In this edition of the News you will find:
Who's Making eTasks?
Professional Development with eTasks
Powers, Indices & Problems
Get to Know a Cameo
... Make A Snake
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- Who's Making eTasks?
Since our last edition these schools and teachers have begun developing their own hands-on problem solving Task Library using the eTask Package.
- Salisbury Primary School, South Australia
- Nunawading Christian College, Victoria
- Remote Schools Support Unit (Central Australia), Northern Territory
- special licence for 24 remote and very small schools to support an upcoming PD program
- Dave, secondary teacher, Victoria
- Penny, primary teacher, South Australia
- The Mathematical Association of Western Australia
- to support current and future PD programs
We look forward to hearing about how you make use of your Task Library.
- Professional Development with eTasks
Professional development is what the eTask Package is about. Sure it involves working together to create a resource used by kids, but that's only part of its purpose. Working together to improve mathematics education in your school is the other part. Professional learning on your home turf.
You aren't just putting together 'stuff' for kids to 'play' with. The library of tasks you create can be used to investigate almost any issue in mathematics education right there in your own classrooms - and we have known for decades that the closer the professional development is to a teacher's classroom, the more effective it is likely to be.
Nominate your issue:
There will be a subset of tasks in the Task Library around which classroom trials can be designed.
- Working with Indigenous learners
- Concrete materials
- Recovering reluctant learners
- Higher order thinking
- Encouraging girls
- Problem solving and reasoning
- Increasing student ownership of learning
- Communicating mathematics
- Encouraging spatial learners
- A particular content strand
- Broader assessment practices
- Fascinating, captivating and absorbing learners
You're not on your own. Mathematics Centre provides teaching notes (Task Cameos) for each and every task (notes which are always open for your contributions); unit plans and models to help you design your trials; teacher stories and research work from classrooms to stimulate and challenge you. All of this collected wisdom is built around learning to work like a mathematician, a task having three lives and teachers taking responsibility for teaching craft. See Link List below.
When you have identified your focus issue, try building professional development sessions around:
Make, Take, Teach, Talk, Repeat.
Decide which tasks will support the exploration of your chosen professional issue. Organise the printing, laminating and materials you need.
- 155 eTasks in the package are easy to make because they use common materials such as dice, counters, blocks, tiles, or other readily obtainable material such as playing cards and only need one page printed and laminated.
- Make 1 task and you can use it to introduce its whole class investigation life, or with two students for a special purpose.
- Make 5 tasks and you have enough for a group in a work station model with say two other stations in the room.
- Make 10 tasks and you have enough for half the class to work on tasks while the other half are involved in some other way.
- Make 20 tasks and you have a class set.
Take your brand new tasks away to a planning space. Take time to use Mathematics Centre to find what others have shared about them.
Take time to prepare.
- Do this by yourself, with a partner or in a team.
- Imagine you and your students working together with the tasks in the timetabled room.
Put your planning into action with the students. Your flexibility might be called on. With a bit of luck it won't go exactly as planned. Hopefully you will be positively challenged by student input.
- If you think of it, take a few photos, or collect a piece or two of student work, or student comments.
||Discussion / Evaluation Session
Meet with colleagues to share classroom experiences. Show your photos or other examples.
- How did you feel? How did they feel?
- What was achieved? What wasn't?
- What could be improved?
- What's next?
||Consolidation / Extension Session
Build on what you have learnt together. Set yourselves a new challenge.
- Learning to teach is a career.
- Powers, Indices & Problems
A mathematician's work begins with an interesting problem. (See Working Mathematically in the Link List below for confirmation of this.) Consequently when the curriculum is seen through the lens of learning to work like a mathematician, a teacher first looks for a problem involving the 'content of the week' and makes teaching craft decisions about how it could be introduced in a way likely to fascinate, captivate and absorb learners.
We are sometimes asked, usually by Year 9 teachers, if we can suggest a problem that will allow the teaching of indices to be entered in this way. The answer is yes and the place to look for problem-based starting point for any of the content involved in our Task Library is the Task Cameo Content Finder (see Link List below). Content is listed in the left column of the Finder. The right column connects that content to problems which include it in some way. Problems in the right column are linked to their Task Cameo teaching notes.
In this case Index Laws isn't listed, but searching on 'Power' produces an entry for 'Powers of 2',
Index law arithmetic is a special arithmetic with a limited set of numbers (powers of 'a', or an). Special in the sense that you want the answers to also be written as powers something like the ones you start with. Essentially this arithmetic has developed because mathematicians have created a shorthand, useful in some situations, that would otherwise produce a long string of multiplications by the same number.
Then they had to ask questions like 'What happens if ... we try to combine these numbers? What will it look like in this shorthand?', which has led to rules and definitions. As it turns out, combining to result in powers can be achieved by multiplication of index/power numbers, but not by addition of them.
Gather students at one table, introduce the story shell as on the card and follow student suggestions for a few moments to begin exploring. When it is clear that students understand the conditions of the problem, they continue exploring with materials in their groups. Share, discuss and record on the board and in journals as appropriate. It will become clear that the Basket Numbers that solve the problem are 1, 2, 4, 8, 16, 32. Discuss why and ask students to explain in their journals using words and pictures.
|So in looking for a problem to start this unit, one choice is The Mushroom Hunt. As it is explored certain numbers become critical to the solution - powers of 2. It is essential of course to allow this to be discovered rather than bringing attention to it too early. Maths300 Lesson 130 (see Link List below) sets this up nicely, but if you don't have this resource, the cameo outlines it like this.
