Bar Modelling: The Basics Explained

Hello, my name is Wendy Leal and I'm here today to talk to you about bar modelling. A little bit about my background. First, I am a teacher and have been teaching for 17 years. Various roles include me a five year six teacher. I've also been a PPA cover teacher which has given me vast experience across nursery to year six. I've also been an assistant head teacher teacher leading in maths and assessment and currently now work as an independent maths consultant and I also work in a school as a maths specialist teacher. So here we are talking about bar modelling today. Very quickly I'm going to give you the content of our webinar. So first of all, what is bar modelling? We're going to talk a little bit more about the different types of models, part whole models, comparison models, the before and after event models and a little bit more about the more advanced bar models a bit later on. So first of all, what is bar modelling? Well, bar modelling was developed in Singapore as part of the CPA approach. The concrete, pictorial and abstract approach was really taken from Brunner's idea of the theories of learning from inactive iconic to the symbolic. It's been adopted in many countries and it involves the use of bars or rectangles to model problems to solve. Within that the different representations allow us to satisfy Zoltan Steen's theory of learning through lots of different representations, not just in a symbolic way. It also involves children learning about the relative proportion of size. One of the best side effects I say that bar modelling gives children is it really develops their spatial awareness along with number sense and their reasoning skills. And we'll talk a little bit more about that a bit later on through one of the questions. I often think of bar modelling as a tool to really eke out the maths in any problem. Very often children are confused by the language, it's too long winded. And barre models actually help children to eke out the step by step operations that a problem may need in order for it to be solved. And finally, I also believe that bar modelling is brilliant for your EAL and SEN learner because it's so visual and children really understand and get to grips with what they're doing through the bars. Okay, so let's move on to part whole models. So what is a part whole model? Here you can see a bar model split into parts and the whole bar make the whole, the whole bar makes a hole. So the partholes are illustrated in the models. Often you can solve your four operations with your PAR models as well. As multi step problems. And so here is an example of a really simple part whole addition problem. So Paul has five stickers, Jesse has eight stickers. And you might think, well, this is really, really simple. Why would I need to use a bar model here? Well, if we introduce the, the concept of bar modelling at a really early stage, it becomes really natural for children to take on board. So when they eventually come to work out harder problems, it becomes second nature to them that they can use bar modelling to help them solve that problem. So in really simplistic form, you've got Paul has five stickers, Jesse has eight stickers. How many stickers do they have all together? So here you can see Paul with five stickers, the bar representing five, Jesse with eight stickers. And then the bracket here shows us how much or what the question is. So how many stickers do they have all together? The question is here, the question mark is here. That's what they're trying to find out. So label up your bars as. So with the part whole model, the you could have two separate bars like this. So this is one version and I'll show you the second version in a moment. So how many stickers do they have? So the question mark is how many stickers they have all together? Five. Paul has five stickers, Jesse has eight stickers. So altogether they have 13 stickers. Now, one of the most important elements of bar modelling is that not only do you label up your bars, but at the end, the result, the answer is always given as a full answer. Okay, another version over here. So in the same way, Paul has five stickers, Jesse has eight stickers altogether, how many stickers do they have? So again, it's the same equation. Five plus eight is 13. So therefore Paul and Jesse have 13 stickers altogether. Okay, so let's move on to a part whole model with subtraction. Okay, so the question is Mr. Pride bakes or baked 28 loaves of bread. Seventeen loaves were white, the rest were brown bread. So that's the information that you're given. How many loaves of brown bread did Mr. Bride bake? Is the question. So again, we model 17 loaves of white bread, we don't know how many loaves of brown bread, but what we do know is that there are 28 loaves of bread altogether. So in this one, the question mark, how many loaves of brown bread did Mr. Pride Break is within that bar