Transcript of measurement and geometry
Speaker 1: Welcome, everybody, to session three of SyllabusPLUS series three. Today we'll be looking at measurement and geometry. I'll start off the presentation and then Chris will take over halfway to focus on the geometry area.
Today we'd like to start by looking at the continuum of key ideas. We'll be looking at length, area, volume, and time in terms of measurement. I'll start by looking at the Stage 3 concepts first, and then move into the Stage 4 and Stage 5 concepts. As you can see here in stage 3 for length, students start by looking at the very early concepts, which is using the kilometre to measure lengths and distances, selecting and using appropriate instruments and units to measure lengths, and record lengths and distances using the abbreviations for kilometre, metre, centimetre, millimetre, as well as finding perimeters of common two-dimensional shapes, and recording the strategies they use.
For part 2, the later part of the concepts, we have students recording lengths and distances using decimal notation to three decimal places, and converting between kilometres to metres to centimetres and millimetres. Students are involved and problem-solving, especially with length and perimeter measurements, and in Stage 4 we then have students moving on to find perimeters of two-dimensional shapes. We want them to establish and use formulas to find the circumference of circles. We'd like students to be able to find arc lengths and the perimeters of quadrants, semicircles, and sectors, followed by problem-solving involving questions that involve both perimeter and circumference.
One thing I'd like you to think about while we go through this session today is how often you actually plan hands-on and practical activities for measurement and geometry, and think about great opportunities for you to actually plan these hands-on lessons. I think in primary school students get to do it quite a bit. As soon as they move to year seven it starts to sort of get cut down a little bit. Please think about the opportunities you have, especially for the measurement chapter, to really include hands-on and practical activities in the work that you plan for students to complete.
Okay, area. Part 1 of area. In Stage 3 students recognise the need for squares, kilometres, and hectares to measure area. They use abbreviations such as kilometres squared, and ha for hectare. They develop strategies to find areas of rectangles and record strategies in their own words.
Part 2 of the area concept, students are expected to develops a strategy to find areas of triangle and record the strategy in words, and then solve problems involving both rectangles, squares, and triangles. As you move on to Stage 4, students are expected to convert between metric units of area, establish and use formulas to find the areas of triangles, special quadrilaterals and circles, and solve problems involving area.
This concept later extends itself to surface area in Stage 5.1, 2, and 3. In 5.1, with solving problems involving areas of composite shapes by dissection into triangles, quadrilaterals, quadrants, semi-circles, and sectors, and solve problems involving the surface area of rectangular and triangular prisms. At level 5.2 we've actually got students solving surface area problems involving right prisms, cylinders, and composite solids, and at stage 5.3 students are solving problems involving surface area of right pyramids, right cones, spheres, and again, related composite solids.
For volume, in Stage 3 students use cubic centimetres and cubic metres to measure and estimate volumes, they select and use appropriate units to measure volume, they record the volumes using the abbreviations centimetre cubed and metre cubed. In Part 2 of the conceptual development, students then connect volume and capacity and their units of measurement. They record volumes and capacities using decimal notations to three decimal places, and convert between millilitres and litres. Also develop a strategy to find volumes of rectangular prisms, and record the strategy in their own words.
From there we connect to the Stage 4 key ideas. For volume we want students to visualise and draw different views of three-dimensional objects, we'd like them to convert between metric units of volume and capacity quite fluently, we want them to establish and use formulas to find volumes of right prisms and cylinders, and solve problems involving volume and capacity. Moving into Stage 5, volume reappears in Stage 5.2, where students solve problems involving volume and capacity of prisms, cylinders, and related composite solids, and for 5.3 pathway, students are solving problems involving volumes involving volume of right pyramids, right cones, spheres, and related composite solids.
It's very clear that when you're teaching volume and capacity, it is really, really important to demonstrate for students exactly what volume and capacity and the links are between them. So in the class, just like you would if you were a science teacher, get out in the front, bring out your fluids, bring out your liquids, in different containers, ask the questions for students to answer, and get them to really investigate the concepts of volume and solids and capacity. I think it's really, really important to incorporate the practical lessons at this point.
