September 06, 2022
Pam Harris
Episode 116

Math is Figure-Out-Able with Pam Harris

Ep 116: Structuring Your Math Class Examples

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Math is Figure-Out-Able with Pam Harris

Ep 116: Structuring Your Math Class Examples

Sep 06, 2022
Episode 116

Pam Harris

What does teaching Real Math look like at your grade level? In the last episode Pam and Kim discussed the One Third/Two Thirds rule to structure math time. In this episode they give specific examples at various grade levels of what it might really look like.

Talking Points:

- 1st Grade - doubling and halving
- 3rd Grade - concepts around 100
- 5th Grade - multiplication and volume
- 7th Grade - rates (see Episode 112 for Unit Rate Problem String)
- Algebra I - exponential functions
- Pre-Calculus - functions

Our workshops are open for registration! Take advantage now to change your math classroom! https://www.mathisfigureoutable.com/workshops

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What does teaching Real Math look like at your grade level? In the last episode Pam and Kim discussed the One Third/Two Thirds rule to structure math time. In this episode they give specific examples at various grade levels of what it might really look like.

Talking Points:

- 1st Grade - doubling and halving
- 3rd Grade - concepts around 100
- 5th Grade - multiplication and volume
- 7th Grade - rates (see Episode 112 for Unit Rate Problem String)
- Algebra I - exponential functions
- Pre-Calculus - functions

Our workshops are open for registration! Take advantage now to change your math classroom! https://www.mathisfigureoutable.com/workshops

Pam Harris:

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague:And I'm Kim.

Pam Harris:And you found a place where math is not about memorizing and mimicking,waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.

Kim Montague:So last week, we talked about how Pam and I view what it could look like to fit in all kinds of great things into a math classroom. And we shared that we'd like to think of a two third/one third split of time as kind of our structure of a math class. This week, we want to dive into some specific examples at a few grade levels to give you a better idea of what we mean.

Pam Harris:Yeah, we did some real general discussing about how it could all work and fit together. And we thought, man,let's give some examples. So here we go, we're gonna dive into a few kinds of representative grades, we won't do them all. But hopefully, this gives you a little bit of an idea of what it could look like.So for example, let's take grade one, if you are thinking about a first grade situation, I might think about a couple of days, I might think about starting off with a string where we work on doubles. And if I'm in first grade, I'm working on small number doubles, we're thinking about helping kids really tackle doubles and get a feel so that it's kind of an automatic. You know, seven and seventh, eight and eight, that they kind of have this sort of sense of doubles. But then we might take from that string, we might dive into a collection, not really a counting collection, because what I want to do is I want to then halve. You want to find halves of things. So we might say, "Hey, we're gonna play a game, and you need half the deck of cards and you need half the deck of cards. So we're gonna work on halving so we all have this bit of sort of better sense of what, how we can find halves." And I might give different sets of things to different students and have them split it in half, like make sure you have half make sure your partner has half and let those kids just dive in. I've done this with first grade students,it's so much fun. It's a really nice way to differentiate because I can give some students who are ready for more experience, I can give them the thumbtacks. Okay, that's maybe not the smartest thing for first grade. I was trying to think of something, there was a lot of.Help me.

Kim Montague:Post-it notes.

Pam Harris:Are there a lot of post it notes in a set?

Kim Montague:Yeah.

Pam Harris:Okay.

Kim Montague:You can buy little stacks, big stacks, yeah.