there. So in this case we've got 17 plus something is 28, therefore 28 using inverse 28 minus 17 is 11. Again, answer in full. So Mr. Pride baked 11 loaves of brown bread. Okay, again, one example of model. Another example would be the whole bar. So 17 white bread, unknown quantity of brown bread, 28 loaves of bread altogether. And again, the equation is the same. Okay, so let's move on to part whole model with multiplication. So there are seven girls and three times as many boys in the class. Very often I find that just the. This part of the question may confuse children somewhat. Three times as many. Well, what does that look like? Some children might ask. So we can simply model it. So what we're trying to find is how many boys are there in the class altogether? So we model girls one part, three times as many boys. So three parts. So where are how many boys altogether in the class? Well, that's this bracket here, this bar here. Okay, so we do know that there are seven girls in the class. That means each bar is seven pupils. So in the same instance, because they're equal parts, that means there are seven, seven, seven, three lots of seven boys here. So therefore, the equation three times seven, 21. Full answer. There are 21 boys in the class altogether. So that's one type of model. The second type of model is the whole bar. So similarly, you've got the girls, you've got the three lots of boys. We know that there are seven girls. We don't know how many boys, but we do know that each equal part represents seven. So therefore, the equation is again same. Three lots of seven, which is 21. Right. Next one we're going to have a look at is the part whole model to do with division. With this one, a whole book of stamps costs £2. All the stamps in the book are the same price. There are five stamps in the book. Well, this one, it makes sense to just use one whole bar. So what is the cost of one stamp? Is the question. So using one whole bar, we can model the whole book of stamps. The cost of it is £2 or 200 pence. And we split them into five equal parts. Because there are five stamps in each book, we don't know what the cost of each stamp is. That's what we are trying to find out. So five times something is equal to 200 pence or 2 pounds. So we can then work out, well, how many. What needs to go into each part to make 200 pence is the question you need to ask your children. Hopefully they'll come. They'll realize that they'll need to divide by five equal parts to work out what Needs to go in each part to make 200 pence. So 200 divided by 5 is equal to 40. So each stamp in the book costs 40 pence. We move on to comparison models then. So comparison models are where you're drawing two or more models in order to solve the problem. Again, could be used with four operations, could be used with multi step. Problems also involved more complex topics like fractions, ratios and percentages. So a simple comparison model for addition. So Ellie made 26 cards, Eva made four less cards than Ellie. How many cards did they make in total? What's important to remember here is that you use two separate bars to model and compare. So you've got ellie, who made 26 cards, and you've got Eva who made an unknown amount of cards. But what we do know is that the difference between the amount of cards Ellie made and the number of cards that Eva made was four. So here's the difference and you can mark it on there on your bar model. Okay, how many cards did they make in total? So here we go. In total, how many cards did they make? So there is your question mark. Notice you also have a question mark here. So in order to work out the total, you'll need to work out first of all, how many cars EVA are made. So Eva Ellie made 26. So Eva made four less. So therefore we need to find the four less than 26 or 26 takeaway for which is 22. So Eva made 22 cards and we can change that to 22 now. Now we can work out the total. So pretty simple. Now we've got 26 cards that Ellie made, 22 cards that Eva made. So altogether 26 plus 22 is 48. Again, full answer. So Ellie and Eva made 48 cards in total. Okay, now we move on to the comparison model to do with subtraction. Okay, we'll go back to a question that we've seen before and we can see how you can use the same question in different models. So Paul has five stickers and Jesse has eight stickers. Again, when you're starting off with using bar models, the idea is to really understand the concepts. So try to make the numbers nice and simple at the beginning so that children understand the concept behind user bar modelling before going on to more complex ideas and complex numbers. So Paul has five stickers, Jesse has eight stickers. How many more stickers does Jesse have than Paul this time? So we model, Paul has five, Jesse has eight. And where is the difference? Well, up to this line here, Jesse and Paul have the same Number of stickers. So this part here is your difference. So