Moving on, the last sort of idea in measurement is time, and numbers of any magnitude. At Stage 3 level, students are converting between 12 and 24 hour time. They determine and compare the duration of events. In Part 2 of the concept they then interpret and use timetables and draw an interpret timelines given a scale. At Stage 4 level we've got students performing operations with time units mentally and with a calculator, and of course solving problems involving the interpretation of the international time zones.
Once you get to stage 5.1 it changes a little bit. We're not talking about time in particular but it's numbers of any magnitude. We want students rounding numbers to a specified number of significant figures, we want students converting between metric units of very large and very small measurements, including measurements of digital information. We also have students expressing numbers in scientific notation using both positive and negative powers of 10.
This image hasn't come up, but this is basically a list of the types of shapes that students deal with. With perimeters we want students finding the perimeters of parallelograms, trapezium, rhombuses, kites, simple composite figures, circumference of the circle, quadrants, semicircles, arc lengths, and perimeter of sectors. In terms of area, in Stage 4 and 5 students work with rectangles, squares, triangles, parallelograms, rhombus, kites, circle, quadrants, semicircles, and sectors.
One thing that I think's really important is to ensure that students have connected the relationship between length and area, and then move on to volume concepts from there. Making that connexion is really, really important. Giving the students an opportunity to really investigate, and communicate their mathematical ideas is absolutely vital.
Here are some teaching ideas form the 2013 NAPLAN strategies. Have a look at this activity here. "How many different rectangles can you make with a perimeter of 24 centimetres?" Having the grid paper there, having the kids start to draw as many shapes as they can. "Can you make different squares with perimeter of 24 centimetres?" You could then move to blank paper and have kids just measuring it out with their rulers, and actively finding as many squares as possible with a perimeter of 24. Then thinking about, "Can you change one of the rectangles to make a different polygon with the same perimeter." Taking it as far as you can to actually get them to think about the relationship between length and area.
Another nice activity you can do with students is something like this, where you have different shapes on grid paper and have students calculate the perimeter and area of each of the shapes. You can have it also done on blank paper with dimensions, and have students investigate the area and perimeter of the shapes by answering questions such as, "Can we remove any squares and keep the perimeter the same? Why?" "Can we remove any squares and keep the area the same?" "Can we move any squares and keep the perimeter the same?" There's quite a bit they can do there and think about and communicate. They can do it in pairs and then share their ideas with the whole group at the end.
That then leads us moving on to the typical sort of questions we give kids in Stage 4, finding composite areas, but getting them to physically divide the shape into rectangles and triangles, and then add the areas together. Follows on beautifully from there.
In our syllabus we basically have quite a bit of information to help support the teaching of area. We need to establish the formulas for areas of rectangles, triangles, and parallelograms, and use them in problem-solving. You can see here that explaining the relationship that multiplying, dividing, squaring, and factoring have the areas of rectangles and squares of integer side lengths is one of the activities that they want students to engage in, especially the communicating component of working mathematically. Explaining the relationship between the formula for the area of rectangles and squares. Comparing areas of rectangles with the same perimeter, and developing with and without the use of digital technologies, and use the formula to find the area of parallelogram and triangles, including triangles for which the perpendicular height needs to be shown outside the shape. This is really important.
Move us on to finding the area of simple composite figures by dissecting into rectangle, square, parallelogram, triangle, finding areas of trapezium, rhombus, and kites, with and without the use of digital technologies, as long as the kids have the x and y lengths of the diagonals. Develop and use the formula for finding the area of a trapezium where the height is the perpendicular height and A and B are the lengths of the parallel sides. Then of course showing them the trapeziums in different orientations, as long as they can identify the particular height first of the trapezium, and work from there. From there basically solving a variety of practical problems is really important, relating to the area of triangles, quadrilaterals, trapeziums, et cetera. And of course converting between metric units of length and area when solving area problems is really important.
