{"version":"1.0.0","segments":[{"speaker":"Pam","startTime":2.52,"endTime":5.94,"body":"Hey, fellow mathematicians!"},{"speaker":"Pam","startTime":2.52,"endTime":5.94,"body":"Welcome to the podcast where"},{"speaker":"Pam","startTime":5.94,"endTime":8.79,"body":"Math is Figure-Out-Able! I'm"},{"speaker":"Pam","startTime":5.94,"endTime":8.79,"body":"Pam."},{"speaker":"Kim","startTime":8.97,"endTime":9.66,"body":"And I'm Kim."},{"speaker":"Pam","startTime":9.96,"endTime":13.26,"body":"And you found a place where"},{"speaker":"Pam","startTime":9.96,"endTime":13.26,"body":"math is not about memorizing and"},{"speaker":"Pam","startTime":13.26,"endTime":16.89,"body":"mimicking, waiting to be told or"},{"speaker":"Pam","startTime":13.26,"endTime":16.89,"body":"shown what to do. But, ya'll,"},{"speaker":"Pam","startTime":16.89,"endTime":20.19,"body":"it's about making sense of"},{"speaker":"Pam","startTime":16.89,"endTime":20.19,"body":"problems, noticing patterns, and"},{"speaker":"Pam","startTime":20.19,"endTime":24.03,"body":"reasoning using mathematical"},{"speaker":"Pam","startTime":20.19,"endTime":24.03,"body":"relationships. We can mentor"},{"speaker":"Pam","startTime":24.03,"endTime":28.02,"body":"students to think and reason"},{"speaker":"Pam","startTime":24.03,"endTime":28.02,"body":"like mathematicians. Not only"},{"speaker":"Pam","startTime":28.02,"endTime":31.32,"body":"are algorithms not really"},{"speaker":"Pam","startTime":28.02,"endTime":31.32,"body":"particularly helpful in teaching"},{"speaker":"Pam","startTime":31.32,"endTime":34.83,"body":"mathematics, but rotely"},{"speaker":"Pam","startTime":31.32,"endTime":34.83,"body":"repeating steps actually keep"},{"speaker":"Pam","startTime":34.83,"endTime":37.53,"body":"students from being the"},{"speaker":"Pam","startTime":34.83,"endTime":37.53,"body":"mathematicians they can be."},{"speaker":"Kim","startTime":38.22,"endTime":40.41,"body":"Today, we're going to be"},{"speaker":"Kim","startTime":38.22,"endTime":40.41,"body":"wrapping up our series on the"},{"speaker":"Kim","startTime":40.41,"endTime":44.73,"body":"area model and taking it a bit"},{"speaker":"Kim","startTime":40.41,"endTime":44.73,"body":"higher grade wise, as we examine"},{"speaker":"Kim","startTime":44.73,"endTime":48.03,"body":"how the area model impacts"},{"speaker":"Kim","startTime":44.73,"endTime":48.03,"body":"middle and high school."},{"speaker":"Pam","startTime":48.39,"endTime":52.35,"body":"Yeah, so area model Middle,"},{"speaker":"Pam","startTime":48.39,"endTime":52.35,"body":"High School. Let's keep going."},{"speaker":"Pam","startTime":52.35,"endTime":55.74,"body":"So, first of all, teachers of"},{"speaker":"Pam","startTime":52.35,"endTime":55.74,"body":"younger students, please use the"},{"speaker":"Pam","startTime":55.74,"endTime":58.77,"body":"area model well. Check out the"},{"speaker":"Pam","startTime":55.74,"endTime":58.77,"body":"last two episodes. Please use it"},{"speaker":"Pam","startTime":58.77,"endTime":62.31,"body":"well because then we can take"},{"speaker":"Pam","startTime":58.77,"endTime":62.31,"body":"that and do marvelous things"},{"speaker":"Pam","startTime":62.31,"endTime":67.11,"body":"with it higher on, higher up,"},{"speaker":"Pam","startTime":62.31,"endTime":67.11,"body":"higher grades. One of the things"},{"speaker":"Pam","startTime":67.11,"endTime":72.24,"body":"is that because the chunking of"},{"speaker":"Pam","startTime":67.11,"endTime":72.24,"body":"the areas on an area model is"},{"speaker":"Pam","startTime":72.24,"endTime":75.51,"body":"based on the distributive"},{"speaker":"Pam","startTime":72.24,"endTime":75.51,"body":"property, we are for sure going"},{"speaker":"Pam","startTime":75.51,"endTime":79.17,"body":"to use that in higher math, and"},{"speaker":"Pam","startTime":75.51,"endTime":79.17,"body":"so we really want to have that"},{"speaker":"Pam","startTime":79.17,"endTime":83.76,"body":"good basis with area, so that we"},{"speaker":"Pam","startTime":79.17,"endTime":83.76,"body":"can take that intuition and we"},{"speaker":"Pam","startTime":83.76,"endTime":87.54,"body":"can build on it. One way we can"},{"speaker":"Pam","startTime":83.76,"endTime":87.54,"body":"do that is at the middle school"},{"speaker":"Pam","startTime":87.54,"endTime":91.71,"body":"level, where we have things like"},{"speaker":"Pam","startTime":87.54,"endTime":91.71,"body":"fractions and decimals, and we"},{"speaker":"Pam","startTime":91.71,"endTime":95.85,"body":"can help students kind of"},{"speaker":"Pam","startTime":91.71,"endTime":95.85,"body":"reason, build spatial reasoning"},{"speaker":"Pam","startTime":96.18,"endTime":99.57,"body":"with, say, fraction"},{"speaker":"Pam","startTime":96.18,"endTime":99.57,"body":"multiplication. So, for example,"},{"speaker":"Pam","startTime":99.78,"endTime":104.01,"body":"if I were to give some kind of"},{"speaker":"Pam","startTime":99.78,"endTime":104.01,"body":"lovely problem like 2 and a 1/2."},{"speaker":"Pam","startTime":104.04,"endTime":109.32,"body":"So, 2 and 1/2 times 1 and 2/5."},{"speaker":"Pam","startTime":104.04,"endTime":109.32,"body":"If that was the problem that"},{"speaker":"Pam","startTime":109.32,"endTime":113.49,"body":"we're going to deal with. When I"},{"speaker":"Pam","startTime":109.32,"endTime":113.49,"body":"originally, initially begin to"},{"speaker":"Pam","startTime":113.49,"endTime":116.4,"body":"start working with students with"},{"speaker":"Pam","startTime":113.49,"endTime":116.4,"body":"that problem, I would want to"},{"speaker":"Pam","startTime":116.4,"endTime":119.97,"body":"build some chunks of areas. I"},{"speaker":"Pam","startTime":116.4,"endTime":119.97,"body":"would want to use the area model"},{"speaker":"Pam","startTime":119.97,"endTime":123.54,"body":"to build intuition. So, I might"},{"speaker":"Pam","startTime":119.97,"endTime":123.54,"body":"say, \"Hey, students? What would"},{"speaker":"Pam","startTime":123.54,"endTime":127.23,"body":"that look like? What are you"},{"speaker":"Pam","startTime":123.54,"endTime":127.23,"body":"thinking about when you see 2"},{"speaker":"Pam","startTime":127.23,"endTime":130.47,"body":"and a 1/2 times 1 and 2/5?\" And"},{"speaker":"Pam","startTime":127.23,"endTime":130.47,"body":"so, we might draw. I'm drawing"},{"speaker":"Pam","startTime":130.47,"endTime":133.68,"body":"on my paper right now something"},{"speaker":"Pam","startTime":130.47,"endTime":133.68,"body":"that's 2 long, and then 1/2 long"},{"speaker":"Pam","startTime":133.89,"endTime":138.27,"body":"by something that's 1 long, and"},{"speaker":"Pam","startTime":133.89,"endTime":138.27,"body":"then 2/5 long. Like, that sort"},{"speaker":"Pam","startTime":138.27,"endTime":142.29,"body":"of estimate there about how long"},{"speaker":"Pam","startTime":138.27,"endTime":142.29,"body":"that would be. And could we"},{"speaker":"Pam","startTime":142.53,"endTime":148.47,"body":"recognize that as an area that"},{"speaker":"Pam","startTime":142.53,"endTime":148.47,"body":"is 2 and a 1/2 by 1 and 2/5? And"},{"speaker":"Pam","startTime":148.47,"endTime":151.2,"body":"so, I might represent that, and"},{"speaker":"Pam","startTime":148.47,"endTime":151.2,"body":"then I might say, \"What are some"},{"speaker":"Pam","startTime":151.2,"endTime":155.28,"body":"chunks you know?\" And early on,"},{"speaker":"Pam","startTime":151.2,"endTime":155.28,"body":"as we're making sense of"},{"speaker":"Pam","startTime":155.67,"endTime":157.47,"body":"(unclear) fraction"},{"speaker":"Pam","startTime":155.67,"endTime":157.47,"body":"multiplication, I would expect"},{"speaker":"Pam","startTime":157.47,"endTime":164.04,"body":"students to cut that into that 2"},{"speaker":"Pam","startTime":157.47,"endTime":164.04,"body":"and a 1/2 by 1 and 2/5 into"},{"speaker":"Pam","startTime":164.07,"endTime":168.21,"body":"probably 4 chunks where they"},{"speaker":"Pam","startTime":164.07,"endTime":168.21,"body":"might say, \"Okay, I've got a 2."},{"speaker":"Pam","startTime":168.21,"endTime":172.56,"body":"I cut the 2 and a 1/2 into 2,"},{"speaker":"Pam","startTime":168.21,"endTime":172.56,"body":"and 1/2. And I would cut the 1"},{"speaker":"Pam","startTime":172.56,"endTime":177.48,"body":"and 2/5 into 1, and 2/5. And"},{"speaker":"Pam","startTime":172.56,"endTime":177.48,"body":"that's going to end up with kind"},{"speaker":"Pam","startTime":177.48,"endTime":180.33,"body":"of... It's going to be similar"},{"speaker":"Pam","startTime":177.48,"endTime":180.33,"body":"to what a Partial Product might"},{"speaker":"Pam","startTime":180.33,"endTime":183.45,"body":"look like, a Partial Place"},{"speaker":"Pam","startTime":180.33,"endTime":183.45,"body":"Value, Partial Product model"},{"speaker":"Pam","startTime":183.45,"endTime":187.23,"body":"might look like in whole"},{"speaker":"Pam","startTime":183.45,"endTime":187.23,"body":"numbers, where I've kind of have"},{"speaker":"Pam","startTime":187.23,"endTime":190.92,"body":"these 4 chunks I'm dealing with."