Transcript of Stage 5 measurement
Speaker 1: Okay. Today's session, we'll be looking at stage 5 mathematics for the Australian curriculum. We'll be focusing on the substrand measurement, in particular numbers of any magnitude include investigating measurements for digital storage, prefixes such as nano, tera, giga, and related prefixes, their meaning and application. We'll also be looking at number and algebra, nonlinear relationships in particular.
Before I get started, I just wanted to recap what stage five programme of study we actually look at in terms of the Australian curriculum. Very similar to what we've done on the past, in stage 5 we have flexible pathways. We have stage 5.1, stage 5.2, stage 5.3, or a combination of two or three of these. Basically stage 5.1 is designed to assist in meeting the needs of students who are continuing to work toward the achievement in stage 4 outcomes when then enter U9. Stage 5.2 builds on the content of stage 5.1 and is designed to assist in meeting the needs of students who have achieved stage four outcomes generally by the year of year eight. Stage 5.3 builds on the content of stage 5.2 and is designed to assist in meeting the needs of students who have achieved stage 4 outcomes before the end of year eight.
Now, most of you are quite experienced in developing stage 5 programmes and looking at the pathways that you want students to take within your classes. Just keep in mind that students study some or all of the content of stage 5.2 also study all of the content of stage 5.1. Students study some or all of the content of stage 5.3 also study all of the content of stage 5.1 and 5.2 and content written in different substages within stage 5 maybe studied continuously. For example, linear relationships, stage 5.1 content followed by 5.2 content, and then you can continue with 5.3 all in a row.
There's a variety of endpoints in stage 5 that are possible so some students may achieve all of stage 5.2 outcomes and some of stage 5.3 by the end of year 10 while other students may complete all of 5.2 and all of 5.3. Just keep in mind when you're planning the kids' learning experiences, consider the stage 6 pathways of studies, what they're intended to do and how many outcomes you need to access in terms to prepare students for their stage 6 pathways.
Okay. Numbers of any magnitude, the first bit. There's quite a bit of new content in here. In stage 5.1 the outcome interprets very small and very large units of measurement. It uses scientific notation and rounds to significant figures. In this case, rounding, approximation, level of accuracy, significant figures is very similar to what you've already done. Truncating or rounding during calculations has on the accuracy of results is another thing you need to investigate. Interpret the meaning of common prefixes such as milli, centi, and kilo. The new bit is in the last two dot points. Interpret the meaning of prefixes for very small and very large units measurement such as nano, micro, mega, giga, and tera. Record measurements of digital information using correct abbreviations, example kilobytes. As you can see, this content now is great in terms of continuum of learning when we go to stage 6 and enter into general mathematics and you can see there's a lead way into there.
Numbers of any magnitude, what are we talking about here? When we're investigating very small or very large numbers, we're looking at rounding, approximation, levels of accuracy, significant figures, interpreting meaning of prefixes like we always have, but we're adding a few more now, nano, micro, giga, and tera. Record measurements of digital information using a abbreviations, example kilobytes. You can have different notations for kilobytes. As you can see listed here, 40kB or 40K or 40k with a small k, or 40KB with two capitals, all mean the same measurement. Investigating the use of the abbreviations and different notations is quite important here.
The next new bit, converting between units of digital measurement. Converting between gigabytes and terabytes, converting between megabytes and kilobytes are all new information and new content we need to teach in stage 5. Describing limits of accuracy of measurement using instruments so +/- 0.5 units of measurement for your level of accuracy. Explain why measurements are never exact. Recognising the importance of significant figures. Terms of expressing numbers in scientific notation. Recognise the need for scientific notation for certain numbers. Explain the difference between 2 to the power of 4 and 2 times 10 to the power of 4. Students often get mixed up with this. Use index laws, order numbers, and solve problems.
We're looking at the new prefixes now. You've got pico, nano, micro, milli, kilo, mega, giga, tera. These are their values as you can see in the table above. You'll need to go through this with the students, explain it, make sure they understand it. One of the more sort of engaging tasks is actually to get kids to go out and research on the internet and find where these prefixes are used and for what purpose.