our question mark is put up there. Okay, so our question mark is the same as 8. Subtract your 5, which leaves 3. So answer in full. Jesse has three more stickers than Paul. Okay. Another subtraction comparison model, this time a little bit more tricky. Nadia measures out some sugar in a bowl. It weighs 260 grams altogether. In an identical bowl, she measures out twice the amount of flour than sugar. This weighs 450, 445 grams. The question is, what is the weight of the bowl? So this is a lot more complex bar model than the one that we used previously. It involves a lot more elements. So let's model this. First of all, we have the bowl and the sugar, and we know that in total this weighs 260 grams. What's important here is to make sure that we put the constants together or line them up together as best as we can in order for us to compare more easily. So if I put the bowl first, and then we know that there's twice the amount of flour than there is sugar. So there's your flour, which is twice the amount of the sugar. And now we know that this weight is 445 grams. So from here you can easily see that the bowl and the sugar, one part of sugar, is the same weight as the bowl and one part of flour. So we need to work out this part here, which is, which can be done quite easily because we know the total weight of this bar and the total weight of this bar. The difference would be the weight of this one part of bar. So that would be 445 minus your 260 grams, which is 185. Therefore, this means that the flour is 185 grams, but also that the sugar is 185 grams. If we know that the sugar is 185 grams, we also know that the total weight of the bowl and the sugar is 260 grams. Therefore, it's simple to work out the weight of the bowl. So 260 grams. Subtract the hundredth grammar. 85 grams gives you 75 grams, which is the weight of the bowl. So let's have a look at a comparison model to do with multiplication. Okay, Katie weighs three times as much as Emmy. Emmy is 22 kilograms lighter than Katie. Their dad weighs twice their total weight. So there's your information. And the question is, how much do their dad weigh? So again, let's model it. What we do know is that Emmy weighs a certain amount and that Katie is three times Emmy's weight. We also know that their dad is twice their total weight. Now, the total weight for Emmy and Katie is 1, 2, 3, 4 parts. Therefore, twice the four parts would mean that dad is eight parts. So 1, 2, 3, 4, 5, 6, 7, 8, 8 parts. Need to be really careful that children are reading the questions really carefully because very often they will miss this part out twice the total weight. So what else do we know? We know that Emmy is 22 kilograms lighter than Katie. So there is a difference between Emmy and Katie, which are these two parts here at 22 kg. So if we know that, now we also need to find out where dads, what dad weighs. So this is the information that we need to work out. Okay, so we can work out the weight in each part size here. As we know that two of these parts is equal to 22 kg. So 22 divided by the two parts is 11. Therefore, each part must mean 11 kg. Now that we know this, we can go ahead and work out dad's weight eight times the 11 being 88. So therefore dad weighs 88 kilograms. Okay, so now a comparison model to do with both multiplication and division. There are 96 children altogether on a school trip. There are five times less the number of boys and the number of girls. How many girls are on the school trip? So first of all, what does five times less actually mean? Well, five times less is the opposite of five times more. So you can think of this as. You can think of these as being flipped around in a way. So here are the boys and there are five times more the number of girls. So that's 96 children altogether on the school trip. So all of these equal parts here make 96 children altogether. What are we trying to find out? How many girls are on the school trip. So that's this bar here. Okay, so first of all, we need to work out the park size. We know that there are 96 children altogether. There are 1, 2, 3, 4, 5, 6 parts. So that be 96 divided by 6, which equals 16. That means each part is 16 children. Now we can work out the girls. There are 1, 2, 3, 4, 5, 5, lots of 16 here. Therefore, that's 80. So there are 80 girls on the trip altogether. This one uses both multiplication and division to work out the answer. Now we go on to slightly more complex models of the comparison model. Using fractions at a concert, 2/3 of the audience are children. Of the adults, there were half the number of women as there were men. There were 160 men. How many children were at the concert? So again, we model. First of all, we can look at, well, what is the smallest part size? The smallest part size can't be the children, because 2/3 of the audience were children. There were half the number of women as there were men. So the women are half the number of men. So the women's part size is the smallest part size. So we can use one part to represent the women there. So the men would be twice the women. And because the adults in total is one third of the audience, which is three part sizes, therefore the children, two thirds would be double that, which would be six part sizes. We also know that there were 160 men altogether. So from here, it becomes really obvious how you can work out the part sizes. The question that we need to find out is how many children were at the concert, which is this bar here, so we can work out what each part represents. We know that two parts is 160. That means one part is worth 80 people. Now we can work out how many children there were at the concert. 