},{"speaker":"Pam","startTime":187.23,"endTime":190.92,"body":"And one of the chunks is that 2"},{"speaker":"Pam","startTime":190.92,"endTime":195.21,"body":"by 1 chunk. So, the area of a 2"},{"speaker":"Pam","startTime":190.92,"endTime":195.21,"body":"by 1 is 2. So, I've got a 2 by"},{"speaker":"Pam","startTime":195.21,"endTime":198.48,"body":"1. In the middle that, I've got"},{"speaker":"Pam","startTime":195.21,"endTime":198.48,"body":"2. I've also got a chunk that is"},{"speaker":"Pam","startTime":198.48,"endTime":201.99,"body":"that 1/2 by 1. And so, I'm"},{"speaker":"Pam","startTime":198.48,"endTime":201.99,"body":"thinking now students are going"},{"speaker":"Pam","startTime":201.99,"endTime":204.96,"body":"to have to think here, \"What is"},{"speaker":"Pam","startTime":201.99,"endTime":204.96,"body":"the area of something that has a"},{"speaker":"Pam","startTime":204.96,"endTime":208.23,"body":"dimension of 1/2 by 1?\" And I"},{"speaker":"Pam","startTime":204.96,"endTime":208.23,"body":"want students to actually have"},{"speaker":"Pam","startTime":208.23,"endTime":211.44,"body":"to think about that. Like, can"},{"speaker":"Pam","startTime":208.23,"endTime":211.44,"body":"they think about a 1 by 1, and"},{"speaker":"Pam","startTime":211.44,"endTime":213.87,"body":"then cut it in half to get a 1/2"},{"speaker":"Pam","startTime":211.44,"endTime":213.87,"body":"by 1? Like, there's lots of"},{"speaker":"Pam","startTime":213.87,"endTime":216.6,"body":"different. I want students to"},{"speaker":"Pam","startTime":213.87,"endTime":216.6,"body":"grapple with that relationship,"},{"speaker":"Pam","startTime":217.2,"endTime":221.19,"body":"that the area of rectangle"},{"speaker":"Pam","startTime":217.2,"endTime":221.19,"body":"that's 1/2 by 1 is indeed 1/2."},{"speaker":"Pam","startTime":221.55,"endTime":223.65,"body":"Then, I want them to grapple"},{"speaker":"Pam","startTime":221.55,"endTime":223.65,"body":"with this other chunk over"},{"speaker":"Pam","startTime":223.65,"endTime":228.45,"body":"there, that would be 2 by 2/5."},{"speaker":"Pam","startTime":223.65,"endTime":228.45,"body":"What does it mean to have a"},{"speaker":"Pam","startTime":228.45,"endTime":232.77,"body":"rectangle that's 2 by 2/5? Is"},{"speaker":"Pam","startTime":228.45,"endTime":232.77,"body":"that 4/5? Like, what is twice"},{"speaker":"Pam","startTime":233.07,"endTime":238.17,"body":"2/5's? Is that 4/5? And then, I"},{"speaker":"Pam","startTime":233.07,"endTime":238.17,"body":"have this little tiny rectangle"},{"speaker":"Pam","startTime":238.17,"endTime":242.82,"body":"in the bottom that would be 1/2"},{"speaker":"Pam","startTime":238.17,"endTime":242.82,"body":"by 2/5. I'm looking at my"},{"speaker":"Pam","startTime":243.03,"endTime":246.9,"body":"drawing. It's not too... The"},{"speaker":"Pam","startTime":243.03,"endTime":246.9,"body":"scale's not too, too bad. But a"},{"speaker":"Pam","startTime":246.9,"endTime":250.62,"body":"1/2 by 2/5? What in the world?"},{"speaker":"Pam","startTime":246.9,"endTime":250.62,"body":"Now, students really have to"},{"speaker":"Pam","startTime":250.62,"endTime":255.66,"body":"think about what the area of a"},{"speaker":"Pam","startTime":250.62,"endTime":255.66,"body":"1/2 of 2/5 is. But I also want"},{"speaker":"Pam","startTime":255.66,"endTime":260.25,"body":"students thinking about a 1/2 of"},{"speaker":"Pam","startTime":255.66,"endTime":260.25,"body":"2/5. Could they think about 1 by"},{"speaker":"Pam","startTime":260.25,"endTime":267.12,"body":"2/5 to cut it in half to get 1/2"},{"speaker":"Pam","startTime":260.25,"endTime":267.12,"body":"by 2/5? So, is the area of 1/2"},{"speaker":"Pam","startTime":267.12,"endTime":272.76,"body":"by 2/5 just 1/2 of 2/5? Is that"},{"speaker":"Pam","startTime":267.12,"endTime":272.76,"body":"like 1/5? So, now in this"},{"speaker":"Pam","startTime":272.76,"endTime":276.66,"body":"rectangle, I've got 4 chunks of"},{"speaker":"Pam","startTime":272.76,"endTime":276.66,"body":"area. And they're not square by"},{"speaker":"Pam","startTime":276.66,"endTime":279.39,"body":"the way. Maybe I should have"},{"speaker":"Pam","startTime":276.66,"endTime":279.39,"body":"said that. This is not... 2 and"},{"speaker":"Pam","startTime":279.39,"endTime":282.45,"body":"a 1/2 by 1 and 2/5 is not"},{"speaker":"Pam","startTime":279.39,"endTime":282.45,"body":"square. It's rectangular. It's"},{"speaker":"Pam","startTime":282.45,"endTime":285.45,"body":"longer than it is wide. Then,"},{"speaker":"Pam","startTime":282.45,"endTime":285.45,"body":"I've got these rectangles in"},{"speaker":"Pam","startTime":285.45,"endTime":287.58,"body":"there that I'm going to add up."},{"speaker":"Pam","startTime":285.45,"endTime":287.58,"body":"So, I've got a 2, and I've got"},{"speaker":"Pam","startTime":287.94,"endTime":292.44,"body":"an area of 2, an area of 1/2, an"},{"speaker":"Pam","startTime":287.94,"endTime":292.44,"body":"area 4/5, an area of 1/5. Well,"},{"speaker":"Pam","startTime":292.44,"endTime":294.99,"body":"now I want to strategically add"},{"speaker":"Pam","startTime":292.44,"endTime":294.99,"body":"those together. Because I don't"},{"speaker":"Pam","startTime":294.99,"endTime":296.79,"body":"know if you caught that,"},{"speaker":"Pam","startTime":294.99,"endTime":296.79,"body":"especially if you're listening"},{"speaker":"Pam","startTime":296.79,"endTime":299.04,"body":"as you're driving in the car."},{"speaker":"Pam","startTime":296.79,"endTime":299.04,"body":"Ya'll, grab a pencil as we're"},{"speaker":"Pam","startTime":299.04,"endTime":303.18,"body":"doing this. I've got a rectangle"},{"speaker":"Pam","startTime":299.04,"endTime":303.18,"body":"that is four-fifths, an area"},{"speaker":"Pam","startTime":303.21,"endTime":304.95,"body":"four-fifths, and I've got a"},{"speaker":"Pam","startTime":303.21,"endTime":304.95,"body":"rectangle that has an area of"},{"speaker":"Pam","startTime":304.98,"endTime":307.5,"body":"one-fifth. Well, what's"},{"speaker":"Pam","startTime":304.98,"endTime":307.5,"body":"four-fifths and one-fifth? Is"},{"speaker":"Pam","startTime":307.5,"endTime":312.0,"body":"that just five-fifths? Bam! So,"},{"speaker":"Pam","startTime":307.5,"endTime":312.0,"body":"there's an area of 1. Then, I've"},{"speaker":"Pam","startTime":312.0,"endTime":316.14,"body":"also got that area of 2, so"},{"speaker":"Pam","startTime":312.0,"endTime":316.14,"body":"that's 3 so far plus that extra"},{"speaker":"Pam","startTime":316.17,"endTime":320.49,"body":"1/2, and so we end up with 3 and"},{"speaker":"Pam","startTime":316.17,"endTime":320.49,"body":"a 1/2 is the area. Now, not all"},{"speaker":"Pam","startTime":320.49,"endTime":324.57,"body":"problems will end up where the"},{"speaker":"Pam","startTime":320.49,"endTime":324.57,"body":"area chunk so nicely, but I want"},{"speaker":"Pam","startTime":324.57,"endTime":327.57,"body":"to, I think, choose some"},{"speaker":"Pam","startTime":324.57,"endTime":327.57,"body":"strategic fraction"},{"speaker":"Pam","startTime":327.57,"endTime":330.6,"body":"multiplication problems and help"},{"speaker":"Pam","startTime":327.57,"endTime":330.6,"body":"really build that sense of"},{"speaker":"Pam","startTime":330.6,"endTime":332.28,"body":"spatial understanding."},{"speaker":"Kim","startTime":332.28,"endTime":332.34,"body":"Yeah."},{"speaker":"Pam","startTime":332.67,"endTime":335.58,"body":"What's actually happening"},{"speaker":"Pam","startTime":332.67,"endTime":335.58,"body":"with the problem? Yeah."},{"speaker":"Kim","startTime":335.61,"endTime":338.67,"body":"Yeah. You know, it's really"},{"speaker":"Kim","startTime":335.61,"endTime":338.67,"body":"nice because I think you can..."},{"speaker":"Kim","startTime":339.09,"endTime":343.86,"body":"As teachers, we're not bound by"},{"speaker":"Kim","startTime":339.09,"endTime":343.86,"body":"certain numbers, right? Like, we"},{"speaker":"Kim","startTime":343.86,"endTime":346.41,"body":"don't have to throw the ugliest"},{"speaker":"Kim","startTime":343.86,"endTime":346.41,"body":"ones up front. So, I love this"},{"speaker":"Kim","startTime":346.41,"endTime":352.11,"body":"problem because the chunks that"},{"speaker":"Kim","startTime":346.41,"endTime":352.11,"body":"you would see students maybe do"},{"speaker":"Kim","startTime":352.11,"endTime":356.01,"body":"place value chunks with, then"},{"speaker":"Kim","startTime":352.11,"endTime":356.01,"body":"doesn't leave them with horrible"},{"speaker":"Kim","startTime":356.01,"endTime":358.83,"body":"addition at the end. And you're"},{"speaker":"Kim","startTime":356.01,"endTime":358.83,"body":"relieving them from that"},{"speaker":"Kim","startTime":358.83,"endTime":362.13,"body":"struggle at the end to really"},{"speaker":"Kim","startTime":358.83,"endTime":362.13,"body":"think about the area."},{"speaker":"Pam","startTime":362.22,"endTime":363.66,"body":"Yeah, to focus on the area."},{"speaker":"Kim","startTime":363.93,"endTime":364.29,"body":"Mmhmm, yeah."