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In 1859, the Irish mathematician Sir William Rowan Hamilton devised a puzzle with a regular dodecahedron made of wood. Here is a dodecahedron:

He labelled each of the vertices with the name of an important city. The challenge was to find a route along the edges of the dodecahedron which visited every city exactly once and returned to the start.

Here is a graph which represents the dodecahedron. Can you see how each of the 20 vertices, 30 edges and 12 pentagonal faces is represented in the graph?

I start my journey in Rio de Janeiro and visit all the cities as Hamilton described, passing through Canberra before Madrid, and then returning to Rio. What route could I have taken?

Can you find any other ways of making this journey?

Here is a simpler network of countries:

How many different ways are there of visiting each of these countries once and only once, beginning and ending at Australia?

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6-week online workshop, 6-week proportional reasoning course, virtual summit, support / help, episode #282: teach your math class using puzzles & games – an interview with gordon hamilton from math pickle.

Apr 22, 2024 | Podcast | 0 comments

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Episode Summary:

How can we engage our students for the entire class with grade level content? Puzzles aligned to grade level standards is one answer. 

In this episode we speak with Gordon Hamilton from Mathpickle.com who designs mathematical puzzles for K-12 classrooms. 

Stick with us and you’ll learn How to use puzzles to teach grade level standards, how to challenge students who tend to move through problems faster than others, and why games are important for mathematical thinking. 

What You’ll Learn:

• How to challenge students who complete “the work” early.
• How to use puzzles to teach grade level standards; 
• Where do I find gems to use in my math class; 
• How to use a puzzle in my math class that can be used for all grade levels; 
• How you can use the content standards to focus on the practice standards;
• How you can use puzzles to create math moments in our classroom;
• Why games are important for mathematical thinking;
• Math Pickle Website

Attention District Math Leaders:

How are you ensuring that you support those educators who need a nudge to spark a focus on growing their pedagogical-content knowledge? 

What about opportunities for those who are eager and willing to elevate their practice, but do not have the support? 

Book a call with our District Improvement Program Team to learn how we can not only help you craft, refine and implement your district math learning goals, but also provide all of the professional learning supports your educators need to grow at the speed of their learning. 

Book a short conversation with our team now . 

Math Game, Learning Styles, and Student Engagement

The team discussed a math-related game symbolizing mathematical concepts and decisions, with a focus on individual mathematical moments and memories. Gordon shared his story of struggling in a structured school system in Northern Ireland and finding success in a personalized learning environment in Canada, emphasizing the importance of allowing students to learn at their own pace. The conversation then shifted to strategies for keeping students engaged in the classroom, emphasizing the need for celebrating successes, having a variety of challenges ready, and allowing students to work at their own pace. Jon expressed a dislike for the lizards involved in the game.

Thinking Styles and Learning Strategies

Jon, Kyle, and Gordon discussed the importance of fostering a variety of thinking styles in classrooms. They emphasized the value of honoring methodical thinkers and the benefits of challenging students to maintain thinking without the need for a final answer. Gordon shared an example of a puzzle that could be used as a teaching tool, highlighting the advantages of these exercises for skill acquisition and engagement among students of varying abilities. The conversation underscored the importance of failure in teaching, as a way to celebrate learning and encourage creative thinking. Gordon also suggested linking such activities to the curriculum and avoiding dividing tasks into introductory, main, and wrap-up tasks, but rather focusing on one continuous task.

Teaching Strategy for Pre-Service Teachers

Gordon shared his teaching strategy for pre-service teachers, which emphasizes engaging students in classroom scenarios without knowing the answers and encouraging flexibility and adaptability. He stressed the importance of assessing for mathematical practice standards rather than skill acquisition and suggested starting with an unsolved problem, like the Graceful Tree Conjecture, to maintain student interest. Kyle, Jon, and other participants discussed the significance of these approaches and agreed on the need to focus on mathematical practice standards.

Puzzles as Teaching Tools in Math Classes

Gordon proposed the use of puzzles as a teaching tool to engage students in classes, suggesting that teachers embrace vulnerability and uncertainty. He emphasized the importance of problem-solving over arithmetic in mathematics classes and suggested starting with more complex puzzles to keep students interested, even if it means teaching advanced concepts like multiplication before they are officially introduced in the curriculum. He highlighted the role of parents in encouraging this type of thinking at home through board games. Kyle expressed excitement about the idea but raised a concern about maintaining intentionality in lessons. Gordon responded by suggesting the use of Math Pickle, his website, which offers these types of puzzles and games.

Board Games in Home Learning: A Team Discussion

The team discussed the benefits of incorporating board games into home learning environments, with Gordon emphasizing the importance of parents engaging with their children in this way. Gordon also expressed his enjoyment of the podcast, highlighting its potential to inspire positive experiences with math and puzzles. The team then extended an invitation to Gordon to speak at their upcoming virtual summit in 2024, which Gordon accepted. The discussion concluded with a mutual agreement to stay in touch and continue the dialogue.

Teachers should visit mathpickle.com to access puzzles and games for their classrooms. Consider incorporating board games into math lessons to promote problem-solving and thinking skills. Teachers should aim to create “infinite pickles” – puzzles that can be scaled up or down based on student ability. 

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FULL TRANSCRIPT

00:00:00:12 – 00:00:17:20 Gordon Hamilton This gives you an example of the type of puzzle that I really respect. And I might start out and you guys might have a number one painted on your bib. Okay, So you’re up at the front of the gymnasium and you’ve got a number one, and I invite up to grade two kids and I put a number two on them.

00:00:17:22 – 00:00:22:00 Gordon Hamilton And then I said, Let’s try to organize yourselves in a line so that.

00:00:22:05 – 00:00:46:05 Jon Orr How can we engage our students for the entire class with grade level content puzzles aligned to grade level standards is actually one answer. And in this episode we speak with Gordon Hamilton from Math Pickle dot com, who designs mathematical puzzles for K to 12 classrooms. So stick with us and you’ll learn how to use puzzles to teach grade level standards.

00:00:46:09 – 00:01:14:00 Jon Orr How to challenge students who tend to move through problem homes faster than others, and how to manage the overflow and how and why games are important for mathematical thinking. Let’s go.

00:01:14:02 – 00:01:18:06 Kyle Pearce Welcome to the Making Math Moments That Matter podcast. I’m Kyle Pearce.

00:01:18:07 – 00:01:21:08 Jon Orr And I’m Jon Orr we have a make that moment scope.

00:01:21:10 – 00:01:31:08 Kyle Pearce This is the only podcast that coaches you through a six step plan to grow your mathematics program whether it’s at the classroom level or at the district level.

00:01:31:09 – 00:01:46:12 Jon Orr And we do that by helping you cultivate and foster your mathematics program like strong, healthy and balanced treats. So if you master the six parts of an effective mathematics program, the impact that you are going to have on your teachers, your students will grow and reach far and wide.

00:01:46:14 – 00:02:10:04 Kyle Pearce Every week you’ll get the insight you need to stop feeling overwhelmed, gain back your confidence and get back to enjoying the planning and facilitating of your mathematics program for the students or the educators that you serve. Hey. Hey there, Gordon. Welcome to the Making Math Moments That Matter podcast. I can’t remember when it was the last time that you and I were on a Zoom call.

00:02:10:04 – 00:02:28:08 Kyle Pearce It was quite a while ago. I know you recently connected with John on another call and we are just so excited to finally have an opportunity to have you on the podcast so folks can learn a little bit more about what you are doing over at Math Pickle. But before we dig in, Gordon, how are you doing, my friend?

00:02:28:08 – 00:02:36:02 Kyle Pearce Where are you coming to us from? And tell folks just a teeny bit about your schtick here in mathematics.

00:02:36:04 – 00:02:48:20 Gordon Hamilton I’m coming from Calgary in Canada, just like you guys and Siri very helpfully told me today. It was started at -15 0% chance of rain. So that was my day, right?

00:02:48:20 – 00:02:53:01 Kyle Pearce Love that. That’s quite chilly, John. I feel like we were above zero again today. I think we’ve been.

00:02:53:01 – 00:02:55:08 Jon Orr Above zero for weeks.

00:02:55:10 – 00:03:16:20 Kyle Pearce Side note last night John and I did a webinar for Idaho. You know, I was going to say Iowa again, Idaho. And we are actually more south than they were in Idaho, which they it blew their minds. We had like bring up the map. So you’re in Canada, you’re at -15, no precipitation. We’re in Canada, no rain.

