{"version":"1.0.0","segments":[{"speaker":"Pam","startTime":0.99,"endTime":4.38,"body":"Hey, fellow mathematicians!"},{"speaker":"Pam","startTime":0.99,"endTime":4.38,"body":"Welcome to the podcast where"},{"speaker":"Pam","startTime":4.38,"endTime":7.2,"body":"Math is Figure-Out-Able. I'm"},{"speaker":"Pam","startTime":4.38,"endTime":7.2,"body":"Pam."},{"speaker":"Kim","startTime":7.53,"endTime":8.25,"body":"And I'm Kim."},{"speaker":"Pam","startTime":8.49,"endTime":12.0,"body":"And you found a place where"},{"speaker":"Pam","startTime":8.49,"endTime":12.0,"body":"math is not about memorizing and"},{"speaker":"Pam","startTime":12.0,"endTime":15.6,"body":"mimicking, waiting to be told or"},{"speaker":"Pam","startTime":12.0,"endTime":15.6,"body":"shown what to do. But y'all it's"},{"speaker":"Pam","startTime":15.6,"endTime":19.44,"body":"about making sense of problems,"},{"speaker":"Pam","startTime":15.6,"endTime":19.44,"body":"noticing patterns, and reasoning"},{"speaker":"Pam","startTime":19.44,"endTime":22.5,"body":"using mathematical"},{"speaker":"Pam","startTime":19.44,"endTime":22.5,"body":"relationships. We can mentor"},{"speaker":"Pam","startTime":22.5,"endTime":26.04,"body":"students to think and reason"},{"speaker":"Pam","startTime":22.5,"endTime":26.04,"body":"like mathematicians. Not only"},{"speaker":"Pam","startTime":26.04,"endTime":29.85,"body":"are algorithms not particularly"},{"speaker":"Pam","startTime":26.04,"endTime":29.85,"body":"helpful in teaching mathematics,"},{"speaker":"Pam","startTime":30.09,"endTime":33.66,"body":"but rotely repeating steps"},{"speaker":"Pam","startTime":30.09,"endTime":33.66,"body":"actually keeps students from"},{"speaker":"Pam","startTime":33.66,"endTime":35.55,"body":"being the mathematicians they"},{"speaker":"Pam","startTime":33.66,"endTime":35.55,"body":"can be."},{"speaker":"Kim","startTime":36.39,"endTime":39.45,"body":"It is so fun to hear from"},{"speaker":"Kim","startTime":36.39,"endTime":39.45,"body":"listeners. Ya'll, do more of"},{"speaker":"Kim","startTime":39.45,"endTime":43.53,"body":"that, please. A few weeks ago,"},{"speaker":"Kim","startTime":39.45,"endTime":43.53,"body":"we did a short series about"},{"speaker":"Kim","startTime":43.53,"endTime":46.53,"body":"fraction division, which was"},{"speaker":"Kim","startTime":43.53,"endTime":46.53,"body":"super fun and has been well"},{"speaker":"Kim","startTime":46.53,"endTime":51.99,"body":"received. After episode 142, we"},{"speaker":"Kim","startTime":46.53,"endTime":51.99,"body":"heard from Christina Stevens,"},{"speaker":"Kim","startTime":52.02,"endTime":56.25,"body":"and she said, \"I love that this"},{"speaker":"Kim","startTime":52.02,"endTime":56.25,"body":"makes sense to me. I trusted the"},{"speaker":"Kim","startTime":56.25,"endTime":60.42,"body":"roles, but I admittedly didn't"},{"speaker":"Kim","startTime":56.25,"endTime":60.42,"body":"get the why. Hooray! I get it!\""},{"speaker":"Kim","startTime":60.57,"endTime":63.51,"body":"That is super exciting, right?!"},{"speaker":"Kim","startTime":60.57,"endTime":63.51,"body":"And it's exactly what we're"},{"speaker":"Kim","startTime":63.51,"endTime":66.72,"body":"hoping for our students. For"},{"speaker":"Kim","startTime":63.51,"endTime":66.72,"body":"adult mathematicians, it's kind"},{"speaker":"Kim","startTime":66.72,"endTime":69.66,"body":"of why we do the podcast. So,"},{"speaker":"Kim","startTime":66.72,"endTime":69.66,"body":"well done, Christina!"},{"speaker":"Pam","startTime":69.72,"endTime":72.6,"body":"Yeah. Nice. It was super"},{"speaker":"Pam","startTime":69.72,"endTime":72.6,"body":"good to get so much feedback,"},{"speaker":"Pam","startTime":72.63,"endTime":73.26,"body":"especially (unclear)..."},{"speaker":"Kim","startTime":73.26,"endTime":73.35,"body":"Yeah."},{"speaker":"Pam","startTime":73.41,"endTime":76.71,"body":"...Christina and everybody"},{"speaker":"Pam","startTime":73.41,"endTime":76.71,"body":"about how much fun everybody was"},{"speaker":"Pam","startTime":76.71,"endTime":79.2,"body":"having. Somebody was telling me"},{"speaker":"Pam","startTime":76.71,"endTime":79.2,"body":"they were doing it in their car"},{"speaker":"Pam","startTime":79.2,"endTime":82.14,"body":"with their nephew, and they were"},{"speaker":"Pam","startTime":79.2,"endTime":82.14,"body":"like, \"It's math in the car!\""},{"speaker":"Pam","startTime":82.14,"endTime":83.01,"body":"I'm like, \"Alright, that's"},{"speaker":"Pam","startTime":82.14,"endTime":83.01,"body":"awesome.\" Yeah, nice."},{"speaker":"Kim","startTime":85.08,"endTime":88.32,"body":"We did get a listener say"},{"speaker":"Kim","startTime":85.08,"endTime":88.32,"body":"don't listen to the podcast in"},{"speaker":"Kim","startTime":88.32,"endTime":92.04,"body":"the car because we often do"},{"speaker":"Kim","startTime":88.32,"endTime":92.04,"body":"Problem Strings and, you know,"},{"speaker":"Kim","startTime":92.04,"endTime":94.74,"body":"you might want to write"},{"speaker":"Kim","startTime":92.04,"endTime":94.74,"body":"something down. Anyway, today"},{"speaker":"Kim","startTime":94.74,"endTime":94.77,"body":"(unclear)."},{"speaker":"Pam","startTime":94.77,"endTime":96.81,"body":"Don't write while you're in"},{"speaker":"Pam","startTime":94.77,"endTime":96.81,"body":"your car. Don't do it. No"},{"speaker":"Pam","startTime":96.81,"endTime":98.85,"body":"writing while you're in your"},{"speaker":"Pam","startTime":96.81,"endTime":98.85,"body":"car. Not while you're driving"},{"speaker":"Pam","startTime":98.85,"endTime":99.57,"body":"anyway. Yeah."},{"speaker":"Kim","startTime":100.14,"endTime":104.31,"body":"Yeah. Okay, so today, we're"},{"speaker":"Kim","startTime":100.14,"endTime":104.31,"body":"diving into what we think is the"},{"speaker":"Kim","startTime":104.31,"endTime":109.05,"body":"most undervalued but incredibly"},{"speaker":"Kim","startTime":104.31,"endTime":109.05,"body":"important operation. So, we hear"},{"speaker":"Kim","startTime":109.05,"endTime":113.01,"body":"a lot about multiplication, and"},{"speaker":"Kim","startTime":109.05,"endTime":113.01,"body":"facts, and division, but not as"},{"speaker":"Kim","startTime":113.01,"endTime":116.97,"body":"often do we hear about"},{"speaker":"Kim","startTime":113.01,"endTime":116.97,"body":"subtraction. And wow, it's so"},{"speaker":"Kim","startTime":116.97,"endTime":120.78,"body":"important in so many ways. So,"},{"speaker":"Kim","startTime":116.97,"endTime":120.78,"body":"let's kick off a few episodes."},{"speaker":"Kim","startTime":120.81,"endTime":124.8,"body":"You know, because we have a ton"},{"speaker":"Kim","startTime":120.81,"endTime":124.8,"body":"to say about this. Few episodes"},{"speaker":"Kim","startTime":124.83,"endTime":126.27,"body":"about subtraction."},{"speaker":"Pam","startTime":127.23,"endTime":130.86,"body":"Now, ya'll, don't stop"},{"speaker":"Pam","startTime":127.23,"endTime":130.86,"body":"listening just because we said"},{"speaker":"Pam","startTime":130.89,"endTime":136.59,"body":"\"subtraction\", you high school"},{"speaker":"Pam","startTime":130.89,"endTime":136.59,"body":"teachers. Hang on. Hang on. Or"},{"speaker":"Pam","startTime":136.59,"endTime":138.75,"body":"anybody who's like, \"Oh,"},{"speaker":"Pam","startTime":136.59,"endTime":138.75,"body":"subtraction, you know, like"},{"speaker":"Pam","startTime":138.75,"endTime":141.9,"body":"that's... You just do the steps,"},{"speaker":"Pam","startTime":138.75,"endTime":141.9,"body":"and you're done.\" Hang on"},{"speaker":"Pam","startTime":141.9,"endTime":144.9,"body":"because there's some important"},{"speaker":"Pam","startTime":141.9,"endTime":144.9,"body":"things that we're going to talk"},{"speaker":"Pam","startTime":144.9,"endTime":148.62,"body":"about with subtraction that will"},{"speaker":"Pam","startTime":144.9,"endTime":148.62,"body":"impact lots and lots of"},{"speaker":"Pam","startTime":148.65,"endTime":153.72,"body":"mathematics. So, Kim, we often"},{"speaker":"Pam","startTime":148.65,"endTime":153.72,"body":"talk about the routine I Have,"},{"speaker":"Pam","startTime":153.75,"endTime":154.53,"body":"You Need."},{"speaker":"Kim","startTime":155.37,"endTime":155.4,"body":"Mmhmm."},{"speaker":"Pam","startTime":155.46,"endTime":159.18,"body":"And so, if I say something"},{"speaker":"Pam","startTime":155.46,"endTime":159.18,"body":"like, \"I have 87, what do you"},{"speaker":"Pam","startTime":159.18,"endTime":160.71,"body":"need to make 100?\" Go ahead."},{"speaker":"Kim","startTime":160.74,"endTime":161.31,"body":"Thirteen."},{"speaker":"Pam","startTime":161.76,"endTime":166.95,"body":"Thirteen. So, 87 and 13 is"},{"speaker":"Pam","startTime":161.76,"endTime":166.95,"body":"100. What does that have to do"},{"speaker":"Pam","startTime":166.95,"endTime":170.04,"body":"with subtraction? (unclear)"},{"speaker":"Kim","startTime":170.19,"endTime":170.7,"body":"So I..."},{"speaker":"Pam","startTime":170.7,"endTime":171.45,"body":"Yeah, go go."},{"speaker":"Kim","startTime":171.48,"endTime":174.93,"body":"Yeah, I know that we often"},{"speaker":"Kim","startTime":171.48,"endTime":174.93,"body":"talk about it. And because of"},{"speaker":"Kim","startTime":174.93,"endTime":178.14,"body":"the way that you say, \"I"},{"speaker":"Kim","startTime":174.93,"endTime":178.14,"body":"Have...\" and then the kids think"},{"speaker":"Kim","startTime":178.14,"endTime":183.93,"body":"about \"You Need...\" it feels a"},{"speaker":"Kim","startTime":178.14,"endTime":183.93,"body":"lot like addition. But this"},{"speaker":"Kim","startTime":183.93,"endTime":189.3,"body":"routine came about because"},{"speaker":"Kim","startTime":183.93,"endTime":189.3,"body":"subtraction across zeros. Early,"},{"speaker":"Kim","startTime":189.3,"endTime":192.69,"body":"early, early days. Subtraction"},{"speaker":"Kim","startTime":189.3,"endTime":192.69,"body":"across zeros. I was like, \"That"},{"speaker":"Kim","startTime":192.69,"endTime":197.16,"body":"is awful. That is a horrible"},{"speaker":"Kim","startTime":192.69,"endTime":197.16,"body":"thing. And if we could just get"},{"speaker":"Kim","startTime":197.16,"endTime":199.8,"body":"kids to know their partners of"},{"speaker":"Kim","startTime":197.16,"endTime":199.8,"body":"100, then that would never be"},{"speaker":"Kim","startTime":199.8,"endTime":204.0,"body":"something they ever had to deal"},{"speaker":"Kim","startTime":199.8,"endTime":204.0,"body":"with.\" So, it very much came out"},{"speaker":"Kim","startTime":204.0,"endTime":204.75,"body":"of a need for"},{"speaker":"Kim","startTime":204.75,"endTime":205.83,"body":"work on subtraction."},{"speaker":"Pam","startTime":206.07,"endTime":207.84,"body":"Yeah. And so, I have to"},{"speaker":"Pam","startTime":206.07,"endTime":207.84,"body":"interrupt you really quickly"},{"speaker":"Pam","startTime":207.84,"endTime":211.29,"body":"because this is just reminding"},{"speaker":"Pam","startTime":207.84,"endTime":211.29,"body":"me of a story when... And I may"},{"speaker":"Pam","startTime":211.29,"endTime":213.15,"body":"have said this on the podcast"},{"speaker":"Pam","startTime":211.29,"endTime":213.15,"body":"before, but I'm going to tell it"},{"speaker":"Pam","startTime":213.15,"endTime":216.27,"body":"again. I was doing a lot of"},{"speaker":"Pam","startTime":213.15,"endTime":216.27,"body":"professional learning with"},{"speaker":"Pam","startTime":216.27,"endTime":218.94,"body":"secondary teachers. We're doing"},{"speaker":"Pam","startTime":216.27,"endTime":218.94,"body":"graphing calculators and, you"},{"speaker":"Pam","startTime":218.94,"endTime":222.09,"body":"know, kind of higher math and"},{"speaker":"Pam","startTime":218.94,"endTime":222.09,"body":"everything. And at one point..."},{"speaker":"Pam","startTime":222.54,"endTime":225.27,"body":"Well, when I got really"},{"speaker":"Pam","startTime":222.54,"endTime":225.27,"body":"interested in younger math"},{"speaker":"Pam","startTime":225.27,"endTime":228.57,"body":"because of my kid and"},{"speaker":"Pam","startTime":225.27,"endTime":228.57,"body":"everything, my kid's school...in"},{"speaker":"Pam","startTime":228.6,"endTime":230.97,"body":"fact, your teaching partner at"},{"speaker":"Pam","startTime":228.6,"endTime":230.97,"body":"the time, Danna Fincher...asked"},{"speaker":"Pam","startTime":231.93,"endTime":233.58,"body":"me if I could do some"},{"speaker":"Pam","startTime":231.93,"endTime":233.58,"body":"professional learning with the"},{"speaker":"Pam","startTime":233.58,"endTime":235.92,"body":"teachers. And I was like,"},{"speaker":"Pam","startTime":233.58,"endTime":235.92,"body":"\"Yeah!\" Like, \"I would really"},{"speaker":"Pam","startTime":235.92,"endTime":241.53,"body":"like to do that. But what would"},{"speaker":"Pam","startTime":235.92,"endTime":241.53,"body":"we do?\" Like, I knew so little"},{"speaker":"Pam","startTime":241.53,"endTime":244.92,"body":"about younger math. And, you"},{"speaker":"Pam","startTime":241.53,"endTime":244.92,"body":"know, I was really clear what I"},{"speaker":"Pam","startTime":244.92,"endTime":247.71,"body":"could help teachers do with"},{"speaker":"Pam","startTime":244.92,"endTime":247.71,"body":"graphing calculators. Wasn't"},{"speaker":"Pam","startTime":247.71,"endTime":249.87,"body":"going to do graphing calculators"},{"speaker":"Pam","startTime":247.71,"endTime":249.87,"body":"with, you know, K-5 educators."},{"speaker":"Pam","startTime":249.87,"endTime":252.6,"body":"And I was like, \"What would be a"},{"speaker":"Pam","startTime":249.87,"endTime":252.6,"body":"thing?\" You know like, \"What do"},{"speaker":"Pam","startTime":252.6,"endTime":255.24,"body":"you guys want to learn? What do"},{"speaker":"Pam","startTime":252.6,"endTime":255.24,"body":"you need help with?\" And in all"},{"speaker":"Pam","startTime":255.24,"endTime":259.41,"body":"seriousness, the response was,"},{"speaker":"Pam","startTime":255.24,"endTime":259.41,"body":"\"Subtraction across zeros!\" And"},{"speaker":"Pam","startTime":259.41,"endTime":263.55,"body":"I said, \"Okay. Absolutely, we"},{"speaker":"Pam","startTime":259.41,"endTime":263.55,"body":"will do that. What is that?