Year ten topics:

• Pythagoras and right angled trig

• Algebra

A numeracy approach

National Numeracy Hui

Michael Drake

Victoria University of Wellington College of Education

Something to think about…• What do you need to know if you

are to learn Pythagoras’ Theorem and right angled trigonometry with understanding?

• Discuss in groups of 3 – 4

So what numeracy stage do you need to be at to deal with:

Pythagoras?

Right angled trig?

Take a look at this one…

Which line is longest?(a) the top (b) the bottom(c) they are both the same

Predict how you think students will answer

Results: • Lots of students think they are the

same length

First year Second year

Third year

51% 50% 46%

By the way – these were English secondary school students, so

year 1 students were 11 – 12 year olds

year 2 students were 12 – 13 year olds

year 3 students were 13 – 14 year olds

Hart, K. (1978). Mistakes in mathematics. In Mathematics teaching. Number 85, December 1978, 38-41

The broken ruler problem

• This problem can take several forms – one is to measure an item with a broken ruler, the other is to measure the length of an object where the object is not aligned with zero.

Here is the Chelsea diagnostic version:• How long is the line in centimetres?

1 2cm 3 4 5 6 7 8

First year Second year

Third year

46% 30% 23%

The answer 7 occurred thus:

7 can be worked out through• starting at one when measuring• failing to consider the distance between

the start and the finish point (just looking at the highest number reached)

What are the implications of what you have just seen for the teaching of Pythagoras and right angled trig?

Implications…

• There are a lot of things we take for granted when we teach. The basics need to be covered for a lot of students, and the basics may not be what we think they are!

Developing similarity• From a numeracy approach, new ideas

are based on developed understanding. As similarity underpins right angled trig, it seems sensible to spend time developing this first

• The second activity (II similar) extends the understanding of similarity to that of calculating a fraction (the scale factor of enlargement) from lengths of comparable sides

Algebra

• What is algebra?

Think…

Discuss in pairs…

• If you ask students (average year 11 types), what would they say?

How would you average year 11 student solve this problem?

• Sian has 2 packs of sweets, each with the same number of sweets. She eats 6 sweets and has 14 left. How many sweets are in a pack?

6 + = 10

57 + = 83

3.64 + = 4½

57 + x = 83

When is a problem a number problem – and when it is an algebra problem?

• So when does algebra become something that is important for students to know and understand?

So how do we use letters in mathematics?

How do we use letters in mathematics?

1)A letter can be used to name somethingIn 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 foundIn a triangle, x is often used for the angle students need to find

3)A letter is really a numberevaluate 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

Developing an understanding of symbols

• A number of pieces of research indicate students have difficulty with understanding the equals sign

Falkner, Levi & Carpenter (1999). Cited in carpenter, Franke & Levi (2003). Thinking mathematically: integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann

8 + 4 = + 5

Results

Grade 7 12 17 12 & 17

1 & 2 5 58 13 8

3 & 4 9 49 25 10

5 & 6 2 76 21 2

• Note that years 5 & 6 are worse than the years 3 & 4

ScenarioYou have discovered that some students in your class have wrong conceptions of the equals sign

• How do you fix it?

• Children may cling tenaciously to the conceptions they have formed about how the equals sign should be used, and simply explaining the correct use of the symbol is not sufficient to convince most children to abandon their prior conceptions and adopt the accepted use of the equals sign

Carpenter Franke & Levi (2003), p. 12

Dealing with misconceptions

• If a student has a misconception – it must be challenged

• Try introducing problems for discussion. What different conceptions exist, and need to be resolved?• This forces students to articulate beliefs that are often left unstated and implicit. Students must justify their principles in a way that convinces others

Try this part of the puzzle…

• How many matches for 5 squares?

• How many matches for n squares?

• How many matches for 100 squares?

• Did you notice that Hamish changed his solution method when dealing with n sides?

• What methods from the flip chart could he describe (or write) without knowing the conventions of algebraic notation?

• So what algebra does a student need to know before they can record these statements correctly?

Sorting out algebra in years 9 and 10

Year 9 emphasis Year 10 emphasis

Letters for naming •Abbreviations•Development of simple formulae•Units

Letters for naming •Substitution into formulae•Algebraic substitution

Letters as specific unknowns•Box equations (letter equations for brighter students) or word problems that can be solved mentally

Letters as specific unknowns •More formal methods based on knowledge of the equals sign as a balance

Letters as a variable•Patterning•Generalisation from number

Letters as a variable •Graphical functions•Manipulation

• Introduction to generalising – thinking beyond the getting the answer

• Learning to express mathematical ideas with symbols – learning the language of mathematics Learning the conventions of symbol sentencesLearning the different meanings of letters

Year 9

• Exploring algebra itself for what we can learn. For example: we have now met letters (algebra) in a variety of situations – let’s study them in more detail to see what happens when we +/-/×/ them – like we did for fractions, decimals, integers …

Year 10

Starter Warmdown

(1) Draw a picture to show that an odd number plus an odd number always gives you an even number

25 = 25(2)

Show that 25 + 36 – 36 = 25• Will this always work?

• How do you know? • If you think yes – can you write a

sentence that shows it will always work, regardless of the number you start with, and the number you add.

• If you think no – how can you prove it doesn’t always work?

(3) What is 4 4?

• What about 16 16?

• 250 250?• Does this always work?

• Can you write this as a rule in words?

• Explain why this works using a drawing or some equipment from the box

• Write a sentence with symbols to show your rule