Pam Harris:Okay, something where there's a lot so that they have to, or at least more. A number that I think that they can really fuss with. But then I might hand a different group of students, the markers, and I,you know, like, "Here's a marker set of eight. And how do you know if you have half?" Again,I'm being judicious to kind of make sure I'm giving students experiences that are right on the edge of their zone of proximal development to really move them forward. And then I'm going to continue to give them better and bigger collections.So that as they have them, they gain more experience, we would do a math Congress about what were some ways that you were thinking about halving. I would definitely want to pull out a strategy, like, "Ooh, Was anybody kind of dealing out? I give you one, and then I get one, and you get one, and I get one." Because that's a strategy of making sure that we have an,anyway, blah, blah, blah. I don't want to get too deep into any one of these. So this idea of an Investigation and a Congress around halving, then we might follow that up with a doubles game. We might have a game where you sort of roll a die and double that amount, and then move that forward on the board, so that kids are like,again, they're sort of messing with this idea of doubles. Do you see how we're kind of doing things that are all kind of around the same idea? I might follow that with a quick image problem string where I am flashing doubles. And we're looking at using the structure of the 10 frame or structure of a rekenrek to help students kind of glance at doubles and see something. So in that kind of two thirds time, I've just described a sequence of tasks that could happen to really help students thinking about doubles.Really, it's about doubles, it's less about halving, it's about doubles. We halve to give them kind of this nice context of game playing. I'm not trying to cinch halving. I'm trying to get that on their horizon to think about, but really, I'm giving them lots of experiences to count where the count matters.

Kim Montague:Right?

Pam Harris:So it's a big idea in first grade. Now, if that's the two thirds time, what am I doing in the 1/3 time? Well in the one third time I'm thinking about past games that we've already played that I want to keep going all year long. So I'm like, bring those back. We might be doing Count Arounds so that we're looking at patterns in the base 10 number system, like what are things that need to be happening all year long, really important things? And I'm going to continue to do those during that time. Yeah, Kim, how does that feel?

Kim Montague:Yeah, it feels good. And I want to mention again, that I think a small piece of what you said that maybe people maybe didn't catch is that you said, this was not like, all in one day. You're not tackling all of the things that you've said in one day. It might be over the course of two days or three days or, you know. I think it can be really easy to say, like, I want to master doubles in a day. Right? And we know that that's not gonna happen.

Pam Harris:It's one standard.That one standard in isolation.Okay, that's Tuesday.

Kim Montague:Yeah, yeah. Yeah,I mean, it can feel like that is, you know, what you have to do. But we're taking this kind of like a wider approach to what can I do to give kids experiences over the course of several days or week or over time. And I love that 1/3 bit of time, because that's when you can cycle back to things that you know, you've approached, but you still want to, you know,doubles are going to be something that you make use of all year long in the future.

Pam Harris:Absolutely.

Kim Montague:So I was thinking about grade three, and I, you know, I kind of was thinking, in two days, what would a two day plan look like for me? And again, it's not that, you know,everything's gonna be mastered in two days. But I thought,okay, if I'm third grade,addition is kind of a big deal in third grade.

Pam Harris:Very much so.

Kim Montague:And we expect kids to come to us as third grade teachers, having some solid addition. But we know that that's not always the case. And then we panic, because, like, we have to tackle multiplication.So I was thinking over the course of a couple of days, for a two thirds time, I might start with I Have, You Need. A whole class, just a routine to kind of get some things started moving in their brains, or they might do it with partners on day one.And I might do it with partners of five and multiples of 10. So fives and 10s. Just to start warming up and saying like,"We're going to start doing some thinking within 100."

Pam Harris:And if you've never heard of I Have, You Need that's just where you say, if it's partner of 10, you're saying, "I have eight, what do you need to make 10?" And then students are finding those partners of 10.

Kim Montague:Yeah, but I'm, but I'm thinking, I really am thinking that I'm going to do partners 100 by 10s, to just bring out the idea of 100, and then just see what's happening.

Pam Harris:Third grade, right.

Kim Montague:Yeah, yeah. Well,I mean, if they've never played ever before, then that might be a nice entry point. So then I think I'm going to do a Rich Task, or maybe an Open Middle activity, where students are working with those same partnerships. If I'm moving towards total of 100, or total of 200. It's really important for me, in third grade to have kids recognize that if I know42, and 58, then I know that 142and 58. Is that next 100. I think that's a really essential thing for kids to make sense of.That there's just these partnerships within hundreds.And so I would absolutely spend some time doing some work around that.

