{"version":"1.0.0","segments":[{"speaker":"Pam","startTime":0.93,"endTime":4.29,"body":"Hey, fellow mathematicians!"},{"speaker":"Pam","startTime":0.93,"endTime":4.29,"body":"Welcome to the podcast where"},{"speaker":"Pam","startTime":4.29,"endTime":7.08,"body":"Math is Figure-Out-Able! I'm"},{"speaker":"Pam","startTime":4.29,"endTime":7.08,"body":"Pam."},{"speaker":"Kim","startTime":7.44,"endTime":8.19,"body":"And I'm Kim."},{"speaker":"Pam","startTime":8.55,"endTime":11.43,"body":"And you found a place where"},{"speaker":"Pam","startTime":8.55,"endTime":11.43,"body":"math is not about memorizing and"},{"speaker":"Pam","startTime":11.43,"endTime":15.18,"body":"mimicking, waiting to be told or"},{"speaker":"Pam","startTime":11.43,"endTime":15.18,"body":"shown what to do. But, ya'll,"},{"speaker":"Pam","startTime":15.18,"endTime":18.24,"body":"it's about making sense of"},{"speaker":"Pam","startTime":15.18,"endTime":18.24,"body":"problems, noticing patterns, and"},{"speaker":"Pam","startTime":18.24,"endTime":22.35,"body":"reasoning using mathematical"},{"speaker":"Pam","startTime":18.24,"endTime":22.35,"body":"relationships. We can mentor"},{"speaker":"Pam","startTime":22.35,"endTime":25.83,"body":"mathematicians, as we co-create"},{"speaker":"Pam","startTime":22.35,"endTime":25.83,"body":"meaning together. Not"},{"speaker":"Pam","startTime":25.83,"endTime":31.29,"body":"haphazardly! We're planning for"},{"speaker":"Pam","startTime":25.83,"endTime":31.29,"body":"it! Not only are algorithms not"},{"speaker":"Pam","startTime":31.29,"endTime":34.77,"body":"particularly helpful in teaching"},{"speaker":"Pam","startTime":31.29,"endTime":34.77,"body":"mathematics, but rotely"},{"speaker":"Pam","startTime":34.77,"endTime":38.19,"body":"repeating steps actually keep"},{"speaker":"Pam","startTime":34.77,"endTime":38.19,"body":"students from being the"},{"speaker":"Pam","startTime":38.19,"endTime":39.72,"body":"mathematicians they can be."},{"speaker":"Kim","startTime":40.32,"endTime":42.45,"body":"So, today, we're going to"},{"speaker":"Kim","startTime":40.32,"endTime":42.45,"body":"finish up our series on the"},{"speaker":"Kim","startTime":42.45,"endTime":46.38,"body":"fantastic mathematical model"},{"speaker":"Kim","startTime":42.45,"endTime":46.38,"body":"called the ratio table. We've"},{"speaker":"Kim","startTime":46.38,"endTime":49.65,"body":"already talked about how to use"},{"speaker":"Kim","startTime":46.38,"endTime":49.65,"body":"it to multiply, to divide, to"},{"speaker":"Kim","startTime":49.65,"endTime":52.98,"body":"solve proportions, and to add"},{"speaker":"Kim","startTime":49.65,"endTime":52.98,"body":"and subtract fractions. And in"},{"speaker":"Kim","startTime":52.98,"endTime":55.92,"body":"this episode, we're going to"},{"speaker":"Kim","startTime":52.98,"endTime":55.92,"body":"finish out by clarifying some"},{"speaker":"Kim","startTime":55.92,"endTime":60.27,"body":"important ideas and solve an"},{"speaker":"Kim","startTime":55.92,"endTime":60.27,"body":"extension problem. So, Pam,"},{"speaker":"Kim","startTime":60.54,"endTime":64.5,"body":"let's start with some"},{"speaker":"Kim","startTime":60.54,"endTime":64.5,"body":"clarifications. Karen Camp, a"},{"speaker":"Kim","startTime":64.5,"endTime":69.33,"body":"fabulous, consistent participant"},{"speaker":"Kim","startTime":64.5,"endTime":69.33,"body":"in MathStratChat said, \"Do kids"},{"speaker":"Kim","startTime":69.36,"endTime":70.86,"body":"also need to erase?\""},{"speaker":"Pam","startTime":70.89,"endTime":74.91,"body":"Yeah, totally. So, in the"},{"speaker":"Pam","startTime":70.89,"endTime":74.91,"body":"first episode of this series"},{"speaker":"Pam","startTime":74.91,"endTime":77.94,"body":"that we've just done. I don't"},{"speaker":"Pam","startTime":74.91,"endTime":77.94,"body":"remember the number of it. One"},{"speaker":"Pam","startTime":77.94,"endTime":81.39,"body":"of those numbers. A few episodes"},{"speaker":"Pam","startTime":77.94,"endTime":81.39,"body":"ago. So, this is the fifth one,"},{"speaker":"Pam","startTime":81.39,"endTime":83.37,"body":"so the first one. I should be"},{"speaker":"Pam","startTime":81.39,"endTime":83.37,"body":"able to do that. That's a fence"},{"speaker":"Pam","startTime":83.37,"endTime":88.26,"body":"post problem, Kim. I'm not even"},{"speaker":"Pam","startTime":83.37,"endTime":88.26,"body":"going there. So, we talked about"},{"speaker":"Pam","startTime":88.26,"endTime":91.83,"body":"how when you facilitate. When,"},{"speaker":"Pam","startTime":88.26,"endTime":91.83,"body":"teachers, when you facilitate a"},{"speaker":"Pam","startTime":91.83,"endTime":95.94,"body":"Problem String using a ratio"},{"speaker":"Pam","startTime":91.83,"endTime":95.94,"body":"table, we suggested that you"},{"speaker":"Pam","startTime":96.03,"endTime":97.68,"body":"strategically erase."},{"speaker":"Kim","startTime":97.74,"endTime":97.95,"body":"Yeah."},{"speaker":"Pam","startTime":97.95,"endTime":101.04,"body":"So, that as you, as the"},{"speaker":"Pam","startTime":97.95,"endTime":101.04,"body":"students are solving problems,"},{"speaker":"Pam","startTime":101.04,"endTime":104.46,"body":"and you're showing that they"},{"speaker":"Pam","startTime":101.04,"endTime":104.46,"body":"scaled times 10. So, if I scale"},{"speaker":"Pam","startTime":104.46,"endTime":107.28,"body":"the number of packs times 10,"},{"speaker":"Pam","startTime":104.46,"endTime":107.28,"body":"and I scale the number of sticks"},{"speaker":"Pam","startTime":107.28,"endTime":114.57,"body":"times 10. Or if I divide those"},{"speaker":"Pam","startTime":107.28,"endTime":114.57,"body":"in half, then I would show that"},{"speaker":"Pam","startTime":114.57,"endTime":118.47,"body":"scaling. Or if I add these two"},{"speaker":"Pam","startTime":114.57,"endTime":118.47,"body":"numbers of packs together to get"},{"speaker":"Pam","startTime":118.47,"endTime":121.86,"body":"these added numbers of sticks"},{"speaker":"Pam","startTime":118.47,"endTime":121.86,"body":"together. That when we do that"},{"speaker":"Pam","startTime":121.95,"endTime":124.86,"body":"there's all this notation that"},{"speaker":"Pam","startTime":121.95,"endTime":124.86,"body":"happens on the board during a"},{"speaker":"Pam","startTime":124.86,"endTime":128.43,"body":"Problem String. We suggested"},{"speaker":"Pam","startTime":124.86,"endTime":128.43,"body":"that there are strategic times"},{"speaker":"Pam","startTime":128.43,"endTime":132.87,"body":"for you to erase, so that what"},{"speaker":"Pam","startTime":128.43,"endTime":132.87,"body":"remains on the board are the"},{"speaker":"Pam","startTime":132.9,"endTime":137.19,"body":"most important, main"},{"speaker":"Pam","startTime":132.9,"endTime":137.19,"body":"relationships that are happening"},{"speaker":"Pam","startTime":137.49,"endTime":140.22,"body":"as you go to the next problem,"},{"speaker":"Pam","startTime":137.49,"endTime":140.22,"body":"so that when you put the next"},{"speaker":"Pam","startTime":140.22,"endTime":144.48,"body":"problem up, kids don't have to"},{"speaker":"Pam","startTime":140.22,"endTime":144.48,"body":"see how we got the last answer."},{"speaker":"Pam","startTime":144.66,"endTime":147.42,"body":"They can now just focus on what"},{"speaker":"Pam","startTime":144.66,"endTime":147.42,"body":"relationships do we have up"},{"speaker":"Pam","startTime":147.42,"endTime":149.55,"body":"there that they could use to get"},{"speaker":"Pam","startTime":147.42,"endTime":149.55,"body":"the next one."},{"speaker":"Kim","startTime":149.85,"endTime":150.18,"body":"Yeah."},{"speaker":"Pam","startTime":150.48,"endTime":155.25,"body":"And Karen is asking, \"But"},{"speaker":"Pam","startTime":150.48,"endTime":155.25,"body":"don't we want students to, like,"},{"speaker":"Pam","startTime":155.25,"endTime":158.52,"body":"notate their work, so we can"},{"speaker":"Pam","startTime":155.25,"endTime":158.52,"body":"follow what they were doing?\""},{"speaker":"Pam","startTime":158.52,"endTime":163.2,"body":"And the answer to that is,"},{"speaker":"Pam","startTime":158.52,"endTime":163.2,"body":"absolutely! So, what we didn't"},{"speaker":"Pam","startTime":163.2,"endTime":167.28,"body":"do at all in that first episode"},{"speaker":"Pam","startTime":163.2,"endTime":167.28,"body":"was parse out when are teachers"},{"speaker":"Pam","startTime":167.28,"endTime":169.86,"body":"erasing and when are students"},{"speaker":"Pam","startTime":167.28,"endTime":169.86,"body":"erasing."},{"speaker":"Kim","startTime":170.04,"endTime":170.22,"body":"Yeah."},{"speaker":"Pam","startTime":170.25,"endTime":174.15,"body":"And, Kim, this really"},{"speaker":"Pam","startTime":170.25,"endTime":174.15,"body":"reminds me of when I. Early in"},{"speaker":"Pam","startTime":174.15,"endTime":176.58,"body":"our work together when I was"},{"speaker":"Pam","startTime":174.15,"endTime":176.58,"body":"diving into the research, and"},{"speaker":"Pam","startTime":176.58,"endTime":179.7,"body":"diving into my kids classrooms,"},{"speaker":"Pam","startTime":176.58,"endTime":179.7,"body":"and you were at my kids campus,"},{"speaker":"Pam","startTime":179.7,"endTime":182.49,"body":"and we were working with open"},{"speaker":"Pam","startTime":179.7,"endTime":182.49,"body":"number lines for the first time"},{"speaker":"Pam","startTime":182.49,"endTime":187.11,"body":"ever, and trying to model"},{"speaker":"Pam","startTime":182.49,"endTime":187.11,"body":"thinking, the teachers asked me."},{"speaker":"Pam","startTime":187.35,"endTime":192.15,"body":"You noticed that I sometimes put"},{"speaker":"Pam","startTime":187.35,"endTime":192.15,"body":"arrows on open number lines. And"},{"speaker":"Pam","startTime":192.42,"endTime":196.77,"body":"sometimes I put plus and minus"},{"speaker":"Pam","startTime":192.42,"endTime":196.77,"body":"if like I was adding 20."},{"speaker":"Pam","startTime":196.77,"endTime":198.99,"body":"Sometimes I would write \"plus"},{"speaker":"Pam","startTime":196.77,"endTime":198.99,"body":"20\". And sometimes I would"},{"speaker":"Pam","startTime":198.99,"endTime":201.24,"body":"write... If I was doing a"},{"speaker":"Pam","startTime":198.99,"endTime":201.24,"body":"subtraction problem, I would"},{"speaker":"Pam","startTime":201.24,"endTime":203.76,"body":"write \"subtract [something]\"."