Multiples of bytes. We've got kilobyte, megabyte, gigabyte, terabyte. Now, our syllabus goes up to terabyte. If you wish to extend the kids further, you can go into the blue rows, but basically, for the Australian curriculum, [inaudible 00:06:15], we go up to terabyte in terms of multiples of byte. The value is there and it's always interesting to understand the binary value as well.
When you're getting kids to actually research these tasks and have a look at the prefixes and where they're used, it's great for them to understand what the prefixes mean to us. One of the greatest applications is actually with some of the Apple iPods that are on sale at the moment in the stores and kids are very familiar with them. The iPod Nano has 16 GB, which is 16 billion bytes. The iPod shuffle, 2 GB, which is 2 billion bytes, iPod classic, 160 GB, iPod Touch comes in 32 GB and 64 GB. Kids can even look at their phones, look at the capacity of storage on their phones, and talk about that and have discussions. It's quite an interesting topic for students and I think they'll relate greatly with that, even if they're just looking at their USB and how much their USB holds. That would be a really great example as well. You've got a few things you can look at here.
Keep in mind critical and creative thinking and how students develop critical and creative thinking. This is a recap from one of our other Adobe sessions. Just go through it one more time before we go into some more of the maps. As they develop critical and creative thinking, students learn to pose insightful and purposeful questions. Now, remember, critical and creative thinking are two different ways of thinking. You combine them together and you've got four strong interactive elements here. Students apply logic and strategies to uncover meaning and make reasoned judgements. Think beyond the immediate situation to consider the big picture before focusing on the detail. They suspend judgement about a situation to consider alternative pathways. Reflect on thinking, actions, and processes. Generate and develop ideas and possibilities. Analyse information logically and make reasoned judgments. Evaluate ideas, create solutions, and draw conclusions. Assess the feasibility, possible risks and benefits in implementing of their ideas. Transfer their knowledge to new situations. You'll find that critical and creative thinking is going to come up a lot in this topic and the next one that we'll be looking at.
I'm just having a look here. These are the key ideas. These are the key ideas for number in algebra. As you can see here, I'm going to look right across non-linear relationships today. We had a look at number in algebra in stage 4 in the previous sessions. Now we're going to look at stage 5. Now, for non-linear relationships in stage 5.1, students are expected to graph simple parabolas, exponentials, and circles on the Cartesian plane using tables of values and digital technologies. How is it different from our current syllabus? Well, circles is new at stage 5.1 they are. At stage 5.2, key ideas for non-linear relationships. Students identify, draw, and compare graphs of parabolas of the form axe squared + c. They identify, graph, and compare exponential curves and circles. In this case, at stage 5.2, your new content is exponentials and circles.
At stage 5.3 for non-linear relationships, we have draw, interpret, and compare graphs of parabolas with the form axe squared + bx + c using a variety of techniques. Determine the equation of the axis of symmetry, the coordinates of the vertex of the parabola, and draw and interpret and compare graphs of hyperbolas, and there is your new content there, exponentials, circles, and simple cubic functions. Below, polynomials, logarithms, functions, and other graphs, and circle geometry are all the optional topics and they're set up for students intending to study extension one maths and really should be considered and programed into teaching programmes for students wanting to access the stage 6 pathway. It's very, very important that you have time to go through these content knowledge and skills with the students. We're going to focus on non-linear relationships in stage 5 now.
Currently, what are we doing in stage 5.1 for non-linear relationships? We're asking students to graph non-linear relationships such as y = x squared, y = x squared + 2, y = 2 to power of x, and determine whether a point lies on a line by substituting into the equation of the line.
What does the new syllabus ask us to do? Something a little bit different, just slightly different. We graph simple non-linear relationships with and without the use of digital technologies, so complete tables with values to graph simple ones, so y = x squared, y = x squared + 2, y = 2 to the power of x. Then compare these with graphs drawn using digital technology.
