Progression and Transition in Algebra do... think... predict... understand An ATM support presentation

The Aims of ATM To support the teaching and learning of mathematics by: Encouraging increased understanding and enjoyment of mathematics; Encouraging increased understanding of how people learn mathematics; Encouraging the sharing and evaluation of teaching and learning strategies and practices; Promoting the exploration of new ideas and possibilities; Initiating and contributing to discussion and developments in mathematics education at all levels.

ATM’s Guiding Principles The ability to operate mathematically is an aspect of human functioning, which is as universal as language itself. Attention needs constantly to be drawn to this fact. Any possibility of intimidating with mathematical expertise is to be avoided. The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner. It is important to examine critically approaches to teaching and to explore new possibilities, whether deriving from research, from technological developments or from the imaginative and insightful ideas of others. Teaching and learning are co-operative activities. Encouraging a questioning approach and giving due attention to the ideas of others are attitudes to be encouraged. Influence is best sought by building networks of contacts in professional circles.

Problem solving & investigation... …..is at the heart of mathematics Outstanding teaching & learning will… allow children to make decisions encourage creativity and invention.....promote discussion and communication involve children seeing and making patterns, connections....... Making and testing conjectures and hypotheses promote reflecting, interpreting, explaining........proving encourage ‘what if’ ..........and ‘what if not’ questions; Fluency and reasoning... are skills which allow problem solving to happen are in effect ‘twin engines’ Algebraic thinking supports both fluency and reasoning

Algebra in Key Stage 2 (Y6) Pupils should be taught to: express missing number problems algebraically use simple formulae expressed in words generate and describe linear number sequences find pairs of numbers that satisfy number sentences involving two unknowns enumerate all possibilities of combinations of two variables National Curriculum, DfE (2013)

Algebra in Key Stage 3 use algebra to formulate mathematical relationships substitute values in expressions and solve equations make connections between number relationships, and their algebraic and graphical representations (linear and simple quadratics) make and test conjectures about patterns and relationships look for proofs or counter-examples

“We didn’t know that was algebra” Algebraic Reasoning in KS1 & 2 number bonds and commutativity inverse relations (e.g. 4 × 5 = 20 and 20 ÷ 5 = 4) relationship between arrays, number pattern, and counting distributivity can be expressed as 3 × (5 + 2) = (3 × 5) + (3 × 2) write statements of equality of expressions (e.g. 39 × 7 = 30 × 7 + 9 × 7) perimeter of rectangle expressed symbolically as 2(length + breadth) angle sum facts and shape properties as missing number problems linear number sequences, including those involving fractions and decimals, and finding/using the term-to-term rule combinations of operations; exploring order of operations; understanding the equals sign Thanks to Anne Watson

Early formulae A game for young children using a “rule” for finding a number Children sit in a circle, each child with a number that all can see Choose a “rule” e.g. Find someone with a number one less than you Each child takes a turn to roll a ball to another child following the rule Make a new rule and play again Adapted from Roll a number game in ATM’s Little People...Big Maths

Missing number problems ? 18 15 13 11 16 Which will you try to find first,......and next? Use this value to find the others Find the values of the ? symbol from the values you have found.

Missing number problems Equations 44 + 38 = 24 81 - 54 34 3 15 How do you do these? How would your children do them? Can we use algebraic thinking to help them become more fluent?

x-box challenge 1 Try some numbers through these linked number machines Starting number × 3 + 1 ? Which number will go through and change to 10? Investigate other number machines What if you reverse the machines? What if you used three linked machines? Can you make linked machines that always get you back to the starting number?

Grab the formula A game for two or more players Each player in turn rolls a dice twice The numbers obtained form the first two terms of a sequence The player then completes the next four terms of the sequence and then “grabs” that sequence by explaining how it is generated....”nth term” The player who grabs most formulae wins. Taken from What Kind of Game is Algebra

Number sequences ...using Multilink Patterns from crosses Picture frames What patterns can you notice? What does the next cross look like?...the next frame? What about the 10th?....12th?....20th?

Chains Mathematical Journeys Choose any number below 25 If it is odd...add 1; if it is even...half it. Repeat this for each new number e.g. 13 14 7 8 4 2 1 Who has the longest chain? What if we allowed greater starting numbers? What happens if you change the rules?

Frogs The objective is to move the red discs to where the blue ones are and the blue discs to where the reds are....one at a time. A move can be either a slide to an empty cell next to it or a jump over one disc to an empty cell. What is the fewest number of moves? Try this again with: 2 reds and 2 blue 4 reds and 4 blue... Find a way to predict the number of moves just from the number of each frog. Try this now with different numbers of each colour. Recommend doing this practically with teachers and chairs

Tidy numbers Tidy numbers Numbers 9, 10 and 11 are tidy 2 + 3 + 4 = 9 1 + 2 + 3 + 4 = 10 5 + 6 = 11 What makes a tidy number? Investigate whether all numbers are tidy? Check out Consecutive Sums in Mathematical Journeys (ATM publications)

3s and 5s Which numbers can be made by adding just 3s and 5s? (e.g. 9=3+3+3, 10=5+5, 11=3+3+5,…..) Start with finding numbers below 30...then try higher numbers. Which numbers are impossible? What is the greatest impossible number? Explain,........prove….. Investigate other pairs. Is there always a greatest impossible number? Look for a rule that will find the greatest impossible number for any pair. Look at Stamps in Mathematical Journeys

Problem solving & investigation... …..is at the heart of mathematics Outstanding teaching & learning will… allow children to make decisions encourage creativity and invention.....promote discussion and communication involve children seeing and making patterns and connections....... making and testing conjectures and hypotheses, promote reflecting, interpreting, explaining........proving encourage ‘what if’ ..........and ‘what if not’ questions Fluency and reasoning... are skills which allow problem solving to happen are in effect ‘twin engines’ Algebraic thinking supports both fluency and reasoning

Association of Teachers of Mathematics Professional support mathematics community......new CPD package Professional development work with teachers...branches Professional identity part of vision for mathematics education Journals six copies per year; articles by teachers & for teachers e-News monthly updates to members buy e-mail ATM website archive of “old” journals, sales,.... Publications 25% discount for members on all ATM publication Annual Easter Conference check the website Branch meetings check the website Tax concessions £70 p.a. Tax deductable....... effectively £1 a week!!! You can join as a personal or institutional member

Thank you For further information www.atm.org.uk Membership Publications Branches Conferences www.atm.org.uk