},{"speaker":"Pam","startTime":364.32,"endTime":367.38,"body":"And that's a big paradigm"},{"speaker":"Pam","startTime":364.32,"endTime":367.38,"body":"shift, I think, for many math"},{"speaker":"Pam","startTime":367.38,"endTime":371.01,"body":"teachers is, I was sort of"},{"speaker":"Pam","startTime":367.38,"endTime":371.01,"body":"raised with this idea that we"},{"speaker":"Pam","startTime":371.01,"endTime":374.73,"body":"have to make sure that they have"},{"speaker":"Pam","startTime":371.01,"endTime":374.73,"body":"this general procedure because"},{"speaker":"Pam","startTime":374.76,"endTime":376.65,"body":"you better deal with these"},{"speaker":"Pam","startTime":374.76,"endTime":376.65,"body":"easier numbers now because"},{"speaker":"Pam","startTime":376.65,"endTime":379.08,"body":"you're going to need it for the"},{"speaker":"Pam","startTime":376.65,"endTime":379.08,"body":"harder numbers later. When in"},{"speaker":"Pam","startTime":379.08,"endTime":382.59,"body":"reality, it's not about easy and"},{"speaker":"Pam","startTime":379.08,"endTime":382.59,"body":"hard. It's about building"},{"speaker":"Pam","startTime":382.62,"endTime":386.19,"body":"students' brains to be able to"},{"speaker":"Pam","startTime":382.62,"endTime":386.19,"body":"reason, so that then they could"},{"speaker":"Pam","startTime":386.19,"endTime":391.26,"body":"judiciously choose what to do."},{"speaker":"Pam","startTime":386.19,"endTime":391.26,"body":"It's such a different paradigm"},{"speaker":"Pam","startTime":391.26,"endTime":394.65,"body":"perspective. So, what could"},{"speaker":"Pam","startTime":391.26,"endTime":394.65,"body":"students do if they were looking"},{"speaker":"Pam","startTime":394.65,"endTime":397.86,"body":"at that 2 and a 1/2 by 1 and 2/5"},{"speaker":"Pam","startTime":394.65,"endTime":397.86,"body":"if they don't just chunk it into"},{"speaker":"Pam","startTime":397.86,"endTime":400.23,"body":"those crazy pieces? We're not"},{"speaker":"Pam","startTime":397.86,"endTime":400.23,"body":"going to spend a lot of time"},{"speaker":"Pam","startTime":400.23,"endTime":402.45,"body":"here, but if you've listened to"},{"speaker":"Pam","startTime":400.23,"endTime":402.45,"body":"the episode from last week,"},{"speaker":"Pam","startTime":402.69,"endTime":405.63,"body":"could you Double and Halve? And"},{"speaker":"Pam","startTime":402.69,"endTime":405.63,"body":"I'm not actually going to do"},{"speaker":"Pam","startTime":405.63,"endTime":408.06,"body":"that. We're going to leave this"},{"speaker":"Pam","startTime":405.63,"endTime":408.06,"body":"to the readers. You know that"},{"speaker":"Pam","startTime":408.06,"endTime":411.21,"body":"line in textbooks used to say,"},{"speaker":"Pam","startTime":408.06,"endTime":411.21,"body":"\"The proof is left to the"},{"speaker":"Pam","startTime":411.21,"endTime":414.45,"body":"reader.\" So, listeners, I would"},{"speaker":"Pam","startTime":411.21,"endTime":414.45,"body":"challenge you to take 2 and a"},{"speaker":"Pam","startTime":414.45,"endTime":419.55,"body":"1/2 times 1 and 2/5, draw the"},{"speaker":"Pam","startTime":414.45,"endTime":419.55,"body":"array, and then think about"},{"speaker":"Pam","startTime":419.58,"endTime":422.67,"body":"doubling and halving and see"},{"speaker":"Pam","startTime":419.58,"endTime":422.67,"body":"what you do with that, see if"},{"speaker":"Pam","startTime":422.67,"endTime":427.68,"body":"you can find maybe some nicer"},{"speaker":"Pam","startTime":422.67,"endTime":427.68,"body":"chunks to deal with. But, Kim, I"},{"speaker":"Pam","startTime":427.68,"endTime":431.34,"body":"have a feeling that if I gave"},{"speaker":"Pam","startTime":427.68,"endTime":431.34,"body":"you 2 and a 1/2 times 1 and 2/5,"},{"speaker":"Pam","startTime":431.49,"endTime":433.77,"body":"you would abandon fractions. Am"},{"speaker":"Pam","startTime":431.49,"endTime":433.77,"body":"I right on that?"},{"speaker":"Kim","startTime":433.77,"endTime":434.94,"body":"I would, I would."},{"speaker":"Pam","startTime":434.94,"endTime":435.69,"body":"Tell us about that."},{"speaker":"Kim","startTime":435.87,"endTime":436.17,"body":"Yeah, I"},{"speaker":"Kim","startTime":436.17,"endTime":441.42,"body":"would. So, 2 and a 1/2 and 1 and"},{"speaker":"Kim","startTime":436.17,"endTime":441.42,"body":"2/5. So, 2 and a 1/2 is just 2.5"},{"speaker":"Kim","startTime":441.6,"endTime":447.48,"body":"for me, and then 1 and 2/5 is"},{"speaker":"Kim","startTime":441.6,"endTime":447.48,"body":"one and four-tenths, 1.4."},{"speaker":"Pam","startTime":447.54,"endTime":448.05,"body":"Mmhmm."},{"speaker":"Kim","startTime":449.1,"endTime":452.28,"body":"So, then, I would. I would"},{"speaker":"Kim","startTime":449.1,"endTime":452.28,"body":"go Double/Halve, but I would"},{"speaker":"Kim","startTime":452.28,"endTime":457.62,"body":"start with 2.5 times 1.4, and"},{"speaker":"Kim","startTime":452.28,"endTime":457.62,"body":"then I would Double/Halve to get"},{"speaker":"Kim","startTime":458.13,"endTime":462.57,"body":"5 times 7/10 or $0.70."},{"speaker":"Pam","startTime":462.57,"endTime":464.28,"body":"Nice, nice, nice, nice."},{"speaker":"Kim","startTime":464.28,"endTime":465.45,"body":"And I get the same 3.50."},{"speaker":"Pam","startTime":465.45,"endTime":469.47,"body":"Woah! So, pretty slick."},{"speaker":"Pam","startTime":465.45,"endTime":469.47,"body":"Part of..."},{"speaker":"Kim","startTime":469.47,"endTime":472.17,"body":"Oh, I said 3.50 because"},{"speaker":"Kim","startTime":469.47,"endTime":472.17,"body":"money. 3.5 Excuse me."},{"speaker":"Pam","startTime":472.35,"endTime":478.23,"body":"Oh, that's funny because I"},{"speaker":"Pam","startTime":472.35,"endTime":478.23,"body":"wrote down 3.50. Nice. So,"},{"speaker":"Pam","startTime":478.23,"endTime":482.4,"body":"ya'll, one of our points is we"},{"speaker":"Pam","startTime":478.23,"endTime":482.4,"body":"do believe that it is a great"},{"speaker":"Pam","startTime":482.46,"endTime":486.42,"body":"use of the area model to help"},{"speaker":"Pam","startTime":482.46,"endTime":486.42,"body":"students make sense of fraction"},{"speaker":"Pam","startTime":486.42,"endTime":488.85,"body":"multiplication, but then we"},{"speaker":"Pam","startTime":486.42,"endTime":488.85,"body":"don't necessarily want to get"},{"speaker":"Pam","startTime":488.85,"endTime":492.66,"body":"stuck there. Let's make sense of"},{"speaker":"Pam","startTime":488.85,"endTime":492.66,"body":"what's happening. We want"},{"speaker":"Pam","startTime":492.66,"endTime":495.33,"body":"students to make sense of the"},{"speaker":"Pam","startTime":492.66,"endTime":495.33,"body":"chunks. But then, we quickly"},{"speaker":"Pam","startTime":495.33,"endTime":497.7,"body":"want to go, \"Hey, but let's"},{"speaker":"Pam","startTime":495.33,"endTime":497.7,"body":"chunk those chunks, let's group"},{"speaker":"Pam","startTime":497.7,"endTime":501.42,"body":"those groups, and let's use the"},{"speaker":"Pam","startTime":497.7,"endTime":501.42,"body":"strategies that we've developed"},{"speaker":"Pam","startTime":501.42,"endTime":503.67,"body":"for whole numbers. Can you"},{"speaker":"Pam","startTime":501.42,"endTime":503.67,"body":"Double and Halve? Well, let's"},{"speaker":"Pam","startTime":503.67,"endTime":507.48,"body":"see. Maybe not. Could you go a"},{"speaker":"Pam","startTime":503.67,"endTime":507.48,"body":"little Over? Well, let's check"},{"speaker":"Pam","startTime":507.48,"endTime":510.81,"body":"that out. Could we find 3 times"},{"speaker":"Pam","startTime":507.48,"endTime":510.81,"body":"1 and 2/5, and then just take"},{"speaker":"Pam","startTime":510.81,"endTime":516.06,"body":"off a 1/2 of 1 and 2/5.\" Like a"},{"speaker":"Pam","startTime":510.81,"endTime":516.06,"body":"half of 1 and 2/5, ya'll. Half"},{"speaker":"Pam","startTime":516.09,"endTime":520.8,"body":"of 1. Half of 2/5. Little thing"},{"speaker":"Pam","startTime":516.09,"endTime":520.8,"body":"you might think about there. So,"},{"speaker":"Pam","startTime":520.8,"endTime":525.24,"body":"again, we might help students"},{"speaker":"Pam","startTime":520.8,"endTime":525.24,"body":"use the area model, use open"},{"speaker":"Pam","startTime":525.24,"endTime":528.69,"body":"arrays to chunk those chunks,"},{"speaker":"Pam","startTime":525.24,"endTime":528.69,"body":"group those groups in ways that"},{"speaker":"Pam","startTime":528.69,"endTime":532.26,"body":"are clever and efficient and"},{"speaker":"Pam","startTime":528.69,"endTime":532.26,"body":"they're making sense of area,"},{"speaker":"Pam","startTime":532.26,"endTime":534.42,"body":"they're building their spatial"},{"speaker":"Pam","startTime":532.26,"endTime":534.42,"body":"reasoning, they're building the"},{"speaker":"Pam","startTime":534.42,"endTime":538.71,"body":"sense of the distributive"},{"speaker":"Pam","startTime":534.42,"endTime":538.71,"body":"property. But then, don't get"},{"speaker":"Pam","startTime":538.71,"endTime":541.83,"body":"stuck in it. We don't want"},{"speaker":"Pam","startTime":538.71,"endTime":541.83,"body":"students to get stuck in the"},{"speaker":"Pam","startTime":541.83,"endTime":545.25,"body":"model. We want to, then, use"},{"speaker":"Pam","startTime":541.83,"endTime":545.25,"body":"everything else we know to solve"},{"speaker":"Pam","startTime":545.25,"endTime":548.67,"body":"things as efficient and clever"},{"speaker":"Pam","startTime":545.25,"endTime":548.67,"body":"as we can. Cool."