DIRECTED NUMBER Lesson 1: The Elevator Challenge

DIRECTED NUMBER Lesson 1: The Elevator Challenge Australian Curriculum: Mathematics Year 6 ACMNA124: Investigate everyday situations that use integers. Locate and represent these numbers on a number line. Lesson abstract Students develop a strong mental model of negative integers by playing a game that involves moving an elevator up and down in a hotel. By interpreting negative numbers as describing both a distance and a direction (below the lobby) they will understand their ordering and be able to reason about their properties (e.g. to find distances between them). Mathematical purpose (for students) There are numbers below zero. Mathematical purpose (for teachers) In this lesson, students develop a mental model of positive and negative integers that they can carry with them to understand later work with negative numbers. This lesson focusses just on the ordering of negative integers and not their size. Students will see patterns that help movement between positive and negative integers, interpreted in the elevator context. At the end of this lesson, students will be able to: Show that there are numbers less than zero. Move between positive and negative integers. Understand how negative numbers are arranged on the number line. Lesson Length 90 minutes approximately Vocabulary Encountered Lesson Materials integer positive negative negative 3 number line Student Sheet 1 - Role Play Script (4 copies) OR Elevator Challenge animation One A3 copy of Student Sheet 3 - Hotel resolve Student Sheet 3 Hotel resolve and Student Sheet 2 Elevator Challenge Rules copied back-to-back (one per pair of students) Student Sheet 4 Elevator Cards sides 1 and 2 copied back-to-back (one per pair of students) 6-sided dice (one for each pair of students) Plastic counters (one for each pair of students) We value your feedback after this lesson via http://tiny.cc/lesson-feedback

Meeting Amy and Barry Introduction to elevators Discuss students experiences with elevators and not being able to get to the correct floor. Teacher shows A3 version of Student Sheet 3 - Hotel resolve. Watch the animation Elevator Challenge. If unable to watch the animation, read through Student Sheet 1 Role Play Script. Choose four students to be the actors in the play. The script (and animation) are an abbreviated version of the task they only require Amy and Barry to perform 3 deliveries each. The full version of the game that the students will play requires them to do 6 deliveries each. A shorter version of the game can be played by handing out only 3 task cards to each player. Throughout this lesson it is important to reinforce the use of the small superscript ( - ) symbol when writing negative numbers (e.g. - 4). The elevator game Read this summary of the task to the class to consolidate what is happening: Amy and Barry are employees at a large hotel. The manager has given them both 6 errands to run on 6 different levels of the hotel. They need to use the elevator to get up and down to complete each task. Amy and Barry will be sharing the elevator. The person who finishes all their jobs first will be Employee of the Month. Briefly discuss the elevator context, the hotel floors and the numbering system used. Why do you think the floors are labelled like this and what do the labels tell us? The negative sign tells you it is below the lobby and the number tells you how far you are from the lobby (which is numbered zero). The lobby is labelled zero because it is zero distance from itself. If there is no negative sign in front of the number, it is a positive number and the floor will be above ground level (unless it is zero). 2

Playing the Game Read through the rules of the Elevator Game with the class. Students play the game once in pairs. If you have an odd number of students, you may need to make one group of three players just add a new character Carlotta to the game (now we have characters A, B and C), divide the cards evenly between them and play using the same rules. Discussion Ask for students responses to the game. Refer to the A3 picture of Hotel resolve during the discussion. Who won the game? How did it play out? Was there some time in the game when you had to think hard? When I didn t want the other player to win; when I had to choose between two outcomes; when I had to go below ground level; when I was already below ground level and had to move up or down. Discussion Did you have to count each level after you rolled the dice to work out where you were going to end up or did you find a strategy to work it out in your head? When I was on Level - 2 and rolled a 5, I knew it was 2 to go up to ground level, so then there were three more levels to move, which took me to Level 3. A quantum shift in the game play Ask the students to play the game in a different way. After you have played the game a few times for fun: o Play again, but this time turn the cards over so that you can only see the Floor Numbers, not the icons. o Fold the diagram of the Hotel resolve elevator in half vertically so that you can only see the Floor Numbers and not the floor names or pictures. o As you play, record the movements up and down of the elevator. You might want to draw this as jumps on a number line, as a diagram, or record as a narrative, as a table of data, or in another way. This is an important shift in the focus of the game, so that students will focus on the numbers, not the pictures. We still draw on the context of the elevator and the floors of the hotel but build our growing knowledge of directed number by just looking at the numbers. 3

