Your students are all part-whole problem pros?? Must be time to enter the world of comparison problems using visual models!
Some teachers are confused by additive and multiplicative comparison problems, thinking of them as just part-whole multi-step problems. The comparison bars you use in the visual models for this type or problem are sometimes called additive comparison bars, but that doesn’t mean you might not add or subtract in the problem. We also have multiplicative comparison problems, which look just as intimidating as they sound, but which also might feature multiplication, division, or even addition and subtraction! Makes sense, right?
The comparison genre of problem solving is not just a part-whole multi-step problem. While they might feature multiple steps, the mark of a comparison problem is that you’re going to be comparing two or more entities.
Additive comparison problems begin as early as 1st grade! Students start off by looking at an amount compared to someone else who has more or less than them, and then try to figure out how many more or less you have than they do.
Language of Additive Comparison Word Problems
The language of an additive comparison problem is very confusing to young learner. When you ask How many more does Jane have than Bill? They look at you like “what??” Am I supposed to add? Or subtract??
No! This is the signal that we’re supposed to compare! And, if we understand visual models, we can help the language of additive comparison problems become much more clear. We start with proportional bars, line up the blocks, and actually count to see how many more or less there are.
Take this sample problem:
Jane had five apples. Bill has two apples. How many more does Jane have than Bill?
There would be five squares to represent Jane’s apples. Bill’s apples go in a second bar underneath Jane’s, not on the same bar. If they were on the same bar, there’s no way we could compare the two numbers. Then, we put a question mark in the space below the three extra squares or apples to indicate that we need to find out what that amount is.
You could also ask that question in a different way.
Jane still has five apples. Bill has two apples. How many less apples does Bill have?
This could be really confusing as students are listening to that language, but drawing the bars can help clarify. Remember, it’s about matching up the two amounts belonging to each character to see where they have the same amount, and counting how many less Bill has. The question mark really goes in the same place, but what it represents – what it means – is different if the question is different.
Traditionally, at the basic level of problem solving, we might have made a t-chart of vocabulary that might go with addition problems – more, some, all together – or subtraction problems – difference, how many more. That kind of cheat sheet just isn’t useful when we are solving comparison problems because the language is much more complex.
As comparison problems progress, the language starts to change from just comparing one entity to another and looking for more or less, and we start to look at solving for different parts, given one of the parts or the total. For example: Jane and Bill have seven apples in all. If Jane has five, how many does Bill have? This gives the total and a part and students have to solve this differently. With this progression in language, we also want kids to progress to using non-proportional representation as well.
The Additive Comparison Bundle (M³ Members – Click Here!) that we have as a download for you to use in the classroom follows this same kind of progression – we start with the idea of more/less and then we move students a bit further.
Connecting Additive Comparison and Algebra
Once we’re beyond just comparing more or less, I think the best way to help kids make the transition to non-proportional representation is by getting them used to drawing a bar of the same length for each character in the story problem, then reading the problem to work their way through it. Once you have those bars of equal size, you can manipulate them to set up the problem.
Let’s say John and Connor had 11 golf balls. John had 3 more golf balls than Connor. How many golf balls did Connor have?
The two bars belonging to John and Connor are “x” (thinking in algebra terms) – we don’t know how many they had. We know that John had three more than Connor, so we’re going to add a non-proportional unit that will make John’s bar grow by three. We have a bracket that shows the total is 11, and then we will put a question mark in Connor’s box.
Once this is set up, I can look at the information I actually have. I know the total is 11. John has 3 more, so I can subtract those 3 from the total, which leaves me with 8. I know I have two units that equal 8, so, I can determine that each of them is worth 4 and Connor has 4 golf balls. If you think about algebra, this problem really looks more like 2x + 3 = 11.
This is one of the biggest epiphanies that I’ve ever had while doing additive comparison problems – the connection to algebra. In middle school, I memorized procedures for concepts I certainly didn’t understand, which is partly why I do what I do today, and I remember writing down formulas, storing them in my graphing calculator, because I thought if I remember that formula, I’d be able to get the answers correct.
Nobody ever drew pictures, helped me use algebra tiles, or made any connection that what I was doing with my second grade problems was actually algebra!! For a problem like 2x + 3=11, she would say to “do the opposite” and subtract the three. Or I “needed to get the X alone” But why? I wondered! I never understood it, but I just did the math as I was told and got the answer.
