This percent Problem String is a model building string. It's purpose is to build the percent bar model to help students reason using the model: whatever you do to the percent, you do to the whole. If you cut the percent in half, from 100% to 50%, you cut the whole in half, from 40 to 20. This multiplicative scaling up and down is fundamental to understanding the relationships. The percent bar model has the potential to help students begin to realize the simultaneous relationships.

Students can also begin to build the strategy of using chunks of percents to find other percents. In the Problem String, students can add up 10% + 5% + 1% + 0.5% to find 16.5%.

The structure of the string is a building structure, where the problems build off of one another. Students can use multiple relationships to find some of the answers. Sharing and making visible these multiple strategies helps to build the percent bar model and the big idea of percentage. 

Use the context of downloading music or a file in gigabytes (or similar size that makes sense).

Draw a very long percent bar. Put the percentages on either the top or bottom and the unit, 40 gigabytes, on the other side. Since each question supplies the percent and is asking for the part of 40 gigabytes, write the given percentages first. Then, based on how the students scale from one percent to the other percent, do the same from the 40 gigabytes to the missing part. For example, to find 25% of 40 gigabytes, first draw out the relationship of 25% to 100%. You can do this by asking students to help you approximate where to draw the line that represents 25%. If they divide the 50% in half, represent that cutting the 50% portion in half. If they divide the 100% by 4, represent cutting the whole into 4 pieces. Do this on the percent bar and then nudge students to use that action to find the same relationship with the 40 gigabytes. Also, represent the scaling using scaling arrows on the equations. 

As you prepare to lead your class through this Problem String, consider the following sample language, strategies to pull out from students, and ideas to highlight:

Have you ever downloaded something? Can you picture the download bar? It's super frustrating when it stalls and you realize you only have part of the file, right?

Put the percentage on the top or bottom. Label the gigabytes on the other side.

Use the relationships between the given percents to reason about finding the missing number of gigabytes.

Represent student's strategies on the percent bar by using your hands and actions to show size to help them realize that what they do to the percents, they also do to the gigabytes.

Represent students' strategies with scaling arrows with the equations so that students can connect their thinking on the percent bar with the equations.

Consider what you could ask at key times in the Problem String to help students build models, strategies, and big ideas:

Where should I draw the 50% How do you know? So then if 50% is half of 100%, what is half of 40? 

Are there other relationships you can see? Multiplication? Division?

Are there chunks you already know that would be helpful? How? 

Problem Strings are facilitated by the teacher by asking each problem, one at a time. Students work to solve the problem, the teacher calls on individuals to answer, and the teacher models (represents) student thinking on the display.

In this Problem String, strategies are represented on a percent bar.

Strategies are also represented using scaling arrows.

If the question demands some think or work time, the teacher circulates to see what students are doing or to ask students what they are thinking. If the teacher cannot see or hear the desired strategy(ies) quickly enough, the teacher can ask a question that draws out that strategy. Questions like:

Are there any patterns you are seeing? Is there anything that's kind of like blinking at you a little bit? Like if I say 50 and 5?

The teacher models and requests a private response signal. Students using a private response signal allow their peers enough work time, are not distracting other students, and give the teacher valuable information about who is done and who needs more time. In the video, Pam verbally requested a private signal, "Give me a thumbs up ..." and she physically demonstrated the signal by holding her thumb up as she asked questions.

Because this is a model building string, where the purpose is to build the percent bar model, Pam asks for multiple strategies for many of the problem.

In this picture, Pam models finding the 25% by dividing the 50% by 2.

In this picture, Pam models finding the 25% by dividing the 100% by 4.

In this picture, Pam models the strategy of cutting the 10% in half to find the 5%.

Here Pam models the strategy of finding the missing factor by multiplying the 5% by 10 to get the 50%.

Here Pam models the reciprocal division relationship of dividing by 10.

As the string progresses, Pam records students' strategies and strategically erases as they go.

In this picture, you can see two strategies for fiding 25% of 40. They are still on the board when Pam asks the next question, 10% of 40.

But before Pam begins to represent strategies for finding 10% of 40 as seen below, notice that she has erased the 2 strategies for the previous problem.

Similarly, the strategy of dividing by 10 to find the 10% is erased in the following picture where you see the dividing by 2 strategy to find the 5%.

And that strategy is erased before representing the times 10, divided by 10 relationship between 50% and 5% as shown below.

And it continues, with just the current relationships being shown.

This strategic erasing helps students focus on the current problem and the possible relationships to the previous problems.

The very first problem, "What is 100% of 40?" sets the stage for the rest of the string. Notice that Pam does not belabor this first problem. It's almost too easy and if the teacher puts too much emphasis on "How do you think about this?" students might be confused. The problem is almost too easy to emphasize too much. 

Pam simply asks it, clarifies what it means with gigabytes, elicits whole class responses, labels the percent bar and moves on.

Pam: Okay, so if that is, if that is the download, where would the 40 be? Like, we download the whole thing, right Kason? So we're going to put 40 over here. What percent is that? Like, we download it all.

Class: One hundred percent. 

Pam: It's a hundred percent. And then what do you think would go here?

Class: Zero.

Pam: Zero. And then how many, 0%? Yeah. How much percent would it be? Would it be 0%? Does that kind of make sense? So a 100% of 40 is 40. Cool.

As Samuel suggests that 10 is 25% of 40, Pam asks, "You think that might be 10. How are you coming up with 10? Can somebody pop in for like, how's Samuel thinking about 10? Anybody talk about like, how are you thinking about 10? Carlos, how were you thinking about 10? How'd you find that 10 there?"

