The Tree Problem
Which tree grew more?
1st measure
Now
B
B
3m
6m
5m
A
9m
A
1
Your Task with the Tree
Problem
• Individually, determine which Tree grew
more
– You have 1 min
• In pairs, labelled Partner A and Partner B:
– Partner A convinces B that tree A grew more
– Partner B convinces A that tree B grew more
– You have 1 min each
Think – Pair - Share
What reasoning did A people use?
What reasoning did B people use?
B
B
3m
A
5m
9m
A
6m
3
Debriefing the Tree Problem
• This is an example of a
Proportional Reasoning question
• The different partners were
arguing two different
perspectives:
• Additive or Absolute growth
• Multiplicative or Relative growth
A student who is functioning at the additive stage
might respond that Tree B grew more as it grew 3 m
and Tree A grew 2 m.
A student who can function at both additive and
multiplicative stages might respond that Tree A grew
more since 2/3 > 3/6
or 5/3 > 9/6 or 6:3 > 9:5
5-3
9-6
B
B
3m
A
5m
9m
A
6m
5
Consolidating Additive vs
Multiplicative
• The use of these approaches
is developmental
– Additive thinking develops first
– Multiplicative thinking needs to
be refined for proportional
reasoning to develop
Things to Notice
• The Tree Problem was selected as a
problem to illustrate Additive vs
Multiplicative thinking
• The Tree Problem is an Open Question.
– Multiple solution paths are possible
– Rich discourse to promote reasoning and sense
making
Tree Problem Questions
• What do you notice about the numbers
chosen for the question?
• If you hear an approach that is additive
what might you say?
Proportional Reasoning – What is it?
Proportional Reasoning involves the
deliberate use of multiplicative
relationships to compare quantities and
to predict the value of one quantity
based on the values of another.
Proportional Reasoning – What is it?
The essence of Proportional Reasoning
is the consideration of number in relative
terms, rather than absolute terms.