Transcript Towradgi Public School Maths Information Night

Presented by John Santos
What is tonight all about?
• The aim of tonight’s Maths session is to clear up some of the confusion
that surrounds the way we teach some maths concepts in today’s
classrooms.
• As parents, we have all experienced the situation where our child has
asked for help with their maths homework and when we showed them
‘the way we did it at school’, they stared at us with a look of total
confusion, or said “That’s not right!”
• After tonight ‘s presentation, I hope you will feel better equipped and have
greater confidence when you next assist your child with their Maths.
• The areas we will focus on are:
Addition and Subtraction
 mental strategies.
 trading
Multiplication and division
 mental strategies.
Multiplication and division algorithms
The Language of Addition and Subtraction
+
plus
minus
add
addition
and
take away
subtraction
difference
total
altogether
added to
sum of
remaining
what’s left
more than
change from $
how much
join groups
combine groups
remove
give away
difference
Addition and Subtraction
Developmental Levels
There are 5 main developmental levels that children
move through on their way to becoming independent
and efficient addition and subtraction problem
solvers.
Addition and Subtraction Developmental Levels
Emergent
A child at this level may or may not be able to count
from 1 to 10. The child cannot count objects
correctly.
1 3 7 4
5
1 2 3 4 5 6 7 8 9 10
Perceptual
A child at this level needs to see or touch the groups
of objects and counts each object one at a time.
1 2 3 4
1 2 3
1 2 3 4 5 6 7
Figurative
A child at this level can build a picture of objects in
his/her head and will count each pictured object one
at a time, starting from one
1 2 3 4
1 2 3
1 2 3 4
5 6 7
Counting On
At this level, a child can add two numbers by
holding the larger number in their head and
counting on by ones e.g. 10 + 5 =
10
11, 12, 13,
14, 15
Using mental strategies
A child at this level counts by
numbers other than one, and uses
strategies such as the Jump, Split
and Compensation.
Friends of Ten
To develop their understanding of our
number system it is very important that
children learn their addition and
subtraction combinations to 10.
This means the numbers that add
together to total ten. We call these the
‘Friends of Ten’, for example 3 and 7, 2
and 8, 5 and 5 etc.
Bridging to Ten
6 + 8 = 14
+4
+4
6
10
14
6 + 4 = 10, 10 + 4 = 14
I know my ‘friends of 10’ so I know that 6+ 4 = 10.
That still leaves 4 to add so I add that to 10 and
get 14.
63 + 29 =
Jump Strategy
+9
+ 20
63
83
92
63 + 20 = 83, 83 + 9 = 92
I kept the 83 whole and split the 29 into 20 and
9. Then I added 20 to 63 and got 83. Then I
added the 9 and got 92.
Bridging to Ten / Jump Strategy
6 + 14 = 20
+4
6
+10
10
20
6 + 4 = 10, 10 + 10 = 20
First I added 4 to the 6 to get 10, then I
added another 10 and got 20.
Jump Strategy / Bridging to Ten
63 + 29 =
+20
63
+7
83
+2
90 92
63 + 20 = 83, 83 + 7 = 90, 90 + 2 = 92
I kept the 83 whole and split the 29 into 20 and
9. Then I added 20 to 63 and got 83. Then I
added 7 because 3 and 7 make a ten and got
90. Then I added the other 2 and got 92.
Split Strategy
63 + 29 =
+20
60
+3
80 83
+9
92
60 + 20 = 80, 3 + 9 = 12, 80 + 12 = 92
I split the 63 into 60 and 3, and the 29 into
20 and 9. Then I added the 60 and the 20
and got 80. Then I added the 3 and the 9 and
got 12. Then I added the 80 and the 12 and
got 92.
Compensation Strategy
63 + 29 =
+ 30
63
-1
92 93
63 + 30 = 93, 93 – 1 = 92
First I added 30 to 63 because 29 is nearly 30
and it’s easier to add tens. I got 93. Then I had
to take one away because 30 is one more than
29 and I got 92.
Addition Algorithm Procedure
63
+ 29
92
¹
We say:
3 plus 9 equals 12, write down the 2 and
trading one 10 to the tens column.
6 plus 2 equals 8, plus the 1 equals 9.
•The older a person gets the more
they will use mental strategies to
solve mathematical problems.
•Accordingly, it is important that
children develop a conceptual
understanding of mental strategies
and place value rather than rely
solely on algorithms.
Subtraction Algorithm Procedures
Decomposition
4 5¹2
- 18
34
We say:
2 minus 8 you can’t do so we trade a ten from
the tens column. Now my 2 is 12. 12 minus 8
you can do. It leaves 4. Write down the 4.
4 minus 1 equals 3. Write down the 3.
Subtraction Algorithm Procedures:
We say:
0 minus 3 you can’t do. So I need to
get a ten from the tens column but
there aren’t any. So I need to get a
hundred from the hundreds column to
give to the tens column but there
aren’t any. So I can get a thousand
from the thousands column to give to
the hundreds column. That leaves 7 in
thousands column and 10 in the
hundreds column. I give one hundred
to the tens column. That leaves 9 in
the hundreds column and 10 in the
tens column. NOW I can give a ten
from the tens column to the ones
column …..
Decomposition with Zeros
9
1
9
1
8000
- 6 73
73 2 7
7
1
10-3=7,
9-7=2,
9-6=3,
7-0=7 ….
Oh forget it! Let’s just use
the compensation strategy
…….
