Distributive rule
Activity type
Investigation/bridging
Level
Junior Cert
Activity aim and
The aim of this activity is to give learners the opportunity to explore the
learning outcome
distributive law by providing them with opportunities to
●
Strand 3
represent solutions to multiplication problems set in context in a
diagram that explains their reasoning
●
understand that equality is a relationship expressing the idea
that two mathematical expressions hold the same value
●
represent the distributive property in its general terms in a
diagram
●
analyse solution strategies to problems
●
begin to look at the idea of mathematical proof.
Starting by looking at models that help think about what the operation
of multiplication does, they progress to multiplication problems with
numbers larger than they can immediately solve. This allows them to
figure out how to break the problem into parts and use those parts to
solve it correctly. By examining these parts for individual problems
learners can come to generalise them; the property of multiplication, in
this general form, is called the distributive property or law.
The Distributive Law tells us that;
a(b+c) = ab + ac
or (a+b) (c+d) = ac + ad + bc + bd
Prior Learning
This lesson builds on previous work that learners have done when they try to make sense of the
operation of multiplication. They approach this work with several models to help them think about
what the operation of multiplication does: skip counting, accumulating groups of equal size and
representing items in arrays.
Learners extend these models by considering large numbers.
Learners attempting this work must understand that equality is a relationship expressing the idea
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that two mathematical expressions hold the same value. Research (Falkner and Levi, 1999),
however, has shown that primary age learners generally do not see the equal symbol as a
relationship but as an operation sign to carry out the calculation from left to right. A lack of such
understanding is one of the major stumbling blocks in moving from arithmetic towards algebra.
Perhaps the best place to start the investigation is to look for evidence of distributivity in learners’
work. Evidence can be found by examining the solution strategies for problems such as …
Yesterday my mum baked 4 batches of buns for my sister’s birthday party. Today she
decided to bake 3 more batches. Each batch had 5 buns. How many buns did my mum
bake?
Allow learners to work individually first, then share their answer with a partner, and then engage in
a group discussion. Learners may have different approaches to solving this problem. Some may
add the batches together first while others may work with yesterday’s amounts and today’s
amounts separately and add the results. An important discussion point is to note whether learners
can see the connections between the various strategies.
Is (4 x 5) + (3 x 5) the same as 7 x5? If so, why?
Arranging numbers in arrays makes the answer to this question very clear for learners.
. . . .
. . . .
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. . .
. . .
. . . .
. . .
. . . .
. . .
. . . .
. . .
....
...
.... + ...=
....
...
....
...
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4 bunches (columns) of 5 things and 3 bunches of 5 things is the same as 7 bunches of 5 things.
Learners are now in a position to explore different strategies.
Is (3 x5) + (4 x 5) the same as (2 x5) + (2 x 5) + (2 x 5) + (1 x 5)?
Once learners have begun to understand that they can break down multiplication problems in such
a way they can explore situations with larger numbers.
Solve 46 x 35 without using a calculator.
Some possible solutions that learners could come up with are
46 x 10 = 460
46 x 10 = 460
46 x 10 = 460
( 46 x 30 ) + (46 x 5 )
1380 + 230
= 1610
40 x 35 = 1400
(40 x30) = 1200
6 x 35 = 210
(6 x30) = 180
x 35
1400 + 210 = 1610 (40 x5) = 200
1610
46 x 5 = 230
(6 x5) = 30
460 + 460 + 460 + 230
1200+180+200+30
= 1610
= 1610
46
Ask questions to determine if learners can see the distributive property in the traditional algorithm
used for multiplication.
46
x 35
1610
Is it evident to learners that this represents (5X6) + (5 x40) + (30 x 6) + (30 x 40)?
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Explore what strategies the learners use when the problem is presented in a story context
In a bakery 136 dozen cakes are made daily. How many cakes are made each day?
Note to teachers: When learners are solving such problems they are often mindful of symbol
patterns only and lose sight of whether a particular method is suited to what the operation does.
