Transcript of What's new – measurement and geometry
Katherin Cartwright Well, good afternoon, everyone. Thanks again for joining us for our Syllabus PLUS K-6 Mathematics Adobe Connect session. Today, we're going to be presenting on what's new for measurement and geometry. This is the fourth session in our five sessions for this year. We'll be running some more next year as well. And just a reminder that these sessions are being recorded so that we can share them with our colleagues that aren't able to be present today, so... stay tuned and hopefully enjoy our session today.
So, as I mentioned, today is about what's new for measurement and geometry, and I'm here with Chris Francis, and we'll be presenting today on parts of the new syllabus around measurement and geometry.
So you might already have an understanding of measurement and geometry or space and geometry, as we had it in our current syllabus, but in this syllabus, they're joined together, and there's a really good reason for that—they have some connecting and interrelationships that are now evident in our new syllabus, so... This is straight from the syllabus, by the way, these quotes we're using today, so measurement, it's about attributes of objects, and we want to be able to identify those so that we can compare and order them, so in length, area, volume, mass. We want to know how the objects are formed so that we can compare them with other objects, and geometry is the study of spatial forms about shapes, size, pattern, position and movement of objects, so you've still got that shape and objects in there, and that's in the three-dimensional world, so you can see that there's a relationship there between measurement, where we're looking at the attributes of objects, and in geometry, seeing how they move and are orientated in the real world, so they do have a connection there.
Again, we wanted to show you the organisation of content image from the syllabus, so you can see there that portion of the concentric circle where it talks about measurement and geometry, and they've been combined together for our new syllabus. And you can see, I've just boxed their angles, so it's technically a new substrand for our syllabus, but we did cover angles as part of two-dimensional space in our current syllabus. They were outcome B, if you remember the outcome coding from our current syllabus, so we haven't lost them and it hasn't gone anywhere. They're just a separate substrand now that sits within measurement and geometry, and you can see there how those measurement and geometry key ideas and substrands develop into Stages 4 and 5. You can see the relationships and where they go.
So some of the important and critical skills, and these, again, are straight from our syllabus, around measurement and geometry, so visualising is really important, being able to describe features. I've got that twice on there. I didn't notice that, Chris. Describing properties, exploring the relationship between 2D and 3D, solving problems around both measurement and geometry, applying number knowledge - that comes in quite a lot as well, particularly in measurement. Recognising different shapes and objects, and manipulating them, seeing what they look like in different ways, and using geometry software, so there's a really strong link with the use of ICT tools and that developing...ICT capability is one of the learning across the curriculum areas in measurement and geometry. And as I mentioned before, they're presented together so that we can recognise and emphasise how they are interrelated.
So just on that, the application of number and geometry within measurement is really important. So measurement involves the application of number and geometry knowledge in practical situations. So these examples below are from measurement, but they link really heavily–one with number and then with geometry - so when you look at the size of the units that we measure in, so centimetres, metres, kilograms, things like that, and the number of units needed, there's a really strong correlation with number sense and developing those sort of strategies, which I'll mention in a moment when we link to the numeracy continuum. There's also a link to geometry within measurement, where we look at length of shapes, the perimeter, we look at the area of shapes, and then we look at the volume of objects. So a lot of these substrands would really work well together in your scope and sequence if you teach them together, even within the same lesson or within a series of lessons, so the students start to see the interrelationship between measurement and geometry.
Purpose and relevance of the substrand. So this is straight from Stage 4. I've mentioned in previous sessions that a lot of Stage 4 substrands have this purpose and relevance of the substrand that sits within the language and the background information, and I find them invaluable. They're excellent information for us as primary school teachers about where these substrands go or these topics develop into, and they make it pretty clear that we want our students to have an understanding and to study two dimensional shapes, three dimensional objects and position, before they move on to the application of angles and the relationships of these properties. So it's really important that they have that sound understanding of each of those individually before they start to apply the relationships and the properties. And then from there, we move to the relationships and properties so they can start to analyse problems. So once you have that knowledge, it's applying it into problem solving and they talk there about deductive reasoning. And where we've got this from, it actually talks then about a few different people's jobs and how they use measurement and geometry, but straightaway, we see that reasoning and problem-solving, they're part of our working mathematically components that are embedded in our syllabus, and so we really want to make sure that we're providing students with opportunities to develop these reasoning skills and problem-solving skills across all strands of our syllabus.
