Transcript Introducing_Algebra

Introducing Algebra
14th November 2006
Algebra involves …
… developing pupils’ understanding of “generalised
arithmetic”
… constructing and transforming expressions and
formulae
… representing problems and their solutions in tabular,
graphical or symbolic form.
… developing reasoning skills
Algebra is …
… a puzzle
Engaging pupils
Engaging pupils
Engaging pupils
Engaging pupils
Two Principles
Use ‘concrete’ examples in contexts familiar to pupils
24 chocolates altogether
Base new work on prior mathematical understanding
Spilt Ink
I’ve knocked a paint pot over my homework and I can’t
remember the number underneath the stain.
What number was it?
6 + 3 = 14
Hidden number
What number am I hiding beneath my finger?
2 x
+ 1 = 13
Find the number
What number does
3 x
stand for?
- 2 = 19
Number sentences with holes in
What numbers could
+
 and  stand for?
6 +

= 14
Puzzle? Maths?
Nrich example
On a farm there were some hens and sheep.
Altogether there were 8 heads and 22 feet.
How many hens were there?
A = 1 B = 2 C = 3 D = 4 E = 5 etc…
Crack the Code
H
8
Code Key
13
M
V
22
1
A
O
15
+5
20
T
C
3
8
H
N
14
19
S
Cryptarithmetic
O N E
+
T W O
S
I X
Cryptarithmetic
S E N D
+
M O R E
M O N E Y
Year 6 lesson
Number Sequences with holes in
Maths Pack 1
Number Sequences
4
7
10
13
16
1st
2nd
3rd
4th
5th
20th ?
GCSE
Write down the next term in the following sequence.
5
8
12
17
………
Maths Foundation Tier June 2004 Paper 1 Q. 4(a)
Next term?
1
2
…
1
2
3
…
1
3
6
…
0
1
1
2
O
E
T
2
T
G
F
6
E
…
F
8
S
G
S
5
E
E
Nine
…
7
…
V
Odd One Out
Function jigsaw
Number machines
Number machines
Maths Pack 2
Number machines
Number machines
Maths Pack 2
Simple formulae expressed
in words
I think of a number, then multiply it by 2.
I get the answer 14.
What number did I think of?
Number Magic
Think of a number between 1 and 10.
Add 7 to it.
Multiply the answer by 2.
Subtract 4 from the result.
Divide the answer by 2.
Take away the number you first thought of.
If A =1, B = 2 etc. Convert your answer into a letter.
Think of an animal whose name starts with this letter.
Number Magic :
How does it work?
Think of a number between 1 and 10.
n
Add 7 to it.
n+7
Multiply the answer by 2.
2n + 14
Subtract 4 from the result.
2n + 10
Divide the answer by 2.
Take away the number you first thought of.
n+5
5
If A =1, B = 2 etc. Convert your answer into a letter. E
Think of an animal whose name starts with this letter. . .
Guess the Rule
My rule multiplies by
something then takes
something away
What answer does 3
give with your rule?
11
Try another
number or
guess my rule
Guess my Number
My rule multiplies by
3 then takes 2 away
What is your answer?
13
Guess the
number I
started with
Finding the rule
Input
X3
Output
+ 2
1
3
5
2
6
8
3
9
11
4
12
14
5
15
17
3
3
3
3
Finding the rule
Input
X3
Output
+ 2
1
3
5
2
6
8
3
9
11
4
12
14
5
15
17
3
3
3
3
An activity
This is a 3-snake.
It takes seven straws to make this snake.
How many straws would it take to make a 100-snake?
Working through the
activity
1-Snake
3 Straws
4-Snake
9 Straws
2-Snake
5 Straws
3-Snake
7 Straws
5-Snake
11 Straws
Snake
X2
Straws
+ 1
1
2
2
4
5
3
6
7
4
8
9
5
10
11
100
X2
200
+ 1
3
201
2
2
2
2
Another Investigation
Solving equations
I think of a number, then add 3 to it.
I get the answer 10.
What number did I think of?
 + 3 = 10
Solving equations
I think of a number, then add 3 to it.
I get the answer 10.
What number did I think of?
 + 3 = 10
Solving equations
I think of a number, then add 3 to it.
I get the answer 10.
What number did I think of?
? + 3 = 10
?
Solving equations
I have a large stone to weigh on a balance.
When I add 5kg to the stone, they balance with 12kg.
What does the stone weigh?
S
+ 5=
12
Solving equations
I think of two numbers.
When I add 5 to the first number I get the answer 12.
When I add 3 to the second number I get the answer 12.
What numbers did I think of?
? + 5 = 12
? + 3 = 12
Solving equations
I think of two numbers.
When I add 5 to the first number I get the answer 12.
When I add 3 to the second number I get the answer 12.