This suggests that it's about thinking and supporting your thinking. 

Other questions that can begin to build a culture of thinking and reasoning include:

What do you think Anthony?

Samuel, What are you thinking about?

How are you coming up with 10? Can somebody pop in for like, how's Samuel thinking about 10? Anybody talk about like, how are you thinking about 10? Carlos, how were you thinking about 10? How'd you find that 10 there?

I'm a little curious.

Daisy you're thinking about 100. Tell us about that.

What do you think?

Go ahead and think about that. Everybody thinking, everybody thinking

What do you guys think of that?

What do you think?

You think it might be 5? You think it might be 10?

Braxton, what are you thinking about?

Bryce, you want to add onto that?

What are you guys thinking about?

What do you think about that? Giovani?

I wonder if that might be helpful? Kind of like your guys' reasoning.

Does that make sense, that that would be 5%. How does that feel?

Are there any patterns you guys seeing? Kason, are you seeing a pattern?

What do you see?

Any ideas where he, where Kason's thinking about? Why this 0.4? Kason, do you think you could give us a hint?

What are these extra zeros? Like? How could we represent that? What's yeah. Daisy. What's going on?

So lots of relationships you could use. Do you got another one?

What are you guys talking about? What do you think?

And so then we talked about like, what is, if one, do you remember Anthony? Where's, how does one relate here?

Can you enlighten the rest of us? Convince us that you guys are correct that it's 0.4.

I'm going to come around and see what you're thinking about. What are you thinking about Braxton?

What are you guys thinking about?

This is the transcript of the video, with bold comments of Pam reflecting on her teacher moves during the Problem String, inserted to highlight purposeful teacher moves. 

- [Pam] We're going to do a quick Problem String today. I want to talk to you guys about downloading. Context! It's so important. And I use it well . . . for much of the string but there are times I will point out that I could have done a better job. Have you guys ever downloaded something and you see it, like if you're a song, maybe you're installing an application or something and all of a sudden it goes, mmmmm. Lots of hand motions here. And then it like stops. And you're like, no, like, mm. Like you want to like, mmmmm. Right? If I told you that you're downloading something, that is, mm, it's like 40. It's 40 whatever, 40 kilobytes, megabyte, whatever it is. It's maybe 40 songs. I don't know, 40. And I asked you, what is 100% of 40? What would that look like? Like if this was, I'm going to do the download ready? Like if it was like, mm, here it is. Mmmmmm. Like all there, like the whole pretend thats straight. Can you pretend thats straight? That's like, see how straight that is. That's really straight, kind of? Ha, drawing a very long straight line is not my best talent. It's ok. It just needs to be good enough. Okay, so if that is, if that is the download, where would the 40 be? Like, we download the whole thing, right Kason? So we're going to put 40 over here. What percent is that? Like, we download it all.

- [Class] One hundred percent. 

- [Pam] It's a hundred percent. And then what do you think would go here?

- [Class] Zero.

- [Pam] Zero. And then how many, 0%? Yeah. How much percent would it be? Would it be 0%? Does that kind of make sense? So a 100% of 40 is 40. Cool. Notice that I did not belabor the point on this first problem. Choral responses from the whole class and just get it up. What is, what is 50% of 40? So gimme a thumbs up, Now, we'll go for thumbs, a private response signal if you can tell me what 50% of 40 is and where 50% of 40 would be. What's 50% of 40? I'm looking, I'm looking. Anthony, what's 50% of 40. From the little time I've spent with Anthony, I am betting that if I can win him over, get him thinking, the rest of the class will also. 

- [Anthony] Um, 10%. I read this as, "I don't understand percents so I'll guess and you'll leave me alone." Nah, you can do it! Let me help you understand the problem.

- [Pam] 50% of 40. Where would that go? If this is, if this is 100%, where do you think 50% would be? Start with the percent first.

- [Anthony] In the middle?

- [Pam] It'd be totally in the middle. Here we go. Ready? Alright. Oh, did I do it? Is that close enough? Record the position of 50% so we can refer to it.

- [Class] Yeah.

- [Pam] Okay. So that's 50%. Hey, if 50% is smack dab in the middle, nicely done, what is 50% of 40? I really wish I would have said 40 gigabytes. I think context would have helped. What would be smack dab in the middle of 40 gigabytes? Use what you just did, splitting the 100% in half to reason about splitting the 40 in half. What do you think Anthony? What's smack dab in the middle of 40 gigabytes.

- [Anthony] Smack dab in the middle of 40?

- [Pam] Zero to 40 gigabytes. Come on Pam! What is right in the middle?

- [Anthony] Fifty.

- [Pam] 50% is what's the corresponding, how many you're trying to download.

- [Anthony] 20.

- [Pam] You're trying to download all 40 gigabytes, but you've only got half of it. You only got 20 gigabytes. Does that make sense to everybody? Cool. So 50% of 40 is 20 nicely done. Way to go Anthony! Take note everyone - thinking is what's celebrated here. Let me see if I can, slightly different marker here. The next question I have for you is what is 25% of 40. I wish I would have kept the context a bit more alive, like "So, what if you were downloading and it got stuck at 25% How many gigabytes have downloaded so far? So I wonder where would you put 25% of 40 up there? And what, what is 25%? So thumbs up 25% of 40? Samuel, what are you thinking about? What's 25% of 40?

- [Samuel] Between zero and 50%.

- [Pam] Okay. Like, like anywhere in between?

- [Samuel] No, in the middle.

- [Pam] Like right the middle? Like how about, is that close?

- [Samuel] Yes.

- [Pam] Yeah. Okay. So there's 25%. And so what is 25% of 40?

