 kittens in a litter usually four to six and then the age that they can no longer have kittens is about 10 years. So using this information, can a cat have 2,000 descendants in what I'd say 18 months? 18 months. So I'm going to give you guys five to seven minutes to just kind of start working on this problem. You can either work on it as a math teacher or you can start to work on it as you think your students would. But take about five to seven minutes to give that a shot and then we will come back together and share strategies then. Questions before I mute myself? Liz you look like you have a question. Yeah I just I had to figure out how to get to my unmute fast enough. Is there when we do this is there anything for how long between pregnancies for a cat? This is the only information given. Okay thank you. Yeah. All right five to seven minutes. I'll start a timer and everything. Hey Marissa. Hi. How's it going? Good how are you? Good. So we are starting with a task. You should be able to see it up on the screen. It's the cat problem. With all of this information can a cat have 2,000 descendants in 18 months? Okay. So we're going to take another like two-ish minutes on it. You could think of it as a math teacher or think of it as one of your kids would think about it. But just take a couple minutes and then we're going to talk about strategies. Okay cool. Awesome. So we off screen share right? Okay good. Okay so finish up your last thoughts on that one. I'm going to type in a google doc while you guys are talking and then I'll share it with everyone after. That way you can all like you can all still see each other and everything. So first off what what did you get as the answer? Do you think it is possible for that cat to have 2,000 descendants? I didn't get an answer yet. Okay that's fine. What were your starting strategies? I agree. I didn't quite get an answer yet either. Okay. I started with somewhat of a timeline table because I thought that's what students would most likely do. And tried to figure out okay every like start with the the first mom will say. Okay. And then how many offspring she would have over the course possibly of 18 months. And then I wouldn't with the first litter. I started going through and saying okay how much could they have over those remaining months. But that takes a lot of time. Tracy you're on mute. So I did the same thing. I made a table but then I made little offshoot tables with each of them. And then kept multiplying because of the litter they can all have about five. I said it was about five in between five and six. And then I made little offshoots for how many that they would have. And I ended up with more than 2,000. Okay. I started trying a tree diagram and it got too complicated. And then I tried to just think through. But I was getting I was confusing myself with the years involved months. So they have to be four months. But then it's a two month gestation period. So it's six months to have a litter. But then the first cat two months later could have another litter and two months later could have another litter if she could have three a year. And then the first litter six months later could have a litter. So I was going with highest counts. So I got you know there was the first mom and then her first litter was six. And then their first litter was 36. And then their first litter would get me to a 216. So 216 plus 36 plus six plus one. And then I didn't get all the branches of mom kept having litters and the first letter had more letters and the second letter had a litter. But I didn't think the third litter could have any letters. Okay. In 18. Any other strategies that you guys used to approach this or strategies also that you think your kids would use if they were given this problem? Drawing pictures. If they were given this as a hard copy piece of paper. Do you think that would be different? Like they would approach it differently than we just did seeing it up on a computer screen? I suppose it depends what was on a paper. This would be an awesome whiteboard problem because they can re-work. Or jumping off of what Tracy said have individual whiteboards and then kind of come together or because then the students if they're working in teams could say okay you take the first mom okay you take the this litter and let's let's break it up. And you could do that with whiteboards or paper of course. You could also have them I think just post it up on a whiteboard or like blackboard, smartboard, whatever and then have them use vertical workspaces in small groups to draw it out. And I should I like to tell you that breaking up you look work on this litter and their sibling their descendants and so forth. But I think it's it was key that there was no like um table already made for us. It basically just a blank space. Yeah let's do whatever. But I could also see if they were working by themselves where they could get frustrated and just be like I know I'm just gonna you just want a yes or no answer I'm just gonna pick one and give it to you and not really keep going because I'm confusing myself because I know in my drawings it's hard to see this but I mean I tried cat diagram I tried bar graphs and I was just going in circles after for sure. Okay so I just shared a doc with all of you guys with the strategies that I heard you saying as I was typing. So I want us to kind of look at that together and try to think about how we could kind of group those strategies into like buckets similar to what we did during the during the summer kind of thinking about like what bigger buckets of like we can kind of classify them as. I'm seeing a couple right away but I'm curious what you guys come up with first and I'll type as you guys talk as well. Any ideas for like how we could categorize those strategies? I would say the timeline and tree diagram pictures those kind of get lumped together as a visual representation of the the growth for sure. I agreed I thought the visual representation and then how the work gets broken up so the grouping of students working together or not working together would kind of be a different bucket. Okay so splitting up the problem or splitting up the students? Both. Okay any other ideas? So if I were to give this to my students on paper I would want them