Transcript Y6/7

Laying the
Foundations for
Algebra
Objectives
• To identify a progression in Algebra.
• To form and solve equations
• To explore patterns, sequences and
rules.
• To generalise number sequences and
express relationships algebraically.
Lancashire Mathematics Team
Algebra in Primary School
Focus for this session:
•
•
•
•
•
Forming equations
Solving equations
Using inverses
Identifying number patterns
Expressing relationships
Forming equations
6+3
=
9
Solving Equations
6+3=
6+=9
Solving Equations
6+3=+4
Solving Equations
6+3=+
Solving Equations
What do we
know about
the missing
number?
What could
we do next
to find the
missing
number?
Using inverses
I think of a number, subtract 10 and
double the result.
The answer is 44. What is my number?
Answer the question and discuss the
strategies you used.
Using inverses
( - 10) x2 = 44
( - 10) = 44 ÷ 2
( - 10) = 22
 = 22 + 10
 = 32
Using inverses
Emily chooses a number. She halved the number then
added 10 to the result. Her answer was 35. What was the
number she started with?
?
+10
Answer 35
Using inverses
Ben thinks of a number. He adds half of the
number to a quarter of the number. The result is
60. What was the number Ben first thought of?
What images might you draw to answer this
question?
Identifying number patterns
Before identifying patterns in number, they need to be
able to identify and make repeating patterns using
shapes and colours
Identifying number patterns
?
?
?
Identifying number patterns
Identifying number patterns
2, 7, 12, 17…..
Can you continue the pattern?
What would the 20th term be?
Patterns, sequences and rules
YR Talk about, recognise and recreate simple repeating
patterns.
Y1 Describe simple patterns and relationships involving
numbers or shapes; decide whether examples satisfy
given criteria.
Y2 Describe patterns and relationships involving numbers
or shapes, make predictions and test these with
examples.
Patterns, sequences and rules
Patterns, sequences and rules
Y3 Identify patterns and relationships involving numbers or shapes,
and use these to solve problems.
Y4 Identify and use patterns, relationships and properties of
numbers or shapes; investigate a statement involving numbers
and test it with examples.
Y5 Explore patterns, properties and relationships and propose a
general statement involving numbers or shapes; identify
examples for which the statement is true or false.
Y6 Represent and interpret sequences, patterns and relationships
involving numbers and shapes; suggest and test hypotheses;
construct and use simple expressions and formulae in words then
symbols.
Y6/7 Generate sequences and describe the general term; use
letters and symbols to represent unknown numbers or variables;
represent simple relationships as graphs.
Patterns, sequences and rules
Make the fourth shape using multilink.
Describe the shape to your partner.
Can you explain how the pattern is
developing?
Expressing relationships
Possible answers:
•
Vertical columns: 1, 2+1, 3+2, 4+3,
•
Pairs added:
1, 1+2, 1+2+2, 1+2+2+2,
•
Complete rectangle
1, (2x2)-1, (2x3)-1, (2x4)-1
• Predict what the 10th shape will look like and how
many cubes will be used to make it.
• Formulae: 2n-1 where n is the number of cubes in the
left hand column.
Patterns, sequences and rules
Here is a sequence of patterns made from squares and circles.
How many squares will be in the pattern that has 25 circles?
Patterns, sequences and rules
No. of squares
No. of circles
1
3
2
5
3
7
Patterns, sequences and rules
No. of squares
No. of circles
1
3
2
5
3
7
4
9
5
11
12
25
Patterns, sequences and rules
No. of squares
No. of circles
1
3
2
5
3
7
4
9
5
11
?
25
Snowflake sequences
Cops and Robbers
Cops and Robbers
Area ITP
Money bags
Ram divided 15 pennies among four small bags.
He could then pay any sum of money from 1p to
15p, without opening any bag.
How many pennies did Ram put in each bag?
“The radical mistake of algebra teaching
is in jumping from Particular Arithmetic
to Symbolic Algebra and omitting work
on Generalised Arithmetic……for
generalised arithmetic is the simplest
form of algebra.”
B. Branford 1908
Generalised Arithmetic
• Work out the following:
3037 - 258
• Use it to find the following:
3037 - 259
3037 – 257
3037 – 268
• Write two calculations with the same answer
as 6214 – 1989, explain what you did to the
numbers.
Algebra times table
B x H = AE
A x H = CF
G x H = FD
FxH=D
CJ x H = HJ
I x H = FJ
E x H = FH
CxH=H
H x H = CE
D x H = AF
Links
The links have these values:
• Red = 2
Yellow = 3
• Blue = 4
Can you make a chain worth 18?
How many different chains worth 18 can you
make?
Links
All of these chains have a total value of 12.
Partner numbers
Every number has a partner
number.
Make up a rule.
Fill in some partner numbers.
Give it to a friend to try to
complete.
9
The Process
Look at it
Try to extend it
Verbalise it
Predict
Generalise
Playtrain
Key Messages
• It is important to lay the foundations for Algebra
from Reception.
• Children need to build upon their previous
experience. It is important to follow a
progression.
• Encourage children to follow the process – look at
a pattern, extend it, verbalise it, predict and
generalise.
Playtrain