- [Samuel] 10.

- [Pam] You think that might be 10. How are you coming up with 10? Can somebody pop in for like, how's Samuel thinking about 10? Anybody talk about like, how are you thinking about 10? Carlos, how were you thinking about 10? How'd you find that 10 there?

- [Carlos] I divided by 2 from 50.

- [Pam] I mean, we had sort of this 50%, we knew 25% was divided by 2. So you could divide 20 by 2? Making a big deal about the fact that you can use what you know about the relationships of the percents to then reason about the corresponding numbers, the gigabytes and doing all of this on the percent bar. Oh, can I, can I record that over here? Now I record on the equations, using scaling arrows, connecting what's happening on the percent bar with the equations. Let's see if this is half, 25% is half. Is 10 also half of 20? Ah, that kind of makes sense. Kind of cool. I'm a little curious. Somebody in last period said something about going from the hundred, like what Carlos said. Sorry. Ha, I'm sort of choking on my throat lozenge. What Carlos said totally makes sense. Daisy you're thinking about 100. Tell us about that.

- [Daisy] Well, you like break it all down. You're breaking it down by like parts. If you used quarters, for example, 25, 50, 75 and 100.

- [Pam] There's like four of these in there. Yeah. So I heard you say quarters. I take her mentioning quarters and model-represent it on the board: when your brain thinks about quarters, it could look like this. So if you divide 100 by 4, yeah. What's 40 divided by 4? Is that 10? So kind of another relationship that we could use. Nicely done. Next problem. Let's see. What did we do? We did 100, we did 50, we did 25. How about 10%? What's 10% of 40? Hmm, go ahead and think about that. Everybody thinking, everybody thinking. Thanks. Good job. Good job. Let me erase a little bit here. 10% of 40. This one's not near as trivial I don't think. Ya'll, when you're ready, turn to your group, have a conversation with your group. What are you thinking about? How are you thinking about 10% of 40? Go. Ten percent is much harder than the previous problems. Let's give everyone a chance to share together to get ideas sparking. I'll circulate and see what kind of thinking is happening.

- [Samuel] When you make the 20 you need half of 10. So then half of 10 would be 5.

- [Pam] What do you think?

- [Rocha] That's what you think?

- [Pam] What do you think?

- [Rocha] I think that like 10% of 40 would be like, I think five, two, maybe a little bit.

- [Pam] You think it might be 5? You think it might be 10?

- [Samuel] No, I said 5. I meant since it was half of, half of 20 would be 10. So half of 10 would be 5.

- [Pam] Half of 20 is 10. Is half of 25 10?

- [Samuel] No. Half of 10 is 5. So I went to 20, since half of that is 10. So then I went to 10. And what's half of that is 5.

- [Pam] Oh, I see what you're saying. You're going 40, 20, 10, 5. Okay. I'm a little curious. 100, 50 that's half. 50, 25 is half, is half of 25, 10?

- [Rocha] Half of 25? No.

- [Samuel] It's 15.

- [Pam] If I were to put 10% up there, look at the percents. I've got 0, 25, 50, 100. Where would 10 go? I finally started talking about percents. I wish I would have emphasized percents and gigabytes more.

- [Pam] Hang on a second. Yeah.

- [Dataveon] A little bit before the 25.

- [Pam] Before the 25. Yeah. How many, 10 percents are there in 100?

- [Rocha] 10.

- [Dataveon] 10.

- [Pam] Are there 10 of them in there?

- [Dataveon] Yeah.

- [Pam] Yeah. Let's start there. Okay. Let's start there. Enough time had passed and I was concerned that sticking with that group longer would be too long for other groups. We had a good start going, we can work with that. Alright. Let's chat a little bit. Just having a conversation with this table back here. Let's position these guys as sense makers, belonging in this reasoning place. You guys were saying that, that there was something about like, if we were to write the 10% up here, if I were to put it up here and oh man, remind me how to say your name. As terrible as it is that I couldn't remember how to pronounce his hame, I'd rather admit and then pronounce it correctly. Names matter.

- [Dataveon] Dataveon.

- [Pam] Dataveon. Sorry about that. Dataveon. Dataveon was the other student I knew would be helpful to get thinking. If he jumped in thinking, others would follow. If I was to put the 10% up here, you said, it would be kind of close to the zero, right? Does everybody agree with that? And then, and then you said there, would be a bunch of, like, we could think about a lot of 10 percents in there and how many, 10 percents are in 100? Ten of them. That's interesting you all! So if I were to break this up into like 10, I don't know, would help me. Like would, would like, like there, is that, does that feel like 10%? Like, there'd be one of them two of them, three of them, four of them, five of them. Oh, 50%. That's five of them, right? And then I'd have the other five? Hey, that'd be 10. I'm doing all of this hand waving and motioning to help students visualize the percent bar and the relationships there. Okay, so that'd be kind of a way, like, ooh, there, that could be 10%. If, you kind of said I'm going to cut that 100%. What did Dataveon just say? I'm going to cut that 100% by 10. I'm going to cut it into 10 chunks. Now I'm modeling the division by 10 with the equations. This is to help students notice that when they do a dividing action on the percent bar, they can do that same dividing with the equations. 

- [Student] Oh yeah.

- [Pam] There's are 10%. Hey, what would it be like to cut that into 10? Did did I just, did I just hear you Jordan, was that you? No. No. Somebody, was that you Bryce? I wasn't sure it was somewhere in this area. Yeah. Bryce, what'd you say.

- [Bryce] It would be 4.

