Transcript of implications and considerations for teaching Stage 5 mathematics
Presenter: Today we'll be looking at designing flexible pathways of learning for Stage 5 Mathematics. We'll look at what our syllabus says in regards to Stage 5 Mathematics. We'll be looking at two examples of year nine scope and sequence. One from Randwick Boys, courtesy of Yota, and one from Heathcote high school, courtesy of Bev, and their magnificent Mathematics teams. We'll be looking at ideas, teaching ideas for teaching Stage five Mathematics. I've got some GeoGebra that you'll be able to download later on when you get to the end of the presentation. You can include them in your programmes as well, which really help with some of the concepts in Stage 5 Mathematics.
When we're designing pathways of learning for Stage 5 there's a couple of major considerations. The first thing we'd like to see is that students have access to all the curriculum. This is the idea that students access as many outcomes as they can and teachers programme the course allowing for opportunities to extend students further where outcomes are readily achieved.
We want you to think of the Stage 5 Mathematics as being content organised into three areas. There's a Stage 5.1 area, 5.2 area, and 5.3 area. Now, what you should be thinking about is the flexibility of programming [cube 00:01:19]. What we want to do is build up the knowledge and skill and understanding of certain concepts. You'll be starting sometimes down in Stage Four for certain students and then going back to their prior knowledge and skills, building from there some of the concepts from Stage 5.1 content. For other students you may be starting at Stage 5.1 level and building on to the concepts at 5.2 level, et cetera.
The whole idea is the content is organised in three areas but your programming should be quite flexible and always aim to build onto concepts until kids really do understand what's happening, and you can take it and extend it as far as you possibly can with the students.
Teaching programmes can be designed in so many ways. The syllabus states that there are a variety of end points in Stage five. I think we got used to that from the old syllabus, and this hasn't changed much either in this case. For example, students may achieve all of Stage 5.2 outcomes and then a selection of Stage 5.3 outcomes, et. cetera.
I think the biggest thing with this is looking at your students, understanding where your students are at in terms of Stage level. You may have a 15-year old, but your 15-year may not have achieved all of the Stage four outcomes and still back in Stage 4 somewhere. What you really need to do is to work out where they're at and programme for them so you start from where they're at and start building the concepts on top of there leading into 5.1, 5.2, if you can et cetera and moving through.
Another thing to consider when you're designing pathways of learning for Stage five when you've got so many different programmes going is trying to develop a scoping sequence. Probably the best one I've seen is where similar topics are taught at approximately the same time but where each group access the appropriate outcomes for their level and to the extent that they can actually achieve, which is fantastic. That usually works the best and today we'll look at a few examples.
In our syllabus, page 38, it actually states that the arrangement of content in Stage 5 acknowledges the wide range of achievement of students in Mathematics by the time they've reached the end of year eight. We have three sub-Stages of Maths in Stage 5, Stages 5.1, 5.2, and 5.3. Having a look at Stage 5.1, it really is designed to assist in meeting the needs of students who are continuing to work towards the achievement of Stage four outcomes when they enter year nine. When you're looking at Stage 5.2, it builds on the content of Stage 5.1 and is designed to assist in meeting the needs of students who have achieved Stage 4 outcomes by the end year of eight. Stage 5.3, however, builds on the content of Stage 5.2. It's designed to assist in meeting the needs of students who have achieved Stage 4 outcomes before the end of year eight. They're the kids who readily achieve outcomes and are ready to move on and be extended.
Basically what that means is, students studying some or all of the content of Stage 5.2 also study all of the content of Stage 5.1. Students studying some or all of the content of 5.3 also study all of the content of Stage 5.1 and 5.2 Mathematics.
If I put it in a little bit of a diagram this is what we're looking at. It's not three distinct groups but it's groups of students who may be at Stage 4 level, still achieving Stage 4 outcomes. We build onto those concepts and start accessing Stage 5.1 outcomes and that could be one of your groups just [inaudible 00:05:16]. You could have another group of students where they've readily achieved Stage 4 outcomes and we start building the Stage 5.1 outcomes and concepts and extend into Stage 5.2 content knowledge and skills and outcomes et cetera. They may not readily achieve all the Stage 5.2 outcomes or they may, but at least they have access to that in their programmes.
