calculate strategies and algebra students must have a deep understanding of numbers, how they work...
TRANSCRIPT
Calculate Strategies and Algebra• Students must have a deep understanding of
numbers, how they work and the relationship between them, to be successful at algebra
• The Mathematics Australian Curriculum is at least 70% algebra in Year 9
• Using an algorithm provides none of the essential understandings needed for algebra
Many Calculate Strategies Require a Thorough Understanding of Place Value
Asked to expand 463 most students will write:
400 + 60 + 3 and it appears they understand place value.
If a child is struggling with the calculate strategies at school, they almost always do not understand place value.
2 6When asked to show where the ‘2’ is, student’s who don’t understand
place value will point to 2 ones
PartitioningMultiply by the tens and then the ones.
e.g. 6 x 27 is 6 x 20 add 6 x 7, so 120 add 42, which is 162
Distributive LawPartitioning to Multiply
16 x 4 =
10 x 4 = 40 6 x 4 = 24+
It is a small step to understanding that 16a is the same as 10a+6a
Multiplying Larger Numbers, The Distributive Law - Partitioning
13 x 16 (10x10 + 10x6) + (3x10 + 3x6) 10 6
10x10 10x6
10
3 3x10 3x6
(a + b) x (a + c) = (a² + ac) + (ba + bc)
Primary students are not expected to apply this law to algebra.
This slide shows the links between the calculate strategy and how it will assist the development of algebra
Commutative LawTurn Arounds
a + b = b + a
(a+b)+c = a+(b+c)
This begins in Kindy when we drop a collection of 5 objects and ask children what they can see.
a - b ≠ b - a
“I can see a 3 and 2 and that’s 5”
“I can see a 2 and 3 and that makes 5”
8 + 4 =
10 + 2 =
Compensate to Add
This strategy might also be called Bridge to Ten
9 + 7 = 10 + 6 =
24 + 18 =
22 + 20 =
Solving Algebraic Equations The Part-Part-Whole Model
• The Part-Part Whole Model is used to develop an understanding about the relationship between addition and subtraction
• Students are exposed to the language of part-part-whole from Kindy and begin to see the model being used in pre-primary. They should understand the model by the end of Year 3.
Developing the Part-Part-Whole Model
7 ?
10
7 3
?
7 + ? = 10
10 – 7 = ?
7 + 3 = ?
If we know the whole amount and one of the parts, we choose subtraction to solve the problem
If we know both parts we choose addition to solve the problem
10 - ? = 7 can be solved by 10 – 7= ?
And
? = 10 -7
? = 3
10 – n = 7
n = 10 – 7
n = 3
Using the Part-Part-Whole Model
Susan had some books on her bookshelf. She bought 7 more books and put them on her bookshelf too. Now there are 25 books on her bookshelf. How many books did Susan have before she went shopping?
7 ?
25 Write the equation that
matches the story.
? + 7 = 25
Write the calculator equation
25 – 7 =
Children solve problems like this in Years 2 and 3.
They solve similar problems using smaller numbers in Pre-Primary
and Year 1
Compensate to Subtract
4 -2 = 5 - 3
Compensate to Subtract
83 – 68 =
85 – 70 =
172 – 94 =
178 - 100
Children need to understand that if you add or subtract the same amount to each number, the answer will be the same.
This understanding takes a long time to develop.
Front Loading
24 + 33 =
20 + 30 = 50
4 + 3 = 750 + 7 = 57
Front Loading
24 + 33 =
24 + 30 = 54
54 + 3 = 57
• Our aim is to teach Mathematics in a way that develops deep understandings. These understandings allow students to work with abstract ideas and be successful in Maths in the long term.
• Teaching mathematical understandings is necessarily slow.