},{"speaker":"Pam","startTime":201.24,"endTime":203.76,"body":"And sometimes I wouldn't. And"},{"speaker":"Pam","startTime":203.76,"endTime":207.54,"body":"they were like \"We can't tell"},{"speaker":"Pam","startTime":203.76,"endTime":207.54,"body":"what's your rhyme or reason for"},{"speaker":"Pam","startTime":207.54,"endTime":210.09,"body":"when you would and when you"},{"speaker":"Pam","startTime":207.54,"endTime":210.09,"body":"wouldn't.\" And we had to kind of"},{"speaker":"Pam","startTime":210.09,"endTime":215.52,"body":"parse that out. But then, the,"},{"speaker":"Pam","startTime":210.09,"endTime":215.52,"body":"strongly was this suggestion of,"},{"speaker":"Pam","startTime":215.82,"endTime":220.47,"body":"\"But students should notate"},{"speaker":"Pam","startTime":215.82,"endTime":220.47,"body":"their work more.\" Like, there"},{"speaker":"Pam","startTime":220.47,"endTime":223.05,"body":"were times where I said \"Ooh,"},{"speaker":"Pam","startTime":220.47,"endTime":223.05,"body":"but if I'm dealing with the"},{"speaker":"Pam","startTime":223.05,"endTime":226.59,"body":"relationship between addition"},{"speaker":"Pam","startTime":223.05,"endTime":226.59,"body":"and subtraction, let's not maybe"},{"speaker":"Pam","startTime":226.59,"endTime":229.53,"body":"put a plus and minus sign up"},{"speaker":"Pam","startTime":226.59,"endTime":229.53,"body":"there because I want students to"},{"speaker":"Pam","startTime":229.53,"endTime":233.25,"body":"be able to see from this number"},{"speaker":"Pam","startTime":229.53,"endTime":233.25,"body":"line, they could write an"},{"speaker":"Pam","startTime":233.25,"endTime":234.99,"body":"addition problem, they could"},{"speaker":"Pam","startTime":233.25,"endTime":234.99,"body":"write a subtraction problem."},{"speaker":"Pam","startTime":234.99,"endTime":239.34,"body":"Like it's these relationships"},{"speaker":"Pam","startTime":234.99,"endTime":239.34,"body":"could represent, could be"},{"speaker":"Pam","startTime":239.34,"endTime":242.76,"body":"represented by many equations."},{"speaker":"Pam","startTime":239.34,"endTime":242.76,"body":"But as soon as we put a"},{"speaker":"Pam","startTime":242.76,"endTime":245.97,"body":"subtraction sign on the jump."},{"speaker":"Pam","startTime":242.76,"endTime":245.97,"body":"Oh, then it's a subtraction"},{"speaker":"Pam","startTime":245.97,"endTime":246.36,"body":"problem."},{"speaker":"Kim","startTime":246.6,"endTime":246.81,"body":"Right."},{"speaker":"Pam","startTime":247.2,"endTime":250.11,"body":"Anyway, I know I'm kind of"},{"speaker":"Pam","startTime":247.2,"endTime":250.11,"body":"going off on. You're like, \"Why"},{"speaker":"Pam","startTime":250.11,"endTime":251.28,"body":"are you going into addition and"},{"speaker":"Pam","startTime":250.11,"endTime":251.28,"body":"subtraction?\""},{"speaker":"Kim","startTime":251.82,"endTime":252.48,"body":"No, it's good."},{"speaker":"Pam","startTime":252.78,"endTime":255.66,"body":"My point is that there are"},{"speaker":"Pam","startTime":252.78,"endTime":255.66,"body":"times where teachers are"},{"speaker":"Pam","startTime":255.66,"endTime":259.59,"body":"strategic about what they"},{"speaker":"Pam","startTime":255.66,"endTime":259.59,"body":"notate. We want students to not"},{"speaker":"Pam","startTime":259.59,"endTime":263.49,"body":"worry about that. What we want"},{"speaker":"Pam","startTime":259.59,"endTime":263.49,"body":"students to do is notate, like"},{"speaker":"Pam","startTime":263.79,"endTime":267.45,"body":"mark up your work, show us your"},{"speaker":"Pam","startTime":263.79,"endTime":267.45,"body":"thinking as much as possible."},{"speaker":"Pam","startTime":267.45,"endTime":269.52,"body":"And when they might go, \"Oh, I"},{"speaker":"Pam","startTime":267.45,"endTime":269.52,"body":"did it in my head.\" Then. we"},{"speaker":"Pam","startTime":269.52,"endTime":271.89,"body":"could say, \"Oh, yeah, I believe"},{"speaker":"Pam","startTime":269.52,"endTime":271.89,"body":"you. I believe you did it in"},{"speaker":"Pam","startTime":271.89,"endTime":275.16,"body":"your head. I can't see what's in"},{"speaker":"Pam","startTime":271.89,"endTime":275.16,"body":"your head. Can you show me?\" You"},{"speaker":"Pam","startTime":275.16,"endTime":277.56,"body":"know like, \"Can you help me?"},{"speaker":"Pam","startTime":275.16,"endTime":277.56,"body":"Help me see what's in your head?"},{"speaker":"Pam","startTime":277.56,"endTime":279.24,"body":"Help me see what's happening in"},{"speaker":"Pam","startTime":277.56,"endTime":279.24,"body":"your brain?\""},{"speaker":"Kim","startTime":279.66,"endTime":279.9,"body":"Yeah."},{"speaker":"Pam","startTime":279.93,"endTime":282.72,"body":"Remember, Do, Say,"},{"speaker":"Pam","startTime":279.93,"endTime":282.72,"body":"Represent. This might be the"},{"speaker":"Pam","startTime":282.72,"endTime":284.7,"body":"point where a student is like,"},{"speaker":"Pam","startTime":282.72,"endTime":284.7,"body":"\"I don't know. I don't know how"},{"speaker":"Pam","startTime":284.7,"endTime":288.09,"body":"to show you what's in my head.\""},{"speaker":"Pam","startTime":284.7,"endTime":288.09,"body":"At that point, then we say \"Oh,"},{"speaker":"Pam","startTime":288.09,"endTime":290.37,"body":"well tell me. Tell me what's"},{"speaker":"Pam","startTime":288.09,"endTime":290.37,"body":"going on in your head. Here,"},{"speaker":"Pam","startTime":290.4,"endTime":294.78,"body":"I'll help make your thinking"},{"speaker":"Pam","startTime":290.4,"endTime":294.78,"body":"visible this time. Next time,"},{"speaker":"Pam","startTime":294.78,"endTime":298.23,"body":"you give it a go. Does this show"},{"speaker":"Pam","startTime":294.78,"endTime":298.23,"body":"what was happening in your head?"},{"speaker":"Pam","startTime":298.23,"endTime":302.43,"body":"Cool, cool. Alright, now you've"},{"speaker":"Pam","startTime":298.23,"endTime":302.43,"body":"seen how to put on paper what"},{"speaker":"Pam","startTime":302.43,"endTime":304.77,"body":"was in your head, how to make"},{"speaker":"Pam","startTime":302.43,"endTime":304.77,"body":"visible what you're thinking."},{"speaker":"Pam","startTime":304.83,"endTime":308.16,"body":"Next time, you show me what's"},{"speaker":"Pam","startTime":304.83,"endTime":308.16,"body":"going on in your head.\" Because"},{"speaker":"Pam","startTime":308.64,"endTime":311.1,"body":"that's what's important, right?"},{"speaker":"Pam","startTime":308.64,"endTime":311.1,"body":"It's like the relationships"},{"speaker":"Pam","startTime":311.1,"endTime":315.63,"body":"you're using are important. And"},{"speaker":"Pam","startTime":311.1,"endTime":315.63,"body":"yes, Karen, we need to see what"},{"speaker":"Pam","startTime":315.63,"endTime":319.14,"body":"students are doing as much as"},{"speaker":"Pam","startTime":315.63,"endTime":319.14,"body":"possible, so I'm not asking"},{"speaker":"Pam","startTime":319.14,"endTime":320.1,"body":"students to erase. Yeah."},{"speaker":"Kim","startTime":320.19,"endTime":320.52,"body":"Yeah."},{"speaker":"Kim","startTime":320.52,"endTime":323.13,"body":"I'm so glad that you said that"},{"speaker":"Kim","startTime":320.52,"endTime":323.13,"body":"because I could hear people. The"},{"speaker":"Kim","startTime":323.13,"endTime":326.55,"body":"part about helping students."},{"speaker":"Kim","startTime":323.13,"endTime":326.55,"body":"Because I could hear people"},{"speaker":"Kim","startTime":326.58,"endTime":331.23,"body":"saying, \"But my kids don't know"},{"speaker":"Kim","startTime":326.58,"endTime":331.23,"body":"how to represent quite yet.\" And"},{"speaker":"Kim","startTime":331.23,"endTime":334.86,"body":"this actually came up. We had a"},{"speaker":"Kim","startTime":331.23,"endTime":334.86,"body":"conversation not too long ago"},{"speaker":"Kim","startTime":335.19,"endTime":340.23,"body":"with one of our stellar teachers"},{"speaker":"Kim","startTime":335.19,"endTime":340.23,"body":"in Journey membership site. And"},{"speaker":"Kim","startTime":340.32,"endTime":343.98,"body":"do you remember? She shared some"},{"speaker":"Kim","startTime":340.32,"endTime":343.98,"body":"work in the Facebook group with"},{"speaker":"Kim","startTime":344.01,"endTime":348.57,"body":"her students, and she was really"},{"speaker":"Kim","startTime":344.01,"endTime":348.57,"body":"excited because brilliantly, she"},{"speaker":"Kim","startTime":348.57,"endTime":350.94,"body":"noticed that her students were"},{"speaker":"Kim","startTime":348.57,"endTime":350.94,"body":"doing some great thinking with"},{"speaker":"Kim","startTime":350.97,"endTime":353.43,"body":"really sophisticated strategies"},{"speaker":"Kim","startTime":350.97,"endTime":353.43,"body":"for multiplication."},{"speaker":"Pam","startTime":353.52,"endTime":355.29,"body":"Way to go Adina! Way to go!"},{"speaker":"Pam","startTime":353.52,"endTime":355.29,"body":"Woohoo!"},{"speaker":"Kim","startTime":355.29,"endTime":357.75,"body":"But they didn't really"},{"speaker":"Kim","startTime":355.29,"endTime":357.75,"body":"represent their thinking well,"},{"speaker":"Kim","startTime":357.75,"endTime":361.98,"body":"so she. Like, genius. She like"},{"speaker":"Kim","startTime":357.75,"endTime":361.98,"body":"moved into the space, and she"},{"speaker":"Kim","startTime":361.98,"endTime":364.23,"body":"asked students what they were"},{"speaker":"Kim","startTime":361.98,"endTime":364.23,"body":"thinking about, and then she"},{"speaker":"Kim","startTime":364.35,"endTime":368.07,"body":"model represented for them. And"},{"speaker":"Kim","startTime":364.35,"endTime":368.07,"body":"that's what we would expect that"},{"speaker":"Kim","startTime":368.07,"endTime":371.37,"body":"the use of ratio tables would"},{"speaker":"Kim","startTime":368.07,"endTime":371.37,"body":"continue. We can't expect that"},{"speaker":"Kim","startTime":371.37,"endTime":375.09,"body":"students will see us make use of"},{"speaker":"Kim","startTime":371.37,"endTime":375.09,"body":"a ratio table like a few times,"},{"speaker":"Kim","startTime":375.09,"endTime":377.85,"body":"and then automatically know how"},{"speaker":"Kim","startTime":375.09,"endTime":377.85,"body":"to represent all their thinking,"},{"speaker":"Kim","startTime":377.97,"endTime":381.15,"body":"you know, beautifully, clearly."