Developing Our Understanding Understanding the game Conclude the games and introduce a class discussion on the Elevator Game. Share how you recorded the movement of the elevator. Students records of the elevator movement might look like any of these four - a diagram, a number sentence (more on this later), words, or a table. Starting level 0 2 5-1 - 4-2 - 6 Roll 2 3 6 3 2 4 2 New level Up to 2 Up to 5 Down to - 1 Down to - 4 Up to - 2 Down to - 6 Up to - 4 Discussion How did playing the game without the pictures change what you had to do? You still had to do the same thing but this time without looking at pictures but I could remember them anyway. I needed to think more about how far I had to move; I wasn t thinking in terms of the different levels of the hotel anymore; I was looking more at the number values and the position of the numbers on the number line. Discussion Did you develop any strategies to help you see where the elevator moved to? I was able to see how far I was away from zero and then break my dice roll into two parts the part that took me to zero and the part that went past zero; the numbers above zero were just adding and subtracting, butfor the numbers below zero I counted by ones to make sure. 4

Inquiry question Imagine that you have just landed at Level - 2 The Car Park. What might your dice roll have been to get there? What level did you come from? I was at The Café (Level 1) and rolled a 3; I was at the Restaurant (Level 4) and rolled a 6; etc. Extending Prompt What are all the possibilities for how you got to the car park? Show that you have found them all. If I rolled a 1 then I must have been at the gym or the kitchen. If I rolled a 2 then I was at the lobby or the laundry. If I rolled a 3 then I came from the café or the deliveries. If I rolled a 4 then I was at the storage or the business centre. If I rolled a 5 I could only have been at the guest rooms. If I rolled a 6 then I had to start at the restaurant. I couldn t have got to the car park from the spa or the pool in one roll. Going down to car park Going up to car park Dice Roll 6 5 4 3 2 1 x 1 2 3 4 Level you came from 4 3 2 1 0-1 - 2-3 - 4-5 - 6 If students choose to represent their ideas in ways other than a table, ask them to explain how their chosen method is systematic. Their method needs to show that they have considered all possibilities. Drawing the above table in a vertical orientation links back to the hotel and the elevator. 5