to mark up the problem kind of circle the important information annotate a little bit. Anyone else with that one? It's not one that we said and that's kind of what I was trying to I don't know if that was because we were doing it on the computer and so there wasn't so much that we could like physically mark up but I think if we were to have had it in front of us I think a lot of us would have been circling and underlining and annotating. And I think working on the whiteboards we're implying that they're going to be writing and marking drawing. Yeah definitely especially if they're on the whiteboards. Awesome okay so thinking through those if you look back on the handout I think there are some student responses if you scroll down there's Alice there's Wayne there's Ben and that's it so if you look at those three did those three students use any of the strategies that we talked about or did they use strategies completely separate? We have Wayne with a tree diagram I think. Yep that's what it looks like. Oh Tracy good you got to talk. Okay did Alice use one of those strategies? I would say Alice used either a timeline or I guess I think it's got tape diagram or such yeah maybe more so like a timeline. Definitely one of those visual representations though and it seems like Ben tried to although I'm having a hard time following his work on that one but definitely there was a thought process there that he was trying to get across. His visual representation is a little pretty he drew a tree but it's not really like kind of getting to the the underlying math of it quite as much as I think some of the others are and I think that our students would as well. Okay so looking at these strategies I want you guys to take a take a second and look at the tile problem it's page six of the handout document. We're going to take only like three minutes but if you read through that problem how could you use one of our big bucket strategies to approach that problem? Those buckets being pictures splitting up the problem into maybe smaller pieces and labeling problem and like marking up and annotating. So we'll take like three minutes how could you use those and then we'll share out in a second right. My dog is now in my lap she wanted a little bit of attention but I can still type while you guys are sharing so this this tile problem what are some ways that you could use those strategies we'll just share I'm not going to take that but what are some ways that you could share those like use those strategies or encourage your students to use those strategies for this type of problem. While splitting it up into smaller problems like you could go from the 20 by 20 to maybe a 30 by 30 I don't know if that counts but okay go step by step. How would you encourage them to use pictures? Not a trick question might be kind of obvious. Okay Tracy saying pictures and tile paper yeah definitely. I think this is one where you could get all kids drawing pretty easily. Definitely drawing because I think some of them would might just look at it and be like oh well it's 20 by 20 now so 40 by 40 means double it. Yeah realizing with area you have to quadruple it so to make sure to see that you need four of those pictures not two of those pictures. Yeah definitely. I also think that annotation would be beneficial to figure out just all the pieces to the the actual problem would be beneficial for all the words. Yeah I definitely wanted to like write on mine the length of each half and the length of each quarter and things like that. Anything else? Right so we spent about the first half of this doing the problems and actually trying out the strategies on our own the second half of the of the session is thinking about how we could implement this into our own classrooms so both well no I'm not going to give anything away. So thinking about your own classrooms and what we've done for the last half hour what could you do what would you do what are some areas of concern you have with implementation because obviously it's a little bit trickier with kids than with us but what are some ideas that you guys have for implementation. Call it out to everyone. I think for problems like this I like to do what we did this summer and I do random cards to put them in groups and then let them just work. The other thing I did like this summer was everyone's Ronald they only gave us the problem up on the screen you didn't have paper so you had to sit and keep talking to each other to force them to start talking and I think it depends on the type of group of students you have if you need to use that strategy or if you could give them the actual handout for them to be marking up and I think that's just if it's later in the year you know that about your students if it's at the beginning of the year you don't really know that yet. And then just to piggyback on that to give it out loud to to actually read it out loud so that you they all start at the same place instead of the slower readers it takes a little bit why you know a little while for them to figure out what the problem is actually asking and everyone else has moved on to the problem solving strategies so just by reading it out loud then you put them all on the same page at once. I think these type of problems would be great at the beginning of the unit especially to kind of engage them for whatever the unit is going to be covering because these type of problems are engaging it forces them to talk to each other and and I agree with I believe it was Liz who mentioned about not giving them the certain informations or certain visuals because it's really amazing what some of the students can come up with on their own and a lot of times I end up learning new strategies of how to approach a problem from them. Definitely what about encouraging students to come up with these types of strategies like I didn't tell you guys any strategies to try at the beginning I just said here's this problem be ready to share your strategies how could you get students to come up with their own strategies and then how could you collect them as a class to get students to come up with a strategy or to share their strategy both I guess for me for sharing it's here's your piece on the board you have to go put something up and everyone's going to walk