Pam Harris:Let me give another example. So you're saying like,if I know if 32 and 68, partner to 100. I also know 332 and 68,partner to 400.Yeah, I think it's a really big idea. And so I might

Kim Montague:Yeah.I Have, You Need, and then I might do an Investigation where kids are drawing out possible combinations. And it could be like an Open Middle and we treat it like a really rich time where kids are messing with what can they find? And are they looking for any patterns? So that might be like the end of day one for me. And then in my 1/3 time, on that day, I might be doing something where they're playing Close to 20. Or they might be doing a, you know it might be something that I'm I know that I'm eventually going to play Close to 100, some might be introducing some games in a small group type setting. So the next day, I might come back together. And I might say, let's do a Get to a Friendly Number string with some numbers where I'm having them look at 27 and saying how much more to the next friendly number? So we're still looking for partnerships of 10s and hundreds. And then we're going to have a Congress to pull out the patterns that the kids noticed in the thing that they did the day before. Or maybe I can see playing Close to 100,which is one of our favorite games. Maybe during the one third time, like I said, I'm playing close to 20 or 100, even1000 with different students,because I know they need something different.Which is brilliant, right?Because all the kids are playing, quote unquote, the same game.Yeah.

Pam Harris:But you've differentiated the total that allows them to gain more experience where they need more

Kim Montague:Yeah, yeah. Or in my 1/3 time, I might be, you experience.know, like I mentioned in the last episode, sometimes looking back at work, or sometimes doing a problem string with slightly different numbers. You know, if I haven't heard the voice of some of my students in whole group, then I might pull half the class and do a problem string and so that I can, like more focus in on what some of my kids are thinking. So you know,it might all be centered around100 for a little bit, a couple of days, but we're like you said the first grade doubles is kind of the big idea, and how they can use doubles. You're tackling it in a couple of different ways and I think 100 is crucial in third grade. So,

Pam Harris:Yeah, hey, really quick when you said at the end of the, golly, I think it was the 1/3 time. Okay, I'm not even sure when it was. But when you were talking about that you might pull a small group in prep for the next day.

Kim Montague:Oh, yeah.

Pam Harris:Can you say more about that?

Kim Montague:Sure. So if I know that I'm going to play, I'm going to introduce Close to 100,which is, you know, one of our favorites. And I know that I'm going to play that game. And I'm going to introduce it maybe whole class. Because we like to do that. Play against the class.So they really understand how to play. But it's a game that I'm going to play all year long.It's not like a one shot game.

Pam Harris:You start playing,sorry, start playing against the class. So they really understand how to play, so that then you

Kim Montague:You can send them in partners. Yeah, but there can see can-might be times, there might be classes, there might be kids who I know, maybe there's a lot of instruction to the game, or maybe notating their thinking might be something that they're not really sure how to do. It might be that there's a whole host of reasons why I might pull a small group and say, "Hey,tomorrow, we're going to play this game. And I want you guys to be my experts. Like, I really want you to know how to play this game." And it just boosts a little bit of confidence. Maybe?Or maybe, you know, it's not the kid who's always the expert in the thing. So there are times where I might pull a group the week before or a couple of days before and give them some experience before we play as a class.

Pam Harris:Just sort of front load their experience.

Kim Montague:Yeah.

Pam Harris:Yeah. And then when you're doing it whole class,because of that they're able to join in. What a wonderful way to give them that experience that just a little extra experience that they need to then be able to join in. That's nice.

Kim Montague:Yeah.

Pam Harris:And I loved it how you couched it as you guys are gonna be my experts.

Kim Montague:Yeah.

Pam Harris:That's cool.

Kim Montague:Everybody wants to be an expert at some point,right?

Pam Harris:Yeah, absolutely.All right. I really liked that third grade plan centered around a really big idea. Yeah, nicely done.

Kim Montague:Okay. So let me give you a little bit about fifth grade, too.

Pam Harris:Yep. All right,fifth grade. Go ahead.