These are the types of formulas that students need to develop with you on the board, and understand and memorise for later on. Moving from there, the syllabus asks us then to look at the circle, and it always starts really well if you start with the circle features. The radius, circle centre, the diameter, semicircle. I would introduce the tangent and the chord at this point as well, get the kids familiar with the terminology. The arc, the segment, the sector, and of course circumference of the circle is very important to go through here.
Moving from there we look at the formula, C = 2 pi r, for the circumference, or pi times the diameter, and the area of the circle as well, pi r squared. Talking about pi and using pi as a fraction, as a decimal, and as a calculated value is really important, and you can see here in the syllabus content, it asks you to investigate the relationship between features of circles such as the area and the radius, and use formulas to solve problems. Find the radii of circles given the circumference, find the area of quadrants, semicircles, and sectors, and solve a variety of practical problems involving circles and parts of circles, giving an exact answer in terms of pi, as well as an approximate answer using a calculator approximation of pi. You need to to give both problems consideration when you're planning tasks for students.
With the circle it's great to discuss the relationship between the diameter and the radius. You know if the radius of a circle is 16 centimetres what's the diameter. If the diameter of a circle is 15 metres, what is the radius? Use examples when you answer as a whole number as well as decimals, and model for students how to calculate the area of a sector, I think is really important. If the area of a circles is 24 centimetres squared, what is the area of a sector? We'll choose an angle size of 90 degrees.
Again with the circle, get students in pairs solving familiar problems, and similar problems which you've already modelled for them. Model how to calculate the length of an arc, I think's important. If the circumference of a circle is 60 centimetres what is the arc length of a sector which has an angle size of 150? Providing as many examples as possible, I think, is important in getting students to actually draw the sector on each circle, measure and label the angle. Then show the working out to find the solution of the problems, I think is really important, whether they do it in pairs or by themselves. Both are quite effective.
Moving on to volume now, these are the type of images that students see in primary school. I don't think they see it as much in high school, but they should. Presenting volume to them using actual physical concrete materials is absolutely vital for them to understand and have that visualisation in their minds. Talking about what the volume of an object actually is, and then leading that into the fact that objects with the same volume may have a different shape. So cubes stacked three in a row, and then two, and one, can have the same volume of another shape with the cubes stacked in a different orientation with the same number of cubes.
Moving on from there, looking at capacity, so the amount a container can actually hold in terms of fluid or liquid, is very important, and looking at the visuals. Pouring the liquid in front of the kids and measuring out different size containers, and seeing how much the capacity of the containers are, and looking at the units of measurement there in terms of representing the fluid is really important.
In primary school they also, in Stage 3, look at displacement. They actually put objects in the water so that you can see the water being displaced, and talking about that concept is really important. Here the marbles are submerged and the water moves up because the marbles have displaced the water. Talking about the fact that one centimetre cubed is equal to a mL, and a 1,000 centimetres cubed is equal to a litre, is really important here, so one centimetre cubed will displace 1 mL of water. Making that connexion, and knowing the fact that displacement is actually used for finding the volume of irregular objects, I think's really important here as well. Even a little research task here is great for looking at Archimedes' principle, I think is quite relevant.
Looking at your volume facts is really important, and if you're going to use centicubes for any practical activities, go through the facts about centicubes. Talk it through with students so the understand the length, the breadth, the height of the centicube, and how the volume works from there, and the connexion. Activities like this, which model was made from the most cubes, which has the largest volume, which is made from the least cubes, which has the smallest volume, is worth discussing and having kids look at in terms of activities.
Calculating volumes, so students can use centicubes to construct a rectangular prism, which is three centimetres by two centimetres and one centimetre height. Discuss how many centicubes you would use, what is the volume. Then start adding the layers to the prism. There's a nice table that they can fill out over the page here, so you've the length, the breadth, the height, number of centicubes, and then the volume in centimetres cubed. They keep stacking on the next layer and the next layer, and increase the size of their solid, and look at the way the volume increases as well.