So, what's new for measurement and geometry? Just a reminder that we're not going into absolutely every aspect of the new syllabus - we only have half an hour today - but, again, in this presentation and the file pod at the end of the presentation, I've included my Excel spreadsheet that gives you a very clear breakdown of where content is new or where it's moved from, so if you want more information, please go there at the end of our session today.
So in Early Stage 1, there's a little bit of a change with comparing length indirectly, so we've done a lot of direct comparison. And this does link really closely to our aspect 7 of the numeracy continuum, so I've just chopped a little bit out of the continuum there. And that actually has a hyperlink to it, that section. If you've not been to the numeracy continuum website before, you can click on that link on that picture. So it talks in our continuum about students developing an ability to directly align objects, and then they move into this transitive comparison where they're comparing them indirectly, so having to copy them or not being able to put them on top of one another, so they have to be able to look at them and visualise and compare them that way, so that's sort of been strengthened within our syllabus. There's also a bit of a change in mass. The equal arm balance, that's been moved to Stage 1. So we don't need to work with that. I'm sure if your kids are up to that on their level of learning and along the continuum, then feel free to introduce it, but just to let you know that it's moved into Stage 1, the use of an equal arm balance. We still focus on hefting and things like that in Kindergarten. And also in two dimensional space, there's a focus now on being able to identify and draw closed shapes without tracing, so it's just getting them to explore, "How do I create shapes? What are they?" And then being able to explain, so give some reasons about what closed shapes are.
So our little activity for this, that Chris made, which is fantastic, we just get the students to create a background. They can do a computer-generated background or they can do it on art paper if they like, you know, using some patterns or lines, and then we get them to draw some lines and curves and just sort of make some of those closed shapes by overlapping their lines and curves and discuss and identify, where are the closed shapes in this design? Where are the lines? Where are the curves? You could obviously get the students to use zigzags, straight lines if you want. Chris has used curved lines for this activity. It's up to you. And once they've done that, once they've had a bit of a discussion about, "Well, why do you think that circle in the middle is a closed shape? What tells you it's a closed shape or a shape at all?" That the lines join up to make it a shape.
We then have the students cut the shapes out and use them to create a picture, which Chris has done a beautiful job of, and it marries up into our syllabus there. It talks about the students being able to identify and draw the straight and curved lines, to compare them and describe closed shapes and open lines and to be able to do it without tracing and to then be able to explain it, so there's a whole lot of depth of questioning that you can go into with that little activity, and it really gives them an understanding of how to make shapes, and I think even just that task of cutting them out, that they can see that that's how I make a shape, so, generally, we talk about shapes as being something you draw. Obviously, as you start cutting them out, they sort of become 3D objects, but at this level of understanding for our students, it is good for them to cut them out and see how the shape actually works.
So in the previous session we did, we had a few examples of learning development, if you weren't there, just to explain that - it's not necessarily all within the one substrand of the learning continuum but just shows you how something develops throughout the syllabus, particularly with emphasis from our new syllabus, so this one is actually all from area. But just to show you that that word 'compare' has really been strengthened in this new syllabus. So in Early Stage 1, we're talking about comparing areas directly. And you can see my little pictures there, because there's a new term that some people may be familiar with but others not, so most people understand 'superimposing', which is where you put the shape or the object on top of the other to compare them, and then there's 'superpositioning' of two surfaces, where you actually align it to a corner, like you can see I've done there with the squares, so you're almost using a measurement or a line, an area visual to work out which shape is the larger and which is the smaller. So just for your understanding, that's what 'superpositioning' is referring to. In Stage 1, it talks about comparing - that indirectly comes out there. In Stage 1, in the second part of area, it then leads them into ordering, so not only are they comparing but they're ordering areas. In Stage 2, they have to compare the areas or regular and irregular shapes. That's another strong part of our syllabus now, is looking at irregular shapes as well, so, what's regular and what's not, and comparing objects with familiar metric units. And then into Stage 3, they start to compare areas of triangles and compare the areas of rectangles that either have the same area or perimeter, so it's really developing those skills of reasoning and problem-solving and... you know, answering some of those questions or, "Do all rectangles that have the same perimeter have the same area and vice versa?" And really getting the kids to explore those topics.
So, what's new for measurement and geometry Stage 1? There's a focus on recording, so... I'm not saying we didn't record previously, from our current syllabus, but the wording is there. It now says they need to record using drawings, numerals and words. That comparing comes out strongly in Stage 1. There's also a focus on quarter-hour in time, the increased use of mathematical language, which you'll see in the next slide, and that angles have been moved to Stage 1. Sorry, have been moved to Stage 2 from Stage 1. So we don't deal with angles at all in Stage 1 now. They've been moved to Stage 2. And as I mentioned before, they are now their own substrand.