00:03:16:20 – 00:03:22:00 Gordon Hamilton That’s it was introduced to me. No rain, no rain. And that’s what my friend told me this morning.

00:03:22:06 – 00:03:40:09 Kyle Pearce Oh, got it. Got it. I see, I see what’s happening. But there could be icicles coming from the sky, so look out. Yeah, I totally missed it. But we are definitely quite a bit further south than the folks in Calgary. But, hey, my friend, let’s keep rolling along. Tell us a little bit about you. What do you have going on in the math space?

00:03:40:09 – 00:03:46:22 Kyle Pearce We know about you, but let’s make sure the math moments community knows a bit about you. What’s your role? What are you up to in mathematics?

00:03:46:24 – 00:04:11:12 Gordon Hamilton So I got my Ph.D. in mathematics and it was going along, enjoying myself and then I was my wife got pregnant. I thought, I’m going to volunteer at a math start to see what education has in store for my son. This was 18 years ago, and I walked into this math fair and I cheaper. These kids had been working for two weeks on their puzzle.

00:04:11:12 – 00:04:30:09 Gordon Hamilton I walked to trying to solve one puzzle, solved another puzzle, then stood in front of this grade five girl, and she showed me a puzzle and I looked at, Oh, I do. I saw this. And she knew that I was a mathematician. I was just going, Oh, my, no. And the more that I struggle, the more this little evil grin went across her face.

00:04:30:11 – 00:04:52:14 Gordon Hamilton And I never solved it. There I saw it, but I left there with midlife euphoria. I just thought, Oh my gosh, I can take my board game design skills and throw them into designing puzzles for the math classroom. And I know this is going to work and it’s going to be wonderful.

00:04:52:16 – 00:05:01:13 Jon Orr Awesome. That was how you got into your current kind of role right now, working with puzzles, working at Math Pickle. Is that the case?

00:05:01:15 – 00:05:21:04 Gordon Hamilton That’s right. So I don’t have any background in education, but I hit the ground running because my mentor was Richard Guy. He was the oldest practicing mathematician at 103 years old. Never. He died a couple of years ago. He was my mentor and his book and solve problems and number theory. Oh my gosh, I just love that book.

00:05:21:06 – 00:05:46:06 Gordon Hamilton So I just hit the ground with very poor pedagogy, probably. But going into classrooms and introducing kids to unsolved problems and saying, okay, let’s struggle with this together. And of course, right away you realize you don’t want to be doing that and then just laughing at them. You want to be coming up with vignettes of success along the way to them discovering these beautiful gems of mathematics.

00:05:46:08 – 00:06:11:02 Kyle Pearce Wow, that is awesome. And you saying that and giving me this glimpse into sort of a bit of your past and how you’ve gotten into this world really gets me excited for the question. The question that we ask every guest that comes on this podcast. And it’s about your math moment. I’m so curious with someone who is I love how you described it as your midlife euphoria.

00:06:11:04 – 00:06:37:11 Kyle Pearce You had this euphoric moment by going to that fair and you had obviously had a love for mathematics, had you Ph.D. in mathematics. So when you go back and you think about the math moment that you remember from your we call it kind of like childhood, but really it’s through your education, through K through 12, something that pops into your mind when we say mathematics or math class, what is it for you?

00:06:37:11 – 00:06:58:07 Kyle Pearce Because I know that there’s so many people out there that have varied memories, and for some it’s actually hard memories and maybe moments that aren’t so, so pleasant. I’m really curious about yours. I’m curious to see where you go. Is it was it a positive one? Was it one that maybe taught you something? What’s the math moment that sticks out to you?

00:06:58:09 – 00:07:29:01 Gordon Hamilton Age three I was introduced to Mesas in Northern Ireland. That’s where I’m from. By grade five, I’m in Canada and I’m failing school. There’s 60 students in my class, there’s three teachers. It’s university school in Calgary, and I’m taken out with two other boys and one is Slow Learners Group. And I hated school and I went home and I would be designing these elaborate three dimensional marble mazes.

00:07:29:03 – 00:07:55:11 Gordon Hamilton So I end up repeating grade five. My mother took me out of that school, put me in another school. So I’m very careful about children and looking at them and really trying to celebrate slow, methodical problem solvers. I want to protect them. I don’t have hands raised in my class ever. Whenever I’m going around, I ask every kid for giving something to the puzzle that we’re doing on the whiteboard.

00:07:55:17 – 00:08:01:14 Gordon Hamilton And hands are never raised, but those kids are contributing. So that becomes their puzzle.

00:08:01:16 – 00:08:22:11 Jon Orr Right? Right. Now I’m going to dig here a little bit. Gordon Like I’m curious about that experience that you just shared thinking like in grade five, you repeated you went to a different school, but then what was different? What was at one school, but then different in the other school on that. And I’m buried here. So you became a mathematician and having to repeat the fifth grade.

00:08:22:11 – 00:08:33:05 Jon Orr So that is a very interesting story because you don’t hear of like, I’m a mathematician and I had to repeat the fifth grade. So fill us in on the details because I think all the listeners right now are like, wait a minute.

00:08:33:05 – 00:08:36:07 Kyle Pearce Yeah, nobody was betting that it’s coming. You know what I mean?

00:08:36:09 – 00:08:43:13 Jon Orr What was the difference between those 2/5 grade years? Or was there some moment that kind of changed for you?

00:08:43:15 – 00:09:09:11 Gordon Hamilton It was coming over from Northern Ireland. Northern Ireland was a very structured system and then I was put into this very loose system where kids were encouraged to learn by themselves and at their own pace, and I was just totally lost. So I got back into a structured school, was an average student until grade nine, and then Jim McConnell, who runs Stratford Hall in Vancouver, he was my science teacher.

00:09:09:13 – 00:09:32:13 Gordon Hamilton And he I can remember the lesson where he’s describing the bore model of the atom. And every day we have to do at scientific experiment. And I had to write up a hypothesis about depressing Kubrick oxide, and I just hypothesized that to pass oxide was the touching of two circles at a point, and two brass oxide was touching at two points.

00:09:32:13 – 00:09:45:19 Gordon Hamilton Anyway, it was totally wrong. He just marked it. Oh, what an interesting idea. And he marched it like a really high mark, which she hadn’t done before. I was thrilled. So that was me becoming a scientist first, before I’m a mathematician.

00:09:45:21 – 00:10:09:19 Jon Orr Right? Right. So now you’re getting into classrooms. Now you’ve taken this experience to say, I’m conscious of the slow, methodical thinker that brings up that slow part of my brain, fast part of my brain, and thinking about making sure that we honor those thinkers that take time to kind of postulate, think in our classrooms when I think so much of our classrooms and we think about math classrooms, like you said, no hand raising.

00:10:09:19 – 00:10:28:08 Jon Orr And it’s like I think so many classrooms, the teachers are very quick to be like, here’s a question, who’s the fastest to answer? Instead of thinking like we need that methodical portion of our class. So now you’re in classrooms these 18 years working with students on puzzles. Paint us a picture like, what does that look like when you’re going in?

00:10:28:10 – 00:10:38:18 Jon Orr What are some of the benefits you’re seeing with students engaging in the puzzles you’re working with in those classrooms? And then, like I said, paints a picture of like what a puzzle might look like when you’re working in those classrooms.

00:10:38:23 – 00:11:07:05 Gordon Hamilton So an example of a puzzle, this one’s not mine, but it’s one that I did in Collingwood School in Vancouver to the whole assembly of the school. So this is like hundreds of kids multi-age group and this gives you an example of a type of puzzle that I really respect and I might start out and you guys might have a number one painted on your bib, okay, so you’re up at the front of the gymnasium and you’ve got a number one, and I invite up to grade two kids and I put a number two on them.

00:11:07:07 – 00:11:24:08 Gordon Hamilton And then I said, Let’s try to organize ourselves in a line so that between those of you who have once is a single person and between those of you who have two, there’s two people. Okay, well, if you go to one, one, two, well, then the twos are happy, but the ones have got no one in between them.

00:11:24:08 – 00:11:42:07 Gordon Hamilton So that’s not a good one, two, one, two. Well, and the ones are happy because there’s just a single person in between them. But the two are not. Okay, let’s get up some grade threes. So the grade three come up. And is this going to work with grade threes? So let’s see then what? Then we have the kids, sir.

