by Josh Rappaport
In my tutoring, I use a variety of ways to help students connect to math. One of the techniques I use involves an imaginative approach: together, the student and I create a fictional situation and imbue it with mathematical content. Then the student and I pretend the situation is happening as we speak. And as we talk about it, we explore the mathematical ideas.
So for this month’s article I’d like to show what this approach looks like, in hopes that it may spark some of you to try it. Of course, given that homeschoolers often have their children learn math through everyday life situations, this technique dovetails with that approach. So I would imagine that many — if not most of you — use a similar approach to math. This article just shows how you can do this in a semi-structured way as well as in an informal way.
What follows is an exchange that recently occurred while I was helping a student learn two early percent concepts. The concepts were: 1) that if a quantity gets split into two parts, the two parts add up to the whole quantity. And 2) that just as the two constituent parts sum to the whole, the two percents that correspond to those parts sum to 100%.
In the dialogue that follows, I, of course, am the TUTOR, and the CHILD is an 11-year-old girl whom I’ve been tutoring for about seven months.
TUTOR: So Toya, suppose that you own a little ranch where you raise and sell rabbits. (This child had previously told me that she loves bunnies.) Should we give the ranch a name?
CHILD: Could we call it Toya’s Rabbit Ranch?
TUTOR: Sure. That’s perfect. So let’s imagine that Toya’s Rabbit Ranch starts out with a bunch of rabbits. What do you think would be a good number to start out with?
CHILD: Can we start with … I don’t know … 40?
TUTOR: That sounds good. So let’s say that those 40 rabbits, since they’re what we start out with, make up the total. So that means that the 40 rabbits are 100% of the rabbits. Does that make sense?
CHILD: I guess so. No, yes, it does.
TUTOR: O.K., so now imagine that some of the rabbits are old enough to be sold. How many do you think might be ready to be sold after a month at the ranch?
CHILD: How about …. ten? Ten cute little rabbits.
TUTOR: O.K., ten. That works. Now if you sell ten, how many rabbits would the ranch have after the people leave with —
CHILD: Wait! Do we have to sell that many?
TUTOR: It’s all up to you. How many do you want to sell?
CHILD: (pause) Uh, is eight o.k.?
TUTOR: That’s fine. So, after you sell eight, how many would the ranch have left?
CHILD: So that would be forty minus eight, right?
TUTOR: Right.
CHILD: (scribbles 40 – 8 on paper, works the problem). Thirty-two?
TUTOR: Exactly, good work. Now one other thing. What do you think you’d get when you add those two numbers: the number of rabbits you sell, and the number of rabbits you’re left with?
CHILD: I’m not sure.
TUTOR: Well, can you just add the numbers?
CHILD: The number I sell, and … what’s the other one?
TUTOR: … and the number the ranch is left with.
CHILD: (writing as she speaks) So I sell 8, and I’m left with … how many?
TUTOR: It’s over here (pointing to her subtraction problem answer).
CHILD: Thirty two. O.K., so let’s see (adding 8 and 32 on paper). 40, we get 40.
TUTOR: And that’s the number you started with, right? 40 rabbits.
CHILD: Right.
TUTOR: So when you add the number of rabbits you sell to the number of rabbits you still have after the sale, you get the number you started with. Does that make sense?
CHILD: Sort of.
TUTOR: Well, think of it this way. If you start with a certain amount, and you split it into two groups, when you put the two groups back together, shouldn’t you get what you started with?
CHILD: As long as you don’t lose any.
TUTOR: Right. And at Toya’s Rabbit Ranch will we be careless and lose rabbits?
CHILD: No!!!
TUTOR: So will the amount sold and the amount you end up with add up to the 40 you started with?
CHILD: Yes!
TUTOR: O.K., then. So now, one other thing, and it’s an important thing. When you keep records for Toya’s Rabbit Ranch, do you keep careful records, or sloppy records?
CHILD: Good records.