- [Pam] You think 40 divided by 10 is 4? Yeah. What do you guys think of that? Yeah. Samuel's nodding yes. Does 4 feel right here? Like if I've got 0, 10, 25, 0, 4, 10, like does 4 feel like it would fit kind of in there, a little bit? Let's make sure that your intuition is matching. Does anybody have any reasons why you like, I mean, it kind of goes in order, but is there some, yeah. Braxton, what are you thinking about?

- [Braxton] Um, my math right here, like a lot of kind messed up, but I did 10 over 25 divided by 5. That gave me 2 and 5 and I times it by 2 and it gave me 4 over 10. In that moment, I was not sure what Braxton was doing but I had a sense that it was rule, step by step based. I wanted to stay in sense making land, so I kind of acknowledged it and kept asking about sense.

- [Pam] Lots of stuff. I'm not sure I followed all that. I have a question. With all of that does it make sense that this 4 right here, does it fit, that it would kind of like, like when you look up here, do you go, whoa, that can't go there or do you go, yeah? Like that 4, that feels right. What feels right about it? For example, Rocha, when we were talking back there, you guys said something about you wished you said 40, 20, 10, you wished this was five. Right? Cause you're like, hey, we're following that pattern. But then Rocha, you said, but half of 25 is not 10. Is this exactly, did I put that half in here? I tried not to. Because it's not half. Right? So I tried to put it a little bit this way. If it was half, would that where's, somebody pick it? What do you think about Kason?

- [Kason] A little bit to the right.

- [Pam] Would be what? That'd be the, and that'd be the 5. Yeah. So, oh, it feels like that 4 kind of belongs there. Bryce, you want to add onto that?

- [Bryce] Well, 10 times 10 is 100 and 4 times 10 is 40.

- [Pam] So you're kind of going this way. Yeah. 10 times 10 is 100, 4 times 10 is 40. Huh? Nice. So we kind of have this multiply pattern, my hands go up from 10 to 100% to show the multiply pattern, this divide pattern my hands go down to show the 100 to 10% pattern. Super cool. Next question. How about if I asked you for 5% of 40 gigabytes, come on Pam, stay in context!? 5% of 40, go ahead and work on that ya'll. Find 5%. Where would it go? Over there? What is it here? Feel free to talk to the people next to you. 5% of 40 gigabytes. 5% of 40. Let him think, let him think. What are you guys thinking about? I'll get you in a second Bryce. What are you guys thinking about? Adam?

- [Jordan] Five is half of 10. And think about what's half of 4 and it's 2.

- [Pam] So that's what Adam said. That's what you said. It was two. What do you think about that? Giovani?

- [Pam] You think it might be? Can you find 5% up there? Adam? Can you, if I asked you, if I give you the marker and I was like, dude, walk up there and put the 5%, where would you put it?

- [Pam] Hang on a second. Go ahead and

- [Giovanni] Next to the 4?

- [Pam] Next to that 4? How come?

- [Giovanni] Because 5 comes after 4. Ah, I'm clear now that they aren't sure if we're talking about the percents or the gigabytes. Sure wish I would have stayed in context. Ok, I'll go back there now...

- [Pam] Oh, nice. So check it out, the numbers on the top are not the percents. That's like the, the kilobytes that you're downloading. Numbers on the, yeah, five is a percent so it's got to go on the bottom. 

- [Pam] Yeah, how come? How does it, how like, like five, you got 0, 10, 25. Where would that 5 fit?

- [Giovanni] Right in the middle there.

- [Pam] Right in the middle? Yeah. Yeah. Can I put it up there? I wonder if that might be helpful? Kind of like your guys' reasoning, You guys are stopping and talking way too fast. Alright. Let's go. So let's see, we're having a conversation and Giovanni, you were telling me where I could sort of put 5% over here. Right? You're like, hey, we're we're talking about 5%. All of that is 40 let's sort of, and I think you said 5% goes, where'd you tell me to put it? It's like right in the middle. Yeah. You said right in the middle. So if I put is, is if that's zero and that's 10%, does that make sense, that that would be 5%. How does that feel? 5%? Yeah. Cool, and then how did that help you Jordan or Adam? Adam, go ahead. How did that help you know what this is?

- [Adam] 'cause half of 10 is 5 and half of 4 is 2.

- [Pam] Half of 4 is 2. Sure. Can I do that over here? Now that we've made sense on the percent bar, let's represent that with equations. Can we go like, hey, that 5, half of that 10 is 5, half of that 4, what is half of 4, everybody? Ah, sure enough, there's the 2. Does that make sense? Hey, is there any patterns that you see? I'm just kind of curious. If I were to erase this divided by 2 here and you kind of look up here. Are there any patterns that you see? Thanks Samuel. Are there any patterns you guys seeing? Kason, are you seeing a pattern?

- [Kason] I see it's multiples of 2.

- [Pam] Tell me where.

- [Kason] So the answer, the bottom one's 2, the next one's 4, the next one's 10, 20 and 40.

- [Pam] Those are all even numbers. Those are all multiples of 2. Nice. Anybody see any other patterns? Nice pattern. Samuel, you first. I'll get you Bryce. What do you think Samuel?

- [Samuel] So I did, so 100% divided by 2 for each one.

- [Pam] Oh.

- [Samuel] That 100 divide by 2 is 50 and 50 divide by 2 is 25, and 25 divide hour. And it goes 15.

- [Pam] Or half of 25. What is half of 25?

- [Immanuel] 12.5

- [Pam] Immanuel saying it might be 12.5. Does that sound right? 12.5. Yeah. Yeah. And then you could go half again here. Half here. Does anybody know, oh, Bryce, I almost forgot your pattern, sorry. What, what do you see a pattern up here?