},{"speaker":"Kim","startTime":377.97,"endTime":381.15,"body":"But in time, seeing their"},{"speaker":"Kim","startTime":381.15,"endTime":384.93,"body":"thinking on paper, it can become"},{"speaker":"Kim","startTime":381.15,"endTime":384.93,"body":"a tool for thinking as well."},{"speaker":"Pam","startTime":385.23,"endTime":388.95,"body":"Absolutely. And then, now"},{"speaker":"Pam","startTime":385.23,"endTime":388.95,"body":"that it's a tool for them, then"},{"speaker":"Pam","startTime":388.95,"endTime":390.96,"body":"they're going to be more"},{"speaker":"Pam","startTime":388.95,"endTime":390.96,"body":"naturally notating they're"},{"speaker":"Pam","startTime":390.96,"endTime":393.96,"body":"thinking. And all of a sudden"},{"speaker":"Pam","startTime":390.96,"endTime":393.96,"body":"all the thinking is happening,"},{"speaker":"Pam","startTime":394.26,"endTime":398.28,"body":"and we're able to see it. They"},{"speaker":"Pam","startTime":394.26,"endTime":398.28,"body":"can make it visual because"},{"speaker":"Pam","startTime":398.43,"endTime":401.61,"body":"they're actually using the ratio"},{"speaker":"Pam","startTime":398.43,"endTime":401.61,"body":"table as a tool. They're not"},{"speaker":"Pam","startTime":401.61,"endTime":404.01,"body":"just, \"Ooh, what happened in my"},{"speaker":"Pam","startTime":401.61,"endTime":404.01,"body":"head? Here, let me put that"},{"speaker":"Pam","startTime":404.01,"endTime":406.41,"body":"down.\" But they're actually"},{"speaker":"Pam","startTime":404.01,"endTime":406.41,"body":"like, I can't tell you the"},{"speaker":"Pam","startTime":406.41,"endTime":409.68,"body":"number of times that we see"},{"speaker":"Pam","startTime":406.41,"endTime":409.68,"body":"students now where we give them"},{"speaker":"Pam","startTime":409.68,"endTime":412.23,"body":"a multiplication problem, and"},{"speaker":"Pam","startTime":409.68,"endTime":412.23,"body":"they throw down a ratio table"},{"speaker":"Pam","startTime":412.23,"endTime":415.32,"body":"and start messing with values."},{"speaker":"Pam","startTime":412.23,"endTime":415.32,"body":"In that way, it's actually a"},{"speaker":"Pam","startTime":415.32,"endTime":419.28,"body":"tool that they're using in time"},{"speaker":"Pam","startTime":415.32,"endTime":419.28,"body":"to think with. Yeah, that's"},{"speaker":"Pam","startTime":419.28,"endTime":423.33,"body":"super cool. In fact, it might be"},{"speaker":"Pam","startTime":419.28,"endTime":423.33,"body":"worth parsing out right now"},{"speaker":"Pam","startTime":423.33,"endTime":426.63,"body":"something I almost wish that I"},{"speaker":"Pam","startTime":423.33,"endTime":426.63,"body":"had thought of to say in the"},{"speaker":"Pam","startTime":426.63,"endTime":431.04,"body":"first episode on ratio tables."},{"speaker":"Pam","startTime":426.63,"endTime":431.04,"body":"But hey, let's wrap up our"},{"speaker":"Pam","startTime":431.04,"endTime":435.9,"body":"series on ratio tables by making"},{"speaker":"Pam","startTime":431.04,"endTime":435.9,"body":"this point clear. What a ratio"},{"speaker":"Pam","startTime":435.9,"endTime":442.2,"body":"table is not is some funky, new"},{"speaker":"Pam","startTime":435.9,"endTime":442.2,"body":"math. \"Oh, now we're going to do"},{"speaker":"Pam","startTime":442.2,"endTime":446.88,"body":"this different weird thing.\""},{"speaker":"Pam","startTime":442.2,"endTime":446.88,"body":"It's actually much more like"},{"speaker":"Pam","startTime":446.88,"endTime":453.54,"body":"this. That what we've done is"},{"speaker":"Pam","startTime":446.88,"endTime":453.54,"body":"interview mathy people. This is"},{"speaker":"Pam","startTime":453.54,"endTime":456.09,"body":"my work. I don't know. I don't"},{"speaker":"Pam","startTime":453.54,"endTime":456.09,"body":"know that I've clearly said this"},{"speaker":"Pam","startTime":456.09,"endTime":458.04,"body":"very well on the podcast, so I'm"},{"speaker":"Pam","startTime":456.09,"endTime":458.04,"body":"going to be a little more clear"},{"speaker":"Pam","startTime":458.04,"endTime":463.92,"body":"today. Part of my work is to"},{"speaker":"Pam","startTime":458.04,"endTime":463.92,"body":"interview mathy people and"},{"speaker":"Pam","startTime":463.92,"endTime":469.11,"body":"figure out how do mathematicians"},{"speaker":"Pam","startTime":463.92,"endTime":469.11,"body":"actually think and reason about"},{"speaker":"Pam","startTime":469.11,"endTime":472.5,"body":"math? What do they actually do?"},{"speaker":"Pam","startTime":469.11,"endTime":472.5,"body":"Because, ya'll, they don't use"},{"speaker":"Pam","startTime":472.5,"endTime":477.06,"body":"algorithms most of the time."},{"speaker":"Pam","startTime":472.5,"endTime":477.06,"body":"They come up with algorithms"},{"speaker":"Pam","startTime":477.06,"endTime":480.03,"body":"because they want to generalize"},{"speaker":"Pam","startTime":477.06,"endTime":480.03,"body":"relationships, but they don't"},{"speaker":"Pam","startTime":480.03,"endTime":483.69,"body":"actually use them when they're"},{"speaker":"Pam","startTime":480.03,"endTime":483.69,"body":"solving problems. So, what I do"},{"speaker":"Pam","startTime":483.69,"endTime":488.79,"body":"is I dive into the math, and I"},{"speaker":"Pam","startTime":483.69,"endTime":488.79,"body":"dive into people's brains, and I"},{"speaker":"Pam","startTime":488.82,"endTime":492.69,"body":"pull out the thinking, like"},{"speaker":"Pam","startTime":488.82,"endTime":492.69,"body":"what's actually happening as"},{"speaker":"Pam","startTime":492.69,"endTime":496.68,"body":"mathy people are solving"},{"speaker":"Pam","startTime":492.69,"endTime":496.68,"body":"problems, and then we make that"},{"speaker":"Pam","startTime":496.68,"endTime":500.46,"body":"accessible to the rest of the"},{"speaker":"Pam","startTime":496.68,"endTime":500.46,"body":"world. Then, we say, \"Oh, hey,"},{"speaker":"Pam","startTime":500.82,"endTime":504.72,"body":"solve that proportion. Solve"},{"speaker":"Pam","startTime":500.82,"endTime":504.72,"body":"that multiplication. Oh, those?"},{"speaker":"Pam","startTime":504.99,"endTime":507.78,"body":"Those relationships? They can"},{"speaker":"Pam","startTime":504.99,"endTime":507.78,"body":"look like this on a ratio table."},{"speaker":"Pam","startTime":508.02,"endTime":511.11,"body":"Ah, that's how you mathy people"},{"speaker":"Pam","startTime":508.02,"endTime":511.11,"body":"have always been thinking about"},{"speaker":"Pam","startTime":511.11,"endTime":514.71,"body":"division? Hmm, we could make"},{"speaker":"Pam","startTime":511.11,"endTime":514.71,"body":"that thinking visible on a ratio"},{"speaker":"Pam","startTime":514.71,"endTime":517.8,"body":"table.\" And then, the rest of us"},{"speaker":"Pam","startTime":514.71,"endTime":517.8,"body":"now have access to it."},{"speaker":"Kim","startTime":518.25,"endTime":518.49,"body":"Yeah."},{"speaker":"Pam","startTime":518.49,"endTime":522.48,"body":"So, the ratio table is one"},{"speaker":"Pam","startTime":518.49,"endTime":522.48,"body":"way that we are able to take"},{"speaker":"Pam","startTime":522.48,"endTime":527.61,"body":"what people have naturally done"},{"speaker":"Pam","startTime":522.48,"endTime":527.61,"body":"to solve math problems and"},{"speaker":"Pam","startTime":528.3,"endTime":532.05,"body":"represent those relationships"},{"speaker":"Pam","startTime":528.3,"endTime":532.05,"body":"that they have sort of naturally"},{"speaker":"Pam","startTime":532.05,"endTime":534.63,"body":"been using that the rest of us"},{"speaker":"Pam","startTime":532.05,"endTime":534.63,"body":"didn't know existed because we"},{"speaker":"Pam","startTime":534.63,"endTime":537.54,"body":"were just doing the steps. We're"},{"speaker":"Pam","startTime":534.63,"endTime":537.54,"body":"over there mimicking the steps"},{"speaker":"Pam","startTime":537.54,"endTime":540.78,"body":"we were told to do. But what we"},{"speaker":"Pam","startTime":537.54,"endTime":540.78,"body":"weren't doing was being able to"},{"speaker":"Pam","startTime":540.78,"endTime":544.56,"body":"see inside their brains. Well,"},{"speaker":"Pam","startTime":540.78,"endTime":544.56,"body":"that's what we're helping you"},{"speaker":"Pam","startTime":544.56,"endTime":549.39,"body":"with, with ratio tables. We're"},{"speaker":"Pam","startTime":544.56,"endTime":549.39,"body":"helping you go, \"Oh, when sort"},{"speaker":"Pam","startTime":549.39,"endTime":552.72,"body":"of mathematicians, mathy people"},{"speaker":"Pam","startTime":549.39,"endTime":552.72,"body":"have been kind of naturally"},{"speaker":"Pam","startTime":552.72,"endTime":556.62,"body":"using their whatever talent to"},{"speaker":"Pam","startTime":552.72,"endTime":556.62,"body":"solve problems, these are the"},{"speaker":"Pam","startTime":556.62,"endTime":559.23,"body":"relationships they've been"},{"speaker":"Pam","startTime":556.62,"endTime":559.23,"body":"using.\" And we can make them"},{"speaker":"Pam","startTime":559.23,"endTime":563.79,"body":"visible in a ratio table. And"},{"speaker":"Pam","startTime":559.23,"endTime":563.79,"body":"then, we can help that ratio to"},{"speaker":"Pam","startTime":563.79,"endTime":567.99,"body":"become a tool, so that you can"},{"speaker":"Pam","startTime":563.79,"endTime":567.99,"body":"think the same way, you can"},{"speaker":"Pam","startTime":567.99,"endTime":571.8,"body":"start to use the same kinds of"},{"speaker":"Pam","startTime":567.99,"endTime":571.8,"body":"thinking to solve multiplication"},{"speaker":"Pam","startTime":571.8,"endTime":575.04,"body":"problems, the division problems,"},{"speaker":"Pam","startTime":571.8,"endTime":575.04,"body":"and proportions like we did in"},{"speaker":"Pam","startTime":575.04,"endTime":579.18,"body":"the third episode. And last week"},{"speaker":"Pam","startTime":575.04,"endTime":579.18,"body":"when we did in solving addition"},{"speaker":"Pam","startTime":579.18,"endTime":581.16,"body":"and subtraction of fractions by"},{"speaker":"Pam","startTime":579.18,"endTime":581.16,"body":"finding equivalent fractions."