Next, explore the connexion between algebraic and graphical representations of relations such as simple quadratics, circles, and exponentials using digital technologies as appropriate. Using digital technologies, and we're talking about, for example GeoGebra. We want to graph simple quadratics, exponentials, and circles. For example, y = x squared, y = -x squared, y = x squared + 1, x squared -1, 2 to the power of x, 3 to the power of x, y = 4 to the power of x. We want those exponentials graphed using digital technology, and of course, the circle, x squared + y squared = 1, x squared + y squared = 9 or 4. Now we want students to engage in critical and creative thinking when they're describing and comparing a variety of simple non-linear relationships. Connecting the shape of a non-linear graph with the distinguishing features of its equation. We want students to be able to do all that using technology. That's taking them up a notch from what they were doing currently at stage 5.1.
We're talking about graphing simple non-linear equations using tables of values and compare to graphs used using technology, so your simple parabola y = x squared, raising your parabola one unit so y = x squared plus 1, exponentials y = 2 to the power of x. Very similar to what you can see on the board here now.
Graph simple non-linear relationships using digital technologies. I want to connect the shape with the distinguishing features of its equation y = x squared, y = -x squared, y = 1 + x squared, y = -x squared - 2, y = x squared + 2, etc. Making the connexion between the shape and the features of the graph.
Again, using technology, graphing very simple non-linear relationships at stage 5.1, so y = 2 to the power of x, 3 to the power of x, 4 to the power of x, and I suppose promoting discussions about the shape of the graph in relation to the actual equation in front of us is really important. I know a lot of you have other mathematical software, graphing software, that you like to use. It doesn't matter which one you use as long as you promote the discussions and the kids can see clearly what's happening in terms of the change of the equation and the shape of the graph.
Okay, graphing again circles using digital technologies, connecting the shape with the distinguishing features of the equation. Circle centre, the radius, and making those connexions, I think, is really important.
Stage 5.2, currently what are we doing? We draw and interpret graphs including simple parabolas and hyperbolas. In the new New Southwest syllabus, the new bit is the exponentials and we're asking students to graph simple, non-linear relationships with and without the use of digital technologies and solve simple related equations. So, we want them to graph parabolic relationships of the form y = axe squared, y = axe squared + c with and without technology. Identify the shapes in the environment, so where do they see parabolas in the environment in the world around us. Describe the effect on the graph of y = x squared, of multiplying x squared by different numbers including negative numbers, or of adding different numbers including negative numbers to x squared. So, y = x squared -2, y = -x squared + 3, etc. Determine equation of a parabola given a graph of the parabola with the main features clearly indicated.
Determine the x coordinate of a point on the parabola given the y coordinate of the point, and sketch, compare and describe with and without the use of digital technologies, the key features of simple exponential curves. Example, sketch and describe the similarities and differences of the following family of graphs: y = 2 to the power of x, y = -2 to the power of x, y = 2 to the power of -x, y = -2 to the power of -x, y = 2 to the power of x + 1, etc, so, describing similarities and differences. We've got a bit of literacy happening here and a nice explanation would be good even if the kids draw a table and list the similarities and differences, that would be fantastic there. Describe exponentials in terms of what happens to the y values as the x value becomes very large or very small and the y value for x = 0 is a good one to discuss as well.
Where do we find parabolic shapes in the world around us and in our environment? Get the students to find images and share them with the class. I think that's the best way. Here's one. A bouncing ball captured with a flash at 25 images per second. Some of you may have seen this before. Parabolas are found all over the world. You've got in the USA, the St. Louie Gateway Arch and the Millennium Bridge in London, or right here close to home in Hyde Park. If you look at the water stream which follows a parabola as well for its path of motion. You've got Olympic Park Fountain, again following the streams of water. It follows the path of a parabola.
Graphing parabolas with or without technology. Again, any graphing software you'd like, but GeoGebra is really simple to use and all the kids have it on their DR laptops. Graphing y = 2x squared, y = 4x squared, half x squared. Go to negative numbers. Negative 2x squared, etc and have that discussion with the kids explaining what the graph looks like in relation to the equation, what the equation looks like in relation to the graph, etc.
Negative values when the coefficient of x squared is negative and how that looks. Y = -2x squared. Y = -4x squared. Negative a half x squared, etc.