around and look at them right so they each have to kind of have ownership and go post some you have to put something on the wall because everybody's going to be able to see it so make sure they have something to come up with and I guess I do tend to guide students to be like when I don't know what to do I draw a picture to get them thinking but I don't know if that's too guiding in a situation like this I think it definitely depends on the kids with that one and I think it also depends on how long they've been working on it you know if they're stuck and they're frustrated they just need that prompt to to start so whatever the but if they're starting on their own you know kind of leave them alone I think if you had them on vertical white boards it'd be easy to just be like look over there what was their strategy or and you wouldn't have to do as much prodding to get them to share other thoughts or ideas on implementation you can always just give them a hint also in terms of you know remember remember some of the strategies that we've gone over before that we've used before in past classes and just sort of put that bug in their ear without giving them any any hints of what the strategies they could use yeah definitely one of the things I probably would add to the question is to have them justify their reasoning especially with the cat problem because some of them would say yes or no right because I guess or you know I estimated or something like that but then to justify it mathematically to be able to communicate it and explain it so like that one kid's problem where it was all kind of a mess um even asking him will somebody be able to follow your reasoning just by looking at your your strategy well I thought that Ben won yeah the tree was a mess but if you really read what it was like he was really thinking about it he said like oh she you know gets pregnant in June she can have the baby in August and then she can have the next litter and like he did try to get some tree diagram out obviously he took that literally but he did have three branches of three litters with six branches on each there was something there I wouldn't have been like he has no idea what he's doing he just needed a little push yeah yeah for sure right so we started to kind of mention them but what are some uh concerns that you have some areas uh like misconceptions that might come up about uh either these kinds of strategies or how to approach problems without being given a strategy like what are some uh concerns or misconceptions or um issues that you see arising with classroom implementation I think some students just kind of give up quickly they don't have that perseverance and something like this you do have to push through and and actually think so I think that that's uh I could see that being an issue in in my class other concerns around implementation they might ask like how do I do this so sometimes I'll have students say that well how do I how do I do this and just giving them the response of what do you think about it what do you think would work they're not they're not so happy with that response no they're definitely not but I give that response a lot I don't know would students I think that timing can be an issue like too much time versus too little time like if you give me too short of a time I get frustrated because like I could have gotten there and now you I was working and you stopped me and then it's the same on the other end like I'm stuck I don't want to do I'm bored and now I'm going to get bored and start doing something different so finding that right balance of time for struggle and time to intervene I think could be a challenge and it's different with every group too yeah it's all the groups are at different places so one of them may be struggling and frustrated and the other one needs more time so it's tricky if you have a whole class so then what are some ways that we could uh oh hi someone else joined hi Alex um so what are some ways that we could try to fix these these concerns what are some things that we could do either proactively or when they arise to help students with this idea of problem solving strategies so students give up I need to think through it we can start there so like what are some ways that that we can either proactively or in the moment help them to not give up I think it's important that students know it's okay to not get an answer right away like they have to understand the idea of struggling with something is legit and it happens but when they're getting frustrated to kind of give them a hint or point them in the right direction so they don't just shut them and that's kind of what Tracy was saying earlier you have to create that culture in your class from the get go what are some ways that you do that anyone what are some ways to create that start like create that culture of like it's okay to struggle it's okay to not give up it's okay to like come up with strategies and problem solve and all of that what are some ideas I verbally tell them I I mean it gives legitimacy to to their struggle but also encouragement to say no but you know push through and so when a student tries to take some more time to actually figure out the problem I praise them for that for that you know push versus getting the right answer you know I'm not so much concerned with the answer I I want to see how you got there is what I usually say and from the start of the year just doing problems where they're not going to get an answer right away so that they see that this is the norm and it's okay I read them that uh butterfly story about the kid the boy who gets the chrysalis and is watching it and the butterfly is struggling to get out just the butterfly is not going to get out and so he takes scissors it cuts open the cocoon and then the butterfly kind of flops down because it needed that struggle to to get out of the cocoon and and I tell them I want them to fly so I'm not going to cut out their cocoon they're going to have to figure out a way to get out that's good I like that oh my gosh can you share that story yeah sure sure I kind of right do the same thing I encourage them at the beginning like it's okay if we don't know something and it's okay if we don't get to the answer as long as we're working on getting