“Ah, I see it now!” We hear that a lot during our model drawing trainings as teachers start to understand how to actually do model drawings. However, when they leave the comfort and structure of the training and start to apply the strategies they’ve learned to the problems in their math book, they often struggle.
There are many different types of problems that students face in elementary school, and during our trainings, I work to help the teachers develop an understanding of them so they can identify the problems as they encounter them and help students learn to understand what the problem is actually asking. We also assign a code to each problem type.
Why Code Story Problems?
Simply put, coding story problems allows teachers to feel successful when they go back to their classroom and try to apply a model drawing strategy to a story problem from their district-purchased text. The books don’t often come with coded problems, and as I’ve gone through several math series in our project schools, I start to question the thought process behind the writing of these texts. Is anyone taking into consideration the types of story problems the child has experienced in the previous year? Are the problems in the books sorted to help children be successful? In my experience, the answer is no!
For example, in second grade, one particular math program starts with an addition story problem, then goes to a subtraction problem, a multi-step problem and then goes back to an additive comparison problem and then back to a part-whole addition problem.
When your book skips around that dramatically to so many different types of problems and model drawings, it is very difficult to teach strategies. In a perfect world, we could have kids learn each of the different types of story problems, and then give them a mixed review of the problems and they would be able to identify the different bar models. However, in reality, it takes about 40 days of modeling to get kids to understand the step-by-step model drawing process to do it to the point of independence. There just isn’t time!
The Model Drawing Protocol Booklet we’ve developed helps remind us – students AND teachers – that model drawings are not so much about what the words say (something was taken away or added on to) as much as they are about the checking and chunking of the problem. Kids read the problem, chunk it, and then add that part on to the model drawing. If approached analytically, like coding the problems helps us do and we will explore in the rest of this post, students CAN be successful in model drawing and story problems!
For only $2, this digital download includes a list of the different types of model drawings that kids would experience by grade, a page with all of the codes and what they mean along with an sample problem for each, the step-by-step model drawing checklist that we use, and a fill-in journal has a place for the problem and the code it is assigned. The Model Drawing Protocol Booklet is formatted so that teachers can create a half-page flipbook as a tool to help them in the classroom. So many of our teachers have used this tool to help them be successful with model drawings in their classrooms! Get yours now!
We want teachers to understand that these part-whole problems have specific characteristics, or clues, in them to help us know that the bar model/unit bar/tap diagram model drawing will look a certain way. Part-whole problems are typically given starting in first grade and are usually some variation of parts given in the problem and asking for the whole. There are actually quite a few different types of problems within that part-whole family.
Part-Whole Addition (PWA) – Laura collected some marbles. On Tuesday, she found 36. On Friday she found 14. How many did she find all together?
Obviously part-whole addition is really easy to see when you’re given a part and a part and asked to find the whole. This is probably one of the easier types of problems to be given. Of course, the numbers could be higher, and the unit bar could look more complicated, or it could have three addends, or three parts, to add.
Part-Whole Subtraction (PWS) – Emma had $215 to spend at the mall. She spent $109 on clothes. How much money did she have left?
This type of problem is very distinctive of something being taken away, eaten, spent, dropped, etc. The model drawing is also very distinctive where it gives you the total, but it has a diagonal slash because something is being taken away. So, in this problem, you would use subtraction to solve it, but that doesn’t mean a child couldn’t use Springling (a character coming soon!) to count up and solve it as well. PWS is not always indicative of which operation you would use in the computation, but rather the idea of taking something away, represented by a diagonal slash in the model drawing.
Part-Whole Missing Addends (PWMA) – There are 87 chickens on the farm. 52 of them are in the barn, how many of them are in the field?
It is easy to get PWS and PWMA problems confused when you’re looking at subtraction. However, in the PWMA problems, there isn’t anything actually being taken away, so there won’t be a diagonal slash in the model drawing. The problem isn’t saying the chickens disappeared, just that they are in a separate location. There is a distinct total of 87 chickens and the problem gives you information about one part that makes up that total and the other part up to you to figure out.
Part-Whole Multi-Step (PWMS) – Connor has $523 dollars in his savings account. He was going on a ski weekend and so he went and bought skis for $225 and bought new snow gear for $102 dollars. How much money does he have left?
This one is probably the hardest for our teachers. The part-whole multistep could first have you add and then divide, or you could add first, then subtract…basically, you are doing multiple operations within one problem. With this example, it’s on one bar, but it’s giving you more that one part. Here, you have to remember that the actual computation is up to the student, but we know that it’s going to involve more than just one step. Step A might be that the student adds $225 and $102. Step B might be that they subtract that total from the $523 to get the difference. Or, a student might choose to do Step A as $523 minus $225. For Step B, they might take that difference and then take off $102. It helps to label the problem with A and B to show the parts of the computation they’re doing.