Then we're looking at the parabola y = x squared + c, so y = x squared + 2, y = x squared + 1, and looking at that family of parabolas and how the curve changes with the equation.
There are so many things you can do. Get them to investigate the shape of graphs y = axe squared and y = axe squared + c for positive and negative values of a. Class discussions. GeoGebra investigations. Have graphs. Get the kids to match the graphs to the equations and the equations to the graphs. I think they'll find that quite engaging and really help them connect the concepts together.
At stage 5.2 the other new area is circles. Students are going to use Pythagoras' theorem to establish the equation of a circle with the centre, the origin, and the radius. Recognise and describe equations that represent circles with the centre of the origin. Sketch circles of the form x squared + y squared = r squared. You do not have to move where the circle centre is out of the origin at this stage. Keep it at the origin and just work with your basic equations. Explore the connexion between algebraic and graphical representations of relationships such as simple quadratic circles and exponentials using digital technologies as appropriate. Identify graphs and equations and straight lines, parabolas, circles, and exponentials. Match the graphs of the lines, parabolas, circles and exponentials to the appropriate equation. Sort and classify different types of graphs. You can have a whole hip of graphs all mixed up and the kids have to sort them out and then match the graph to an equation and justify each of their choices. It would be great to whip up something there for them that they can use in terms of this outcome.
Matching graphs of exponentials to equations. Very easy. Open up any graphing file. Throw in your equation and you'll generate a beautiful graph. Arrange them on a slide show or however else you want to do it. Put them on a worksheet. Put them on the [inaudible 00:19:51]. Get the equations there and get the kids matching and it should work well.
Again, matching graphs of non-linear relationships, so a nice selection there from all the different families of graphs, non-linear relationships, and get the kids to match them with the graphs as well. You can see that they'll be well prepared for stage 6 if they're delving into all these equations at this level.
Now, the other thing that I found in the syllabus that is at stage 5.2, I'll just read it here. It says that this substrand provides opportunities for mathematical modelling. For example, y = 1.2 to the power of x where x is greater than or equal to 0 models the growth of a quantity beginning at 1 and increasing 20% for each unit increase in x. You can even go toward looking at a bit of the modelling as well at this level for an application. There's a fair bit for you to do here in terms of the content knowledge and skills for the students, but it's good to give that real life application as well with the mathematical modelling of the exponentials.
Currently at stage 5.3, we have draws and interprets a variety of graphs including parabolas, cubics, exponentials, and circles, and applies coordinate geometry techniques to solve problems. What's new now with our new [inaudible 00:21:17] syllabus for the Australian curriculum is that we have now added the hyperbola in this case. If we have a [inaudible 00:21:27], we've got describes, interprets, and sketches parabolas, hyperbolas, circles, and exponential functions and their transformation. Find x and y intercepts where appropriate for the graph of y = axe squared + bx + c, like we've always done. Graph a variety of parabolas including where the equation is given in the form of y = axe squared + bx + c for various values of a, b, and c. Then use digital technologies to investigate and describe features of the graphs of parabolas given in t following forms for both positive and negative values of a and k, so axe squared, y = axe squared + k, y = (x + a) squared, and y = (x + a) squared + k. Describe features of a parabola by examining its equation.
Again, getting out your graphing software here for the students, and whether you're giving them these to do for homework, in class the discussion needs to be generated. They need to explain what's happening and describe the features of the graph and the changes in the graph in relation to the equation. Use digital technologies to investigate and describe features of the graphs of the parabolas y = axe squared + k, so y = x squared, y = x squared -3, x squared + 3, y = x squared + 2, or y = x squared -2. There's a fair bit that you could do there with lots of examples.
Again, using digital technologies investigate and describe features of the graphs of the parabola y = (x + a) squared and have them look there at how the graph touches the x axis and the double root there. Describe features of each graph by examining the equation and having that discussion with the students and relating the features to the equation is really good.
Use digital technologies to investigate and describe features of the graphs of the parabola y = (x + a) squared + k for both positive and negative values of a and k. Again making the connexion between the equation and the features of the graph and seeing if the kids can actually match the graphs to the equations [inaudible 00:23:41] is really important.
This goes into parabola as well. Determine the equation of the axis of symmetry. This is not new information. Using x = -b on 2a. Find the coordinates of the vertex. Identify features of the parabolas and their equations to assist in sketching quadratic relationships, x and y intercepts, vertex, axis of symmetry, and concavity.
What is new is this next bit. You've got determine quadratic expressions to describe particular number patters, etc. and then, here it comes, graph hyperbolic relationships of the form y = k on x for integer values of k. Describe the effect on the graph of y = 1 on x of multiplying 1 on x by different constants. You can see there's critical and creative thinking cogs there. Can you see those two little cogs at the end of the word communicating? Explain what happens to the y values of the points on the parabola y = k on x as x values become very large or close to 0. Explain why it may be useful to choose both small and large numbers when constructing a table of values for a hyperbole. You want that communication and the reasoning to come out there. It's very, very important.
It moves on further. Finding asymptotes is new to this level. Graph a variety of hyperbolic curves including where the equation is given in the form y = k on x + c or y = k over x -b for integer values of x, b, and c. Determine the equation of the asymptotes of a hyperbole in this form. Identify features of hyperbolas from their equations to assist in sketching the graphs, and identifying asymptotes orientation, x and y intercepts, and where they exist. This is really important. Describe how hyperbolas in terms of what happens to the y values of the points on the hyperbola as x becomes very large or very small and whether there is a y value for every x value, and what occurs near or at x = 0. A lot of discussion here. There should be a lot of graphs being drawn. Kids communicating their ideas. Identifying features of the equation with features of the graph, and going from there. There's quite a bit of critical and creative thinking, a lot of use of technology, as you can see, and a lot of investigations by students.
Moving into the cubic, students have to describe, interpret, and sketch cubics, other curves, and their transformation. We're looking at y = axe cubed first and then graphing and comparing that to features of y = axe cubed + d or y = (axe -r)(x-s)(x-t), so you've got the polynomial happening. Then describing the effect on the graph of different values of a, d, r, s, and t. Graph a variety of equations of the form y = axe n for n and an integer. Now n is greater than or equal to 2 and describing the effect of n being odd or even on the shape of the curve. Graph curves of the form y = axe n + k from curves of the form y = axe to the nth for n an integer, again greater than or equal to 2, but this time using vertical transformations and then using horizontal transformations for y = a (x-r) to the power of n, and again n has to be greater than or equal to 2. There's quite a fair bit to do here, a little bit more prescriptive than before.
When graphing and comparing features of graphs as you can see above, you've got critical and creative thinking happening. You've got literacy because you want kids to communicate reason and actually explain why my graph looks the way it is when my equation is as follows, etc., so connecting the features of the equation of the graph to what we're actually seeing here, which was created actually using GeoGebra. So, something like y = x cubed, y = x cubed + 2, y = x(x + 1)(x-2). Having a look at where the graph cuts the x axis. Looking at the turning points. Describing the shape. A fair bit of discussion that can evolve from all of this.
Explain what's happening to the graph as the values a, r, s, and t change in the equation y = a(x-r)(x-s)(x-t). So having that discussion, maybe graph separately, and then graph them together on the one number plane so they can actually see the difference and what happens when a changes, etc.
Graph a variet of equations of the form y = axe to the power of n. Again, for n being odd or n being even. I've graphed y = x squared, y = x to the power of 4, y = x to the power of 6, comparing it to x cubed or x to the power of 5 and you can see the distinct difference in the graphs and relate that to the power, whether it be odd or even for the equation y = axe to the power of n.
Okay, so that's all for our session for today. 4:00 comes on really fast, but I hope that has helped and gives you a little bit more insight into what the syllabus is asking us to do and what we need to teach our kids at stage 5 in measurement and terms of new content and in number and algebra in terms of new content. I hope you do find that useful. Next week we'll be looking at bivariate data analysis in stage 5 including other areas as well as statistics that are new. I thank you for joining us today. You've been absolutely fantastic and I hope you enjoyed it. Bye for now.