What numbers did I think of?
a + 5 = 12
b + 3 = 12
Solving equations
I think of a number, then add 3 to it.
I get the answer 10.
What number did I think of?
a + 3 = 10
a
Loop Cards
Solving equations
S + 3
=
10
Solving equations
S + 3
=
10
7
Take 3 from each side
Solving equations
2S + 3
=
15
Solving equations
2S + 3
=
15
12
Take 3 from each side
Solving equations
2S
=
16
Take 3 from each side
Halve each side
http://nlvm.usu.edu/en/nav/vm1_asid_324.html
Solving equations
2a + 3 = 15
a
x by 2
2a
+3
2a + 3
6
 by 2
12
-3
15
Solving equations :
Trial & Improvement
2a + 3 = 26
Try a = 10
x by 2
20
+3
23
Too small
Try a = 11
x by 2
22
+3
25
Too small
Try a = 12
x by 2
24
+3
27
Too big
Try a = 11.5 x by 2
23
+3
26

Number Relationships
I think of two numbers.
When I add them together, I get the answer 10.
What numbers did I think of?
x + y = 10
“Y to the sky”
Connect 4 Coordinates
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Connect 4 Coordinates
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
My
table
is 4s100
90
Tables Grid
My
table
is 3s
80
70
60
50
40
7 x 3 is 21
6 x 4 is 24 30
20
10
0
0 1
2 3
4 5
6 7
8 9 10
Odd One Out
(3,6)
(2,5)
(5,10)
(4,8)
(1,2)
Odd One Out … Coordinates
10
(3,6)
(2,5)
9
8
7
6
(5,10) 5
4
(4,8)
(1,2)
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Odd One Out … Coordinates
10
(3,6)
(2,5)
9
8
7
6
(5,10) 5
4
(4,8)
(1,2)
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Partner numbers
6
4
13
3
11
19
5
16
Partner numbers : Rule
?
x by 3 then + 1
n
3n + 1
x + y = 10
10
9
8
Could be …
7
2 and 8
6
2nd Number
y
5
4
3
2
1
0
0
1
2
3
4
5
1st Number
6
x
7
8
9
10
x + y = 10
10
9
8
Could be …
7
6 and 4
6
2nd Number
y
5
4
3
2
1
0
0
1
2
3
4
5
1st Number
6
x
7
8
9
10
x + y = 10
10
9
8
Could be …
7
9 and 1
6
2nd Number
y
5
4
3
2
1
0
0
1
2
3
4
5
1st Number
6
x
7
8
9
10
x + y = 10
10
9
8
Could be …
7
… etc
6
2nd Number
y
5
4
3 ½ and 6 ½
3
7.4 and 2.6
1
2
0
0
1
2
3
4
5
1st Number
6
x
7
8
9
10
Patterns in Coordinates
10
9
(3,8)
(5,8)
(3,4)
(5,4)
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Coordinates & Translations
10
9
(4,8)
8
7
6
5
4
(6,5)
(1,4)
(4,5)
3
2
(3,1)
1
(1,1)
0
0 1 2
3
4
5
6
7
8
9
10
Graphs and Equations
3
x
x–
2=
y
3
x
x–
2
3
4
2=
y
10
9
x= 1
y= 1
8
7
6
5
4
3
2
1
0
0
1
5
6
7
8
9
10
3
x
x–
2
3
4
2=
y
10
9
x= 2
y= 4
8
7
6
5
4
3
2
1
0
0
1
5
6
7
8
9
10
3
x
x–
2
3
4
2=
y
10
9
x= 3
y= 7
8
7
6
5
4
3
2
1
0
0
1
5
6
7
8
9
10
a = 4
b = 3
“What else do I know?”
3a + 3b 2a + 2b 3a + b
= 21
= 14
= 15
a + b
= 7
a + 2b
= 10
a - b
= 1
2a =
8
3a - b
= 9
Year 6 Plenary
Sum and Product
Substitution Dice
White
Dice
Operation
2x
7x
3x + 1
4x -1
5x + 2
8x - 6
On your whiteboard …
I think of a number.
When I multiply it by itself I get the answer 42.
Estimate the number I thought of.
Developing Thinking
Find me a formula
that has the value 7
when a = 2 and b = 3
a
b =7
24 Game
Four 4s
+ - x
Estimation Golf
1st Hole : Par 4
273 < 19 x b < 275
2nd Hole : Par 5
160 < x2 < 162
What’s It Worth?
Maths Pack 3
Simplifying
Which expressions are the same as 7b?
Algebra Loop Card Game
Algebra Addition Squares
Algebra Addition Squares
Algebra Addition Squares
Algebra Addition Squares
Follow the rule
Follow the rule
Find the rule
nrich
http://www.nrich.maths.org.uk/public/index.php
“Real Life Graphs”
Like terms
I have 3 apples and 2 bananas in one bag and 4 apples
and 3 bananas in another bag.