},{"speaker":"Kim","startTime":548.7,"endTime":550.71,"body":"Right. Can I interrupt for"},{"speaker":"Kim","startTime":548.7,"endTime":550.71,"body":"just a second because I know"},{"speaker":"Kim","startTime":550.71,"endTime":553.74,"body":"we're talking about the area"},{"speaker":"Kim","startTime":550.71,"endTime":553.74,"body":"model? I think this is where"},{"speaker":"Kim","startTime":553.74,"endTime":560.13,"body":"sometimes this particular model"},{"speaker":"Kim","startTime":553.74,"endTime":560.13,"body":"gets a bad rap because the"},{"speaker":"Kim","startTime":560.13,"endTime":563.67,"body":"message that sometimes parents"},{"speaker":"Kim","startTime":560.13,"endTime":563.67,"body":"are hearing is, \"Your child must"},{"speaker":"Kim","startTime":563.67,"endTime":566.91,"body":"draw an area model for every"},{"speaker":"Kim","startTime":563.67,"endTime":566.91,"body":"problem,\" and sometimes kids are"},{"speaker":"Kim","startTime":566.91,"endTime":572.88,"body":"like, \"Not for that problem."},{"speaker":"Kim","startTime":566.91,"endTime":572.88,"body":"That's not a four chunk, two"},{"speaker":"Kim","startTime":572.88,"endTime":576.24,"body":"chunk... Like whatever I'm"},{"speaker":"Kim","startTime":572.88,"endTime":576.24,"body":"thinking about, so now I have to"},{"speaker":"Kim","startTime":576.24,"endTime":581.82,"body":"draw this thing.\" And so, the"},{"speaker":"Kim","startTime":576.24,"endTime":581.82,"body":"area model should be a tool, and"},{"speaker":"Kim","startTime":581.82,"endTime":584.94,"body":"if we're just forcing kids to go"},{"speaker":"Kim","startTime":581.82,"endTime":584.94,"body":"back and sketch it out after"},{"speaker":"Kim","startTime":584.94,"endTime":587.52,"body":"they're already thinking about"},{"speaker":"Kim","startTime":584.94,"endTime":587.52,"body":"it, after they've already solved"},{"speaker":"Kim","startTime":587.52,"endTime":591.12,"body":"it, then it's... We're kind of"},{"speaker":"Kim","startTime":587.52,"endTime":591.12,"body":"missing the point a little bit."},{"speaker":"Pam","startTime":591.15,"endTime":595.17,"body":"Yeah, absolutely. It's sort"},{"speaker":"Pam","startTime":591.15,"endTime":595.17,"body":"of an effort that we that"},{"speaker":"Pam","startTime":595.2,"endTime":597.78,"body":"doesn't need to be expended. I'm"},{"speaker":"Pam","startTime":595.2,"endTime":597.78,"body":"sure there's a better way to say"},{"speaker":"Pam","startTime":597.78,"endTime":599.73,"body":"that. We don't want to force"},{"speaker":"Pam","startTime":597.78,"endTime":599.73,"body":"kids to do stuff they don't"},{"speaker":"Pam","startTime":599.73,"endTime":602.34,"body":"need. It's not going to be"},{"speaker":"Pam","startTime":599.73,"endTime":602.34,"body":"helpful. And so, yeah, we would"},{"speaker":"Pam","startTime":602.34,"endTime":605.85,"body":"encourage you to look at if"},{"speaker":"Pam","startTime":602.34,"endTime":605.85,"body":"you're ever saying, \"Show your"},{"speaker":"Pam","startTime":605.85,"endTime":608.94,"body":"work,\" and you're like. Yes, I"},{"speaker":"Pam","startTime":605.85,"endTime":608.94,"body":"want to see what's happening in"},{"speaker":"Pam","startTime":608.94,"endTime":612.75,"body":"your brain. But to force kids to"},{"speaker":"Pam","startTime":608.94,"endTime":612.75,"body":"draw, especially if you're"},{"speaker":"Pam","startTime":612.75,"endTime":615.9,"body":"forcing them to draw the Place"},{"speaker":"Pam","startTime":612.75,"endTime":615.9,"body":"Value Partial Products all the"},{"speaker":"Pam","startTime":615.9,"endTime":620.64,"body":"time for every problem, that's"},{"speaker":"Pam","startTime":615.9,"endTime":620.64,"body":"not. Like, let's have a purpose."},{"speaker":"Pam","startTime":620.82,"endTime":623.67,"body":"Let's make sure that it's a"},{"speaker":"Pam","startTime":620.82,"endTime":623.67,"body":"helpful purpose. Yeah, nicely"},{"speaker":"Pam","startTime":623.67,"endTime":626.7,"body":"said. I do agree with you that"},{"speaker":"Pam","startTime":623.67,"endTime":626.7,"body":"is one place where we are"},{"speaker":"Pam","startTime":626.7,"endTime":631.65,"body":"missing the boat a little bit,"},{"speaker":"Pam","startTime":626.7,"endTime":631.65,"body":"and wreaking... Not wreaking"},{"speaker":"Pam","startTime":631.65,"endTime":636.87,"body":"havoc but causing havoc. Like"},{"speaker":"Pam","startTime":631.65,"endTime":636.87,"body":"inviting controversy when we"},{"speaker":"Pam","startTime":636.87,"endTime":639.84,"body":"don't need it. Because it's not"},{"speaker":"Pam","startTime":636.87,"endTime":639.84,"body":"particularly helpful to force"},{"speaker":"Pam","startTime":639.84,"endTime":643.44,"body":"kids. And in fact, let me just"},{"speaker":"Pam","startTime":639.84,"endTime":643.44,"body":"be clear, when I'm using an area"},{"speaker":"Pam","startTime":643.44,"endTime":649.35,"body":"model, using it in class, often"},{"speaker":"Pam","startTime":643.44,"endTime":649.35,"body":"it's myself. I am actually"},{"speaker":"Pam","startTime":649.35,"endTime":651.72,"body":"representing. As the teacher,"},{"speaker":"Pam","startTime":649.35,"endTime":651.72,"body":"I'm representing student"},{"speaker":"Pam","startTime":651.72,"endTime":655.71,"body":"thinking. We'll do plenty of"},{"speaker":"Pam","startTime":651.72,"endTime":655.71,"body":"Problem Strings where students"},{"speaker":"Pam","startTime":655.71,"endTime":659.37,"body":"aren't drawing anything at all,"},{"speaker":"Pam","startTime":655.71,"endTime":659.37,"body":"but as they say the relationship"},{"speaker":"Pam","startTime":659.37,"endTime":661.56,"body":"they're thinking about, we are"},{"speaker":"Pam","startTime":659.37,"endTime":661.56,"body":"making that relationship"},{"speaker":"Pam","startTime":661.56,"endTime":664.56,"body":"visible, so that then we can"},{"speaker":"Pam","startTime":661.56,"endTime":664.56,"body":"point at it and talk about it,"},{"speaker":"Pam","startTime":664.56,"endTime":667.56,"body":"but we're not actually forcing"},{"speaker":"Pam","startTime":664.56,"endTime":667.56,"body":"students to draw at all. We will"},{"speaker":"Pam","startTime":667.56,"endTime":670.05,"body":"sometimes say to students,"},{"speaker":"Pam","startTime":667.56,"endTime":670.05,"body":"especially when we're doing"},{"speaker":"Pam","startTime":670.08,"endTime":673.05,"body":"area, we might say, \"You know,"},{"speaker":"Pam","startTime":670.08,"endTime":673.05,"body":"like, area. We're really focused"},{"speaker":"Pam","startTime":673.05,"endTime":676.44,"body":"on area.\" Like, \"What would that"},{"speaker":"Pam","startTime":673.05,"endTime":676.44,"body":"look like?\" But we're not going"},{"speaker":"Pam","startTime":676.44,"endTime":679.05,"body":"to get stuck in it. Is that a"},{"speaker":"Pam","startTime":676.44,"endTime":679.05,"body":"way to say that? Cool."},{"speaker":"Kim","startTime":679.05,"endTime":679.14,"body":"Yeah."},{"speaker":"Pam","startTime":679.14,"endTime":682.02,"body":"I'm glad you brought that"},{"speaker":"Pam","startTime":679.14,"endTime":682.02,"body":"up. Alright. Now, as we move"},{"speaker":"Pam","startTime":682.02,"endTime":686.88,"body":"from thinking about kind of"},{"speaker":"Pam","startTime":682.02,"endTime":686.88,"body":"middle school-y fractions, and"},{"speaker":"Pam","startTime":686.88,"endTime":690.6,"body":"decimals, and using an area"},{"speaker":"Pam","startTime":686.88,"endTime":690.6,"body":"model or an open array. We are"},{"speaker":"Pam","startTime":690.6,"endTime":695.43,"body":"definitely going to continue to"},{"speaker":"Pam","startTime":690.6,"endTime":695.43,"body":"build things on that basis. Area"},{"speaker":"Pam","startTime":695.43,"endTime":699.03,"body":"happens a lot in calculus. I"},{"speaker":"Pam","startTime":695.43,"endTime":699.03,"body":"mean, in calculus, we find the"},{"speaker":"Pam","startTime":699.03,"endTime":702.75,"body":"area under a curve. We can use"},{"speaker":"Pam","startTime":699.03,"endTime":702.75,"body":"that to find integration rates"},{"speaker":"Pam","startTime":702.75,"endTime":706.05,"body":"based on area. There's lots of"},{"speaker":"Pam","startTime":702.75,"endTime":706.05,"body":"things. I mean, just the"},{"speaker":"Pam","startTime":706.05,"endTime":709.5,"body":"definition of a derivative,"},{"speaker":"Pam","startTime":706.05,"endTime":709.5,"body":"Riemann sums and everything."},{"speaker":"Pam","startTime":709.5,"endTime":715.35,"body":"Like, we often model that with"},{"speaker":"Pam","startTime":709.5,"endTime":715.35,"body":"area. So, as we think about a"},{"speaker":"Pam","startTime":715.35,"endTime":718.98,"body":"reason to use area models in the"},{"speaker":"Pam","startTime":715.35,"endTime":718.98,"body":"younger grades, we absolutely"},{"speaker":"Pam","startTime":718.98,"endTime":722.64,"body":"want to develop a sense of area."},{"speaker":"Pam","startTime":718.98,"endTime":722.64,"body":"What we don't want is for kids"},{"speaker":"Pam","startTime":722.64,"endTime":725.4,"body":"to go, \"Area? Perimeter? Mmm..."},{"speaker":"Pam","startTime":722.64,"endTime":725.4,"body":"Let's see one of them you add,"},{"speaker":"Pam","startTime":725.4,"endTime":728.28,"body":"one of them you multiply. Which"},{"speaker":"Pam","startTime":725.4,"endTime":728.28,"body":"ones this?