},{"speaker":"Pam","startTime":581.19,"endTime":585.63,"body":"You can use ratio tables to"},{"speaker":"Pam","startTime":581.19,"endTime":585.63,"body":"think in those ways. And then,"},{"speaker":"Pam","startTime":585.63,"endTime":588.78,"body":"we can all think more and more"},{"speaker":"Pam","startTime":585.63,"endTime":588.78,"body":"like mathematicians. Does that"},{"speaker":"Pam","startTime":588.78,"endTime":591.12,"body":"make sense, Kim? Did I say that?"},{"speaker":"Pam","startTime":588.78,"endTime":591.12,"body":"I'm thinking I'm still parsing"},{"speaker":"Pam","startTime":591.12,"endTime":591.99,"body":"out how to kind of say all that."},{"speaker":"Kim","startTime":592.02,"endTime":599.88,"body":"Yeah, yeah. No, I"},{"speaker":"Kim","startTime":592.02,"endTime":599.88,"body":"appreciate that what you do is"},{"speaker":"Kim","startTime":599.94,"endTime":604.95,"body":"really try to make sense of"},{"speaker":"Kim","startTime":599.94,"endTime":604.95,"body":"mathematics for the masses, in"},{"speaker":"Kim","startTime":604.95,"endTime":609.24,"body":"that these are things that are"},{"speaker":"Kim","startTime":604.95,"endTime":609.24,"body":"happening, and it's just"},{"speaker":"Kim","startTime":609.27,"endTime":612.81,"body":"sometimes hidden, sometimes like"},{"speaker":"Kim","startTime":609.27,"endTime":612.81,"body":"trapped inside people's brains"},{"speaker":"Kim","startTime":612.81,"endTime":615.63,"body":"because they don't have the"},{"speaker":"Kim","startTime":612.81,"endTime":615.63,"body":"language maybe or definitely"},{"speaker":"Kim","startTime":615.63,"endTime":618.81,"body":"don't have a way. That was it"},{"speaker":"Kim","startTime":615.63,"endTime":618.81,"body":"for me, right? I kind of"},{"speaker":"Kim","startTime":618.81,"endTime":621.03,"body":"stumbled my way through trying"},{"speaker":"Kim","startTime":618.81,"endTime":621.03,"body":"to figure out how to even talk"},{"speaker":"Kim","startTime":621.03,"endTime":624.18,"body":"about it, but I didn't have any"},{"speaker":"Kim","startTime":621.03,"endTime":624.18,"body":"way of putting stuff on paper to"},{"speaker":"Kim","startTime":624.18,"endTime":628.62,"body":"make sense of it for others. And"},{"speaker":"Kim","startTime":624.18,"endTime":628.62,"body":"so, it was just not a thing. We"},{"speaker":"Kim","startTime":628.62,"endTime":631.59,"body":"just didn't talk about it. We"},{"speaker":"Kim","startTime":628.62,"endTime":631.59,"body":"just didn't. You know, I didn't"},{"speaker":"Kim","startTime":631.59,"endTime":633.81,"body":"share strategies. I certainly"},{"speaker":"Kim","startTime":631.59,"endTime":633.81,"body":"didn't have a way of like"},{"speaker":"Kim","startTime":633.81,"endTime":636.15,"body":"learning more of the"},{"speaker":"Kim","startTime":633.81,"endTime":636.15,"body":"sophisticated strategies that I"},{"speaker":"Kim","startTime":636.15,"endTime":639.06,"body":"wasn't using. So, you know, it's"},{"speaker":"Kim","startTime":636.15,"endTime":639.06,"body":"good work."},{"speaker":"Pam","startTime":639.54,"endTime":642.72,"body":"Yeah, super. I appreciate"},{"speaker":"Pam","startTime":639.54,"endTime":642.72,"body":"you saying that. Cool. So, in"},{"speaker":"Pam","startTime":642.72,"endTime":645.93,"body":"this episode, we'd like to"},{"speaker":"Pam","startTime":642.72,"endTime":645.93,"body":"finish out by giving an example"},{"speaker":"Pam","startTime":645.93,"endTime":649.5,"body":"of where ratio tables can go in"},{"speaker":"Pam","startTime":645.93,"endTime":649.5,"body":"higher math. What do we do even"},{"speaker":"Pam","startTime":649.53,"endTime":652.89,"body":"after the work that we've done"},{"speaker":"Pam","startTime":649.53,"endTime":652.89,"body":"so far with solving proportions"},{"speaker":"Pam","startTime":652.89,"endTime":656.97,"body":"and adding and subtracting"},{"speaker":"Pam","startTime":652.89,"endTime":656.97,"body":"fractions? Well, first of all,"},{"speaker":"Pam","startTime":657.33,"endTime":661.8,"body":"let's acknowledge that if we've"},{"speaker":"Pam","startTime":657.33,"endTime":661.8,"body":"done the work, if we can get"},{"speaker":"Pam","startTime":661.8,"endTime":664.17,"body":"everybody to do the work that"},{"speaker":"Pam","startTime":661.8,"endTime":664.17,"body":"we've talked about so"},{"speaker":"Pam","startTime":664.17,"endTime":667.29,"body":"far...getting kids multiplying"},{"speaker":"Pam","startTime":664.17,"endTime":667.29,"body":"and dividing and solving"},{"speaker":"Pam","startTime":667.29,"endTime":670.38,"body":"proportions, finding equivalent"},{"speaker":"Pam","startTime":667.29,"endTime":670.38,"body":"fractions, ratios using ratio"},{"speaker":"Pam","startTime":670.41,"endTime":674.16,"body":"tables...that's a huge win"},{"speaker":"Pam","startTime":670.41,"endTime":674.16,"body":"because students will be"},{"speaker":"Pam","startTime":674.16,"endTime":678.51,"body":"thinking and reasoning"},{"speaker":"Pam","startTime":674.16,"endTime":678.51,"body":"proportionally. And getting them"},{"speaker":"Pam","startTime":678.51,"endTime":681.27,"body":"to think and reason"},{"speaker":"Pam","startTime":678.51,"endTime":681.27,"body":"proportionally is a super huge"},{"speaker":"Pam","startTime":681.27,"endTime":686.13,"body":"win because now we can build on"},{"speaker":"Pam","startTime":681.27,"endTime":686.13,"body":"that. Everything from rates,"},{"speaker":"Pam","startTime":686.16,"endTime":690.93,"body":"rates of change, to scaling to"},{"speaker":"Pam","startTime":686.16,"endTime":690.93,"body":"fractions and ratios, to then"},{"speaker":"Pam","startTime":690.93,"endTime":696.99,"body":"rational functions all are going"},{"speaker":"Pam","startTime":690.93,"endTime":696.99,"body":"to be based on, build on"},{"speaker":"Pam","startTime":697.23,"endTime":701.28,"body":"reasoning proportionally. And"},{"speaker":"Pam","startTime":697.23,"endTime":701.28,"body":"all of those things are going to"},{"speaker":"Pam","startTime":701.28,"endTime":704.79,"body":"be more of a natural outcome"},{"speaker":"Pam","startTime":701.28,"endTime":704.79,"body":"that we'll have more access to,"},{"speaker":"Pam","startTime":705.09,"endTime":706.92,"body":"if we've gotten students"},{"speaker":"Pam","startTime":705.09,"endTime":706.92,"body":"thinking and reasoning"},{"speaker":"Pam","startTime":706.92,"endTime":709.59,"body":"proportionally. And one of the"},{"speaker":"Pam","startTime":706.92,"endTime":709.59,"body":"super good ways to do that is"},{"speaker":"Pam","startTime":709.59,"endTime":713.76,"body":"using a ratio table. But there's"},{"speaker":"Pam","startTime":709.59,"endTime":713.76,"body":"also a fun side effect that I"},{"speaker":"Pam","startTime":713.76,"endTime":716.61,"body":"did not anticipate, and it's"},{"speaker":"Pam","startTime":713.76,"endTime":716.61,"body":"super cool the more and more"},{"speaker":"Pam","startTime":716.61,"endTime":720.06,"body":"that I see it as more and more"},{"speaker":"Pam","startTime":716.61,"endTime":720.06,"body":"people are using ratio tables to"},{"speaker":"Pam","startTime":720.06,"endTime":723.78,"body":"build up proportional reasoning."},{"speaker":"Pam","startTime":720.06,"endTime":723.78,"body":"This super fun side effect about"},{"speaker":"Pam","startTime":723.78,"endTime":727.53,"body":"writing the equation of a line."},{"speaker":"Pam","startTime":723.78,"endTime":727.53,"body":"And we did a little bit of this"},{"speaker":"Pam","startTime":727.56,"endTime":731.67,"body":"in an earlier episode where..."},{"speaker":"Pam","startTime":727.56,"endTime":731.67,"body":"And we're going to do it again."},{"speaker":"Pam","startTime":731.7,"endTime":737.1,"body":"So, this is super fun. So, even."},{"speaker":"Pam","startTime":731.7,"endTime":737.1,"body":"Kim's laughing."},{"speaker":"Kim","startTime":737.13,"endTime":737.55,"body":"Super fun."},{"speaker":"Pam","startTime":738.0,"endTime":741.0,"body":"Even if this is not your"},{"speaker":"Pam","startTime":738.0,"endTime":741.0,"body":"content, ya'll, dive in because"},{"speaker":"Pam","startTime":741.0,"endTime":744.39,"body":"I think especially if you've"},{"speaker":"Pam","startTime":741.0,"endTime":744.39,"body":"been using ratio tables, you're"},{"speaker":"Pam","startTime":744.39,"endTime":748.38,"body":"going to be amazed at what's"},{"speaker":"Pam","startTime":744.39,"endTime":748.38,"body":"going to pop for you as we go."},{"speaker":"Pam","startTime":748.38,"endTime":752.49,"body":"So, Kim, I'm going to give you a"},{"speaker":"Pam","startTime":748.38,"endTime":752.49,"body":"couple of points, and I'm going"},{"speaker":"Pam","startTime":752.49,"endTime":755.34,"body":"to ask you to consider. Well,"},{"speaker":"Pam","startTime":752.49,"endTime":755.34,"body":"I'm going to walk you through"},{"speaker":"Pam","startTime":755.34,"endTime":756.18,"body":"kind of what we're going to do,"},{"speaker":"Pam","startTime":755.34,"endTime":756.18,"body":"so."},{"speaker":"Kim","startTime":756.18,"endTime":756.45,"body":"Okay."},{"speaker":"Pam","startTime":757.08,"endTime":761.67,"body":"Here we go. If I give you"},{"speaker":"Pam","startTime":757.08,"endTime":761.67,"body":"the point: (1, 14). So, 1 comma"},{"speaker":"Pam","startTime":761.67,"endTime":768.27,"body":"14. X is 1. Y is 14. And a"},{"speaker":"Pam","startTime":761.67,"endTime":768.27,"body":"point: (2, 28)."},{"speaker":"Kim","startTime":768.54,"endTime":768.84,"body":"Okay."},{"speaker":"Pam","startTime":769.2,"endTime":771.36,"body":"Before we get into writing"},{"speaker":"Pam","startTime":769.2,"endTime":771.36,"body":"the equation of a line or"},{"speaker":"Pam","startTime":771.36,"endTime":773.79,"body":"anything, could you picture"},{"speaker":"Pam","startTime":771.36,"endTime":773.79,"body":"sticks of gum?"},{"speaker":"Kim","startTime":774.81,"endTime":775.38,"body":"Yeah."},{"speaker":"Pam","startTime":776.01,"endTime":777.84,"body":"Could you tell me about"},{"speaker":"Pam","startTime":776.01,"endTime":777.84,"body":"that context?"