\""},{"speaker":"Pam","startTime":264.09,"endTime":267.66,"body":"Like, I had no idea. I literally"},{"speaker":"Pam","startTime":264.09,"endTime":267.66,"body":"was like, \"What does that mean?\""},{"speaker":"Pam","startTime":267.66,"endTime":270.42,"body":"So, what is it? Kim, tell us."},{"speaker":"Pam","startTime":267.66,"endTime":270.42,"body":"What does it mean? When you say"},{"speaker":"Pam","startTime":270.42,"endTime":272.04,"body":"subtraction across zeros, what"},{"speaker":"Pam","startTime":270.42,"endTime":272.04,"body":"does that mean?"},{"speaker":"Kim","startTime":272.7,"endTime":278.61,"body":"Oh, gosh. So, when you have"},{"speaker":"Kim","startTime":272.7,"endTime":278.61,"body":"100. So, if you are teaching"},{"speaker":"Kim","startTime":278.61,"endTime":282.51,"body":"with an algorithm, subtraction"},{"speaker":"Kim","startTime":278.61,"endTime":282.51,"body":"across zeros is always a beast"},{"speaker":"Kim","startTime":282.51,"endTime":287.19,"body":"because you would write 100"},{"speaker":"Kim","startTime":282.51,"endTime":287.19,"body":"minus 87 below it. And then"},{"speaker":"Kim","startTime":287.43,"endTime":287.46,"body":"(unclear)."},{"speaker":"Pam","startTime":287.46,"endTime":289.8,"body":"Line it up, right? Make"},{"speaker":"Pam","startTime":287.46,"endTime":289.8,"body":"sure you line it up correctly."},{"speaker":"Pam","startTime":290.13,"endTime":290.16,"body":"Mmhmm."},{"speaker":"Kim","startTime":290.52,"endTime":294.57,"body":"Yep. Super abstract for"},{"speaker":"Kim","startTime":290.52,"endTime":294.57,"body":"kids because you're doing things"},{"speaker":"Kim","startTime":294.57,"endTime":298.68,"body":"like 0 minus 7. And you"},{"speaker":"Kim","startTime":294.57,"endTime":298.68,"body":"\"can't...\" I'm air quoting"},{"speaker":"Kim","startTime":298.68,"endTime":302.73,"body":"\"can't do that\". So, then, you"},{"speaker":"Kim","startTime":298.68,"endTime":302.73,"body":"have to go to the 10's to"},{"speaker":"Kim","startTime":302.73,"endTime":305.94,"body":"borrow. Oh, wait. There's none"},{"speaker":"Kim","startTime":302.73,"endTime":305.94,"body":"there, and so you keep going."},{"speaker":"Kim","startTime":305.94,"endTime":309.39,"body":"And there's all this like"},{"speaker":"Kim","startTime":305.94,"endTime":309.39,"body":"crossing out and, \"Is it a 10 or"},{"speaker":"Kim","startTime":309.39,"endTime":315.63,"body":"is it a 9.\" And it's so little"},{"speaker":"Kim","startTime":309.39,"endTime":315.63,"body":"to do with place value because"},{"speaker":"Kim","startTime":315.63,"endTime":318.42,"body":"kids are just following steps."},{"speaker":"Kim","startTime":315.63,"endTime":318.42,"body":"They don't have a clue most of"},{"speaker":"Kim","startTime":318.42,"endTime":319.86,"body":"the time, and they're just"},{"speaker":"Kim","startTime":318.42,"endTime":319.86,"body":"trying to mimic what's"},{"speaker":"Kim","startTime":319.86,"endTime":324.33,"body":"happening. And so it always. If"},{"speaker":"Kim","startTime":319.86,"endTime":324.33,"body":"you are a third grade teacher"},{"speaker":"Kim","startTime":324.33,"endTime":327.21,"body":"starting to talk about"},{"speaker":"Kim","startTime":324.33,"endTime":327.21,"body":"subtraction across zeros, you"},{"speaker":"Kim","startTime":327.21,"endTime":331.86,"body":"know the pain of mimicking those"},{"speaker":"Kim","startTime":327.21,"endTime":331.86,"body":"steps (unclear)."},{"speaker":"Pam","startTime":331.86,"endTime":332.67,"body":"Lots and lots of pain,"},{"speaker":"Pam","startTime":331.86,"endTime":332.67,"body":"yeah."},{"speaker":"Kim","startTime":332.67,"endTime":336.3,"body":"(unclear) and demonstrating"},{"speaker":"Kim","startTime":332.67,"endTime":336.3,"body":"100 million times. And then,"},{"speaker":"Kim","startTime":336.3,"endTime":339.99,"body":"when you get to maybe later"},{"speaker":"Kim","startTime":336.3,"endTime":339.99,"body":"third grade and it's 1,000,"},{"speaker":"Kim","startTime":340.02,"endTime":344.16,"body":"you're like, \"Oh, here we go"},{"speaker":"Kim","startTime":340.02,"endTime":344.16,"body":"again.\" So, yeah, that was. You"},{"speaker":"Kim","startTime":344.16,"endTime":348.39,"body":"know, in my very first year when"},{"speaker":"Kim","startTime":344.16,"endTime":348.39,"body":"I taught pretty traditionally,"},{"speaker":"Kim","startTime":348.45,"endTime":351.54,"body":"it was like, \"Oh, gosh. I can't."},{"speaker":"Kim","startTime":348.45,"endTime":351.54,"body":"This is not working.\""},{"speaker":"Pam","startTime":351.84,"endTime":354.9,"body":"You know, and you just said"},{"speaker":"Pam","startTime":351.84,"endTime":354.9,"body":"that there's no place value as"},{"speaker":"Pam","startTime":354.9,"endTime":355.59,"body":"they're doing that."},{"speaker":"Kim","startTime":356.13,"endTime":356.94,"body":"Not for the kids."},{"speaker":"Pam","startTime":357.18,"endTime":359.07,"body":"Maybe I'll tweak it to say"},{"speaker":"Pam","startTime":357.18,"endTime":359.07,"body":"there's no place value"},{"speaker":"Pam","startTime":359.07,"endTime":362.49,"body":"understanding happening because"},{"speaker":"Pam","startTime":359.07,"endTime":362.49,"body":"there's a lot of value embedded"},{"speaker":"Pam","startTime":362.49,"endTime":363.06,"body":"in that algorithm."},{"speaker":"Kim","startTime":363.06,"endTime":364.14,"body":"Oh, one hundred percent."},{"speaker":"Kim","startTime":363.06,"endTime":364.14,"body":"Yeah."},{"speaker":"Pam","startTime":364.14,"endTime":367.2,"body":"And that's why it works."},{"speaker":"Pam","startTime":364.14,"endTime":367.2,"body":"But the understanding, we try."},{"speaker":"Pam","startTime":367.23,"endTime":369.69,"body":"You know, \"Okay, you're in the"},{"speaker":"Pam","startTime":367.23,"endTime":369.69,"body":"10's column, so you're borrowing"},{"speaker":"Pam","startTime":369.69,"endTime":371.61,"body":"a 10, and you're putting the 10"},{"speaker":"Pam","startTime":369.69,"endTime":371.61,"body":"over here, and so you're"},{"speaker":"Pam","startTime":371.61,"endTime":374.31,"body":"crossing that out. It's becoming"},{"speaker":"Pam","startTime":371.61,"endTime":374.31,"body":"a 9, but you're in the 10's.\""},{"speaker":"Pam","startTime":374.64,"endTime":376.77,"body":"\"What's a 10 in the 10's...\""},{"speaker":"Pam","startTime":374.64,"endTime":376.77,"body":"Like it's..."},{"speaker":"Kim","startTime":376.77,"endTime":377.22,"body":"Yeah."},{"speaker":"Pam","startTime":377.22,"endTime":379.35,"body":"It's so abstract."},{"speaker":"Pam","startTime":377.22,"endTime":379.35,"body":"Especially for those young"},{"speaker":"Pam","startTime":379.35,"endTime":380.01,"body":"learners. Yeah."},{"speaker":"Kim","startTime":380.85,"endTime":381.21,"body":"Yeah."},{"speaker":"Pam","startTime":381.42,"endTime":386.49,"body":"So, when I just said, \"I"},{"speaker":"Pam","startTime":381.42,"endTime":386.49,"body":"have 87, what do you need?\" It's"},{"speaker":"Pam","startTime":386.49,"endTime":389.52,"body":"kind of like that missing add in"},{"speaker":"Pam","startTime":386.49,"endTime":389.52,"body":"problem, right? 87 plus"},{"speaker":"Pam","startTime":389.52,"endTime":393.78,"body":"something is 100. And then, you"},{"speaker":"Pam","startTime":389.52,"endTime":393.78,"body":"said 13. But it's also the"},{"speaker":"Pam","startTime":393.78,"endTime":398.49,"body":"answer to 100 minus 87 is 13. Or"},{"speaker":"Pam","startTime":393.78,"endTime":398.49,"body":"we could turn it around 100"},{"speaker":"Pam","startTime":398.49,"endTime":404.19,"body":"minus 13 is 87. So, subtraction"},{"speaker":"Pam","startTime":398.49,"endTime":404.19,"body":"across zeros kind of becomes a"},{"speaker":"Pam","startTime":404.19,"endTime":409.32,"body":"non issue if we get students"},{"speaker":"Pam","startTime":404.19,"endTime":409.32,"body":"working with these partners of"},{"speaker":"Pam","startTime":409.65,"endTime":412.65,"body":"100, partners of 1,000. And you"},{"speaker":"Pam","startTime":409.65,"endTime":412.65,"body":"might be like, \"Well, do they"},{"speaker":"Pam","startTime":412.65,"endTime":415.5,"body":"need to have partners of 200?\" I"},{"speaker":"Pam","startTime":412.65,"endTime":415.5,"body":"mean, if they have partners of"},{"speaker":"Pam","startTime":415.53,"endTime":419.22,"body":"100, then they're already"},{"speaker":"Pam","startTime":415.53,"endTime":419.22,"body":"thinking about that 100 in that"},{"speaker":"Pam","startTime":419.22,"endTime":424.05,"body":"200 to use for subtraction."},{"speaker":"Pam","startTime":419.22,"endTime":424.05,"body":"Maybe we'll parse that a little"},{"speaker":"Pam","startTime":424.05,"endTime":430.53,"body":"bit more later. But it can be"},{"speaker":"Pam","startTime":424.05,"endTime":430.53,"body":"super helpful to have kids own"},{"speaker":"Pam","startTime":430.53,"endTime":433.62,"body":"those partners, and we can use"},{"speaker":"Pam","startTime":430.53,"endTime":433.62,"body":"the routine I Have, You Need to"},{"speaker":"Pam","startTime":433.62,"endTime":437.64,"body":"build those. And like you said,"},{"speaker":"Pam","startTime":433.62,"endTime":437.64,"body":"the reason that we started to"},{"speaker":"Pam","startTime":437.64,"endTime":441.36,"body":"work on that, it was because"},{"speaker":"Pam","startTime":437.64,"endTime":441.36,"body":"subtraction was difficult. But"},{"speaker":"Pam","startTime":441.36,"endTime":445.44,"body":"not only just subtraction"},{"speaker":"Pam","startTime":441.36,"endTime":445.44,"body":"problems. The more that you and"},{"speaker":"Pam","startTime":445.44,"endTime":450.45,"body":"I built our numeracy for"},{"speaker":"Pam","startTime":445.44,"endTime":450.45,"body":"multiplication and division, we"},{"speaker":"Pam","startTime":450.45,"endTime":455.52,"body":"also started to see that the"},{"speaker":"Pam","startTime":450.45,"endTime":455.52,"body":"sticking point, the tricky part"},{"speaker":"Pam","startTime":455.52,"endTime":458.13,"body":"came not so much in thinking"},{"speaker":"Pam","startTime":455.52,"endTime":458.13,"body":"about the multiplication and"},{"speaker":"Pam","startTime":458.13,"endTime":463.35,"body":"division for kids, but in doing"},{"speaker":"Pam","startTime":458.13,"endTime":463.35,"body":"the subtraction. Like, let's"},{"speaker":"Pam","startTime":463.35,"endTime":466.29,"body":"give an example maybe. So, if I"},{"speaker":"Pam","startTime":463.35,"endTime":466.29,"body":"were to say something like, Kim,"},{"speaker":"Pam","startTime":466.29,"endTime":470.22,"body":"I want to find 99 times 87."},{"speaker":"Pam","startTime":466.29,"endTime":470.22,"body":"We've been just dealing with 87."},{"speaker":"Pam","startTime":470.22,"endTime":470.73,"body":"Let's stay with 87's."},{"speaker":"Kim","startTime":470.82,"endTime":471.15,"body":"Yeah."},{"speaker":"Pam","startTime":471.96,"endTime":475.38,"body":"So, if I said ninety-nine"},{"speaker":"Pam","startTime":471.96,"endTime":475.38,"body":"87's, what are you thinking"},{"speaker":"Pam","startTime":475.38,"endTime":475.71,"body":"about?"},{"speaker":"Kim","startTime":476.34,"endTime":480.09,"body":"It's really close to 100,"},{"speaker":"Kim","startTime":476.34,"endTime":480.09,"body":"so I'm going to say one hundred"},{"speaker":"Kim","startTime":480.36,"endTime":490.32,"body":"87's is 8,700. But then, because"},{"speaker":"Kim","startTime":480.36,"endTime":490.32,"body":"I have one too many 87's, then I"},{"speaker":"Kim","startTime":490.32,"endTime":494.16,"body":"have to subtract 99 from the"},{"speaker":"Kim","startTime":490.32,"endTime":494.16,"body":"8,700."},{"speaker":"Pam","startTime":494.61,"endTime":494.88,"body":"Wait."},{"speaker":"Kim","startTime":494.91,"endTime":495.51,"body":"So, then..."},{"speaker":"Pam","startTime":496.11,"endTime":496.92,"body":"Wait, subtract (unclear)."},{"speaker":"Kim","startTime":496.95,"endTime":498.09,"body":"I'm sorry, subtract 87."},{"speaker":"Pam","startTime":499.11,"endTime":500.64,"body":"You only want 99. You only"},{"speaker":"Pam","startTime":499.11,"endTime":500.64,"body":"want ninety-nine 87's."},{"speaker":"Kim","startTime":501.48,"endTime":501.9,"body":"Yeah."},{"speaker":"Pam","startTime":501.99,"endTime":504.36,"body":"So, you're going to"},{"speaker":"Pam","startTime":501.99,"endTime":504.36,"body":"subtract. Yeah, subtract 87 from"},{"speaker":"Pam","startTime":504.63,"endTime":505.38,"body":"8,700. Mmhmm."},{"speaker":"Kim","startTime":505.59,"endTime":509.13,"body":"So, again, if you're"},{"speaker":"Kim","startTime":505.59,"endTime":509.13,"body":"teaching pretty traditionally,"},{"speaker":"Kim","startTime":509.25,"endTime":512.61,"body":"even if you're starting to"},{"speaker":"Kim","startTime":509.25,"endTime":512.61,"body":"dabble, then your kids would be"},{"speaker":"Kim","startTime":512.61,"endTime":517.11,"body":"thinking, \"Oh, I have 8,700"},{"speaker":"Kim","startTime":512.61,"endTime":517.11,"body":"minus 87.\" And there's all those"},{"speaker":"Kim","startTime":517.11,"endTime":521.04,"body":"zeros again, right? So, that's"},{"speaker":"Kim","startTime":517.11,"endTime":521.04,"body":"not what I would do. I would"},{"speaker":"Kim","startTime":521.04,"endTime":528.48,"body":"subtract 100 from 8,700, which"},{"speaker":"Kim","startTime":521.04,"endTime":528.48,"body":"would be 8,600. And then, I"},{"speaker":"Kim","startTime":528.48,"endTime":533.49,"body":"subtracted too much. 100 was too"},{"speaker":"Kim","startTime":528.48,"endTime":533.49,"body":"much, so then I add back 13"},{"speaker":"Kim","startTime":533.52,"endTime":535.11,"body":"because it was 13 too much."},{"speaker":"Pam","startTime":536.1,"endTime":540.15,"body":"Because you know the"},{"speaker":"Pam","startTime":536.1,"endTime":540.15,"body":"partner of 87 to 100 is 13."},{"speaker":"Kim","startTime":540.15,"endTime":540.21,"body":"Yeah."},{"speaker":"Pam","startTime":540.63,"endTime":544.95,"body":"Absolutely. So, now, you've"},{"speaker":"Pam","startTime":540.63,"endTime":544.95,"body":"added back 13 to 8,600. And"},{"speaker":"Pam","startTime":544.95,"endTime":549.09,"body":"that's the hard question, right?"