It was not until I understood additive comparison models that I realized that we were solving for that was the X! It all made sense – why do I need to subtract the three? Why do I need to get X alone? So I can figure out how many golf balls Connor had!
Of course, that’s what my teacher was saying, but visually, I had zero understanding of what she was talking about. I’m sure I would have understood algebra a whole lot better if I had a picture in my head, which is certainly why algebra tiles and things like visual models are really helpful for kids to understand this type of problem.
The whole premise for additive comparison problems is to lay the foundation for early algebraic understanding. Students can actually write algebraic equations, or, as I’ve done with my audiences, they can create a model drawing based on the context of a story, or they could solve the problem based on seeing the visual model.
Enter: Professor Barble.
He was designed to help your students understand the different types of problems and make sense of them. Having a Professor Barble Additive Comparison Poster hanging in your classroom would be a great reference for students to use, especially since additive comparison problems are probably the number one word problem genre that stumps kids because they don’t understand what is being asked.
Multiplicative Comparison – the Next Step
If you thought the language for additive comparison problems was tricky, just the name of this next type of problems is enough to make you scratch your head. Most people, especially students, have no idea what “multiplicative comparison” means.
However, if kids have a foundation of understanding additive comparison problems, they know more than they think!
Multiplicative comparison bars actually start as early as third grade, and a problem will read something like this: If A is 2, and B is 2 times as big as A, how big is B? Kids hear that kind of language and think what??
One of the best tools that you can use to help students understand the language in multiplicative comparison is to use one inch square tiles, which you’ll see in the tutorial video of our Multiplicative Comparison Bundle (M³ Members – Click Here!).
The most effective visual I’ve used with 3rd, 4th or 5th graders to help them really get this concept of multiplication is a copy machine. We take a class field trip to the copy machine (which they think is pretty cool!), and we talk about how I have a unit that has 2 and I want to see two times as many. I put two one-inch square tiles on the glass, close the lid, and type in 2 copies, which I explain means 2 times. I ask them what they think will come out – some said 2, some just didn’t understand. But when I pushed the button, and they saw the copies come out, it was like a light bulb went off!
We weren’t talking about two more. We were talking about two TIMES as many, or two copies as many.
We did another example with 1 one-inch square tile on the glass. Let’s say A is 1, and B is 3 times as big as A. How many copies do I want? 3! So I hit the button three times, and the kids were able to predict right away that three units would be created.
Making sure your students understand the language of the problems before we start to enter into scenarios is super important. As students start to get this, they’ll start to understand it with the language and, of course, they can always use the one-inch square tiles if they need to.
Let’s look at another problem: Molly has three times as many books as Julie. All together, they have 20 books. How many does Molly have?
Kids have to be able to think about what that problem looks like – how can we make Molly have three times as many? Well, I have to be able to put in one unit for Julie, but then replicate that three times from Molly. So Molly has 3 units and Jane has 1. That represents easily what the words are saying.
Now, I know that the total is 20. I know that all 4 of those units equal 20. Therefore, how much does one unit equal? One unit must equal five. So, pretty easily, we can see that Molly has three units, each is worth 5, and so she has 15 books.
“Milking the Problem“
I always love to milk the problem for all it’s worth.
My friend, Char Forsten, author of Step by Step Model Drawing, always says “milk the problem for all it’s worth.” I could extend that question about Molly and her friend and ask how many more books does Molly have than her friend? That’s putting the question mark in a different spot before you put in the question mark at the end of Molly.
Now we’re really comparing Molly and her friend to see how many more she has. In this case we, know the answer is 10.
Using the Word Problem Bundles
Show parts of the video tutorial included in the word problem bundles to your class, then have them use the one-inch square tiles and the exercises that are all laid out for you. Make the word problems come to life in your classroom! You can download the PowerPoint presentation and even edit it to put in different problems or problems that meet the needs of your particular students. If you’re part of the M3 Membership, you’ve got it made – just download the bundles from the Membership resource library.
Teaching virtually? One-inch square tiles are super easy to create if your students don’t have them at home. Or, have them use virtual manipulatives, like the square tiles from Didax. Or, have them use cereal, crackers or even post-it notes – anything so that they can replicate the understanding of multiplicative comparison.
The last two genres of word problems, when we combine additive and multiplicative comparisons, and then fractions, will be featured later in the school year due to their complexity. For now, download the Additive Comparison and Multiplicative Comparison Bundles – and go build your students’ foundation of problem solving!