Kim Montague:So a lot of this depends on the time of the year,because, you know, both of us talked about Problem Strings and games and some Rich Tasks. And those are, I think, essential Jey, before we go to one third.for the whole year. But there are times where we're getting towards test taking I mentioned or we're getting towards, I need to, I you know, I've introduced a topic. And I feel like I need to spend more time with you on it. So fifth grade, depending on the time of the year, I might start with a multiplication string to just get gears turning. And then we might do an Investigation about, say, like,how many boxes could they make with 24 chocolates, which of course leads to conversation about volume and later on surface area. And we have a rich Investigation. I might follow up that with a Problem String about prime factorization, or another one of our favorites, the Product Game. In the 1/3 time, I might be-Yeah.

Pam Harris:So that sounds like you're really maybe focusing around either multiplication or volume. Like that sounds like a really fifth grade thing to do that we're really getting more sophisticated in multiplication.So let's use volume to do that.Oh, and volume is this thing we need to do anyway.

Kim Montague:Yeah.

Pam Harris:So we're just sort of kind of everything, you're doing different things, but they're all building toward facility with multiplication and volume.

Kim Montague:Yeah, since they're so connected, right?There's so many strategies for multiplication that are going to help so that it's not, here's this formula for volume.

Pam Harris:Yeah.

Kim Montague:So in that third time, which is kind of extra,could be set aside from the two thirds or could roll right together, I might be setting up a game or again, maybe an Open Middle puzzle for the class to work on. I might be saying,"Okay, today, you guys are gonna play the Product Game or play,"or whatever, pick a game that I've already introduced. And that's, you know, we've gone over the expectations for how that's played. And, you know,here's your partners, or pick your partners or whatever. And I'm going to pull small groups of students during that time. So you know, for 20 minutes you guys are going to play, they know how to play because we've gone over it together. I'm pulling small groups of kids, or maybe I'm free, and they're coming to me one on one with,"Hey, I had a chance to look at this assignment that you passed back and I want to talk about whatever this this thing that I wasn't sure about," or "I want to talk to you about this thing that I thought I understood and maybe I need some more support."or "Hey, look at me, I did so fabulous on this and, and I just want to celebrate it." So it might be that I'm freer, and they're, they just have some kind of back and forth flow between the game and meeting with me.

Pam Harris:And that game or that task, whatever they're doing at that point is something or centered around a thing you know, it's important all year long. And so that's going to be one that you're going to continue to bring back all year.

Kim Montague:Yeah.

Pam Harris:So the kids were like, oh, yeah, like this is a good engaging thing. While, it frees you up. I'm going to mention one other kind of thing that you might do well, it frees you up. It frees you up to circulate and listen in.

Kim Montague:Yeah.

Pam Harris:Right? So you get a chance to kind of hear where students are. And that gives you really good information about what you do next, because now you know, "Oh, everybody's really mastered this game.There's not a lot of thinking going on anymore. Bam, it's time to move up."

Kim Montague:Yep, for sure.

Pam Harris:Yeah. Cool. Hey, it seems like we're kind of in an odd grade habit here, because I'm gonna go to grade seven.

Kim Montague:Okay. All right.

Pam Harris:Let's stay odd a little bit here, 1, 3, 5, 7. All right. So in grade seven, what could it look like? I'm thinking that rates are really important.Often non unit rates are really important in grade seven. And so I might think about doing a rate string. We did a rate string on the podcast not too long ago, I don't know a couple months ago,maybe we'll put that in the show notes of when I did that rate string. But, you know, if you're covering 20 feet in four seconds, how fast are you going?And then 19 feet in four seconds, a string like that,where kids are kind of starting from non unit rate to a unit rate, and back and forth, and forth and back. They're also getting some really nice division experience as they do that they're getting the sense of, of a speed. And what that means. And all of that is sort of front loading for what I know, they're going to do later with slope of a line. So this rate stuff. So I might start with that rate string. And then to continue to do that kind of an idea, I might have them do a motion detector Investigations.I love motion detectors.

Kim Montague:Yay you do.

