Many mathematicians are good at searching for patterns in numbers, however, an area that I think we all need to continue to explore is Visualizing.
Instead of just looking for procedural rules, or numeric patterns I encourage you to take one of the following and actually VISUALIZE what is going on.
Pick one of the above that interests you. Answer some of these questions:
- What relationships do you notice here?
- What are you curious about?
- What visual might be helpful to represent this/these relationships?
- Will these relationships work in other instances? When will it work/ when won’t it work?
- How might a visual help others see the relationships you’ve noticed?
I’d love to hear some answers. You can respond here below, or via Twitter @MarkChubb3
Thinking Mathematically 
I was looking at the equation “11=6+5=6^2-5^2. I tried the next one – 13=7+6=7^2-6^2 and it worked! So, I thought about a visual for it. I envisioned a 7×7 grid with the 6×6 shaded inside it and then thought of what was left. It’s the top and right side border of the 7×7, which would be a row of 7 and a column of 6 on the right below the last tile of the top row. .
(My picture wouldn’t load – sorry)
This led me to realize that this would always work for 2 consecutive numbers. The visual is what made it obvious to me that it would work every time.
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Happy to hear the visual was helpful. To some, the relationships pop out, but for many of us….and many of our kids, the visuals both help us make these conjectures and share our thinking with others.
Thanks for participating. Maybe another problem might be equally interesting.
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Would you ever give these to students in a class setting?
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Yes I would. But I would have specific purposes in mind. I’ve used the fraction operations problem a few times now.
Really, what I’d be aimed at, is solidifying specific concepts by making sure my students can visually represent the concepts.
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I think kids have trouble visualizing in many settings. I teach gifted third graders who have to create a new solution to a problem. It’s been a while so details have become hazy, but one group was doing something with a car. I suggested they enact the steps to getting in a car and starting it. They skipped many steps. I can’t say it was the context. This problem has come up in many situations. They have the start and the finish, but miss the middle that makes completing the act possible!
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