},{"speaker":"Pam","startTime":544.95,"endTime":549.09,"body":"What is 8,600 and 13. Ha, ha,"},{"speaker":"Pam","startTime":549.09,"endTime":549.15,"body":"ha."},{"speaker":"Kim","startTime":549.24,"endTime":549.3,"body":"Yeah."},{"speaker":"Pam","startTime":549.3,"endTime":549.78,"body":"It's 8,613."},{"speaker":"Kim","startTime":549.87,"endTime":550.26,"body":"Yep."},{"speaker":"Pam","startTime":550.56,"endTime":554.01,"body":"And so, that's a great"},{"speaker":"Pam","startTime":550.56,"endTime":554.01,"body":"strategy for multiplying. Use"},{"speaker":"Pam","startTime":554.01,"endTime":555.36,"body":"that Over strategy. What were"},{"speaker":"Pam","startTime":554.01,"endTime":555.36,"body":"you going to say?"},{"speaker":"Kim","startTime":555.36,"endTime":555.75,"body":"Yep."},{"speaker":"Kim","startTime":556.02,"endTime":559.47,"body":"Yeah, I was going to say, you"},{"speaker":"Kim","startTime":556.02,"endTime":559.47,"body":"know I... You've mentioned more"},{"speaker":"Kim","startTime":559.47,"endTime":563.97,"body":"than one time that I'm a super"},{"speaker":"Kim","startTime":559.47,"endTime":563.97,"body":"big Over fan. And I like to"},{"speaker":"Kim","startTime":563.97,"endTime":567.42,"body":"think \"more than\" and backup in"},{"speaker":"Kim","startTime":563.97,"endTime":567.42,"body":"lots of operations. And so,"},{"speaker":"Kim","startTime":567.42,"endTime":572.91,"body":"subtraction, has become really"},{"speaker":"Kim","startTime":567.42,"endTime":572.91,"body":"important, particularly in that"},{"speaker":"Kim","startTime":572.94,"endTime":573.66,"body":"strategy."},{"speaker":"Pam","startTime":574.02,"endTime":577.26,"body":"Yeah. Nice. Nice. And I"},{"speaker":"Pam","startTime":574.02,"endTime":577.26,"body":"might just... Another way to"},{"speaker":"Pam","startTime":577.26,"endTime":582.57,"body":"kind of think about that 8,700"},{"speaker":"Pam","startTime":577.26,"endTime":582.57,"body":"minus 87 is... And maybe this is"},{"speaker":"Pam","startTime":582.57,"endTime":585.03,"body":"just kind of another way of"},{"speaker":"Pam","startTime":582.57,"endTime":585.03,"body":"envisioning. But I just wrote"},{"speaker":"Pam","startTime":585.03,"endTime":588.48,"body":"down on my paper a number line"},{"speaker":"Pam","startTime":585.03,"endTime":588.48,"body":"where I have 8,600 on the left"},{"speaker":"Pam","startTime":588.48,"endTime":592.29,"body":"and 8,700 on the right, and I'm"},{"speaker":"Pam","startTime":588.48,"endTime":592.29,"body":"kind of thinking about that 100"},{"speaker":"Pam","startTime":592.29,"endTime":595.83,"body":"in between there. And I'm not"},{"speaker":"Pam","startTime":592.29,"endTime":595.83,"body":"really thinking about"},{"speaker":"Pam","startTime":595.83,"endTime":598.62,"body":"subtracting the 100, and then"},{"speaker":"Pam","startTime":595.83,"endTime":598.62,"body":"adding the 13. I'm kind of just"},{"speaker":"Pam","startTime":598.62,"endTime":602.79,"body":"thinking about there's that 100"},{"speaker":"Pam","startTime":598.62,"endTime":602.79,"body":"in there. And I need to subtract"},{"speaker":"Pam","startTime":602.82,"endTime":609.6,"body":"87, so I've kind of drawn that"},{"speaker":"Pam","startTime":602.82,"endTime":609.6,"body":"jump of 87 back from 8,700, and"},{"speaker":"Pam","startTime":609.6,"endTime":613.47,"body":"that leaves me with that little"},{"speaker":"Pam","startTime":609.6,"endTime":613.47,"body":"bit of space between 8,600, and"},{"speaker":"Pam","startTime":613.47,"endTime":614.55,"body":"that jump back."},{"speaker":"Kim","startTime":614.94,"endTime":615.24,"body":"Yep."},{"speaker":"Pam","startTime":615.24,"endTime":618.12,"body":"And then I can say, \"Ooh,"},{"speaker":"Pam","startTime":615.24,"endTime":618.12,"body":"what is the partner of 87?\""},{"speaker":"Pam","startTime":618.45,"endTime":623.46,"body":"That's 13. And so, now, I have"},{"speaker":"Pam","startTime":618.45,"endTime":623.46,"body":"this. In that space between"},{"speaker":"Pam","startTime":623.49,"endTime":629.1,"body":"8,600 and 8,700, I have a jump"},{"speaker":"Pam","startTime":623.49,"endTime":629.1,"body":"of 13 and a jump of 87, and I"},{"speaker":"Pam","startTime":629.1,"endTime":633.48,"body":"want to land in that where those"},{"speaker":"Pam","startTime":629.1,"endTime":633.48,"body":"two jumps meet. Or \"meet\"?"},{"speaker":"Kim","startTime":633.48,"endTime":633.54,"body":"Yeah."},{"speaker":"Pam","startTime":633.9,"endTime":638.97,"body":"Is that the right word? And"},{"speaker":"Pam","startTime":633.9,"endTime":638.97,"body":"so, that is 8,613. That's sort"},{"speaker":"Pam","startTime":638.97,"endTime":639.63,"body":"of in there."},{"speaker":"Kim","startTime":640.08,"endTime":641.7,"body":"Yeah, I think it's really"},{"speaker":"Kim","startTime":640.08,"endTime":641.7,"body":"important that you mentioned"},{"speaker":"Kim","startTime":641.7,"endTime":645.21,"body":"that because if you're an older"},{"speaker":"Kim","startTime":641.7,"endTime":645.21,"body":"grades teacher, and you hear"},{"speaker":"Kim","startTime":645.21,"endTime":647.43,"body":"about I Have, You Need, then"},{"speaker":"Kim","startTime":645.21,"endTime":647.43,"body":"you're like, \"Cool, but that's"},{"speaker":"Kim","startTime":647.43,"endTime":651.93,"body":"just 100. That's super low"},{"speaker":"Kim","startTime":647.43,"endTime":651.93,"body":"numbers compared to what my kids"},{"speaker":"Kim","startTime":651.93,"endTime":655.08,"body":"are working with.\" And you"},{"speaker":"Kim","startTime":651.93,"endTime":655.08,"body":"mentioned the 200 earlier. Do"},{"speaker":"Kim","startTime":655.08,"endTime":658.59,"body":"you do I Have, You Need with"},{"speaker":"Kim","startTime":655.08,"endTime":658.59,"body":"200. But really, once you've"},{"speaker":"Kim","startTime":658.59,"endTime":662.22,"body":"established that partnership"},{"speaker":"Kim","startTime":658.59,"endTime":662.22,"body":"with 100, then I do think that"},{"speaker":"Kim","startTime":662.22,"endTime":666.03,"body":"you bring in other amounts of"},{"speaker":"Kim","startTime":662.22,"endTime":666.03,"body":"hundreds and help kids"},{"speaker":"Kim","startTime":666.03,"endTime":672.36,"body":"understand that a 400 is a 300"},{"speaker":"Kim","startTime":666.03,"endTime":672.36,"body":"and a 100. And then, it's always"},{"speaker":"Kim","startTime":672.36,"endTime":675.75,"body":"that partner of 100 work. And"},{"speaker":"Kim","startTime":672.36,"endTime":675.75,"body":"then, you come back and tack on"},{"speaker":"Kim","startTime":675.75,"endTime":680.49,"body":"that 300 at the end. So, there's"},{"speaker":"Kim","startTime":675.75,"endTime":680.49,"body":"always that 100 within any"},{"speaker":"Kim","startTime":680.49,"endTime":682.68,"body":"amount of hundreds that you can"},{"speaker":"Kim","startTime":680.49,"endTime":682.68,"body":"find the partner for."},{"speaker":"Pam","startTime":682.71,"endTime":683.16,"body":"Yeah, so"},{"speaker":"Pam","startTime":683.16,"endTime":687.6,"body":"like in this problem that 100 is"},{"speaker":"Pam","startTime":683.16,"endTime":687.6,"body":"within that 8,600 and 8,700. And"},{"speaker":"Pam","startTime":687.6,"endTime":690.84,"body":"so, I deal with what I know"},{"speaker":"Pam","startTime":687.6,"endTime":690.84,"body":"about 100 in there, or the 13"},{"speaker":"Pam","startTime":690.84,"endTime":693.9,"body":"and the 87. Then, I just tack on"},{"speaker":"Pam","startTime":690.84,"endTime":693.9,"body":"that 8,600 back on the"},{"speaker":"Pam","startTime":693.9,"endTime":694.14,"body":"beginning."},{"speaker":"Kim","startTime":694.14,"endTime":694.8,"body":"Yeah, yep."},{"speaker":"Pam","startTime":694.8,"endTime":700.14,"body":"Yeah, nicely said. So,"},{"speaker":"Pam","startTime":694.8,"endTime":700.14,"body":"subtraction is super helpful in"},{"speaker":"Pam","startTime":700.14,"endTime":702.87,"body":"things like this crazy"},{"speaker":"Pam","startTime":700.14,"endTime":702.87,"body":"multiplication problem we just"},{"speaker":"Pam","startTime":702.87,"endTime":706.44,"body":"did. 99 times 87. And we could"},{"speaker":"Pam","startTime":702.87,"endTime":706.44,"body":"do similar things with division"},{"speaker":"Pam","startTime":707.1,"endTime":710.73,"body":"where the problem actually"},{"speaker":"Pam","startTime":707.1,"endTime":710.73,"body":"becomes like, \"Can I think about"},{"speaker":"Pam","startTime":710.73,"endTime":716.76,"body":"subtraction in the midst of this"},{"speaker":"Pam","startTime":710.73,"endTime":716.76,"body":"higher math problem?\" So, let's"},{"speaker":"Pam","startTime":716.76,"endTime":720.99,"body":"parse out a little bit about two"},{"speaker":"Pam","startTime":716.76,"endTime":720.99,"body":"ways of thinking about this 87"},{"speaker":"Pam","startTime":720.99,"endTime":725.22,"body":"plus what is 100? When I said,"},{"speaker":"Pam","startTime":720.99,"endTime":725.22,"body":"\"I have 87, what do you need?\""},{"speaker":"Pam","startTime":725.22,"endTime":727.98,"body":"You said, \"13.\" That's a way of"},{"speaker":"Pam","startTime":725.22,"endTime":727.98,"body":"thinking about kind of the"},{"speaker":"Pam","startTime":727.98,"endTime":730.95,"body":"distance between 87 and 100,"},{"speaker":"Pam","startTime":727.98,"endTime":730.95,"body":"right?"},{"speaker":"Kim","startTime":730.98,"endTime":731.25,"body":"Yep."},{"speaker":"Pam","startTime":731.25,"endTime":735.06,"body":"I'm at 87, would I need to"},{"speaker":"Pam","startTime":731.25,"endTime":735.06,"body":"make 100? 87 plus something is"},{"speaker":"Pam","startTime":735.06,"endTime":738.33,"body":"100. I can look at a number"},{"speaker":"Pam","startTime":735.06,"endTime":738.33,"body":"line. I'm at 87. I've got to"},{"speaker":"Pam","startTime":738.33,"endTime":741.87,"body":"jump up to 100. That's a way of"},{"speaker":"Pam","startTime":738.33,"endTime":741.87,"body":"thinking about distance or"},{"speaker":"Pam","startTime":741.87,"endTime":747.84,"body":"difference. But I can also think"},{"speaker":"Pam","startTime":741.87,"endTime":747.84,"body":"about 100 minus 87. I can think"},{"speaker":"Pam","startTime":747.84,"endTime":752.28,"body":"about removing 87. That looks"},{"speaker":"Pam","startTime":747.84,"endTime":752.28,"body":"less like starting at 87 and"},{"speaker":"Pam","startTime":752.28,"endTime":756.96,"body":"going up to 100, and that looks"},{"speaker":"Pam","startTime":752.28,"endTime":756.96,"body":"more like starting at 100 and"},{"speaker":"Pam","startTime":756.96,"endTime":763.08,"body":"going back 87. And I might think"},{"speaker":"Pam","startTime":756.96,"endTime":763.08,"body":"about going back 80 to 20, and"},{"speaker":"Pam","startTime":763.08,"endTime":767.19,"body":"then going back 7 more to 13. I"},{"speaker":"Pam","startTime":763.08,"endTime":767.19,"body":"might also think about going"},{"speaker":"Pam","startTime":767.19,"endTime":771.36,"body":"back 90 and landing on 10, and"},{"speaker":"Pam","startTime":767.19,"endTime":771.36,"body":"then I went too far, so I'd go"},{"speaker":"Pam","startTime":771.36,"endTime":776.4,"body":"back up 3 to get to 13. But in"},{"speaker":"Pam","startTime":771.36,"endTime":776.4,"body":"that way, I'm really thinking"},{"speaker":"Pam","startTime":776.4,"endTime":780.42,"body":"about removing. I'm thinking"},{"speaker":"Pam","startTime":776.4,"endTime":780.42,"body":"about removing 87 from 100,"},{"speaker":"Pam","startTime":780.57,"endTime":787.29,"body":"going back from 100 to land on"},{"speaker":"Pam","startTime":780.57,"endTime":787.29,"body":"13. Those two types of thinking,"},{"speaker":"Pam","startTime":787.41,"endTime":791.76,"body":"one of them's sort of distance"},{"speaker":"Pam","startTime":787.41,"endTime":791.76,"body":"going from the 87 up. One of"},{"speaker":"Pam","startTime":791.76,"endTime":797.79,"body":"them's removing, going from the"},{"speaker":"Pam","startTime":791.76,"endTime":797.79,"body":"100 back. Those are the two ways"},{"speaker":"Pam","startTime":797.79,"endTime":802.23,"body":"that we need to help kids think"},{"speaker":"Pam","startTime":797.79,"endTime":802.23,"body":"about subtraction. Maybe we"},{"speaker":"Pam","startTime":802.23,"endTime":804.75,"body":"could call it the two"},{"speaker":"Pam","startTime":802.23,"endTime":804.75,"body":"interpretations of subtraction"},{"speaker":"Pam","startTime":804.78,"endTime":808.89,"body":"that we can actually think about"},{"speaker":"Pam","startTime":804.78,"endTime":808.89,"body":"the problem 100 minus 87 as"},{"speaker":"Pam","startTime":809.16,"endTime":813.84,"body":"\"What do I need to do to get"},{"speaker":"Pam","startTime":809.16,"endTime":813.84,"body":"from 87 up to 100?\" Or as \"What"},{"speaker":"Pam","startTime":813.84,"endTime":819.57,"body":"do I get when I remove 87 from"},{"speaker":"Pam","startTime":813.84,"endTime":819.57,"body":"100.\" Now, if we're stuck in the"},{"speaker":"Pam","startTime":819.57,"endTime":822.87,"body":"traditional algorithm, we often"},{"speaker":"Pam","startTime":819.57,"endTime":822.87,"body":"haven't built that intuition"},{"speaker":"Pam","startTime":822.9,"endTime":825.39,"body":"about distance, about"},{"speaker":"Pam","startTime":822.9,"endTime":825.39,"body":"difference, that we haven't"},{"speaker":"Pam","startTime":826.35,"endTime":830.16,"body":"built that part of it. If we're"},{"speaker":"Pam","startTime":826.35,"endTime":830.16,"body":"stuck in the subtraction"},{"speaker":"Pam","startTime":830.16,"endTime":834.6,"body":"algorithm, we might only ever"},{"speaker":"Pam","startTime":830.16,"endTime":834.6,"body":"think about removing."},{"speaker":"Kim","startTime":835.11,"endTime":835.53,"body":"Yeah."},{"speaker":"Pam","startTime":835.62,"endTime":837.24,"body":"And when you were doing all"},{"speaker":"Pam","startTime":835.62,"endTime":837.24,"body":"that stuff with subtraction"},{"speaker":"Pam","startTime":837.24,"endTime":841.47,"body":"across zeros. 0 take away 7, you"},{"speaker":"Pam","startTime":837.24,"endTime":841.47,"body":"can't do that, so then we have"},{"speaker":"Pam","startTime":841.47,"endTime":843.39,"body":"to go to the next one. It's"},{"speaker":"Pam","startTime":841.47,"endTime":843.39,"body":"always \"take away\", right? It's"},{"speaker":"Pam","startTime":843.39,"endTime":847.02,"body":"always, \"Can I remove this"},{"speaker":"Pam","startTime":843.39,"endTime":847.02,"body":"number from that one?