\" That's not. We want"},{"speaker":"Pam","startTime":728.28,"endTime":731.94,"body":"to develop a far more robust"},{"speaker":"Pam","startTime":728.28,"endTime":731.94,"body":"sense of what it means to"},{"speaker":"Pam","startTime":731.94,"endTime":735.18,"body":"measure linearly and what it"},{"speaker":"Pam","startTime":731.94,"endTime":735.18,"body":"means to measure that sort of"},{"speaker":"Pam","startTime":735.18,"endTime":739.14,"body":"two dimensional square unit,"},{"speaker":"Pam","startTime":735.18,"endTime":739.14,"body":"that span. And so that's super,"},{"speaker":"Pam","startTime":739.14,"endTime":742.86,"body":"super important. But then, we're"},{"speaker":"Pam","startTime":739.14,"endTime":742.86,"body":"going to kind of rely on that"},{"speaker":"Pam","startTime":742.86,"endTime":748.32,"body":"intuition a little bit to help"},{"speaker":"Pam","startTime":742.86,"endTime":748.32,"body":"motivate us using what I'm not"},{"speaker":"Pam","startTime":748.32,"endTime":751.02,"body":"going to now call an area model,"},{"speaker":"Pam","startTime":748.32,"endTime":751.02,"body":"what is going to become a bit"},{"speaker":"Pam","startTime":751.02,"endTime":755.22,"body":"more of a graphic organizer to"},{"speaker":"Pam","startTime":751.02,"endTime":755.22,"body":"do some other things. So, there"},{"speaker":"Pam","startTime":755.22,"endTime":759.96,"body":"is a distinction between an area"},{"speaker":"Pam","startTime":755.22,"endTime":759.96,"body":"model, and an array, and a"},{"speaker":"Pam","startTime":759.96,"endTime":765.21,"body":"graphic organizer, and we can"},{"speaker":"Pam","startTime":759.96,"endTime":765.21,"body":"use them at different times to"},{"speaker":"Pam","startTime":765.21,"endTime":767.94,"body":"do different things. So, let's"},{"speaker":"Pam","startTime":765.21,"endTime":767.94,"body":"just get some some clarity on"},{"speaker":"Pam","startTime":767.94,"endTime":771.06,"body":"where we can use the"},{"speaker":"Pam","startTime":767.94,"endTime":771.06,"body":"understanding that we build in"},{"speaker":"Pam","startTime":771.06,"endTime":774.09,"body":"the younger grades, the"},{"speaker":"Pam","startTime":771.06,"endTime":774.09,"body":"intuition and the real sense,"},{"speaker":"Pam","startTime":774.09,"endTime":777.45,"body":"the spatial reasoning that we"},{"speaker":"Pam","startTime":774.09,"endTime":777.45,"body":"build. Using an area model and"},{"speaker":"Pam","startTime":777.45,"endTime":781.95,"body":"an open array can then help us"},{"speaker":"Pam","startTime":777.45,"endTime":781.95,"body":"do some things with like,"},{"speaker":"Pam","startTime":782.28,"endTime":786.39,"body":"polynomial multiplication."},{"speaker":"Pam","startTime":782.28,"endTime":786.39,"body":"What's... Yeah. Woah! Just got"},{"speaker":"Pam","startTime":786.39,"endTime":789.09,"body":"higher math. So, with"},{"speaker":"Pam","startTime":786.39,"endTime":789.09,"body":"polynomials if I'm looking at"},{"speaker":"Pam","startTime":789.09,"endTime":793.92,"body":"something like x minus 3 times x"},{"speaker":"Pam","startTime":789.09,"endTime":793.92,"body":"plus 2, sometimes people will"},{"speaker":"Pam","startTime":793.92,"endTime":797.19,"body":"say, \"Oh, I'm going to use an"},{"speaker":"Pam","startTime":793.92,"endTime":797.19,"body":"area model to do that.\" Well,"},{"speaker":"Pam","startTime":797.52,"endTime":800.94,"body":"let's be clear. As soon as we've"},{"speaker":"Pam","startTime":797.52,"endTime":800.94,"body":"gotten negatives involved, we're"},{"speaker":"Pam","startTime":800.94,"endTime":804.21,"body":"really not measuring anymore. We"},{"speaker":"Pam","startTime":800.94,"endTime":804.21,"body":"don't really have negative"},{"speaker":"Pam","startTime":804.21,"endTime":806.64,"body":"lengths. We don't have negative"},{"speaker":"Pam","startTime":804.21,"endTime":806.64,"body":"area. That would be kind of like"},{"speaker":"Pam","startTime":806.64,"endTime":810.03,"body":"what a black hole or, I don't"},{"speaker":"Pam","startTime":806.64,"endTime":810.03,"body":"know, anti-matter. I mean,"},{"speaker":"Pam","startTime":810.03,"endTime":815.52,"body":"that's maybe off in space"},{"speaker":"Pam","startTime":810.03,"endTime":815.52,"body":"somewhere. As soon as we have"},{"speaker":"Pam","startTime":815.52,"endTime":818.46,"body":"negative numbers, we're not"},{"speaker":"Pam","startTime":815.52,"endTime":818.46,"body":"really having positive"},{"speaker":"Pam","startTime":818.46,"endTime":822.69,"body":"measurement anymore. But we can"},{"speaker":"Pam","startTime":818.46,"endTime":822.69,"body":"use the sense that we've sort of"},{"speaker":"Pam","startTime":822.69,"endTime":828.03,"body":"built about the distributive"},{"speaker":"Pam","startTime":822.69,"endTime":828.03,"body":"property and about kind of how"},{"speaker":"Pam","startTime":828.03,"endTime":831.39,"body":"we could take an area model and"},{"speaker":"Pam","startTime":828.03,"endTime":831.39,"body":"turn it into a graphic organizer"},{"speaker":"Pam","startTime":831.96,"endTime":835.62,"body":"to do things like, I could say,"},{"speaker":"Pam","startTime":831.96,"endTime":835.62,"body":"\"Well, if I'm thinking about the"},{"speaker":"Pam","startTime":835.62,"endTime":841.95,"body":"product of x minus 3 times x"},{"speaker":"Pam","startTime":835.62,"endTime":841.95,"body":"plus 2, then I could kind of"},{"speaker":"Pam","startTime":841.95,"endTime":845.34,"body":"draw a graphic organizer. \" And"},{"speaker":"Pam","startTime":841.95,"endTime":845.34,"body":"now, I am drawing a square, and"},{"speaker":"Pam","startTime":845.34,"endTime":849.9,"body":"I'm putting x on the left hand"},{"speaker":"Pam","startTime":845.34,"endTime":849.9,"body":"side, and then negative 3, sort"},{"speaker":"Pam","startTime":849.9,"endTime":852.96,"body":"of below that. And then across"},{"speaker":"Pam","startTime":849.9,"endTime":852.96,"body":"the top, I'm putting x, and then"},{"speaker":"Pam","startTime":852.96,"endTime":857.16,"body":"positive 2, and I am splitting"},{"speaker":"Pam","startTime":852.96,"endTime":857.16,"body":"that into 4 equal chunks. And"},{"speaker":"Pam","startTime":857.16,"endTime":861.69,"body":"I'm using this graphic organizer"},{"speaker":"Pam","startTime":857.16,"endTime":861.69,"body":"of this square cut into 4 equal"},{"speaker":"Pam","startTime":861.69,"endTime":864.0,"body":"chunks to help me think about"},{"speaker":"Pam","startTime":861.69,"endTime":864.0,"body":"the distributive property, to"},{"speaker":"Pam","startTime":864.0,"endTime":866.94,"body":"help me make sure I don't miss"},{"speaker":"Pam","startTime":864.0,"endTime":866.94,"body":"any parts. And I can think about"},{"speaker":"Pam","startTime":866.94,"endTime":870.99,"body":"x times x, and that gives me"},{"speaker":"Pam","startTime":866.94,"endTime":870.99,"body":"x^2, a product of x^2. And then,"},{"speaker":"Pam","startTime":871.65,"endTime":874.02,"body":"that negative 3 times x, and"},{"speaker":"Pam","startTime":871.65,"endTime":874.02,"body":"that gives me a product of"},{"speaker":"Pam","startTime":874.02,"endTime":878.4,"body":"negative 3x. And then, I've got"},{"speaker":"Pam","startTime":874.02,"endTime":878.4,"body":"x times 2 and I'm filling in the"},{"speaker":"Pam","startTime":878.43,"endTime":881.76,"body":"pieces of this graphic"},{"speaker":"Pam","startTime":878.43,"endTime":881.76,"body":"organizer. X times 2 is 2x, and"},{"speaker":"Pam","startTime":881.76,"endTime":884.76,"body":"then that final square in the"},{"speaker":"Pam","startTime":881.76,"endTime":884.76,"body":"graphic organizer, negative 3"},{"speaker":"Pam","startTime":884.76,"endTime":887.91,"body":"times 2 is negative 6. And then,"},{"speaker":"Pam","startTime":884.76,"endTime":887.91,"body":"I could gather all those terms"},{"speaker":"Pam","startTime":887.91,"endTime":891.24,"body":"together. So, I had an x"},{"speaker":"Pam","startTime":887.91,"endTime":891.24,"body":"squared, a 2x, a negative 3x,"},{"speaker":"Pam","startTime":891.24,"endTime":893.97,"body":"and a negative 6. When I gather"},{"speaker":"Pam","startTime":891.24,"endTime":893.97,"body":"those together, I've got x^2."},{"speaker":"Pam","startTime":894.42,"endTime":898.44,"body":"The negative 3x and the 2x is"},{"speaker":"Pam","startTime":894.42,"endTime":898.44,"body":"negative x, negative 1x. And"},{"speaker":"Pam","startTime":898.44,"endTime":902.19,"body":"then, I've got that negative 6"},{"speaker":"Pam","startTime":898.44,"endTime":902.19,"body":"or minus 6. So, in other words,"},{"speaker":"Pam","startTime":902.19,"endTime":906.3,"body":"the product of the quantity x"},{"speaker":"Pam","startTime":902.19,"endTime":906.3,"body":"minus 3 times the quantity x"},{"speaker":"Pam","startTime":906.3,"endTime":913.38,"body":"plus 2 is x^2 minus x minus 6."},{"speaker":"Pam","startTime":906.3,"endTime":913.38,"body":"And I can help find that, I can"},{"speaker":"Pam","startTime":913.38,"endTime":916.38,"body":"help keep track of the"},{"speaker":"Pam","startTime":913.38,"endTime":916.38,"body":"distributive property, using"},{"speaker":"Pam","startTime":916.38,"endTime":920.07,"body":"something that kind of resembles"},{"speaker":"Pam","startTime":916.38,"endTime":920.07,"body":"the area model that we built"},{"speaker":"Pam","startTime":920.07,"endTime":924.12,"body":"this intuition around with"},{"speaker":"Pam","startTime":920.07,"endTime":924.12,"body":"rectangles and area of whole"},{"speaker":"Pam","startTime":924.12,"endTime":926.58,"body":"numbers, and fractions, and"},{"speaker":"Pam","startTime":924.12,"endTime":926.58,"body":"decimals. And now I'm going to"},{"speaker":"Pam","startTime":926.58,"endTime":929.28,"body":"use that intuition to help me"},{"speaker":"Pam","startTime":926.58,"endTime":929.28,"body":"make sense of the distributive"},{"speaker":"Pam","startTime":929.28,"endTime":932.88,"body":"property with these variables in"},{"speaker":"Pam","startTime":929.28,"endTime":932.88,"body":"polynomial multiplication."