moving forward in the problem that it's not the end result that I care about and that's what Aisha was saying too it's not the end result but showing me your problem solving in your process of what you're thinking about to get to wherever you got thank you Tracy thank you um so something that I do I do a lottery tickets for group work and so when they're working in the group if I hear a great strategy or if I hear some great math vocabulary I give tickets for that more so than if a kid raises their hands so like off the right answer I don't necessarily reward that right away so that kind of goes with praising for the struggle um something else that I did recently was I had kids make a instead of this do this uh so we made two different chart papers and kids would raise their hands and give things that they currently do when they are stuck or struggling uh so they said like giving up or going to the bathroom or uh getting up and walking around and so that was kind of cool to see them owning all those things that they do that we know they do when they are giving up but we sometimes wonder if they are aware that's why they're doing it and then we said okay so instead of raising your hand asking to go to the bathroom what could you do instead and they made this whole list of all these great ideas and they made them on their their own I just wrote them down I didn't prompt them in any way and then hanging those up in the room now when they get stuck I could be like okay instead of shutting down pick one of those things instead that you can do um and so that was kind of cool to see them make that list on their own and own those strategies uh themselves as opposed to me just telling them like hey instead of giving up you should try harder uh they kind of had their own ideas. Ariel what were some of the things that they came up with themselves for what they could do differently um they said ask their group members they said uh read the problem again they said like skip it and come back to it um and we made a point to differentiate between just skip it all together that was on the instead of doing this instead of skipping it all together you could skip it and then come back to it when you're ready uh so that was one of the good ones um I could take pictures of those and and show those out with you guys too I still have the posters I just didn't embrace them but yeah there were some there were some good ones on there they did not come up with the three kind of or they came up with annotating and marking up the problem that was a something they came up with they did not come up with like splitting up the problem or splitting up the group to work on parts of the problem uh and they don't really think they talked about pictures I'm a big fan of pictures and tables and they did not jump right to the that conclusion uh their answers to that one any other thoughts on uh implementation concerns or fixes this is more so uh extension but how do you use them uh these type of problems do you use them at the beginning of the unit the end of the unit and then depending on what you say so I don't use them nearly as much as I wish I did uh DC public schools has a really strict curriculum and I'm still trying to figure out how to best fit good problem solving uh into that um at my old school what I would do though is I would try to do them kind of towards the end as a review uh so kind of give the the big challenge problems and then use that as an opportunity to pull smaller groups to work on skills that they're missing uh which had pros and cons because it allowed me to time to pull those kids but then those kids didn't get to work on the challenge you problem so uh definitely not a perfect solution to that one but I'm curious what other people's answers are to your question as well see I tend to do them more at the beginning because I know that at the end I'm going to be catching up so it's like the only time that I can get myself to do them without feeling bad about it I tend to do one in the beginning and then uh and then branch off of that for the unit so it's just I was curious in terms of linking uh these type of problems specifically to the standards of that particular unit but my follow-up question in a sense then for for you doing it at the end of the unit do you find that students make those connections on their own of what they have learned not just in that unit but in general do you see them making those connections not as much as I would like uh there always are those couple of great kids that are like oh we learned this let me flip back in my notes and figure that out uh and then there are a lot of kids that don't necessarily see the direct connection with the content and the skills that we just learned and the challenging problem that goes along with it so ideas on that from you guys would be great as well and how to like kind of merge the two together we've been doing more investigations in class but they're not quite so open and did they definitely lead to what I want them to know by the end for example right now we're doing dilations and so we gave investigations about like well here's the pre-image here's the image what is the scale factor what did you discover about the coordinates for the scale factor and it was very much like hey look multiply by the scale factor on the coordinates as opposed to them really struggling with that on their own I do these kind of problems instead on what we call throwaway days like if there's a fire drill or if it's the last day before break um I I don't ever show movies I tell them I'm not movie teacher and so we do that instead of instead of nothing really so that way I feel like we're doing something they're engaged they're they're thinking um and it's that's what I do to deal with because we have curriculum too that I've got to get through and it's tricky to fit those in but on the throwaway days I feel like I can use those yeah definitely I feel like I'm the question person I sorry I always have questions but uh because I feel for you guys in terms of the curriculum because I'm in the same boat especially this year versus prior years um I don't I feel like I don't have as much autonomy as I used to and um when I made the switch over to our our middle school or junior high