You need counters, blocks or red Poly Plug as mushrooms and paper plates or A4 sheets of paper as the baskets in each group. You could even put the text book to good use as in this Year 9 class at Wade High School, NSW.
If you use groups of 3 or 4 the larger table space will be just right for laying out the problem.
These Basket Numbers are what helps bridge into powers and index laws.
It's probably in the next lesson that you begin with the Basket Numbers. Write them down the board with 64 at the top. If it didn't come up as an extension in the previous lesson, discuss whether 64 could be a basket number and if it is, what would be the largest 'normal number' that could be made by combining these 7 baskets. This step reinforces:
- the doubling pattern students have recognised
- that the basket numbers are being combined by addition.
Starting with 64 ask how it is built by doubling. Write the equation: 64 = 2x2x2x2x2x2x2. Repeat with the other Basket Numbers, without making any big deal about 2 and 1.
- Mathematicians realised that there were occasions when they needed to write long strings of multiplications using the same number just like these Basket Numbers. So they invented a short way of doing it. Would you like to see what it is? Or perhaps there is someone here who already knows.
This leads into introducing index notation and language and discussing and agreeing that 2 = 21 and 1 = 20, as far as possible with the students deciding this for themselves from the pattern.
- Mathematicians have to define these things this way or there is no pattern.
Consolidate by asking what happens if our problem was based on a tripling, or quadrupling or 10 times bigger pattern. Apply the new notation and discuss again the index forms 31 and 30, 41 and 40 etc. Generalise on the basis of the evidence that any number to the power of one or zero - a1 and a0 - will be 'a' or 1 respectively.
Now notation has been introduced, operation can be looked at. Mathematicians might now ask What happens if...?.
- We know from our mushroom hunt that if basket numbers are added to basket numbers, the answer is a 'normal number'. The same would happen if they were subtracted. But what happens if basket numbers are multiplied or divided by basket numbers.
Time to explore again. You might begin by asking for two basket numbers (say 8 and 4) and writing something like this on the board as the students guide you:
8 x 4 = 32 ... Hey that's another basket number! That means we can write the numbers as powers of 2.
It doesn't take long for students to discover the addition rule for multiplying index numbers. It is important that after sharing discoveries orally and annotating the board, students record their own explanation in their journals. Division follows in a similar way.
23 x 22 = 25
Keep investigating and see what you find out. You can use other bases too if you want.
It is also important to discover that if basket numbers are multiplied or divided by non-basket numbers, the result is not a basket number. Power of 2 arithmetic is closed for multiplication and division because the answers are still power of 2 numbers. It's a small step to discover this is true for power of 3 numbers and any other base - albeit whole number bases at this stage.
So, what do you think?
- the curriculum is learning to work like a mathematician and
- content is uncovered by beginning with a problem which creates context and
- teachers make deliberate pedagogical choices to interest kids in the problem
what benefits do you see for student learning?
- Sets of exercises can still be used within and following the exposition above.
- The Mushroom Hunt is also a doorway into Binary Arithmetic.
- The Task Cameo Content Finder lists several other problems related to powers of 2, at least one more of which you could include to indicate the importance of powers and index laws.
- Get to Know a Cameo
Task 5, Make A Snake
|Mungo the Maths Snake is born with one coloured ring around his body - Colour A. In the next season, as Mungo grows, each Colour A ring is replaced by ABA, where B is a second colour. Each season after that, as Mungo continues to grow, each A again changes to ABA. Any Bs on Mungo's body never change - they just stay on his body. If the birth year is Year 0, investigate any patterns that show up during Mungo's growth.
Not so easy to state without having materials to represent the story, but after a little making and drawing of 'snake body patterns' most students move towards tabulating results, finding patterns and using the patterns to predict the number of rings after ten seasons.
At one level the patterns involve odd and even numbers, at another doubles and near doubles, and at another, powers of 2. The notes suggest reasons for the patterns which are verbalised as a step before symbolising. They also offer several extensions which include exploring graphs of the various sets of ordered pairs involved and thereby uncovering a curve representing y = an, where a is a constant.
In the eTask Package this task is in the 'easy to make' set because it only needs a collection of counters, tiles or blocks in two colours. The whole class investigation life of the task is best managed with students in groups of 3 or 4 using 2 or 3 Poly Plug sets, but it can be successfully explored if you have sufficient numbers of blocks, counters or tiles.
Task 56, Challenge
The focus of this task is reasoning, selecting and using problem solving strategies and communicating mathematics. These processes are central to a mathematician's work. The only textbook-style content involved is difference between two numbers, and then only from the set 1 to 8. It's a fabulous opportunity for students to display their ability to work like a mathematician without needing significant skill knowledge to begin.
Blocks can be placed on any space of the network drawing provided the difference between any two blocks joined by a line is not one. Find a solution. Find another.
- How many solutions are there?
- How do you know when you have found them all?
At this point we don't know the answer to the total number of solutions, but the cameo lists several (which become data) and explores strategies for finding out. Multiple solutions make the task perfect for a whole class investigation because there are enough solutions for most pairs to find one. That then produces a broader range of data to examine further.
In the eTask Package this task is in the 'easy to make' set because it only needs eight numbered blocks or tiles which fit the spaces on the card. Blocks are best because there is a lot of picking up and putting down involved in the task.
- Did you miss the Previous News?
If so you missed information about:
- Fractions, Algebra, Measurements & More
- Planning Units of Work
- What's The Chances With Coke?
- Get to Know a Cameo
... Scale Drawing, Crossing The River 2
- ...and more...
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