},{"speaker":"Kim","startTime":932.94,"endTime":933.66,"body":"You're speaking my"},{"speaker":"Kim","startTime":933.66,"endTime":934.92,"body":"language, Pam, because that's..."},{"speaker":"Pam","startTime":934.95,"endTime":935.22,"body":"Bam!"},{"speaker":"Kim","startTime":935.61,"endTime":938.28,"body":"...exactly what Cooper is"},{"speaker":"Kim","startTime":935.61,"endTime":938.28,"body":"starting to work on."},{"speaker":"Pam","startTime":938.55,"endTime":939.09,"body":"Are you serious?"},{"speaker":"Kim","startTime":939.51,"endTime":943.53,"body":"Yeah. And so, he is just"},{"speaker":"Kim","startTime":939.51,"endTime":943.53,"body":"kind of... There's no models,"},{"speaker":"Kim","startTime":943.56,"endTime":945.66,"body":"right, and so when we started"},{"speaker":"Kim","startTime":943.56,"endTime":945.66,"body":"(unclear)"},{"speaker":"Pam","startTime":945.66,"endTime":947.13,"body":"That's unfortunate, that's"},{"speaker":"Pam","startTime":945.66,"endTime":947.13,"body":"unfortunate."},{"speaker":"Kim","startTime":947.16,"endTime":948.0,"body":"Well, yeah."},{"speaker":"Pam","startTime":948.0,"endTime":949.62,"body":"We're not liking there's no"},{"speaker":"Pam","startTime":948.0,"endTime":949.62,"body":"models. Okay, keep going."},{"speaker":"Kim","startTime":949.77,"endTime":950.7,"body":"So, I"},{"speaker":"Kim","startTime":950.73,"endTime":954.0,"body":"will sketch, \"Oh, it's like"},{"speaker":"Kim","startTime":950.73,"endTime":954.0,"body":"this.\" And he looks at me, and"},{"speaker":"Kim","startTime":954.0,"endTime":958.92,"body":"he's like, \"Okay...\" and it's"},{"speaker":"Kim","startTime":954.0,"endTime":958.92,"body":"connecting to stuff that he's"},{"speaker":"Kim","startTime":958.92,"endTime":960.78,"body":"done before, which is really"},{"speaker":"Kim","startTime":958.92,"endTime":960.78,"body":"cool. You know?"},{"speaker":"Pam","startTime":961.35,"endTime":963.99,"body":"Yeah. And that's the big"},{"speaker":"Pam","startTime":961.35,"endTime":963.99,"body":"point. So, you, when we say,"},{"speaker":"Pam","startTime":964.02,"endTime":966.3,"body":"\"Know your content, know your"},{"speaker":"Pam","startTime":964.02,"endTime":966.3,"body":"kids,\" you obviously know your"},{"speaker":"Pam","startTime":966.3,"endTime":969.93,"body":"kid, and you know the content,"},{"speaker":"Pam","startTime":966.3,"endTime":969.93,"body":"and so you're looking at this"},{"speaker":"Pam","startTime":969.96,"endTime":972.75,"body":"higher math that he's doing, and"},{"speaker":"Pam","startTime":969.96,"endTime":972.75,"body":"you're like, \"Oh, dude, we can"},{"speaker":"Pam","startTime":972.75,"endTime":975.93,"body":"base this on what we built"},{"speaker":"Pam","startTime":972.75,"endTime":975.93,"body":"before. Like, check it out. This"},{"speaker":"Pam","startTime":975.93,"endTime":977.97,"body":"is... Remember, this thing we've"},{"speaker":"Pam","startTime":975.93,"endTime":977.97,"body":"done before? Look how this"},{"speaker":"Pam","startTime":977.97,"endTime":982.53,"body":"relates to that.\" And that is"},{"speaker":"Pam","startTime":977.97,"endTime":982.53,"body":"exactly the point. So, middle"},{"speaker":"Pam","startTime":982.53,"endTime":984.81,"body":"school, high school teachers,"},{"speaker":"Pam","startTime":982.53,"endTime":984.81,"body":"I'm recommending that you don't"},{"speaker":"Pam","startTime":984.81,"endTime":988.02,"body":"call that graphic organizer an"},{"speaker":"Pam","startTime":984.81,"endTime":988.02,"body":"area model because it's not."},{"speaker":"Pam","startTime":988.05,"endTime":990.99,"body":"It's not representing area, but"},{"speaker":"Pam","startTime":988.05,"endTime":990.99,"body":"it is representing the product"},{"speaker":"Pam","startTime":990.99,"endTime":994.65,"body":"of two factors. Just like"},{"speaker":"Pam","startTime":990.99,"endTime":994.65,"body":"earlier we used the area model"},{"speaker":"Pam","startTime":994.65,"endTime":997.71,"body":"to represent the product of the"},{"speaker":"Pam","startTime":994.65,"endTime":997.71,"body":"two dimensions of factors that"},{"speaker":"Pam","startTime":997.71,"endTime":1000.74,"body":"were representing dimensions and"},{"speaker":"Pam","startTime":997.71,"endTime":1000.74,"body":"area. We can sort of use that."},{"speaker":"Pam","startTime":1000.95,"endTime":1004.37,"body":"Similarly, we can use the same"},{"speaker":"Pam","startTime":1000.95,"endTime":1004.37,"body":"kind of thing to think about"},{"speaker":"Pam","startTime":1004.37,"endTime":1007.58,"body":"polynomial division. There's"},{"speaker":"Pam","startTime":1004.37,"endTime":1007.58,"body":"actually a paper out there"},{"speaker":"Pam","startTime":1007.58,"endTime":1011.03,"body":"somewhere that says, \"Students"},{"speaker":"Pam","startTime":1007.58,"endTime":1011.03,"body":"must learn long division with"},{"speaker":"Pam","startTime":1011.03,"endTime":1012.62,"body":"whole numbers because we're"},{"speaker":"Pam","startTime":1011.03,"endTime":1012.62,"body":"going to need it for our"},{"speaker":"Pam","startTime":1012.62,"endTime":1016.61,"body":"polynomial long division.\" To"},{"speaker":"Pam","startTime":1012.62,"endTime":1016.61,"body":"which I say, \"Nope. No, we do"},{"speaker":"Pam","startTime":1016.61,"endTime":1021.26,"body":"not. We can absolutely use that"},{"speaker":"Pam","startTime":1016.61,"endTime":1021.26,"body":"same graphic organizer to reason"},{"speaker":"Pam","startTime":1021.29,"endTime":1025.16,"body":"about polynomial division and"},{"speaker":"Pam","startTime":1021.29,"endTime":1025.16,"body":"have far less opportunity for"},{"speaker":"Pam","startTime":1025.16,"endTime":1028.61,"body":"error because you don't have"},{"speaker":"Pam","startTime":1025.16,"endTime":1028.61,"body":"that crazy subtracting negatives"},{"speaker":"Pam","startTime":1028.61,"endTime":1031.43,"body":"that happens if...\" And I know"},{"speaker":"Pam","startTime":1028.61,"endTime":1031.43,"body":"I'm talking straight to high"},{"speaker":"Pam","startTime":1031.43,"endTime":1033.65,"body":"school teachers right now. But,"},{"speaker":"Pam","startTime":1031.43,"endTime":1033.65,"body":"high school teachers, you know"},{"speaker":"Pam","startTime":1033.65,"endTime":1035.96,"body":"when you do polynomial long"},{"speaker":"Pam","startTime":1033.65,"endTime":1035.96,"body":"division, that once you've got"},{"speaker":"Pam","startTime":1036.08,"endTime":1038.51,"body":"that second row on there, and"},{"speaker":"Pam","startTime":1036.08,"endTime":1038.51,"body":"now you've got to subtract."},{"speaker":"Pam","startTime":1038.51,"endTime":1040.43,"body":"You've got all these negatives"},{"speaker":"Pam","startTime":1038.51,"endTime":1040.43,"body":"floating around, and you're"},{"speaker":"Pam","startTime":1040.43,"endTime":1044.09,"body":"subtracting negatives. And I've"},{"speaker":"Pam","startTime":1040.43,"endTime":1044.09,"body":"heard you say to students, \"Just"},{"speaker":"Pam","startTime":1044.12,"endTime":1046.7,"body":"put a plus sign, and then change"},{"speaker":"Pam","startTime":1044.12,"endTime":1046.7,"body":"all the signs.\" And, like,"},{"speaker":"Pam","startTime":1047.51,"endTime":1051.05,"body":"\"Ahh!\" I mean, that is a way to"},{"speaker":"Pam","startTime":1047.51,"endTime":1051.05,"body":"get correct answers, but not a"},{"speaker":"Pam","startTime":1051.05,"endTime":1054.56,"body":"lot of reasoning happening"},{"speaker":"Pam","startTime":1051.05,"endTime":1054.56,"body":"there. So, when students are"},{"speaker":"Pam","startTime":1054.56,"endTime":1057.68,"body":"doing polynomial long division,"},{"speaker":"Pam","startTime":1054.56,"endTime":1057.68,"body":"if you've done polynomial"},{"speaker":"Pam","startTime":1057.68,"endTime":1061.01,"body":"multiplication in that graphic"},{"speaker":"Pam","startTime":1057.68,"endTime":1061.01,"body":"organizer, you'll start to"},{"speaker":"Pam","startTime":1061.01,"endTime":1066.5,"body":"notice some patterns, like the"},{"speaker":"Pam","startTime":1061.01,"endTime":1066.5,"body":"like terms are happening in the"},{"speaker":"Pam","startTime":1066.5,"endTime":1070.25,"body":"diagonals. And if you notice"},{"speaker":"Pam","startTime":1066.5,"endTime":1070.25,"body":"that the diagonals are having"},{"speaker":"Pam","startTime":1070.73,"endTime":1072.71,"body":"when you're collecting like"},{"speaker":"Pam","startTime":1070.73,"endTime":1072.71,"body":"terms those are happening in the"},{"speaker":"Pam","startTime":1072.71,"endTime":1076.4,"body":"diagonals. When you undo that,"},{"speaker":"Pam","startTime":1072.71,"endTime":1076.4,"body":"then you can just reason about"},{"speaker":"Pam","startTime":1076.4,"endTime":1079.91,"body":"what you need, based on the part"},{"speaker":"Pam","startTime":1076.4,"endTime":1079.91,"body":"of the diagonal you have needs"},{"speaker":"Pam","startTime":1079.91,"endTime":1082.4,"body":"to add to that term. And I know"},{"speaker":"Pam","startTime":1079.91,"endTime":1082.4,"body":"I'm talking kind of all crazy."