- [Bryce] Yeah. It's going by 5s, that percent.

- [Pam] Oh, lots of 5s. So just like we saw lots of 2s over here. You're seeing lots of 5s over here. Nice. I see lots of 5s. Hey, if I were to just say, look at maybe this one and that one. Is there anything that's kind of like like blinking at you a little bit? Like if I say 50 and 5? Daisy, you got this look on your face. 50 and 5, 20 and 2, are those related? I mean, you guys were totally just telling me how half a 10 got you to 5, half here. Kason you're excited. What do you see?

- [Kason] They are the same equation, it just has the zero.

- [Pam] Yeah. What are these extra zeros? Like? How could we represent that? What's yeah. Daisy. What's going on?

- [Daisy] I was going to say that all you had to do was add 10 and you get, wait, you add, multiply by 10 and you get.

- [Pam] Oh. So if you multiply by 10 here, 5 times 10, is that right? And what's 2 times 10. Huh? Look at that. Interesting. Hey, I wonder Bryce, you had your thumb up, were you thinking times 10? I'm curious if anybody was thinking sort of the reverse of divided by 10.

- [Bryce] Divided by 10 being times too.

- [Pam] Like there's both this kind of, guys, that's interesting. Do you think when you divide by 10 it will always sort of shift like that? Like sort of that zero kind of goes away and that, that like that, huh? Like this divided by 10 over here. I don't know. Is that a pattern that might be helpful? A lot of you guys to find that 5% you're like, whoa, how am I? Ooh, I could do it from here. But if you already had the 50, could you have found the 5 just by dividing by 10? Daisy, what do you want to, go ahead?

- [Daisy] I was just going to say that you could also do a reverse where you can do 50 divided by, 5 divided by 50. You could get to 10. That's how you know that.

- [Pam] Oh, nice. So like you, like from the 50, you could divide by 5 to get to the 10? That would be kind of cool. Is that true you guys? Is 50 divided by 5, 10? What's 20 divided by 5? Four. Bam. Look at that. Nice. So lots of relationships you could use. Do you got another one?

- [Bryce] Yeah.

- [Pam] No way.

- [Bryce] 4 times 10 is 40 and 10 times 10 is 100. Like doing it that way.

- [Pam] I'm not sure where you are help me.

- [Bryce] So where the 4 is? You multiply it by 10. It would get 40.

- [Pam] Like, like from here to here?

- [Bryce] Yeah.

- [Pam] Okay.

- [Bryce] 10 Times 10 would be 100.

- [Pam] Oh, nice. So you make me want to erase these really fast. You're saying that we can sort of go from that 10% up to the 100. Is that kind of what you were saying?

- [Bryce] Yeah.

- [Pam] Oh, I can do it. Like that's times 10. And so is that. Whoa, all of these relationships kind of popping out. Cool. And you get to choose whichever one you want to use. Yes! It's all about relationships! Next problem. Oh, forget I did that. Here we go. Ready? What is 1% of 40? Whoa. 1%. Where's 1% of 40? Now if you're like, I don't even know where to start, see if you could find 1%. Where would you put 1% on that percent bar? Use the bar to find the percent first, then reason about the gigabytes. Maybe think about that just a little bit. Let's see, I don't know if I've been over here for a minute. What you guys thinking about? You are fabulous today. Thank you. You think you think 1% and you think it might be 1. What do you guys think? You think 1% is 1? You're not sure. You don't think so. He's not sure.

- [Kason] Cause I think of it as one thirds or something like that.

- [Pam] I ask you a question. Oh, thanks. Up there you got 2, right? And 2, can you see how 2 is like next to 5%? So if you cut that 2 in half to get 1, what's 5 in half?

- [Carlos] 2.5.

- [Pam] Yeah. Is it 2.5? So 2.5% is 1, but we only want 1%. Here is another place I should have stayed in context. So 2.5% is 1 gigabyte, but we only want 1%. It's got to be smaller than 1? What? What kind of number is smaller than one?

- [Anthony] Zero point five.

- [Pam] Oh, so some sort of decimal. Nice intuition! Let me honor that. You think it might be 0.4? How'd you get 0.4?

- [Kason] Because it's 10 divided by 1 or it is something related to that, just minus the zero.

- [Pam] Ooh, you're using the pattern. He's using the pattern. Can you show him which pattern you're using? Let's see what you guys are doing. What are you guys talking about? What do you think?

- [Daisy] We were saying that if you put 1%, like before the 2 in here equals still 1, because like...

- [Nariyan] And you can't go farther

- [Daisy] And you can't go into the negatives.

- [Pam] Hey, Hey Anthony. They're they're doing exactly what you were doing. Do you think you could help them understand where you went wrong? And it's my fault, at least somewhat, because I wasn't using context as well as I could have. Anthony and I just had this great conversation. Let's see if we can do it together. Well, let's do it together. Alright ya'll, if you don't mind. Hey Anthony and I were having this great conversation over there where he said, I think it might be, let me choose a color here, I think you said, I think it might be 1, right? And you guys just said, I think it might be 1 gigabyte?. And so then we talked about like, what is, if one, do you remember Anthony? Where's, how does one relate here?

- [Anthony] Two?

- [Pam] It it's half of 2, right? It's right in there in half of 2. What was half? What's halfway in between that zero and that 5 percent? Kason, you were,

- [Anthony] 2.5.