Subtraction Algorithm Procedures: Compensation
Change the 8000 into 7999 + 1.
7999
- 673
7326
7326 + 1 = 7327
Subtraction Algorithm Procedures
Equal Addends
5¹2
- ¹1 8
34
We say:
2 minus 8 you can’t do so we add a ten to the ones
column in the top number and a ten to the tens column
in the bottom number.
Now my 2 is 12. 12 minus 8 you can do. It leaves 4. Write
down the 4.
5 minus 2 equals 3. Write down the 3.
The Language of Multiplication and Division
X
÷
multiply
equal groups
times
multiples
factors
equal rows
array
double, triple
product
divide
equal share
equal groups
equal parts
quotient
remainder
equal rows
array
fraction
percentage
Multiplication and Division Levels
Forming Equal Groups
A child at this level will be able to form the
objects into equal groups but will then count each
object by ones, to find the total number of
objects.
1 2 3 4 5 6 7 8 9
Multiplication and Division Levels
Perceptual
A child at this level will be able to form the
objects into equal groups and will skip count
or rhythmically count while looking at or
touching the counters, to find the total.
1 2 3
3
4 5 6
6
9
7 8 9
Multiplication and Division Levels
Figurative Units
A child at this level will not need to see the
individual objects but will need a marker for
each group.
They will be able to skip count or rhythmic count
to find the total number of objects represented
by the group markers.
1 2 3
4 5 6
7 8 9
3
6
9
Multiplication and Division Levels
Repeated Abstract Composites
A child who is at this level will use known facts
or doubles.
8x7=
2 x 7 = 14, 2 x 14 = 28, 2 x
28 = 56
4 x 7 = 28, so 8 x 7 = 56
I know 7 x 7 = 49, so 7 x 8 = 56
Multiplication and Division Levels
Multiplication & Division as Operations
7 x 8 = 56
Arrays
3 rows of 4
makes 12
4 + 4 + 4 = 12
4 x 3 = 12
3 x 4 = 12
12 ÷ 4 = 3
12 ÷ 3 = 4
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x
1
2
3
4
5
6
7
8
9 10
1
= = = = = = = = = =
2
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3
= = = = = = = = = =
4
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5
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7
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9
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10 = = = = = = = = = =
Mental Strategies: Multiplication and Division
Repeated Addition
+26
0
26 x 4 =
+26
26
+26
52
+26
78
104
26 + 26 = 52, 52 + 26 = 78, 78 + 26 = 104
I added 26 and 26 and got 52.
Then I added another 26 and got 78. Then
I added the 4th 26 and got 104.
Mental Strategies: Multiplication and Division
Doubling
26 x 4 =
Double 26
0
26
Double 52
52
Double 26 = 52. Double 52 = 104
I doubled 26 and got 52.
Then I knew I needed another 2 26s
which I knew was another 52
so I doubled 52 and got 104.
104
Mental Strategies: Multiplication and Division
Compensation
Strategy
26 x 4 =
4 x 25
0
100 104
4 x 25 = 100, 100 + 4 = 104
I knew that 25 times 4 is 100.
Then I needed 1 more 4 to make 26 4s.
So 100 plus 4 made 104.
Mental Strategies: Multiplication and Division
Split Strategy
26 x 4 =
4 x 20
0
4x6
80
104
4 x 20 = 80, 4 x 6 = 24 then 80 + 24 = 104
I knew that 26 was made of 20 plus 6
20 times 4 is 80
6 times 4 is 24.
80 plus 24 is 104
The Multiplication Algorithm:
Extended Form
26
X 4
24
80
104
We say:
4 times 6 equals 24,
write down the 24.
We write a zero
in the ones column.
Then we say 4 times 2 equals 8,
and write it in the tens column.
We then add 4 and 0 to equal 4
and 2 and 8 to equal 10.
The Multiplication Algorithm:
Contracted Form
We say:
² 26
4 times 6 equals 24,
write
down
the
4.
X 4
and carry the 2.
104
4 times 2 equals 8,
plus the 2 equals 10.
Write down the 10.
When solving an algorithm, we treat each digit
as a ‘one’, even the ‘tens’ and ‘hundreds’!
The Division Algorithm Extended Form
)
0 2 6
4 1 0 4
-8
2 4
-2 4
0
4 into 1 goes 0 times, write down the 0
4 into 10 goes 2. Write down the 2.
Check that division fact using
multiplication: 2 x 4 = 8.
Write down the 8 below the 10.
Subtract the 8 to find the remainder:
10 – 8 = 2. Write it below the 8.
Bring down the next number
which is 4.
4 into 24 goes 6. Write 6 above the 4.
Check that division fact using
multiplication: 6 x 4 = 24.
Write it below the other 24.
Subtract the 24 to find the remainder:
24 – 24 = 0.
The Division Algorithm Contracted Form
4
)
026
2
104
4 into 1 goes 0 times,
write down the 0.
4 into 10 goes 2.
Write down the 2 above the 10.
2 x 4 = 8 so there are 2 left over,
write it in front of the 4.
4 into 24 goes 6,
write 6 above the 4.
When solving an algorithm, we treat each digit as a
‘one’, even the ‘tens’ and ‘hundreds’! A reliance on the
algorithm limits children’s conceptual understanding of
division and place value.
What Happens If this Doesn’t
Work?
When we have taken students through this
exhaustive process and tried our very best to
teach them Maths, what do we turn to when
all else fails?????
Mr O’Connor