},{"speaker":"Pam","startTime":1082.43,"endTime":1084.95,"body":"We'll do a whole podcast episode"},{"speaker":"Pam","startTime":1082.43,"endTime":1084.95,"body":"on polynomial long division at"},{"speaker":"Pam","startTime":1084.95,"endTime":1089.09,"body":"some point. But for today, I"},{"speaker":"Pam","startTime":1084.95,"endTime":1089.09,"body":"just want to note that if we use"},{"speaker":"Pam","startTime":1089.09,"endTime":1092.15,"body":"the area model well for whole"},{"speaker":"Pam","startTime":1089.09,"endTime":1092.15,"body":"numbers and decimals, then we"},{"speaker":"Pam","startTime":1092.15,"endTime":1095.63,"body":"can build into polynomial"},{"speaker":"Pam","startTime":1092.15,"endTime":1095.63,"body":"multiplication and polynomial"},{"speaker":"Pam","startTime":1095.63,"endTime":1096.77,"body":"division. (unclear)"},{"speaker":"Kim","startTime":1096.77,"endTime":1097.43,"body":"I just want to..."},{"speaker":"Pam","startTime":1097.64,"endTime":1098.03,"body":"Oh, keep going."},{"speaker":"Kim","startTime":1098.03,"endTime":1098.24,"body":"Go ahead."},{"speaker":"Pam","startTime":1098.24,"endTime":1099.35,"body":"Go ahead. Well, I'll"},{"speaker":"Pam","startTime":1098.24,"endTime":1099.35,"body":"finish."},{"speaker":"Kim","startTime":1099.35,"endTime":1099.56,"body":"I was... Okay."},{"speaker":"Pam","startTime":1099.83,"endTime":1102.02,"body":"Don't forget. I'm going to"},{"speaker":"Pam","startTime":1099.83,"endTime":1102.02,"body":"go. I'm going to go. High school"},{"speaker":"Pam","startTime":1102.02,"endTime":1105.86,"body":"teachers, the Problem Strings"},{"speaker":"Pam","startTime":1102.02,"endTime":1105.86,"body":"I've written in the advanced"},{"speaker":"Pam","startTime":1105.86,"endTime":1109.79,"body":"algebra Problem String book"},{"speaker":"Pam","startTime":1105.86,"endTime":1109.79,"body":"actually has you start with"},{"speaker":"Pam","startTime":1110.21,"endTime":1114.11,"body":"thinking about an area model"},{"speaker":"Pam","startTime":1110.21,"endTime":1114.11,"body":"with whole numbers into a"},{"speaker":"Pam","startTime":1114.11,"endTime":1118.43,"body":"graphic organizer with"},{"speaker":"Pam","startTime":1114.11,"endTime":1118.43,"body":"multiplying polynomials into a"},{"speaker":"Pam","startTime":1118.43,"endTime":1121.43,"body":"graphic organizer with then"},{"speaker":"Pam","startTime":1118.43,"endTime":1121.43,"body":"dividing polynomials. Okay, Kim,"},{"speaker":"Pam","startTime":1121.43,"endTime":1121.7,"body":"go."},{"speaker":"Kim","startTime":1121.73,"endTime":1125.6,"body":"Yeah. Well, I just want to"},{"speaker":"Kim","startTime":1121.73,"endTime":1125.6,"body":"give a shout out to my kid's"},{"speaker":"Kim","startTime":1125.6,"endTime":1130.73,"body":"former teacher because Luke"},{"speaker":"Kim","startTime":1125.6,"endTime":1130.73,"body":"recently was watching a video"},{"speaker":"Kim","startTime":1130.82,"endTime":1133.58,"body":"about long division, and I kind"},{"speaker":"Kim","startTime":1130.82,"endTime":1133.58,"body":"of walked up, and I was, \"What"},{"speaker":"Kim","startTime":1133.79,"endTime":1139.43,"body":"are you doing?\" And he said,"},{"speaker":"Kim","startTime":1133.79,"endTime":1139.43,"body":"\"Oh, this was the video for"},{"speaker":"Kim","startTime":1139.43,"endTime":1143.57,"body":"today in math.\" And so I said,"},{"speaker":"Kim","startTime":1139.43,"endTime":1143.57,"body":"\"Why? What?\" And he said, \"Well,"},{"speaker":"Kim","startTime":1143.6,"endTime":1147.65,"body":"we're about to start doing this"},{"speaker":"Kim","startTime":1143.6,"endTime":1147.65,"body":"other thing, and we needed to"},{"speaker":"Kim","startTime":1147.65,"endTime":1151.34,"body":"review long division first"},{"speaker":"Kim","startTime":1147.65,"endTime":1151.34,"body":"before we could do this new"},{"speaker":"Kim","startTime":1151.34,"endTime":1155.48,"body":"thing.\" And I was like, \"Okay.\""},{"speaker":"Kim","startTime":1151.34,"endTime":1155.48,"body":"And Luke's never done long"},{"speaker":"Kim","startTime":1155.48,"endTime":1155.96,"body":"division."},{"speaker":"Pam","startTime":1156.32,"endTime":1156.98,"body":"With whole numbers."},{"speaker":"Kim","startTime":1157.37,"endTime":1159.86,"body":"With whole numbers, right."},{"speaker":"Kim","startTime":1157.37,"endTime":1159.86,"body":"So, he's always just kind of"},{"speaker":"Kim","startTime":1161.12,"endTime":1163.04,"body":"stayed under the radar and never"},{"speaker":"Kim","startTime":1161.12,"endTime":1163.04,"body":"done it."},{"speaker":"Kim","startTime":1163.13,"endTime":1163.79,"body":"And so..."},{"speaker":"Pam","startTime":1164.09,"endTime":1166.01,"body":"Meaning...sorry, let me"},{"speaker":"Pam","startTime":1164.09,"endTime":1166.01,"body":"just be clear...he's always"},{"speaker":"Pam","startTime":1166.01,"endTime":1169.64,"body":"reasoned through division. He"},{"speaker":"Pam","startTime":1166.01,"endTime":1169.64,"body":"hasn't done a series of steps"},{"speaker":"Pam","startTime":1169.64,"endTime":1171.95,"body":"that he memorized under a house"},{"speaker":"Pam","startTime":1169.64,"endTime":1171.95,"body":"trying to do long division."},{"speaker":"Kim","startTime":1171.95,"endTime":1172.1,"body":"Correct."},{"speaker":"Pam","startTime":1172.13,"endTime":1172.58,"body":"Keep"},{"speaker":"Pam","startTime":1172.58,"endTime":1173.18,"body":"going. Mmhmm."},{"speaker":"Kim","startTime":1173.18,"endTime":1177.35,"body":"Correct. And so, anyway."},{"speaker":"Kim","startTime":1173.18,"endTime":1177.35,"body":"When he showed me a problem, the"},{"speaker":"Kim","startTime":1177.35,"endTime":1182.96,"body":"polynomial division problem, and"},{"speaker":"Kim","startTime":1177.35,"endTime":1182.96,"body":"I said, \"Oh, this is a division"},{"speaker":"Kim","startTime":1182.96,"endTime":1187.82,"body":"problem. So, that's kinda like"},{"speaker":"Kim","startTime":1182.96,"endTime":1187.82,"body":"we know one of the factors,"},{"speaker":"Kim","startTime":1188.09,"endTime":1190.49,"body":"and...\" I said area because I"},{"speaker":"Kim","startTime":1188.09,"endTime":1190.49,"body":"didn't know what else to say."},{"speaker":"Kim","startTime":1190.49,"endTime":1193.04,"body":"So, now I'll do better, but. I"},{"speaker":"Kim","startTime":1190.49,"endTime":1193.04,"body":"said, \"That's kind of...\" And I"},{"speaker":"Kim","startTime":1193.04,"endTime":1196.73,"body":"said, \"It's not really area, but"},{"speaker":"Kim","startTime":1193.04,"endTime":1196.73,"body":"it's kind of like, here's the"},{"speaker":"Kim","startTime":1196.73,"endTime":1199.28,"body":"product of that. So, all you're"},{"speaker":"Kim","startTime":1196.73,"endTime":1199.28,"body":"trying to figure out is the"},{"speaker":"Kim","startTime":1199.28,"endTime":1202.82,"body":"other dimension.\" And he was"},{"speaker":"Kim","startTime":1199.28,"endTime":1202.82,"body":"like, \"Oh, that's it?\" I was"},{"speaker":"Kim","startTime":1202.82,"endTime":1202.97,"body":"like, \"Yeah.\""},{"speaker":"Pam","startTime":1202.97,"endTime":1204.05,"body":"Yeah, that's it. Yeah."},{"speaker":"Kim","startTime":1204.08,"endTime":1208.94,"body":"So, he goes on his merry"},{"speaker":"Kim","startTime":1204.08,"endTime":1208.94,"body":"way to just figure out and, you"},{"speaker":"Kim","startTime":1208.94,"endTime":1213.38,"body":"know, it's brilliant because he"},{"speaker":"Kim","startTime":1208.94,"endTime":1213.38,"body":"has a picture in his mind of the"},{"speaker":"Kim","startTime":1213.38,"endTime":1218.6,"body":"way you consider the two factors"},{"speaker":"Kim","startTime":1213.38,"endTime":1218.6,"body":"and the product, and the inverse"},{"speaker":"Kim","startTime":1218.6,"endTime":1222.53,"body":"of that is just what if I know"},{"speaker":"Kim","startTime":1218.6,"endTime":1222.53,"body":"one of the dimensions, then"},{"speaker":"Kim","startTime":1222.53,"endTime":1222.56,"body":"(unclear)."},{"speaker":"Pam","startTime":1222.92,"endTime":1224.24,"body":"And you know the product."},{"speaker":"Kim","startTime":1224.39,"endTime":1224.69,"body":"Yeah."},{"speaker":"Pam","startTime":1225.02,"endTime":1227.12,"body":"Then, you could find the"},{"speaker":"Pam","startTime":1225.02,"endTime":1227.12,"body":"other factor. Oh, that's"},{"speaker":"Pam","startTime":1227.12,"endTime":1229.55,"body":"brilliant. Way to go, Luke! You"},{"speaker":"Pam","startTime":1227.12,"endTime":1229.55,"body":"know, you're reminding me of a"},{"speaker":"Pam","startTime":1229.55,"endTime":1233.42,"body":"story when my second was taking"},{"speaker":"Pam","startTime":1229.55,"endTime":1233.42,"body":"either second or third semester"},{"speaker":"Pam","startTime":1233.42,"endTime":1236.63,"body":"calculus. So, my number two was"},{"speaker":"Pam","startTime":1233.42,"endTime":1236.63,"body":"at university, and he's taking"},{"speaker":"Pam","startTime":1236.66,"endTime":1238.52,"body":"their second or third semester"},{"speaker":"Pam","startTime":1236.66,"endTime":1238.52,"body":"calculus. I can't remember which"},{"speaker":"Pam","startTime":1238.52,"endTime":1241.22,"body":"one. And he said, \"Hey, Mom, I"},{"speaker":"Pam","startTime":1238.52,"endTime":1241.22,"body":"did your thing.