So going a little bit deeper here, there's just some of the differences between the current syllabus and the new syllabus, so that focus on language there, you can see "Using terms 'flat surface' and 'curved surface'". And I'll go into a little bit more detail about that in a moment. The use of the word 'vertex' instead of 'corner'. So, obviously, you still colloquially use the word 'corner' for the corner of the room, but we're definitely increasing students' understanding of using the word 'vertex' from an earlier age to talk about where two lines meet or where... Oh, sorry, two or more lines meet. They also need to do that comparison between 3D and 2D. Tessellations has moved into Stage 2. There is a little bit of symmetry still there, but tessellations has moved, and flip, slide and turn was what we already dealt with in Stage 1. We still do that but now has been added to full, half and quarter turns of shapes, so not just one movement but possibly two movements, and that idea of quarter-turns and when I mentioned about quarter-hour on the time, there's some really good links you can make there and understanding it obviously links to fractions as well.
So an example of the learning progression from Stage 1 is that slide and flip, the flip, slide and turn - we just thought we'd take that through this sort of same process. So from Stage 1, it talks about just being able to identify the shapes in different orientations, and then it talks about being able to perform a one-step slide or flip with a single shape and that recognising that when we do this, it doesn't change the size of the shape or its features. We do still have students coming through thinking that a square on its point, or on its vertex, is now all of a sudden a diamond, which, for people that know me well, know that 'diamond's a dirty word, and if anything, it would be a rhombus, if it was that sort of shape, but a square on its point, or on its vertex, is still a square. And there's easy ways to explain that to students, around things like show them a triangle and show them a triangle upside down - they generally still see that it's a triangle, so... There's a lot of work around visualising that needs to happen in Stage 1 to build that understanding.
In Stage 2, they now have to show special quadrilaterals in different orientations, and also they have to start talking about that language of translating, reflecting and rotating. Now, that's currently in our syllabus as well, but it's just that it's there with a bit more emphasis because of this half, quarter, full turn that's been introduced as well, so just be mindful of that, and they have to use them to create patterns and designs. In Stage 3, they're still focusing on that translating, reflecting or rotating. And, again, it's that... bringing it back, that understanding that they don't change, that the properties of the shapes don't change when they're moved in that sort of way, so they still have the same interior angles when it's on the side or when you rotate it around. And they also are encouraged to construct patterns using computer software and off computer tasks as well, so... There's that ICT tools link as well. So a lot of work around visualising and what happens to shapes when we move them and what they look like in different orientations.
So a little example here for Stage 1 we've got using the geoboard, from count me In too, so that's a hyperlink. That count me in too wording will take you straight to the count me in too page, and the image there of the geoboard will also take you straight to the geoboard task on count me in too. So, obviously, you might still have the geoboards in your school, with the rubber bands, but if you want to do something that involves the computer, this is a great little activity. The students just can pick up the little rubber bands, and they connect them across two points, and they can stretch them, so for this activity, we thought we might ask the students to create regular and irregular hexagons, so...you know, I talked about the fact that we now focus a lot more on irregular shapes as well, and a lot of students don't think hexagons that aren't regular are hexagons at all, so we need to expose them to what that looks like and that it's just talking about the number of sides, not what it looks like visually when we think about common shapes or regular shapes. So, and just looking at them in different orientations. So you can always make these and then talk about them with the students or have the students then manipulate someone else's drawing to make a similar shape in a different orientation. Again, you could always, you know, do a screen grab of that when the kids are... doing the activity and then print it off and cut them out, move them around, do flips, slides and turns with them. There's a number of things you can do with the geoboard. But it's just nice doing it online instead of having the hundreds of boards with rubber bands in your classroom, so... That's just a little activity that we still recommend from count me in too. If you're not used to the count me in too website, it's a fantastic place to go, and there's plenty of learning objects on there that don't just deal with measurement and geometry but the other strands as well, and it links really clearly to our syllabus as well, this activity about making representations of two-dimensional shapes and being able to draw them in different orientations with and without digital technologies.