\" And so,"},{"speaker":"Pam","startTime":847.02,"endTime":850.53,"body":"we might have kids stuck in only"},{"speaker":"Pam","startTime":847.02,"endTime":850.53,"body":"that one interpretation of"},{"speaker":"Pam","startTime":850.53,"endTime":854.01,"body":"subtraction and never actually"},{"speaker":"Pam","startTime":850.53,"endTime":854.01,"body":"build intuition about the other"},{"speaker":"Pam","startTime":854.01,"endTime":858.03,"body":"interpretation of subtraction."},{"speaker":"Pam","startTime":854.01,"endTime":858.03,"body":"And we need both."},{"speaker":"Kim","startTime":858.51,"endTime":858.78,"body":"Yeah."},{"speaker":"Pam","startTime":858.78,"endTime":863.04,"body":"One of the ways that we can"},{"speaker":"Pam","startTime":858.78,"endTime":863.04,"body":"do that is we can use the open"},{"speaker":"Pam","startTime":863.04,"endTime":866.01,"body":"number line to represent what"},{"speaker":"Pam","startTime":863.04,"endTime":866.01,"body":"students are doing, the way"},{"speaker":"Pam","startTime":866.01,"endTime":869.49,"body":"they're thinking about things."},{"speaker":"Pam","startTime":866.01,"endTime":869.49,"body":"That's a great tool. We like to"},{"speaker":"Pam","startTime":869.49,"endTime":873.39,"body":"use it. And actually, as I was"},{"speaker":"Pam","startTime":869.49,"endTime":873.39,"body":"just talking about both of those"},{"speaker":"Pam","startTime":873.39,"endTime":877.74,"body":"ways of thinking about 100 minus"},{"speaker":"Pam","startTime":873.39,"endTime":877.74,"body":"87, I was... If you could see my"},{"speaker":"Pam","startTime":877.74,"endTime":881.07,"body":"hands. I had my hands kind of"},{"speaker":"Pam","startTime":877.74,"endTime":881.07,"body":"spanning. So, when I was talking"},{"speaker":"Pam","startTime":881.07,"endTime":885.81,"body":"about 87 plus what is 100, I"},{"speaker":"Pam","startTime":881.07,"endTime":885.81,"body":"literally had my hands kind of"},{"speaker":"Pam","startTime":885.81,"endTime":890.73,"body":"spanning 100. My hands are going"},{"speaker":"Pam","startTime":885.81,"endTime":890.73,"body":"back and forth. Here's 0 to 100."},{"speaker":"Pam","startTime":890.73,"endTime":892.5,"body":"I wonder if you can hear that."},{"speaker":"Pam","startTime":890.73,"endTime":892.5,"body":"Can you hear that over the"},{"speaker":"Pam","startTime":892.5,"endTime":898.86,"body":"microphone? No, it's like air."},{"speaker":"Pam","startTime":892.5,"endTime":898.86,"body":"Anyway, so 0 to 100. And then,"},{"speaker":"Pam","startTime":898.89,"endTime":902.19,"body":"I'm thinking to myself, \"But In"},{"speaker":"Pam","startTime":898.89,"endTime":902.19,"body":"between that span of 0 to 100,"},{"speaker":"Pam","startTime":902.19,"endTime":905.52,"body":"here's 87.\" And I've just put my"},{"speaker":"Pam","startTime":902.19,"endTime":905.52,"body":"hand at 87. And I'm thinking,"},{"speaker":"Pam","startTime":905.82,"endTime":909.36,"body":"\"If my hands here at 87, what's"},{"speaker":"Pam","startTime":905.82,"endTime":909.36,"body":"that distance to get up to 100?\""},{"speaker":"Pam","startTime":909.39,"endTime":913.08,"body":"I'm literally have this number"},{"speaker":"Pam","startTime":909.39,"endTime":913.08,"body":"line in front of me that's"},{"speaker":"Pam","startTime":913.08,"endTime":916.56,"body":"either in my head, or it's"},{"speaker":"Pam","startTime":913.08,"endTime":916.56,"body":"between my hands. Or on the"},{"speaker":"Pam","startTime":916.56,"endTime":920.1,"body":"other way of starting at 100,"},{"speaker":"Pam","startTime":916.56,"endTime":920.1,"body":"and then I'm backing up. And"},{"speaker":"Pam","startTime":920.1,"endTime":923.52,"body":"now, my hands kind of backing up"},{"speaker":"Pam","startTime":920.1,"endTime":923.52,"body":"from 100 to the 87, and I can"},{"speaker":"Pam","startTime":923.52,"endTime":928.05,"body":"kind of think about... Or sorry."},{"speaker":"Pam","startTime":923.52,"endTime":928.05,"body":"Backing up. Ha. Not to the 87."},{"speaker":"Pam","startTime":928.05,"endTime":932.04,"body":"That would still be difference."},{"speaker":"Pam","startTime":928.05,"endTime":932.04,"body":"But backing up 80 and 7. Like,"},{"speaker":"Pam","startTime":932.04,"endTime":935.79,"body":"removing that 87. So, both of"},{"speaker":"Pam","startTime":932.04,"endTime":935.79,"body":"those are kind of happening on"},{"speaker":"Pam","startTime":935.79,"endTime":938.61,"body":"an open number line, whether I'm"},{"speaker":"Pam","startTime":935.79,"endTime":938.61,"body":"writing it or it's kind of"},{"speaker":"Pam","startTime":938.61,"endTime":940.95,"body":"what's happening in the space in"},{"speaker":"Pam","startTime":938.61,"endTime":940.95,"body":"my head. So, ya'll the open"},{"speaker":"Pam","startTime":940.95,"endTime":944.61,"body":"number line is not a picture"},{"speaker":"Pam","startTime":940.95,"endTime":944.61,"body":"that we're dumbing down to help"},{"speaker":"Pam","startTime":944.61,"endTime":948.75,"body":"your kids, I don't know, do"},{"speaker":"Pam","startTime":944.61,"endTime":948.75,"body":"fuzzier math. No, no, no. It's"},{"speaker":"Pam","startTime":948.75,"endTime":951.87,"body":"literally what's been happening"},{"speaker":"Pam","startTime":948.75,"endTime":951.87,"body":"in mathematicians' minds this"},{"speaker":"Pam","startTime":951.87,"endTime":955.23,"body":"whole time. We're just making"},{"speaker":"Pam","startTime":951.87,"endTime":955.23,"body":"that thinking visible. Go ahead."},{"speaker":"Kim","startTime":955.44,"endTime":957.6,"body":"Yeah. And I think this is"},{"speaker":"Kim","startTime":955.44,"endTime":957.6,"body":"really important that you're"},{"speaker":"Kim","startTime":957.6,"endTime":960.54,"body":"talking about it because"},{"speaker":"Kim","startTime":957.6,"endTime":960.54,"body":"especially with younger grade"},{"speaker":"Kim","startTime":960.54,"endTime":965.25,"body":"students, there is intuition in"},{"speaker":"Kim","startTime":960.54,"endTime":965.25,"body":"kids to find the distance"},{"speaker":"Kim","startTime":965.25,"endTime":968.22,"body":"between numbers. I think you'd"},{"speaker":"Kim","startTime":965.25,"endTime":968.22,"body":"be hard pressed to find a kid"},{"speaker":"Kim","startTime":968.43,"endTime":973.02,"body":"who when you said 100 minus 99,"},{"speaker":"Kim","startTime":968.43,"endTime":973.02,"body":"they couldn't come up with. I"},{"speaker":"Kim","startTime":973.02,"endTime":976.38,"body":"mean, depending on their age."},{"speaker":"Kim","startTime":973.02,"endTime":976.38,"body":"But I think many kids would just"},{"speaker":"Kim","startTime":976.38,"endTime":981.09,"body":"say 1. They're not thinking"},{"speaker":"Kim","startTime":976.38,"endTime":981.09,"body":"about remove 99 until we put a"},{"speaker":"Kim","startTime":981.09,"endTime":984.96,"body":"problem on the page in a"},{"speaker":"Kim","startTime":981.09,"endTime":984.96,"body":"stacking format and say,"},{"speaker":"Kim","startTime":985.05,"endTime":985.74,"body":"\"Subtract.\""},{"speaker":"Pam","startTime":986.01,"endTime":987.24,"body":"\"Thou shalt...\" Yeah."},{"speaker":"Kim","startTime":987.27,"endTime":990.9,"body":"Absolutely. I think we"},{"speaker":"Kim","startTime":987.27,"endTime":990.9,"body":"schooled them out of some of"},{"speaker":"Kim","startTime":990.9,"endTime":995.04,"body":"that intuition. And so, if we"},{"speaker":"Kim","startTime":990.9,"endTime":995.04,"body":"can help them recognize that"},{"speaker":"Kim","startTime":995.04,"endTime":999.57,"body":"that's a thing and say, \"Oh,"},{"speaker":"Kim","startTime":995.04,"endTime":999.57,"body":"here's a representation of this"},{"speaker":"Kim","startTime":999.57,"endTime":1004.1,"body":"distance, this gap, that space"},{"speaker":"Kim","startTime":999.57,"endTime":1004.1,"body":"that you just called out as 1.\""},{"speaker":"Kim","startTime":1005.66,"endTime":1008.24,"body":"They could take that"},{"speaker":"Kim","startTime":1005.66,"endTime":1008.24,"body":"representation and do so many"},{"speaker":"Kim","startTime":1008.24,"endTime":1009.53,"body":"more things with it, but."},{"speaker":"Pam","startTime":1010.43,"endTime":1014.27,"body":"And you can almost picture"},{"speaker":"Pam","startTime":1010.43,"endTime":1014.27,"body":"that ornery boy that's like,"},{"speaker":"Pam","startTime":1014.39,"endTime":1015.92,"body":"\"I'm not going to do that. I'm"},{"speaker":"Pam","startTime":1014.39,"endTime":1015.92,"body":"not going to write down your"},{"speaker":"Pam","startTime":1015.92,"endTime":1019.04,"body":"steps, teacher. I just know 100"},{"speaker":"Pam","startTime":1015.92,"endTime":1019.04,"body":"minus 99 is 1.\" And then, I can"},{"speaker":"Pam","startTime":1019.04,"endTime":1021.14,"body":"picture that well meaning"},{"speaker":"Pam","startTime":1019.04,"endTime":1021.14,"body":"teacher that goes, \"Yeah, but"},{"speaker":"Pam","startTime":1021.14,"endTime":1023.69,"body":"you're going to need this for"},{"speaker":"Pam","startTime":1021.14,"endTime":1023.69,"body":"harder problems, so you got to"},{"speaker":"Pam","startTime":1023.69,"endTime":1024.29,"body":"show me all these steps.\""},{"speaker":"Kim","startTime":1024.38,"endTime":1024.5,"body":"I mean,"},{"speaker":"Kim","startTime":1024.5,"endTime":1026.48,"body":"we have some of those boys."},{"speaker":"Pam","startTime":1026.96,"endTime":1030.02,"body":"You and I. You and I"},{"speaker":"Pam","startTime":1026.96,"endTime":1030.02,"body":"birthed some of those boys. That"},{"speaker":"Pam","startTime":1030.02,"endTime":1032.6,"body":"would look at that teacher and"},{"speaker":"Pam","startTime":1030.02,"endTime":1032.6,"body":"go, \"No, I'm not doing all those"},{"speaker":"Pam","startTime":1032.6,"endTime":1035.54,"body":"steps for that crazy problem.\""},{"speaker":"Pam","startTime":1032.6,"endTime":1035.54,"body":"But that will meaning teacher"},{"speaker":"Pam","startTime":1035.54,"endTime":1037.82,"body":"that says, \"No, no. No, really."},{"speaker":"Pam","startTime":1035.54,"endTime":1037.82,"body":"You're going to find harder"},{"speaker":"Pam","startTime":1037.82,"endTime":1040.49,"body":"subtraction problems. So, we"},{"speaker":"Pam","startTime":1037.82,"endTime":1040.49,"body":"gotta get these steps down with"},{"speaker":"Pam","startTime":1040.49,"endTime":1041.87,"body":"these easier...\" And so, what we"},{"speaker":"Pam","startTime":1040.49,"endTime":1041.87,"body":"are respectfully saying is,"},{"speaker":"Pam","startTime":1041.87,"endTime":1042.92,"body":"\"Let's not try to beat that"},{"speaker":"Pam","startTime":1041.87,"endTime":1042.92,"body":"intuition out of kids.\" Let's"},{"speaker":"Pam","startTime":1042.92,"endTime":1044.27,"body":"actually say, \"Yeah, yeah, yeah."},{"speaker":"Pam","startTime":1042.92,"endTime":1044.27,"body":"Let's think about the fact that"},{"speaker":"Pam","startTime":1044.27,"endTime":1045.26,"body":"100 minus 99 is just the"},{"speaker":"Pam","startTime":1044.27,"endTime":1045.26,"body":"difference between 99 and 100."},{"speaker":"Pam","startTime":1045.26,"endTime":1045.83,"body":"I mean, Pam. I'm sorry, go"},{"speaker":"Pam","startTime":1045.26,"endTime":1045.83,"body":"ahead."},{"speaker":"Pam","startTime":1045.829,"endTime":1046.369,"body":"Let's use that intuition."},{"speaker":"Kim","startTime":1046.369,"endTime":1047.539,"body":"I'm having a hard time"},{"speaker":"Kim","startTime":1046.369,"endTime":1047.539,"body":"keeping my brain stopping."},{"speaker":"Pam","startTime":1048.05,"endTime":1048.71,"body":"Let's use"},{"speaker":"Pam","startTime":1049.73,"endTime":1062.12,"body":"that intuition whenever we can."},{"speaker":"Pam","startTime":1049.73,"endTime":1062.12,"body":"Now, what were you going to say"},{"speaker":"Pam","startTime":1062.12,"endTime":1062.69,"body":"Go, go, go."},{"speaker":"Kim","startTime":1062.72,"endTime":1065.93,"body":"Well, I was going to say I"},{"speaker":"Kim","startTime":1062.72,"endTime":1065.93,"body":"would love to be in a third"},{"speaker":"Kim","startTime":1065.93,"endTime":1069.74,"body":"grade classroom right now"},{"speaker":"Kim","startTime":1065.93,"endTime":1069.74,"body":"saying, \"At what point?\" Like,"},{"speaker":"Kim","startTime":1069.77,"endTime":1073.13,"body":"\"Where's the point where you"},{"speaker":"Kim","startTime":1069.77,"endTime":1073.13,"body":"say, 'You know, if it's 99 and"},{"speaker":"Kim","startTime":1073.13,"endTime":1075.74,"body":"100,' they're going to say 1.\""},{"speaker":"Kim","startTime":1073.13,"endTime":1075.74,"body":"Right, it's just 1. And you"},{"speaker":"Kim","startTime":1075.74,"endTime":1078.47,"body":"model the number line? And I"},{"speaker":"Kim","startTime":1075.74,"endTime":1078.47,"body":"would love to say, \"What's the"},{"speaker":"Kim","startTime":1078.47,"endTime":1083.51,"body":"threshold?\" Like, \"At what point"},{"speaker":"Kim","startTime":1078.47,"endTime":1083.51,"body":"do you switch from finding the"},{"speaker":"Kim","startTime":1083.51,"endTime":1086.69,"body":"distance between them to...\""},{"speaker":"Kim","startTime":1083.51,"endTime":1086.69,"body":"It's like if you said, 100 minus"},{"speaker":"Kim","startTime":1086.69,"endTime":1089.84,"body":"1, then they're going to remove."},{"speaker":"Kim","startTime":1086.69,"endTime":1089.84,"body":"And I think it's different for"},{"speaker":"Kim","startTime":1089.84,"endTime":1093.74,"body":"every kid based on their"},{"speaker":"Kim","startTime":1089.84,"endTime":1093.74,"body":"familiarity and like different"},{"speaker":"Kim","startTime":1093.74,"endTime":1097.37,"body":"strategies. But I almost want to"},{"speaker":"Kim","startTime":1093.74,"endTime":1097.37,"body":"like plunk numbers on a number"},{"speaker":"Kim","startTime":1097.37,"endTime":1101.57,"body":"line and say, or ask them"},{"speaker":"Kim","startTime":1097.37,"endTime":1101.57,"body":"questions and say, \"Okay, what"},{"speaker":"Kim","startTime":1101.57,"endTime":1108.98,"body":"do you want to do when it's 100"},{"speaker":"Kim","startTime":1101.57,"endTime":1108.98,"body":"minus 20? What about 23? 