Transcript Mathematics Workshop

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TEN (Targeted Early Numeracy) is a program that
explicitly teaches students in K-2 the fundamental
skills of addition and subtraction for problem
solving.
It targets students identified at risk in 5 week cycles
by the classroom teacher to improve their ability to
add, subtract and recognise numbers in many
different learning situations.
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Focus groups have 3-4 students only in them.
Skills and strategies are targeted through intense ten
minute sessions every day, using games and
strategies that promote success.
Students are assessed constantly to ensure
appropriate intervention happens.
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The program focuses on Aspect 2 of the Numeracy
Continuum, Counting As a Problem Solving Process.
TENS
Addition and Subtraction 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
TENS
Addition and Subtraction Levels
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
TENS
Addition and Subtraction Levels
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
TENS
Addition and Subtraction Levels
Counting on and back
A child at this level will keep the greater number
in his/her head and count on or back the lesser
number.
34 + 7 …..
34 35 36 37 38 39 40 41
TENS
Addition and Subtraction Levels
Facile
A child at this level counts by numbers other
than one, and may use strategies such as the
Jump, Split and Compensation.
34 + 7 …..
I know 4 and 6 makes 10 so
that’s 40 and 1 which makes
41
Working Mathematically
Addition and subtraction Strategies
Once a child gets to Year 3 they should be
able to or are starting to count by numbers
other than one, and use strategies such as
the Jump, Split and Compensation.
We like to get the students to share their
strategies with each other and say how they
solved the problem.
Mental Strategies for Addition and Subtraction
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.
Mental Strategies for Addition and Subtraction
63 + 29 =
Jump
+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.
Mental Strategies for Addition and Subtraction
63 + 29 =
Jump
+9
+ 20
63
83
92
63 + 20 = 83, 83 + 9 = 92
I kept the 63 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.
Mental Strategies for Addition and Subtraction
63 + 29 =
Split
+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.
Mental Strategies for Addition and Subtraction
63 + 29 =
Compensation
+ 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
¹
63
+ 29
92
We say:
3 plus 9 equals 12, write down the 2 and
add one 10.
6 plus 2 equals 8, plus the 1 equals 9.
When solving an algorithm,
we treat each digit as a ‘one’,
even the ‘tens’!
A reliance on the algorithm
limits children’s conceptual
understanding of mental
strategies and place value.
Mental Strategies for Addition and Subtraction
52 – 18 =
Number Line:
Numbers:
Words:
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.
Subtraction Algorithm Procedures
Decomposition
4 5¹2
-18
34
We say:
2 minus 8 you can’t do so we get 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: Decomposition with Zeros
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 …..
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
What is Multiplication?
joining equal groups together to see
how many altogether.
repeated addition
What is Division?
splitting / sharing a group into smaller
equal groups.
repeated subtraction
Multiplication and Division are inverse
operations.
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
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In the early years students will use objects to
make groups or arrays to understand what
multiplication and division really is and the
process.
4 groups of 3 is….. Or 4 X 3 =
Multiplication Strategies
A child may 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
Mental Strategies: Multiplication and Division
26 x 4 =
Repeated Addition
+26
0
+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
26 x 4 =
Doubling
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
26 x 4 =
Compensation Strategy
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
26 x 4 =
Split Strategy
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
10 4
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:
4 times 6 equals 24,
²26
write down the 4.
X 4
and carry the 2.
4 times 2 equals 8,
10 4
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 Multiplication Algorithm:
Extended Form - 2 digits by 2 digits.
1 2
46
X 24
18 4
920
11 0 4
1
4 times 6 equals 24,
write down the 4 and carry the 2.
4 times 4 equals 16,
plus the 2 equals 18.
Write down the 18.
We write a zero in the ones column.
Then we say 2 times 6 equals 12,
write down the 2 and carry the 1.
2 times 4 equals 8,
plus the 1 equals 9.
Write down the 9.
We then add 4 plus 0 equals 4.
8 plus 2 equals 10,
write down the 0 and carry the 1.
1 plus 9 equals 10,
plus the 1 we carried equals 11.
Mental Strategies for Multiplication
and Division
104 ÷ 4 =
Repeated Subtraction
-26
0
-26
26
-26
52
-26
78
104
Repeatedly subtracting 4 from 104 is inefficient –
students mix with other strategies such as:
104 – 26 = 78, 78 – 26 = 52, 52 – 26 = 26,
26 – 26 = 0.
I knew 25 x 4 was 100 so 26 x 4 was going to be 104.
I demonstrated on a number line by keeping on
subtracting 26.
Mental Strategies for Multiplication
and Division
104 ÷ 4 =
Halving
Halve 52
0
26
Halve 104
52
104
Halve 104 = 52. Halve 52 = 26. I halved 104 and got
52.
Then I knew I needed to halve again because
dividing by 4 is like finding a quarter and a quarter is
half of a half.
So I halved 52 and got 26.
Mental Strategies for Multiplication
and Division
104 ÷ 4 =
Compensation Strategy
100 ÷ 4 = 25
0
25
50
-4
75
100 104
100 - 4 = 100, 100 ÷ 4 = 25, 4 ÷ 4 = 1, 25 + 1 = 26
I took 4 away from 104 and got 100.
Then I did 100 ÷ 4 = 25.
I still had the 4 that I took away and 4 ÷ 4 is 1, I
added the 1 to the 25 and it was 26.
Mental strategies increases children’s conceptual
understanding of multiplication, division and place
value.
The Division Algorithm
0 2 6
4 )1 0 4
2
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.
The Division Algorithm
0 2
4 )1 0
-8
2
-2
6
4
4
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.
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Students start with informal units and
comparison before moving on to formal units.
Students don’t start with formal units until the
end of year 2 or into year 3.
Students should not use informal units if they
have not understood the concepts for
measuring with informal units.
A rectangle shape is not
a good shape to
measure area because it
may be arranged in more
than one way eg
vertically or horizontally.
This makes consistent
measurement difficult.
The pattern made when measuring area with
squares is an array.
This is the same as the array children use in
multiplication and division and fractions.
The array structure provides the
understanding for rectangular area to be
calculated using multiplication.
1 row of 12
12cm x 1cm
3 rows of 4
3 cm x 4 cm
2 rows of 6
2 cm x 6 cm
½ cm x 24 cm
4 rows of 3
4 cm x 3 cm
6 rows of 2
6 cm x 2 cm
12 rows of 1
12 cm x 1 cm
Rectangular
Prism
6 sides,
opposite faces equal,
8 vertices,
all vertices equal
Cube
6 sides,
all faces equal,
8 vertices,
all vertices equal
The volume of the
rectangular prism
with 1 layer is ____
cubic centimetres.
The volume of the
rectangular prism with
3 layers is ____ cubic
centimetres.
The volume of the
rectangular prism with 2
layers is ____ cubic
centimetres.
The volume of
the rectangular prism
with 4 layers is ____
cubic centimetres.
Are there any
questions?