},{"speaker":"Kim","startTime":777.9,"endTime":778.41,"body":"Sure. So,"},{"speaker":"Kim","startTime":778.41,"endTime":781.38,"body":"I'm going to. I actually"},{"speaker":"Kim","startTime":778.41,"endTime":781.38,"body":"(unclear) because we're talking"},{"speaker":"Kim","startTime":781.38,"endTime":784.32,"body":"about ratio tables. I actually"},{"speaker":"Kim","startTime":781.38,"endTime":784.32,"body":"put that on a ratio table, and I"},{"speaker":"Kim","startTime":784.32,"endTime":791.13,"body":"put 1 to 14 in the first entry."},{"speaker":"Kim","startTime":784.32,"endTime":791.13,"body":"And then, I put 2 to 28 in the"},{"speaker":"Kim","startTime":791.13,"endTime":795.27,"body":"second. And it completely makes"},{"speaker":"Kim","startTime":791.13,"endTime":795.27,"body":"me think of 1 pack has 14"},{"speaker":"Kim","startTime":795.27,"endTime":798.21,"body":"sticks, 2 packs has 28 sticks."},{"speaker":"Pam","startTime":798.27,"endTime":802.23,"body":"Brilliant. So, here's a"},{"speaker":"Pam","startTime":798.27,"endTime":802.23,"body":"necessary step in order to get"},{"speaker":"Pam","startTime":802.23,"endTime":805.62,"body":"my final thing to work is that"},{"speaker":"Pam","startTime":802.23,"endTime":805.62,"body":"we would need to have done"},{"speaker":"Pam","startTime":805.65,"endTime":808.68,"body":"sticks of gum. Which obviously"},{"speaker":"Pam","startTime":805.65,"endTime":808.68,"body":"you're like, \"Yep, clearly."},{"speaker":"Pam","startTime":808.71,"endTime":811.59,"body":"Clearly, that's pinging for you"},{"speaker":"Pam","startTime":808.71,"endTime":811.59,"body":"right now.\" But we would also"},{"speaker":"Pam","startTime":811.59,"endTime":815.07,"body":"needed to have done a little bit"},{"speaker":"Pam","startTime":811.59,"endTime":815.07,"body":"of an extra step where we would"},{"speaker":"Pam","startTime":815.07,"endTime":819.27,"body":"graph those points as ordered"},{"speaker":"Pam","startTime":815.07,"endTime":819.27,"body":"pairs. So, over 1, up 14. Over"},{"speaker":"Pam","startTime":819.27,"endTime":823.29,"body":"2, up 28. And we would have"},{"speaker":"Pam","startTime":819.27,"endTime":823.29,"body":"written the equation of the line"},{"speaker":"Pam","startTime":823.29,"endTime":826.74,"body":"that contains those points. So,"},{"speaker":"Pam","startTime":823.29,"endTime":826.74,"body":"we would have wanted to say, \"If"},{"speaker":"Pam","startTime":826.77,"endTime":831.57,"body":"I want to find any number of"},{"speaker":"Pam","startTime":826.77,"endTime":831.57,"body":"sticks based on the number of"},{"speaker":"Pam","startTime":831.57,"endTime":837.09,"body":"packs, what  equation of a line"},{"speaker":"Pam","startTime":831.57,"endTime":837.09,"body":"would that be? And so, what"},{"speaker":"Pam","startTime":837.09,"endTime":840.69,"body":"equation of a line would work"},{"speaker":"Pam","startTime":837.09,"endTime":840.69,"body":"for these two points?\""},{"speaker":"Kim","startTime":840.96,"endTime":843.21,"body":"So, it would be 14x."},{"speaker":"Pam","startTime":845.64,"endTime":846.12,"body":"Yeah."},{"speaker":"Kim","startTime":846.3,"endTime":848.91,"body":"So, y would equal 14x."},{"speaker":"Pam","startTime":849.21,"endTime":851.76,"body":"Yeah. And if we stayed in"},{"speaker":"Pam","startTime":849.21,"endTime":851.76,"body":"sticks of gum, then we would say"},{"speaker":"Pam","startTime":851.76,"endTime":855.15,"body":"the number of sticks is"},{"speaker":"Pam","startTime":851.76,"endTime":855.15,"body":"equivalent to 14 times the"},{"speaker":"Pam","startTime":855.15,"endTime":855.99,"body":"number of packs."},{"speaker":"Kim","startTime":856.02,"endTime":859.41,"body":"Yeah, so in mine, I"},{"speaker":"Kim","startTime":856.02,"endTime":859.41,"body":"actually made a third entry, and"},{"speaker":"Kim","startTime":859.41,"endTime":864.51,"body":"I put x in the top, and I put"},{"speaker":"Kim","startTime":859.41,"endTime":864.51,"body":"14x in the bottom."},{"speaker":"Pam","startTime":864.99,"endTime":868.29,"body":"Oh, nice. Nice. So, you can"},{"speaker":"Pam","startTime":864.99,"endTime":868.29,"body":"kind of see that that's the"},{"speaker":"Pam","startTime":868.32,"endTime":871.32,"body":"relationship, the output will be"},{"speaker":"Pam","startTime":868.32,"endTime":871.32,"body":"14 times the number of sticks."},{"speaker":"Pam","startTime":871.65,"endTime":875.19,"body":"Cool. So, listeners, we would"},{"speaker":"Pam","startTime":871.65,"endTime":875.19,"body":"want to get students to be able"},{"speaker":"Pam","startTime":875.19,"endTime":879.63,"body":"to write the equation of a"},{"speaker":"Pam","startTime":875.19,"endTime":879.63,"body":"proportional relation where it"},{"speaker":"Pam","startTime":879.63,"endTime":882.93,"body":"would fit in a sticks of gum"},{"speaker":"Pam","startTime":879.63,"endTime":882.93,"body":"scenario, where it includes the"},{"speaker":"Pam","startTime":882.93,"endTime":886.17,"body":"origin and all of the ratios are"},{"speaker":"Pam","startTime":882.93,"endTime":886.17,"body":"equivalent. 1 to 14 is"},{"speaker":"Pam","startTime":886.17,"endTime":889.89,"body":"equivalent to 2 to 28. So, when"},{"speaker":"Pam","startTime":886.17,"endTime":889.89,"body":"I have a proportional relation,"},{"speaker":"Pam","startTime":889.92,"endTime":892.8,"body":"we'd want students to be able to"},{"speaker":"Pam","startTime":889.92,"endTime":892.8,"body":"write the equation of that line."},{"speaker":"Pam","startTime":893.13,"endTime":897.51,"body":"In this case, y equals 14x. We"},{"speaker":"Pam","startTime":893.13,"endTime":897.51,"body":"would want to do that a bit, so"},{"speaker":"Pam","startTime":897.51,"endTime":901.02,"body":"that students kind of have this"},{"speaker":"Pam","startTime":897.51,"endTime":901.02,"body":"feel for proportional relations,"},{"speaker":"Pam","startTime":901.05,"endTime":904.56,"body":"not only operating in a ratio"},{"speaker":"Pam","startTime":901.05,"endTime":904.56,"body":"table but also being able to"},{"speaker":"Pam","startTime":904.56,"endTime":907.86,"body":"write the equation of the line"},{"speaker":"Pam","startTime":904.56,"endTime":907.86,"body":"that fits those points. When we"},{"speaker":"Pam","startTime":907.86,"endTime":913.44,"body":"do that, then I think it follows"},{"speaker":"Pam","startTime":907.86,"endTime":913.44,"body":"fairly naturally for me to give"},{"speaker":"Pam","startTime":913.5,"endTime":915.87,"body":"a set of points. I'm going to"},{"speaker":"Pam","startTime":913.5,"endTime":915.87,"body":"give you another set of points,"},{"speaker":"Pam","startTime":915.87,"endTime":918.84,"body":"Kim, and I'm going to ask you if"},{"speaker":"Pam","startTime":915.87,"endTime":918.84,"body":"you see any connection, or if"},{"speaker":"Pam","startTime":918.84,"endTime":922.29,"body":"anything pings for you. So, what"},{"speaker":"Pam","startTime":918.84,"endTime":922.29,"body":"if I give you the points (1,"},{"speaker":"Pam","startTime":922.32,"endTime":924.3,"body":"13). 1 comma 13."},{"speaker":"Kim","startTime":924.75,"endTime":925.17,"body":"Okay."},{"speaker":"Pam","startTime":925.26,"endTime":926.94,"body":"And (2, 27)."},{"speaker":"Kim","startTime":927.93,"endTime":928.44,"body":"Okay."},{"speaker":"Pam","startTime":929.79,"endTime":932.22,"body":"Is there anything that"},{"speaker":"Pam","startTime":929.79,"endTime":932.22,"body":"pings for you based on....This"},{"speaker":"Pam","startTime":932.55,"endTime":935.73,"body":"is a separate ratio table. I"},{"speaker":"Pam","startTime":932.55,"endTime":935.73,"body":"should have said that. We're not"},{"speaker":"Pam","startTime":935.73,"endTime":940.02,"body":"adding to the first ratio table."},{"speaker":"Pam","startTime":935.73,"endTime":940.02,"body":"So, new set of points. Does"},{"speaker":"Pam","startTime":940.02,"endTime":941.61,"body":"anything ping for you as you?"},{"speaker":"Kim","startTime":941.61,"endTime":945.33,"body":"Well, yeah. Yeah, it's like"},{"speaker":"Kim","startTime":941.61,"endTime":945.33,"body":"the first ratio table because"},{"speaker":"Kim","startTime":946.08,"endTime":950.7,"body":"instead of 1 to 14, it's 1 of"},{"speaker":"Kim","startTime":946.08,"endTime":950.7,"body":"the 13. Instead of 2 to 28, it's"},{"speaker":"Kim","startTime":950.7,"endTime":959.04,"body":"2 to 27. So, it's like that same"},{"speaker":"Kim","startTime":950.7,"endTime":959.04,"body":"14 sticks per pack, but you took"},{"speaker":"Kim","startTime":959.04,"endTime":964.98,"body":"away a stick. So, 14 per pack,"},{"speaker":"Kim","startTime":959.04,"endTime":964.98,"body":"minus 1."},{"speaker":"Pam","startTime":966.06,"endTime":970.11,"body":"Because there's still. 13"},{"speaker":"Pam","startTime":966.06,"endTime":970.11,"body":"plus 14 is still 27, right?"},{"speaker":"Pam","startTime":970.11,"endTime":975.03,"body":"There's still 14 sticks in"},{"speaker":"Pam","startTime":970.11,"endTime":975.03,"body":"between the two points. Like, if"},{"speaker":"Pam","startTime":975.03,"endTime":979.68,"body":"we were to look at from 13 to"},{"speaker":"Pam","startTime":975.03,"endTime":979.68,"body":"27, it's still 14. So, there's"},{"speaker":"Pam","startTime":979.68,"endTime":985.02,"body":"still that change of 14 every"},{"speaker":"Pam","startTime":979.68,"endTime":985.02,"body":"time, 14x. But we've subtracted"},{"speaker":"Pam","startTime":985.02,"endTime":989.01,"body":"1. And I will never forget the"},{"speaker":"Pam","startTime":985.02,"endTime":989.01,"body":"day that Anne Roman said to me,"},{"speaker":"Pam","startTime":989.61,"endTime":993.84,"body":"\"I just see the 14, 28 just"},{"speaker":"Pam","startTime":989.61,"endTime":993.84,"body":"popping out.\" It's just, like,"},{"speaker":"Pam","startTime":993.84,"endTime":997.56,"body":"sitting there.\" I've dealt with"},{"speaker":"Pam","startTime":993.84,"endTime":997.56,"body":"ratio tables so often that I"},{"speaker":"Pam","startTime":997.56,"endTime":1001.64,"body":"just see the 14, 28. But it's"},{"speaker":"Pam","startTime":997.56,"endTime":1001.64,"body":"just... It's just shifted just a"},{"speaker":"Pam","startTime":1001.64,"endTime":1002.96,"body":"little. It's just off 1.