Represent remainders in different ways depending on the context of the problem (Part 2)

Key Stage 2

Maths

Multiplication and division

﻿Hello it is Mr. Whitehead, ready for your maths lesson. I know I'm ready because the room is quiet. I'm free of distractions. Are you as well? If you're not, you need to pause. Take yourself to a quieter space, away from any siblings away from any tablets, notifications or televisions. Somewhere that you can focus on your learning for the next 20 minutes. Press pause, go and get yourself sorted, then come back when you're ready to start. In this lesson, we will be interpreting remainders using the context of a problem. So we'll be looking at division, there'll be remainders and the context of the problem will tell us what to do with those remainders. We're going to start off with a number line activity. Then we're going to explore a problem, respond to a problem, and it will leave you ready for your independent task. Things that you will need, pen or pencil, a ruler, and some paper, a pad, or a book. Press pause, collect the items, come back and we'll start. Here we go. Your number line activity to get started. Take a look. You've got six arrows pointing to parts of the number line. Speaking of parts, the number line has been divided into equal parts. Depending on which divisions you're looking at, the number of equal parts will change. Are you looking at the longest divisions, the medium or shortest? Think really carefully about what each of them are worth as you start to identify the value of the numbers pointed to by the arrows. Press pause, have a go, then come back and we'll check. Are you ready to have a look? Well, can I have a look at yours first of all? Hold up your paper, let me see how you've approached it. What have you been recording? Fantastic, let me share with you the solutions then, working along the bottom first of all. We've got 5,700. We know that those longer divisions are worth 500. The shortest divisions are worth 50. So two of them would be worth 100 and that medium-sized division is worth 250. So 5,700, 50 short of 5,750. Next, 6,900. 100 away from the next big division, 7,000. Then we've got 8,750, 250 away from 9,000. Along the top 6,500, 8,000 and then 9,750. How are you feeling off that activity? Got smiles on our faces, some thumbs up. Brilliant. Let's keep going. Here's a problem for you. I'd like you to read it with me. 654 guests are on an Oak Travel cruise ship. The life jackets are arranged into packs of seven. How many packs will the cruise ship need to carry to ensure that everyone has one? Pause, three questions for you. What do you know from reading the problem? What do you not to know? And are there any connections to knowledge and skills that you already have that you can use here to solve the problem? Come back when you're ready. So what do you know from reading the problem? Call it out to me. Anything else? Brilliant. What do you not know? Let's take a look then. Here, I'm just going to highlight those things that we know. There are 654 guests. We know that the life jackets are arranged in packs of seven groups of seven, sets of seven. What we don't know is how many packs of seven the boats will need to carry, that the ship will need to carry, so there's one for everyone. To help us make those connections, the knowledge and skills that we already have, which will help us to answer the problem, a bar model is our next step. Press pause and to have a go at drawing a bar model to represent the information from the problem. Come back when you're ready. Can I have a look? Hold up your bar models. Looking good. I can see the parts of the problem represented. I can see what we're trying to find out as well. Compare it to mine. There might be some slight differences. That's okay, as long as the models are revealing the matter to us. So I started by representing the number of guests. There are 654, and I know that for the number of life jackets, we've got packs of seven. And we're working out for all of the guests, how many packs we will need. So I want to know how many packs of seven I can make from 654. One person, one life jacket. How many packs of seven? How many equal groups of seven can we make from 654? What connections are you seeing from this and your bar model to the knowledge and skills that you have that you're going to need to use here? Fantastic, some division. 654 divided by 5 into equal groups of-- By 5? 7. 654 divided by 7 into equal groups of 7, thanks for catching me with that mistake. I'm going to work through with you now a short division. Not a mental approach. We've got a range of mental strategies that we could use, but I want us to use short division. If you'd like to pause and work independently on your short division and possibly push yourself to representing that short division with some place value counters as well, press pause now. Otherwise let's work through at the same time. So you can be filling in on your own paper as we work through this problem. Let's start with an estimate. 654 divided by 7. Any connections coming through to help with your choice of estimate? 63, 630, seven lots of 90. Seven 9's 63, seven 90's 630. So I'm expecting my quotient to be around 90. Let's represent 654 with counters and start our division. How many groups of seven can we make from six hundreds? We can't make any. We're one short. So let's exchange. Our six hundreds for 60 tens. 30, 40, 50, 60. So now we've got 65 tens. How many groups of seven can we make from 65 tens? Excellent. We can make nine-- Oh that was a bit quick. Should I do that again? We can make nine groups of seven using 63 of the tens. Nine, seven, 63. So we've used 63 tens to make nine groups of seven with two tens remaining that we're going to exchange for 20 ones. Did you see that happen? Watch again. The 20 ones. How many ones do we now have? 24 ones. How many groups of seven can we make from 24 ones? We can make three using 21 of them. We have a remainder of three. 