The big mathematical content picture – algebraic thinking Michael Drake SNP national resource coordinator

What is algebra? Think… Discuss in pairs… Collate feedback – store on board… Discuss in pairs…

Try this problem 1000 - 647 What do you notice? Does this always work? - 647 Do problem 1000 -647 353 Idea that 9999+1=10000 (rolls over) What do you notice? Does this always work? Explore…

Algebraic Thinking Algebraic thinking can be defined as what happens when “students demonstrate they can use and understand principles that are generally true and do not relate to particular numbers.” Algebra arises throughout mathematics whenever generalisations are made. This may be when a student is looking at how a simple pattern is being created by adding threes, or recognising that it is quicker to find the area of a rectangle by multiplying the number of rows by the number of columns, instead of skip counting or counting squares

Algebra is a language The language is based on symbols Students need to learn how to use this language. If they are not taught how to use this language – they will make their own sense of the symbols

What does the equals sign mean in each of these situations? 7 + 8 + 9 = Some will have already seen Marg Horne’s presentation in Auckland and may have Charles Darr’s set article. Forgive the repeat if you already know this… 7+8+9 = =means work out the answer X=3 assigning a value 4 + 5=+3 a balance Darr article of 300 students at a large intermediate, 49.2% of year 7 and 35.2% of year 8s put 9 as the answer (40.7% of Y7 and 60.7% put 6 and 6.8% of Y7 put 12) Higher % of students incorrect at lower ages and over 20% still incorrect in years 9 and 10 Can’t understand algebra unless have the concept of an equals sign as a balance… % of students put 9 as the answer here, some then continued to write =12 Dealing with this Before moving onto algebra, it is essential for students to have the understanding that = means that the two sides are balanced, the same or identical. To facilitate this: when recording number problems with students, discuss (or invent) with your class a new symbol to mean ‘is’ in running arithmetic - and practice its use. Work with and discuss situations when an equal sign would not make a true statement. For example, with the equation  + 2 = 5, what number(s) would make this true? What would happen if you used another number? (Working towards the use of the greater and less than signs). Likewise when solving equations invent with your class a symbol to mean “I worked out that out…” (In some problems, an easy way to do this is to use the word “so”.) References Darr, C. (2003). The meaning of equals. in Set 2, 2003. Horne, M. (2006). Presentation at the national Numeracy hui, Auckland, 14 – 16 Feb., 2006 x = 3 4 + 5 =  + 3

John has 54 marbles. He buys another packet of them and ends up with 83 marbles. How many marbles were in the packet? How did you solve this problem? Put these up on board – split one side into additions and the other into subtractions 83 - 54 54 +  = 83

What understandings of the language do students need to have? What generalised understandings do children need to be able to solve this problem? Generalised understandings + and – are related operations… (Language/reading and understanding issues but will park those – focus is on mathematics Language understandings  Is an unknown number to be found (If can record the problem as 54 + = 83 doing high level algebra – this sort of thing still causes problems in year 11 for NCEA) = as a balance Need to address the understanding of how the symbols work – before start using them without reference to this (assuming people know…) E g  is a number. In terms of the symbols, what does one more than the number look like? ( + 1) If need scaffolding put up 6 – what would one more look like – if I am not allowed to rub out the 6?) Play Symbol loopy Use after some initial teaching has been done What understandings of the language do students need to have?

It’s a race… Have a clean piece of paper and a pen at the ready Lolly for the winner… Malcolm – you hold the lolly for the winner… Finishing it off How did you do the problem? Put up various solutions Lolly for the winner… Set… Add up all the numbers from 1 to 100

How do we use letters in mathematics? 1) A letter can be used to name something In the formula for the area of a rectangle, the base of the rectangle is often named b 2) A letter can be used to stand for a specific unknown number that needs to be found In a triangle, x is often used for the angle students need to find 3) A letter is really a number evaluate a + b if a = 2 and b = 3 4) A letter can be used as a variable that can take a variety of possible values 5) In the sequence, (n, 2n+1), n takes on the values of the natural numbers – sequentially How do we use letters in mathematics? Throw out the question or think-pair share (depends on time) The big 4 content areas Equations Patterns – one example of patterning already done. Traditional patterning very powerful if done properly Formulae Generalisation from algebra – already introduced Then bring up rest of slide

The development of formulae Area = base × height A = b × h A = bh 3cm 1) Letters have a naming function …so we use letters as abbreviations. Sometimes they are abbreviations of concepts – Area or temp (often use capital here – but not always – t for time Sometimes they label things – base etc Sometimes they stand for units – usually use lower case here – but not always – named for someone – but not always – Mr Litre Confusing – and needs to be addressed carefully Story of the student with the area of a parallelogram (b is not the base…) Note that students seem do develop an understanding of algebra as used in formulae more easily than other contexts – regardless of the complexity A = 5 × 3 A = 15cm2 5cm

Equations Sian has 2 packs of sweets. She eats 6 sweets and is left with 14 sweets. How many sweets are in a pack? Traditionally at secondary school, students in years 9-11 (in ALL of these years) have been given problems like this and expected to write an algebraic equation for it… Then solve it… WHY? Begs another question (next slide) How did you solve this?

When is a problem a number problem – and when it is an algebra problem? 6 +  = 10 Bring on and ask for each 6 +  = 10 basic fact 57 +  = 83 Triggers the use of a strategy. Size of the number is often the trigger to the generalising – not the problem. 57 + x = 83 the move from boxes to letters can be a very small step… 3.64 +  = 4½ can’t solve in a mixed number system – must translate into one system or the other 57 +  = 83 57 + x = 83 3.64 +  = 4½

Can you draw a picture to show the problem? Ameeta has 3 packs of biscuits, and 4 extra loose biscuits. Sam has one pack of biscuits and 16 loose biscuits. If they both have the same number of biscuits, how many biscuits are in a pack? Can you draw a picture to show the problem?

4 16 Talk through the representation  For unknown Trick of making lengths line up to show they are equal Show solution of the problem This method is much easier than 3x + 4 = x + 16 (form 4 equation solving, or form 5 simultaneous eqns)

Generalisation from numbers Materials Imaging Property of numbers Algebra comes from working with symbols Algebraic thinking and generalisation comes from thinking beyond the answer to the problem, to looking for patterns in the answers… Generalisation from numbers

What is algebra? Pause. Leave this up and count to 10 Final thoughts Algebra is not something that can/should be taught in a single ‘block’ – the different interpretations need their own space Algebra and algebraic thinking puts the challenge and the thinking into maths. Should be used with all students. In particular, older students with low skills need this form of challenge – but it needs to be appropriate to the level of understanding of the student Challenge How are you as facilitators giving the message that algebraic thinking and algebra are something that teachers should not be afraid of? And that all teachers (primary and secondary) should look at including every day as part of numeracy?