TUTOR: Right. So you know, when you keep good records, it’s often really helpful to figure things out in percents as well as numbers. Using percents just helps you see patterns in how your ranch operates. So we want to figure things out in percents, too, o.k.?
CHILD: O.K. But I don’t know how.
TUTOR: Well, it works a lot like the numbers, except that instead of using the numbers, we turn the numbers into percents, and we use those percents. So to start out, the total of 40 rabbits, since it’s the total, we call that 100%. Remember?
CHILD: Uh …
TUTOR: O.K., do you remember how when we first started talking about percents, we said that whatever a whole amount is, we call that amount 100%? We were talking about a bucket of ice cream. And we said that when the bucket is full of ice cream, it holds 100% of what it can hold. Remember the raspberry swirl ice cream?
CHILD: Oh yeah. That’s so good!
TUTOR: True. So all I’m saying now is that since we start out with 40 rabbits, those 40 rabbits are all of the rabbits — like all of the ice cream — so the 40 rabbits make 100% of the rabbits.
CHILD: Oh o.k., I get it.
TUTOR: So what we want to do next is figure out, for the rabbits you sell, what percent of all of the rabbits are they?
CHILD: The eight I sell?
TUTOR: Right, the eight. But it’s not easy to figure that out as a percent immediately. But we can do it. Do you remember how we said you could find 10% of a number, when the number ends in zero?
CHILD: You said just take off the zero.
TUTOR: Exactly, so then can you figure out 10% of the 40 rabbits? 10% of 40?
CHILD: (writing down 40 and crossing out the zero). Four!
TUTOR: That’s right. So that means that 10% of the 40 rabbits you started out with would be 4 rabbits. But what we really want to figure out is what percent 8 rabbits would be.
CHILD: Would it be 80%?
TUTOR: Well, it doesn’t work like that. But you can figure it out. Try this. We know how much 10% of the rabbits is, right?
CHILD: Right, you said it’s 4.
TUTOR: And we want to know how much 20% is, right?
CHILD: Yes.
TUTOR: So here’s the question: if you have 10, and you need to multiply it by something to get 20, what do you multiply it by? In other words: 10 x WHAT= 20?
CHILD: 10 x 2 = 20, so times two.
TUTOR: Right. Times two. So if you know that 10% of the rabbits is 4 rabbits, how could we find 20% of the rabbits?
CHILD: Eight, it would be eight.
TUTOR: That’s great. But how’d you get it?
CHILD: I just ‘timesed’ it by two. I ‘timesed’ four by two.
TUTOR: Very good, but I think you mean you “multiplied” it by two? Don’t you?
CHILD: Oh right. Whatever …
TUTOR: Well anyhow, let’s write down what we’ve just figured out. (Writing as we talk …)
100% of the rabbits = 40 rabbits
10% of the rabbits = 4 rabbits
20% of the rabbits = 8 rabbits
CHILD: It keeps going up by 4.
TUTOR:. Very observant. Every time you add 10%, the number of rabbits goes up by 4. So let’s keep going with the table. (We continue till we get to: 80% = 32).
CHILD: Why are we stopping?
TUTOR: Remember just a little while ago we figured out how many rabbits we were left with at the ranch after you sold the 8 rabbits?
CHILD: It was … (searching her sheet) … 32!
TUTOR: So now look, we just found out that 32 rabbits is 80 percent of the rabbits.
CHILD: So … is that, like, really important?
TUTOR: It is, actually. Do you remember how the two groups: 8 and 32, added up to 40?
CHILD: Yes. That was easy.
TUTOR: So now we want to figure out what their percents add up to. Remember? The number of rabbits sold was 20%, and the number of rabbits we end up with is 80%. So now, can you figure out what 20% and 80% add up to?
CHILD: Do percents add just like numbers?
TUTOR: Just like numbers.
CHILD: (writing) 20 plus 80 is … 100.
TUTOR: Exactly. And that’s good.
CHILD: What’s good?