00:11:42:09 – 00:12:10:15 Gordon Hamilton The kids at the front. It’s difficult for them to think, okay, so I would expect them to think. I just want them to get in line. Okay. Just forget thinking, just get in a line and of course they’re going to mess up. But I love to teach through failure, so I want them to mess up. Right. This is more interesting for the audience, for them to say, Oh, you’ve got three people between the grade three kids, you’ve got two people between the grade two kids, but you don’t have one person between the grade one kids failure.

00:12:10:17 – 00:12:41:09 Gordon Hamilton So I celebrate failure all the time in kindergarten. I call it failure, but we celebrate it with a light heart. And let’s give some examples of okay, So one example of a failure is this puzzle with bubbling cauldrons. This is an unsolved problem from 1917, it was a Jewish mathematician working in Berlin, ended up having a pretty tragic life escaping to Palestine, where he died in poverty on his 66th birthday.

00:12:41:15 – 00:13:05:13 Gordon Hamilton So I’d like to honor this man, I say. Sure. So anyway, back to this beautiful puzzle. I have it bubbling cauldrons. Okay. And I have two bubbling cauldrons. And we are going to put in a lizard into one cauldron. Number one, number one lizard. There we go. Okay, John, you’re holding one of these. Kyle, you’re holding the other pot.

00:13:05:13 – 00:13:06:09 Gordon Hamilton So there you.

00:13:06:09 – 00:13:12:04 Kyle Pearce Go. People are going to have to see this on YouTube now. Yeah, my pot’s better than yours. John. Yeah, my cauldron.

00:13:12:06 – 00:13:20:05 Gordon Hamilton Okay. John has lizard number one in his pot. Lizard number two. Kyle, where would you like it to go in John’s pot or yours?

00:13:20:07 – 00:13:23:10 Kyle Pearce Oh, I want a lizard myself. So I like number two.

00:13:23:13 – 00:13:30:22 Gordon Hamilton Two. Okay, you’ve got number two. John Where do you want lizard number three? Do you want to give it to Kyle or do you want to keep it yourself selfishly.

00:13:30:24 – 00:13:31:13 Kyle Pearce I don’t really.

00:13:31:13 – 00:13:33:20 Jon Orr Want the lizard. So, Kyle, you take this list.

00:13:33:22 – 00:13:35:19 Kyle Pearce You know, was there. There’s two.

00:13:35:22 – 00:13:46:03 Gordon Hamilton And three. Two and three are in Kyle’s pot. And John, I got one. Number one. Number four. Okay, Where are you going? To put number four. Kyle.

00:13:46:05 – 00:13:47:15 Jon Orr Take them. I don’t want it.

00:13:47:17 – 00:13:51:18 Kyle Pearce You. I didn’t miss whether lizards are good or bad, did I?

00:13:51:18 – 00:13:53:17 Jon Orr Yeah, I’m not. We’re not sure of the goal here. Okay.

00:13:53:17 – 00:13:56:20 Kyle Pearce Okay. Just so. So we’re just saying we’re playing. This is.

00:13:56:20 – 00:13:59:06 Gordon Hamilton Good. Use the rules of the puzzle.

00:13:59:07 – 00:14:02:17 Kyle Pearce Yes, I love it. I’m going to keep them. I’m going to keep on. I love.

00:14:02:17 – 00:14:07:16 Gordon Hamilton It. Three and four, three and four. I got one. I got one. Number five.

00:14:07:18 – 00:14:08:14 Jon Orr I get to pick.

00:14:08:16 – 00:14:09:20 Gordon Hamilton You’re going to give it away right?

00:14:10:00 – 00:14:12:18 Jon Orr I’m totally giving it away. Okay. Take this take.

00:14:12:18 – 00:14:14:01 Gordon Hamilton Unfortunately for.

00:14:14:01 – 00:14:16:00 Jon Orr Kyle, there’s no rule two.

00:14:16:02 – 00:14:37:02 Gordon Hamilton Plus three. You’ve got two, three and four in your pot. Kyle. Yeah, yeah. Two plus three is equal to five. Whenever that lizard drops in my boom, you get grouped because two plus three is equal to five. The lizard that you are dropping in is equal to the sum of two lizards that you have in your pot.

00:14:37:08 – 00:14:38:17 Kyle Pearce And what happens?

00:14:38:19 – 00:14:39:13 Gordon Hamilton He gets grouped.

00:14:39:13 – 00:14:40:12 Kyle Pearce And then boom.

00:14:40:14 – 00:14:44:09 Gordon Hamilton No, that’s failure. But what grade two kid doesn’t want to be grouped?

00:14:44:11 – 00:14:48:17 Kyle Pearce Yeah, totally. The cauldron. So we’re done?

00:14:48:21 – 00:14:51:17 Jon Orr Yeah. You’re saying we can’t put it in there or the game?

00:14:51:17 – 00:14:55:02 Gordon Hamilton You put it in there, it explodes. Back to zero. Okay, Try to do it again.

00:14:55:03 – 00:14:55:19 Jon Orr Okay. We it.

00:14:55:19 – 00:15:14:20 Gordon Hamilton Again. It’s also important that we do not just try to find acid and say, okay, let’s take it out again. No, the whole dynamic of the classroom, you want to have an epic fail, the thing explodes. You’re right back to square one. Okay, you got up to five. Can you guys get up to six? Can you get up to seven?

00:15:14:22 – 00:15:25:22 Kyle Pearce I love it. And we both want to get as high as we can right? So we’re kind of like, collaboratively here. It’s not me trying out boop or goop or dupe John. You don’t.

00:15:25:22 – 00:15:26:07 Gordon Hamilton Want to do.

00:15:26:07 – 00:15:29:20 Jon Orr That. And we’re taking turns choosing. We were a team. We’re a team.

00:15:29:22 – 00:15:35:10 Gordon Hamilton 13, 13. So it’s a matter of sending off the kids. They’re in two pairs. They’re off in their own desks.

00:15:35:10 – 00:15:36:08 Jon Orr Let’s figure this out.

00:15:36:08 – 00:15:55:18 Gordon Hamilton They’re trying to get up to eight. I do tell them that eight is the objective here. Grade two classrooms. Some of those kids will get up to eight in 5 minutes. Some will take 20 minutes. How do you deal with that? So this is a typical example of what I call an infinite pickle puzzle, ones that just keep on going.

00:15:55:20 – 00:16:13:09 Gordon Hamilton You just out of pocket. I can you get the three pots? So you want to shut down those kids as soon as you get that celebration. Right. We saw that. We saw that. We saw this. Okay. You do not want everyone around to be demotivated. So you send those kids off. You work on three pots, you send another group off.

00:16:13:11 – 00:16:36:08 Gordon Hamilton Oh, you guys work with odd numbers. Okay. Just stay with two pots. You guys work with odd numbers. Next group, you guys work with even numbers. How high can you get? So I love that you’re all the time kind of cutting against the assessment that you have of student ability. I don’t want my top kids to be ego basking in front of the whole class.

00:16:36:11 – 00:16:38:03 Gordon Hamilton I want to shut that down.

00:16:38:05 – 00:16:54:19 Jon Orr It reminds me we always get asked that question. GOURD So it’s like when we talk with teachers here on the podcast, it’s one very common question is when I’m, you know, engaging my students in problem solving at the boards or wherever or whatever they’re working on, it’s like some groups are done early, some groups not. How do I challenge the ones that are done early?

00:16:54:19 – 00:17:13:00 Jon Orr We’ve always said you got to have this kind of arsenal of extensions or we call them like these little things in our back pocket. We take out and keep the challenge going, keep that thinking going, keep them in flow. And your idea of let’s just add another plot is like, what are those? We have to have those in our back pocket every time we have a problem like that, right?

00:17:13:01 – 00:17:32:14 Jon Orr It’s like, what are those next step? Things that keep the thinking going, but also raise the bar, raise the challenge a little bit. And what I love about your problem is that not everybody has to get to adding the extra pot. That’s not the goal. The goal is to just keep thinking going. It’s not like, hey, we got to make sure everyone gets to the three pots or even numbers.

00:17:32:14 – 00:17:36:14 Kyle Pearce And then once you do, you get to stop. No, it’s we’re going to keep going. We’re going to keep thinking.

