Download Multiplication and Division 29, Patterns and

Multiplication and Division 29, Patterns and Algebra 29_ Guided and Independent Investigations
(Year 6) ACMNA122, NSW MA3-4NA
Identify and explain square and triangular numbers.
GUIDED INVESTIGATION
Children learn
how to investigate the concept by following teacher’s instructions until
they are ready to investigate the concept independently.
INDEPENDENT INVESTIGATION
Children investigate and explain independently over many lessons at their current level
of understanding informing both themselves and the teacher of their current level of understanding
Resources: counters, squares, pencil, paper
What could we do?
Children:
1. sit in pairs
2. make square arrays and equilateral
and right angled triangles using the
counters, as guided by the teacher
3. explain why the number is a square
number
4. explain why the number is a
triangular number
What language could we use to
ask questions and explain?
 Can we make a square
array using these
counters?
 Can we make an
equilateral triangle using
these counters?
 Is this a square number?
What language could we use
to explain?
What could we do?
1. sit in pairs
 I made a square array
using ... counters
2. make square arrays and equilateral and right
angled triangles using the counters that are neither
too challenging, nor too easy
 I made an equilateral
triangle using ...
counters
3. explain why the number is a square number
 ... is a square number
Children:
explain why the number is a triangular number
 ... is a triangular number
Is this a triangular
number?
REFLECTION Before, during and after lessons, children discuss then record
responses to reflection questions to inform themselves and the teacher of their current
level of understanding
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
What is a square number?
What is a triangular number?
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1
If you can’t explain it
simply, you don’t
understand it well enough.
CONGRUENT INVESTIGATIONS
These investigations allow children to investigate and explain the concept in new and varied situations, providing formative assessment
data for both the child and the teacher.
‘Doing’ mathematics is not enough and is not a good indicator of understanding.
Anyone who has
Investigation takes time as children develop both the capacity and meta-language to explain mathematical concepts
never made a
at their current level of understanding.
mistake has never
tried anything new.
As they investigate, allow children to experience confusion (problematic knowledge) and to make mistakes to develop
resilience and deep understanding,
Identify square
numbers.

In pairs, children create square arrays, identifying the square number.
Construct square
numbers.

In pairs, children create square numbers starting with 1. They add counters to make 2 rows of 2 and
record 4 as a rectangular number. They add counters to make 3 rows of 3 and record 9 as a rectangular
number. They add counters to make 4 rows of 4 and record 16 as a rectangular number.
Square number
patterns.

In pairs, children make a list of the square numbers in order. They look for patterns in the sequence. For example, the square
numbers are 1, 4, 9, 16, 25, 36... We are adding 3, then 5, then 7, then 9, then 11, etc
Square number
statements.

In pairs, children investigate the following statements:
►
A square number can end only with digits 0, 1, 4, 6, 9, or 25
►
Every square number is the sum of 2 triangular numbers
►
Every square number is the sum of 2 or more square numbers
►
Squares of even numbers are even
►
Squares of odd numbers are odd

In pairs, add sequential odd numbers and investigate the pattern they find. For example, 1 + 3 = 4 (4 is a square number), 1 +
3 + 5 = 9 (9 is a square number), 1 + 3 + 5 + 7 = 16 (16 is a square number)...
(1 investigation
on each page)
Adding odd
number
sequences to
make square
numbers.
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
YouTube: A Learning Place A Teaching Place
Facebook: A Learning Place
2
Odd number
sequences to
make square
numbers.

In pairs, children make a list of square numbers set out as, for example,
Square and
triangular
numbers on dot
paper.

In pairs, children use dot paper to create square and triangular numbers, for example,
Square numbers
from triangular
numbers.

In pairs, children add triangular numbers to make square numbers, for example,
Website: http://www.alearningplace.com.au
Email: [email protected]
Twitter: @learn4teach
YouTube: A Learning Place A Teaching Place
10 + 6 = 16
Facebook: A Learning Place
3
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Select counters and make square arrays.
Count the counters.
Explain why the number is a square number.
http://www.alearningplace.com.au
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Create square numbers starting with 1.
Add counters to make 2 rows of 2 and record 4 as a rectangular number.
Add counters to make 3 rows of 3 and record 9 as a rectangular number.
Add counters to make 4 rows of 4 and record 16 as a rectangular number.
http://www.alearningplace.com.au
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Make a list of the square numbers in order.
Look for patterns in the sequence.
For example, the square numbers are 1, 4, 9, 16, 25, 36... We are adding 3, then 5,
then 7, then 9, then 11, etc
http://www.alearningplace.com.au
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Investigate the following statements:
►
A square number can end only with digits 0, 1, 4, 6, 9, or 25
►
Every square number is the sum of 2 triangular numbers
►
Every square number is the sum of 2 or more square numbers
►
Squares of even numbers are even
►
Squares of odd numbers are odd
http://www.alearningplace.com.au
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Add sequential odd numbers and investigate the pattern you find.
For example, 1 + 3 = 4 (4 is a square number),
1 + 3 + 5 = 9 (9 is a square number),
1 + 3 + 5 + 7 = 16 (16 is a square number)...
http://www.alearningplace.com.au
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Make a list of square numbers set out as, for example,
http://www.alearningplace.com.au
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Use dot paper to create square and triangular numbers, for example,
http://www.alearningplace.com.au
Investigating Square and Triangular Numbers
MULTIPLICATION AND DIVISION 29 PATTERNS AND ALGEBRA 29 Identify and explain square and triangular numbers
Add triangular numbers to make square numbers, for example,
10 + 6 = 16
http://www.alearningplace.com.au
http://www.alearningplace.com.au