1, 2, 3, 4, 5, 6. Part 6 times 80 is 480. Therefore, there were 480 children at the concert. Okay, let's go on to a percentage model now. Now the percentage model can start to get a little bit more complex, especially if children cannot do not understand the link between percentages and fractions. Having the ability to understand that 25% is the same as one quarter of the whole, and that 50% is 1/2 of the whole, and that 20% is 1/5 of the whole is really important for children to be able to split the bars here. Okay, so let's have a look at this question. In the sale, Susanna bought a toy that was reduced by 30%, but Penny bought the same toy at 50% off. The original cost of the toy was 36 pounds. So the question is, what was the difference in the price they both paid? Now, there are multiple ways to explore this, and we'll go into that, into a little bit more detail in a moment. But let's have a. Just have a quick look at this way first. Okay, so we can model Susanna, who bought a toy. So the whole bar represents that 100%. The original cost of that toy was £36. We know that Susanna only paid 70% of that for the toy, which meant that 30% was the discount. Okay, in the same way, we have Penny, the same size bar representing the 100% cost of the toy, but she only paid 50% of the toy, which meant that it was reduced by 50%, a 50% discount. I tend to either shade in or use dotted lines to show that actually, this part doesn't really exist in the question. Okay. Now, up until this point, so we're looking at the difference in the price they both paid. Up until this point, they both paid the same amount. So the difference is between these. This point here, essentially, this bar, this part of the bar here. Okay, that is what we're trying to find out. Now, your more advanced learner or somebody who really understands percentages may be able to look at this and go straight away, oh, that's 20% of the whole. So you could approach it using that technique. Some children might need to take a little bit more time and actually work out what 70% is as a price and what 50% is as a price and find the difference by doing a subtraction. So there is that root as well. Now, with percentages, we can think of 100% being 100 equal parts. That means each part would represent 1%. Now, in this situation, though, because we have multiples of 10 here, we wouldn't really need to divide it by 100. Although if you had an a more obscure fraction like 63%, you may want to go to that route of dividing it by 100 and then timesing it by 63. But in this sense, because we've got multiples of 10, we can think of splitting the whole into 10 equal parts, meaning that each part would be 10% of the original. So if that's how we're going to think about it, we need to work out first of all, well, how much did Susanna pay for her toy? So the cost of the toy originally was 36 pounds. If we find 10% of it, we need to divide it by 10. And then if we want to find 70% of it, we then need to multiply it by seven to find 70%. So 36 divided by 10 times by seven is equal to 25.2. Therefore, Susanna paid 25 pounds and 20 pence for her toy. Now we can work out Penny's price that she paid. With Penny's bar, it is already split into two equal parts, so it's nice and easy. 36. The cost of the original cost of the toy divided it by 2 will give us 18 pounds. A penny's cost of the toy is 18 pounds. So the difference then becomes the subtractions between 25 pounds, 20 and 18 pounds. Therefore, 25.2 subtract the 18 is 7, which means that the difference in price that they paid was 7 pounds 20. Okay, now ratio. I love using bar models with ratio because it really makes the problem really clear and really real to children. And we can see exactly what, where each part fits in. So let's have a quick look at a very basic model first. So purple paint is made by mixing blue and red paint together using a ratio of 2 to 5 tins. Assad needs 21 tins of purple paint altogether. The question is, how many tins of each paint does he need to buy? So we can model blue paint, two tins, red paint, five tins. So the ratio of two to five is represented in your bar models here. We also know that Asad needs 21 tins all together. Therefore, how