00:17:36:19 – 00:18:02:08 Gordon Hamilton Top problem solvers, top kids in your class, they’re always deflected into problem solving, never a waste of time. The kids at the bottom who are working on skill acquisition, you can go round and you can help individually with skill acquisition, but your top kids are engaged because this is an interesting problem. If we go back to the other problem where you have kids now in grades one, two and three, okay, let’s just say that they were in grade three.

00:18:02:08 – 00:18:34:02 Gordon Hamilton This is you’ve got 25 kids and you have got some kids who have solved three. You guys can figure out three You can solve. You can see you’ve got two threes tutus, two ones, and you can figure out a way to match those. And then you might have really great kids, smart kids coming up and solving it. And then you shut them down with, okay, you try it for six different students and then a slow group later on you actually tell them so that the other group can hear you guys be the first in the class to work on seven.

00:18:34:07 – 00:18:57:20 Gordon Hamilton Okay, So all the time trying to muddy the water. So you give those the kids that are not used to getting that the higher level activity you give them publicly in a subtle kind of way. You give them publicly and even tougher looking activity. In this case, it’s not that much tougher, but in fact, I think six is impossible to solve and seven isn’t so is clever little things you can do at people.

00:18:57:24 – 00:19:15:17 Kyle Pearce If you’re listening right now, friends, you have to go on YouTube because the look on your face. Gordon you were talking about the kids having them, the mischievous little grin there at that math festival that you were at, but you’ve got the same one, which I love. I love that about this right now. I’m so happy we have video on.

00:19:15:19 – 00:19:36:10 Jon Orr I’m completely reminded of when you said that you got to be clever. You got to have these little kind of nudges in, kind of like to stick it to them almost sometimes to keep that engagement going. Often reminds me of not Banting. Kyle from Saskatoon, Saskatchewan. He’s got a teaching move that he’s always like, Oops, I forgot. He’s kind of saying like, he gives the instructions out, but he didn’t give them all.

00:19:36:15 – 00:19:52:01 Jon Orr And then all of a sudden, like they’re working around and he’s like, You know what? The kids solve a problem. Like, Oh, I forgot. It’s supposed to be an even number. You have to use. And then they’re like, Oh, it’s everyone’s back to the drawing board, right? And then they’ll do it again. He’s like, Oh, wait a minute.

00:19:52:01 – 00:19:56:24 Jon Orr And then they’re all like, Oh, come on, Mr. Banting. I’ve used that in my class. It is gold.

00:19:57:01 – 00:19:59:11 Gordon Hamilton Yeah, I love that.

00:19:59:13 – 00:20:30:17 Kyle Pearce I love it. I love it. This is fantastic. So you the thing we were talking before we hit record and we’re talking about, there’s so many similarities of some of the things we talk about. There’s likely a lot of differences as well, right, in terms of the approaches. But I definitely love that similarity there where although you’re using, we’ll call it maybe not in all cases with not all puzzles, but I’m sure that many of these puzzles are maybe considered non curricular puzzles because they may not directly connect to, say, the curriculum that they’re working on.

00:20:30:17 – 00:20:54:22 Gordon Hamilton I’m quite specific with curriculum. I really try to link it up. So very rarely do I introduce a puzzle that is non curricular. There’s no need. There’s such a wealth of beautiful puzzles, there’s no need to go for a non curricular. So listening through some of the old podcasts I have Peter Lynch I’ll say which tasks are non curricular and I just don’t get the I don’t understand.

00:20:55:00 – 00:21:19:18 Gordon Hamilton For me, there’s no reason that rich tasks cannot be non curricular. I don’t have an introductory task. And then the main task and then a wrap up. I have one task and if I was to say one thing for, if I’m going to be teaching some pre-service teachers coming up and I’m just going to be throwing them into a classroom unprepared, I don’t want them to know the answers.

00:21:19:20 – 00:21:42:15 Gordon Hamilton I don’t want them to be burdened. Teaching bats how to fly. I want them to go in and solve it with the kids. And I don’t want them. If the kids are engaged, I don’t want them wrapping up that lesson with a nice little bow tie. If the kids are engaged, there is nothing that you can say as an educator that is useful for a vast majority of those students.

00:21:42:15 – 00:21:57:14 Gordon Hamilton Yes, you might hit 20%. 30%. Do not interrupt your class. Go right to the buzzer. So very different what I’m teaching whenever I’m in Alberta, very different from what I see many of your other educators teaching. Mm.

00:21:57:16 – 00:22:21:08 Kyle Pearce This is really interesting. And, and you had mentioned about the whole tying it up, right? And I think that’s a really interesting one because when there is really great thinking going on, I think I agree. You know and John, I’d be curious to kind of hear your thinking as well. And this is where I think it gets tricky for educators because sometimes they hear one thing and then they think it’s all in on that one thing.

00:22:21:08 – 00:22:44:07 Kyle Pearce So, for example, never maybe consolidating. But what I think I heard you say, and maybe you can reiterate this or confirm or deny this is like if students are still engaged in the actual thinking and the actual learning that’s going on and they’re engaged in that, that we don’t want to sort of cut that short in order to wrap it up.

00:22:44:07 – 00:22:46:18 Kyle Pearce Is that what you’re saying? Or are you saying that.

00:22:46:20 – 00:23:02:13 Gordon Hamilton I would use that as a technique if I see that the class needs a breather, I would use what you would call consolidation at any time during the lesson to say, okay, I see that now I got down to 80% engagement. Not good. I want to get the class just a breather and then back in.

00:23:02:15 – 00:23:25:08 Kyle Pearce I love it. Yeah, very awesome. And I’m curious, so with some of these tasks that are you said vast majority are curricular, John and I also have always sort of advocated for the idea of, especially at the beginning of a school year, sometimes some educators think of bringing in different tasks that are non curricular because it’s easy to get students into.

00:23:25:14 – 00:23:56:14 Kyle Pearce But John and I have always sort of felt differently about that. We’ve always felt that we can use mathematics to engage students so that they’re actually learning mathematics, right? Math that’s going to assist them throughout this year and beyond. So we’ve always kind of advocated for that for you. If you’re in a classroom and let’s say and let’s pretend that you’re not just dropping into a classroom, like let’s say it’s your math classroom, what in your mind does it look like to help students?

00:23:56:14 – 00:24:19:09 Kyle Pearce Sort of when we say consolidate, when we say almost like tying the loose ends to make sure that they sort of got what they got out of the problem. So what I mean by that is sometimes we solve problems, but we’re not necessarily sure of what that means or how I might apply it somewhere else. So what might that look like to Gordon?

00:24:19:11 – 00:24:50:05 Gordon Hamilton Okay, so I am talking about elementary school. I am skeptical about. I think that there’s a problem between assessing and getting the classroom culture that you guys want. These things look to me like they are in conflict. I would much rather an elementary school be assessing for the standards for mathematical practice rather than anything else. So if I see a kid really try to work, yeah, that’s great.

00:24:50:07 – 00:24:55:05 Gordon Hamilton That’s what I think I would want to be assessing, not skill acquisition.

00:24:55:07 – 00:25:21:18 Jon Orr Yeah, I think we’re in complete agreement with that for sure. We often say that we would love for the refocus of what’s happening in our classrooms across North America. And I guess the world is to say that I would love the focus to be the math practice standards and the vehicle to get there is the curriculum or the content standards we should be using that as a way to unfold the math practices instead of the other way around.

00:25:21:19 – 00:25:43:07 Jon Orr Most teachers are like, We got to get the content in focus. This content and these other things kind of trickle in. If I structure my lesson right, I’ll get those things in, in, in instead of going. The point is to get those things in and we’re going to use the content to help us get there. I think that is been a big push that we’ve had with the districts that we work with and also the classrooms we’ve been teaching in.

00:25:43:09 – 00:26:09:13 Kyle Pearce Gordon I’m wondering, you know, with the work that you’re doing, what does that look like and sound like? We have some educators who have visualized, some of them have visualized, John and I holding cauldrons. Some of them are visualizing students in front of the gym. And what I’m getting is they’re probably picturing in their minds are like, Wow, I wish I was better at doing stuff like that.

00:26:09:15 – 00:26:33:01 Kyle Pearce So my wonder is for those teachers who are listening going, I wish that I had a little bit more of what Gordon’s got going on. What would you say would be a good first step for them? How can they maybe take a step in that direction if maybe they haven’t felt that they’ve maybe they’re just unsure? How do I begin this journey?