TUTOR: That we end up with 100%, because getting 100% means that we know what happened to all of the rabbits. 20% got sold, and 80% got kept by the ranch. And since those two percents adds up to 100%, that means all of the rabbits are safe and accounted for. See how percents help you figure that out?
CHILD: So 100% is like all of them?
TUTOR: Yes, exactly, when you get 100%, that means you have all of the rabbits.
CHILD: I don’t totally get it, but I sort of get it. Percents are still kind of weird.
TUTOR: Well, percents can be confusing, especially at first. But you are starting to understand. One of the main things to keep in mind is that 100% always means all of what you have, no matter how much you have, and no matter what kind of thing you’re talking about.
CHILD: O.K., but can we keep talking about the ranch?
There are several things I like about this educational experience. For one, it shows that mathematical understanding does not necessarily all come in a single burst. Rather, it takes time for a concept — such as the concept of percent — to first register, then sink in a bit, and then sink in still deeper. Notice how Toya goes in and out of grasping that 100% of something means all of that thing. This concept is just starting to gel for her.
This dialogue also shows, how talking through a situation — especially a situation that a child finds interesting — can help children explore mathematical ideas in a fluid sort of way. Because the approach is relaxed, the student feels free to ask questions, express what she does and does not grasp, and in so doing, gradually understand the ideas better.
In presenting a lesson on percent, I do not mean to suggest that this is the only concept that lends itself to this kind of conversational, situational approach. In the course of my tutoring I’ve used this same sort of approach with a variety of topics. And I’m sure that you can do the same.
Basically all you do is decide what concept you want to teach or explore. And then, with the child’s help, come up with a situation that will let you explore the concept. Then just start coming up with numbers as you talk about the situation. Then ask a lot of questions. Also encourage your child to ask questions by making the situation one that the child likes to think about. That is very important. Make the situation relevant to your child’s interests.
Anyhow, this can be a fun and interesting way to explore math. I encourage you to try it and see how it works for your and your children.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
In my tutoring, I use a variety of ways to help students connect to math. One of the techniques I use involves an imaginative approach: together, the student and I create a fictional situation and imbue it with mathematical content. Then the student and I pretend the situation is happening as we speak. And as we talk about it, we explore the mathematical ideas.
So for this month’s article I’d like to show what this approach looks like, in hopes that it may spark some of you to try it. Of course, given that homeschoolers often have their children learn math through everyday life situations, this technique dovetails with that approach. So I would imagine that many — if not most of you — use a similar approach to math. This article just shows how you can do this in a semi-structured way as well as in an informal way.
What follows is an exchange that recently occurred while I was helping a student learn two early percent concepts. The concepts were: 1) that if a quantity gets split into two parts, the two parts add up to the whole quantity. And 2) that just as the two constituent parts sum to the whole, the two percents that correspond to those parts sum to 100%.
In the dialogue that follows, I, of course, am the TUTOR, and the CHILD is an 11-year-old girl whom I’ve been tutoring for about seven months.
TUTOR: So Toya, suppose that you own a little ranch where you raise and sell rabbits. (This child had previously told me that she loves bunnies.) Should we give the ranch a name?
CHILD: Could we call it Toya’s Rabbit Ranch?
TUTOR: Sure. That’s perfect. So let’s imagine that Toya’s Rabbit Ranch starts out with a bunch of rabbits. What do you think would be a good number to start out with?
CHILD: Can we start with … I don’t know … 40?
TUTOR: That sounds good. So let’s say that those 40 rabbits, since they’re what we start out with, make up the total. So that means that the 40 rabbits are 100% of the rabbits. Does that make sense?
CHILD: I guess so. No, yes, it does.
TUTOR: O.K., so now imagine that some of the rabbits are old enough to be sold. How many do you think might be ready to be sold after a month at the ranch?
CHILD: How about …. ten? Ten cute little rabbits.