76?\""},{"speaker":"Kim","startTime":1109.43,"endTime":1114.32,"body":"Like, there's... At some point,"},{"speaker":"Kim","startTime":1109.43,"endTime":1114.32,"body":"before we mess with kids too"},{"speaker":"Kim","startTime":1114.32,"endTime":1118.58,"body":"much, they have this gut"},{"speaker":"Kim","startTime":1114.32,"endTime":1118.58,"body":"reaction to where they want to"},{"speaker":"Kim","startTime":1118.58,"endTime":1122.69,"body":"start removing, and where they"},{"speaker":"Kim","startTime":1118.58,"endTime":1122.69,"body":"want to find the distance. I"},{"speaker":"Kim","startTime":1122.69,"endTime":1124.61,"body":"just think it would be a fun"},{"speaker":"Kim","startTime":1122.69,"endTime":1124.61,"body":"exercise and like, \"What are you"},{"speaker":"Kim","startTime":1124.61,"endTime":1128.03,"body":"thinking about? And what numbers"},{"speaker":"Kim","startTime":1124.61,"endTime":1128.03,"body":"do it for you in either way?\""},{"speaker":"Pam","startTime":1128.54,"endTime":1131.78,"body":"Mmhmm. Yeah, absolutely."},{"speaker":"Pam","startTime":1128.54,"endTime":1131.78,"body":"Yeah, that's super cool. I want"},{"speaker":"Pam","startTime":1131.78,"endTime":1133.88,"body":"to hang with the open number"},{"speaker":"Pam","startTime":1131.78,"endTime":1133.88,"body":"line for just a second, and then"},{"speaker":"Pam","startTime":1133.88,"endTime":1135.5,"body":"I want to go back to what you"},{"speaker":"Pam","startTime":1133.88,"endTime":1135.5,"body":"were just saying. Let's see if"},{"speaker":"Pam","startTime":1135.5,"endTime":1137.21,"body":"we can do both of those things."},{"speaker":"Kim","startTime":1137.24,"endTime":1138.41,"body":"Stay focused. Okay."},{"speaker":"Pam","startTime":1138.44,"endTime":1141.11,"body":"Hanging with the open"},{"speaker":"Pam","startTime":1138.44,"endTime":1141.11,"body":"number line for a minute. You"},{"speaker":"Pam","startTime":1141.11,"endTime":1143.57,"body":"know, teachers of higher math,"},{"speaker":"Pam","startTime":1141.11,"endTime":1143.57,"body":"you might be like, \"Okay, Pam."},{"speaker":"Pam","startTime":1143.57,"endTime":1147.05,"body":"You told us to listen to this"},{"speaker":"Pam","startTime":1143.57,"endTime":1147.05,"body":"episode because it's important"},{"speaker":"Pam","startTime":1147.05,"endTime":1149.99,"body":"some way.\" Yeah, let me just"},{"speaker":"Pam","startTime":1147.05,"endTime":1149.99,"body":"take that a little bit. Because"},{"speaker":"Pam","startTime":1150.14,"endTime":1153.32,"body":"as we think about the"},{"speaker":"Pam","startTime":1150.14,"endTime":1153.32,"body":"difference. Say, I've got"},{"speaker":"Pam","startTime":1153.32,"endTime":1158.9,"body":"something like 105 minus 97. Do"},{"speaker":"Pam","startTime":1153.32,"endTime":1158.9,"body":"we really want to remove 97 from"},{"speaker":"Pam","startTime":1158.93,"endTime":1164.84,"body":"105? Or would it make more sense"},{"speaker":"Pam","startTime":1158.93,"endTime":1164.84,"body":"to kind of think in your mind,"},{"speaker":"Pam","startTime":1164.87,"endTime":1169.34,"body":"\"Where's 97? Where's 105? How"},{"speaker":"Pam","startTime":1164.87,"endTime":1169.34,"body":"far apart are those?\" As we do"},{"speaker":"Pam","startTime":1169.34,"endTime":1172.04,"body":"that kind of action, we're"},{"speaker":"Pam","startTime":1169.34,"endTime":1172.04,"body":"building this sense of, \"I can"},{"speaker":"Pam","startTime":1172.07,"endTime":1178.04,"body":"use subtraction to represent"},{"speaker":"Pam","startTime":1172.07,"endTime":1178.04,"body":"that.\" So, from that 97 up 3,"},{"speaker":"Pam","startTime":1178.13,"endTime":1182.78,"body":"and then up another 5. So, I can"},{"speaker":"Pam","startTime":1178.13,"endTime":1182.78,"body":"think about 105 minus 97 as that"},{"speaker":"Pam","startTime":1182.81,"endTime":1186.53,"body":"distance of 8, that 3 and then"},{"speaker":"Pam","startTime":1182.81,"endTime":1186.53,"body":"that 5 in between there. But I"},{"speaker":"Pam","startTime":1186.53,"endTime":1192.14,"body":"can write that as 105 minus 97."},{"speaker":"Pam","startTime":1186.53,"endTime":1192.14,"body":"That subtraction symbol can mean"},{"speaker":"Pam","startTime":1192.41,"endTime":1196.61,"body":"\"Find out how far apart they"},{"speaker":"Pam","startTime":1192.41,"endTime":1196.61,"body":"are.\" That so relates to stuff"},{"speaker":"Pam","startTime":1196.61,"endTime":1200.27,"body":"in higher math, that we can see"},{"speaker":"Pam","startTime":1196.61,"endTime":1200.27,"body":"places in higher math where"},{"speaker":"Pam","startTime":1200.27,"endTime":1204.74,"body":"there is a subtraction symbol"},{"speaker":"Pam","startTime":1200.27,"endTime":1204.74,"body":"and it doesn't mean \"remove\". It"},{"speaker":"Pam","startTime":1204.74,"endTime":1206.93,"body":"means \"Find the difference"},{"speaker":"Pam","startTime":1204.74,"endTime":1206.93,"body":"between the numbers.\" And I'm"},{"speaker":"Pam","startTime":1206.93,"endTime":1209.09,"body":"just going to say a couple of"},{"speaker":"Pam","startTime":1206.93,"endTime":1209.09,"body":"examples, and we'll do some"},{"speaker":"Pam","startTime":1209.09,"endTime":1212.0,"body":"episodes later where I get into"},{"speaker":"Pam","startTime":1209.09,"endTime":1212.0,"body":"these more deeply. But if you"},{"speaker":"Pam","startTime":1212.0,"endTime":1215.66,"body":"think about finding the slope of"},{"speaker":"Pam","startTime":1212.0,"endTime":1215.66,"body":"a line between two points. If"},{"speaker":"Pam","startTime":1215.66,"endTime":1219.68,"body":"I'm thinking about y2 minus y1,"},{"speaker":"Pam","startTime":1215.66,"endTime":1219.68,"body":"does that subtraction symbol"},{"speaker":"Pam","startTime":1219.68,"endTime":1225.41,"body":"mean \"subtract y1 from y2.\" Or"},{"speaker":"Pam","startTime":1219.68,"endTime":1225.41,"body":"does it actually mean \"How far"},{"speaker":"Pam","startTime":1225.41,"endTime":1230.42,"body":"apart are those y values?\" So,"},{"speaker":"Pam","startTime":1225.41,"endTime":1230.42,"body":"I'm going to let teachers of"},{"speaker":"Pam","startTime":1230.42,"endTime":1233.66,"body":"higher math think about that a"},{"speaker":"Pam","startTime":1230.42,"endTime":1233.66,"body":"little bit. X2 minus x1. Does"},{"speaker":"Pam","startTime":1233.66,"endTime":1238.37,"body":"that mean, \"remove x1 from x2.\""},{"speaker":"Pam","startTime":1233.66,"endTime":1238.37,"body":"Or can it actually mean, \"How"},{"speaker":"Pam","startTime":1238.37,"endTime":1242.09,"body":"far apart are those x values?\""},{"speaker":"Pam","startTime":1238.37,"endTime":1242.09,"body":"Because is that rate of change"},{"speaker":"Pam","startTime":1242.09,"endTime":1245.54,"body":"formula talking about the"},{"speaker":"Pam","startTime":1242.09,"endTime":1245.54,"body":"distance, the difference between"},{"speaker":"Pam","startTime":1245.54,"endTime":1248.96,"body":"the y values and the ratio of"},{"speaker":"Pam","startTime":1245.54,"endTime":1248.96,"body":"that, ratio of the distance to"},{"speaker":"Pam","startTime":1248.96,"endTime":1251.78,"body":"the y values to the distance of"},{"speaker":"Pam","startTime":1248.96,"endTime":1251.78,"body":"the x values, the difference"},{"speaker":"Pam","startTime":1251.78,"endTime":1254.81,"body":"between those. I would suggest"},{"speaker":"Pam","startTime":1251.78,"endTime":1254.81,"body":"it's far more about that and not"},{"speaker":"Pam","startTime":1254.81,"endTime":1258.92,"body":"at all about removing things."},{"speaker":"Pam","startTime":1254.81,"endTime":1258.92,"body":"And we could go even higher than"},{"speaker":"Pam","startTime":1258.92,"endTime":1261.23,"body":"that. We could talk about"},{"speaker":"Pam","startTime":1258.92,"endTime":1261.23,"body":"calculus, and the way that we"},{"speaker":"Pam","startTime":1261.23,"endTime":1263.63,"body":"represent intervals when we're"},{"speaker":"Pam","startTime":1261.23,"endTime":1263.63,"body":"trying to find the area under a"},{"speaker":"Pam","startTime":1263.63,"endTime":1265.7,"body":"curve. When we represent"},{"speaker":"Pam","startTime":1263.63,"endTime":1265.7,"body":"intervals, we often use the"},{"speaker":"Pam","startTime":1265.7,"endTime":1268.73,"body":"subtraction symbol to represent"},{"speaker":"Pam","startTime":1265.7,"endTime":1268.73,"body":"an interval. An interval is the"},{"speaker":"Pam","startTime":1268.73,"endTime":1272.18,"body":"distance between two points."},{"speaker":"Pam","startTime":1268.73,"endTime":1272.18,"body":"It's not about removing one from"},{"speaker":"Pam","startTime":1272.18,"endTime":1274.88,"body":"the other, but we're using"},{"speaker":"Pam","startTime":1272.18,"endTime":1274.88,"body":"subtraction. If I was finding"},{"speaker":"Pam","startTime":1274.88,"endTime":1279.11,"body":"the area between two curves, I'm"},{"speaker":"Pam","startTime":1274.88,"endTime":1279.11,"body":"going to subtract f of x minus g"},{"speaker":"Pam","startTime":1279.11,"endTime":1282.38,"body":"of x, and then take the integral"},{"speaker":"Pam","startTime":1279.11,"endTime":1282.38,"body":"from a to b of that difference."},{"speaker":"Pam","startTime":1282.38,"endTime":1285.89,"body":"But it's... I just said"},{"speaker":"Pam","startTime":1282.38,"endTime":1285.89,"body":"\"difference\". The difference"},{"speaker":"Pam","startTime":1285.89,"endTime":1288.62,"body":"between those two functions,"},{"speaker":"Pam","startTime":1285.89,"endTime":1288.62,"body":"between the two y values of"},{"speaker":"Pam","startTime":1288.62,"endTime":1291.32,"body":"those curves. It's the"},{"speaker":"Pam","startTime":1288.62,"endTime":1291.32,"body":"difference between the y's of"},{"speaker":"Pam","startTime":1291.32,"endTime":1294.14,"body":"those curves. It's not about"},{"speaker":"Pam","startTime":1291.32,"endTime":1294.14,"body":"removing a function from another"},{"speaker":"Pam","startTime":1294.14,"endTime":1296.6,"body":"function. Anyway, there's a"},{"speaker":"Pam","startTime":1294.14,"endTime":1296.6,"body":"couple of examples of where"},{"speaker":"Pam","startTime":1296.6,"endTime":1299.18,"body":"we're actually building"},{"speaker":"Pam","startTime":1296.6,"endTime":1299.18,"body":"intuition towards that, so that"},{"speaker":"Pam","startTime":1299.18,"endTime":1303.05,"body":"when we get them in higher math,"},{"speaker":"Pam","startTime":1299.18,"endTime":1303.05,"body":"we have both interpretations of"},{"speaker":"Pam","startTime":1303.05,"endTime":1307.22,"body":"subtractions. And, Kim, I got to"},{"speaker":"Pam","startTime":1303.05,"endTime":1307.22,"body":"mention one more. Not to mention"},{"speaker":"Pam","startTime":1307.25,"endTime":1308.87,"body":"integers like integers."},{"speaker":"Kim","startTime":1309.32,"endTime":1309.35,"body":"Oh,"},{"speaker":"Kim","startTime":1309.56,"endTime":1312.17,"body":"yeah, I was going to say, \"Wait"},{"speaker":"Kim","startTime":1309.56,"endTime":1312.17,"body":"a second, you missed a big one!\""},{"speaker":"Kim","startTime":1312.17,"endTime":1316.07,"body":"Because there's all these rules,"},{"speaker":"Kim","startTime":1312.17,"endTime":1316.07,"body":"right, for when do you add, when"},{"speaker":"Kim","startTime":1316.07,"endTime":1320.03,"body":"do you subtract. And I have been"},{"speaker":"Kim","startTime":1316.07,"endTime":1320.03,"body":"living in that with my sixth"},{"speaker":"Kim","startTime":1320.03,"endTime":1326.36,"body":"grader because he's doing things"},{"speaker":"Kim","startTime":1320.03,"endTime":1326.36,"body":"like 37 minus negative 42. And"},{"speaker":"Kim","startTime":1326.36,"endTime":1330.23,"body":"I'm like, \"Oh, you're not"},{"speaker":"Kim","startTime":1326.36,"endTime":1330.23,"body":"removing.\" He's like, \"Oh. Ohhh,"},{"speaker":"Kim","startTime":1330.26,"endTime":1334.49,"body":"those are 79 apart on the number"},{"speaker":"Kim","startTime":1330.26,"endTime":1334.49,"body":"line.\" I'm like, \"There you go.\""},{"speaker":"Kim","startTime":1334.52,"endTime":1337.01,"body":"There's your other"},{"speaker":"Kim","startTime":1334.52,"endTime":1337.01,"body":"interpretation of subtraction."},{"speaker":"Pam","startTime":1337.43,"endTime":1340.97,"body":"Yeah. And what's"},{"speaker":"Pam","startTime":1337.43,"endTime":1340.97,"body":"interesting. And we'll do a"},{"speaker":"Pam","startTime":1340.97,"endTime":1343.52,"body":"whole episode on just integer"},{"speaker":"Pam","startTime":1340.97,"endTime":1343.52,"body":"subtraction soon, but what's"},{"speaker":"Pam","startTime":1343.52,"endTime":1346.73,"body":"interesting about integer"},{"speaker":"Pam","startTime":1343.52,"endTime":1346.73,"body":"subtraction is we actually need"},{"speaker":"Pam","startTime":1346.76,"endTime":1349.97,"body":"both interpretations to make"},{"speaker":"Pam","startTime":1346.76,"endTime":1349.97,"body":"sense of integer subtraction"},{"speaker":"Pam","startTime":1349.97,"endTime":1352.43,"body":"problems when we subtract a"},{"speaker":"Pam","startTime":1349.97,"endTime":1352.43,"body":"negative. We have to think about"},{"speaker":"Pam","startTime":1352.43,"endTime":1354.89,"body":"the difference, the distance"},{"speaker":"Pam","startTime":1352.43,"endTime":1354.89,"body":"between the two numbers, and"},{"speaker":"Pam","startTime":1354.89,"endTime":1358.07,"body":"then we have to consider from"},{"speaker":"Pam","startTime":1354.89,"endTime":1358.07,"body":"that first number, \"Am I"},{"speaker":"Pam","startTime":1358.07,"endTime":1361.55,"body":"subtracting something smaller"},{"speaker":"Pam","startTime":1358.07,"endTime":1361.55,"body":"than it?