00:26:33:03 – 00:26:58:19 Gordon Hamilton Yeah. So I would take a beautiful unsolved problem. One of my favorite ones is from 1967. It’s called The Graceful Tree Conjecture. I call it animal subtraction on math. Pickleball come and just take that. Understand how it works. Very simple rules. If you want to have minimal words in a math classroom, all of these things that have got this much text, I don’t get it.

00:26:58:20 – 00:27:23:21 Gordon Hamilton This is not esthetically pleasing mathematics to me. I want to have a minimal set of rules that opens up a universe of interesting possibilities. This is an unsolved problem since 1967, but the kids can solve. I’ll just describe it. You’ve got five circles, okay? They’re connected in a row and you’re going to put odd consecutive integers into those circles.

00:27:23:23 – 00:27:44:14 Gordon Hamilton So go ahead. Maybe your first attempt is one, three, five, seven, nine. Okay, Just in a row. Well, you fail, okay? Because you’re trying to get the differences between those neighboring circles. You’re trying to get those differences different. So one, two, three. The difference between those is two, three and five. The difference is also two. Okay, You’ve already failed.

00:27:44:16 – 00:28:09:18 Gordon Hamilton So that’s the question. How do you do that? But then you can reorganize those five circles. You could, for example, have two circles touching or three circles touching one circle. So you can do that. And then all the touching circles, the differences between those have to be different. It’s easier to see this one in real life, but if you Google graceful tree conjecture, you’ll see a number file video of me working with it.

00:28:09:18 – 00:28:33:07 Gordon Hamilton And that’s a nice example, but that is an example of a puzzle that is pretty well guaranteed to keep your kids entertained and engaged for the entire class. So this problem about how to wrap up classes that I don’t think it’s going to happen a lot if you go in and you don’t have to prepare, I don’t want you to know the answers, okay?

00:28:33:07 – 00:28:58:17 Gordon Hamilton Just relax the night before. Go get have green tea and celebrate life with your friends. I don’t want you knowing the answers. I want you to go in and yes, you feel vulnerable. Okay. That vulnerability is working through that, I think is the most wonderful, liberating experience for teachers, because once you get through that and you’re okay, okay, what’s the worst that’s going to happen?

00:28:58:19 – 00:29:15:22 Gordon Hamilton The worst that can happen is you might think that one of your students outshines you. Okay, well, a big deal like that happens. Another embarrassment might be that you end up not being able to solve it as a as well. What does that mean? That means math classes or math classes become exciting then. Is it going to work?

00:29:15:22 – 00:29:29:16 Gordon Hamilton Is it not going to work? Are we going to completely blow up altogether and not end up with anything beautiful? Yeah, it makes math class interesting to go in with that rawness of I don’t know if this is going to work or not.

00:29:29:18 – 00:29:56:13 Kyle Pearce I love it. I’m excited. I’m kind of you’ve got me giddy over puzzles here, but I’m now trying to think and I’m trying to think of the listener out there who’s saying, I love this idea. I want to dig in. I want to learn more. But I’m wondering and I’m 100% in the line, it’s almost like an a belief piece where I say, yes, that truly problem solving with your students side by side is such a magical opportunity, right?

00:29:56:13 – 00:30:15:19 Kyle Pearce When we talk about math, moments like think of the moments that you can have, but then the other part of me, and this is maybe the devil’s advocate kind of coming out saying, you know, Kyle, I’ve got this curriculum and I have to achieve, whether we like it or not, or whether it should be done that way or not.

00:30:15:21 – 00:30:48:13 Kyle Pearce What would you say to those who are thinking like, how can I be intentional about, say, the teaching and what I’m trying to pull out now? I’m totally on board. I totally get the idea that, yes, we want the math practices, their process expectations, whatever you want to call them, from whichever curriculum we’re looking at. But when it comes to the actual, let’s say, content that we are uncovering in a course, whether it’s elementary or high school, how do we as educators sort of feel like the intentionality of the lesson?

00:30:48:15 – 00:31:02:08 Kyle Pearce Is there, especially if maybe I’m not exactly sure, or do the teachers know what the intentionality is of the puzzle, But we’re still going to solve it all together in the class on the fly.

00:31:02:14 – 00:31:24:07 Gordon Hamilton That’s right. So if you want to work on multiplication today before you teach multiplication, because if you you know that 20% of your kids already know multiplication before you begin, okay? So you don’t start out with something boring. You start out with a really interesting puzzle right away. Okay? The objective of math class is not to teach easy.

00:31:24:09 – 00:31:47:09 Gordon Hamilton It’s to teach engaged. And that means you’re going to be presenting a tougher puzzle than the kids would otherwise be expecting. Just seeing multiplication for the first time. But it’s an engaging puzzle for everyone. So Mondrian Art Puzzles is an example of a beautiful. There we go. There’s one solution of managing our process for those of you who are watching the video.

00:31:47:11 – 00:32:11:17 Gordon Hamilton So the objective for the kids who are learning multiplication is to learn, okay, I need to have all of these rectangles of different dimensions. Okay, I’m going to be covering a square. All of my rectangles must be a different dimension. So I can’t have a two by six and a three by four. Oh, yeah, I can, because those are different dimensions, even though the area is the same.

00:32:11:19 – 00:32:42:07 Gordon Hamilton And then I’m going to score my beautiful piece of artwork that you’re helping Mondrian to paint. I’m going to score it your largest area, minus your smallest area, and you want that score to be as small as possible. Okay. So this is a really interesting and beautiful puzzle, Kate. Those kids, you know that you’re going to have 15, 20% of kids who are they’re not excited by math class, but they’re excited about making a beautiful artifact.

00:32:42:12 – 00:33:02:20 Gordon Hamilton And for them, they can color it if they’ve got a messy puzzle sheet. I have extras because it is never a waste of time for me to give them an extra puzzle sheet and for them to make something beautiful that they’re proud of. Yeah, always celebrating beauty. If I have this kid has to solve the bass like they’ve got a really low answer.

00:33:03:01 – 00:33:11:22 Gordon Hamilton But this kid is is not as high achieving in mathematics, but they’ve got something beautiful. Oh, I’ll hold up this one. Look at this. Oh, that’s nice. Okay.

00:33:11:24 – 00:33:38:03 Jon Orr Yeah. What I’m hearing you say, I think throughout this recording in our talk is again, kind of going back to the intentionality can be and should be some of the time because strand a in our document here in Ontario now is about the practice standards so it’s like we can make the intentionality specifically about engaging students in those practice standards instead of the content and the contents coming along for the ride.

00:33:38:03 – 00:34:09:07 Jon Orr So I completely agree with you on that. It’s like this is the point of the lesson is to engage students in problem solving gaze students in reflection engage students and conjecturing engage students in in reasoning that right there is is can be the intentionality of some of your lessons. And I think teachers get hung up on the like I need to have the intentionality on making sure that we understand the algorithm of two digit by to do multiplication or I need to make sure that we can add these two things consistently this way like that.

00:34:09:08 – 00:34:32:12 Jon Orr Yes, we want to be doing some of that. But yes, we also want to be doing some of what you’re talking about in focusing on those standards, because now here in Ontario, it is part of the curriculum in a more direct way than it used to be. It’s not just front matter anymore. It’s building right in. So your puzzles are math pick Ockham If you go there, you’re there’s videos that walk you through how to kind of think about that puzzle.

00:34:32:12 – 00:34:49:19 Jon Orr Get started in that puzzle kind of like shows you a little bit of the rules a little bit when you kind of get in. Gordon I think it’s going to be a lot of people going over there and kind of checking things. Give us a snapshot of grade level content. What can they expect? We have primarily teachers listening to this episode, so they’re going over there.

00:34:49:21 – 00:34:58:16 Jon Orr What are they looking for in a quick synopsis? And then soon as you share that, give us what big kind of takeaway you want teachers to take away from our chat here today.

00:34:58:18 – 00:35:29:24 Gordon Hamilton Okay. Whenever they go over that, there’s going to be two tabs, There’s going to be puzzles and games. We haven’t talked about games. So just let me quickly say that the number one message of teachers to parents in elementary school is that you should be doing no worksheets with your children at home. Your only job is to adopt a board game and to play a board game with your child or to get the child playing with another child and that is because our number one objective in elementary school mathematics classrooms, it is problem solving.

