Using Open Ended Tasks in the
Mathematics Classroom
Glenroy West Primary School
19th May, 2010
Northern Metropolitan Region, DEECD
Adrian Berenger
Part 1: Types of Questions
Closed Questions are those which
require an answer or response to
be given from memory.
Open Questions require students
to think more deeply and to give a
response which involves more
than recalling a fact or reproducing
a skill. Usually more than one
possible answer.
Mother’s Day Bill
• $711 was the total bill
• Mum & Dad, Brother and his Wife and 2
children, Sister and her Boyfriend, and me
• Who pays what?
Buying Shoes
• You go with a friend to buy shoes. Each pair
cost $80
• The sale offers $100 off if you buy 3 pairs.
• Your friend gets 2 pairs and you get only one.
• Who pays what?
Making Shapes
• Minimum Properties of Shapes
• Construct a quadrilateral that has two pairs of
adjacent sides equal.
• What shapes are possible?
Closed to Open
12 cm
3 cm
Calculate the Perimeter and
Area of the rectangle.
I want to make a garden in the shape
of a rectangle. I have 30 metres of
fence for my garden. What might be
the area of the garden?
Garden paving
• Peter wants to pave an area using 16 square
paving stones and 16 straight border pieces.
• Each stone is 1m2 and each border piece is 1m.
• What different arrangements are possible?
• On the way home, Peter carelessly breaks
several of the paving stones. What is the
maximum number of broken stones possible in
order to still be able to use all his 16 border
pieces?
• Draw all possible arrangements.
‘Our goals in education are for
our students to think, to learn,
to analyse, to criticise and to
be able to solve unfamiliar
problems.’
Peter Sullivan and Doug Clarke
Bloom’s Taxonomy
Can the student GENERATE new products,
ideas or ways of viewing things?
Can the student JUSTIFY a decision or
course of action?
Can the student DIFFERENTIATE between
constituent parts?
Can the student USE the new
knowledge in another familiar situation?
Can the student EXPLAIN ideas or
concepts?
Can a student RECALL
information?
Why do we use Open Tasks?
• To draw out misconceptions
• To encourage deeper thinking
• To develop problem solving
• To cater for mixed ability levels
Creating Open-Ended Tasks
‘Any closed questions can be
reformulated to create an open
ended questions using one of two
methods’.
(Sullivan & Lilburn 1997)
Misconceptions with =
• There are 11 more
8+3=?+4
students enrolled at
the school this term. • 7 because 8 and 3
equals 11 and so
What’s possible?
does 4 and 7
• 7 because 4 is one
• How many whole
more than 3 so ?
number pairs sum to
must be one less
11?
than 8
• 11
Totals
10
11
•
•
•
•
•
•
11 + 0
10 + 1
9+2
8+3
7+4
6+5
•
•
•
•
•
•
10 + 0
9+1
8+2
7+3
6+4
5+5
9
•
•
•
•
•
9+0
8+1
7+2
6+3
5+4
8
•
•
•
•
•
8+0
7+1
6+2
5+3
4+4
7
•
•
•
•
7+0
6+1
5+2
4+3
6
•
•
•
•
6+0
5+1
4+2
3+3
Other Questions with Numbers
• A maths teacher always
likes to have his class
working together in
groups but no matter if
he suggests his students
work in groups of 2, 3 or
4 there is always one
student on their own.
How many students
might there be in this
teacher’s class?
Other Questions with Numbers
10
7
3
1
What scores are
possible from 4
darts?
Method 1
• Omit enough information so that, although
the answer remains the same, the digits
required to achieve the answer becomes
variable.
TRADITIONAL
249
+ 173
OPEN-ENDED
2**
+ *7*
422
Method 1 (continued)
TRADITIONAL
OPEN-ENDED
Two fifths of 250 students
borrow books from the library
each day. Calculate the
number of students who
borrow books each day.
Two fifths of the students in a
school borrow books from the
library each day. How many
students might there be in the
school and how many of them
borrow books each day?
Find the missing angle on this
trapezoid
40◦
What might the angles on this
trapezoid be?
140◦
Method 2
• Work backwards from the answer. Begin with a
closed task. Calculate the answer, then work
backwards and using the context of the question,
create a question that would allow multiple
responses to achieve the same answer.
TRADITIONAL
OPEN-ENDED
The following numbers
represents the temperature of
5 consecutive days in
Melbourne: 44◦C, 42◦C, 36◦C,
22◦C, 29◦C. Find the average
temperature.
The average temperature over
five consecutive days in
Melbourne was 35◦C. The
highest temperature was 44◦C.
What might the temperature
have been on the other days?
Method 2 (continued)
TRADITIONAL
OPEN-ENDED
35.0
X 0.5
What is the volume of
the cylinder?
4cm
6 cm
The answer is 17.5
What might the
question be?
What might the
dimensions of a
cylinder prism that has
a volume of 300 cm3?
Summing Up!
• Resources
Open Ended
Maths Activities
2nd Edition
Thinking Tools for
the Mathematics
Classroom.
Peter Sullivan
and Pat Lilburn
Oxford
Sue Gunningham
Hawker Brownlow
Education
Part 2: Creating Open-Ended Questions
• 8 closed questions to open questions in pairs
• Hot dots
Video: 7X13=28