93. Good close to my estimate of 90. 93, remainder three. Let's go back to the problem. 93, remainder three. So we can make 93 remainder three packs of seven. That does not sound right. 93 remainder three packs of seven. How many packs will the cruise ship need to ensure that everyone has one? Ah, this type of problem is one where we now need to make a choice between 93 packs and 94 packs. We're keeping with that whole part of the quotient, 93, or we're rounding to the next whole number. Which do you think, 93 packs or 94 packs? Why do you think that? If we picked 90-- This is-- And I shouldn't be laughing, this is a serious matter. If we picked 93 packs, we wouldn't have enough life jackets. If we picked 94 packs, we will have enough with four spare. 94 packs so that everyone on the cruise ship is going to be safe. Here's another problem for us to have a look at. Let's give it a read. Florian has £585. Lucky him. He wants to work out how many friends he could share his money with equally. However, if he cannot share the money to give each person an exact amount, sorry, an exact equal amount, he will rule out that amount of friends. Can you help him to solve his problem? What do we know? What do we not know? Are there any connections starting to come through? Press pause and have a go at those questions. So tell me what you know. Yeah, anything else? And what do you not know? Okay, let me represent those thoughts then. So we know that he has £585. We know that he cannot share-- Sorry, if he can't share the money equally between a number of friends, he won't use that number of friends. We don't know how many friends he could share his money with equally. So there's quite a bit there that-- I mean we definitely know about the 585, but we don't know how many friends he is sharing it between. So perhaps to get started, here are some tips. Try sharing that money equally between two friends, three friends, four friends. See what happens, see what you find and whether or not you continue from there. Press pause and have a go. Use some short division to divide £585 pounds between that many friends at a time. Come back when you're ready. Should we take a look? So, I've started off with dividing that amount between two friends. I'm not going to use place value counters. That's a stepping stone to helping us understand short division. I think we can understand it. We can show we understand it even if we're using the language of place value. So for example, how many groups of two can we make from five hundreds. That's showing our understanding of what's happening. We can make two groups of two from five hundreds, with 100 remaining that we can exchange for 10 tens. How many groups of two can we make from 18 tens? Tell me. Good, nine. Nine groups of two from 18 tens. How many groups of two can we make from five ones? Two groups with one remaining. One one that we can exchange for 10 tenths. How many groups of two can we make from 10 tenths? Five. We have divided 585 by two. We're working with money. So we're thinking for two friends, that's £292.50 each. Each friend would get an equal part of 585. Okay, so that's worked. Let's move on. Could you try three friends next? And we should have one group of three from five hundreds remainder two hundreds. 28 tens now. How many groups of three from 28 tens? Nine remaining one. One remaining. How many groups of three can we make from 15 ones? Five. We finished the division. We've shared 585 between three friends. They each get 195 pounds. Okay. So he could share his money with two or with three. Those friends would get an equal amount and that was his rule. If they don't get an equal amount, he won't share with that many friends. So let's carry on. Four. How much did you get as your quotient here? Tell me. Okay, let's compare. How many groups of four from five hundreds? One with 100 to exchange for 10 tens. How many groups of four from 18 tens? Four, with how many to regroup or exchange sorry? With two tens to exchange for 20 ones. How many groups of four from 25 ones. Six, with how many ones remaining? So let's not leave it as a remainder. This is money. Let's exchange that one for 10 tenths and continue dividing. How many groups of four from 10 tenths? Two with two tenths to exchange for 20 hundredths. How many groups of four there? Five. The division is complete. With for friends they would each get £146.25. An equal amount each. Did you stop or carry on? Me too? I tried it with five friends next. Fives. How many groups of five in five hundreds. One. How many groups of five and eight tens? One with three tens to exchange for 30 ones. How many groups of five from 35 ones? Seven. Oh, that's a nice division. We haven't got anything to no exchange into our tenths. So five friends, £117 each. Okay. It's looking like he can share his money with two, three, four or five friends. Hang on. I didn't start with one friend. How much would that one friend? They would get it all. 585 pounds. Let's continue. Six. Did you do that? If you didn't and you want to pause and work out six, do that now because I'm going to share it next. Six friends. How many groups of six can we make from five hundreds? Let's exchange. 