Pam Harris:One of my favorite things to do in Rich Tasks. And so I might set them off to do some kind of Rich Task about how do I know what a fast or slow rate looks like? If I'm walking fast in front of the car, if I'm walking slow in front of the CBR, what do those look like?And how do I know? And how do we make sense of that? And we might have a math Congress on that,that sort of leads to generalizing things about rates.But then I want to kind of get a little bit more numeric. And so then I'm going to follow that either, I can think of a couple of different things I might do.But I think I might follow that with a rate string that gets a little bit more picky about the rates that they were actually just walking. So instead of 20feet in four seconds, nobody's gonna walk 20 feet in front of a CBR, because you're out of range. And so now I'm gonna get kind of picky, "Hey, well, if you just in the four seconds that you were just walking, but you only cover two feet, then how fast were you going?" So now I can do rates that are under one. So it was, in that first string, I was sort of playing with rates that are greater than one now that I'm playing with rates that are less than one.And so now we're kind of dealing with fractions, which we know is a big deal in middle school.That could potentially be a follow up to that. So if that's happening in the two thirds time, I might in the 1/3 time be thinking about, "Well, in order for them to sort of cover distance, or think about timing/" So later, I might want to say, "Well, if I started at two feet from the CBR, and I walked away to the eight foot mark?" Now they have to think about how much distance they've traveled. Well, that's kind of subtraction. But sort of looking at the meaning of subtraction,the interpretation of subtraction, that's the distance or difference between the numbers. So because of that, I might actually do a subtraction string, where we're thinking about the fact that subtraction has these two interpretations.And we're messing around with subtraction so that as we do that differencing in finding rates that they have that kind of sense of subtraction happening. But also, I'm really aware in seventh grade that we're trying to think and reason proportionally. So I might think, sort of out of what we're doing and say, "Wow, in order to do all of this proportional reasoning, I need kids who have better facility with their multiplication facts." So we actually might play the Product Game here or the Factor Game. I do some gaming, so that kids,they did build their multiplication sets.

Kim Montague:Yeah.

Pam Harris:All right. So there's a great seven.A look at what might happen. Oh, and I'm gonna

Kim Montague:Okay.mention again, just like you said, that's not all happening in one day. That's over a couple of days. You can't do all that in one day. I don't think so.Even that Investigation, the Investigation itself, should take most of the class period,especially if you do the Congress after it.

Pam Harris:Then the strings are coming on either ends of those.