When you're looking at another group of students their programmes, when they're studying at the Stage 5.3 level, really must address all the 5.1 outcomes, all the 5.2 outcomes and then extend into reaching the Stage 5.3 outcomes and extend into reaching as many as you possibly can with these students keeping in mind that they will need certain outcomes to proceed in Stage 6 to higher level courses.
Some of the major considerations for programming is where are my students now, what do I want my students to learn and how will students get there. Assessment of students' prior knowledge skills and understanding is absolutely vital in your programme so there must be a mention of that. That's something I think teacher's do automatically when they're in the classroom, but it's very important that's there in your programmes as well. Accessing the continuum of learning and having a look at the concept before the one you're trying to teach and making sure students do have, or possess, a certain amount of knowledge in that area before you start building on that concept. Assessing students' prior knowledge is absolutely vital if you're going to be successful with what you're trying to do at Stage 5 level.
What do I want my students to learn? Well, there are basically Stage 5 outcomes, concepts, knowledge, skills and understandings. You know where you want to take kids and you know where you want them to be leading towards and what you want them to achieve. How will your students get there? Well, there are the differentiated teaching and learning activities. What am I going to ask kids to do in the classroom? What does my lesson look like? How am I going to differentiate activities so all students can access the outcomes.
Then again, looking at tasks for gathering evidence of learning, they don't always have to be a pen and paper test. There are many, many ways we can give students tasks for them to show us evidence of their learning, and I think this is really important, especially at Stage 5 level now. You no longer have the school certificate there in front of you. There are so many ways to gather evidence of learning, and I think this is a perfect opportunity, when we're programming for Stage 5, to include some tasks there that are quite different to what you've done before that gives students the opportunity to show what they've learned, to problem solve and to really extend their thinking.
Another consideration, which I think is really important when we're writing new programmes, is student engagement, connecting concepts and ideas to their life. You'll notice that in your syllabus there's an area for relevance and background knowledge. That's just touching on the surface. Take that information and use it. Have a look in your syllabus and go and use your own knowledge. Chat with other teachers and science [inaudible 00:08:24], easy creative arts, and try to get other learning concepts and try to integrate ideas.
If you have an abstract Mathematical concept find the real life application to it and use that in your teaching. I think it is really, really important. Set tasks that will improve student learning and tasks that students will enjoy completing. I think when you take an abstract idea and apply it to a real life example it gives a whole new meaning to students and they really do tend to understand it a lot better. This is really, really important. Get their heads out of their textbooks so there's room for textbook work, there is room for investigations and there is room for real life applications. I think it'd be great for teachers to consider this in their programmes.
Another consideration for the different groups that are going through in Stage 5 is their Stage 6 pathways. If you know that your top group are aiming towards the high level Extension 1 Mathematics make sure that you include in your programmes the optional topic. I think you will see through your syllabus the following icons. If you see the diamond they're basically recommending that students who would like to move on and do general Mathematics at Stage 6 level should ensure that they cover these topics. If you see these squiggly little icon, that's intended for students who would like to continue on in Stage 6 and to learn the Mathematics level.
Then there's the Mathematics Extension 1, that's a hash. If you see a hash next to some of the options topics it means it's optional but recommended that students who do these do study these areas, which to continue to say Mathematics Extension 1. Have a look at these symbols in your syllabus. I think they're very important. They're on page 39, where the actual page definitions are and there's recommendations in the syllabus of what students should be studying at the Stage 5 level in order to be well prepared with foundation concepts to build up on when they get to Stage 6. This is a very important part of your syllabus as well as should be included in your programming so that students do get to continue and finish the option topics for example in year ten who proceed to study Extension One in Stage 6.
