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Marathon Training

﻿Hi. It's Mr. Whitehead, ready for a maths lesson, A maths lesson all about... Can you hear that? Listen. What is it? Yes, it is a maths lesson, all about money. I wonder where you keep your money, your coins. In a piggy bank, in a purse, hidden away somewhere safe. We are going to be looking at using certain coins to pay for some stuffs. Before we get started with this math lesson and this money problem, can you check that you are free of distractions. The coins in my hands are distracting me already. So I've put them down. Move yourself away from any distractions that might be around you. Press pause while you do that. As soon as you feel like you are able to focus for about 20 minutes on some learning, then press play again and we'll get started. Press pause now, and I'll see you again in a moment. In this lesson, we are developing strategies for planning and solving problems. We are going to start with a short task called money in my pocket. Then we will explore the problem that we are focusing on for the lesson. I'll give you some time to respond to the problem, and it will leave you ready to complete the problem independently. Things you're going to need; a pen or pencil, a ruler if you have one, and some paper, or a book from school to write into. Press pause if you need to go and collect any of those items, then come back when you're ready to start. Okay. Our first problem, just to get us started. Mike has four coins in his pocket. How much money could these coins total? Four coins. I've got a picture there of the coins that he could have. Would he have all of them? He only has four, four coins. The coins are all different. He has four different coins in his pocket. How much money could they total? Press pause, and have a go at this problem. Come back when you've got some solutions. How did you get on? Did you find one possible solution, more than one? You think there'll be quite a few. This would be a challenging problem to find all the possibilities for. It would take us a long time. There are just so many possible solutions. So instead, let's just look at a few. Here are some that I found. The chances aren't I, that what I show you now is what you found, because like I said, there were so many choices. But, I thought maybe, he had these four coins. One pound coin, a 5P, 2P, and a 1P. And if he did, the total would be one pound eight. He could have had these four. Again, they're all different, but totaling, a different amount of one pound 27, or these four two pound coin, five P, two P, one P, two pound eight. Tell me, did you make a total more than what I've shown you here? Did anyone have a total less than my one pound eight? And just tell me another total that you managed to make with your four coins. Good. Let's have a look at the problem that we are focusing on in this session. It's called stamps. Meg is sending a letter to China. She has to pay 85P. If she pays using 20P coins, 10P coins and 5P coins, how many different ways can she pay? Now, that's the question you'll be answering by the end of the session, in the independent task. But, these next few questions are going to get you ready for that. So here's my first question. How much is Meg paying to send her letter? Have a look again at those sentences in the problem and see if you can spot the answer. Have you found it? She has to pay 85P. That's how much she is paying to send her letter. Second question. What coins did she use to pay for it? Have a look again. Found it? She is paying using 20P, 10P and 5P coins. She uses at least one of each of those coins. 20Ps, 10Ps, 5Ps. No other coins. Number three. Did she use all the different types of coin? Should be a quick answer. No. She only used three types of 5P, 10P, 20P. She used more than one of some of those coins. She used at least one of each. She makes 85P in total. She doesn't use any other type of coin. Just 20Ps, 10Ps, 5Ps. Okay. Here's one for you to have a go at with a little pause. Imagine Meg used two 20P coins, how many different ways could she have paid 85P? So she's definitely got two 20P coins. How could she make her 85P total? In which different ways with which different coins? What is the least amount of coins she could have used to make 85P? And what is the most amount of coins she could have used to make 85P? Press pause and to have a go at those three questions, and come back and we'll share what we found. Let's take a look, shall we? How did you get on? Did you manage to solve all three of those problems? Let's look at the first problem. How many different ways could she have paid 85P if two of the coins were 20Ps? Hold up with your hands on your fingers, how many ways did you find? Okay. Should we check? So definitely two 20P coins. The rest could have been, 10Ps with one 5P. That's one way to make 85. 20, 40, 50, 60, 70, 85P. That's one way. She could have changed one of those 10Ps for two more 5Ps. So now we've got two twenties, three tens, three fives. That's the second way. We could have switched and had five 5Ps, two tens, two twenties. That's way three. Do you see what's going to happen next? We could have had seven 7Ps, hm! Seven 5Ps, one 10P, two twenties. She's got one of each coin represented. So there are four ways, that Meg could have made 85P using 10Ps, 5Ps, 20Ps. What is the most amount of coins that she could have used? Still using at least one of each. But how could we make it so she has used the most possible? How many coins did you come up with? Hold up your paper if you need to. Maybe you've drawn some pictures of the coins. Okay. So, let's see. She needs at least one, at least one to 20, at least one 10. She'll need at least one five. And now those fives, if we want there to be lots of coins, she needs to use coins with the lowest value. So, with 30Ps so far, and the rest 5Ps, 10, 20, 30, 40, 55P plus 30P... 85P, made out of 11 to 13 coins. Let's have a think now about that main problem. So Meg is sending a letter to China. She has to pay 85P. If she pays using 20Ps, 10Ps, 5Ps, how many different ways can she pay? How could we organise our results as we start solving this problem so that we know we have found all of the different combinations, all the possibilities for solving the problem? Let me share with you a table that you might like to use. Number of 20P coins, number of 10P coins, number of 5P coins. It's a good idea to work systematically if you want to find all the possibilities. So starting for example, with one 20P coin, I could then have six 10P coins. That's 80P in total, and one 5P coin, 85P. Next, I'm going to keep the number of 20P coins the same and change the number of tens and fives. Five 10P coins, and three 5P coins. 20+50, 70+15, 85. Next, keep the 20P the same, but reduce the number of 10Ps increasing the 5Ps. Now we could work with coins like this, drawings of coins like this, or represent with a number. One six and one. One 20, five tens, three fives. One 20, four tens, five fives. As long as you are keeping a track of what each of those numbers represents, and the 85 is still being made, this can be a useful way to work. Especially, if you start to spot some patterns. Can you see any patterns there already? If you spot patterns, it helps you to know when you have found all the possible combinations. So, the problem is set. You've got an option here for how to record. And if you are ready for a challenge, explain how you know. You have found all the possible combinations and spot and describe the patterns you notice within your results. Press pause. How did you get on? Did you find all the possible combinations? So how many were there? How many different ways were there for her to pay 85P. Let's take a look. So I started you off and gave you a suggestion for how to work systematically. If you carried on keeping one 20P coin, did you get this same set of results with that pattern through the number of 10P coins you would be using compared to the number of 5P coins? Notice, as the number of 10Ps decreases by one, the number of 5Ps increases by two. How did you know when you had one 20, one 10 and 11 fives, that that was a point to stop? We couldn't have zero 10Ps. So, we've tried with one 20, all of the options of a number of 10Ps. And therefore, all of the options for a number of 5Ps, continuing then working systematically increasing the number of 20Ps. Let's start with two. Let's look at the maximum number of 10Ps we could have, and therefore how many 5Ps we would need. And again, look at the pattern. As the number of 10Ps decreased by one, the number of 5Ps increased by two. Again, when I had one 10P with my two twenties, I knew it was a point to stop. I couldn't have zero 10Ps because each coin must be represented. I then increased my 20Ps to three. I only had two options here. I could have two tens and one five, or one 10 and three fives. But again, once I had one 10, that was my place to stop because I couldn't have zero. So, in total, how many different ways can she pay? There are 12 different combinations of 20P, 10P and 5P coins, of making 85P. I've talked through that, as I put to my results, part of the ready for a challenge too. And I wonder how that compared to you. Give me a thumbs up if you tried the ready for a challenge. And I wonder how you were explaining, how you know, or how you knew that you had found all the possibilities. And, how did you describe the patterns in the results table that you spotted. If you weren't able to spot any patterns, I wonder if working more systematically would help. Changing one thing at a time like I did working through those lists of coins. If you'd like to share your results table or any more of your work from this session with Oak National, please ask your parents or carer, to share your work on Twitter tagging @OakNational and #LearnWithOak. What a fantastic session. I've really enjoyed working with you problem solvers, problem planners, and problem solvers. You've worked really hard. Thank you for participating. And of course, thank you for helping me to reach a solution to the problem as well. I'm going to go and clear my money away, back into my money box. I look forward to seeing you again soon for some more maths learning. If you've got any more lessons lined up for the day, then I hope you enjoy them, work hard, keep smiling, and I look forward to seeing you again soon. Bye.