Extension task Imagine you have just started a game. You start at the lobby. You notice that after three turns, you are back at the lobby. What might the three rolls have been? When you have finished, imagine your own example. Tell your partner how many turns they have and where they need the elevator to end up the destination doesn t need to be the lobby. Ask your partner to list several different ways that they could get there and what numbers they need to roll. I was at the lobby. I rolled a 2 and went up to the Business Centre. Then I rolled a 3 and went down to the gym. Then I rolled a 1 and went back to the lobby. It is important that students refer to the dice rolls needed not just say where they stop on their journey. Below are all the possible rolls for starting at the lobby and ending back at the lobby, recorded with number sentences. Note that one of the throws has to equal the sum of the other two throws. The number sentences are to be read as elevator movements. For example, 0+2+1-3 = 0 describes the journey "Start at the lobby (0), move up 2, then up 1, and then down 3. The elevator then arrives back at 0 (lobby). The addition and subtraction indicates the direction the elevator moves. Students will probably not write the journeys with number sentences at this stage, nor find them all. Some of the roll combinations will take you below ground level. These are all the possible combinations of dice rolls. 0+1+1-2 = 0 0+2-1-1 = 0 0+3-1-2 = 0 0+4-1-3 = 0 0+5-1-4 = 0 0+6-1-5 = 0 0-1-1+2 = 0 0-2+1+1 = 0 0-3+1+2 = 0 0-4+1+3 = 0 0-5+1+4 = 0 0-6+1+5 = 0 0+1+2-3 = 0 0+2+1-3 = 0 0+3+1-4 = 0 0+4+1-5 = 0 0+5+1-6 = 0 0+6-2-4 = 0 0-1-2+3 = 0 0-2-1+3 = 0 0-3-1+4 = 0 0-4-1+5 = 0 0-5-1+6 = 0 0-6+2+4 = 0 0+1-2+1 = 0 0+2+2-4 = 0 0+3+2-5 = 0 0+4-2-2 = 0 0+5-2-3 = 0 0+6-3-3 = 0 0-1+2-1 = 0 0-2-2+4 = 0 0-3-2+5 = 0 0-4+2+2 = 0 0-5+2+3 = 0 0-6+3+3 = 0 0+1+3-4 = 0 0+2-3+1 = 0 0+3-2-1 = 0 0+4+2-6 = 0 0+5-3-2 = 0 0+6-4-2 = 0 0-1-3+4 = 0 0-2+3-1 = 0 0-3+2+1 = 0 0-4-2+6 = 0 0-5+3+2 = 0 0-6+4+2 = 0 0+1-3+2 = 0 0+2+3-5 = 0 0+3+3-6 = 0 0+4-3-1 = 0 0+5-4-1 = 0 0+6-5-1 = 0 0-1+3-2 = 0 0-2-3+5 = 0 0-3-3+6 = 0 0-4+3+1 = 0 0-5+4+1 = 0 0-6+5+1 = 0 0+1+4-5 = 0 0+2+4-6 = 0 0+3-4+1 = 0 0+4-5+1 = 0 0+5-6+1 = 0 0-1-4+5 = 0 0-2-4+6 = 0 0-3+4-1 = 0 0-4+5-1 = 0 0-5+6-1 = 0 0+1-4+3 = 0 0+2-4+2 = 0 0+3-5+2 = 0 0+4-6+2 = 0 0-1+4-3 = 0 0-2+4-2 = 0 0-3+5-2 = 0 0-4+6-2 = 0 0+1+5-6 = 0 0+2-5+3 = 0 0+3-6+3 = 0 0-1-5+6 = 0 0-2+5-3 = 0 0-3+6-3 = 0 0+1-5+4 = 0 0+2-6+4 = 0 0-1+5-4 = 0 0-2+6-4 = 0 0+1-6+5 = 0 0-1+6-5 = 0 6

Going Further with a Number Line Playing variations of the elevator challenge game Place all 12 cards on the hotel at the appropriate levels. Players collect the cards when they land on a level. Winner is player with the most cards. Have 3 players Amy, Barry and Carlotta. Deal out the cards but players don t reveal to each other what cards they have. That way, if you don t have the card for a level you will be uncertain which of your opponents has it. Deal out a dummy hand that no-one sees so that there are some safe floors but you don t know what they are. Formalising the number line Draw a number line with 21 positions but only fill in 7 numbers (0-6) on it. What numbers should be written on the empty spaces? - 10, - 9, - 8, - 7, - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10 Some students might wonder about the existence of negative zero and where it goes on the number line. As it has the same value as positive zero it doesn t get its own point on the line. The number - 0 is the level which is zero levels below the lobby this is still at the lobby. - 0 = 0. Skip counting on the number line Count on the number line: o start at 10 and count down by 1s. o start at - 10 and count up by 2s. o start at 8 and count down by 3s. o teacher (or students) decide on other starting points and skipping rules. Skip counting can reinforce position on the number line. It is an activity that everyone is familiar with and provides low floor, high ceiling access. When they are ready, students can close their eyes and visualise the number line as they count up and down, rather than looking at it. 7

What Did We Learn? Moving on the number line Draw a number line and put in the integers from - 10 to 10 Write a skip counting pattern going backwards from 20 by 2, by 3, by 6 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0, - 2, - 4, - 6, - 8, - 10, - 12, - 14, - 16, - 18, - 20 20, 17, 14, 11, 8, 5, 2, - 1, - 4, - 7, - 10, - 13, - 16, - 19 20, 14, 8, 2, - 4, - 10, - 16, - 22 Discussion How are negative numbers positioned on the number line? How does the example of the elevator explain this? From left to right on a horizontal line: - 10-9 - 8-7 - 6-5 - 4-3 - 2-1 The level numbers tell you how far you are from the lobby (which is zero). The negative sign tells you that you are below ground level. Related Ideas Make your own number line using: o a long rope o a ball of string o a roll of paper o cracks in the footpath or playground o construction blocks. Make sure that the spaces between the integers are equal. Why is this important? 8

