Transcript Number and algebra 7-9 - CED-Mxteachers-news-site

Number and Algebra yr 7-9
Alison Fagan Maths advisor
Outline of course
• Curriculum changes from MINZC to NZC
• Mathematics standards Number and Algebra
yrs 7-8
• Number
• Algebra
Curriculum changes from MINZC to
NZC
Handout changes document
Identify one major change from MINZC to
NZC
Number
•
•
•
•
•
adding and multiplying strategies
decimals
fractions
percentages
ratio
Students learn through
Material
Imaging
Knowledge
Materials appropriate for this level
• Numberlines
• Arrays
Using number lines
The number line is the “material”
These can become images for the student
Using tidy numbers
( bridging through 10)
For addition
eg 18 + 6 = 18 +2 + 4
+2 + 4
20
= 24
What numeracy stage
must students be at before
they can do this??
Stage 5,
able to part-whole
Using tidy numbers for subtraction
33-16 = 33 – 3 -10 – 3
= 17
-3 -10 -3
20
30
Rounding and compensating
For addition
eg 423 + 189 = 423 +200 – 11
(use an “empty number line for large
numbers)
612
423
623
What happens for subtraction??
374 – 59 = 374 – 60 + 1
314
315
374
A special one for subtraction
Reversibility
(Don’t subtract ADD)
Eg 36 -17 =
,
Find 17 +
= 36
+10
+3
17
+6
20
3 +10 + 6 = 19,
So 36 – 17 = 19
30
36
Write a word problem for one of
Using tidy numbers ( through 10’s)
Rounding and compensating
Don’t subtract ADD
Multiplying
Use skip counting as a visual link for
multiplying
Eg 4 x 5 is 4 skips of 5 along number line
0
10
20
Dividing
Dividing is the reverse ( inverse)
process to multiplication,
So to find 30 ÷ 3 ,
how big is each skip if it takes me 3
skips to get to 30?
The first step might be to…..
Using a strip of paper tape folding it and
marking the fold lines.
Remember to talk about the meaning of
numerator and denominator
0
1
2
1
1
2
Then progress to double number lines
0
1
4
1
2
3
4
1
To find the fraction of a quantity..
eg. one quarter of the class of 32 students travel
to school by bus,
how many of the class travel by bus?
0
1
4
8
1
2
3
4
1
16
24
32
So 8 students travel by bus
These lines can be simplified to
0
0
1
4
1
8
32
What stage do students need to be to
do this?
Multiplicative?
Arrays
Arrays are a more powerful tool for
multiplication
This array could be used to show
7 + 7+ 7 + 7
or 4 x 7
To show 6 x 7
These arrays can be simplified to box
diagrams eg for 13 x 12
x
10
3
100
10
100
30
+30
+20
+6
= 156
2
20
6
And even more useful for decimals
eg 1.3 x 1.2
x
1
0.2
1
0.3
1
0.3
0.2
????
How do most teachers explain that the answer to 0.3 x
0.2 = 0.06??
Does this array help?
This can also be explained using a
version of the decimat.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2 x 0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Each of these small squares are hundredths so, we have 6
hundredths =0.06
Can also be used to demonstrate
multiplication of fractions
eg ½ x ½ = ¼
0
½
1
0
½
1
BUT…what does the side of the
square represent now???
We can extend this idea to a series of
puzzles which start numerically and
can lead to algebraic factorising
x
a
10
b
5
100
a 10
100
50
+70
+10
c2
20
10
What do a, b and c =?
180
x
a
b
400
a
400
120
160
+12
c
40
12
What do a, b and c =?
572
x
a
b
100
a
100
100
+16
c
16
What do a, b and c =?
216
Other uses of arrays
They can be used for ratios,
eg sharing in a given ratio
Eg at school there are 2 girls for every 3
boys
This can be written 2:3
This could be shown as
So if there were 10 children altogether
We could show them like this
This can be extended to find the number
of girls and boys in a class of 30,
Or in a school of 300?
This helps students understand the
algorithm of sharing in a given ratio
What would be a rich mathematical task to
give to year 7- 9 students involving
sharing in a given ratio?
Ratio and Measurement
Use the proportion shapes
How does this relate to area???
Algebra
• What is algebra?
a language
a way of thinking
a tool for problem solving
patterns, relationships, generalities,
modelling
Algebra
•
introduction
• generalising
• patterns
• equations
Algebra for students
algebra is “letters” or “hard”
• letters have a naming function
• letter as an unknown number
• letters as generalised numbers
• letters are variables
How do we use letters in
mathematics?
A letter can be used to name
something
In the formula for the area of a rectangle,
the base of the rectangle is often named b
A letter can be used to stand for
a specific unknown number that
needs to be found
In a triangle, x is often used for the angle
students need to find
A letter is really a number
evaluate a + b if a = 2 and b = 3
In the sequence, (n, 2n - 1),
n takes on the values of the natural
numbers
– sequentially
Generalising
Develops from number properties
eg multiplication 3 x 9 = 3x 10 – 3x 1
leading to 4 x 99 = 4 x 100 – 4 x 1
And then 4 x (a + 1) = 4 x a + 4 x 1
Patterns
Sequential
Recursive
Rule can be generalised or found from previous term
What’s coming up next?
1
1
1
0
2
3
2
1
4
3
2
5
5
3
6
8
4
7
7
8
9
13
21
34
6
15
8
Write a rule
for this
pattern
Write a rule
for this
pattern
Figure it Out
Building Borders
FIO, Link, Algebra, Book One,
Building Borders, pages 12-13
Three bean salad
Equations
We can adapt double number lines to “block
diagrams “ to solve algebra problems.
This helps to enhance the connection between
number and algebra for students
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?
How would a student solve this?
How would you solve it?
The “traditional” way??
2 p – 6 = 14
2 p = 20
p = 10
Add 6 to
both sides
Divide both
sides by 2
10 sweets
in each
packet
Or using “numberlines” ??
20 ÷ 2 = 10
p
14
p
6
14 + 6 = 20
A more complicated example..
Amy has 3 packets of biscuits and 4
loose ones.
Sam has one packet and 16 loose ones.
If they both have the same number of
biscuits
how many are in a packet?
The traditional way??
Take p
from both
sides
3 p + 4 = p + 16
2 p + 4 = 16
2 p = 12
p=6
Take 4
from both
sides
Divide both
sides by 2
6 biscuits
in each
packet
Or using “numberlines”??
p
p
p
p
4
16
p
p
6
6
4
2p + 4 =16
2p =12
p=6
Or more simply, as students will
p
p
p
4
p
6
16 6
4
Write your own word problem
Swap with a partner
Solve using a number line
HOMEWORK !!
Take one of these activities and USE it
Feedback at our next workshop on
Its success/ disaster
How you would change/ adapt it
Next workshop
Contact me at
[email protected]
With ideas/ suggestions/ requests for next
workshop