Computation methods suitable for addition may need to be re-thought for subtraction, multiplication
and division. Consider the following:
27 + 35 = (20 + 30) + (7 + 5 ) = 50 + 12 = 62
Is 27 x 35 = (20 x 30) + (7 x 5)? If not, why not?
Making representations helps learners to figure out why the distributive property works and to
create arguments explaining how they know it will work for other numbers.
Learners can now try to generalise their observations or show that they are true for all numbers. In
mathematics, making a generalisation means making it explicit, and when a generalisation
becomes explicit, it becomes available as a strategy for solving problems in a variety of situations.
Encourage learners to make representations of their problem; when discussing and explaining
these representations they can move towards a generalisation.
If we have one bunch of groups (B groups) and another bunch of groups (C groups), and each
group has A things in it, then we can work out how many things we have altogether in two different
ways.
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AXB
AXC
A
B
C
The first is to multiply A times the total number of groups (B + C). The second is to multiply each
number of groups by A and then add the two products: (A X B) + (A X C)
Once learners are proficient at doing these types of calculations involving the distributive property,
the teacher can present them with opportunities to challenge their understanding of the property
and stimulate their curiosity to do more thinking about it.
The work above gives learners the opportunity to consolidate an understanding of the
fact that a(b + c ) = (a x b) + (a x c)
Asking learners to make sense of questions of the form (a + b) (c + d) and what they would equal
using the distributive property, raises their level of thinking. Learners working on such a problem
are laying the foundation for continued work with the distributive property when they are later
asked to multiply algebraic expressions such as (x+5) (x+7).
When introducing the distributive property in the form (a + b) (c + d) it is a good idea to embed the
problem in a story context this will engage the learners and encourage them to think at the level
required.
Consider the following
You work for a campsite owner. He wants to sell bays in his campsite and wants to include
parking for the camper’s car beside their tent. The owner wants the parking bay to be
suitable for different sized cars and so wants the bays configured as follows.
The length of a bay is 5m longer than the width of the camper’s car. The width of the bay is
2m longer than the width of the camper’s car.
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The bay will be paved. To determine the amount of paving, you know you need to multiply
the dimensions of the bay. You show the following expression to your boss and tell him
that all he has to do is to substitute the width of a car for w and he will have the area that
needs to be paved. The owner asks “What’s another way to write (w+5) (w+2) if we do not
know the width?” What would you tell him? How can you check to see if you are correct?
As learners work through this problem they may need to review their previous work
Is (10 + 9) x (10 + 4) = (10 x10) + (9 x 4)?
Learners who use a rectangular array to justify their answer to this question will be in a position to
move to the justification of the more abstract problem of the campsite.
10 X10
10 X 4
9 X10
9X4
10
9
10
4
Learners can see that (10 x 10) + (9 x 4) is only part of the solution and that (9 x 10) + (10 x 4) are
also part of the solution.
In moving from the concrete to the abstract learners are beginning to deal with the idea of
mathematical proof. At this level, proof is acceptable if the following criteria are met:
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meaning of the operation is represented in diagrams or story contexts
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representations accommodate a class of instances, e.g. are true for all whole numbers, etc.
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representations show how conclusions of the claim follow from the structure of the
representation.
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The campsite problem
wXw
5Xw
wX2
5X2
w
5
w
2
In General terms
aXc
bXc
aXd
bXd
a
b
c
d
Draft Questions
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Q1. JCFL
Jo and Charlotte were finding the products of large numbers without using a calculator.
They used the model illustrated below to show the product of 24 x 52.
20
4
What product does each rectangle represent?
Show how Jo and Charlotte used this model to find the product of 24 x 52.
50
2
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Use a similar model as Jo and Charlotte to find the product of 127 x 325
Q2. JCHL
Write an expression in its simplest form that gives the area of the three shaded rectangles in the
picture below
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Q3. JCHL
Will 37 x 150 produce the same answer as (37 x 95) + (37 x 55)?
Explain your thinking
Q4. JCHL
Show that in all cases (a+b) (c+d) = ac+ bc +ad + bd
Q5. JCHL
What happens to the product of two numbers if you cut one number in half and double the other
number? Show that this is true for all numbers.
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