Mathematics

Kua whiti te rà, he rangi hou anò.

The sun has risen, a new day is born.

Matariki ahunga nui Ka p ērā hoki te taurangi Matariki provider of plentiful food So too does algebra.

What Building confidence with algebraic skills through developing understanding Why Algebra can be a block to mathematics learning when misunderstood How Through the use of multiple representations and contexts

Some Food for Thought!

What do we mean by 2x? Does it mean x + x, 2 × x, (x + x) or doesn’t it matter?

Work through the following: 1. What does 2x mean?

2. What is the value of 2x when x = 3 ?.

3. What is the value of 12 × 2 × 3 ?

4. What is the value of 12 ÷ 2 × 3 ?

5. What is the value of 12 ÷ 2 × x when x = 3 ?

6. What is the value of 12 ÷ 2x when x = 3 ?

1.

4.

2 × x or similar expression Did you get these answers?

2. 6 18 (convention tells us to work from left to right) 5. 18 3. 72 6. Most people give the answer 2, different to that of question 5. What does it tell us about 2x and 2 × x? Nothing is quite as simple as it first appears, perhaps. Perhaps the moral of the story here is to use brackets everywhere where there might possibly be ambiguity

nzmaths / algebra

A patterned approach vs a generalised arithmetic approach.

Is this a 6,4,4,4,4,4,. . . or a 2,4,4,4,4,4,4,4,4, . . . pattern To get <6, 10, 14, 18, 22, . . .> What’s the rule? Total = 6 + 4(n - 1) or Total = 4n + 2

Ka ngaere mai nga ngaru ki te one.

The rollers came tumbling up the beach.

Taniko Niho Taniwha

http://whakaahua.maori.org.nz/tukutuku.htm

The Ngaru Nui represent the waves of theNgatokimatawhaorua. The zig zag part are the waves. The rectangle part is the waka

Algebra Strand

Algebra gives us a vehicle to reason why things are.

algebra applets http://www.active maths.co.uk/algebra/investigations/polygon1.html

A Message from Lewis Carroll

The red Queen says to Alice, “

What’s one and one and one and one and one and one and one and one and one and one?” “I don’t know”

, said Alice, “

I lost count.”

“

She can’t do addition”

, said the Queen.

Are there similarities for Number and Algebra?

Are there similarities for mathematical Processes – Logic and Reasoning and Algebra?

First we state a law that all parents will agree with: Teenagers = Time × Money so Teenagers = Money × Money = Money 2 because Time is Money Because money is the root of all evil

Money = (root of all evil) 1/2

So Money 2 = root of all evil and hence: Teenagers = root of all evil

Looking for patterns.

Draw / display examples to give you some ideas Plotting points on a graph to display your examples

Graphically

What type of graph is it?

Straight Line Linear?

Other?

Making links Algebraically generalising

Numerically Generating a numerical pattern / sequence What is the rule?

" Choose any two digit number, add together both digits and then subtract the total from your original number. When you have the final number look it up on the chart and find the relevant symbol. "

mysticalball

1 5 6 2 3 7 4 8 9 10 11 12 13 14 15 16 Make the magic square so that all of the rows, columns and major diagonals add to 34 16 5 4 9 2 11 7 14 3 10 13 8 6 15 12 1

What’s coming up next?

1 2 3 4 5 6 7 8 9 1 1 2 3 5 8 13 21 34 0 1 2 3 4 7 6 15 8

At this stage in the exploring patterns progression, students are able to copy given elements in a pattern, work out the next element in the pattern, and show it in some way.

Copy a pattern and create the next element At this stage in the exploring patterns progression, students are able to use systematic counting to continue a pattern. This allows them to work out the next element in the pattern more efficiently and accurately.

Use a systematic approach to continue a pattern and find number values

Algebra: Level 3 Achievement Objectives

Exploring patterns and relationships  describe in words, rules for continuing number and spatial sequential patterns;  make up and use a rule to create a sequential pattern;  state the general rule for a set of similar practical problems;  use graphs to represent number, or informal, relations.