and so um how have your one do you share these type of tasks with your colleagues and two how do they take it do are they uh on board or they push back and say no I I'm sticking by the book for me I definitely share with um with my colleagues and we definitely all work together and we're on the same page about what we wish we could do and then it's just kind of like working together to figure out okay how do we actually do that I share just probably goes back a little too far that I always do these types of activities for the opener for like a unit so I've been teaching calculus this year so to find area under curve starting with integration I gave them just a random curve that I drew I don't know the equation of the graph I don't know what the actual area is and I said approximate the area and that's all I said to them now I had some students that have taken calculus before so they kind of knew and I had some that started looking in their books I'm like what are you doing no like just just this is all you get you don't have to look at anything else yeah and by making them do that I thought it helped them appreciate doing Riemann sums and then getting to the actual fundamental theorem of calculus and doing integrations but it helped them understand the limit definitions so much better because they could break it down and visually compare what they did with what they then had to do so I find I like to use those things to help them appreciate the quote-unquote short cuts of math to help them solve problems so that's and they've come to know that that's what I do at the beginning of every single unit so they kind of look forward to it they look forward to being in groups and working instruction okay so with 15 ish minutes left what are some next steps like are you going to go in on Monday and give the cat problem I or like you know how especially with this weird little couple of weeks between now and winter break what are some ways that we can try to implement within the next couple of weeks or maybe when they come back from break is a great time to reset and make sure that they're good with that sort of thing what are some ideas that you guys have for next steps for yourselves and for your kids I actually really like that cat problem I teach them at community college I do have a lot more flexibility in my curriculum and so one of the courses I'm redesigning for the campus which is math for elementary education so I basically am teaching six and seven grade math to them and so we're starting patterns and equations on Monday so I actually think your cat problem might be perfect to get them started with and the tile problem because we just finished geometry so thanks Ariel those are fantastic I'll be using I will definitely be using the cat problem we start exponential functions in our integrated one and our eighth grade math so that's like the perfect opener for that and especially my eighth grade math groups really need to work on persisting through problems so this would be a good conversation starter I get to my exponentials next semester so I'm definitely saving it in my back packet and likewise with me I'd like to give them that problem maybe halfway through the the unit just to see what they do or the end as you suggested but also I wouldn't mind giving them the tile problem in my algebra class or even my math eight class just for them to grapple with it to see how a pattern because that's actually what we're working on right now is how a pattern grows at a constant rate so just to have them visualize and see this as a different pattern with a different growth rate would be beneficial for I like how Marissa said she's putting in your back pocket I find my back pocket gets really overstuffed sometimes with things and I like forget what's back in there but yeah yeah yeah I agree with that too which is why I really like these reconnect sessions because there's definitely so many things that we talked about this summer that I went and then jumped into the curriculum and completely forgot about it and then we started talking about them and like oh yeah I remember learning about that I should totally do that so I definitely appreciate the coming back together and reminding and I'm hoping that next weekend's PCMI and DC helps with that as well any other thoughts questions concerns last ideas from anyone just thank you this has been wonderful great ideas and and thinking thank you for and reflection yeah this was great thank you so much Ariel I love those ideas I'm excited that I have a week's worth of lessons now but that's what I did I had like a whole worksheet of patterns but I like these even better just to get them struggling a little bit and then I dive like I said I give them something bigger dive into it and then I keep going so thank you so much I wish one or all of you had been my math teachers while I was going through school thanks so much for letting me listen then yeah of course thank you guys all for getting up early I appreciate it and I apparently didn't know time zones so sorry I was late okay not a problem and Tracy good job being up super early again so thanks our west coast representative I love it you're both west coasts thanks all right well thanks everybody thank you ladies this was fantastic we have another one in January you know after all our fun festivities of holidays and mid semesters and midterms come back so uh I'll be looking forward to setting up those emails and I hope to see everybody go thank you can you stay on for a minute I want to ask you something about I got a request from someone at a community college the other day I just want to run something by you absolutely and did we also want to take a screenshot of the pictures like we did last time was yeah that's a good idea do you want to do it uh do you know how to does anyone have a PC because I'm on a Mac right now and I always forget what the control something something I think Ian just took a photo the man shift for yeah man shift for thank you all right so uh command are we still doing our thumbs up or do you guys want to come up with a new little logo last time we all did thumbs up you know on our nose not it all right well but whatever you decide to do in three and two and one thank you awesome thanks everybody guys