Meg uses 2p coins (as well as 20p, 10p and 5p coins) to total £0.85.  What is the largest number of coins she could use to make the total?

Meg uses 2p coins (as well as 20p, 10p and 5p coins) to total £0.85.  What is the smallest number of coins she could use to make the total?

Meg makes £0.85 using 17 coins.  If she used 20p, 10p, 5p and 2p coins, how many 5ps did she use?

Stamps

In this lesson, we will investiate a money problem involving calculating the number of combinations of different types of coin we could have that add to the same total.

Maths

Y4 Maths

Key Stage 2

Year 4

Solving measure and money problems

Number

THEME

UNIT

Units of Measure

Converting Lengths (Part 1)

Converting Lengths (Part 2)

Converting Measurements

Ordering Measurements

Ribbons

Money Bags 1

Money Bags (Part 2)

Two point seven

Fruit

Souvenirs

Flag Pole 1

Flag Pole (Part 2)

Tim buys some ribbons.  Green measure 80 cm, blue measure 1.4 m and red measure 90 cm. He buys one of each colour.  What is the total length of the ribbons bought?

Which unit of measure would you use to measure the mass of a television?

Which is the longest length?

How would you write 4 cm and 3 mm?

Which measurement is equivalent to 750 ml?

Which measurement is equivalent to 0.8 km?

Which measurement is equivalent to 900 g?

Which measurement is equivalent to 1250 m?

Choosing 4 weights (one of each type), what is the heaviest mass you can build?  Choices: 1 kg, 500 g, 100 g, 50 g, 20 g, 10 g

Apples cost £0.25, Oranges cost £0.35 and Lychees cost £0.45.  Emily buys three fruits.  What is the most expensive total cost?

Ryan makes a flag pole.  From lengths of one metre, half a metre and quarter of a metre, he chooses 6.  What is the longest length his flag pole could be?

Lesson overview:Stamps

Solving measure and money problems: