### Structuring numeracy lessons to engage all students: R * 4

```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
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
• 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
• keep trying even if it is difficult (it is meant to
be)
• 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
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
• 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