00:35:30:02 – 00:35:54:15 Gordon Hamilton It’s not arithmetic. Arithmetic is number two. But number one. I want my kids to think and there is no more joyous excuse to think than a good quality board game. Not all board games are created equal. So that’s why I’ve got some of my picks on the Games section. But this is an enormous this can really help to have parents really get their kids into thinking and hope.

00:35:54:15 – 00:36:16:22 Gordon Hamilton So very important message. On the puzzle side, you will see my wide range of puzzles. There is most of them are. I’m trying to create these what I’m calling infinite pixels. These are the puzzles that you can dial up with a single sentence. So if you solve the Mondrian puzzle, you think you’ve got a good answer for five by five.

00:36:16:24 – 00:36:42:01 Gordon Hamilton Well, then you just tell the child, okay, go for six. By six, you guys go for six by six. If you guys have solved six by six, then try eight by eight. And so it gives you such classroom management power to have a really good puzzle behind you, because so many of these tasks that I look at on many, many different websites, I would describe them as low floor, low ceiling.

00:36:42:03 – 00:37:07:03 Gordon Hamilton Are no way are they anything that I would see. For me, a high ceiling means I’m interested as a mathematician, I can take any of my grade one puzzles and you’ll see on there my book, The Infinite Pickle. I can take any of those puzzles and I can present them to a graduate level mathematics classroom so they can be understood by grade one, Grade two, grade three students, but they are really ceiling forget ceiling.

00:37:07:03 – 00:37:28:09 Gordon Hamilton Let’s say it’s a starry sky out there. We don’t need a ceiling head that’s the kind of puzzle that is worthy of the math classroom. And that’s we should expect nothing less whenever we decide to do this type of puzzle and math. Pickel is not a full curriculum. You go there to get gems of mathematics, not to get a full curriculum.

00:37:28:11 – 00:37:49:23 Kyle Pearce I love that. So it sounds like scratch that itch that some of these people may have as they’re listening going, Where do I get started? They’ve got a wealth of resources and opportunities over at math pixel dot com, so friends definitely head on over there. I love that idea. Why even have the ceiling? You know, it’s a starry sky.

00:37:49:23 – 00:38:07:17 Kyle Pearce What an awesome quote. That’s maybe my big takeaway for this episode. But Gordon, if we flip it back to you, if there’s just one thing that you hope that the make math moments community, the math moment makers out there are taking away from this particular conversation, what are you hoping that they leave here.

00:38:07:17 – 00:38:13:12 Gordon Hamilton With number one message to parents get your parents playing board games in the home environment.

00:38:13:14 – 00:38:29:05 Jon Orr I love it. Awesome. Thanks for that, Gordon. And we want to thank you for joining us here on the podcast and we’re going to put your Web site in our show Notes page. Is there anywhere else folks can kind of, you know, reach out to you or ask you more questions or where’s the best spot to find it?

00:38:29:07 – 00:38:36:14 Gordon Hamilton Yeah, good at math pickle dot com is my email and the website has everything else, including that.

00:38:36:16 – 00:38:43:05 Kyle Pearce Awesome. Thanks so much Gordon. It’s been great to see you. Stay warm and don’t bring the umbrella out. You won’t need it, right?

00:38:43:07 – 00:38:55:05 Gordon Hamilton Yeah. A bonus. Bonus high fives for you guys. That was an incredible two week discovery of your podcast. I’m up to number 28 now and I enjoy in it so much.

00:38:55:08 – 00:38:58:12 Kyle Pearce One that’s so fantastic to hear. We love it.

00:38:58:14 – 00:39:07:24 Gordon Hamilton I just want to say, Peter, I mentioned something that I didn’t like about him, but boy, I really like what I hear there. So yeah, a wonderful one here is.

00:39:07:24 – 00:39:27:11 Kyle Pearce A fantastic, fantastic read. I think you should get along with him really well. You guys would really hit it off. So hopefully we’ll all get together at maybe a live conference at some point soon. I know we bump into Peter quite a bit. If you haven’t been to some of the math conferences, I would recommend Gordon get proposals in there.

00:39:27:11 – 00:39:29:11 Kyle Pearce I think your message is such a great one.

00:39:29:16 – 00:39:35:11 Gordon Hamilton Being a puzzle designer, finances are not the strength.

00:39:35:13 – 00:39:42:05 Gordon Hamilton I have not been good at getting to conferences and they’re not cheap. Yeah, so that’s not coming up in the near future.

00:39:42:07 – 00:40:05:24 Kyle Pearce Yeah. Well, hopefully what I’m hoping for is that some folks out there, we know that you’re listening, they’re friends. There’s lots of people that are conference organizers, including ourselves. The virtual summit next year, I think, sounds like a great opportunity for you to be speaking at our What would it be, John, our sixth annual virtual summit. So take this as an official invite.

00:40:06:02 – 00:40:23:01 Kyle Pearce Gordon, we would love to have you as a part of that this coming November and 2024. So let’s stay in touch about that and those who are running conferences. Hey Gord at Math Pexels.com guess what? If you can get them there, I’m sure you’d show up.

00:40:23:03 – 00:40:27:16 Gordon Hamilton Sure he would. Thanks, Gord. Okay.

00:40:27:18 – 00:40:46:20 Jon Orr Okay. I hope you loved that interview with Gordon just as much as we did. And when we think about our math tree, you know, our math classroom tree or even our district level planning tree, a lot of times, actually, Kyle and I get asked about questions about like, what should we focus on first? Should I try to bump up my math pedagogical knowledge?

00:40:46:22 – 00:41:04:01 Jon Orr Should I think about how to strengthen my roots of my tree? Should I think about how to strengthen my own content knowledge and the models and the strategies around concepts I teaching before I start thinking about pedagogy, Some districts are asking us what curriculum should I be choosing? Should I be choosing this curriculum? If we’re valuing these in our classroom, should we be choosing this?

00:41:04:01 – 00:41:20:23 Jon Orr How do we choose our curriculums. And this is a case where I think, think about Gordon’s work. It’s almost like we’re start at what you think is important at the time, but also go, Hey, you know what? I have this resource in front of me. I want to use the filter of what I think is important using this, but now it might spark the next thing.

00:41:20:23 – 00:41:38:19 Jon Orr So for example, Gordon’s talking about puzzles, which is the leaves of our tree. These are the resources we do, the lessons, these are the things that we can grab and bring into our classroom. We call those the leaves of our tree. But once you start thinking about a leaf of a tree or resource, sometimes that will dictate the next moves.

00:41:38:19 – 00:41:58:24 Jon Orr If I grapple and I want to use this resource, what are the things that I need to strengthen around my tree so that I can effectively use this resource? Sometimes it’s like I have to strengthen the roots of my tree. I actually have to dive into the mathematics behind this. So I get understand it myself and the tools in the strategies to unpack it for my students.

00:41:59:01 – 00:42:19:00 Jon Orr Sometimes using a resource, we have to go. We have to strengthen our pedagogical moves, our branches of the tree, and that kind of leads down a pathway to kind of go down, see what moves should I be doing in the classroom, Do I need to kind of perfect a move, need to try a move in the classroom so that I can unpack this resource or this puzzle, as Gordon as outlined here today in the classroom?

00:42:19:00 – 00:42:36:05 Jon Orr And what are some of the moves around that? So I like to kind of think about those things as I grab different resources around the Internet or I hear on podcasts like this one. What are some of those next steps? What are the ways that we use the things that we have at our fingertips to strengthen our classroom tree?

00:42:36:05 – 00:42:50:10 Jon Orr I want you to consider that here today. Do we need to kind of if you’re going to use this, do you need to go down the pedagogical route, the branches? Do we need to strengthen the roots, root That’s your kind of thinking point here today. That’s my big takeaway. So thanks for listening to the Make Math Moments That Matter podcast.

00:42:50:15 – 00:43:14:00 Jon Orr This was episode 282. You can find the show notes over on make map moments dot com for episode 282. And again, thanks for listening and if you have not yet say rated or reviewed the show, go ahead, do that and hit the follow button so that you can get to the new episode as they come out every single Monday morning and we can be in your ears on your way to work doing the dishes.