You can move to sort of your standard diagrams, and have kids create the solid and calculate the volume of the solid. There's a nice activity there that they can do as well, attached to that sheet.
Having students go out and find a box or an object and try to estimate its volume is really important as well. We don't stop at estimation in primary school, I think a lot of it should be done in high school as well. They can explain how they calculated the volume, how they estimated it, and then they could check to see whether or not they're correct by actually measuring.
In terms of capacity, discussions with students about the conversions, discussions with students about which liquids are usually measured in millimetres, which ones would be measured in litres and kilolitres, is really important. Having them actually associate a picture with each of these units of measurement will help them remember in the future that millilitres is small, litres is larger, like a litre of milk, and kilolitres is what we're talking about when we're talking about how much water is in a pool for example.
Having students research capacity of Sydney Harbour, Warragamba Dam, or anything else that you've got in your local area. A petrol tanker for example, and think about the amount in kilolitres that each would hold, is a really great way to report back to the class, and compare the different type of capacities of different areas, objects, et cetera. Again, we have to look at conversion of units for capacity. Having the typical diagram that most of us use I think's really important, and modelling it, modelling the process for students, is really important, with an image to help them visualise it a little bit better.
Then again, moving to numeristic problems, I think that's absolutely vital. Worded problems, where they're using volume and capacity concepts to actually solve the problem. Get them to draw a picture or include a picture for them, abut help them unpack those problems, I think is absolutely important at this point.
My point here really is that you need to link area to volume and volume to capacity, and make sure all the links are there: perimeter, area, volume, and capacity, in the work that you do, and the activities that you give students. Even though we tend to teach them sort of a little bit separately, I think it's really important to make those connexions where possible, and give them as many activities as you can where they are finding the area, then they're finding volume, and then they're finding the capacity of the actual containers. This is a really nice activity where they're matching equivalent quantities, and they can see that they actually understand how to do the conversions.
Just to finish up for measurement, homework viewing's really important. I think use YouTube to its max. Have a look, these are just some examples of little clips kids can see for Stage 5 surface area. In each of these the presenter actually unpacks a question with the students. I think it's great for homework viewing, for revision, and for consolidation. This is just one that I picked up on the internet, but surf the internet and have a look at what's available. You can pick a presenter that you prefer for your students, just be careful and make sure that it's centimetres and metres, et cetera, that they're working in. Give the URL to your students and set it up for homework for them. I think it's really, really important that they do practise, and they see it from different presenters.
I'll pass over to Chris now, who will continue and work on the geometry section.
Thank you, [Nagla 00:19:57]. The presentation in the syllabus of measurement and geometry as a single strand recognises and emphasises their interrelationship. I think few of us would argue that the study of geometry is indeed about representation of the shape, size, pattern, position, and movement of objects, the investigation of three-dimensional objects and two-dimensional shapes, as well as the concepts of position, location, and movement.
What changes have occurred in the movement from the old syllabus, or from the current syllabus we've used [inaudible 00:20:36], and the new syllabus? In Stage 4, students studied the properties of solids, identified, and named angels formed by the intersection of straight lines, studied the properties of geometrical figures, including the properties of triangles and quadrilaterals, and identified congruent and similar two-dimensional figures.
Notably, of course, in the old syllabus there was no geometry in Stage 5.1. In Stage 5.2 students developed and applied results related to the angle sum of interior and exterior angles for any convex polygon, and developed and applied results for proving that triangles are congruent or similar. Then in Stage 5.3 came the deductive geometry component, with students constructing arguments to prove geometrical results and then study the circle geometry.
In Stage 4 of the new syllabus, you would have noticed that students no longer study the properties of solids. This content moving to Stage 3, nor do they study similar two-dimensional figures, this content moving to Stage 5.1. They do, however, classify, describe, and use the properties of triangles and quadrilaterals, and determine congruent triangles to find unknown sides, side lengths, and angles. They identify new angle relationships, including those related to transversals, on sets of parallel lines.