\""},{"speaker":"Kim","startTime":1003.2,"endTime":1003.5,"body":"Yeah."},{"speaker":"Pam","startTime":1003.53,"endTime":1006.56,"body":"So, then, what's the"},{"speaker":"Pam","startTime":1003.53,"endTime":1006.56,"body":"equation of the line for 1, 13"},{"speaker":"Pam","startTime":1006.56,"endTime":1008.78,"body":"and 2, 27? (unclear)."},{"speaker":"Kim","startTime":1008.78,"endTime":1010.79,"body":"(unclear) Minus 1. Yeah."},{"speaker":"Pam","startTime":1010.79,"endTime":1014.3,"body":"Yeah, so y = 14x - 1. Let's"},{"speaker":"Pam","startTime":1010.79,"endTime":1014.3,"body":"do another one. One more"},{"speaker":"Pam","startTime":1014.63,"endTime":1017.93,"body":"example. Alright, so this time,"},{"speaker":"Pam","startTime":1014.63,"endTime":1017.93,"body":"I'm going to give you... Again,"},{"speaker":"Pam","startTime":1017.93,"endTime":1018.8,"body":"two ordered pairs."},{"speaker":"Kim","startTime":1019.1,"endTime":1019.37,"body":"Okay."},{"speaker":"Pam","startTime":1019.4,"endTime":1023.3,"body":"The first one is (1, -6)."},{"speaker":"Kim","startTime":1024.349,"endTime":1024.979,"body":"Mmm, okay."},{"speaker":"Pam","startTime":1025.19,"endTime":1028.64,"body":"Second point (2,-12)."},{"speaker":"Kim","startTime":1029.93,"endTime":1035.06,"body":"Okay, so this one is"},{"speaker":"Kim","startTime":1029.93,"endTime":1035.06,"body":"interesting to me because you"},{"speaker":"Kim","startTime":1035.06,"endTime":1039.02,"body":"could think about, like sticks"},{"speaker":"Kim","startTime":1035.06,"endTime":1039.02,"body":"of gum 1 to 6, 2 to 12. But"},{"speaker":"Kim","startTime":1039.02,"endTime":1047.27,"body":"then, it's negative. But I also"},{"speaker":"Kim","startTime":1039.02,"endTime":1047.27,"body":"was thinking more like, one time"},{"speaker":"Kim","startTime":1047.27,"endTime":1051.92,"body":"something happened, I lost"},{"speaker":"Kim","startTime":1047.27,"endTime":1051.92,"body":"$6.00. And (unclear) happened, I"},{"speaker":"Kim","startTime":1051.92,"endTime":1057.92,"body":"lost two 6s. (unclear) like a"},{"speaker":"Kim","startTime":1051.92,"endTime":1057.92,"body":"debt, or like I owed $6.00 each"},{"speaker":"Kim","startTime":1057.92,"endTime":1058.91,"body":"time something happened."},{"speaker":"Pam","startTime":1059.15,"endTime":1062.48,"body":"Sure enough. So, nice"},{"speaker":"Pam","startTime":1059.15,"endTime":1062.48,"body":"flexibility. You can kind of"},{"speaker":"Pam","startTime":1062.48,"endTime":1065.96,"body":"think of that two ways. So, what"},{"speaker":"Pam","startTime":1062.48,"endTime":1065.96,"body":"would be the equation of a line"},{"speaker":"Pam","startTime":1065.96,"endTime":1067.01,"body":"that would represent those?"},{"speaker":"Kim","startTime":1067.46,"endTime":1069.83,"body":"y = - 6x."},{"speaker":"Pam","startTime":1070.34,"endTime":1073.64,"body":"Cool. And I'll just focus"},{"speaker":"Pam","startTime":1070.34,"endTime":1073.64,"body":"because we've been talking about"},{"speaker":"Pam","startTime":1073.64,"endTime":1076.58,"body":"ratio table. So, I like how you"},{"speaker":"Pam","startTime":1073.64,"endTime":1076.58,"body":"did the $6.00 things. But I'll"},{"speaker":"Pam","startTime":1076.58,"endTime":1079.7,"body":"focus on what you said first."},{"speaker":"Pam","startTime":1076.58,"endTime":1079.7,"body":"That if we've got students"},{"speaker":"Pam","startTime":1079.7,"endTime":1084.17,"body":"thinking about 6 sticks of gum."},{"speaker":"Pam","startTime":1079.7,"endTime":1084.17,"body":"12 sticks of gum and 2 packs,"},{"speaker":"Pam","startTime":1084.56,"endTime":1087.2,"body":"then could they write the"},{"speaker":"Pam","startTime":1084.56,"endTime":1087.2,"body":"equation of line y = 6x. And"},{"speaker":"Pam","startTime":1087.2,"endTime":1091.76,"body":"they can even picture that line,"},{"speaker":"Pam","startTime":1087.2,"endTime":1091.76,"body":"sort of going through the first"},{"speaker":"Pam","startTime":1091.76,"endTime":1095.12,"body":"quadrant, and then say to"},{"speaker":"Pam","startTime":1091.76,"endTime":1095.12,"body":"themselves, \"But it's the"},{"speaker":"Pam","startTime":1095.12,"endTime":1098.57,"body":"opposite. I've got to reflect"},{"speaker":"Pam","startTime":1095.12,"endTime":1098.57,"body":"everything. And so, now, it's"},{"speaker":"Pam","startTime":1098.57,"endTime":1102.11,"body":"going to go through the second"},{"speaker":"Pam","startTime":1098.57,"endTime":1102.11,"body":"and fourth quadrant. It's y ="},{"speaker":"Pam","startTime":1102.14,"endTime":1105.23,"body":"-6x.\" So, just a different way."},{"speaker":"Pam","startTime":1102.14,"endTime":1105.23,"body":"We would want students to own"},{"speaker":"Pam","startTime":1105.23,"endTime":1107.78,"body":"both of the ways that you just"},{"speaker":"Pam","startTime":1105.23,"endTime":1107.78,"body":"thought about that. I'm so super"},{"speaker":"Pam","startTime":1107.93,"endTime":1112.88,"body":"glad that you do. Cool. Then, if"},{"speaker":"Pam","startTime":1107.93,"endTime":1112.88,"body":"we've got sort of that"},{"speaker":"Pam","startTime":1112.88,"endTime":1116.12,"body":"relationship kind of pinging for"},{"speaker":"Pam","startTime":1112.88,"endTime":1116.12,"body":"kids, could we give them problem"},{"speaker":"Pam","startTime":1116.42,"endTime":1120.23,"body":"points like this? (1,-5);"},{"speaker":"Pam","startTime":1116.42,"endTime":1120.23,"body":"(2,-11)."},{"speaker":"Kim","startTime":1120.23,"endTime":1120.32,"body":"Okay."},{"speaker":"Pam","startTime":1121.34,"endTime":1132.59,"body":"So, (1,-5), (2,-11). And"},{"speaker":"Pam","startTime":1121.34,"endTime":1132.59,"body":"wonder if anything pings?"},{"speaker":"Kim","startTime":1133.52,"endTime":1139.82,"body":"Yeah, so it's going to be"},{"speaker":"Kim","startTime":1133.52,"endTime":1139.82,"body":"similar to the one that you just"},{"speaker":"Kim","startTime":1139.82,"endTime":1149.72,"body":"gave me where it's -6x, but it's"},{"speaker":"Kim","startTime":1139.82,"endTime":1149.72,"body":"not quite as negative each time."},{"speaker":"Kim","startTime":1151.88,"endTime":1155.96,"body":"So, it's 1 away. It's not quite"},{"speaker":"Kim","startTime":1151.88,"endTime":1155.96,"body":"as negative by 1. So, plus 1."},{"speaker":"Kim","startTime":1155.96,"endTime":1158.18,"body":"So, -6x plus 1."},{"speaker":"Pam","startTime":1158.33,"endTime":1159.8,"body":"So, y e= -6x +1."},{"speaker":"Kim","startTime":1160.4,"endTime":1160.64,"body":"Yeah."},{"speaker":"Pam","startTime":1160.64,"endTime":1164.15,"body":"Super. Yeah, because we've"},{"speaker":"Pam","startTime":1160.64,"endTime":1164.15,"body":"just. We can see... It's almost"},{"speaker":"Pam","startTime":1164.15,"endTime":1168.44,"body":"like you can see the -6, -12,"},{"speaker":"Pam","startTime":1164.15,"endTime":1168.44,"body":"but it's not quite as negative."},{"speaker":"Pam","startTime":1168.47,"endTime":1172.01,"body":"I like it. It's been shifted up"},{"speaker":"Pam","startTime":1168.47,"endTime":1172.01,"body":"1. And I actually just sort of"},{"speaker":"Pam","startTime":1172.01,"endTime":1174.17,"body":"took that line that I had kind"},{"speaker":"Pam","startTime":1172.01,"endTime":1174.17,"body":"of drawn in the air, and I"},{"speaker":"Pam","startTime":1174.35,"endTime":1180.56,"body":"shifted it up 1 to get -6x + 1."},{"speaker":"Pam","startTime":1174.35,"endTime":1180.56,"body":"Super cool. So, what we're"},{"speaker":"Pam","startTime":1180.56,"endTime":1184.58,"body":"suggesting is that there's this"},{"speaker":"Pam","startTime":1180.56,"endTime":1184.58,"body":"natural outcome that once"},{"speaker":"Pam","startTime":1184.58,"endTime":1188.3,"body":"students really start thinking"},{"speaker":"Pam","startTime":1184.58,"endTime":1188.3,"body":"and reasoning with ratio tables"},{"speaker":"Pam","startTime":1188.57,"endTime":1191.78,"body":"that they begin to see them in"},{"speaker":"Pam","startTime":1188.57,"endTime":1191.78,"body":"places that we might not have"},{"speaker":"Pam","startTime":1191.78,"endTime":1194.54,"body":"seen them before if we hadn't"},{"speaker":"Pam","startTime":1191.78,"endTime":1194.54,"body":"already kind of owned those"},{"speaker":"Pam","startTime":1194.54,"endTime":1197.15,"body":"relationships. And if we can"},{"speaker":"Pam","startTime":1194.54,"endTime":1197.15,"body":"write the equation of a line for"},{"speaker":"Pam","startTime":1197.15,"endTime":1199.58,"body":"proportional relation, and we"},{"speaker":"Pam","startTime":1197.15,"endTime":1199.58,"body":"can see that this non"},{"speaker":"Pam","startTime":1199.58,"endTime":1203.9,"body":"proportional relation is just a"},{"speaker":"Pam","startTime":1199.58,"endTime":1203.9,"body":"little bit off, that we decide"},{"speaker":"Pam","startTime":1203.9,"endTime":1207.14,"body":"how far off is it. Oh, bam! And"},{"speaker":"Pam","startTime":1203.9,"endTime":1207.14,"body":"then, write the equation of the"},{"speaker":"Pam","startTime":1207.14,"endTime":1211.79,"body":"line based on just those"},{"speaker":"Pam","startTime":1207.14,"endTime":1211.79,"body":"noticings. Kim, when I say in"},{"speaker":"Pam","startTime":1211.79,"endTime":1215.12,"body":"the beginning of the podcast"},{"speaker":"Pam","startTime":1211.79,"endTime":1215.12,"body":"that we can... I'm trying to go"},{"speaker":"Pam","startTime":1215.12,"endTime":1218.87,"body":"back to my script here, so I can"},{"speaker":"Pam","startTime":1215.12,"endTime":1218.87,"body":"say it correctly. When we notice"},{"speaker":"Pam","startTime":1218.87,"endTime":1221.78,"body":"patterns and reason using"},{"speaker":"Pam","startTime":1218.87,"endTime":1221.78,"body":"mathematical relationships."},{"speaker":"Pam","startTime":1222.02,"endTime":1224.99,"body":"That's what I mean. That's what"},{"speaker":"Pam","startTime":1222.02,"endTime":1224.99,"body":"I mean when I say that. That"},{"speaker":"Pam","startTime":1224.99,"endTime":1228.02,"body":"it's about noticing the pattern"},{"speaker":"Pam","startTime":1224.99,"endTime":1228.02,"body":"like this ratio table. It's like"},{"speaker":"Pam","startTime":1228.02,"endTime":1232.1,"body":"letting it ping for you. \"Oh,"},{"speaker":"Pam","startTime":1228.02,"endTime":1232.1,"body":"but it's just off a little bit.