Exploring equations and expressions

 solve problems of the type: +

15 = 39

Algebra: Level 4 Achievement Objectives

Exploring patterns and relationships find and justify a word formula which represents a given practical situation;  solve simple linear equations such as:

2 + 4 = 16

.

Exploring equations and expressions

find a rule to describe any member of a number sequence and express it in words;  use a rule to make predictions;  sketch and interpret graphs on whole number grids which represent simple everyday situations.

At this stage in the exploring patterns progression, students are able to recognise relationships between successive elements in a pattern and may be able to use a table to list values.

Predict values using relationships between successive elements At this stage in the exploring patterns progression, students are able to explain a rule to predict the value of any given element in a pattern. They no longer need to rely on knowing the previous element to work out any given element.

Predict values using rules

Algebra: Level 5 Achievement Objectives

Exploring patterns and relationships generate patterns from a structured situation, find a rule for the general term, and express it in words and symbols;  generate a pattern from a rule;  sketch and interpret graphs which represent everyday situations;  graph linear rules and interpret the slope and intercepts on an integer co ordinate system.

Exploring equations and expressions

evaluate linear expressions by substitution;  solve linear equations;  combine like terms in algebraic expressions;  simplify algebraic fractions;  factorise and expand algebraic expressions;  use equations to represent practical situations.

At this stage in the exploring patterns progression, students are able to state and use an algebraic expression for a relationship. They are able to use symbols and the variable "n" to express their rule.

Find an algebraic expression for a relationship Students are now able to use equations for a pattern to solve a range of problems related to that pattern. For example, they might use equations to solve the problem: "If 53 red tiles are used, how many blue tiles are used?" Solve linear equations related to patterns

Copy a pattern and create the next element Predict relationship values by continuing the pattern with systematic counting Predict relationship values using recursive methods e.g. table of values, numeric expression Predict relationship values using direct rules e.g.

? x 3 + 1 Express a relationship using algebraic symbols with structural understanding e.g. m = 6f + 2 or m = 8 + 6(f – 1)

http://arb.nzcer.org.nz/nzcer3/keywordm.htm

Fill in the grid so that every row, every column, and every 3x3 box contains the digits 1 through 9.

9 i 9 i

1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 6 9 12 15 18 21 24 27 4 8 12 16 20 24 28 32 36 5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81  Vedic numbers and patterns – completing the multiplication table and converting to vedic then connect up all of the number 1’s with lines OR 5’s etc to link with the GEOMETRY and NUMBER strand.

1 2 3 4 5 6 7 8 9 2 4 6 8 1 3 5 7 9 3 6 9 3 6 9 3 6 9 4 8 3 7 2 6 1 5 9 5 1 6 2 7 3 8 4 9 6 3 9 6 3 9 6 3 9 7 5 3 1 8 6 4 2 9 8 7 6 5 4 3 2 1 9 9 9 9 9 9 9 9 9 9

Algebra: Number

• Algebra allows analysis and descriptions of number patterns.

2, 4, 6, 8, ……..

99, 96, 93, 90, …… 1, 4, 9, 16, …..

Algebra: Measurement

• Algebra allows analysis and descriptions of relationships between things we can measure. • e.g. height and weight, length of foot and hand span, how we feel over time, savings over time, • Algebra also provides formulae for common measurement problems. e.g. area = base x height

Algebra: Geometry

• Algebra allows investigation and descriptions of relationships between geometrical properties e.g. the number of sides of a polygon and the number of diagonal lines or the number of sides of a polygon and the number of degrees in its angles.

Algebra: Statistics

• Algebra allows analysis and descriptions of data and graphs that show trends.

• Category data (e.g. favourite breakfast cereal, eye colour, languages we speak) is often unable to be described in terms of algebraic relationships.

?

Algebra: Mathematical Processes

• Mathematical Processes - the A.O.’s link closely with the Algebra A.O.’s.

e.g. use words and symbols to describe and continue patterns, record and talk about the results of mathematical exploration, interpret information and results in context

• http://www.nzmaths.co.nz/algebra/Back ground.htm

A good platform for students to move forward to algebra is provided by a sound knowledge of the properties of number and of the four basic operations. • understanding equality; • understanding operations; • using a wide range of numbers; • describing patterns.

What is Jane’s pattern?