- [Pam] 2.5. So that's two, I can do it, that's 2.5%. So we could add to our list here. You guys are like, hey, we know another one. We know one, you didn't even ask for. You guys are telling me that 2.5% of 40, you're telling me right now that that is 1 gigabyte. Well, we don't want 2.5%. We only want 1%. It's got to be, it's got to be what? Bigger or smaller?

- [Anthony] Smaller than 0, which is zero point five.

- [Pam] Which is crazy. And you're like, hey, 0.5 is smaller than 1. And then Kason was like, I think it might be, Kason, what did you think? It might be.

- [Kason] 0.4.

- [Pam] Kason thinks it might be 0.4. Is that also less than one Anthony? Totally less than one. What do you guys think? Point four? Where'd you get that? That seems really random. Any ideas where he, where Kason's thinking about? Why this 0.4? Kason, do you think you could give us a hint?

- [Kason] I though of one third for the, so we figured out that it was 1.

- [Pam] We figured out that, what was 1?

- [Kason] When I think you said it was 2.5 for 1, I thought of it as one third. So it had to be smaller than one.

- [Pam] Okay.

- [Kason] Had to go to the decimals.

- [Pam] Oh, not, and that's what you said. It's got to be point something, right? It's got to be point something. It's got to go in the decimals. Okay.

- [Kason] And then basically it it's like 10 of 40 is 4. And so I just removed the zero and added the decimal to the 4, in front of it.

- [Pam] Whoa. Who followed that? Did you follow that Jordan? I followed it, but let's bring in some other voices.

- [Jordan] A little bit.

- [Pam] A little bit.

- [Jordan] That's what I got at least.

- [Pam] You got 0.4 as well.

- [Jordan] Yeah.

- [Pam] And did you follow what Kason was saying?

- [Jordan] Pretty much, yeah.

- [Pam] Okay. Can you enlighten the rest of us? Convince us that you guys are correct that it's 0.4.

- [Jordan] I think 0.4 times something is 40, but I don't know what to multiply by.

- [Pam] Like you're looking up. Are you looking at the 100, to see, oh I wonder if it's related to the 1? Kason I don't think you were looking at the 100. Were you looking at the 10?

- [Kason] Mm-hmm. Kason just said something about the 10. You all are these related at all? How do you get from 10 to 1? So a few times we're going to have to go this way, right? 1 times 10, is 10 this way. 1 times 10, not that arrow, 1 times 10 is 10. What do you have to do over here you guys?

- [Class] Multiply by 10, What is 0.4 times 10?

- [Student] Four.

- [Pam] Is that four? Hey, there's that zero thing happening again. But Jordan, we didn't or Kason, we didn't know this one. Right? So if we don't know it, we can't just say times 10 of something we don't know. We didn't know this one. What is that relationship, ya'll? If this is times 10 Kason is saying this is, divided by 10? Does that work? Oh, so then the question becomes, what is 4 divided by 10? What is 4 divided by 10?

- [Class] 0.4.

- [Pam] It's 0.4? Huh? What do you want to add on?

- [Bryce] 0.4 times 100 would be 40. And 1 times 100 would be 100.

- [Pam] You're saying that we can do something here.

- [Bryce] Yeah.

- [Pam] What do you guys think? You got 100, 40, 40, 10, 40, 4, 1, 40, 0.4.

- [Bryce] You can multiply them by 100.

- [Pam] Is that true? Huh? So a way we could think about it is if we know 100%, could you divide by 100 to get 1? Divide 40 by 100 to get 0.4? Interesting. Huh? Nice relationships. I'm going to go ahead and get rid of this extra one that we had just for a second. I think it might make the next thing we're doing a little bit. Purposeful erasing. Let's keep the board showcasing the relationships. Alright. So let's see, we've got 100%. We've got 50, 25, 10, 5, 1%. The next question I have for you is, we are so close to almost being done here you guys. Ready? I think, I think you can handle this one. This might be the hardest one all day. I think you can do it. You guys are up to it. This was my attempt to focus their attention. We've been going for a while. Ready? What is one half percent of 40? Uh oh. Oh, I don't know. Yeah. Get some thinking going on. Talk to people next to you. What are you thinking about? How do you pronounce your name?

- [Nariyah] Nariyah.

- [Pam] Nariyah, good night. What is wrong with me? Okay. I'm going to call on you.

- [Nariyah] Wait what?

- [Pam] What are you thinking about? Tell me, tell me what you're thinking about.

- [Nariyah] When you said one half and half of 40 is 20.

- [Pam] Say that again, sorry.

- [Nariyah] Half is 20.

- Half of 40 is 20. Another moment when I wish I had been in context. We're looking for a half a percent of 40 gigabytes. That's a little bit different. If we know that the whole thing. So if I'm downloading stuff mmmm, and I download that's a 100% is 40. But if I, if I only downloaded a half a percent, can you picture on that bar? Where would a half a percent be? Would it be like way over there? You said 20, but, but that'd be 50%. We only want half of a percent. Even tinier than a percent. Tell me what you're doing.

- [Nariyah] I don't know . Well, it's interesting. I'm watching your lips move, but I'm not sure what they mean. You got me really curious.

- [Nariyah] This one.