\" That's what we usually"},{"speaker":"Pam","startTime":1361.55,"endTime":1364.01,"body":"do. And if you subtract"},{"speaker":"Pam","startTime":1361.55,"endTime":1364.01,"body":"something smaller than that"},{"speaker":"Pam","startTime":1364.01,"endTime":1366.62,"body":"first number, then you end up"},{"speaker":"Pam","startTime":1364.01,"endTime":1366.62,"body":"with a positive answer. But if"},{"speaker":"Pam","startTime":1366.62,"endTime":1369.29,"body":"you subtract something bigger"},{"speaker":"Pam","startTime":1366.62,"endTime":1369.29,"body":"than it, then you're going to"},{"speaker":"Pam","startTime":1369.29,"endTime":1372.17,"body":"have a negative answer, even"},{"speaker":"Pam","startTime":1369.29,"endTime":1372.17,"body":"though the distance was"},{"speaker":"Pam","startTime":1372.17,"endTime":1375.2,"body":"positive. So, I know I just said"},{"speaker":"Pam","startTime":1372.17,"endTime":1375.2,"body":"that. We'll have to do an"},{"speaker":"Pam","startTime":1375.2,"endTime":1376.91,"body":"episode where we totally build"},{"speaker":"Pam","startTime":1375.2,"endTime":1376.91,"body":"that at some point."},{"speaker":"Kim","startTime":1376.91,"endTime":1376.97,"body":"Yeah."},{"speaker":"Pam","startTime":1377.21,"endTime":1379.79,"body":"So, what's important to"},{"speaker":"Pam","startTime":1377.21,"endTime":1379.79,"body":"know about integers is we have"},{"speaker":"Pam","startTime":1379.79,"endTime":1383.36,"body":"to have both interpretations in"},{"speaker":"Pam","startTime":1379.79,"endTime":1383.36,"body":"order to make sense of integer"},{"speaker":"Pam","startTime":1383.36,"endTime":1387.38,"body":"subtraction. And that's huge"},{"speaker":"Pam","startTime":1383.36,"endTime":1387.38,"body":"because then, if we hadn't made"},{"speaker":"Pam","startTime":1387.38,"endTime":1389.09,"body":"sense of that, what are we left"},{"speaker":"Pam","startTime":1387.38,"endTime":1389.09,"body":"with? Well, we're left with all"},{"speaker":"Pam","startTime":1389.09,"endTime":1391.82,"body":"those rules, right? And then,"},{"speaker":"Pam","startTime":1389.09,"endTime":1391.82,"body":"kids are using Keep, Change,"},{"speaker":"Pam","startTime":1391.82,"endTime":1396.95,"body":"Flip no matter whether they're"},{"speaker":"Pam","startTime":1391.82,"endTime":1396.95,"body":"dividing fractions or they're"},{"speaker":"Pam","startTime":1397.1,"endTime":1400.55,"body":"operating with integers. And"},{"speaker":"Pam","startTime":1397.1,"endTime":1400.55,"body":"neither of those are a good rule"},{"speaker":"Pam","startTime":1400.58,"endTime":1402.32,"body":"because then we don't know when"},{"speaker":"Pam","startTime":1400.58,"endTime":1402.32,"body":"to use them, and we don't know"},{"speaker":"Pam","startTime":1402.32,"endTime":1405.86,"body":"why they work, and we're not"},{"speaker":"Pam","startTime":1402.32,"endTime":1405.86,"body":"reasoning. We're pretending that"},{"speaker":"Pam","startTime":1405.86,"endTime":1408.59,"body":"math isn't figure-out-able,"},{"speaker":"Pam","startTime":1405.86,"endTime":1408.59,"body":"pretending math is rote"},{"speaker":"Pam","startTime":1408.59,"endTime":1412.49,"body":"memorize. Okay, so there was one"},{"speaker":"Pam","startTime":1408.59,"endTime":1412.49,"body":"other thing I wanted to do. What"},{"speaker":"Pam","startTime":1412.49,"endTime":1415.85,"body":"was the other thing, Kim? I said"},{"speaker":"Pam","startTime":1412.49,"endTime":1415.85,"body":"there were two things. Oh, oh. I"},{"speaker":"Pam","startTime":1415.85,"endTime":1421.43,"body":"think I wanted to talk a little"},{"speaker":"Pam","startTime":1415.85,"endTime":1421.43,"body":"bit about like. Kim, I'm going"},{"speaker":"Pam","startTime":1421.43,"endTime":1422.24,"body":"to give you a problem to do."},{"speaker":"Kim","startTime":1422.78,"endTime":1422.99,"body":"Okay."},{"speaker":"Pam","startTime":1422.99,"endTime":1426.44,"body":"And so, if I were to say..."},{"speaker":"Pam","startTime":1422.99,"endTime":1426.44,"body":"Well, actually don't do the"},{"speaker":"Pam","startTime":1426.44,"endTime":1432.11,"body":"problem, please. If I were to"},{"speaker":"Pam","startTime":1426.44,"endTime":1432.11,"body":"say 328 subtract 96. Will you"},{"speaker":"Pam","startTime":1432.11,"endTime":1435.05,"body":"line those up as if you were"},{"speaker":"Pam","startTime":1432.11,"endTime":1435.05,"body":"going to do the algorithm? So,"},{"speaker":"Pam","startTime":1435.05,"endTime":1437.06,"body":"328 minus 96."},{"speaker":"Kim","startTime":1437.06,"endTime":1437.45,"body":"Okay."},{"speaker":"Pam","startTime":1440.18,"endTime":1440.21,"body":"(unclear)."},{"speaker":"Kim","startTime":1440.21,"endTime":1443.78,"body":"Oh, I'm laughing because"},{"speaker":"Kim","startTime":1440.21,"endTime":1443.78,"body":"the last time I told you I"},{"speaker":"Kim","startTime":1443.78,"endTime":1447.2,"body":"attempted to do an algorithm, I"},{"speaker":"Kim","startTime":1443.78,"endTime":1447.2,"body":"totally did it wrong. So, let's"},{"speaker":"Kim","startTime":1447.2,"endTime":1450.89,"body":"see if I can do it. Okay. 328."},{"speaker":"Kim","startTime":1447.2,"endTime":1450.89,"body":"let me write down. Minus 96."},{"speaker":"Pam","startTime":1450.89,"endTime":1451.49,"body":"Three hundred twenty-eight"},{"speaker":"Pam","startTime":1450.89,"endTime":1451.49,"body":"minus 96."},{"speaker":"Kim","startTime":1451.49,"endTime":1451.85,"body":"Okay."},{"speaker":"Pam","startTime":1452.15,"endTime":1454.7,"body":"If you were doing the"},{"speaker":"Pam","startTime":1452.15,"endTime":1454.7,"body":"algorithm. If you were doing the"},{"speaker":"Pam","startTime":1454.7,"endTime":1456.56,"body":"algorithm. Which, I know it's"},{"speaker":"Pam","startTime":1454.7,"endTime":1456.56,"body":"been a hot minute, since you've"},{"speaker":"Pam","startTime":1456.56,"endTime":1457.01,"body":"done that."},{"speaker":"Kim","startTime":1457.13,"endTime":1457.4,"body":"Yeah."},{"speaker":"Kim","startTime":1457.43,"endTime":1461.42,"body":"So, I'm starting in the ones"},{"speaker":"Kim","startTime":1457.43,"endTime":1461.42,"body":"place. So, 8 minus 6 is 2."},{"speaker":"Pam","startTime":1461.69,"endTime":1464.87,"body":"Because? Did you just think"},{"speaker":"Pam","startTime":1461.69,"endTime":1464.87,"body":"about the difference between 8"},{"speaker":"Pam","startTime":1464.87,"endTime":1467.36,"body":"and 6? Or did you subtract? Do"},{"speaker":"Pam","startTime":1464.87,"endTime":1467.36,"body":"you even know? I mean, I don't"},{"speaker":"Pam","startTime":1467.36,"endTime":1468.95,"body":"know if you can think about"},{"speaker":"Pam","startTime":1467.36,"endTime":1468.95,"body":"that. 8 minus 6."},{"speaker":"Kim","startTime":1468.98,"endTime":1472.19,"body":"I think I probably found"},{"speaker":"Kim","startTime":1468.98,"endTime":1472.19,"body":"the distance between them."},{"speaker":"Pam","startTime":1472.64,"endTime":1473.06,"body":"Because?"},{"speaker":"Pam","startTime":1473.39,"endTime":1476.42,"body":"Like, why? What about 8 and 6"},{"speaker":"Pam","startTime":1473.39,"endTime":1476.42,"body":"made you think about distance"},{"speaker":"Pam","startTime":1476.42,"endTime":1476.78,"body":"between them?"},{"speaker":"Kim","startTime":1476.81,"endTime":1479.09,"body":"Yeah, they're really close."},{"speaker":"Kim","startTime":1476.81,"endTime":1479.09,"body":"(unclear)."},{"speaker":"Pam","startTime":1479.6,"endTime":1481.76,"body":"So, the distance is small,"},{"speaker":"Pam","startTime":1479.6,"endTime":1481.76,"body":"right? The distance between them"},{"speaker":"Pam","startTime":1481.76,"endTime":1484.58,"body":"is small. So, yeah. Might as"},{"speaker":"Pam","startTime":1481.76,"endTime":1484.58,"body":"well just find that distance. I"},{"speaker":"Pam","startTime":1484.58,"endTime":1485.84,"body":"didn't mean to put words in your"},{"speaker":"Pam","startTime":1484.58,"endTime":1485.84,"body":"mouth. Sorry."},{"speaker":"Kim","startTime":1486.02,"endTime":1489.35,"body":"No, that's okay. So, then,"},{"speaker":"Kim","startTime":1486.02,"endTime":1489.35,"body":"I'm in the 10's place."},{"speaker":"Pam","startTime":1489.35,"endTime":1492.29,"body":"Ooh, but before you move"},{"speaker":"Pam","startTime":1489.35,"endTime":1492.29,"body":"on. Sorry. But could you imagine"},{"speaker":"Pam","startTime":1492.29,"endTime":1494.87,"body":"that we might have kids that"},{"speaker":"Pam","startTime":1492.29,"endTime":1494.87,"body":"were. If we're teaching them. If"},{"speaker":"Pam","startTime":1494.87,"endTime":1496.67,"body":"we're saying, \"Thou shalt do"},{"speaker":"Pam","startTime":1494.87,"endTime":1496.67,"body":"this algorithm.\" Could you"},{"speaker":"Pam","startTime":1496.67,"endTime":1501.11,"body":"imagine that the kid is saying,"},{"speaker":"Pam","startTime":1496.67,"endTime":1501.11,"body":"\"Okay, 8 minus x. Minus means"},{"speaker":"Pam","startTime":1501.11,"endTime":1505.73,"body":"take away. So, I've got to take"},{"speaker":"Pam","startTime":1501.11,"endTime":1505.73,"body":"away 6. 8, 7, 6, 5...\" And"},{"speaker":"Pam","startTime":1505.73,"endTime":1508.1,"body":"they're counting on their"},{"speaker":"Pam","startTime":1505.73,"endTime":1508.1,"body":"fingers until they tell you. 6"},{"speaker":"Pam","startTime":1508.1,"endTime":1510.56,"body":"fingers up, and they land on a"},{"speaker":"Pam","startTime":1508.1,"endTime":1510.56,"body":"2. And they're like, \"Okay, so"},{"speaker":"Pam","startTime":1510.56,"endTime":1511.79,"body":"it's 2.\" Alright."},{"speaker":"Kim","startTime":1511.79,"endTime":1512.03,"body":"Yep."},{"speaker":"Pam","startTime":1512.06,"endTime":1514.31,"body":"So, I've written down the"},{"speaker":"Pam","startTime":1512.06,"endTime":1514.31,"body":"2. Now, carry on with the"},{"speaker":"Pam","startTime":1514.31,"endTime":1514.55,"body":"algorithm"},{"speaker":"Kim","startTime":1514.55,"endTime":1516.8,"body":"And, then, maybe if you're"},{"speaker":"Kim","startTime":1514.55,"endTime":1516.8,"body":"lucky, they'll look at the whole"},{"speaker":"Kim","startTime":1516.8,"endTime":1519.05,"body":"problem again, but probably not"},{"speaker":"Kim","startTime":1516.8,"endTime":1519.05,"body":"because then they're just moving"},{"speaker":"Kim","startTime":1519.05,"endTime":1520.04,"body":"into the next column."},{"speaker":"Pam","startTime":1520.1,"endTime":1521.57,"body":"Yeah, the trying to"},{"speaker":"Pam","startTime":1520.1,"endTime":1521.57,"body":"remember the next step. Yeah."},{"speaker":"Pam","startTime":1521.57,"endTime":1522.23,"body":"Okay, next column. Mmhmm."},{"speaker":"Kim","startTime":1522.23,"endTime":1526.55,"body":"Yep. So, 2 minus 9. Which"},{"speaker":"Kim","startTime":1522.23,"endTime":1526.55,"body":"they will say, \"Can't do that.\""},{"speaker":"Kim","startTime":1526.7,"endTime":1533.57,"body":"So, you got to go to the 3,"},{"speaker":"Kim","startTime":1526.7,"endTime":1533.57,"body":"cross out the 3, make the 2 a"},{"speaker":"Kim","startTime":1534.74,"endTime":1537.83,"body":"12. And then, 12 minus 9."},{"speaker":"Pam","startTime":1538.07,"endTime":1540.86,"body":"And as you think about that"},{"speaker":"Pam","startTime":1538.07,"endTime":1540.86,"body":"now 12 because you brought that"},{"speaker":"Pam","startTime":1540.86,"endTime":1541.67,"body":"10 over there."},{"speaker":"Both Pam and Kim","startTime":1542.0,"endTime":1542.63,"body":"Yeah."},{"speaker":"Pam","startTime":1543.2,"endTime":1546.89,"body":"Again, minus 9. And so,"},{"speaker":"Pam","startTime":1543.2,"endTime":1546.89,"body":"often, now kids are going to"},{"speaker":"Pam","startTime":1546.89,"endTime":1552.17,"body":"say, \"Alright. 12, 11, 10...\""},{"speaker":"Pam","startTime":1546.89,"endTime":1552.17,"body":"And they count backwards. The 9,"},{"speaker":"Pam","startTime":1552.59,"endTime":1557.51,"body":"right. Even though 12 and 9, if"},{"speaker":"Pam","startTime":1552.59,"endTime":1557.51,"body":"I think about those on a number"},{"speaker":"Pam","startTime":1557.51,"endTime":1560.75,"body":"line, it'd be so much easier to"},{"speaker":"Pam","startTime":1557.51,"endTime":1560.75,"body":"think about those as the"},{"speaker":"Pam","startTime":1560.75,"endTime":1566.45,"body":"distance between 9 and 12,"},{"speaker":"Pam","startTime":1560.75,"endTime":1566.45,"body":"right, to get that 3. So, it's"},{"speaker":"Pam","startTime":1566.45,"endTime":1571.13,"body":"just interesting that if we"},{"speaker":"Pam","startTime":1566.45,"endTime":1571.13,"body":"pigeonhole kids into, \"Here are"},{"speaker":"Pam","startTime":1571.13,"endTime":1576.59,"body":"the steps that thou shalt do,\""},{"speaker":"Pam","startTime":1571.13,"endTime":1576.59,"body":"even in those digit oriented"},{"speaker":"Pam","startTime":1576.59,"endTime":1580.01,"body":"problems, where we've got the 8"},{"speaker":"Pam","startTime":1576.59,"endTime":1580.01,"body":"minus 6, and then 12 minus 9. If"},{"speaker":"Pam","startTime":1580.01,"endTime":1583.55,"body":"we're saying minus, and we're in"},{"speaker":"Pam","startTime":1580.01,"endTime":1583.55,"body":"that algorithm, kids are only"},{"speaker":"Pam","startTime":1583.55,"endTime":1588.53,"body":"thinking about removal. They're"},{"speaker":"Pam","startTime":1583.55,"endTime":1588.53,"body":"not thinking about distance or"},{"speaker":"Pam","startTime":1588.53,"endTime":1591.98,"body":"difference, which would make so"},{"speaker":"Pam","startTime":1588.53,"endTime":1591.98,"body":"much more sense for the exact"},{"speaker":"Pam","startTime":1591.98,"endTime":1593.12,"body":"numbers that we just did, right?"},{"speaker":"Kim","startTime":1593.96,"endTime":1593.99,"body":"Mmhmm."},{"speaker":"Pam","startTime":1594.02,"endTime":1595.58,"body":"Okay. Now, I'm going to let"},{"speaker":"Pam","startTime":1594.02,"endTime":1595.58,"body":"you think. How would you"},{"speaker":"Pam","startTime":1595.58,"endTime":1598.07,"body":"actually do 328 minus 96?"},{"speaker":"Kim","startTime":1599.33,"endTime":1607.67,"body":"So, 328 minus 100 is 228,"},{"speaker":"Kim","startTime":1599.33,"endTime":1607.67,"body":"but I took off too much by 4, so"},{"speaker":"Kim","startTime":1607.67,"endTime":1611.48,"body":"I'm going to add 4 back to get"},{"speaker":"Kim","startTime":1607.67,"endTime":1611.48,"body":"232."},{"speaker":"Pam","startTime":1611.9,"endTime":1615.71,"body":"Because 4 to that 228 you"},{"speaker":"Pam","startTime":1611.9,"endTime":1615.71,"body":"landed on is 232. Nice."},{"speaker":"Kim","startTime":1616.16,"endTime":1616.52,"body":"Yep."