many tins does each paint? How many tins of each paint does he need to buy? We need to work out how many tins of blue and how many tins of red. So we can see that there are seven equal parts here, the total being 21. So therefore each part is 21 divided by seven. So three. So three tins of paint should go into each part. Okay. The blue paint we can now work out is two lots of three tins, which is six tins of paint. So we need, he needs six tins of blue paint. And similarly, 1, 2, 3, 4, 5. Five tins of three red paint is 15 tins of red paint. So Asad needs six tins of blue paint and 15 tins of red paint to make 21 tins of purple paint. Within ratio models, and within any model actually, you can work, there are many kind of questions that you can eke out from the model. So for instance, here you could work out if the question was, what's the difference in the number of blue and the number of red paint that Assad needs? You could see the difference here. So that would be nine tins of red paint would be the difference. Okay, so let's have a very quick look at the before and after model. Okay. And these ones we compare models before an event and after an event. And sometimes it can be multiple events. So depending on what the question is, you may have more than two models. Okay. Again, used to solve problems within four operations, multi step problems. And again can include problems to do with fractions, percentages and ratios. Okay, so simple addition, before and after model would look like this. So last January, Zara measured 122 cm tall. This January she was 5 cm taller. So very basic. How tall was Zara this January? So before Zara's height is represented by this bar and she was 122cm tall. And then afterwards, we've got Zara's height, the original 122, but we also know that she grew from centimeters taller. So therefore the question how tall is Zara this January is this whole bar here. So 122/ the 5 makes 127cm. So Zara is 127cm this January. Okay, to do with subtraction, on Friday, Tom put £184 into his savings account. He now has £1,461 in his account. The question is how much money was in his account before Friday. So again, we model before. So this is the original amount of money that Tom had in his savings account. Afterwards, we've got the original amount of money he had plus the 184 pounds that he put in. Okay, we also know that this total here is £1,461. What we're trying to find out is how much money was in his account before Friday. So before the model, how much money was this part worth? So again, we know that the total of these two parts is 1,461. Therefore, this part we can do a simple subtraction calculation. 1461 subtract. The 184 pounds is 1277. Therefore, Tom had 1277 pounds in his savings account before Friday. Okay, a before and after model to do with fractions. One third of the apples in the fruit shop was sold on Monday. On Tuesday, three quarters of the remaining apples were sold. The shop sold 45 apples on Tuesday. So the question is, how many apples did the shop start off with on Monday? So we model, on Monday the shop had this amount of apples. We know that one third of the apples were sold. Now, I put it in dotted lines because I'd like to remind myself that this part doesn't exist anymore. After the apples were sold. And I also know that there were two thirds that were left. Now, on Tuesday, this part that was left, 3/4 of it was sold. So the part that was left on Tuesday, sorry, on Monday is this 2/3 here, which I've taken down and represented with this bar. I have now split the bar into quarters to represent three quarters sold and one quarter of the apples that were left. Okay, I also know that on Tuesday 45 of the apples were sold. So these three parts here represent 45 apples. The question is, how many apples did the shop start off with on Monday? So on Monday, these were the apples that he started off with. So this is My question mark. Now, from here, we can see that three parts is 45 apples. So therefore, we can work out what one part is. 45 apples were sold on Tuesday. This means that each part was 45 divided by three, which makes 15. So each part is 15 apples. If we know that each part is 15 apples, then the whole of this bar would be four lots of 15, which would be 60 apples, which is the same as these two bars. So these two bars here are 62. Now, if we know that these two bars are 60, then we can work out what the whole bar is, because one part, one part here would be 30, 60 divided by two, and we've got three of them. So that would be 90 apples. 