In my opinion there is a welcome return of geometry to Stage 5.1, and as mentioned earlier, Stage 5.1 students now describe and apply the properties of similar figures and scale drawings. In Stage 5.2 students calculate the angle sum of any polygon, and use minimum conditions to prove triangles are congruent or similar. Finally in Stage 5.3, students prove triangles are similar, they use formal geometric reasoning to establish properties of triangles and quadrilaterals, and then apply deductive reasoning to prove circle theorems and to solve related problems.
Before I go on, a little aside. You'll be pleased to know, I think, that a mathematics K-10 continuum of key ideas poster has been developed for the new syllabus. A special thanks to [Nagla 00:23:31] for the tenacity she has displayed in pushing that one through. Copies will be distributed to schools soon, so stay tuned for further information.
Taking a look at the key ideas in the early Stage 1 to Stage 3 part of the syllabus, there is a greater language demand, with an emphasis on using current mathematical terms earlier. I'm sure you will all feel that this is a positive change. [Ag rules 00:24:02] is a separate substrand in this part of the syllabus. Of course the content was previously taught as part of the two-dimensional space. When the new syllabus is fully implemented, and all students K-10 are working from the same syllabus, we can expect that Stage 3 students will understand the difference between regular and irregular shapes. Describe, identify, and predict translations and rotations with shapes. Identify and determine the number of parallel faces of 3D objects. Visualise and name 3D objects from their notes. Describe and compare the properties of prisms and pyramids. Draw different views of an object constructed from connecting cubes in isometric grid paper. Investigate, with and without the use of digital technologies, adjacent angles that form a right angle, straight line, and angles at a point. And finally, use the properties of adjacent angles and vertically opposite angles to find the size of unknown angles in diagrams. In summary, in the early Stage 1 to Stage 3 syllabus, there has been a general move from identifying and describing features to understanding properties.
To the Stage 4 to Stage 5.3 part of the continuum, as we have said on a number of occasions in this series, be aware of the lag in the staggered implementation of the syllabus, because this will have important implications for your programming for at least the next two years. In particular we need to be aware that properties of solids has moved from Stage 4 to Stage 3, identifying line and rotational symmetries has moved from Stage 3 to Stage 4, and similar figures has moved from Stage 4 to Stage 5.1.
The geometry substrand, of course, presents many opportunities to teach students to reason and communicate mathematically. In the previous SyllabusPlUS session, Peter Gould showed us how to develop reasoning skills in our students through a number of examples drawn directly from geometry content in the syllabus. In his presentation Peter described a process for teaching students to communicate their reasoning, which relied on them first convincing themselves, then convincing a friend, and finally convincing an enemy, or at least a person that is less likely to accept the truth of what they are attempting to communicate. If you haven't viewed Peter's presentation then do yourself and your students a favour by doing so before you teach your next geometry unit in learning. Follow the link on the bottom of this page, or in the web links part on the final screen of the presentation.
Following the link between geometry and teaching mathematical reasoning, I was interested to read recently some research by a couple of academics, a husband and wife team, in fact, by the surname of Hiele, or Hiele, I'm not quite sure how to pronounce that correctly. Anyway, this focused on the developmental levels of geometric reasoning. According to the so-called Hiele model there are five developmental levels that students progress through:
At the entry level students view objects as entire entities, not noticing individual components or properties. At Level 1 students begin to recognise that geometric shapes have parts and special properties. At Level 2 students comprehend the connexion between the properties within geometric figures, and from one set of figures to another. At Level 3 students can construct a geometric proof and understand the connexion between postulates and theorems. And at the most advanced level, students see geometry in the abstract and can move between different geometric systems and compare and contrast them.