},{"speaker":"Pam","startTime":1616.52,"endTime":1619.94,"body":"So, we could think about"},{"speaker":"Pam","startTime":1616.52,"endTime":1619.94,"body":"that. Could you also think"},{"speaker":"Pam","startTime":1619.94,"endTime":1622.04,"body":"about... And that's a great"},{"speaker":"Pam","startTime":1619.94,"endTime":1622.04,"body":"strategy. Could you also think"},{"speaker":"Pam","startTime":1622.04,"endTime":1625.31,"body":"about if I had sort of that 96"},{"speaker":"Pam","startTime":1622.04,"endTime":1625.31,"body":"on a number line, and I had the"},{"speaker":"Pam","startTime":1625.34,"endTime":1629.06,"body":"328 on a number line, and I"},{"speaker":"Pam","startTime":1625.34,"endTime":1629.06,"body":"wanted to find the difference"},{"speaker":"Pam","startTime":1629.06,"endTime":1630.56,"body":"between them? Could you talk"},{"speaker":"Pam","startTime":1629.06,"endTime":1630.56,"body":"through that? What would that"},{"speaker":"Pam","startTime":1630.56,"endTime":1630.89,"body":"look like?"},{"speaker":"Kim","startTime":1631.37,"endTime":1636.95,"body":"Yeah, so if I was at 96,"},{"speaker":"Kim","startTime":1631.37,"endTime":1636.95,"body":"then I'd probably add that 4"},{"speaker":"Kim","startTime":1636.95,"endTime":1642.11,"body":"first, instead of... When I did"},{"speaker":"Kim","startTime":1636.95,"endTime":1642.11,"body":"Over, I added it at the end. But"},{"speaker":"Kim","startTime":1642.11,"endTime":1646.13,"body":"I'd add that first to get to"},{"speaker":"Kim","startTime":1642.11,"endTime":1646.13,"body":"100. Because then once I'm at a"},{"speaker":"Kim","startTime":1646.13,"endTime":1650.9,"body":"friendly number, I know the"},{"speaker":"Kim","startTime":1646.13,"endTime":1650.9,"body":"space between 100 and 328 is"},{"speaker":"Kim","startTime":1651.02,"endTime":1655.91,"body":"228. So, 228 and the 4 is 232."},{"speaker":"Pam","startTime":1656.21,"endTime":1658.97,"body":"Which is then a really fun"},{"speaker":"Pam","startTime":1656.21,"endTime":1658.97,"body":"conversation you can have with"},{"speaker":"Pam","startTime":1658.97,"endTime":1662.39,"body":"students if you compare those"},{"speaker":"Pam","startTime":1658.97,"endTime":1662.39,"body":"two strategies. Because in both"},{"speaker":"Pam","startTime":1662.39,"endTime":1667.37,"body":"cases, you ended up adding 228"},{"speaker":"Pam","startTime":1662.39,"endTime":1667.37,"body":"plus 4. You just did it in a"},{"speaker":"Pam","startTime":1667.37,"endTime":1669.32,"body":"different order. Now, your"},{"speaker":"Pam","startTime":1667.37,"endTime":1669.32,"body":"thought process was completely"},{"speaker":"Pam","startTime":1669.32,"endTime":1672.56,"body":"different, but that's a"},{"speaker":"Pam","startTime":1669.32,"endTime":1672.56,"body":"fascinating conversation to have"},{"speaker":"Pam","startTime":1672.56,"endTime":1674.12,"body":"with students, right?"},{"speaker":"Kim","startTime":1674.12,"endTime":1674.21,"body":"Yep."},{"speaker":"Pam","startTime":1674.21,"endTime":1676.55,"body":"Because in the one case,"},{"speaker":"Pam","startTime":1674.21,"endTime":1676.55,"body":"you subtracted 100, and you"},{"speaker":"Pam","startTime":1676.76,"endTime":1679.79,"body":"landed on 228. So, then, you had"},{"speaker":"Pam","startTime":1676.76,"endTime":1679.79,"body":"to add the 4 back because you"},{"speaker":"Pam","startTime":1679.79,"endTime":1682.7,"body":"subtracted too much. So, 220"},{"speaker":"Pam","startTime":1679.79,"endTime":1682.7,"body":"plus 4. But in the other case,"},{"speaker":"Pam","startTime":1682.7,"endTime":1685.94,"body":"she added the 4 first to get up"},{"speaker":"Pam","startTime":1682.7,"endTime":1685.94,"body":"to the 100, and then you just"},{"speaker":"Pam","startTime":1685.94,"endTime":1688.61,"body":"like, \"Well, and now I still"},{"speaker":"Pam","startTime":1685.94,"endTime":1688.61,"body":"have 228 left to get up to the"},{"speaker":"Pam","startTime":1688.64,"endTime":1694.34,"body":"328.\" In both cases, you ended"},{"speaker":"Pam","startTime":1688.64,"endTime":1694.34,"body":"up adding 4 and 228. But in a"},{"speaker":"Pam","startTime":1694.34,"endTime":1696.89,"body":"different order and for"},{"speaker":"Pam","startTime":1694.34,"endTime":1696.89,"body":"different reasons. But, ya'll,"},{"speaker":"Pam","startTime":1696.92,"endTime":1701.24,"body":"that's making sense of what's"},{"speaker":"Pam","startTime":1696.92,"endTime":1701.24,"body":"happening with place value, and"},{"speaker":"Pam","startTime":1703.01,"endTime":1705.44,"body":"the order of numbers, and the"},{"speaker":"Pam","startTime":1703.01,"endTime":1705.44,"body":"magnitude of numbers, and you're"},{"speaker":"Pam","startTime":1705.44,"endTime":1710.6,"body":"having kids actually think about"},{"speaker":"Pam","startTime":1705.44,"endTime":1710.6,"body":"numbers like 328 and 96, and"},{"speaker":"Pam","startTime":1710.6,"endTime":1714.17,"body":"what they're close to,"},{"speaker":"Pam","startTime":1710.6,"endTime":1714.17,"body":"neighborhood and nearness. Not"},{"speaker":"Pam","startTime":1714.29,"endTime":1720.02,"body":"3-2-8 and 9-6. And 8 minus 6."},{"speaker":"Pam","startTime":1714.29,"endTime":1720.02,"body":"And 2 minus 9. \"Oh, can't do"},{"speaker":"Pam","startTime":1720.02,"endTime":1723.71,"body":"that. Kill the 3, make it a 12.\""},{"speaker":"Pam","startTime":1720.02,"endTime":1723.71,"body":"You know like, all of those"},{"speaker":"Pam","startTime":1723.74,"endTime":1727.73,"body":"super small numbers that can get"},{"speaker":"Pam","startTime":1723.74,"endTime":1727.73,"body":"(unclear). Alright, cool. So,"},{"speaker":"Pam","startTime":1727.76,"endTime":1730.43,"body":"some things, Kim, that we could"},{"speaker":"Pam","startTime":1727.76,"endTime":1730.43,"body":"think about as we leave this"},{"speaker":"Pam","startTime":1730.43,"endTime":1735.59,"body":"podcast. Super young kids, we"},{"speaker":"Pam","startTime":1730.43,"endTime":1735.59,"body":"need to help them count"},{"speaker":"Pam","startTime":1735.59,"endTime":1736.46,"body":"backwards."},{"speaker":"Kim","startTime":1736.73,"endTime":1737.48,"body":"Absolutely."},{"speaker":"Pam","startTime":1737.63,"endTime":1743.96,"body":"They need to be able to"},{"speaker":"Pam","startTime":1737.63,"endTime":1743.96,"body":"think about minus 1 repeatedly"},{"speaker":"Pam","startTime":1743.96,"endTime":1747.71,"body":"is sort of counting backwards."},{"speaker":"Pam","startTime":1743.96,"endTime":1747.71,"body":"So, you can. Now, don't only,"},{"speaker":"Pam","startTime":1747.74,"endTime":1750.05,"body":"teachers of young kids, don't"},{"speaker":"Pam","startTime":1747.74,"endTime":1750.05,"body":"only go, \"Okay. Everybody ready?"},{"speaker":"Pam","startTime":1750.08,"endTime":1751.58,"body":"10, 9, 8...\""},{"speaker":"Kim","startTime":1751.58,"endTime":1751.67,"body":"Yeah."},{"speaker":"Pam","startTime":1751.7,"endTime":1753.08,"body":"That's a fine one. You can"},{"speaker":"Pam","startTime":1751.7,"endTime":1753.08,"body":"do that."},{"speaker":"Kim","startTime":1753.08,"endTime":1753.2,"body":"Yep."},{"speaker":"Pam","startTime":1753.32,"endTime":1755.18,"body":"But also say, \"Hey, we're"},{"speaker":"Pam","startTime":1753.32,"endTime":1755.18,"body":"going to count back today."},{"speaker":"Pam","startTime":1755.18,"endTime":1759.71,"body":"Ready?\" Start from 8. \"8, 7...\""},{"speaker":"Pam","startTime":1755.18,"endTime":1759.71,"body":"Or, \"Let's start from 12. 12...\""},{"speaker":"Pam","startTime":1759.74,"endTime":1763.79,"body":"Or start from 20. Start from 18."},{"speaker":"Pam","startTime":1759.74,"endTime":1763.79,"body":"Like, pick a (unclear) number,"},{"speaker":"Pam","startTime":1763.79,"endTime":1766.85,"body":"and let them count backwards."},{"speaker":"Pam","startTime":1763.79,"endTime":1766.85,"body":"Did I hear you say the other day"},{"speaker":"Pam","startTime":1766.85,"endTime":1769.31,"body":"that you actually had some kids"},{"speaker":"Pam","startTime":1766.85,"endTime":1769.31,"body":"counting backwards by 2's?"},{"speaker":"Kim","startTime":1770.33,"endTime":1774.47,"body":"Yeah. So, I think we were"},{"speaker":"Kim","startTime":1770.33,"endTime":1774.47,"body":"talking about a kindergarten"},{"speaker":"Kim","startTime":1774.47,"endTime":1778.79,"body":"classroom that I was in. And we"},{"speaker":"Kim","startTime":1774.47,"endTime":1778.79,"body":"talked about some Count Arounds,"},{"speaker":"Kim","startTime":1778.85,"endTime":1784.64,"body":"and starting with numbers and"},{"speaker":"Kim","startTime":1778.85,"endTime":1784.64,"body":"counting just back by 1's and"},{"speaker":"Kim","startTime":1784.64,"endTime":1787.73,"body":"writing kind of off to the side"},{"speaker":"Kim","startTime":1784.64,"endTime":1787.73,"body":"\"minus 1, minus 1, minus 1,"},{"speaker":"Kim","startTime":1787.73,"endTime":1791.99,"body":"minus 1\" and looking at"},{"speaker":"Kim","startTime":1787.73,"endTime":1791.99,"body":"patterns. But then, also"},{"speaker":"Kim","startTime":1791.99,"endTime":1796.58,"body":"counting back by 2's to"},{"speaker":"Kim","startTime":1791.99,"endTime":1796.58,"body":"recognize the repeating digits"},{"speaker":"Kim","startTime":1796.58,"endTime":1797.33,"body":"that happened."},{"speaker":"Pam","startTime":1798.95,"endTime":1800.51,"body":"Yep, nice. Nice."},{"speaker":"Kim","startTime":1800.72,"endTime":1804.32,"body":"And, you know, you and I"},{"speaker":"Kim","startTime":1800.72,"endTime":1804.32,"body":"saw a Problem String not too"},{"speaker":"Kim","startTime":1804.32,"endTime":1810.41,"body":"long ago where some second grade"},{"speaker":"Kim","startTime":1804.32,"endTime":1810.41,"body":"students were having some... I"},{"speaker":"Kim","startTime":1810.41,"endTime":1813.77,"body":"don't know if it's difficult."},{"speaker":"Kim","startTime":1810.41,"endTime":1813.77,"body":"They're grappling with the idea"},{"speaker":"Kim","startTime":1813.77,"endTime":1819.17,"body":"of if you're at 80, and you"},{"speaker":"Kim","startTime":1813.77,"endTime":1819.17,"body":"subtract 1 is that 81? Is that"},{"speaker":"Kim","startTime":1819.59,"endTime":1824.63,"body":"89? Is that 79? So, that's a"},{"speaker":"Kim","startTime":1819.59,"endTime":1824.63,"body":"really. It's a real situation."},{"speaker":"Kim","startTime":1824.75,"endTime":1827.36,"body":"And so, giving kids more"},{"speaker":"Kim","startTime":1824.75,"endTime":1827.36,"body":"experience to deal with like"},{"speaker":"Kim","startTime":1827.36,"endTime":1832.64,"body":"that decade before. You know, we"},{"speaker":"Kim","startTime":1827.36,"endTime":1832.64,"body":"do a lot of counting up and a"},{"speaker":"Kim","startTime":1832.64,"endTime":1834.65,"body":"lot of moving forward with"},{"speaker":"Kim","startTime":1832.64,"endTime":1834.65,"body":"addition, but (unclear)."},{"speaker":"Pam","startTime":1834.77,"endTime":1837.14,"body":"Well, and kids get used to"},{"speaker":"Pam","startTime":1834.77,"endTime":1837.14,"body":"that pattern, right? \"76, 77,"},{"speaker":"Pam","startTime":1837.17,"endTime":1840.68,"body":"78, 79...\" And they have to"},{"speaker":"Pam","startTime":1837.17,"endTime":1840.68,"body":"think about what that next"},{"speaker":"Pam","startTime":1840.68,"endTime":1843.11,"body":"decade is. \"80.\" And then, they"},{"speaker":"Pam","startTime":1840.68,"endTime":1843.11,"body":"kind of continue (unclear) \"81,"},{"speaker":"Pam","startTime":1843.14,"endTime":1845.78,"body":"82...\" And so, they get kind of"},{"speaker":"Pam","startTime":1843.14,"endTime":1845.78,"body":"used to that. But we need them"},{"speaker":"Pam","startTime":1845.78,"endTime":1853.43,"body":"to get used to \"82, 81, 80... Is"},{"speaker":"Pam","startTime":1845.78,"endTime":1853.43,"body":"that 89 that comes next? Is that"},{"speaker":"Pam","startTime":1853.79,"endTime":1858.35,"body":"71 that comes next?\" Like,"},{"speaker":"Pam","startTime":1853.79,"endTime":1858.35,"body":"that's a tricky... What's the"},{"speaker":"Pam","startTime":1858.35,"endTime":1861.92,"body":"word I want? Tricky landmark to"},{"speaker":"Pam","startTime":1858.35,"endTime":1861.92,"body":"bump over."},{"speaker":"Kim","startTime":1861.92,"endTime":1861.98,"body":"Yeah."},{"speaker":"Pam","startTime":1862.97,"endTime":1865.61,"body":"Not so much forward. I mean"},{"speaker":"Pam","startTime":1862.97,"endTime":1865.61,"body":"they do that a little more"},{"speaker":"Pam","startTime":1865.61,"endTime":1867.59,"body":"readily. So, we've got to give"},{"speaker":"Pam","startTime":1865.61,"endTime":1867.59,"body":"them experience doing that"},{"speaker":"Pam","startTime":1867.59,"endTime":1872.72,"body":"backward as well. Yeah. Nice."},{"speaker":"Pam","startTime":1867.59,"endTime":1872.72,"body":"Super. So, another thing that we"},{"speaker":"Pam","startTime":1872.72,"endTime":1875.45,"body":"have to help kids when they're"},{"speaker":"Pam","startTime":1872.72,"endTime":1875.45,"body":"super young wrap their brains"},{"speaker":"Pam","startTime":1875.45,"endTime":1882.44,"body":"around is that when they first"},{"speaker":"Pam","startTime":1875.45,"endTime":1882.44,"body":"run into addition, subtraction"},{"speaker":"Pam","startTime":1882.44,"endTime":1885.92,"body":"problems, they're with action."},{"speaker":"Pam","startTime":1882.44,"endTime":1885.92,"body":"And so, we often have problems"},{"speaker":"Pam","startTime":1886.16,"endTime":1888.11,"body":"with addition, where we'll say"},{"speaker":"Pam","startTime":1886.16,"endTime":1888.11,"body":"things like, \"I've got 5"},{"speaker":"Pam","startTime":1888.11,"endTime":1891.8,"body":"puppies. 3 puppies crawl in the"},{"speaker":"Pam","startTime":1888.11,"endTime":1891.8,"body":"room.