Here's our first example of a scope and sequence. This one was done by Randwick Boys, and as you can see you've got the levels here. We've got Stage 5.1, 5.2 and 5.3. They basically have roughly three groups that go through. They've got their tasks at the bottom and it's broken into terms, term one, term two, and then on the other page I'll show you term three and four.
Week one is usually lost with just kids turning up in assemblies, et cetera. They've taken that into account right here. They start by reviewing number, fractions, decimals, integers, percentages, et cetera. As you can see, we step down to some of the Stage 4 outcomes and then start moving on into the Stage 5 outcomes.
As you can see here, the first group for algebraic techniques, for example, will be accessing Stage 4 outcomes first, [inaudible 00:11:46] algebra, and as you can see, that the next group of students that it could be a couple of classes, they'll be learning Stage 5.1 outcomes and accessing 5.2 outcomes. They'll be doing operating with algebraic expressions and then move it on to algebraic fractions, binomial products, quadratic trinomials, et cetera.
The very top group will then move up to algebraic techniques into more sophisticated 5.3 content. They're working Mathematically outcomes, so MA 5.3-1 and accessing 5.3 and 5.2 outcomes for algebra, and they've got their tasks at the bottom. This just keeps your whole faculty moving together. Basically, each class is studying the same area but accessing the outcomes that they can for that group of students.
There's lots of flexibility here. You've got a group that's going to move quite fast and do quite a bit in the three weeks, the next group, move a little bit slower and do a little bit less, and then the next group also move a little bit slower and are accessing Stage 4 outcomes as well as 5.1 outcomes.
This is an example of term three and term four. You can see here what's happening as well, in terms of their learning. It works quite nice. They all get examined at the same time, but they are accessing different outcomes at each level. That's where the differentiation occurs in their scope and sequence. Heathcote High School, and just to mention that I have attached, with permission from the schools, these files at the end of Adobe Professional Learning session so that you can download them. I couldn't put all of the paperwork for Heathcote High, but if you would like it just email me and I will send you any files that you need.
Heathcote High, in their scope and sequence, as you can see, this is term one, we've got our substrains, our strains. We've got our outcomes and then we've got all the Stage 4 ideas listed, the Stage 5.1 and then 5.2 and moving on to 5.3. As teachers are teaching they have everything in front of them and they know, well if my group are accessing Stage 4 outcomes because they didn't achieve them then we start here and move on as far as we can. Start accessing 5.1 outcomes, maybe touch on 5.2, maybe not. The next group of students may be accessing just a couple of Stage 4 pieces of content and then moving straight into Stage 5.1 and proceeding through the 5.2 outcomes.
You can see the pink arrow. That'll be for other classes who are readily achieved Stage 4. They're starting at Stage 5.1, moving on to Stage 5.2 outcomes and continuing to access Stage 5.3 outcomes for as far as they can possibly go. This continues right across, so this is your term two scoping sequence. There's been time allowed for yearly exams, time allowed for when kids will pulled out of class for [inaudible 00:14:54] plan, et cetera. Make sure all that information is included as well in your scoping sequence so you don't lose any time and that everything is quite accurate.
This is term three. This is area and surface area. They do their own two projects for Spreadsheets. There's rates and ratios and moving on to factorization and as you can see, Stage 5.1, 5.2, 5.3 continuing on as far as students can in terms of accessing outcomes. This is their term four scoping sequence. Again, they give a week for revision and a week for the yearly exams, but you can see how they start with linear relationships, from Stage 4, plotting linear relationships, moving on to patterns, down to 5.1 content where we're finding midpoint, gradient, length of an interval, graphing linear relationships. Moving on to 5.2 content where they're applying gradient [inaudible 00:15:49] equation of a line and properties of parallel and perpendicular lines on the [cartesian 00:15:54] plane. That flow of knowledge is great for teachers to see so they can access as many outcomes as they can.
In addition to this, each sequence of topics comes with a spreadsheet that looks like this. Teachers then have the actual content listed. You can see here, these are the classes. There are six maths classes. 9M6, for example, will start accessing the content from here. We're talking about Stage 4 content at this stage. As they progress through they'll then move into the Stage 5.2 content. 9M4, they've roughly start them over here, which is midway through the Stage 4 outcomes, and then they move on and progress into 5.1 and later into 5.2. You can see very clearly that even 9M6 come down here and access a couple of outcomes that they need to do in Stage 5 that are important for them as well even though they won't be accessing other outcomes.