30, 60, 90. So the shop started off with 90 apples on Monday, Before and after model to do with percentages. Okay, now again, as I said before, it's really important that children understand the relationship between percentages. Simple percentages like 25% being a quarter, 10% being a tenth, and so on. So let's have a look at this question. In 2016, there were 20% more visitors to a festival than in 2015. In 2017, there was a rise of 25% compared to 2016, there were 78,000 visitors in 2016. The question is, how many visitors were there in 2015? And also how many visitors were there in 2017? So two parts to this question. So let's have a look at the first part. So we model 2015, we know that there were a number of visitors. We also know that in 2016, there were the same number of visitors plus 20% more, 20% of the original 2015. So if a pupil knows that 20% is the same as 1 55th, then they can go ahead and divide their bar into five equal parts, and they know that one of those parts, 20%. So therefore, they can put that onto 2016. Essentially, 2016 being 120% of 2015. Now, we know that in 2016, there were 78,000 visitors. Therefore, the whole bar represents 78,000 people. And there are 1, 2, 3, 4, 5, 6 parts there? So therefore, each part is worth 78,000, divided by six is worth 13,000 people. So each part is worth 13,000 people. Coming back to the question, how many visitors were there in 2015? So here's 2015. This is what we need to find. We now know that each part is worth 13,000 people. There are five parts. So therefore, five times 13,000 is equal to 65,000. So there were 65,000 visitors in 2015. So let's move on to the next part. So how many visitors were there in 2017? So again we model 2016, the bar represents the total number of visitors. Now, this time, because there was a rise of 25%. Again, pupils need to understand that 25% is the same as one quarter. So we would split the bar into quarters in 2017, there was the original number of visitors as 2016, but now there's an increase of that 25%. One extra part, we know that 2016 was 78,000 visitors. So we know that this 2017 is the same as 78,000 visitors plus the additional 25%. So how do we work out the 25%? Well, if 78,000 is equal to four equal parts here, then one equal part would be 19,500. So therefore, 19,500 people are in each equal part. If we know that, then we can either add together so we can add 78,000 plus the 19,500 to give us 97,500 people, or if we know that each part is representing 19,500, 500 people and there are 1, 2, 3, 4, 5 parts, we can do a multiplication and work out that it's 97,500 people. So all together, there were 97,500 visitors to the festival in 2017. Okay, so now let's have a quick look at what more advanced bar models look like and what they involve. Okay, so they are definitely going to be more multi step. Sometimes it involves cutting up the bar, sometimes it involves moving different parts of the bar around. Again involving fractions, percentage and ratios. But definitely more complex and using a different mixture of the different types of models as well. So I'll give you a quick taster of what one more advanced bar model looks like. So I've chosen a ratio problem. The problem is, or the information is the ratio of the number of Sam's football cards to Julia's football cards is 3 to 1. Sam then gave Julia three 35 cards and Julia brought another nine cards herself. After this, Sam still had five more cards than Julia. The question is, how many cards did they both start out with? So this is a lot more complex. As you can see, it involves a lot more steps. So before we know that Sam had three, the ratio of Sam's cards to Julia's cards is three to one. So Sam had three parts, Julia has one part. Okay, and what we want to find out is how many cards they started out with. Okay, so afterwards, so Sam had his amount of cards, but he gave 35 cards to Julia. So we can move that 35 down here. So I've used a dotted line again to remind myself that that part doesn't exist anymore. We also know that Julia bought nine cards herself. So there's a nine represented. Now, the other piece of information we're given is that after the this, Sam still had five more cards than Julia. So essentially, it's the difference between Sam's amount of cards here and Julia's amount of cards here. And it's a difference of five cards. So it's that little gap here, which is worth five cards. Now, we also know that this missing part here, in order to make two original parts here, this missing part here is the same as this part here. So we can put that part in. Okay, that doesn't mean that it's there. It's just there to help us work out and solve the problem. Now, we can see that the total of this part here, with all the numbers in, with all the known values now, is the same as two equal parts, the original two equal parts parts. Okay? So this totals 84. We know that there are two parts there. So therefore, each part would be worth 42 cards. So each original part is worth 42 cards. Now that we know that, we can then go ahead and work out the question. So how many cards does Sam have? What? Well, each part is worth 42. So three lots of 42 is 126. So Sam started off with 26 cards. And of course, Julia has one lot of 42. So she started off with 42 football cards. Okay, and that is it.