Part-Whole Multiplication (PWM) – I saw five spiders. Each spider has eight legs. How many legs do they have all together?
Pretty straightforward multiplication – five groups of eight. PWM (different than multiplicative comparison) is introduced in 3rd grade where kids are doing multiplication, and it usually involves taking and multiplying something in a group or groups.
The model drawing for a PWM problem could get more complicated if we did a different kind of problem. Say someone wants to line up rows of chairs. They want 25 rows with 8 chairs in each row. In this model drawing, you’re not going to make 25 different bars, but you could have one bar, labeled Row 1, then have an ellipses […] and then a bar labeled Row 25 to represent that there were 25 rows. Then you can break those bars up into eight to show the groups of eight chairs, and then proceed to solve.
Part-Whole Division (PWD) – If Scott had $15 and he wanted to go to Starbucks and buy his colleagues a cup of coffee, and each cup of coffee costs $2.25, how many cups of coffee could he buy?
This type of problem is a little more challenging. You’ll have one bar, with a total at the end, and then we chunk the bar, write a [….] and then $2.25 to show that we want to know how many equal groups of $2.25 are in $15. Again, we have to remind kids this is reading comprehension, so it’s supposed to be a picture of what the story is asking.
Sometimes, they might give you the number of groups in the problem. For example, Scott has $15 and he wants to divide it up between four friends. How much will each friend get? In this case, you can make the bar have four parts to it, and then put the question mark under it. The question mark would represent how much money is each box worth?
Elapsed Time (ET) – The party starts at 10:15 AM and goes until 2:31 PM. How long did it last?
This type of problem can fall under the part-whole family. We usually do elapsed time on the open number line, beginning with the start time and then going to the ending time, trying to end on a friendly time. We try to never make the minutes go over 60 because then we have to do more converting. The idea would we that we’d do large mountains to show the hours and a little hill for the minutes.
Once kids have a good grasp on part-whole problems, additive comparison problems make up the next largest problem type, and they are probably some of the most complex students will encounter.
Additive Comparison (AC) – Shannon and Kathleen have some cards. Shannon has 15 cards. Kathleen has 8 more than Shannon.
Additive comparison problems ask you to compare two people’s amounts. These problems always start by giving an equal bar to each person in the problem. Then you can add on to the bar or take away depending on the story. In this example, we start by showing that Shannon and Kathleen have the same amount, and then we can add the additional amount that Kathleen has (eight) to her bar.
It also works the other way. Each person might have a bar, but we might say that Shannon has eight LESS than Kathleen. So in this case, Kathleen’s bar will still be longer because you’ll still add more to Kathleen’s bar to show that Shannon has less.
Additive comparisons have a direct link to algebra. Take this example: Shannon and Kathleen have 11 pieces of candy. Shannon has three more pieces than Kathleen. How many does Kathleen have. This problem is really representative of 2X + 3 = 11. Even though we’re doing these problems as early as first grade, it’s really important for us to look at additive comparison problems and try to figure out how we can label these problems and really teach them so that students will have the necessary foundation of skills to build upon later.
Once students get to 3rd grade, they’ll experience multiplicative comparison problems. These are even more complex than additive comparisons and they also tend to confuse students. You can usually tell when a problem will be multiplicative comparison because of the language used – it will often say that somebody “has five times as many” or “two times as many.”
Multiplicative Comparison (MC) – Sean and Molly have 135 books. Sean has four times as many as Molly.
In the drawing, Molly is given one unit bar that we don’t know the value of and Sean’s bar is divided into four units because he has four times as many as Molly. (You could also represent that by giving Molly two bars, and then Sean would have eight bars.) Then, you can see that those five units equal 135, so you can figure out how much each unit is worth.
Occasionally, though not much before fourth or fifth grade, students will encounter a combination of additive and multiplicative comparison, which might be labelled AC/MC. This helps us know that we will need to do multiple operations within that problem.
Fraction (F) or Fraction Comparison (FC) – Professor Barble went to buy ice cream cups for his class. He bought 48 ice cream cups. One third of them were chocolate, ¾ of the remainder were strawberry and the rest were vanilla.
This type of problem uses the bar to make the fractional parts that the child is looking for.
Ratios and proportions will make an appearance in the middle school years, but the ability to solve problems involving those concepts stems from a solid understanding of solving these particular problem types from elementary school.
Your Step-by-Step Guide!
Crystal clear, right?? Don’t worry! We have a great booklet that is a step-by-step guide to model drawings that can be your go-to resource in the classroom. Get yours now!