Kim Montague:Yeah.Okay, so staying odd, I'm gonna go to a typical ninth grade or an Algebra I course. So whenever you teach Algebra I, it might be an eighth grade, we do a lot of the similar anyway, it's Algebra I. But I think you could also do what I'm going to describe in Algebra II course, depending on how in depth you go, and how ugly the numbers are. I might think about the fact that we are about to write our first exponential function. And so I might say to myself, "What needs to happen before we have kids writing exponential functions?And what kind of data do I want them using to write an exponential function? Do I want it to be messy data that they kind of are approximating? Or do I want it to be exactly?" I have to make all these decisions. So I'm gonna go ahead and decide that I want to start with messy data. Or at least in this for this thing I'm going to do it's going to be messy data. And so I might say to myself, on the Investigation, I'm going to work around the the Rich Task I'm going to work around, is I'm gonna have students drop a ball under a motion detector or just drop a ball against a ruler. And I'm going to have them either use that motion detector or that ruler to measure the max heights. How high is the ball coming up after it bounces? When it bounces, what's the height of the bounce? So we drop a ball,maybe a racquetball or basketball or something that bounces, and we measure the max height of each bounce. And we record that. So now we have for bounce one we have a height, for bounce two we have a height, for bounce three we have a height.Well if you can picture a ball bouncing up (claps slowly,faster, faster).Like as it bounces, it bounces more often right? And then lower and lower, right? It comes up not as high every time. So we have sort of this decreasing function. And then I'd ask students, you know, like, "What,what do you think that is? Then let's write a function for it."But to get into that, I'm going to be really aware that the multiplier is less than one,because every time that bounce height is decreasing, and that's weird for kids. That's weird to say that it bounced up eight feet the first time, I'm just gonna use a nice number. And if it only bounced 75% of the time,75% of the height, then that's like a multiplier of point seven five, that's weird. And so I might say, I need to front load that experience with giving the kids just this experience of having them know that a sequence can have a multiplier, but the terms are decreasing, the terms are getting lower every time. So I might do a string to start off with where the multipliers decreasing. And I might just say, "Hey, we're just, it's just a sequence string." I just give them a sequence of numbers where the numbers are decreasing. And we find the multiplier and we do some interesting, make the numbers interesting so that they have to like think about it and,but they have this, they already have percolating in their minds,that you can have a sequence where the terms are decreasing,and it can be multiplicative.And so that multiplayer can be less. I know, I'm explaining a lot. But that's a big sticking point. And so I want to, like have that makes sense. So do the string. So that's like,percolating in their minds, then we do the Investigation, they actually go drop the balls,measure the things as I circulate, and listen to what they're thinking about. They write their first exponential function, we graph it, we have a Congress about what not really how they found them. But I actually put up all the ones they found. So like, each, this group had the tennis ball, and this group had the racquetball,and this group had the basketball and I might say,"Here are the exponential functions that you've found. Now I'd like you to sort them. Which one do you think represents which ball? And how do you know?" And so now they're kind of have to ask each other,"Well, did you just bounce really high? Did you drop it from a high starting point?" So communication happens, kids are making sense of which number in the function means which thing,whether the start height or the return percentage. They're having to make sense of that.And then I might follow that with, "Hey, y'all yesterday, or whenever we did it yesterday,you did this investigation where y'all were finding these different functions? Remind me,remind me with this function?What did this number mean? What did that number mean? What it looked like in the graph? What did it what happened with the ball? Okay, all right, we got it. What if I changed it this way?" And I just give the new function. And I say, "What, what what happened in the graph?Let's check it out." We look at the graph, "Well, huh? Well, if we change the what was that in the scenario, if we drop it from a higher bounce? Check out what,how that affected the graph,check out how that affected the function. Okay, here's a new problem three." So doing a problem string that cinches what the different parameters meant in that Investigation. So that's an example of what could happen in a higher grade, Where again we're using the same sort of same sort of lesson structures,Problem Strings, Rich Tasks,Congresses, Problem Strings to cinch in a way that kind of helps students. I'm aware of our time that You just dropped some lesson plans for some Algebra I teachers. That's awesome.Bam! Okay, so let me do the same thing for Pre Cal. Are you ready? All right, if you're a Precalculus class or someone that's teaching Trig, one of my favorite things to teach ever was Trig. Could I actually do something sort of similar where I think about if the main investigation that I'm going to do is we're going to take