- [Pam] So maybe, maybe, can we go back to downloading stuff? If I'm downloading something and 100% of it is all over here, that's 40. But if you've only downloaded a half a percent, where is the 1%? If I've downloaded 1%, where was that up there? Can you find that? It's not very clear. Is that the pink one? I think that's actually two and a half. Here's when I realized that I had not drawn in the 1% on the percent bar. Drat! Oh well, let's do it now. You know what ya'll, I did something. I messed up just a little bit. And when we mess up, we admit it, fix it, and move on. It's all good. Oh, can you hang on just a second, Anthony, but we don't have the 1% up here. I'm feeling bad about that. Like this, this 1, that's not the percent right? That one. That's two and a half percent. Where's and this over here, that's the 0% right here. Where's the 1%? Like, if, if this is, oh, sorry, wait, if this is the 1% you guys told me that 1% was like 40 cents. I don't even have, like, I got like, there's the zero. This right here, that's the 40% Ooh, I misspoke here, I meant 0.4. So if 1%, I didn't even make this big enough. We can't even see the half percent. If 1% is that little tiny slice, a tiny one. Where's the half percent? Where is it Anthony?

- [Anthony] Are we on question 0.5?

- [Pam] Yeah, like I'm trying to find like, where is the half percent, the 0.5. Is it way out? Is it out here? Is it? [Anthony] Can I tell you the answer? You want to tell me this answer? Sure. Go for it.

- [Anthony] 0.2. Way to go Anthony!

- [Pam] You think it might be 0.2? What? Where did he come up with that?

- [Anthony] It's 0.4, but you can divide by 2.

- [Pam] Nariyah, why would you divide that by 2?

- [Nariyah] Because 4 divided by 2 is 2.

- [Pam] Sure enough. What's what's how are these two related? How is a half related to one? Is that just, like if I had one and I divide it by two, do you have a half? So you take one divided by two. Nicely done Anthony. Cool. So if I was to like, stick this here somewhere, golly, I don't even think I can do it. You guys can. I I'm not even sure I could write that half in there.

- [Class] Erase that a little bit and then make it bigger.

- [Pam] Like make it a little bit bigger. That might like change the whole, like maybe we should blow it up. Like if we blew it up, like, if we like, hey, let's take this little area right here. Does that look like a, that's trying to be that thing. you know, like if like we'll blow that up. If this is the 0% and this is the 1%, what was 0%? What was 1%. Oh, but it's got to be right in the middle. Bam. Right in the middle. So that's the 0.5%. And what is in the middle of zero and 0.4? 0.2. Notice the order here. Put in the percent, then the gigabytes. We know the percent, use it to figure the gigabytes. Nice. Yeah. Super cool. Well done. Okay. This is the easy one. Ready? Last problem you got. You guys just did the hard one. Whew. Here's the easy one. What is 16.5% of 40. You got this. You totally got this. Hey, all you got to do. All you got to do is find some chunks to get 16.5. Do, do you know some pieces that you could put together? I bet you do. I bet you do. I'm going to come around and see what you're thinking about. What are you thinking about Braxton? How can you get to 16?

- [Braxton] It's 20% and 5%.

- [Pam] You got a great start. I'm walking away. Yeah. A good start. I'm walking away. I don't need to stay to hear him out. He's on his way to a fine strategy. In fact, let me see if I can nudge him toward using a different set of relationships. Hey, can you go back from the 25?  What are you guys thinking about? You think what? Why?

- [Jordan] Cause add the 5 to the 10 and then the 1.

- [Pam] To that. So maybe so the 10 and the 5 and the 1. That sounds like 16, but you need 16 and half. So you're close. I'm going to let you guys add those up. Did you follow that?

- [Giovanni] Ah.

- [Pam] That was, that was kind of fast, huh?

- [Giovanni] Yeah.

- [Pam] Hey Jordan, can you write down the pieces that you guys are talking about?

- [Jordan] Yeah.

- [Pam] And, and, and, and Giovanni you check out those pieces he's talking about, write it down, write down. Many students can follow if they can see the relationships instead of just hearing about them. What are you guys thinking?

- [Rocha] Well, I was thinking, it says 16.5%.

- [Pam] Yeah.

- [Rocha] I'm thinking it would be close. It would be in between the 10% and 25%.  Nice relationship!

- [Pam] Oh, okay. So if you started with 10, you could start with 10 and then tack some more on.

- [Pam] Because, and it says 16 it's 16 and 0.5. It's not going to be exactly middle.

- [Pam] It's not 12 and a half. That'd be right in the middle. Yeah. Hey Datavian if we were starting with 10, we know what 10% is, right? 10%. Can you find 10%? There's kind of a lot going on out there. What's 10? It's 10%, 4. 10% of 40 is 4. Can you find that one, 10%? It's kind of hard to find.

- [Dataveon] Oh, yeah.

- [Pam] You found it. So we know 10%, but we got to get to 16. What could you add to 10% to get you closer to 16? I wish I would have said 16 percent. Stay in context Pam!

- [Dataveon] You could add, wait 4 to 12.

- [Pam] Maybe. Maybe let's see. We've got, we've got 10%, but we need to get to 16.

- [Dataveon] We got to find 6.

- [Pam] We got to find 6. Percent! 

- [Pam] Do we have a 6 up there?

- [Dataveon] No.

- [Rocha] No, but we got a 5.

- [Pam] No, we got a 5.

- [Rocha] And we also got a 1.

- [Pam] So can we add 10 plus? Oh golly. Can you draw a 5? That's a terrible 5. You're a lefty, nice. Being left handed is just something I can notice to say I see you. So 10 and 5, that gets us to 15, but we need another 1 to get to your 16. So can you put a 1 there? These are all percents, right? So put a percent after that percent. Now we got a 1%. Okay. What are we at? If we got 10, 5, 1, what are we at to so far?

- [Dataveon] 16.

- [Pam] That's 16. Now we need 16 and a half.

- [Rocha] There's also a 0.5.

- [Pam] We've got the half up there. Okay. So can you put the corresponding things? What is 10% of 40?

- [Dataveon] 10% Of 40.