TUTOR: O.K., ten. That works. Now if you sell ten, how many rabbits would the ranch have after the people leave with —
CHILD: Wait! Do we have to sell that many?
TUTOR: It’s all up to you. How many do you want to sell?
CHILD: (pause) Uh, is eight o.k.?
TUTOR: That’s fine. So, after you sell eight, how many would the ranch have left?
CHILD: So that would be forty minus eight, right?
TUTOR: Right.
CHILD: (scribbles 40 – 8 on paper, works the problem). Thirty-two?
TUTOR: Exactly, good work. Now one other thing. What do you think you’d get when you add those two numbers: the number of rabbits you sell, and the number of rabbits you’re left with?
CHILD: I’m not sure.
TUTOR: Well, can you just add the numbers?
CHILD: The number I sell, and … what’s the other one?
TUTOR: … and the number the ranch is left with.
CHILD: (writing as she speaks) So I sell 8, and I’m left with … how many?
TUTOR: It’s over here (pointing to her subtraction problem answer).
CHILD: Thirty two. O.K., so let’s see (adding 8 and 32 on paper). 40, we get 40.
TUTOR: And that’s the number you started with, right? 40 rabbits.
CHILD: Right.
TUTOR: So when you add the number of rabbits you sell to the number of rabbits you still have after the sale, you get the number you started with. Does that make sense?
CHILD: Sort of.
TUTOR: Well, think of it this way. If you start with a certain amount, and you split it into two groups, when you put the two groups back together, shouldn’t you get what you started with?
CHILD: As long as you don’t lose any.
TUTOR: Right. And at Toya’s Rabbit Ranch will we be careless and lose rabbits?
CHILD: No!!!
TUTOR: So will the amount sold and the amount you end up with add up to the 40 you started with?
CHILD: Yes!
TUTOR: O.K., then. So now, one other thing, and it’s an important thing. When you keep records for Toya’s Rabbit Ranch, do you keep careful records, or sloppy records?
CHILD: Good records.
TUTOR: Right. So you know, when you keep good records, it’s often really helpful to figure things out in percents as well as numbers. Using percents just helps you see patterns in how your ranch operates. So we want to figure things out in percents, too, o.k.?
CHILD: O.K. But I don’t know how.
TUTOR: Well, it works a lot like the numbers, except that instead of using the numbers, we turn the numbers into percents, and we use those percents. So to start out, the total of 40 rabbits, since it’s the total, we call that 100%. Remember?
CHILD: Uh …
TUTOR: O.K., do you remember how when we first started talking about percents, we said that whatever a whole amount is, we call that amount 100%? We were talking about a bucket of ice cream. And we said that when the bucket is full of ice cream, it holds 100% of what it can hold. Remember the raspberry swirl ice cream?
CHILD: Oh yeah. That’s so good!
TUTOR: True. So all I’m saying now is that since we start out with 40 rabbits, those 40 rabbits are all of the rabbits — like all of the ice cream — so the 40 rabbits make 100% of the rabbits.
CHILD: Oh o.k., I get it.
TUTOR: So what we want to do next is figure out, for the rabbits you sell, what percent of all of the rabbits are they?
CHILD: The eight I sell?
TUTOR: Right, the eight. But it’s not easy to figure that out as a percent immediately. But we can do it. Do you remember how we said you could find 10% of a number, when the number ends in zero?
CHILD: You said just take off the zero.
TUTOR: Exactly, so then can you figure out 10% of the 40 rabbits? 10% of 40?
CHILD: (writing down 40 and crossing out the zero). Four!
TUTOR: That’s right. So that means that 10% of the 40 rabbits you started out with would be 4 rabbits. But what we really want to figure out is what percent 8 rabbits would be.
CHILD: Would it be 80%?
TUTOR: Well, it doesn’t work like that. But you can figure it out. Try this. We know how much 10% of the rabbits is, right?
CHILD: Right, you said it’s 4.
TUTOR: And we want to know how much 20% is, right?
