemporary Perspectives on Mathematics in Early Childhood Education p.82, Clements and Sarama

The connection between composing and decomposing shapes and numbers is quite exciting to me. However, I am also very interested in the meeting place between Spatial tasks (composing/decomposing shapes) and Number tasks that involve composing and decomposing.

A few years ago I found a neat little puzzle in a resource called The Super Source called “Cover the Giraffe”. The idea was to cover an image of a giraffe outline using exactly 1 of each size of cuisenaire rods. The task, simple enough, was actually quite difficult for students (and even for us as adults). After using the puzzle with a few different classes, I decided to make a few of my own.

After watching a few classrooms of students complete these puzzles, I noticed an interesting intersection between spatial reasoning, and algebraic reasoning happening…. First, let me share the puzzles with you:

## Objective:

To complete a Cuisenaire Cover-Up puzzle, you need exactly 1 of each colour cuisenaire rod. Use each colour rod once each to completely fill in the image.

Below are the 5 puzzles:

## Assessment Opportunities

Knowing what to look for, helps us know how to interact with our students.

• Which block are students placing first? The largest blocks or the smallest?
• Which students are using spatial cues (placing rods to see which fits) and which students are using numerical cues (counting units on the grid)? How might we help students who are only using one of these cueing systems without over-scaffolding or showing how WE would complete the puzzle?
• How do our students react when confronted with a challenging puzzle?
• Who is able to swap out 1 rod for 2 rods of equivalent length (1 orange rod is the same length as a brown and red rod together)?
• Which of the following strands of proficiency might you be noting as you observe students:

## Questions to Reflect on:

• Why might you use a task like this? What would be your goal?
• How will you interact with students who struggle to get started, or struggle to move passed a specific hurdle?
• How might these puzzles relate to algebraic reasoning? (try to complete one with this question in mind)
• How are you making the connections between spatial reasoning and algebraic reasoning clear for your students to see? How can these puzzles help?
• How might puzzles allow different students to be successful in your class?