\""},{"speaker":"Pam","startTime":1232.1,"endTime":1234.71,"body":"And so, I can write the equation"},{"speaker":"Pam","startTime":1232.1,"endTime":1234.71,"body":"for that proportional, and for"},{"speaker":"Pam","startTime":1234.71,"endTime":1238.13,"body":"this non proportional, I can"},{"speaker":"Pam","startTime":1234.71,"endTime":1238.13,"body":"just tweak the equation to"},{"speaker":"Pam","startTime":1238.13,"endTime":1241.04,"body":"match. It's about noticing"},{"speaker":"Pam","startTime":1238.13,"endTime":1241.04,"body":"patterns and using"},{"speaker":"Pam","startTime":1241.4,"endTime":1247.7,"body":"relationships. Woah! Super cool."},{"speaker":"Pam","startTime":1241.4,"endTime":1247.7,"body":"And let me just note that we"},{"speaker":"Pam","startTime":1247.7,"endTime":1254.18,"body":"also have this shift happening,"},{"speaker":"Pam","startTime":1247.7,"endTime":1254.18,"body":"and so there's also this idea of"},{"speaker":"Pam","startTime":1254.18,"endTime":1258.35,"body":"shifting. And boy, I'm just. I'm"},{"speaker":"Pam","startTime":1254.18,"endTime":1258.35,"body":"reminded of two things. I'm"},{"speaker":"Pam","startTime":1258.35,"endTime":1262.01,"body":"reminded of a geometric shift"},{"speaker":"Pam","startTime":1258.35,"endTime":1262.01,"body":"that when we translate"},{"speaker":"Pam","startTime":1262.01,"endTime":1265.52,"body":"functions. And so, that's an"},{"speaker":"Pam","startTime":1262.01,"endTime":1265.52,"body":"important kind of thing that"},{"speaker":"Pam","startTime":1266.6,"endTime":1269.6,"body":"comes out naturally because of"},{"speaker":"Pam","startTime":1266.6,"endTime":1269.6,"body":"this ratio table connection,"},{"speaker":"Pam","startTime":1270.08,"endTime":1272.6,"body":"that we can use then with"},{"speaker":"Pam","startTime":1270.08,"endTime":1272.6,"body":"students as we talked about"},{"speaker":"Pam","startTime":1272.6,"endTime":1277.25,"body":"transformations of functions."},{"speaker":"Pam","startTime":1272.6,"endTime":1277.25,"body":"There's also... This is like the"},{"speaker":"Pam","startTime":1277.25,"endTime":1281.87,"body":"hardest thing for me to talk"},{"speaker":"Pam","startTime":1277.25,"endTime":1281.87,"body":"about in higher math. When we"},{"speaker":"Pam","startTime":1281.87,"endTime":1284.45,"body":"think of that shift in"},{"speaker":"Pam","startTime":1281.87,"endTime":1284.45,"body":"transformations of functions,"},{"speaker":"Pam","startTime":1284.48,"endTime":1288.41,"body":"the vertical shift is super"},{"speaker":"Pam","startTime":1284.48,"endTime":1288.41,"body":"easy, right? Like, if you have y"},{"speaker":"Pam","startTime":1288.41,"endTime":1291.92,"body":"= x^2 + 1, you're like, \"Duh,\""},{"speaker":"Pam","startTime":1288.41,"endTime":1291.92,"body":"because you're taking all the y"},{"speaker":"Pam","startTime":1291.92,"endTime":1294.74,"body":"values, and you add 1, and so"},{"speaker":"Pam","startTime":1291.92,"endTime":1294.74,"body":"that parabola just shifts up 1."},{"speaker":"Kim","startTime":1294.77,"endTime":1295.07,"body":"Yep."},{"speaker":"Pam","startTime":1295.19,"endTime":1299.18,"body":"If it's y = x^3 - 15, then"},{"speaker":"Pam","startTime":1295.19,"endTime":1299.18,"body":"you take that x^3 function, and"},{"speaker":"Pam","startTime":1299.21,"endTime":1304.01,"body":"all the y values are just down"},{"speaker":"Pam","startTime":1299.21,"endTime":1304.01,"body":"15. So, easy shift. But boy,"},{"speaker":"Pam","startTime":1304.01,"endTime":1308.12,"body":"that shift in the parentheses."},{"speaker":"Pam","startTime":1304.01,"endTime":1308.12,"body":"It's tricky, right? It's tricky."},{"speaker":"Pam","startTime":1308.33,"endTime":1313.67,"body":"When you replace x with x - 5,"},{"speaker":"Pam","startTime":1308.33,"endTime":1313.67,"body":"why doesn't the parabola go to"},{"speaker":"Pam","startTime":1313.67,"endTime":1317.42,"body":"the left 5?! It was x - 5. You'd"},{"speaker":"Pam","startTime":1313.67,"endTime":1317.42,"body":"think subtract, it should be"},{"speaker":"Pam","startTime":1317.42,"endTime":1322.25,"body":"left. But it's not. And I tell"},{"speaker":"Pam","startTime":1317.42,"endTime":1322.25,"body":"you, Kim, when we work with, so,"},{"speaker":"Pam","startTime":1322.28,"endTime":1324.74,"body":"I'm definitely talking to high"},{"speaker":"Pam","startTime":1322.28,"endTime":1324.74,"body":"school teachers now, or high"},{"speaker":"Pam","startTime":1324.74,"endTime":1327.41,"body":"school and above. But when we"},{"speaker":"Pam","startTime":1324.74,"endTime":1327.41,"body":"work with, well, I shouldn't"},{"speaker":"Pam","startTime":1327.41,"endTime":1329.75,"body":"just say that. Anybody who's"},{"speaker":"Pam","startTime":1327.41,"endTime":1329.75,"body":"messed with functions and"},{"speaker":"Pam","startTime":1329.75,"endTime":1331.82,"body":"transformations of functions"},{"speaker":"Pam","startTime":1329.75,"endTime":1331.82,"body":"lately, that's who I'm talking"},{"speaker":"Pam","startTime":1331.82,"endTime":1336.89,"body":"to. When we mess with this"},{"speaker":"Pam","startTime":1331.82,"endTime":1336.89,"body":"proportional relation, and how"},{"speaker":"Pam","startTime":1337.01,"endTime":1341.36,"body":"I'm encouraging us to see the"},{"speaker":"Pam","startTime":1337.01,"endTime":1341.36,"body":"ratio table. The equivalent"},{"speaker":"Pam","startTime":1341.36,"endTime":1346.43,"body":"ratio is just shifted up or"},{"speaker":"Pam","startTime":1341.36,"endTime":1346.43,"body":"down. That is the same feeling I"},{"speaker":"Pam","startTime":1346.43,"endTime":1352.7,"body":"get when I am finally now able"},{"speaker":"Pam","startTime":1346.43,"endTime":1352.7,"body":"to explain that horizontal shift"},{"speaker":"Pam","startTime":1352.76,"endTime":1357.5,"body":"for function transformations."},{"speaker":"Pam","startTime":1352.76,"endTime":1357.5,"body":"Because for me, in my head,"},{"speaker":"Pam","startTime":1357.53,"endTime":1362.54,"body":"what's happening is I'm saying"},{"speaker":"Pam","startTime":1357.53,"endTime":1362.54,"body":"to myself, \"I'm replacing x with"},{"speaker":"Pam","startTime":1362.54,"endTime":1367.46,"body":"x + 2. And when I replace x with"},{"speaker":"Pam","startTime":1362.54,"endTime":1367.46,"body":"x + 2...\" So, x + 2 is in the"},{"speaker":"Pam","startTime":1367.46,"endTime":1371.51,"body":"parenthesis. And that function"},{"speaker":"Pam","startTime":1367.46,"endTime":1371.51,"body":"is going to shift to the left."},{"speaker":"Pam","startTime":1371.99,"endTime":1374.63,"body":"\"What?! But it's plus! Shouldn't"},{"speaker":"Pam","startTime":1371.99,"endTime":1374.63,"body":"it be to the right?!\" It's all"},{"speaker":"Pam","startTime":1374.63,"endTime":1379.34,"body":"about me having to go in the"},{"speaker":"Pam","startTime":1374.63,"endTime":1379.34,"body":"table to where I would normally"},{"speaker":"Pam","startTime":1379.34,"endTime":1384.32,"body":"have gone the x, but I've got to"},{"speaker":"Pam","startTime":1379.34,"endTime":1384.32,"body":"go plus 2, I've got to go down"},{"speaker":"Pam","startTime":1384.32,"endTime":1389.36,"body":"the table 2, grab that y value."},{"speaker":"Pam","startTime":1384.32,"endTime":1389.36,"body":"But then, it's got to shift back"},{"speaker":"Pam","startTime":1389.39,"endTime":1392.9,"body":"up the table to give it where"},{"speaker":"Pam","startTime":1389.39,"endTime":1392.9,"body":"that x was. See, I don't say"},{"speaker":"Pam","startTime":1392.9,"endTime":1395.81,"body":"that very well. Anybody who"},{"speaker":"Pam","startTime":1392.9,"endTime":1395.81,"body":"understands that shift just went"},{"speaker":"Pam","startTime":1395.84,"endTime":1398.33,"body":"\"Yeah, yeah, yeah, yeah. That's"},{"speaker":"Pam","startTime":1395.84,"endTime":1398.33,"body":"totally right!\" And anybody who"},{"speaker":"Pam","startTime":1398.33,"endTime":1399.98,"body":"doesn't is like, \"What?\""},{"speaker":"Kim","startTime":1400.94,"endTime":1407.54,"body":"I can remember when we were"},{"speaker":"Kim","startTime":1400.94,"endTime":1407.54,"body":"filming for Building Powerful"},{"speaker":"Kim","startTime":1407.54,"endTime":1415.61,"body":"Linear Functions. I remember"},{"speaker":"Kim","startTime":1407.54,"endTime":1415.61,"body":"thinking, \"What is going on?\""},{"speaker":"Kim","startTime":1415.64,"endTime":1419.3,"body":"Like, in this moment that you're"},{"speaker":"Kim","startTime":1415.64,"endTime":1419.3,"body":"talking about right now. I was"},{"speaker":"Kim","startTime":1419.3,"endTime":1423.14,"body":"like, you know, \"Oh, it's plus"},{"speaker":"Kim","startTime":1419.3,"endTime":1423.14,"body":"1. It goes this way. It's plus.\""},{"speaker":"Kim","startTime":1423.17,"endTime":1426.56,"body":"And I remember going, \"Wait a"},{"speaker":"Kim","startTime":1423.17,"endTime":1426.56,"body":"second.\" And like really"},{"speaker":"Kim","startTime":1426.56,"endTime":1430.52,"body":"thinking about. And then, when"},{"speaker":"Kim","startTime":1426.56,"endTime":1430.52,"body":"my oldest Luke tackled this, I"},{"speaker":"Kim","startTime":1430.52,"endTime":1434.36,"body":"was like, \"Wait a second.\" Like,"},{"speaker":"Kim","startTime":1430.52,"endTime":1434.36,"body":"it took. Like, I had that minor"},{"speaker":"Kim","startTime":1434.36,"endTime":1437.45,"body":"experience the first time, and I"},{"speaker":"Kim","startTime":1434.36,"endTime":1437.45,"body":"was like. It was real kind of"},{"speaker":"Kim","startTime":1437.99,"endTime":1441.14,"body":"foggy for me. I was just trying"},{"speaker":"Kim","startTime":1437.99,"endTime":1441.14,"body":"to make sense of it. But then,"},{"speaker":"Kim","startTime":1441.29,"endTime":1444.86,"body":"when Luke and I kind of worked"},{"speaker":"Kim","startTime":1441.29,"endTime":1444.86,"body":"through some stuff, I was like,"},{"speaker":"Kim","startTime":1445.61,"endTime":1447.98,"body":"\"Ah!\" It was just one of those"},{"speaker":"Kim","startTime":1445.61,"endTime":1447.98,"body":"moments where I was like, So,"},{"speaker":"Kim","startTime":1447.98,"endTime":1450.17,"body":"I'm following you right now."