00:43:14:02 – 00:43:23:09 Jon Orr Or maybe you’re doing some exercise right now and listening to us, but we can be there giving you tips, strategies, ideas around your math classroom and strengthening your classroom tree up along the way.

00:43:23:13 – 00:43:28:15 Kyle Pearce All right. Their math moment makers. Until next time. I’m Kyle Pearce.

00:43:28:15 – 00:43:30:00 Jon Orr And I’m Jon Orr high.

00:43:30:00 – 00:43:32:04 Kyle Pearce Fives for us.

00:43:32:06 – 00:43:33:09 Gordon Hamilton At a high bar for.

00:43:33:09 – 00:43:34:17 Jon Orr You.

00:43:34:19 – 00:43:35:20 Gordon Hamilton Oh.

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DOWNLOAD THE 3 ACT MATH TASK TIP SHEET SO THEY RUN WITHOUT A HITCH!

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!

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LESSONS TO MAKE MATH MOMENTS

Each lesson consists of:

• TEACHER GUIDE
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Each Make Math Moments Problem Based Lesson consists of a Teacher Guide to lead you step-by-step through the planning process to ensure your lesson runs without a hitch!

Each Teacher Guide consists of:

• Intentionality of the lesson;
• A step-by-step walk through of each phase of the lesson;
• Visuals, animations, and videos unpacking big ideas, strategies, and models we intend to emerge during the lesson;
• Sample student approaches to assist in anticipating what your students might do;
• Resources and downloads including Keynote, Powerpoint, Media Files, and Teacher Guide printable PDF; and,

Each Make Math Moments Problem Based Lesson begins with a story, visual, video, or other method to Spark Curiosity through context.

Students will often Notice and Wonder before making an estimate to draw them in and invest in the problem.

After student voice has been heard and acknowledged, we will set students off on a Productive Struggle via a prompt related to the Spark context.

These prompts are given each lesson with the following conditions:

• No calculators are to be used; and,
• Students are to focus on how they can convince their math community that their solution is valid.

Students are left to engage in a productive struggle as the facilitator circulates to observe and engage in conversation as a means of assessing formatively.

The facilitator is instructed through the Teacher Guide on what specific strategies and models could be used to make connections and consolidate the learning from the lesson.

Often times, animations and walk through videos are provided in the Teacher Guide to assist with planning and delivering the consolidation.

A review image, video, or animation is provided as a conclusion to the task from the lesson.

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At the end of each lesson, consolidation prompts and/or extensions are crafted for students to purposefully practice and demonstrate their current understanding. 

Facilitators are encouraged to collect these consolidation prompts as a means to engage in the assessment process and inform next moves for instruction.

In multi-day units of study, Math Talks are crafted to help build on the thinking from the previous day and build towards the next step in the developmental progression of the concept(s) we are exploring.

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Solve rate and scaling problems - Problem-Solving Investigation - Year 6

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

15 April 2024

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This in-depth maths investigation is an open-ended problem solving activity for Year 6 children. It can be used to support teaching towards the multiplication and division objective: use mental strategies, factors and multiples to solve problems of rate and scaling.

In-depth Investigation: Get to the Root Children use their fluency in mental multiplication to explore the patterns of digital roots in multiplication.

This investigation will develop maths meta-skills, support open-ended questioning and logical reasoning, and enable children to learn to think mathematically and articulate mathematical ideas.

This problem-solving investigation is part of our Year 6 Multiplication and Division block. Each Hamilton maths block contains a complete set of planning and resources to teach a term’s worth of objectives for one of the National Curriculum for England’s maths areas.

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A bundle is a package of resources grouped together to teach a particular topic, or a series of lessons, in one place.

Year 6 Multiplication and Division - Problem-Solving Investigations

These in-depth maths investigations are open-ended problem solving activities for Year 6 children. **In-depth Investigation: Magic Multiplication Squares** Children complete a magic multiplication square using their knowledge of number properties and relationships. They then explore factors and multiples to create a new multiplication magic square. **In-depth Investigation: The Eights Have It** Children multiply numbers starting with 9 by 9 and add single-digit numbers in a decreasing sequence. They identify and describe the patterns and start to explain them. **In-depth Investigation: Awesome Answers** Using a magic square to generate 3-digit numbers, children create divisions with dividends containing specified fractions. **In-depth Investigation: Stunning Squares** Children explore patterns in the squares of numbers with reversed digits to find pairs of ‘stunning squares’. **In-depth Investigation: Geometry Genius** Children use what they know about how to find the areas of triangles and parallelograms to find the areas of rhombi, kites and trapezia. **In-depth Investigation: Get to the Root** Children use their fluency in mental multiplication to explore the patterns of digital roots in multiplication. **In-depth Investigation: Riveting Reversals** Multiply 3-digit numbers with consecutive digits by a 2-digit number; reverse the 3-digit number and repeat. Find the difference between the two answers. **In-depth Investigation: Why is it so?** Children identify a pattern in the division of a total of six numbers created using the same 3 digits. They then use algebra to explain why it is so. These investigations will develop maths meta-skills, support open-ended questioning and logical reasoning, and enable children to learn to think mathematically and articulate mathematical ideas. These problem-solving investigations come from our [Year 6 Maths Blocks](https://www.hamilton-trust.org.uk/maths/year-6-maths/). Each Hamilton maths block contains a complete set of planning and resources to teach a term’s worth of objectives for one of the National Curriculum for England’s maths areas.

Solve rate and scaling problems (Year 6 Multiplication and Division)

This bundle provides three days of teaching that cover the objective: **Use mental strategies, factors and multiples to solve problems of rate and scaling**. *Teaching Presentation* The teaching presentation includes starter activities, whole class teaching, group activities, practice sheets and mastery questions. It can be used on a variety of interactive whiteboards. *Practice Worksheets* The procedural fluency practice worksheets are differentiated for children working towards Age Related Expectations (ARE), at ARE and at greater depth. *Problem-Solving Investigation* This in-depth maths investigation will develop maths meta-skills, and enable children to learn to think mathematically and articulate mathematical ideas. This teaching is part of Hamilton’s [Year 6 Multiplication and Division](https://www.hamilton-trust.org.uk/maths/year-6-maths/multiplication-and-division-2/) block. Each Hamilton maths block contains a complete set of planning and resources to teach a term’s worth of objectives for one of the National Curriculum for England’s maths areas.

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6.4: Hamiltonian Circuits

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• Page ID 22341

• Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier
• Coconino Community College

The Traveling Salesman Problem (TSP) is any problem where you must visit every vertex of a weighted graph once and only once, and then end up back at the starting vertex. Examples of TSP situations are package deliveries, fabricating circuit boards, scheduling jobs on a machine and running errands around town.

Example \(\PageIndex{1}\): Hamilton Path:

Figure \(\PageIndex{1}\): Examples of Hamilton Paths

Not all graphs have a Hamilton circuit or path. There is no way to tell just by looking at a graph if it has a Hamilton circuit or path like you can with an Euler circuit or path. You must do trial and error to determine this. By the way if a graph has a Hamilton circuit then it has a Hamilton path. Just do not go back to home.

Graph a. has a Hamilton circuit (one example is ACDBEA)

Graph b. has no Hamilton circuits, though it has a Hamilton path (one example is ABCDEJGIFH)

Graph c. has a Hamilton circuit (one example is AGFECDBA)

Example \(\PageIndex{2}\): Complete Graphs

Figure \(\PageIndex{2}\): Complete Graphs for N = 2, 3, 4, and 5

In each complete graph shown above, there is exactly one edge connecting each pair of vertices. There are no loops or multiple edges in complete graphs. Complete graphs do have Hamilton circuits.

Example \(\PageIndex{3}\): Reference Point in a Complete Graph

Many Hamilton circuits in a complete graph are the same circuit with different starting points. For example, in the graph K3, shown below in Figure \(\PageIndex{3}\), ABCA is the same circuit as BCAB, just with a different starting point (reference point). We will typically assume that the reference point is A.

Figure \(\PageIndex{3}\): K\(_3\)

Example \(\PageIndex{4}\): Number of Hamilton Circuits

How many Hamilton circuits does a graph with five vertices have?

(N – 1)! = (5 – 1)! = 4! = 4*3*2*1 = 24 Hamilton circuits.