It's quite interesting here, and what they have is this sort of shaded system. It's quite flexible, but at least they've got a bit of an indication of which outcomes the classes will be accessing and moving through towards. Thank you for both schools for lending us their ideas and their work and sharing it with the state.
The next thing I'll be looking at, some teaching ideas. I've taken syllabus outcomes and matched it up with some really nice dynamic geometry software, [applets 00:17:31] ready to go. You just download them, hyperlink them to your programmes, insert them there. Put them in a shared file on your faculty drive, great to use as lesson openers, great for little investigations for students, little dynamic worksheets. We've got heaps more of this to come because we've got two more sessions of applying IT into your programmes, but a great hyper link is all you need and it will be great.
This is properties of geometrical figures for Stage 5. We're looking at level 5.2 and this is what the syllabus actually states. Dynamic geometry software and prepared [applets 00:18:06] are useful tools for investigating the interior and exterior angles sum of polygons allowing students a visual representation of a result. This makes your teaching a lot easier. These are your outcomes down here, working mathematically and then your outcome for geometrical figures. Students or student calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar.
Now, this is the content I'll be looking at, in terms of this certain applet that I've downloaded for you. It's talking about establishing the sum of the exterior angles of any convex polygon. It's 360 degrees and it states explicitly, use dynamic geometry software to investigate that the exterior sum of polygons for different polygons, et cetera, and it puts you in the background information.
This is very important in your syllabus, background/relevance. The concept of the exterior angles sum of a convex polygon may be interpreted as the amount of turning required when completing a circuit of the boundary. I can't make this come alive on Adobe Connect, but if you go and have a play after you download it it really is great. You press play, the car goes around the edge of the polygon and the car, as it travels around the polygon the exterior angles are calculated and they come up here on the worksheet for the students to see. The students are then asked to add up the angles to find and explore that the exterior angles of a polygon is 360 degrees. There are many ways of doing this but this is a very nice lesson opener and great for investigations of the students.
Here's another applet that you'll be able to download at the end as well. I've put the hyperlink here to the GeoGebra site where I've downloaded it from, and they're free for everyone to use and share. This is describe and interpret in sketch cubics and their transformation. This is beautiful because you've got 'a' as your constant is a variable. There's your slider there. You've got your 'k' constant variable and there's your slider and a 'c' constant variable.
At the whole time as you change your slider the equation changes. The basic y=x^3 curve stays in light grey and then your new transformation comes up in green. It comes up really, really nice. That's a great one to add to your programmes and get all your teachers opening it up and sharing with kids and getting them to investigate, and you could have your own set of questions that you build up for them to look at or investigations, et cetera.
Here's a lovely one for when you're doing surface area, especially when we're looking at Stage 5.1. In this applet the cube opens up flat into a net and students can actually visualise all the different sections and the areas that they need to find and add together. The net opens up flat, which is lovely. It's just a little visual but it makes a big impact on student learning when students can see and it's not all abstract, just symbols brackets and number added together. Have a look at this and attach it to your programmes. It's pretty good.
This is a lovely one. If you're trying to teach linear relationships, and I'm going to start right at Stage 4 level and build the concept up to 5.2, et cetera. At the very least level, and now in our new syllabus, plotting coordinates on the Cartesian Plane actually starts at Stage 3. I wouldn't dare teach linear relationships without making sure my kids can plot points regardless of what they've done in Stage 3, but this is a great little applet and again I can't animate it for you on Adobe, but you can have a look at the animation later.
Basically, the battle ship comes up. The students have to select the 'x' ordinate and it comes up here. As soon as they get it right it pops up. Then they have to select the 'y' ordinate using the slider and it will come up as well. Once they get their point they get a really nice ... A big smiley face, congratulations. You're absolutely fantastic and wonderful, but at least you know the kids know how to plot their points and understand what they're doing.