CHILD: Yes.
TUTOR: So here’s the question: if you have 10, and you need to multiply it by something to get 20, what do you multiply it by? In other words: 10 x WHAT= 20?
CHILD: 10 x 2 = 20, so times two.
TUTOR: Right. Times two. So if you know that 10% of the rabbits is 4 rabbits, how could we find 20% of the rabbits?
CHILD: Eight, it would be eight.
TUTOR: That’s great. But how’d you get it?
CHILD: I just ‘timesed’ it by two. I ‘timesed’ four by two.
TUTOR: Very good, but I think you mean you “multiplied” it by two? Don’t you?
CHILD: Oh right. Whatever …
TUTOR: Well anyhow, let’s write down what we’ve just figured out. (Writing as we talk …)
100% of the rabbits = 40 rabbits
10% of the rabbits = 4 rabbits
20% of the rabbits = 8 rabbits
CHILD: It keeps going up by 4.
TUTOR:. Very observant. Every time you add 10%, the number of rabbits goes up by 4. So let’s keep going with the table. (We continue till we get to: 80% = 32).
CHILD: Why are we stopping?
TUTOR: Remember just a little while ago we figured out how many rabbits we were left with at the ranch after you sold the 8 rabbits?
CHILD: It was … (searching her sheet) … 32!
TUTOR: So now look, we just found out that 32 rabbits is 80 percent of the rabbits.
CHILD: So … is that, like, really important?
TUTOR: It is, actually. Do you remember how the two groups: 8 and 32, added up to 40?
CHILD: Yes. That was easy.
TUTOR: So now we want to figure out what their percents add up to. Remember? The number of rabbits sold was 20%, and the number of rabbits we end up with is 80%. So now, can you figure out what 20% and 80% add up to?
CHILD: Do percents add just like numbers?
TUTOR: Just like numbers.
CHILD: (writing) 20 plus 80 is … 100.
TUTOR: Exactly. And that’s good.
CHILD: What’s good?
TUTOR: That we end up with 100%, because getting 100% means that we know what happened to all of the rabbits. 20% got sold, and 80% got kept by the ranch. And since those two percents adds up to 100%, that means all of the rabbits are safe and accounted for. See how percents help you figure that out?
CHILD: So 100% is like all of them?
TUTOR: Yes, exactly, when you get 100%, that means you have all of the rabbits.
CHILD: I don’t totally get it, but I sort of get it. Percents are still kind of weird.
TUTOR: Well, percents can be confusing, especially at first. But you are starting to understand. One of the main things to keep in mind is that 100% always means all of what you have, no matter how much you have, and no matter what kind of thing you’re talking about.
CHILD: O.K., but can we keep talking about the ranch?
There are several things I like about this educational experience. For one, it shows that mathematical understanding does not necessarily all come in a single burst. Rather, it takes time for a concept — such as the concept of percent — to first register, then sink in a bit, and then sink in still deeper. Notice how Toya goes in and out of grasping that 100% of something means all of that thing. This concept is just starting to gel for her.
This dialogue also shows, how talking through a situation — especially a situation that a child finds interesting — can help children explore mathematical ideas in a fluid sort of way. Because the approach is relaxed, the student feels free to ask questions, express what she does and does not grasp, and in so doing, gradually understand the ideas better.
In presenting a lesson on percent, I do not mean to suggest that this is the only concept that lends itself to this kind of conversational, situational approach. In the course of my tutoring I’ve used this same sort of approach with a variety of topics. And I’m sure that you can do the same.
Basically all you do is decide what concept you want to teach or explore. And then, with the child’s help, come up with a situation that will let you explore the concept. Then just start coming up with numbers as you talk about the situation. Then ask a lot of questions. Also encourage your child to ask questions by making the situation one that the child likes to think about. That is very important. Make the situation relevant to your child’s interests.
Anyhow, this can be a fun and interesting way to explore math. I encourage you to try it and see how it works for your and your children.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