Role Play Script Name: Student Sheet 1 Role Play Script Manager (Off screen) So you guys want to be Employee of the Month? Let s see if you deserve it! I m going to give you three errands each to different floors of the hotel. The first person to complete their errands will be the winner. But you know the rules You have to share the elevator. You take it in turns to move the elevator. You will have to roll a dice to see the number of floors you can travel up or down. So what are you waiting for? There s work to be done! Amy Barry Elevator OK Barry, you have been working here longer than me. You can roll first. (Rolls dice and gets a 1..Thinking) Hmm I could move the elevator one floor up or one floor down. But Amy has an errand down at the gym so I should stay away from there. Oh! And I have something to do at the café! Yessss! Floor one Café! (Music plays Happy Birthday ; sounds of cheering; Barry hands cake over to elderly lady.) (Amy rolls dice and gets a 2) Amy Elevator Amy Barry Awesome! I got 2! That means to can go down to the gym to deliver these 100kg weights! One below ground level the Gymnasium! Ok! No weighting around. Your move Barry! (Barry rolls a 6; sound of clock ticking as Barry thinks) (Thinking) I have to move six floors I have to do an errand to the Delivery Dock but that is only four floors down Wait! (Speaking) I will go down to the very bottom floor and then come back up one level to the Deliveries level! (Sound of clown hooter or spring boing to show a failed option) Elevator Rule 57a, Section 3, paragraph 2 You can only move the elevator in one direction per move. Rebounding is not allowed. Barry But but..then I can only move up from here to floor 5! Amy Yesss! The Spa! And this is where I have to deliver these towels. Barry, you re looking a little stressed! Do you need to get out here for a bit of relaxation therapy? Barry Amy Grrrr! OK, my turn to roll now. (Rolls dice and gets 3) Oh no! I cannot go up any higher! I can only move down 3 floors to The Business Centre! Barry Ha-ha! The Business Centre! This is where I need to take these posters! (Barry s turn; he rolls a 5) Oh bad luck again! I cannot go up so I have to move down 5 floors to Amy The Kitchen! Ah! The sweet taste of victory! (Amy gives bag of sugar to the chef and they celebrate) Manager Barry (offscreen) And Amy is our Employee of the Month! (Sadly) Ah well, life has its ups and downs. There s always next month

Elevator Challenge Rules Name: Student Sheet 2 E levator Ch allenge Rule s Amy and Barry are employees at Hotel resolve. The manager has given them both 6 errands to run on 6 different levels of the hotel. They need to use the elevator to get up and down to complete each task. Amy and Barry will be sharing the elevator. The person who finishes all their jobs first will be Employee of the Month. How to Play 1. Deal out the 12 elevator floor cards. These are your 6 errands. 2. Roll a dice to see who goes first. 3. A counter (the elevator) is placed at Level 0 The Lobby. 4. Player 1 rolls the dice and moves up or down as they choose. Whoever holds the card for that floor can discard it. Job done! 5. Player 2 then rolls the dice and moves the elevator from the level where it is, up or down in either direction. Once again, when the elevator stops the owner of the card for that level can discard it. 6. Continue rolling the dice, taking turns to move the elevator up or down. 7. The winner is the first player to complete all their jobs Employee of the Month! 8. You cannot go above Level 6, below Level - 6 or bounce the elevator off the top or bottom. This means if you are on Level 5 and roll a 3, you must go down. Each journey can only be in one direction. You have to move the total of your dice roll.

Student Sheet 3 Hotel reso lve Hotel resolve Name:

Student S heet 4 Elevator Cards Elevator Cards Print this page and the next back-to-back then cut out each card.

Elevator Cards Print this page and the previous back-to-back then cut out each card. - 5-6 6-2 - 3-4 2 1-1 5 4 3