9 · Exploring properties of number operations
9
Exploring properties of number
operations
This session is intended to:
앫 develop an understanding that addition and multiplication
are commutative (a × b = b × a) but subtraction and division
are not;
앫 develop learners’ ability to estimate the result of a
multiplication or a division
For each pair or small group of learners you will need:
앫 a calculator;
앫 Card set A – Always, sometimes or never true?
앫 large sheet of paper, glue stick and felt pens for making
posters.
Display one of the harder statements from Card set A on the
overhead projector, whiteboard or similar. For example
If you multiply 10 by a number, your answer will be greater
than the number.
Ask learners to say whether they think it is a true statement or
not. Typically, learners start by saying that this is obviously
true because ‘multiplying makes things bigger’. Ask questions
about the statements, such as:
Can you give me a value for a that makes the statement
true?
Can you give me another value? And another?
Can you give me a value for a that makes the statement
false?
Can you give me another value? And another?
Try a fraction, a decimal, a negative number...
Can we state precisely when the statement is true and
when it is not?
Explain that in this session learners will be asked to consider a
number of statements in a similar way. Explain that the
objective is for each group of learners to produce a poster that
shows each statement classified according to whether it is
always, sometimes or never true and furthermore:
앫 If it is sometimes true, then to write examples around the
statement to show when it is true and when it is not true;
앫 If it is always true, then to give a variety of examples
demonstrating that it is true, using large numbers,
decimals, fractions and negative numbers if possible;
앫 If it is never true, then to write an explanation of how you
can be sure that this is the case.
Ask learners to work in groups of two or three. Give each
group Card set A, a large sheet of paper, a glue stick and a felt
pen.
Ask learners to divide their sheet into three columns and to
head the columns with the words: ‘Always true’, ‘Sometimes
true’, ‘Never true’.
Learners now take it in turns to place
a card from Card set A in one of the
columns and justify their answer to
their group. Their group should
challenge them if the explanation has
not been clear and complete. When
the pair or group agrees, they should
paste the card down and write
examples around it to justify their choice. This should include
examples and counter-examples.
Learners who
struggle should be
given calculators to
help with the
arithmetic.
It is not necessary
for all groups to
complete all
cards.
Suggesting numbers for
learners to substitute may
help to take their thinking
forward.
Ask learners to display their posters to the rest of the group
and to describe one thing they have learned. Discuss particular
misconceptions that you identified as you listened to learners
working on the activity.
Always true
a+b=b+a
It doesn’t matter which
way round you add,
you get the same answer.
Sometimes true
a–b=b–a
It doesn’t matter which
way round you subtract,
you get the same answer.
True only when a = b
a + 10 > a
If you add 10 to a number,
your answer will be
greater than the number.
a×b=b×a
It doesn’t matter which
way round you multiply,
you get the same answer.
a÷b=b÷a
It doesn’t matter which
way round you divide, you
get the same answer.
True only when a = +b or –b
10a > 10
If you multiply 10 by a
number, your answer will
be greater than 10.
True only when a > 1
10 ÷ a < 10
If you divide 10 by a
number, your answer will
be less than 10.
True only when a < 0 or a >1
10a > a
If you multiply 10 by a
number, your answer will
be greater than the
number.
True only when a > 0
a ÷ 10 < a
If you divide a number by
10, the answer will be
less than the number.
True only when a > 0
Never true
a – 10 > a
If you take 10 away from a
number, the answer will
be greater than the
number.
Card set A – Always, sometimes or never true?
It doesn’t matter which way
round you add, you get the
same answer.
It doesn’t matter which way
round you subtract, you get
the same answer.
a+b=b+a
a–b=b–a
If you add 10 to a number,
your answer will be greater
than the number.
If you take 10 away from a
number, the answer will be
greater than the number.
a + 10 > a
a – 10 > a
It doesn’t matter which way
round you multiply, you get
the same answer.
It doesn’t matter which way
round you divide, you get
the same answer.
a×b=b×a
a÷b=b÷a
If you multiply 10 by a
number, your answer will
be greater than 10.
If you multiply 10 by a
number, your answer will
be greater than the number.
10a > 10
10a > a
If you divide a number by
10, the answer will be less
than the number.
If you divide 10 by a
number, your answer will
be less than 10.
a ÷ 10 < a
10 ÷ a < 10