Jane started adding up odd numbers. She got a surprise when she realised what pattern she was making.

• What was the pattern?

• Can you show why Jane’s pattern works using counters or some other method?

http://www.nzmaths.co.nz/PS/L5/Algebra/Squarea ndTriNos.htm

Magic squares a b c = L = L d g e h i f = L = L = L = L = L = L (a + e + i) + (g + e + c) + (d + e + f) = 3L Rearranging a + d + g + 3e + c + f + i = 3L L + 3e + L = 3L 3e = L

e – d + a e + d – 2a e + a e + d e e – d e – a e – d + 2a e + d - a

Choose a value for the variables e, a and d e.g. e = 15, a = 1 and d = 5

11 20 14 18 15 12 16 10 19

Patterning

1 = 2 + 3 – 4 2 = 3 + 4 – 5 3 = 4 + 5 – 6 4 = 5 + 6 – 7 ……..

n = (n + 1) + ( n + 2) - ( n + 3) generalising or = + 1 + ( + 2) - ( + 3) Using lego blocks to get

• generate patterns from a structured situation, find a rule for the general term, and express it in words and symbols; (paper folding, choose a number, Kava bowl, number chants) • generate a pattern from a rule; • sketch and interpret graphs which represent everyday situations; (a race, reading, flagpole…) • graph linear rules and interpret the slope and intercepts on an integer co-ordinate system. (exposition if time)

• evaluate linear expressions by substitution; • solve linear equations; (jelly beans, rod representations) • combine like terms in algebraic expressions; • simplify algebraic fractions; (exposition if time) • factorise and expand algebraic expressions; • use equations to represent practical situations. (tea leaves, lightning, power prices, cooking a roast…)

Folding a strip of paper: 7 8 9 …… n 3 4 5 6 1 2 Fold number 0 128 256 512 …… 2 n 8 16 32 64 2 4 Sections 1 7 15 31 63 1 3 Creases 0 127 255 511 …….

2 n - 1 What about if the paper was folded into 1 / 3 ‘s

choose a number add 4 multiply by 2 subtract 4 divide by 2 subtract the number you started with and the answer is … N N + 4 2N + 8 2N + 4 N + 2

2

Kava Bowl (Samoa) Pattern (two-variable relationships)

Kava Bowl (Samoa) Pattern Row Number of diamonds 1 2 3 4 5 6 R

Kava Bowl (Samoa) Pattern Row 1 2 3 4 5 6 R Number of cuts

Kava Bowl (Samoa) Pattern 1 2 3 4 5 6 R Diagonal Number of diamonds 8.5

1 1, 2, 1 1, 2, 3, 2, 1 1, 2, 3, 4, 3, 2, 1 1, 2, 3, 4, 5, 4, 3, 2, 1

1 1, 2, 1 1, 2, 3, 2, 1 1, 2, 3, 4, 3, 2, 1 1, 2, 3, 4, 5, 4, 3, 2, 1 What patterns could we analyse from this number chant?

1 1, 2, 1 1, 2, 3, 2, 1 1, 2, 3, 4, 3, 2, 1 1, 2, 3, 4, 5, 4, 3, 2, 1 Row 1 2 3 4 5 6 R Number of actions

1 1, 2, 1 1, 2, 3, 2, 1 1, 2, 3, 4, 3, 2, 1 1, 2, 3, 4, 5, 4, 3, 2, 1 Row Total Number of actions 1 2 3 4 5 6 R

1, 1, 2, 2, 1 1, 2, 3, 3, 2, 1 1, 2, 3, 4, 4, 3, 2, 1 1, 2, 3, 4, 5, 5, 4, 3, 2, 1

1, 1, 2, 2, 2, 1 1, 2, 3, 3, 3, 2, 1 1, 2, 3, 4, 4, 4, 3, 2, 1 1, 2, 3, 4, 5, 5, 5, 4, 3, 2, 1 …

sketch and interpret graphs which represent everyday situations; (a race, reading, and flagpole…)

• Sketch a graph to show the distance travelled as time passed in the story of The Tortoise and the Hare.

Tea leaves, lightning, rugby, and plumbing… (Using equations to represent practical situations.)

