Structuring numeracy lessons to
engage all students: 5-10
Peter Sullivan
Overview
• We will work through three lessons I have taught this
year as part of classroom modelling in years 5-10.
• The lessons are structured to maximise engagement of
all students, especially those who experience difficulty
and those who complete the work quickly.
• I will ask you to examine the commonalities and
differences between the lessons and identify key
teacher actions in supporting this lesson structure.
• I will ask you to reflect upon what implications for
leading whole school Numeracy improvement.
Assumptions
• We do not want to tell the students what to do
before they have had a chance to explore their
own strategy
• We want to step back to allow ALL students to
engage with the task for themselves
• We want them to see new ways of thinking about
the mathematics
• There is no need to hurry
• We want them to know they can learn (as distinct
from knowing they can be taught)
Patterns with remainders
Years 5 - 6
• Some people came for a sports day.
• When the people were put into groups of 3
there was 1 person left over.
• When they were lined up in rows of 4 there
were two people left over.
• How many people might have come to the
sports day?
OLOM Coburg 2013
Multiplication content descriptions
• Year 4: Develop efficient mental and written
strategies and use appropriate digital
technologies for multiplication and for division
where there is no remainder
• Year 5: Solve problems involving division by a one
digit number, including those that result in a
remainder
• Year 6: Select and apply efficient mental and
written strategies and appropriate digital
technologies to solve problems involving all four
operations with whole numbers
OLOM Coburg 2013
Patterns
• Explore and describe number patterns
resulting from performing multiplication
(ACMNA081)
• Solve word problems by using number
sentences involving multiplication or division
where there is no remainder (ACMNA082)
OLOM Coburg 2013
Some “enabling” prompts
• Some people came for a sports day. When they
were lined up in rows of 4 there were two people
left over. How many people might have come to
the sports day?
•
• Some people came for a sports day. When the
people were put into groups of 3 there was noone left over. When they were lined up in rows of
4 there was no-one left over. How many people
might have come to the sports day?
OLOM Coburg 2013
An extending prompt
• Some people came for a sports day. When the
people were put into groups of 3 there was 1
person left over.
• When they were lined up in rows of 4 there
was 1 person left over.
• When they were lined up in columns of 5
there was 1 person left over.
• How many people might have come to the
sports day?
OLOM Coburg 2013
The “consolidating” task
• I have some counters.
• When I put them into groups of 5 there was 2
left over.
• When they were lined up in rows of 6 there
was the same number in each column and
none left over.
• How many counters might I have?
OLOM Coburg 2013
How many fish?
Year 7
In this lesson, I need you to
• show how you get your answers
• keep trying even if it is difficult (it is meant to
be)
• explain your thinking
• listen to other students
Our goal
• The meaning of mean, median and mode
• To explain our thinking clearly
To start
• Write a sentence with 5 words, with the
mean of the number of letters in the
words being 4.
To start
• Write a sentence with 5 words, with the
mean of the number of letters in the
words being 4.
These sets of scores each have a
mean of 5
5, 5, 5
4, 5, 6
3, 5, 7
1, 1, 13
To start
• Write a sentence with 5 words, with the
mean of the number of letters in the
words being 4.
Next
• Seven people went fishing.
• The mean number of fish the people
caught was 5, and the median was 4.
• How many fish might each person have
caught?
Next
• Seven people went fishing.
• The mean number of fish the people
caught was 5, and the median was 4.
• How many fish might each person have
caught?
These sets of scores have a median of
10
10, 10, 10
8, 10, 12
1, 10, 11
9, 10, 200
8, 12, 10
And now
• Seven people went fishing.
• The mean number of fish the people
caught was 5, the median was 4
• How many fish might each person have
caught?
• Seven people went fishing.
• The mean number of fish the people caught
was 5, the median was 4 and the mode was 3.
• How many fish might each person have
caught?
• Seven people went fishing.
• The mean number of fish the people caught
was 5, the median was 4 and the mode was 3.
• How many fish might each person have
caught?
If you are stuck
• A family of 5 people has a mean age of 20.
What might be the ages of the people in the
family?
If you are finished
• How many different answers are there?
• What is the highest number of fish that
anyone might have caught?
Now try this
• The mean age of a family of 5 people is 24.
The median age is 15. What might be the ages
of the people in the family?
Our goal
• To see the meaning of mean, median and
mode
• To explain our thinking clearly
Co-ordinates of squares
Year 8 - 9
Assumptions
• They have had an introduction to placing coordinates
The key task
Four lines meet in such a way as to create a
square. One of the points of intersection is
(-3, 2)
What might be the co-ordinates of the other
points of intersection?
Give the equations of the four lines.
How might you run that class?
• How much would you tell the students?
• What approach do you recommend to
doing this task?
• How much confusion can you cope with?
• When is challenge and uncertainty
productive?
• What is meant by “cognitive activation”?
Quotes from PISA in Focus 37
• When students believe that investing effort in
learning will make a difference, they score
significantly higher in mathematics.
• Teachers’ use of cognitive-activation strategies,
such as giving students problems that require
them to think for an extended time, presenting
problems for which there is no immediately
obvious way of arriving at a solution, and helping
students to learn from their mistakes, is
associated with students’ drive.
Numeracy keynote SA
Where is the point (-2,3)?
Where is the point (-2,3)?
Show all the points which have an x
value of 1
Show all the points which have an x
value of 1
Show all the points which have a y
value of -2.
• What is the equation?
Responses
Four lines meet in such a way as to create a
square. One of the points of intersection is
(-3, 2)
What might be the co-ordinates of the other
points of intersection?
Give the equations of the four lines.
On this sheet draw the letter of your
name and give the co-ordinates of the
points at the ends of each line.
Mark all the points where y is bigger
than x
What is your reaction to those lesson?
What might make it difficult to teach
like that in your school?
In what ways were those lessons
similar?
What actions might you take to
encourage teachers to adopt such
approaches, at least sometimes?