The following is a example of how teachers can use this model to assess the level their students have attained. The activity goes like this. Have your students ort the triangles into as many sets as possible, then ask them to write a paragraph describing how they placed each triangle into a given set. At Level 0 the entry level students divide the triangles into sets based on size. At Level 1 you can expect that students divide the triangles according to one characteristic, most likely focusing on either the lengths of sides or the size of angles. At Level 2 students would observe more than one characteristic of the triangles; for example they will see that there are isosceles right-angled triangles and scalene right triangles. At Level 3 students would use definitions, postulates, or theorems to make connexions and to reason. And finally at Level 4, students would grasp the abstract concepts and apply them through more than one geometric system.
Also from the syllabus strand overview, this description. If geometry is all about the manipulation of a variety of real objects and shapes then, like measurement, it can be described in very simple terms as a hands-on topic. Of course a hands-on approach doesn't need to be an elaborate extravaganza of resources. A couple of examples here of activities that can be undertaken with your students, that require little more in the way of resources than what is readily available in any classroom. I urge you to take a look at both of them.
The implementation of any new syllabus is, of course, a lot of hard work, but adopting a glass-half-full outlook, I think it's also true that it provides us with the opportunity to assess our current practise, and perhaps take a closer look at more interactive pedagogies for delivering the same content. The measurement and geometry strand provides us with great opportunities, for example, for cooperative learning in the classroom. Listed on this and on the following side are examples of the types of activities that could form the basis of cooperative learning lessons in your classroom:
Vocabulary and communication enhancement activities that encourage students to talk about mathematics, for example barrier activities where students work in pairs on either side of a physical barrier, with one student describing to their partner properties of a shape, and the other student sketching the shape from the information given. I think pair share activities where a question is posed to students, which they must first consider alone and then discuss with a neighbour before settling on a final answer. Group investigations, and I guess what could be considered a subset of this, jigsaw activities, where each student in a group has a part of the information, or jigsaw piece if you like, which is essential to the completion and full understanding of the final product.
Of course the syllabus makes special mention of manipulatives. These can be either physical, for example items around the house or teaching aids produced in class, or by commercial bodies, such as geo strips, or they can be virtual manipulatives, meaning digital learning objects. Consider the use of matching activities, squaresaws, and sequencing activities. Venn diagrams or graphic organisers are great to show similarities and differences, and finally dynamic geometry software.
Commonly used dynamic geometry software includes WinGeom, for those of us that are a little bit older, Geometer's Sketchpad, and a presentation by me would not be complete without mention of GeoGebra. So powerful are these tools that I would go so far as to say that, if you are attempting to teach the geometry substrands without them, then you are making life for yourself and for your students harder than it needs to be.
I've listed here, of course, links to GeoGebra resources. As we have mentioned on other occasions, the GIA website is a good place to start for access to locally produced resources, and to those produced by the vast community of international users of GeoGebra. Specific mention of a couple of resources, I produced the Geometer's Warehouse a couple of years ago. It has 70 applets for teaching Stage 4 and Stage 5 geometry, and a resource produced by a former colleague, Damian [inaudible 00:34:41], Geometry gems. A word of caution here, both resources were produced by the old syllabus, and as such contain references to that syllabus only.
[inaudible 00:34:54] have a very quick look at two components form those resources, produced for two different purposes. Firstly a pre-prepared applet from the Geometer's Warehouse to demonstrate or investigate a geometric property, and for this I would just go to the next view. This is an example of one of the applets contained within the Geometer's Warehouse. The intention of this resource was, it to be a collection of resources that one had perhaps used on the interactive Whiteboard, or of course students could have direct access to that. Here the construction has obviously been made previously.
The other way, of course, that students can access this technology is by this method. This activity involves the students in building the applet, and allows for a higher level of investigation with [discovered 00:36:01] learning. My point here is that, if the only access students have to the technology is via a teacher-centered demonstration, that's better than none at all, but obviously there is potentially much better greater learning to be achieved if the technology is directly in the hands of the students.