58 tens. How many groups of six? Good. Nine groups of six using 54 tens. We've got four tens to exchange for 40 ones. How many groups of six here? Good, seven using 42 ones. We've got three ones to exchange for 30 tenths. How many groups of six from 30 tenths? Five. The division is finished. We're working with money. Five tenths, 0.5 is five tenths. So we've got 50 hundredths, 5/10 of a pound, £.50. Six friends, 97.50 each. I wonder, at this point, if he's going to be able to share with any number of friends. It keeps on working. They keep on getting an equal amount. So let's try seven. How many groups of seven can we make from five hundreds? How many from 58 tens? Good, eight. Using 56, good. 56 of those tens with two tens to exchange for 20 ones. How many groups of seven can we make from 25 ones? We can make three using 21 ones. Four to exchange for 40 tenths. How many groups of seven can we make from 40 tenths? Five, using how many of the tenths? Using 35 of them. How many tenths remaining? Let's exchange them for hundredths. How many groups of seven can we make from 50 hundredths? Let's use 49 of them to make seven. Oh. One hundredth remaining to exchange for one thousandth-- Sorry. To exchange for ten thousandths. How many groups of seven can we meet from 10 thousandths? One. How many... We used seven of them to make one group. How many remaining? Three. What? So we're going to exchange into a fourth decimal column? This isn't looking-- Ah! This is looking right for division, but not for money. Sometimes when we are dividing and we continue dividing into decimal places, we will keep on having remaining tenths or hundredths of thousandths to exchange to the next smallest column, and it just won't stop. That's why sometimes some rules are given. And actually this problem is giving us a rule, isn't it? If we're working with money, how many decimal places? What's the maximum number of decimal places we can have when we're working with money? We would have two as a rule. Tenths and hundredths. So if you then have to move into the thousandths, what you would then do is use the thousandths digit to round the hundredths digit. So we've got 83.57. We would look at the one thousandth and we would keep it at £83.57. We wouldn't round to £83.58. Okay. So £83.57. But then if I check that and multiply it by seven, the number of friends there are. Seven lots of £83.57 is equal to £584.99. What's the problem there? He's sharing 585. He's not been able to share 585 equally between seven friends. So seven would be a number of friends that he wouldn't share with. Okay, for a little while I was thinking, this could just continue. Then we tried seven and it's not worked. Yes, we can divide it, but in the context of money, we won't end up with an equal amount of 585 each for seven friends. If you'd like to, you can continue on your own beyond seven. Try eight friends, nine friends. When does it work? When does it not work? At some point, though, you might want to ask yourself, when will this ever end? What's the maximum number of friends that he could have? That's quite a challenging question, but give it a go if you'd like to. Otherwise that aside, it's now time for you to have a practise on your independent task. So press pause, go and complete the activity then come back and look at the solutions. Here we go. I'm just going to take you through each question one at a time. But before I do, hold up your paper, let me see how you got on. Looking good. I can see some bar models. Nice. I can see your division, your short division, and based on the question context, you're making decisions about what to do with the remainder. Well done. So question one was around eggs. We're finding 1/6 of 843. It's 140.5. How many boxes will he needs to hold all the eggs? 140.5. 140 3/6 of a box. We're going to need here I think to think about the next multiple of one, the next whole number because we can't have 140 boxes and half a box. We need to hold all 843 eggs. So we're going to have 141 boxes, although there'll be three empty spaces in the last box, we'll have enough boxes for all of the eggs. Question two, so we're working with money and the money is being divided between eight charities. £311.75 for each charity. We could have thought about that as 311 3/4 because there was a remainder of six, 6/8, but we're working with money. So the most appropriate way to interpret the remainder is using a decimal representation. Question three. So we've got 765 sweets and we are looking at how many packs of seven we can create from those 765 sweets. How many full packs? So dividing, we ended up with 109 remainder 2. So we can have 109 remainder 2 packs. It doesn't make sense. This is an example where we're either going to keep it at 109 or round to the next whole number, 110. Which do you think, which did you pick? Why? Question asked how many full packs can they make? They've only got 765 sweets so they can only make 109 full packs. If they had five more sweets they'd be able to make 110 packs. But they don't, so they can make 109 packs. If you would like to share any of your learning from this session with Oak National, please ask your parents or carer to share your work on Twitter tagging @oaknational and #learnwithoak. Thank you so much for joining me for that lesson. Once again, you've left me feeling incredibly proud from your participation to your activity responses. Big smiles for me and I hope it's the same for you. I don't know if you noticed at the start of the lesson, it said on the slide, lesson 20 of 20. Now some of you who have been here for all of those, some of you will have been here just for this session. Either way, you've all been incredibly welcome. That said, 20 lessons, if you've been here for all of them, that is quite an achievement. If you've got more learning lined up for the day, I hope that you enjoy that as well. Hopefully I'll see you again soon for some more maths learning. Until then look after yourselves. Bye.

Represent remainders in different ways depending on the context of the problem (Part 2)

Represent remainders in different ways depending on the context of the problem (Part 2)

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UnitMaths / Multiplication and division

Maths