algebra and the secondary numeracy project. find the general rule linking the number of matches...

Algebra and the

Secondary Numeracy Project

Find the general rule linking the number of matches needed to the number of squares built.

To find the number of matches needed multiply the number of squares by 3 and add 1.

In algebraic shorthand m = 3s + 1

In this example we are doing algebra because we are making general statements in patterns that apply to any numbers and not just particular numbers.

This is what distinguishes algebra from arithmetic.

w

b

A = wb

Another example: A = area.

To be able to cope with this kind of algebra we children need to be advanced multiplicative (Stage 7) thinkers.

Examples of Advanced Multiplicative Thinking:

Calculate 89 x 5

90 x 5 is 450 then take away 5 to give 445

Or 44.5 x10 = 445

Or 80 x 5 + 9 x 5 = 445

Or ……………..

Work Out 72 ÷ 5

72 ÷ 5 is the same as 144 ÷ 10 which is 14.4

Or …….

Number sense shows 73 ÷ 0.97 = 70.81is wrong.

How?

73÷ 0.97 is just a little bit more than 73 since division by a number less than 1 has an answer that is bigger

Number Properties as Generalisation

Stage 6 Part of Student SheetWork these out.

663 + 199 = 669 + 197 =

117 + 398 = 698 + 127 =

272 + 296 = 397 + 184 =

Moving to the Use of Letters

Part of Student SheetFor each of the following equations write True or False in the box. Do not add up the numbers on each side

97 + 47 = 100 + 44 77 + 95 = 76 + 9

85 + 56 = 89 + 60 63 + 72 = 66 + 67

Part of a Student Sheet

For each of the following fill in the box without adding the numbers up on each side

85 + 34 = 86 + 40 + = 42 + 5

34 + 88 = 39 + 55 + = 52 + 53

Write three more equations using different numbers.

57 + = 54 + 57 + = 54 +

57 + = 54 +

Complete the statement: The number in the right box is always 3 than the number in the left box

Fill in the empty boxes. Each letter stands for any number.

75 + n = 72 + 96 + n = 99 +

k + 45 = + 42 m + 300 = + 40

Letters Only

Complete down these equations.

a + c + b - c = x + y + w - y =

s + y + 11 - y = 18 - k + h + k =

Stages 5 and Earlier

What algebra is there at these stages?

Jean works out 57 + 8

Step 1: 57 +3 = 60

Step 2: 60 + 5 = 65

(Early Part-Whole: Stage 5)

Advanced Counting Stage 4

Find the ninth number in the pattern

3, 8, 13, 18 …

Students’ Major Misunderstandings of the

Meaning of Letters in Algebra

Letter Needs a ValueLetter is seen as a numerical value instead of being treated as an unknown or generalised number. Sally in Peanuts says in a maths test “x equals 7, x is always 7.”Letter is an ObjectLetter is seen as an object. For example, 2a + 3a + a = 6a because 2 apples + 3 apples + 1 apple = 6 apples.

Students’ Correct Understandings of the Meaning of Letters in Algebra Letter Used in Number GeneralisationsFor example, the correct generalisation the order of addition of two numbers does not affect the answer can be expressed in letters:a + b = b + a where a and b can be any numbers.

Letter Used in Pattern GeneralisationsFor example, the nth term in the sequence 4, 8, 12, 16,… is 4n.

Assessing Understanding of Letters

Levels Description of Level

0, 1,2

At best students solve problems without realising letters represent any numbers. Students use particular numbers for letters, or regard letters as objects. Students at these levels effectively understand no algebra

Levels Description of Level

3,4Students treat letters as specific unknowns or generalised numbers. These students have understanding of algebra

Percentage of English Children at Algebra Levels (Hart)

Levels 13 years 14 years 15 years

0, 1, 2 83% 65% 58%

3, 4 17% 35% 42%

Algebraic Thinking Is Not An Optional Extra

I have a number problem. Can I work

out the answer mentally?

Yes No

I work out the answer. I am using algebraic thinking

I choose to work out the answer by pencil and

paper or calculator. I am not using algebraic

thinking

I must estimate the answer as a check. I am using algebraic

thinking

NCEALevel 1 = Old School Cert – Year 11Level 2 = Old Sixth Form Cert – Year 12Level 3 = Old Bursary – Year 13

Level 2 2003 Algebra results a disaster500 teacher secondary pilot in 2005

NCEA Results 2003(NZQA, 2003)

Level 1 Mathematics (Algebra): Use straightforward algebraic methods to solve equations

“Achieved” Questions

Q1 Simplify 3x4.2x3 Q2 Expand and simplify 3(x + 1) + 2(x - 3)

Q3 Josh is estimating the area of a circle by using A = 3r2. What is the area if r = 5?

Q4Josh increases his donations to World Vision by $2 each month.

Month (m) Donations (d)

1

2

3 $12

4

5 $16

6

7

Write a rule for the amount Josh donates after m months.

Q5 Solve the equations:a 2x(x + 3) = 0b 3x + 5 = x + 6

c 2x =

5

3 7

Achieved Criteria

3 out of 4 of the first four questions and2 out of 3 from Q5

A Merit QuestionJosh bought mini pizzas for his flat one

night.

The Supreme pizza was 50 cents more than the Hawaiian pizza.

3s + 2h = 20.75 s = h + 0.5

Solve these equations to find the price of one Supreme pizza.

Results

n = 39,067

Not Achieved Achieved

Merit Excellence

50.1% 32.8% 12.7% 4.4%

Summary

• The Numeracy Project is not about computation

• The Numeracy Project is about Algebraic Thinking

• The Numeracy Project is about making generalisations• Algebraic Thinking in Primary &

Intermediate is really important

Purpose

Looking to the future the good measure of the long-term success of the Numeracy Projects will be the improvements in algebra performance in NCEA.

Some Examples of Stage 8 Advanced Proportional Thinking