},{"speaker":"Kim","startTime":1447.98,"endTime":1450.17,"body":"Like, I have a clue what you're"},{"speaker":"Kim","startTime":1450.17,"endTime":1453.47,"body":"talking about because I've had"},{"speaker":"Kim","startTime":1450.17,"endTime":1453.47,"body":"an experience where I had to"},{"speaker":"Kim","startTime":1453.47,"endTime":1455.27,"body":"make sense of that myself."},{"speaker":"Pam","startTime":1456.05,"endTime":1460.37,"body":"Super cool. Oh, I love"},{"speaker":"Pam","startTime":1456.05,"endTime":1460.37,"body":"Luke. Nice. Alright. So, why"},{"speaker":"Pam","startTime":1460.37,"endTime":1466.52,"body":"this model? Why the ratio table"},{"speaker":"Pam","startTime":1460.37,"endTime":1466.52,"body":"model? In summary, we cannot"},{"speaker":"Pam","startTime":1466.52,"endTime":1469.43,"body":"only multiply and build"},{"speaker":"Pam","startTime":1466.52,"endTime":1469.43,"body":"multiplicative reasoning with"},{"speaker":"Pam","startTime":1469.43,"endTime":1473.15,"body":"division, and throw long"},{"speaker":"Pam","startTime":1469.43,"endTime":1473.15,"body":"division away because we can use"},{"speaker":"Pam","startTime":1473.15,"endTime":1476.18,"body":"ratio tables to think and reason"},{"speaker":"Pam","startTime":1473.15,"endTime":1476.18,"body":"to division. But then, it can"},{"speaker":"Pam","startTime":1476.18,"endTime":1479.63,"body":"extend to solving proportions."},{"speaker":"Pam","startTime":1476.18,"endTime":1479.63,"body":"It can extend to thinking about"},{"speaker":"Pam","startTime":1479.63,"endTime":1482.75,"body":"equivalent ratios and fractions"},{"speaker":"Pam","startTime":1479.63,"endTime":1482.75,"body":"and everything to do with"},{"speaker":"Pam","startTime":1482.75,"endTime":1486.71,"body":"proportional reasoning. And we"},{"speaker":"Pam","startTime":1482.75,"endTime":1486.71,"body":"can use it as a tool for"},{"speaker":"Pam","startTime":1486.71,"endTime":1489.86,"body":"reasoning in higher math like"},{"speaker":"Pam","startTime":1486.71,"endTime":1489.86,"body":"finding the equation of a line."},{"speaker":"Pam","startTime":1490.04,"endTime":1493.13,"body":"We can use it to reason about"},{"speaker":"Pam","startTime":1490.04,"endTime":1493.13,"body":"rational functions. There's so"},{"speaker":"Pam","startTime":1493.13,"endTime":1496.64,"body":"many different places that it's"},{"speaker":"Pam","startTime":1493.13,"endTime":1496.64,"body":"going to help us reason"},{"speaker":"Pam","startTime":1496.64,"endTime":1501.8,"body":"proportionally and beyond in"},{"speaker":"Pam","startTime":1496.64,"endTime":1501.8,"body":"mathematics. And remember, it is"},{"speaker":"Pam","startTime":1501.8,"endTime":1506.03,"body":"not a strategy. It's not... When"},{"speaker":"Pam","startTime":1501.8,"endTime":1506.03,"body":"someone says, \"How did you solve"},{"speaker":"Pam","startTime":1506.03,"endTime":1510.02,"body":"it?\" The answer isn't, \"I did a"},{"speaker":"Pam","startTime":1506.03,"endTime":1510.02,"body":"ratio table.\" The answer is,"},{"speaker":"Pam","startTime":1510.11,"endTime":1513.41,"body":"\"Oh, I use these relationships"},{"speaker":"Pam","startTime":1510.11,"endTime":1513.41,"body":"in a ratio table.\" Like, \"This"},{"speaker":"Pam","startTime":1513.41,"endTime":1518.63,"body":"is... I did... I was thinking"},{"speaker":"Pam","startTime":1513.41,"endTime":1518.63,"body":"about an Over strategy.\" Or, \"I"},{"speaker":"Pam","startTime":1518.63,"endTime":1521.66,"body":"was saying about Five is Half of"},{"speaker":"Pam","startTime":1518.63,"endTime":1521.66,"body":"Ten.\" Or, \"I was scaling up, and"},{"speaker":"Pam","startTime":1521.66,"endTime":1524.51,"body":"then.\" It's the idea of what"},{"speaker":"Pam","startTime":1521.66,"endTime":1524.51,"body":"were the relationships that"},{"speaker":"Pam","startTime":1524.51,"endTime":1528.74,"body":"we're using. Remember, the ratio"},{"speaker":"Pam","startTime":1524.51,"endTime":1528.74,"body":"table is a model. It's a super"},{"speaker":"Pam","startTime":1528.74,"endTime":1533.09,"body":"cool model because it can be"},{"speaker":"Pam","startTime":1528.74,"endTime":1533.09,"body":"used as a tool for reasoning."},{"speaker":"Kim","startTime":1533.27,"endTime":1538.01,"body":"Yeah, Ya'll. I'm super"},{"speaker":"Kim","startTime":1533.27,"endTime":1538.01,"body":"excited because we have a"},{"speaker":"Kim","startTime":1538.01,"endTime":1540.32,"body":"challenge coming up. We are"},{"speaker":"Kim","startTime":1538.01,"endTime":1540.32,"body":"super excited."},{"speaker":"Pam","startTime":1540.98,"endTime":1541.01,"body":"Woo-hoo!!"},{"speaker":"Kim","startTime":1541.22,"endTime":1545.6,"body":"Three times a year, we run"},{"speaker":"Kim","startTime":1541.22,"endTime":1545.6,"body":"a free challenge for math"},{"speaker":"Kim","startTime":1545.6,"endTime":1548.21,"body":"teachers everywhere. Everyone's"},{"speaker":"Kim","startTime":1545.6,"endTime":1548.21,"body":"invited. We'd love for you to"},{"speaker":"Kim","startTime":1548.21,"endTime":1552.47,"body":"join us. It's going to be"},{"speaker":"Kim","startTime":1548.21,"endTime":1552.47,"body":"January 18th through 20th."},{"speaker":"Pam","startTime":1552.95,"endTime":1555.95,"body":"And I may be even more"},{"speaker":"Pam","startTime":1552.95,"endTime":1555.95,"body":"excited than you are. It is"},{"speaker":"Pam","startTime":1555.95,"endTime":1560.81,"body":"super fun! It is like a free"},{"speaker":"Pam","startTime":1555.95,"endTime":1560.81,"body":"webinar on steroids because it's"},{"speaker":"Pam","startTime":1560.81,"endTime":1564.71,"body":"not just that one hour that we"},{"speaker":"Pam","startTime":1560.81,"endTime":1564.71,"body":"get together. But we get three"},{"speaker":"Pam","startTime":1564.74,"endTime":1570.74,"body":"hours! It's three days, one hour"},{"speaker":"Pam","startTime":1564.74,"endTime":1570.74,"body":"each where we dive in. It's"},{"speaker":"Pam","startTime":1570.74,"endTime":1574.58,"body":"totally free. I teach about"},{"speaker":"Pam","startTime":1570.74,"endTime":1574.58,"body":"something very timely, something"},{"speaker":"Pam","startTime":1574.58,"endTime":1577.4,"body":"that's on my mind that I'm"},{"speaker":"Pam","startTime":1574.58,"endTime":1577.4,"body":"seeing happening as I'm working"},{"speaker":"Pam","startTime":1577.4,"endTime":1579.38,"body":"with teachers all around the"},{"speaker":"Pam","startTime":1577.4,"endTime":1579.38,"body":"world that I'm like, \"Oh, we"},{"speaker":"Pam","startTime":1579.38,"endTime":1581.81,"body":"gotta dive into this thing. This"},{"speaker":"Pam","startTime":1579.38,"endTime":1581.81,"body":"is going to be super helpful for"},{"speaker":"Pam","startTime":1581.81,"endTime":1584.81,"body":"everybody.\" We give you short,"},{"speaker":"Pam","startTime":1581.81,"endTime":1584.81,"body":"actionable things, so that you"},{"speaker":"Pam","startTime":1584.81,"endTime":1588.83,"body":"can get quick wins and really"},{"speaker":"Pam","startTime":1584.81,"endTime":1588.83,"body":"learn a ton all at once. Things"},{"speaker":"Pam","startTime":1588.83,"endTime":1591.98,"body":"that you can try immediately"},{"speaker":"Pam","startTime":1588.83,"endTime":1591.98,"body":"with people right next to you"},{"speaker":"Pam","startTime":1591.98,"endTime":1594.89,"body":"and try immediately in your"},{"speaker":"Pam","startTime":1591.98,"endTime":1594.89,"body":"classrooms. And, ya'll, it's"},{"speaker":"Pam","startTime":1594.89,"endTime":1598.7,"body":"just a way for us to get"},{"speaker":"Pam","startTime":1594.89,"endTime":1598.7,"body":"together and do some exciting"},{"speaker":"Pam","startTime":1598.73,"endTime":1602.84,"body":"math for teaching. We always"},{"speaker":"Pam","startTime":1598.73,"endTime":1602.84,"body":"have a special guest, and we"},{"speaker":"Pam","startTime":1603.44,"endTime":1606.62,"body":"have a super fun Facebook group,"},{"speaker":"Pam","startTime":1603.44,"endTime":1606.62,"body":"and all the things. It's just a"},{"speaker":"Pam","startTime":1606.62,"endTime":1610.82,"body":"blast. Really, I want to invite"},{"speaker":"Pam","startTime":1606.62,"endTime":1610.82,"body":"you cordially to be part of our"},{"speaker":"Pam","startTime":1610.82,"endTime":1611.87,"body":"upcoming challenge."},{"speaker":"Kim","startTime":1611.99,"endTime":1614.69,"body":"And you will be able to"},{"speaker":"Kim","startTime":1611.99,"endTime":1614.69,"body":"register for this coming"},{"speaker":"Kim","startTime":1614.69,"endTime":1618.83,"body":"challenge next week, January"},{"speaker":"Kim","startTime":1614.69,"endTime":1618.83,"body":"9th, 2023."},{"speaker":"Pam","startTime":1619.4,"endTime":1621.26,"body":"And if you're happening to"},{"speaker":"Pam","startTime":1619.4,"endTime":1621.26,"body":"listen to this episode after"},{"speaker":"Pam","startTime":1621.26,"endTime":1625.64,"body":"that date, you can always check"},{"speaker":"Pam","startTime":1621.26,"endTime":1625.64,"body":"mathisfigureoutable.com/change"},{"speaker":"Pam","startTime":1625.67,"endTime":1629.45,"body":"to see when our next challenge"},{"speaker":"Pam","startTime":1625.67,"endTime":1629.45,"body":"will be coming up. We would love"},{"speaker":"Pam","startTime":1629.45,"endTime":1633.08,"body":"to have you join us. And thanks"},{"speaker":"Pam","startTime":1629.45,"endTime":1633.08,"body":"for joining us today on the"},{"speaker":"Pam","startTime":1633.53,"endTime":1637.07,"body":"podcast! Thanks for tuning in"},{"speaker":"Pam","startTime":1633.53,"endTime":1637.07,"body":"and teaching more and more Real"},{"speaker":"Pam","startTime":1637.07,"endTime":1639.59,"body":"Math! To find out more about the"},{"speaker":"Pam","startTime":1637.07,"endTime":1639.59,"body":"Math is Figure-Out-Able"},{"speaker":"Pam","startTime":1639.62,"endTime":1643.73,"body":"movement, visit"},{"speaker":"Pam","startTime":1639.62,"endTime":1643.73,"body":"mathisfigureoutable.com. Let's"},{"speaker":"Pam","startTime":1643.73,"endTime":1646.58,"body":"keep spreading the word that"},{"speaker":"Pam","startTime":1643.73,"endTime":1646.58,"body":"Math is Figure-Out-Able!"}]}