a data collector, and we're going to collect tuning fork pitches. So I'm going to, or not tuning fork pitch, the sound from a tuning fork. So I'm going to ping a tuning fork, it's going to admit sound. Now a tuning fork is specific, because it emits just one frequency of a tone. And I ping it, "ping." And when it pings, we collect that data with this really cool recording device. And it sends that data into a graphing calculator. And I can see literally the wave. So picture what you think a sound wave looks like. And I literally can see that wave. And I say,"Alright, y'all we've been studying sine functions, write a sinusoidal function to represent this data." And now students have to like think about,"Alright, what do I know about amplitude? What do I know about period? And what do I, how can I mess with the parameters of a sine function to represent this data?" Well, that's the Rich Task that I want to do with students, then I might think about, "Well, what are the things students are going to have to mess with?" Well, one of the things they're gonna have to mess with is finding the amplitude. And they're gonna have to do a little bit of subtraction to do that. So we might do a subtraction string. I might do subtraction string with numbers that are kind of kind of come up in the Investigation.But I also, that's a possibility. But I also might think about, so different possibility, I don't know that I do both of these. I might do one or the other. I might think about the fact that when they write that sinusoidal function,they're gonna have to choose between a sine function and a cosine function. And both of those functions have different,what's the word I want,advantages to using when we write, when we fit them to data.And so I might want to do something, a Problem String that helps students think about the difference between sine and cosine, specifically the y intercept, so that then when they see the y intercept of their, of the data, then they can sort of make sense of which function might make more sense to use as they model that relationship. So do the string to help them think about the differences between the two functions, collect the data,write the function that actually fits the data. And then I might think about a string kind of similar to the one we did with the exponential function with the ball drop thing. I might say to them, "Okay, like, here's a function that you wrote yesterday to match this tuning forks data." By the way, when I taught Precalculus, I was used to called pitchfork, like, not on purpose, I would say, "Okay,your pitchfork," and they would all laugh at me. And I was like,"Okay, tuning fork, like,Whatever, whatever the fork thing is called," anyway. So then I might say, "Yesterday,you all that had this tuning fork, you had this data." We'd show the data on the graph, and we'd show the function over the data. And then we, and then I might say, "Remind me, what is this number? Where do I see that in the graph? Where do I see that? And then how do I know it's this frequency? Okay, okay,what if we change this number?How would that affect the graph?How would that affect the pitch?How would that affect, like all the things?" Well, then we would discuss that, "Well, what if what if we change this number?How would that ?" And we again,we're sort of like cementing or cinching what the different numbers, the different parameters in the function mean.Okay, so if that's happening in the two thirds, oh, I forgot to talk about what would happen in the 1/3 for Algebra I. Sorry. So let me stay with Precalculus then I'll go back to Algebra I.So in Precalc, we're doing this big idea of writing sinusoidal functions to match data. Then in the 1/3 time, I might think to myself, "What is the big idea that needs to be getting continual experience for kids all year long? Man, they need to have those unit circle relationships." It's not about memorizing all the unit surgical stuff, it's about having a couple of them. And then how can I use those from there? How can I know something about those special angles and how, or even just the y intercepts and the x intercepts. How do I, if I know something about those, how can I reason about sine, cosine and tangent about the rest of those values on the unit circle? So similarly, if I was in Algebra I, what's what are some relationships, I want to keep going all year long? Well, in that 1/3 time, I might have students doing something with factoring quadratics, which I think I mentioned in the last episode, where I might do this thing, where I'm saying, "Hey,if I give you these two numbers,what could, what numbers could you find that would multiply to this number as a sum, but also simultaneously add to this or the sum to this, add or subtract to this sum? So how can I sum to get these from same two numbers and multiply to get this product. And then that leads towards when we put all the variables and stuff for kids being able to sort of factor quadratics. So those would be some of the things that would be happening in that kind of all year time, where I want to keep those things going.Really kind of the heart of the 1/3 time for me is that we're recycling back or we're getting some some little,maybe pre-teaching or revisiting things. And I love that you're talking about doing that at seventh and high school courses as well. So we just tackled sixth grade levels. And that's a lot. And we know that maybe listeners aren't able to do everything that we just talked about. We know that everyone's got different demands, and they maybe have different expectations from their schools.But we think that there's an entry point for everyone, right?