Tea leaves, lightning, rugby, and plumbing…

Define the variables.

Express the rule in words.

Express the rule as an equation.

To check the equation is reasonable, try the equation for an example or two and relate the answers back to the context.

Tea leaves, lightning, rugby, and plumbing…

One teaspoon for each person and one for the pot. How much tea is needed?

T = total number of teaspoons needed P = the number of people having tea

Tea leaves, lightning, rugby, and plumbing…

One teaspoon for each person and one for the pot. How much tea is needed?

T = total number of teaspoons needed P = the number of people having tea T = P + 1

Tea leaves, lightning, rugby, and plumbing…

One teaspoon for each person and one for the pot. How much tea is needed?

T = total number of teaspoons needed P = the number of people having tea T = P + 1 If there is one person, T = 1 + 1 = 2 If there are five people, T = 5 + 1 = 6

Tea leaves, lightning, plumbing… rugby, and

How far away is the lightning?

Define the variables.

Express the rule in words.

Express the rule as an equation.

To check the equation is reasonable, try the equation for an example or two and relate the answers back to the context.

Tea leaves, lightning, rugby, plumbing… and

How many points does the team score?

Tea leaves, lightning, rugby, and plumbing …

A plumber charges $50 for a call out and $60 for each hour she works.

Write an equation for the amount of the invoice.

Solving Equations

(Finding the number(s) which make the equation true.) Jelly beans Rod Representations

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

+ =

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

+ = C + 5 = 8 c = 3

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cups?

+ =

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cups?

+ = 2c + 3 = 11 c = 4

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

+ = +

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

+ =

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

=

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

+ = + 2c + 3 = c + 13

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

+ = c + 3 = 13

Solving Equations

(Finding the number(s) which make the equation true.) How many jelly beans are in the cup?

= c = 10

Solving Equations

(Finding the number(s) which make the equation true.) Rod Representations X + 5 = 17

Solving Equations

(Finding the number(s) which make the equation true.) Rod Representations X + 5 = 17 X 5 17

Solving Equations

(Finding the number(s) which make the equation true.) Rod Representations 2x + 5 = 21

Solving Equations

(Finding the number(s) which make the equation true.) Rod Representations 2x + 5 = 21 2x 5 21

Solving Equations

(Finding the number(s) which make the equation true.) Rod Representations 3x + 5 = x + 12

Solving Equations

(Finding the number(s) which make the equation true.) Rod Representations 3x + 5 = x + 12 x x x 5 x 12

Solving Equations

(Finding the number(s) which make the equation true.) Rod Representations 3x + 5 = x + 12 x x x 5 x 12

Use the jelly beans model or the rod representation to show your answers to:

1. x + 11 = 23 2. 2x + 3 = 51 3. 5x + 1.3 = 3.8

4. 2x + 5 = x + 9 5. 4x + 11 = 3x + 18

Forward maintenance

What is 50% of $200?

What is 10% of 140 kilograms?

Number sense:

• Ability to judge reasonableness of numerical outcomes • Looking for and finding links between new information and acquired number knowledge.

Ministry of Education. (2004). Teaching number sense and algebraic thinking, book 8. Wellington: Ministry of Education.

Number Sense

Ability to use a range of mental strategies e.g. knowing 12 x 15 is the same as 6 x 30, knowing 65 - 38 is the same as 67 – 40, using part whole thinking, Understanding relative size of numbers e.g. 1/3, ¼ 123, 1250 5.4, 0.54

Ability to estimate e.g. 27% of 389 is roughly 30% of 400 Belief that numbers make sense and that s/he is capable of working out number problems McChesney & Biddulph (1994) Number Sense. In J. Neyland (Ed).

Mathematics Education Volume 1

. Wellington. The Wellington College of Education.

Kay buys a CD player costing $141. She receives a 9.9% discount.

The shop assistant uses her calculator and asks for $133.98.

Kay asks her to work it out again because she has estimated….

Develop mental strategies:

Between and through phases • Using materials • Using imaging • Using number properties

Swaps and Switches:

Understanding Commutativity • Is 4 plus 3 the same as 3 plus 4?

• What about 4 minus 3 and 3 minus 4?