How many apples and bananas do I have altogether?
Like terms
I have 3 apples and 2 bananas in one bag and 4 apples
and 3 bananas in another bag.
How many apples and bananas do I have altogether?
3 a and 2 b and 4 a and 3 b
7 a and 5 b
Like terms
I have 3 apples and 2 bananas in one bag and 4 apples
and 3 bananas in another bag.
How many apples and bananas do I have altogether?
3 a + 2 b + 4 a + 3 b
7 a + 5 b
Algebra Magic Square
Formulae in symbolic form
I bought 2 boxes of sweets from one shop and 3
boxes of the same sweets from another shop.
How many boxes of sweets did I buy?
2 boxes + 3 boxes = 5 boxes
If I then had 45 sweets altogether, how many sweets
were in each box?
5 boxes
= 45
1 box
= 45
5
= 9
Formulae in symbolic form
I bought 2 boxes of sweets from one shop and 3
boxes of the same sweets from another shop.
How many boxes of sweets did I buy?
2 boxes + 3 boxes = 5 boxes
If I then had 45 sweets altogether, how many sweets
were in each box?
5 boxes
= 45
1 box
= 45
5
= 9
Formulae in symbolic form
The cost of hiring a limousine is £30 plus £2 for
every mile travelled.
What would be the total cost of hiring the limousine
if you travelled 6 miles in it?
Total cost
=
£30 plus £2 for each of the 6 miles
=
£30 plus £12
=
£42
Formulae in symbolic form
The cost of hiring a limousine is £30 plus £2 for
every mile travelled.
What would be the total cost of hiring the limousine
if you travelled 6 miles in it?
C
=
30 + 2 x 6
=
30 + 12
=
£42
Formulae in symbolic form
The cost of hiring a limousine is £30 plus £2 for
every mile travelled.
What would be the total cost of hiring the limousine
if you travelled 6 miles in it?
C
=
30 + 2 x 6
=
30 + 12
Cost =
£42
Formulae in symbolic form
The cost of hiring a limousine is £30 plus £2 for
every mile travelled.
What would be the total cost of hiring the limousine
if you travelled 8 miles in it?
C
=
30 + 2 x 8
=
30 + 16
Cost =
£46
Formulae in symbolic form
The cost of hiring a limousine is £30 plus £2 for
every mile travelled.
What would be the total cost of hiring the limousine
if you travelled m miles in it?
C
=
30 + 2 x m
Formulae in symbolic form
The area of a rectangle is calculated by multiplying
the length times the width.
What would be the area of a rectangle of length 6 cm
and width 4 cm?
Area =
=
=
Length x Width
6 cm x
4 cm
24 cm2
Formulae in symbolic form
The area of a rectangle is calculated by multiplying
the length times the width.
What would be the area of a rectangle of length 6 cm
and width 4 cm?
Area =
=
=
Length x Width
6 cm x
4 cm
24 cm2
Formulae in symbolic form
The area of a rectangle is calculated by multiplying
the length times the width.
What would be the area of a rectangle of length 6 cm
and width 4 cm?
Area =
Length x Width
Formulae in symbolic form
The cooking time for a chicken is ½ hour per kg plus
an extra ½ hour. It is also suggested to allow a
further ¼ hour before serving.
At what time should I put the chicken into the oven
if I want to eat at 1 p.m. and the chicken weighs 4
kg?
Cooking Time
=
½x4+½+¼
=
2 +½+¼
=
2¾
hours
hours
hours
etc…
Formulae in symbolic form
The cooking time for a chicken is ½ hour per kg plus
an extra ½ hour. It is also suggested to allow a
further ¼ hour before serving.
At what time should I put the chicken into the oven
if I want to eat at 1 p.m. and the chicken weighs w
kg?
Cooking Time
=
½xw+½+¼
hours
etc…
Brackets
I’ll have 3 fish and 2 lots of chips, please.
3 fish + 2 chips
3 fish + 2 chips
I’ll have fish and chips twice, please.
2(fish + chips)
2(fish + chips)
Grid method for 34
x
27
30
4
20
600
80
680
7
210
28
238 +
=
918
Grid method for (x + 2)(x + 3)
x
+2
x
2
x
+2x
+3
+3x
+6
=
x2 +5x +6
MicroSMILE : Algebra
Equation Buster
Flag Diagrams
Another sequence
What is the next number in this list :
1
Another sequence
What is the next number in this list :
1
2
Another sequence
What is the next number in this list :
1
2 4
Another sequence
What is the next number in this list :
1
2 4 7
Another sequence
What is the next number in this list :
1
2 4 7 23
…