Pam Harris:Absolutely. And so if you're listening to this podcast, and you're like, "Whoa,I mean, that sounds interesting.I'd like, hand that to me, and I'll go do it." And you don't want to create it all. We don't think that teachers should be expected to be master teachers and master curriculum writers at the same time.

Kim Montague:Yeah.

Pam Harris:So to where might you start? What might you, if you want to dip your toes in and give something a try? Just consider carving out some time.Find some time once a week that

you can commit:I'm going to do a Problem String. Start there.Start with, "Hey, y'all" like Fun Friday, Marvelous Monday,Terrific Tuesday, whatever it is, a time of the week where you feel like, "I can get 15minutes, I'm going to commit,I'm going to do this thing." A place to start is to start doing Problem Strings. Just quickly,the reason to carve out that

time is A:so you'll actually do it. But B: it's also, by carving it out, I also mean, not just like the time itself, but actually the space that you set up this time and space with students so that they know during that time and space,you're doing something different, you're doing this thing, and they know their ex,they know what their role is,they know the expectations during that time. And they're able to dive in and do thinking and reasoning. That allows them to feel safe during that time that they're like, "Yes, I can,I don't have to worry about anything. I just like dive in to actually think. I know my job here is to use what I know." But it also protects you a little bit, because kids aren't going to be going, "Hey, miss, I thought we were supposed to like do that thing." It's not you saying, "Alright, forever after we're going to do this from now on." And then you slip, you forget, you, you know, go back into I do,we do, you do just naturally, because that's sort of what you're used to. And now the kids are giving you this hard time or whatever. It gives you this carved out time where you're like, "During this time,this is going to happen." You get better at it, they get used to it, then it can start to permeate more and more of what you do. You know, it's interesting, one of the fascinating comments that I keep getting in our message boards.So you know, we have these online workshops that go on. And I'm in the message board,interacting with people,answering their questions. I keep getting this interesting comment that teachers and leaders are saying that they finally get to experience all the things, not just hear people talk about them. They say like,"Pam, people talk about growth mindset, they talk about equity,and they talk about discourse,"and you know, blah, blah, all the things. And they're like,"But Pam, we actually experience it in your workshop." Well, I really appreciate that. Because they say that it actually is happening naturally that what they don't see is me saying,"Y'all equity is important and discourse is important. And growth mindset is important."But they experienced that the way I'm teaching is growth mindset, that it is equitable,that you are seeing discourse happen. And then we have this really cool module in our workshops, where we sort of step outside, and we rewatch some of the clips from what they've just experienced. And we point out,"Hey, did you notice this was actually a purposeful move I did to have equity happen. This,this was actually a purposeful move I did that created that discourse that allowed the sense and a feel of what was happening so that this discourse happened.Or that I just did that right now see how that encourage the growth mindset that then right here you see a teacher having,because of what we just set up."So I think it's a really unique part of our online workshops,where you do the online workshop, and you learn the math. And then we take a look back at certain specific things.And we call out and identify really important things that happen kind of naturally in the workshop. And now you can make it happen because you're like more, you've experienced it.

Kim Montague:You're aware.

Pam Harris:Now you're aware of the things that you have control over as a teacher to instigate so these things can actually happen in your classroom. And I'm particularly happy about that.

Kim Montague:And can I just add that it's an experience that we don't often get to have in our classrooms, because time is passing by as we're making the moment to moment decisions. And so you don't get often a chance to look back. But in the workshops, you participate in the math, and then you get to look back and you get to like internalize what's happening a little bit and go, "Ah, like, I believe that I want to do that."

Pam Harris:And it just cements it all you're in all of us and all of us get better and better.So remember, if you're interested to learn more registration for my online workshops is currently open. But it's only open until next week,Friday. Eeek! The workshops are jam-packed with the content and the guidance you need to change your math class. So if you're dedicated to the Math is Figure-Out-Able movement, this is your next step. Y'all Building Addition for Young Learners, pre K-2. Building Powerful Multiplication, anyone who teaches multiplication, so grades three and up. Building Powerful Division, anyone who teaches division, so grades three and up. Building Powerful Proportional Reasoning for anyone who teaches anything to do with proportional reasoning.So really grade six and up, but I want grade five, too as well.And then Building Powerful Linear Functions, boom, my signature work up to this point has just launched we're about to I guess we're about to just launch because we're done with it. But now it's coming out soon as registration is over. So y'all, if you teach eighth grade or higher, you are really going to want to look at how we Build Powerful Linear Functions.

Kim Montague:Right. And so registration is only open until next Friday. Remember, we only open registration three times a year, so you need to get in now so that you don't miss out. You can find out more when you head over to mathisFigureOutAble.com/workshops.

Pam Harris:And that's where you register. Y'all thank you for tuning in and teaching more and more Real Math to find out more about the Math is Figure-Out-Able movement visit mathisFigureOutAble.com Let's keep spreading the word that Math is Figure-Out-Able

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