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Today Math was LOUD and a load of fun. Why wouldn’t it be when you mix the Cartesian Plane and chocolate eggs. I’m not usually one for sweets in class but if my students are bringing them in to share with each other (as predicted by me at this time of year) then I may as well plan to make it into Math.
Constructing Understanding or Surface Learning
I took a quick visit to the curriculum outcomes to see how much I could make out of this Math moment. It is always enjoyable when you look at the wall of words inside curriculum documents with a fresh motivation to see how you could make new engagements with the Math - especially in an interactive way. So that is exactly where we began our Constructing Understanding or Surface Learning, as Hattie identifies this stage of learning, by sharing these learning intentions and success criteria developed from the curriculum.
Learning Intention: Recognise the Cartesian coordinate system using all four quadrants.
Success Criteria:
There is always a flurry of questions when we read the Learning Intentions and Success Criteria. It is one of my favourite parts of the inquiry process. My students know this and don’t hold back. What does number plane mean? Which is the first quadrant and which direction are they ordered? What is the point of origin? What is the difference between axes and axis? What does intervals mean? How do your order coordinates? There ensues an explosion of discussion where students are posing questions to and answering questions for each other. I step into the “background” to observe prior knowledge and monitor the communication, providing acknowledgements and only offering clarification if warranted.
Learning Intention: Recognise the Cartesian coordinate system using all four quadrants.
Success Criteria:
- Identify and label the number plane with the point of origin.
- Identify and label both axes and each axis with intervals of one.
- Recognise the order of coordinates is important when locating points on a number plane.
- Plot coordinates to locate points on the number plane.
There is always a flurry of questions when we read the Learning Intentions and Success Criteria. It is one of my favourite parts of the inquiry process. My students know this and don’t hold back. What does number plane mean? Which is the first quadrant and which direction are they ordered? What is the point of origin? What is the difference between axes and axis? What does intervals mean? How do your order coordinates? There ensues an explosion of discussion where students are posing questions to and answering questions for each other. I step into the “background” to observe prior knowledge and monitor the communication, providing acknowledgements and only offering clarification if warranted.
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We started with a Class Cartesian Construction. I guided the class through a whole group task following these basic steps:
- On the floor using tape, construct a giant class Cartesian plane. We used tape for the each axis and two sided counters (green/red with all green facing up) to mark each point across the plane.
- Label the point of intersection of the two axes as the origin with the coordinates (0,0)
- Identify the horizontal axis (x-axis).
- Label the horizontal axis from left to right with -6 to +6 as intervals of one.
- Identify the vertical axis (y-axis).
- Label the vertical axis from top to bottom with +6 to -6 as intervals of one.
- Using - + die and regular die roll for the first coordinate with the first roll always representing the location along the x-axis.
- Move to the location on the x-axis.
- Roll the dice for the second coordinate representing the location along the y-axis.
- Remaining the location on the x-axis, now move to the location in relation to the y-axis.
- Mark the point (by turning over the green counter to show the red side
- Write the coordinates ( x, y ) We did this on a whiteboard. You can involve every student every turn by providing mini whiteboards.
- Continue for all students to have a turn.
Differentiation with Enabling and Extending Prompts
I was also considering the need to differentiate using prompts to enable and to extend students who needed this. However, first, I wanted to provide all students with a step-it-out-in-order experience and the scaffold of writing all points with either a - or + symbol to avoid any confusion but so as to make it explicit that when there is an absence of any symbol it is presumed positive and therefore not negative. To enable students to explore this concept I could colour code the x-axis green for go and then y-axis red for stop. They could have a “walking” partner. I could have other students predict the quadrant so that this enabled students to know the general direction in which to head. I could also position these students halfway through the whole group task so they could repeatedly observe the solutions of other students before they had a turn. To extend students within this concept and into other concepts I posed questions. How many points on the plane with a range of -6 to +6 for both axis? How can you record these with formal mathematical notation? If we extended the range each time by one in all four axis directions, what would the pattern be? What are the chances of rolling the same point twice? What are the chances of rolling the same coordinate twice?