Building on that, we then move to the abstracts. We've seen in real life, kids are having a bit of a play at the 5.1 level we're plotting two points and finding the midpoint of the points. Get the kids to plot the two points and then plot the midpoint visually. Have a look. What would the midpoint be. Look at the interval that you've got and use the process for calculating the mean of the 'x' values and the 'y' values to find the midpoint. Then plot and join the two points to form an interval and form a right angled triangle.
As you're building the concept we join the interval, we've created our right angled triangle, and it explicitly states ... This is what it says in the syllabus. Drop an interval down vertical side from the higher point and a horizontal side from the lower point et cetera. You have your rise, your run, you've got your midpoint. You've got your hypotenuse, make sure you make that quite explicit, and then we get the concept of the gradient being the rise over the run.
Moving from there, we ask students to find the distance between the two points using a range of strategies. One of which is Pythagoras' theorem, the other is using graphing software. You continue in 5.1 where they sketch linear graphs using coordinates of two points and you can do that on paper and you can do it digital technology. You can use GeoGebra. They graph a variety of linear equations. They graph vertical and horizontal lines and also solve problems, very basic ones, involving parallel lines and determining that parallel lines have equal gradients.
Here's a nice little applet that I've included as well. It shows you Pythagoras' theorem and it demonstrates it. The actual square of three here falls into here and the square of four falls into here, and it shows you that these areas combined give you the area of the hypotenuse here. It's a nice little visual and these are the sliders that you slide across and they collapse the areas into each other and then you can see the total area. I know there's a few more out there as well that are really good, but this is just one you can add to your programmes if you don't have one.
Moving on to 5.2 linear relationships, it gets a little more sophisticated. We're talking about the gradient-intercept form to interpret and graph linear relationships. We're graphing lines with equations in the gradient-intercept form. We're recognising that equations of the form actually have an 'x' coefficient as the gradient and the constant or the y-intercept. We've rearranged the equation into general form and then we graph a series of equations using technology find the gradient, find he y-intercept, solve problems including parallel and perpendicular lines.
I found a great app for you, which is really nice. Here, you get to move the gradient on the slider and you can see the line will move with the gradient and you've got 'b' so you get to transform the equation as well shifting it up and down, depending on the y-intercept here, which is 'b'. This is a great lesson opener. You can have a whole heap of investigative questions or you can have a dynamic worksheet where kids are actually doing this at their desks or on a laptop etc. You'll find that at the end of the session as well.
Don't forget our smart notebook. Everybody's got this downloaded on their classroom computers, but in the gallery ... If you go into gallery, select mathematics, patterns and relations, there are interactive multimedia objects. This is one for the slope of a line, ready-made. All you have to do is vary the gradient, vary the slope of the line, vary the constant and you will see this line shift and moving it graphed, and it looks fabulous. It actually calculates the gradient as well, so here's another one that you can add to your programmes and make sure that your other teachers know about it as well.
Moving on to Trigonometry and Pythagoras' theorem, as you can see, we have the little squiggly line here, so, intended Stage 6 pathway is Mathematics level. Remember the hash is Extension 1 and that really does refer to the option topics like polynomials and logarithms and so called geometry, et cetera. Here we're talking about trigonometry, Pythagoras' theorem, 3D trig, applies Pythagoras' theorem, Trigonometric relationships, sine rule, cosine rule, area rule to solve problems including involving 3D dimensional trigonometry.
I'm going to look in particular today at the fact that as you move through here this is very much what you did with the last syllabus in terms of trigonometry but there's this extra bit here, which I think is so important, recognise you've got prove the tangent ratio, can be expressed as ratio of sign and cosine, which we already do. Use the unit circle and digital technologies to investigate sine, cosine and tangent ratios. Compare the features of trigonometric graphs looking at the period and the symmetry. Describe how the value of each trigonometric ratio changes as the angle increases, so angles of any magnitude, and as well recognise that trigonometric functions can be used to model natural or physical phenomena, example tide.