\" And I was like,"},{"speaker":"Pam","startTime":1241.25,"endTime":1243.05,"body":"\"What?\" And he goes, \"Yeah, did"},{"speaker":"Pam","startTime":1241.25,"endTime":1243.05,"body":"your thing today in calculus.\" I"},{"speaker":"Pam","startTime":1243.05,"endTime":1244.91,"body":"was like, \"Oh, dude, send me a"},{"speaker":"Pam","startTime":1243.05,"endTime":1244.91,"body":"picture.\" And he sent me a"},{"speaker":"Pam","startTime":1244.91,"endTime":1248.63,"body":"picture of a full page. Because,"},{"speaker":"Pam","startTime":1244.91,"endTime":1248.63,"body":"you know, calculus problems kind"},{"speaker":"Pam","startTime":1248.63,"endTime":1251.39,"body":"of take a bit of work, right?"},{"speaker":"Pam","startTime":1248.63,"endTime":1251.39,"body":"So, there's one problem, took"},{"speaker":"Pam","startTime":1251.39,"endTime":1254.3,"body":"the full page, and he circled"},{"speaker":"Pam","startTime":1251.39,"endTime":1254.3,"body":"his little bit on the top."},{"speaker":"Pam","startTime":1254.3,"endTime":1258.2,"body":"Because as part of this entire"},{"speaker":"Pam","startTime":1254.3,"endTime":1258.2,"body":"huge calculus problem, part of"},{"speaker":"Pam","startTime":1258.2,"endTime":1261.35,"body":"what he had to do was a little"},{"speaker":"Pam","startTime":1258.2,"endTime":1261.35,"body":"bit of polynomial division, and"},{"speaker":"Pam","startTime":1261.35,"endTime":1264.77,"body":"he had, literally, he had kind"},{"speaker":"Pam","startTime":1261.35,"endTime":1264.77,"body":"of that graphic organizer, looks"},{"speaker":"Pam","startTime":1264.77,"endTime":1266.87,"body":"like an area model kind of"},{"speaker":"Pam","startTime":1264.77,"endTime":1266.87,"body":"thing, and he had just a little"},{"speaker":"Pam","startTime":1266.87,"endTime":1269.42,"body":"bit. Like, he didn't even fill"},{"speaker":"Pam","startTime":1266.87,"endTime":1269.42,"body":"it all out because once he had"},{"speaker":"Pam","startTime":1269.54,"endTime":1272.54,"body":"put enough in it, then he just"},{"speaker":"Pam","startTime":1269.54,"endTime":1272.54,"body":"moved what he knew was going to"},{"speaker":"Pam","startTime":1272.54,"endTime":1275.15,"body":"happen to the party needed for"},{"speaker":"Pam","startTime":1272.54,"endTime":1275.15,"body":"the rest of the calculus"},{"speaker":"Pam","startTime":1275.15,"endTime":1275.45,"body":"problem."},{"speaker":"Kim","startTime":1275.45,"endTime":1275.6,"body":"Yep."},{"speaker":"Pam","startTime":1275.87,"endTime":1277.7,"body":"And he kind of put little"},{"speaker":"Pam","startTime":1275.87,"endTime":1277.7,"body":"smiley face next to that little"},{"speaker":"Pam","startTime":1277.7,"endTime":1281.0,"body":"part, and it was it was. Anyway,"},{"speaker":"Pam","startTime":1277.7,"endTime":1281.0,"body":"so, yeah. Because we can just"},{"speaker":"Pam","startTime":1281.03,"endTime":1284.9,"body":"use what we know to solve"},{"speaker":"Pam","startTime":1281.03,"endTime":1284.9,"body":"problems more and more advanced,"},{"speaker":"Pam","startTime":1284.93,"endTime":1288.38,"body":"if we actually own it and we"},{"speaker":"Pam","startTime":1284.93,"endTime":1288.38,"body":"actually are using relationships"},{"speaker":"Pam","startTime":1288.38,"endTime":1292.1,"body":"to solve problems. And I'll just"},{"speaker":"Pam","startTime":1288.38,"endTime":1292.1,"body":"mention as a last here, an"},{"speaker":"Pam","startTime":1292.13,"endTime":1294.95,"body":"application for higher math,"},{"speaker":"Pam","startTime":1292.13,"endTime":1294.95,"body":"that if we've got students"},{"speaker":"Pam","startTime":1294.95,"endTime":1298.85,"body":"really reasoning, using this"},{"speaker":"Pam","startTime":1294.95,"endTime":1298.85,"body":"graphic organizer, then we can"},{"speaker":"Pam","startTime":1298.85,"endTime":1301.97,"body":"kind of bring back area a little"},{"speaker":"Pam","startTime":1298.85,"endTime":1301.97,"body":"bit, at least a sense of area,"},{"speaker":"Pam","startTime":1302.18,"endTime":1306.23,"body":"and have a super nice way of"},{"speaker":"Pam","startTime":1302.18,"endTime":1306.23,"body":"completing the square. And so,"},{"speaker":"Pam","startTime":1306.26,"endTime":1308.93,"body":"completing the square is a thing"},{"speaker":"Pam","startTime":1306.26,"endTime":1308.93,"body":"that we do. And I think the best"},{"speaker":"Pam","startTime":1308.93,"endTime":1312.11,"body":"reason to complete the squares"},{"speaker":"Pam","startTime":1308.93,"endTime":1312.11,"body":"to be able to find the centers"},{"speaker":"Pam","startTime":1312.11,"endTime":1314.96,"body":"of conics, that if we have a"},{"speaker":"Pam","startTime":1312.11,"endTime":1314.96,"body":"conic section, and we're trying"},{"speaker":"Pam","startTime":1314.96,"endTime":1318.53,"body":"to like find the center of a"},{"speaker":"Pam","startTime":1314.96,"endTime":1318.53,"body":"circle, or a hyperbola, or an"},{"speaker":"Pam","startTime":1318.53,"endTime":1321.77,"body":"ellipse, then it can be really"},{"speaker":"Pam","startTime":1318.53,"endTime":1321.77,"body":"helpful to complete the square,"},{"speaker":"Pam","startTime":1321.77,"endTime":1325.13,"body":"so that we can tell where the"},{"speaker":"Pam","startTime":1321.77,"endTime":1325.13,"body":"the vertex is going to be, or"},{"speaker":"Pam","startTime":1325.13,"endTime":1327.98,"body":"the center is going to be. And"},{"speaker":"Pam","startTime":1325.13,"endTime":1327.98,"body":"that's a super nice. By the way,"},{"speaker":"Pam","startTime":1327.98,"endTime":1330.17,"body":"I'll just say it out loud, I"},{"speaker":"Pam","startTime":1327.98,"endTime":1330.17,"body":"don't really like completing the"},{"speaker":"Pam","startTime":1330.17,"endTime":1333.47,"body":"square for solving equations. I"},{"speaker":"Pam","startTime":1330.17,"endTime":1333.47,"body":"know. Someone's going to be like"},{"speaker":"Pam","startTime":1333.47,"endTime":1335.93,"body":"not happy about that, but I"},{"speaker":"Pam","startTime":1333.47,"endTime":1335.93,"body":"don't think it's a great time. I"},{"speaker":"Pam","startTime":1336.32,"endTime":1340.01,"body":"don't think it's a great time to"},{"speaker":"Pam","startTime":1336.32,"endTime":1340.01,"body":"bring in that. I just don't"},{"speaker":"Pam","startTime":1340.01,"endTime":1342.71,"body":"think we need it for solving"},{"speaker":"Pam","startTime":1340.01,"endTime":1342.71,"body":"equations. But I do think it's a"},{"speaker":"Pam","startTime":1342.71,"endTime":1345.89,"body":"fine thing that we can build. We"},{"speaker":"Pam","startTime":1342.71,"endTime":1345.89,"body":"can actually understand what's"},{"speaker":"Pam","startTime":1345.89,"endTime":1349.82,"body":"happening, completing the"},{"speaker":"Pam","startTime":1345.89,"endTime":1349.82,"body":"square. If we use this sense and"},{"speaker":"Pam","startTime":1349.82,"endTime":1352.85,"body":"understanding from an area model"},{"speaker":"Pam","startTime":1349.82,"endTime":1352.85,"body":"into the graphic organizer, we"},{"speaker":"Pam","startTime":1352.85,"endTime":1356.84,"body":"can literally help complete the"},{"speaker":"Pam","startTime":1352.85,"endTime":1356.84,"body":"square graphically physically,"},{"speaker":"Pam","startTime":1356.84,"endTime":1360.26,"body":"spatially, and it's a super nice"},{"speaker":"Pam","startTime":1356.84,"endTime":1360.26,"body":"way. So..."},{"speaker":"Kim","startTime":1360.32,"endTime":1360.62,"body":"Yep."},{"speaker":"Pam","startTime":1361.13,"endTime":1362.48,"body":"...we like the area model."},{"speaker":"Kim","startTime":1362.72,"endTime":1366.29,"body":"It's so cool, right? So,"},{"speaker":"Kim","startTime":1362.72,"endTime":1366.29,"body":"teachers of younger grades,"},{"speaker":"Kim","startTime":1366.32,"endTime":1370.22,"body":"talking to you here. Let's use"},{"speaker":"Kim","startTime":1366.32,"endTime":1370.22,"body":"it really well in our grades, so"},{"speaker":"Kim","startTime":1370.22,"endTime":1372.98,"body":"that these teachers of older"},{"speaker":"Kim","startTime":1370.22,"endTime":1372.98,"body":"students can build on that"},{"speaker":"Kim","startTime":1372.98,"endTime":1376.01,"body":"intuition to do all of these"},{"speaker":"Kim","startTime":1372.98,"endTime":1376.01,"body":"really cool higher math things."},{"speaker":"Pam","startTime":1376.16,"endTime":1378.68,"body":"Absolutely. Alright, y'all,"},{"speaker":"Pam","startTime":1376.16,"endTime":1378.68,"body":"thanks for tuning in and"},{"speaker":"Pam","startTime":1378.68,"endTime":1381.47,"body":"teaching more and more real"},{"speaker":"Pam","startTime":1378.68,"endTime":1381.47,"body":"math! To find out more about the"},{"speaker":"Pam","startTime":1381.47,"endTime":1384.89,"body":"Math is Figure-Out-Able movement"},{"speaker":"Pam","startTime":1381.47,"endTime":1384.89,"body":"visit mathisfigureoutable.com."},{"speaker":"Pam","startTime":1386.06,"endTime":1389.15,"body":"Let's keep spreading the word"},{"speaker":"Pam","startTime":1386.06,"endTime":1389.15,"body":"that Math is Figure-Out-Able!"}]}