- [Pam] Is four. So put that 4 right here and then find the 5% find the . .  Here's the trail off move. Say a little and then trail off and let them take it, you guys, you're grooving. You're grooving, go, go, go, go. Alright. You guys got something? Yeah. Carlos, you hard, that you're like, yes, no? Maybe I should come over and chat. Do you guys, do you have some way to get to 16? Oh, not yet. You did something, are you talking to them?

- [Kason] Yeah.

- [Pam] Was he talking to you?

- [Carlos] Mm-hmm.

- [Pam] Oh, okay. Alright. Well, what was he saying?

- [Carlos] That is would be 16 or 16 point. Yeah, we need to get 16. Don't we? Okay. How can we get 16 guys? 16. We need to get to 16. Well, actually we need to get to 16.5.

- [Anthony] How do you write backwards?

- [Pam] Yeah, it's hard to do. I'm not doing it very well today. Is that a 5?

- [Carlos] Yeah.

- [Pam] Yeah. Let's try. Let's hope so. So what do we have up there that's close to 16?

- [Kason] 10, 5, 1.

- [Pam] We got a 10. Can you see 10% up there? What do you still need if we got 10% to get up to 16?

- [Anthony] 6.5.

- [Pam] We need, you need, we need 6.5. We don't have 6.5. Do you have some pieces to get to 6.5?

- [Carlos] I have two.

- [Pam] Ooh, you want to go 2? And he wants to go, do we have 2? We have 2 gigabytes, but we don't have 2%. We're trying to get all the percent to get to a percent. Hey guys, one of the things that's going on, let's focus for just a second. One of the things that I'm hearing from almost everybody in the room is sometimes you guys are looking over here and you're choosing these numbers. And sometimes you're looking over here and you're choosing these numbers. How do you decide when you're choosing these numbers and when you're choosing those numbers? Cause I think this is kind of important. Well, if Pam would have stayed in context, that might have been helpful.....How do you know? Like Immanuel, you just said this, look on your face, Like, what are you thinking about? How do you know? Should I ask you more specific question? When, if I, if we're trying to find 16 and half percent, are you looking at trying to get pieces to add the 16 and a half in these percent numbers? Or are you looking over here in these gigabyte numbers?

- [Immanuel] The percent?

- [Pam] Why is Immanuel looking in the percent numbers? Go ahead, go ahead, Anthony.

- [Anthony] Cause those are by itself and don't show percent. Those do.

- [Pam] We're looking for percent. Right? So we got to add up all these percents. Alright. Rocha, what, what were the, what was the, the what'd you guys decide to add together? Go.

- [Rocha] Well, we like.

- [Pam] Go, well, tell me what you got. What'd you get?

- [Rocha] Oh, 16.5 of 40?

- [Pam] Yeah.

- [Rocha] But we didn't finish it.

- [Pam] Oh, you didn't finish it. Let let's let's do it together. What were the pieces that you were messing with to get to 16 and half? Do you remember? I think, I think, I can say your name. Datavian, I think Datavian started with 10. I think you're said you were like, let's start with 10%. You guys is 10% a decent start for 16 and a half? That's a pretty good start. All right. Keep going. Datavian what, what did we add to that? Do you remember?

- [Dataveon] We added five.

- [Pam] Bam. Because we have 5%. Check it out. We got that. Nice. And then what did we add?

- [Dataveon] And then we added the one.

- [Pam] Nice. And?

- [Dataveon] It was that to 16.

- [Pam] Is it, ya'll.

- [Dataveon] 0.5.

- [Pam] Is that 16? Anthony? Anthony is this 16%. 10, 5. It is. And so then like Datavian just said, we've got to add that extra half a percent. Alright. What is 10% of 40, guys?

- [Students] Four.

- [Pam] Is that four? What is 5% of 40? Two? What is 1% of 40?

- [Class] 0.4.

- [Pam] 0.4. And what is help me? What's last.

- [Class] 0.2. What is that half percent of 40?

- [Class] 0.2.

- [Pam] Is that 0.2? Okay. Can we add $4, $2, 40 cents and 20 cents?

- [Student] $6.60.

- [Pam] You think it might be $6 and 60 cents? What you guys think? Hey, check it out. Way to go powerful solvers. You guys just found 16 and a half percent of 40, thinking and reasoning and using what you know. Nice. Cool. Oh yeah! Take that in! Hey Bryce, did I, I kind of poked you a little bit? I was like, hey, can you go from the 25? I'm curious. Did you?

- [Bryce] What I did was I did 10 and then took the 25, but 10 times 2 is 20.

- [Pam] And so you like found 20?

- [Pam] No. So I had 20% and 4 times 2 is 8. So I was like in between.

- [Pam] I'm going to write that down. I'm going to write that down. You found 20% of 40 and you said that's 8. Did everybody follow that?

- [Bryce] So I looked in between 4 and 8 and 6 and, and I did all that other stuff.

- [Pam] Nice. What is in between 10 and 20% guys? What's right in between that? Is that 15%. So I'm thinking that Bryce was like, I can find 15% because it's going to be right in between 10 and 20 and that was 6. And so then he said, oh, I can use that 15%. And then you could add the one and a half percent. Nice. That would totally work. Good work guys. Whew.

Reflections:  I was super pleased to get this last period of the day class that was all boys except 2 girls to think and reason! They were diving in and making sense. They were noticing patterns and using the patterns to find other relationships. If you could have seen this group the day before when I met them, you would be extra proud of them. So many of them were so sure that they could not be successful. Because of that, I was not really sure how well filming was going to go. I had wondered if it would take a bit more time to help them realize that they just had to think. But they did!