How to solve a Traveling Salesman Problem (TSP):

A traveling salesman problem is a problem where you imagine that a traveling salesman goes on a business trip. He starts in his home city (A) and then needs to travel to several different cities to sell his wares (the other cities are B, C, D, etc.). To solve a TSP, you need to find the cheapest way for the traveling salesman to start at home, A, travel to the other cities, and then return home to A at the end of the trip. This is simply finding the Hamilton circuit in a complete graph that has the smallest overall weight. There are several different algorithms that can be used to solve this type of problem.

A. Brute Force Algorithm

• List all possible Hamilton circuits of the graph.
• For each circuit find its total weight.
• The circuit with the least total weight is the optimal Hamilton circuit.

Example \(\PageIndex{5}\): Brute Force Algorithm:

Suppose a delivery person needs to deliver packages to three locations and return to the home office A. Using the graph shown above in Figure \(\PageIndex{4}\), find the shortest route if the weights on the graph represent distance in miles.

Recall the way to find out how many Hamilton circuits this complete graph has. The complete graph above has four vertices, so the number of Hamilton circuits is:

(N – 1)! = (4 – 1)! = 3! = 3*2*1 = 6 Hamilton circuits.

However, three of those Hamilton circuits are the same circuit going the opposite direction (the mirror image).

The solution is ABCDA (or ADCBA) with total weight of 18 mi. This is the optimal solution.

B. Nearest-Neighbor Algorithm:

• Pick a vertex as the starting point.
• From the starting point go to the vertex with an edge with the smallest weight. If there is more than one choice, choose at random.
• Continue building the circuit, one vertex at a time from among the vertices that have not been visited yet.
• From the last vertex, return to the starting point.

Example \(\PageIndex{6}\): Nearest-Neighbor Algorithm

A delivery person needs to deliver packages to four locations and return to the home office A as shown in Figure \(\PageIndex{5}\) below. Find the shortest route if the weights represent distances in miles.

Starting at A, E is the nearest neighbor since it has the least weight, so go to E. From E, B is the nearest neighbor so go to B. From B, C is the nearest neighbor so go to C. From C, the first nearest neighbor is B, but you just came from there. The next nearest neighbor is A, but you do not want to go there yet because that is the starting point. The next nearest neighbor is E, but you already went there. So go to D. From D, go to A since all other vertices have been visited.

The solution is AEBCDA with a total weight of 26 miles. This is not the optimal solution, but it is close and it was a very efficient method.

C. Repetitive Nearest-Neighbor Algorithm:

• Let X be any vertex. Apply the Nearest-Neighbor Algorithm using X as the starting vertex and calculate the total cost of the circuit obtained.
• Repeat the process using each of the other vertices of the graph as the starting vertex.
• Of the Hamilton circuits obtained, keep the best one. If there is a designated starting vertex, rewrite this circuit with that vertex as the reference point.

Example \(\PageIndex{7}\): Repetitive Nearest-Neighbor Algorithm

Suppose a delivery person needs to deliver packages to four locations and return to the home office A. Find the shortest route if the weights on the graph represent distances in kilometers.

Starting at A, the solution is AEBCDA with total weight of 26 miles as we found in Example \(\PageIndex{6}\). See this solution below in Figure \(\PageIndex{7}\).

Starting at B, the solution is BEDACB with total weight of 20 miles.

Starting at C, the solution is CBEDAC with total weight of 20 miles.

Starting at D, the solution is DEBCAD with total weight of 20 miles.

Starting at E, solution is EBCADE with total weight of 20 miles.

Now, you can compare all of the solutions to see which one has the lowest overall weight. The solution is any of the circuits starting at B, C, D, or E since they all have the same weight of 20 miles. Now that you know the best solution using this method, you can rewrite the circuit starting with any vertex. Since the home office in this example is A, let’s rewrite the solutions starting with A. Thus, the solution is ACBEDA or ADEBCA.

D. Cheapest-Link Algorithm

• Pick the link with the smallest weight first (if there is a tie, randomly pick one). Mark the corresponding edge in red.
• Pick the next cheapest link and mark the corresponding edge in red.
• Continue picking the cheapest link available. Mark the corresponding edge in red except when a) it closes a circuit or b) it results in three edges coming out of a single vertex.
• When there are no more vertices to link, close the red circuit.

Example \(\PageIndex{8}\): Cheapest-Link Algorithm

Suppose a delivery person needs to deliver packages to four locations and return to the home office A. Find the shortest route if the weights represent distances in miles.

Step 1: Find the cheapest link of the graph and mark it in blue. The cheapest link is between B and E with a weight of one mile.

Step 2: Find the next cheapest link of the graph and mark it in blue. The next cheapest link is between B and C with a weight of two miles.

Step 3: Find the next cheapest link of the graph and mark it in blue provided it does not make a circuit or it is not a third edge coming out of a single vertex. The next cheapest link is between D and E with a weight of three miles.

Step 4: Find the next cheapest link of the graph and mark it in blue provided it does not make a circuit or it is not a third edge coming out of a single vertex. The next cheapest link is between A and E with a weight of four miles, but it would be a third edge coming out of a single vertex. The next cheapest link is between A and C with a weight of five miles. Mark it in blue.

Step 5: Since all vertices have been visited, close the circuit with edge DA to get back to the home office, A. This is the only edge we could close the circuit with because AB creates three edges coming out of vertex B and BD also created three edges coming out of vertex B.

The solution is ACBEDA or ADEBCA with total weight of 20 miles.

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We provide Hamilton Year 4 maths both as weekly plans (below) and as  short blocks . We will eventually be phasing out the plans, as we believe our short blocks offer you all of the same advantages and more. Find out more about the  advantages of Hamilton's short blocks .

Use place value to add or subtract to and from 4-digit numbers. Place 4-digit numbers on a line. Round 4-digit numbers to the nearest 10, 100 or 1000. Count on and back in steps of 25 and 1000. Write Roman numerals to 100.

Use written subtraction, expanded then compact decomposition to subtract pairs of 3 and 4-digit numbers. Use mental subtraction by counting up (Frog) to subtract pairs of 4-digit numbers. Choose a strategy to subtract pairs of 4-digit numbers depending on the numbers involved.

Written subtraction using decomposition to subtract any pair of four-digit numbers, including those needing 3 moves. Written addition using compact column addition to add any pair of 4-digit numbers. Add and subtract near multiples of 10, 100 and 1000 to or from 3- and 4-digit numbers using place value, Choose mental or written methods to add and subtract. Solve word problems involving addition and subtraction. Includes bar model examples.

Find area of rectilinear shapes by counting squares. Find perimeter of rectilinear shapes in centimetres by counting. Calculate perimeter in centimetres and metres of rectangles. Use co-ordinates in the first quadrant and join to draw posited polygons.

Mark numbers with 1 decimal place on an Empty Number Line and round to the nearest whole. Know what each digit stands for in numbers with 2 decimal places. Multiply and divide by 10 and 100 to give tenths and hundredths. Know equivalent 0.1s and 1/10s, and 0.01s and 1/100s. Write place value related additions and subtractions for numbers with 2 decimal places.

Compare and order number with 2 decimal places. Place numbers with 2 decimal places on landmarked lines (marked in 0.1s). Add and subtract 0.1 or 0.01 to or from numbers with 2 decimal places. Count on or back in tenths and hundredths. Add or subtract multiples of 0.1 or 0.01. Solve simple measure problems using place value in lengths in metres with 2 decimal places.

Find factors of numbers less than 50. Use factors to carry out mental multiplication. Find the product of 3 single-digit numbers using commutativity to help. Use times tables and place value for mental division of multiples of 10. Solve scaling problems (by whole number factors). Convert from centimetres to metres. Solve correspondence problems.

Complete shapes with respect to a line of symmetry. Recognise and compare acute and obtuse angles and angles of 90 degrees. Compare and classify triangles and quadrilaterals, based on properties including types of angles.

Read the 24-hour clock converting times to am and pm, both digital and analogue formats. Find time intervals using 24-hour clock. Read, interpret, draw and describe a time graph. Convert between units of time.

Identify equivalent fractions, including decimals. Find non-unit fractions of amounts. Solve fraction word problems. Written division by chunking of two-digit numbers by single-digit numbers, answers less than 30. Includes bar model examples.

Use the written ladder method to multiply 3-digit numbers by single-digit numbers, estimating answers first. Choose mental or written methods to solve addition, subtraction, division or multiplication word problems and calculations. Includes bar model examples.

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