I've got a little applet that does this for you today and it's a really nice one. Investigate, of course, the graph of sine, cosine, tangent functions for angles of any magnitude, including negative angles, and this continues. There's a lot more to that. Here's one of the apps that I've included on your download page. It calculates sine of any angle. It then can calculate cosine. It can show other angles as well. As you move through it shows you the obtuse angle as well as the acute angle. You can see it will go positive, negative, et cetera. It'll help you when you're trying to discuss with kids all stations to central and it goes through and calculates. Kids might start to discover things for themselves when they can play on an applet like this.
In terms of the application for the tides, this is great, it actually spins. You can't see it here, obviously, because we're in Adobe, but as the globe spins the moons spins, the light house is going round and as it approaches the line of the moon on this side you see the water level rise and drop, rise and drop. It is fantastic. It just shows them what's happening at the lighthouse with the tide as well as what's happening in the bigger picture with the globe spinning and the moon and it coming towards the pull of the gravitational force of the moon.
With it comes another app. You can actually see what's happening here. As the moon comes closer you see the sine curve or the waves or the tide get higher. It shows you the relationship between the sine as well as the gravitational force and the pull. Then you see the moon move away and then the sine curve sort of ... The amplitude gets reduced et cetera. I've put a few links there. If you want to get kids to go and research a little bit before they come to class and then you can show them some of the applications. Very interesting, gives it all a little bit of meaning and a little bit of relevance to the students, much more engaging.
Remember again, on smart notebook you have another trigonometric function, Grapha. You can put your sine rule in. You can put the cosine rule in and you can vary your different 'a', 'b', and 'c'. This is great for Stage 6 as well I must say. I've included this applet as well, where kids can actually grasp y cos, tan, sine, cosics, a combination of them or all of them together. You can look at sine next to cos, and this is part of the Stage 5 Mathematics content and really important. If you use it for Stage 6 you can easily graph sine, with cosic, or tan and see the difference in the functions.
Again, with smart notebook, you have a great grapha for parabolas and looking at the 'a', 'b' and 'c' values. It calculates the roots of the quadratic and finds the vertex of the parabola. If you go into Gallery, Mathematics, Patterns and Relations, go into Interactive and Multimedia, you will find the parabola multimedia function there too. Don't forget your exact trig ratios that kids need to learn. Give them a nice diagram, easy enough to do. Get the kids to draw out what the ratios are. If you go to your smart notebook they also have this as well, which is the units circle and degrees, which you can then write on and add to and have your exact trig ratios on the side.
It then moves and puts the exact trig coordinates in, which I thought was interesting and needs to be used in a proper lesson, but they're nice little objects that you can use. Then it moves into radians. If you're teaching Stage 6 it goes into radians and then, again, adds all of them on together. Useful little things to know about and to know that they actually exist. You'll find that under trigonometry in the Mathematics gallery.
I've included this app too, because this is a beautiful one. This is Bart Simpson surfing, and he surfs y=x^3. It's just beautiful. He moves around as a slider. You can talk about the zero gradient, the positive gradient, again, another positive gradient here. You can extend it in two Stage 6 concepts as well, but just the basics of what gradient looks like and having Bart there surfing the curves is quite nice for kids to see.
I've put here a wall of assessment ideas. This was put together by a series of consultants in our office and it's got some great stuff in it, so ideas for you to think about varying types of assessment and ways to gather evidence.
There's wall posts for discussions that questions about a problem that they're trying to solve, presentations, think of peer share activities, mind mapping, modelling, problem-based learning, brain storming, assessment portfolios are fantastic. Visual creative assessments, student self assessments, peer assessments, just a little reminder of what's out there and what you can do Problem solving journal, which is a fabulous idea to collect evidence of learning and take away the anxiety from a pen and paper test and get them to really solve problems properly whether by themselves, in pairs, in groups, encounter some great stuff. I thought I'd put that up for you, and it's all got links so it takes you to other websites to help you explore those areas further. That's all we've got time for today. Thank you for joining us.