So, what's new measurement and geometry for Stage 2? So these are some of the key points for Stage 2. Again, that focus on starting with concrete and moving to abstract, and that's pretty much across the whole of Early Stage 1 to Stage 3, but we just wanted to mention it again. We want you to really focus on building concepts, so teaching length, then area, then volume, in that order, so that they see length, one-dimensional, area, two-dimensional, volume, three-dimensional. They're building on how they develop their understanding. We've also got there that length now includes temperature, so a blast from the past, back from our blue and white syllabus, and I think it's a great opportunity to delve into negative numbers and see how negative numbers are used in the real world, so I don't think that's going to be a problem for most people, teaching temperature again within mathematics, and you'll probably link it to science anyway. We also want to just bring your attention that there's a focus on that progression of strategies - that is from our numeracy continuum. Again, I'm pretty sure that image is hyperlinked to the numeracy continuum for you. And so you can see, and you probably can't read all of that information from the numeracy continuum, but... They start out with using actual objects to align end to end without gaps, and then they start using one object that they mark. They make, mark and move to measure a length, say, and then they start to use the one, maybe, paperclip and move it and then be able to start estimating length and then get into formal units and then comparing formal units, so there's a development, there's a progression of their strategies, and that's really clear, and we want to make sure that people start to use that sort of thinking in their scope and sequence development so that I need to, in my term, or my half a term, that I'm planning, I might want to do length, area and volume in together. OK, more important that they're done together in that order than mass and time, 'cause these ones all relate to each other, and they then have really clear links to both two dimensional shapes and three dimensional objects. There's also an increased focus on that interpret, compare and record. These are words you're going to see a lot in our new syllabus.
So there's just a little bit more drilling down into what was current and what's new, so I've talked about some of those parts already. Cross sections have now been bumped up to Stage 3. We're now using...going back to using isometric dot paper, which I really love using, so I don't have a problem with that one. It's about drawing and interpreting 3D models. And that focus there on irregular shapes that I mentioned before.
So our little activity here is getting sketchy! Some people from the Sydney region, who I know some of them are on today, we gave this task to last week, and they had a good fun time trying to work it out, so this is just from nrich.maths.org. Great website to give you little problems the students can solve, and they'll often have a cheat sheet or an answer sheet, so do not fear if you'd like to know how many other arrangements of those four cubes you can find. But this is a great introduction to getting kids to explore using isometric dot paper or dotty paper, whatever you'd like to call it, and how do I actually represent that three dimensional object in a two dimensional fashion? A lot of our students have trouble drawing things like pyramids or even cubes in a two dimensional way when it's a three dimensional object, so I think that that dot paper actually helps in that sort of area. And this is a great little investigation, so the students just have to actually get the cubes out and make the different models they can and then draw them. And you could even get them to draw them from different orientations if you like and then compare with someone else. So I think that's a great activity. And it links really closely there to our Stage 2 outcome and dot points about drawing objects from different views using the isometric grid paper and interpreting those drawings that they're making.
So I mentioned before, this focus on curved and flat surfaces, so it's something that we needed to clarify, and it has been clarified into our new syllabus, so just so that you know, and this is straight from the language and background information section in our syllabus, is that 'face' refers to a flat surface with straight edges only, so you use the word 'face' when you're describing prisms and pyramids, and prisms includes cubes. OK, when you're talking about cylinders, cones and spheres, they've got curved surfaces. They're not faces. OK, we're going to make that quite clear. So similarly, flat surfaces with a curved boundary, such as the circular face of a cylinder, where there's two of those, or a cone, that has one, they're not faces. You can call them a base, because 'base' is quite inclusive of both curved and straight, but it's a flat surface, is the best way to describe it, and we want to make sure that we're consistent with this message, so that's why it's in our syllabus.
Here's a little diagram to help if you need to. So on one side, you've got your prisms, pyramids, and prisms includes cubes. The other side's cones, cylinders and spheres. It's really important that you don't try and create property lists that include all of these together. OK? Prisms, pyramids and cubes, and particularly prisms, have formula around face, edges and vertices that is a pattern that you can see, doesn't match up with cones, cylinders and spheres. And we don't want students thinking that cones, cylinders and spheres have nothing, that they still have surfaces–they're just not faces. So instead of saying that a sphere has no faces and putting it on, like, a table, we want them to be able to explain that it has a curved surface. OK? So just... You need to keep those separate. We don't really want to encourage people to make up tables of geometric features anyway. We want to investigate it with students and get them to understand what these objects actually look and feel like, instead of just writing out lists of properties, but just so that you know, we're trying to separate them. So you can see there, my prism, it has rectangular faces, and it has two identical parallel bases. OK? They're the wording we use for prisms. The cylinder... OK, you can call those circular bases or you can call them a flat surface. And then it has a curved surface that is around. OK? You can think about that. So the lines that are down the side of my cylinder, they're not actually lines. If you think about what a cylinder looks like, it's a curve. OK? The lines when I look at them on the prism, they're actually the edges. OK? So... And just seeing them in a picture doesn't always clarify it for students, so get out the concrete materials.