\" We've got puppies joining"},{"speaker":"Pam","startTime":1891.8,"endTime":1895.7,"body":"the other puppies. But we also"},{"speaker":"Pam","startTime":1891.8,"endTime":1895.7,"body":"might have 7 balloons, and I"},{"speaker":"Pam","startTime":1895.7,"endTime":1899.66,"body":"popped 2 balloons. And we sort"},{"speaker":"Pam","startTime":1895.7,"endTime":1899.66,"body":"of have that removal action. And"},{"speaker":"Pam","startTime":1899.66,"endTime":1903.02,"body":"so, that removal action is a"},{"speaker":"Pam","startTime":1899.66,"endTime":1903.02,"body":"first thing that happens. And we"},{"speaker":"Pam","startTime":1903.02,"endTime":1906.68,"body":"do want kids to think about"},{"speaker":"Pam","startTime":1903.02,"endTime":1906.68,"body":"subtraction as removing, as take"},{"speaker":"Pam","startTime":1906.68,"endTime":1909.23,"body":"away. Those balloons are"},{"speaker":"Pam","startTime":1906.68,"endTime":1909.23,"body":"popping, or I'm eating the"},{"speaker":"Pam","startTime":1909.23,"endTime":1915.2,"body":"M\u0026M's. Or, I'm... Help me."},{"speaker":"Pam","startTime":1909.23,"endTime":1915.2,"body":"Popping, eating... Give me"},{"speaker":"Pam","startTime":1915.2,"endTime":1916.34,"body":"another action, Kim. I'm..."},{"speaker":"Kim","startTime":1916.37,"endTime":1920.21,"body":"[Kim laughs] I can't tell"},{"speaker":"Kim","startTime":1916.37,"endTime":1920.21,"body":"you what I just was thinking."},{"speaker":"Pam","startTime":1920.84,"endTime":1924.38,"body":"Oh, okay. I'm grabbing"},{"speaker":"Pam","startTime":1920.84,"endTime":1924.38,"body":"marbles from my friend's pile of"},{"speaker":"Pam","startTime":1924.38,"endTime":1927.89,"body":"marbles. Like, whatever it is."},{"speaker":"Pam","startTime":1924.38,"endTime":1927.89,"body":"So, that is a first thing. But"},{"speaker":"Pam","startTime":1927.89,"endTime":1930.89,"body":"then, we want to help them make"},{"speaker":"Pam","startTime":1927.89,"endTime":1930.89,"body":"those connections between the"},{"speaker":"Pam","startTime":1930.89,"endTime":1934.31,"body":"addition relationships, so they"},{"speaker":"Pam","startTime":1930.89,"endTime":1934.31,"body":"can start to think about more of"},{"speaker":"Pam","startTime":1934.31,"endTime":1937.55,"body":"a part, part, whole"},{"speaker":"Pam","startTime":1934.31,"endTime":1937.55,"body":"relationship. So, if I've got 3"},{"speaker":"Pam","startTime":1937.55,"endTime":1940.97,"body":"red marbles. Are marbles red? I"},{"speaker":"Pam","startTime":1937.55,"endTime":1940.97,"body":"don't even know. I've got 3 red"},{"speaker":"Pam","startTime":1940.97,"endTime":1943.85,"body":"marbles, and you give me 2 blue"},{"speaker":"Pam","startTime":1940.97,"endTime":1943.85,"body":"marbles, how many total marbles"},{"speaker":"Pam","startTime":1943.85,"endTime":1946.04,"body":"do I have? I want to start"},{"speaker":"Pam","startTime":1943.85,"endTime":1946.04,"body":"connecting that with, \"Well,"},{"speaker":"Pam","startTime":1946.04,"endTime":1948.53,"body":"what if I had those 5 total"},{"speaker":"Pam","startTime":1946.04,"endTime":1948.53,"body":"marbles, and I got rid of those"},{"speaker":"Pam","startTime":1948.53,"endTime":1952.01,"body":"3 marbles? Ooh.\" So, there's"},{"speaker":"Pam","startTime":1948.53,"endTime":1952.01,"body":"this relationship between those"},{"speaker":"Pam","startTime":1952.01,"endTime":1955.04,"body":"removal and those combining"},{"speaker":"Pam","startTime":1952.01,"endTime":1955.04,"body":"problems."},{"speaker":"Kim","startTime":1955.04,"endTime":1955.1,"body":"Yeah."},{"speaker":"Pam","startTime":1955.37,"endTime":1955.76,"body":"And at"},{"speaker":"Pam","startTime":1955.76,"endTime":1956.69,"body":"first. Oh, go ahead."},{"speaker":"Kim","startTime":1957.26,"endTime":1960.11,"body":"Oh, well, I was just going"},{"speaker":"Kim","startTime":1957.26,"endTime":1960.11,"body":"to say. And that is super"},{"speaker":"Kim","startTime":1960.11,"endTime":1962.66,"body":"helpful where you could do the"},{"speaker":"Kim","startTime":1960.11,"endTime":1962.66,"body":"work with a number line"},{"speaker":"Kim","startTime":1962.66,"endTime":1962.69,"body":"(unclear)."},{"speaker":"Pam","startTime":1963.05,"endTime":1964.61,"body":"Because we can see both of"},{"speaker":"Pam","startTime":1963.05,"endTime":1964.61,"body":"those parts."},{"speaker":"Kim","startTime":1964.64,"endTime":1969.38,"body":"Yeah, and it's not just"},{"speaker":"Kim","startTime":1964.64,"endTime":1969.38,"body":"writing and rewriting fact"},{"speaker":"Kim","startTime":1969.38,"endTime":1970.73,"body":"families over and over again."},{"speaker":"Pam","startTime":1970.82,"endTime":1971.12,"body":"Yeah."},{"speaker":"Kim","startTime":1971.12,"endTime":1971.15,"body":"(unclear)."},{"speaker":"Pam","startTime":1971.33,"endTime":1974.39,"body":"Well, and you could do the"},{"speaker":"Pam","startTime":1971.33,"endTime":1974.39,"body":"same kind of work with a number"},{"speaker":"Pam","startTime":1974.39,"endTime":1977.6,"body":"rack because I've got those 5"},{"speaker":"Pam","startTime":1974.39,"endTime":1977.6,"body":"beads, right? And I can see the"},{"speaker":"Pam","startTime":1977.6,"endTime":1980.54,"body":"3, and I can see the 2 beads."},{"speaker":"Pam","startTime":1977.6,"endTime":1980.54,"body":"So, I can have the 5, and I"},{"speaker":"Pam","startTime":1980.54,"endTime":1985.37,"body":"could remove 2. But I can have"},{"speaker":"Pam","startTime":1980.54,"endTime":1985.37,"body":"the 3, and I can bring in 2. I"},{"speaker":"Pam","startTime":1985.37,"endTime":1988.19,"body":"think what you were going to say"},{"speaker":"Pam","startTime":1985.37,"endTime":1988.19,"body":"is it's not just saying to kids,"},{"speaker":"Pam","startTime":1988.49,"endTime":1991.19,"body":"\"This is a fact family.\" And,"},{"speaker":"Pam","startTime":1988.49,"endTime":1991.19,"body":"\"Here, write it these 4 ways."},{"speaker":"Pam","startTime":1991.19,"endTime":1993.44,"body":"No, not that way. That's the"},{"speaker":"Pam","startTime":1991.19,"endTime":1993.44,"body":"dumb way. That doesn't...\" It's"},{"speaker":"Pam","startTime":1993.44,"endTime":1996.98,"body":"not about just memorizing ways"},{"speaker":"Pam","startTime":1993.44,"endTime":1996.98,"body":"of writing equations. It's"},{"speaker":"Pam","startTime":1996.98,"endTime":2000.34,"body":"really about understanding the"},{"speaker":"Pam","startTime":1996.98,"endTime":2000.34,"body":"relationship between addition"},{"speaker":"Pam","startTime":2000.34,"endTime":2002.98,"body":"and subtraction. Cool. Yep. And"},{"speaker":"Pam","startTime":2000.34,"endTime":2002.98,"body":"what I was going to say is,"},{"speaker":"Pam","startTime":2002.98,"endTime":2005.92,"body":"that's where CGI research comes"},{"speaker":"Pam","startTime":2002.98,"endTime":2005.92,"body":"in, and it's so helpful to help"},{"speaker":"Pam","startTime":2005.92,"endTime":2009.1,"body":"us recognize that we want to"},{"speaker":"Pam","startTime":2005.92,"endTime":2009.1,"body":"give kids different kinds of"},{"speaker":"Pam","startTime":2009.1,"endTime":2013.3,"body":"problems as they are ready for"},{"speaker":"Pam","startTime":2009.1,"endTime":2013.3,"body":"them, as it's on the edge of"},{"speaker":"Pam","startTime":2013.3,"endTime":2015.31,"body":"their zone of proximal"},{"speaker":"Pam","startTime":2013.3,"endTime":2015.31,"body":"development, that's when we want"},{"speaker":"Pam","startTime":2015.31,"endTime":2019.03,"body":"to give them. We want to start"},{"speaker":"Pam","startTime":2015.31,"endTime":2019.03,"body":"with those action oriented,"},{"speaker":"Pam","startTime":2019.24,"endTime":2022.33,"body":"combining or separating"},{"speaker":"Pam","startTime":2019.24,"endTime":2022.33,"body":"problems. But then, we want to"},{"speaker":"Pam","startTime":2022.33,"endTime":2024.49,"body":"give them those part, part,"},{"speaker":"Pam","startTime":2022.33,"endTime":2024.49,"body":"whole problems where there's not"},{"speaker":"Pam","startTime":2024.49,"endTime":2027.22,"body":"so much action. And then, we"},{"speaker":"Pam","startTime":2024.49,"endTime":2027.22,"body":"want to give them those"},{"speaker":"Pam","startTime":2027.22,"endTime":2030.7,"body":"comparison problems because in"},{"speaker":"Pam","startTime":2027.22,"endTime":2030.7,"body":"those comparison problems,"},{"speaker":"Pam","startTime":2030.7,"endTime":2034.0,"body":"that's really where that"},{"speaker":"Pam","startTime":2030.7,"endTime":2034.0,"body":"difference, the distance meaning"},{"speaker":"Pam","startTime":2034.0,"endTime":2038.32,"body":"really comes in. Now, I've got 5"},{"speaker":"Pam","startTime":2034.0,"endTime":2038.32,"body":"marbles. You've got 3 more"},{"speaker":"Pam","startTime":2038.32,"endTime":2040.66,"body":"marbles than I do. How many"},{"speaker":"Pam","startTime":2038.32,"endTime":2040.66,"body":"marbles do you have? Or, \"I've"},{"speaker":"Pam","startTime":2040.72,"endTime":2044.38,"body":"got 5 marbles, and you've got 8"},{"speaker":"Pam","startTime":2040.72,"endTime":2044.38,"body":"marbles. How many more marbles"},{"speaker":"Pam","startTime":2045.07,"endTime":2048.46,"body":"do you have than I have?\" Those"},{"speaker":"Pam","startTime":2045.07,"endTime":2048.46,"body":"comparison problems really lend"},{"speaker":"Pam","startTime":2048.46,"endTime":2052.72,"body":"themselves to that distance or"},{"speaker":"Pam","startTime":2048.46,"endTime":2052.72,"body":"that difference meaning of"},{"speaker":"Pam","startTime":2052.72,"endTime":2057.55,"body":"subtraction. And so, we really,"},{"speaker":"Pam","startTime":2052.72,"endTime":2057.55,"body":"really want to emphasize that"},{"speaker":"Pam","startTime":2057.55,"endTime":2061.39,"body":"those hardest kinds of"},{"speaker":"Pam","startTime":2057.55,"endTime":2061.39,"body":"problems... No, let me say this."},{"speaker":"Pam","startTime":2061.57,"endTime":2064.36,"body":"We really want to emphasize that"},{"speaker":"Pam","startTime":2061.57,"endTime":2064.36,"body":"subtraction, we need to have"},{"speaker":"Pam","startTime":2064.36,"endTime":2068.05,"body":"both of those meanings"},{"speaker":"Pam","startTime":2064.36,"endTime":2068.05,"body":"happening. We need to build the"},{"speaker":"Pam","startTime":2068.05,"endTime":2071.83,"body":"connection between finding the"},{"speaker":"Pam","startTime":2068.05,"endTime":2071.83,"body":"difference between numbers and"},{"speaker":"Pam","startTime":2071.83,"endTime":2075.34,"body":"removing, and that both of those"},{"speaker":"Pam","startTime":2071.83,"endTime":2075.34,"body":"have everything to do with"},{"speaker":"Pam","startTime":2075.34,"endTime":2079.0,"body":"subtraction. So, Kim, I think we"},{"speaker":"Pam","startTime":2075.34,"endTime":2079.0,"body":"wanted to end with some things"},{"speaker":"Pam","startTime":2079.0,"endTime":2081.82,"body":"that teachers can do with their"},{"speaker":"Pam","startTime":2079.0,"endTime":2081.82,"body":"students."},{"speaker":"Kim","startTime":2082.06,"endTime":2085.27,"body":"Yeah, so we mentioned quite"},{"speaker":"Kim","startTime":2082.06,"endTime":2085.27,"body":"a few within, so I'll just bring"},{"speaker":"Kim","startTime":2085.27,"endTime":2088.18,"body":"them back out again. Counting"},{"speaker":"Kim","startTime":2085.27,"endTime":2088.18,"body":"backwards. Super important"},{"speaker":"Kim","startTime":2088.21,"endTime":2091.69,"body":"particularly with really young"},{"speaker":"Kim","startTime":2088.21,"endTime":2091.69,"body":"students. Play I Have, You Need."},{"speaker":"Kim","startTime":2092.05,"endTime":2095.08,"body":"Talk about both types of"},{"speaker":"Kim","startTime":2092.05,"endTime":2095.08,"body":"subtraction. The distance,"},{"speaker":"Kim","startTime":2095.11,"endTime":2098.26,"body":"difference representation, and"},{"speaker":"Kim","startTime":2095.11,"endTime":2098.26,"body":"removal. So, this is both"},{"speaker":"Kim","startTime":2098.26,"endTime":2099.13,"body":"interpretations."},{"speaker":"Pam","startTime":2100.03,"endTime":2100.06,"body":"Mmhmm."},{"speaker":"Kim","startTime":2100.36,"endTime":2103.87,"body":"Represent subtraction on a"},{"speaker":"Kim","startTime":2100.36,"endTime":2103.87,"body":"number line and rekenrek to give"},{"speaker":"Kim","startTime":2103.87,"endTime":2108.46,"body":"kids a feeling about subtraction"},{"speaker":"Kim","startTime":2103.87,"endTime":2108.46,"body":"not just the fact families."},{"speaker":"Pam","startTime":2108.46,"endTime":2108.49,"body":"Mmhmm."},{"speaker":"Kim","startTime":2108.85,"endTime":2112.33,"body":"And consider where"},{"speaker":"Kim","startTime":2108.85,"endTime":2112.33,"body":"subtraction impacts other"},{"speaker":"Kim","startTime":2112.33,"endTime":2116.62,"body":"operations in your grade because"},{"speaker":"Kim","startTime":2112.33,"endTime":2116.62,"body":"those might be the places where"},{"speaker":"Kim","startTime":2116.62,"endTime":2119.98,"body":"your kids get hung up. Not in"},{"speaker":"Kim","startTime":2116.62,"endTime":2119.98,"body":"the multiplication, not in the"},{"speaker":"Kim","startTime":2119.98,"endTime":2122.98,"body":"division, but that little bit of"},{"speaker":"Kim","startTime":2119.98,"endTime":2122.98,"body":"subtraction at the end. And"},{"speaker":"Kim","startTime":2122.98,"endTime":2124.63,"body":"then, they get a wrong answer."},{"speaker":"Kim","startTime":2122.98,"endTime":2124.63,"body":"And then, we think they don't"},{"speaker":"Kim","startTime":2124.63,"endTime":2127.66,"body":"know what's going on. When"},{"speaker":"Kim","startTime":2124.63,"endTime":2127.66,"body":"really they just need more"},{"speaker":"Kim","startTime":2127.66,"endTime":2128.44,"body":"subtraction work."},{"speaker":"Pam","startTime":2128.77,"endTime":2131.65,"body":"Yeah, nicely said. Alright,"},{"speaker":"Pam","startTime":2128.77,"endTime":2131.65,"body":"ya'll, this was a little bit of"},{"speaker":"Pam","startTime":2131.65,"endTime":2135.07,"body":"a long one. Thank you for tuning"},{"speaker":"Pam","startTime":2131.65,"endTime":2135.07,"body":"in and teaching more and more"},{"speaker":"Pam","startTime":2135.07,"endTime":2137.41,"body":"real math. To find out more"},{"speaker":"Pam","startTime":2135.07,"endTime":2137.41,"body":"about the Math is"},{"speaker":"Pam","startTime":2137.44,"endTime":2141.7,"body":"Figure-Out-Able movement, visit"},{"speaker":"Pam","startTime":2137.44,"endTime":2141.7,"body":"mathisfigureoutable.com. Let's"},{"speaker":"Pam","startTime":2141.7,"endTime":2144.37,"body":"keep spreading the word that"},{"speaker":"Pam","startTime":2141.7,"endTime":2144.37,"body":"Math is Figure-Out-Able!"}]}