• Does 2 times 5 give the same as 5 times 2?

Anthony, G. & Walshaw, M. (2002). Swaps and switches: students’ understanding of commutativity.

Mathematics Education in the South Pacific:

Proceedings of the 25 th annual conference of the Mathematics Education Research Groups of Australasia, (pp91-99). Sydney:MERGA

• Most year 4 and year 8 students could explain why 4 + 3 was the same as 3 + 4 • Many students thought that 4-3 is the same as 3-4 • These students could not model the problem with materials

“The inability to directly model such elementary problems with concrete materials will invariably hinder the student's propensity for visualisation and abstraction deemed necessary for algebraic thinking” Anthony and Walshaw (2002), p 95 Positives and negatives card game

Multiplication and Commutativity • Most used additive reasoning to show 2 fives is the same as 5 twos.

• Multiplicative thinking and commutative understanding shown in array

Making links

• From experiences in arithmetic we hope that children will form structures which support generalisations about numbers and operations • Algebraic thinking thus provides another level of abstraction above the operation-based strategies Ell, F.(2001). Strategies and thinking about number in children aged 9-11 years. Technical report 17. Retrieved 20 November, 2004 from http://www.tki.org.nz/r/asttle/pdf/tech_report_17_ell.pdf

‘Guided Discovery’ as a useful pedagogy to link number skills and generalisations What can we discover about finding 10% of a number?

question 10% of 20 result 2 10% of 140 10% of 750 10% of 1000 10% of 59.7

‘Guided Discovery’ as a useful pedagogy to link number skills and generalisations question 10% of 20 10% of 140 10% of 750 10% of 1000 10% of 59.7

result 2 14 75 100 5.97

‘Guided Discovery’ as a useful pedagogy to link number skills and generalisations question 10% of 20 10% of 25 10% of 750 10% of 1000 10% of 59.7

result 2 2.5

75 100 5.97

How can we find 10% of any number?

‘Guided Discovery’ as a useful pedagogy to link number skills and generalisations If we can find 10% of any number, can we use what we know to find 20% of any number? How about 30%? How about 1%? How about 150%?

If we know how to find 1% of any number can we find x% of that number?

Advanced additive part-whole

Number properties: 478 + 998 = (478 - 2) + (998 + 2) Generalise to 478 + 998 = (478 - n) + (998 + n) or 337 - 98 = (337 + 2) - (98 + 2) Generalise to 337 - 98 = (337 + n) - (98 + n)

• 4 + 4 + 4 + 4 + 4 + 4 = 6 x 4 leads to: p + p + p + p + p + p = 6 x p = 6p • (10 x 6) - 6 = 54 generalises to 10n - n = 9n • 6s - 2s = s + s + s + s + s + s - s - s = 4s develops to 45j - 20j = 25j

Advanced Multiplicative

Number property generalisations: 5 = 2 becomes = 5 - 2 a = b 9 + = 16 becomes becomes = a - b = 16 - 9 a + = b - 16 = 7 - a = b becomes becomes becomes = b - a = 7 + 16 = b + a

Using student strategies: • 6 x 24 = 6 x (25 – 1) = 6 x 25 – 6 x 1 • 81 – 36 = 9 x 9 – 9 x 4 = 9(9 – 4) = 9 x 5 • 12m – 6 = 6 x 2 x m – 6 x 1 = 6(2m – 1)

Linking number sense to algebraic thinking: • I am using hexagonal pavers surrounded by a wooden border are used to make a path. The diagram shows how the path will be made.

• Find a formula that will help me to work out how many pieces of wood I need to buy for a path that has 24 pavers.

What is algebraic thinking?

What is algebraic thinking?

• The ability to generalise patterns, trends and relationships about numbers, shapes and measures, using symbols, graphs, and diagrams.

Number and algebra

Number involves calculating and estimating through the flexible use of appropriate mental, written, or machine calculation methods, knowing how and when to use approximation, and having an alertness to the reasonableness of results. Algebra involves generalising and representing patterns and relationships about numbers, shapes and measures, using symbols, graphs, and diagrams. Draft Mathematics and Statistics Curriculum statement, November, 2007