So that terminology develops through this example of a learning development. From Early Stage 1, they use those terms 'flat' and 'curved' as general, everyday language to describe these 3D objects. In Stage 1, we then get into that discussion we just had about flat surfaces and curved surfaces, and that face is the flat surface with straight edges, so it's quite clear in there. Into Stage 2, we then look at the curved and flat surfaces, and you can do a bit of a compare, so you can look at cylinders and cones in comparison to the types of features that prisms and pyramids have. And then in Stage 3, we really get into some of those more detailed geometry properties. We're looking at pairs of parallel faces of three dimensional objects, so the language is there, and we want the students to be using the language of... the language of measurement and geometry.
So, what's new for Stage 3? I just mentioned that features and properties there, on the right there, so the feature is something that you can observe, so I can look at a triangle and know that it has straight sides, but a property is something I've gained from knowledge, so I know that a triangle has interior angles that add to 180. That's not something you can just tell from looking at it. So just to notice that it develops from features to properties. And that's quite clear in our syllabus as well. We want them to be able to apply their measurement skills into problem solving and use appropriate units. We wanted them to investigate rotational symmetry, and there's that strong links with number again, so being able to recognise equivalents of whole number and decimal representations of measurement. So really strong link there to fractions and decimals. We want them to use words, so we don't want to teach them algebraic formula. They develop that into Stage 4, and it's from a development and investigative point of view. It's just not learning formula. So we want them to gain an understanding of the concept before they get to that formula for working it out. That focus on regular and irregular shapes. And there's some focus there on angle construction and adjacent angles. We don't have time today to go into adjacent angles, but I've popped in the file pod some more activities from GeoGebra that are used in Stage 4 that Chris Robinson made. It's called the geometer's warehouse. They're excellent little observed investigations around what adjacent angles are, and I quite like them, and I could see how I could use them as a little whole-class activity in my classroom to sort of get their heads around what adjacent angles look like, 'cause you can move the angle lines, so do have a look at those.
So there's just a bit more drilling down for Stage 3. So there's that comparing of areas of triangles and rectangles. There's real focus on that and that visualising and naming 3D objects from their nets, so not just creating nets but being able to visualise and name them from the net, OK, from just looking at it, and that describing, identifying, predicting with that translations and rotations that we mentioned in the example of a learning progression. So an activity for Stage 3-–we got this straight from the 'Teaching measurement Stage 2 and 3' book. Hopefully, people will still use these with the new syllabus. We are looking to update them with the outcome numbers that match the new syllabus. I haven't got a release date for that. But you can still use them. Most schools have them in their school. This is a lesson on length and breadth, and we've attached the lesson for this to the pod as well, the file pod at the end of the presentation. So this explores that idea around, let's make some different rectangles, each with the area of 24 square centimetres. Let's then investigate their perimeter as well. So it's getting them to actually cut out the areas, make them, stick them on. Sometimes if we just show them drawings and say, "Find the area," or, "Find the perimeter," they don't see the difference. So I like that this activity actually gets them to cut it out and it's got the grid paper, they know I'm talking about area, when I'm looking at that 24 square centimetres, and then you can write the perimeter round the outside. So a nice little investigative task. And these activities that have the full lesson plan in this book, in 'Teaching measurement Stage 2 and 3', and in the 'Early Stage 1 and 1' book as well, have really great questioning - it's a real strong focus on working mathematically.
So please have a look at that one as well. So that's it for today. They're our contact details, again, if you'd like to ask any further questions. I've been hearing Chris madly typing today, so there's obviously been a lot going on in the chat. I try not to look over there. So please email us if you have further questions around the new syllabus.
Just to let you know, there's the files down there. There are four files in our file pod today. Just some information about where I got today's music from, if people enjoyed it, and to note that our next session's actually next Monday, so it's very fast, so tomorrow or Friday, I'll be sending out the details, but if you've been coming regularly to our sessions, you'll know that it's the same meeting room every week. And we will be asking some questions on our poll next week about future sessions as well. I know we asked one today, but we might ask a few more specific questions, so... I know that's very brief and there's a lot to talk about with measurement and geometry, particularly around the geometry side of our new syllabus. We are happy to continue the conversation and to help us to gain that shared knowledge and understanding, so thanks, everyone, for coming and for tuning in. We hope it's been helpful. And we'll see you next week. Thank you.