Transferring Meaning into Symbols or Deep Learning
We revised what we have found out about how the order of coordinates indicate the order of movement to get to the point or location on the number plane. I think we were ready to move into Transferring Meaning into Symbols or Deep Learning, as Hattie identifies this stage of learning, by analysing given points that may appear similar and distinguishing them from each other through order of coordinates on a number plane.
 | This is where the chocolate eggs rolled in. We hopped onto a game called Whose Egg? Students gathered in teams of four. Each member chose red, yellow, green or blue. Using a class Cartesian plane visible on the IWB, I marked with a corresponding coloured dot the first (red), second (blue), third (green) and fourth (yellow) quadrant points (but did not reveal the coordinates) using the same digits such as ( 2, 3 ) and ( 2, -3 ) and ( -2, -3 ) and ( -2, 3 ). |
I called out one of these coordinates randomly generated by two -+ dice. Simultaneously each student in the team calls Hop, if it is not his or her coordinate, or Egg, if it is his or her coordinate. Then the egg is given to the student who correctly, and only if he or she, calls Egg. To great delight and loads of laughter this was repeated around the classroom until all teams had a turn. I changed the points to maintain the challenge. The repetition allowed other students to listen for what they believed to be the correct response. In doing so, they provided immediate peer feedback which was usually in very audible gasps or congratulatory applause or air pumps.
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Applying With Understanding or Transfer Learning
 | Then the stakes got raised or raided as it were. They were ready to move into Applying With Understanding or Transfer Learning, as Hattie identifies this stage of learning, by acting on their learning. We repurposed the same resource to play Raid The Nest! Students formed four teams. Each team chose one colour red, blue, green or yellow. |
Essentially it was the same steps above. After I called out one coordinate shown on the IWB class Cartesian plane, simultaneously each team calls Hop, if it is not their coordinate, or Egg, if it is their coordinate. Any team, or member thereof, that makes a wrong call has their entire nest of eggs raided. We continue gaining eggs and raiding eggs until time is up.
In fact that is just what happened, time was up. One of the best points of feedback I can ever receive as a teacher is that when the bell goes my students will point blank refuse to go out to break. How do you tell students that the “time” for learning has finished? My only recourse is that a healthy snack, fresh air and a run about is essential for more learning.
In fact that is just what happened, time was up. One of the best points of feedback I can ever receive as a teacher is that when the bell goes my students will point blank refuse to go out to break. How do you tell students that the “time” for learning has finished? My only recourse is that a healthy snack, fresh air and a run about is essential for more learning.
To continue with Applying With Understanding or Transfer Learning, I would/will finish with an Easter Egg Hunt (which at this juncture of chocolate overload, would be best to switch and play with counters). This would put the recording into the hands of each student, build on the prior learning and follow these basic steps:
- Work with a partner.
- Collect a A3 piece of grid paper.
- Label the point of intersection of the two axes as the origin with the coordinates (0,0)
- Identify the horizontal axis (x-axis) and label the it from left to right with -6 to +6 as intervals of one.
- Identify the vertical axis (y-axis) and label it from top to bottom with +6 to -6 as intervals of one.
- Place each Easter egg on a point and record the points using ( x, y )
- Using dice roll the first coordinate which represents the location along the x-axis.
- Using dice roll the second coordinate which represents the location along the y-axis.
- Did you find your egg? Yes - keep your egg. No - try again after your partner’s turn.
Differentiation with Enabling and Extending Prompts
Once again I would consider using prompts to enable and to extend students. Although I would like to observe and respond to the individual student’s understanding, knowledge or skill need, starting with a differentiated mindset and a plan to draw from is wise. I could possibly enable students through working together and/or by reducing the required amount where they find one and I find another. I could modify the Cartesian plane to only include two quadrants or a shorter range from -3 to +3. To extend students I could ask: What is the distance between your two points? How would you record this? What are the multiple methods and tools you can use to find this out? How can you use your knowledge of transformations to describe the journey. How can you explain the direction of the second point in relation from the first point? How can you measure or calculate the angle between your two points?
Oh and to top off what was a highly engaging learning experience were the creative puns that students kept cracking!
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