First and Second Years
Introduction
This lesson is broken into four stages. Following the lesson the students should have been
introduced to the concept of logic and had an opportunity to work with logical tools to solve
a puzzle. This will open the door to the possibility of applying mathematics to problem
solving. The lesson plan is designed so that you may extract sections to teach or to use the
information to build other lessons around the information provided.
Objectives
 To introduce George Boole
 To explain the basis of logic
 To link logic to George Boole’s theories and computer programming
 To operate and use a simple truth table in the application of logic to solve the puzzle
Lesson Time
40 minutes
1
Section 1 – Who is George Boole?


A brief explanation of George Boole and his importance to our modern
world. (For a more comprehensive overview of his life see Appendix 1).
Encourage the students to imagine what life was like in Ireland in 1849
when he became UCC’s first Professor of Mathematics: the famine, the
politics, and the availability of education...
George Boole (2 November 1815 – 8 December 1864) was an English mathematician and
philosopher. He is remembered today as the inventor of Boolean Logic, which is the basis of
modern digital computer logic. Without his ideas we would not be able to communicate in
the way we do through the use of mobile phones, fiber optic cables or even email.
He was awarded the first ever Royal Medal in pure mathematics for a research paper he
wrote in 1844. Although he never attended university, his brilliance won him universal
recognition and only 5 years later he was appointed as the first Professor of Mathematics at
Queen's College Cork (now UCC). While in Ireland he wrote his most famous work called An
Investigation of the Laws of Thought.
November 2nd 2015 is his 200th birthday and UCC is celebrating the genius of George Boole.
His advances in mathematics provided essential groundwork for modern mathematics,
microelectronic engineering and computer science.
As his work laid the foundations of the information age he has become known as the
forefather of the information age.
Today you are all joining in the celebration as we explore some of the mathematical areas
that he helped to develop. We are going to do some puzzles and games which will introduce
Boolean logic and mathematical truth tables.
Section 2 - What is Logic?



Begin by explaining what logic is.
Use statements about the school/class neighbourhood to gain an
understanding of the logical conclusion based on statements.
Encourage the students to make up their own statements and draw a
conclusion.
What is Logic?
The term "logic" came from the Greek word logos, which can be translated as "opinion",
"expectation", "reason", or "argument". Hence logical reasoning is a process in which
statements are collected together in an argument, with the intention of providing support
or evidence for some conclusion. Logic as a science is concerned with the evaluation of
arguments, namely with deciding whether they are correct or not, as well as being
consistent, complete and sound.
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For example, consider the following –
Mary is human,
Humans are mortal
Therefore Mary is mortal.
Assumptions.
Conclusion.
This is obviously a correct argument in the sense that the conclusion follows from the
assumptions. If the assumptions of the argument are true, the conclusion of the argument
must also be true.
What is Boolean Logic?

Explain Boolean logic and the TRUE or FALSE element of the logic.

Challenge the children to make up their own propositions after the Boolean Logic
Practice Exercise.
Boolean logic works only with statements that, at any given moment in time, are either TRUE or
FALSE. These statements cannot be anything in between (this is called the law of the
excluded middle). Such statements are called Propositions.
Practice Exercise: Decide which of the following statements is a proposition and which is
not:
(1) What time is it?
(2) John’s T-shirt is red.
(3) This statement you’re reading just now is false.
Solution:
Only (2) is a proposition, since at any moment in time, it can be either TRUE or FALSE.
(1) A question is neither TRUE nor FALSE.
(3) leads to a contradiction: If we assume (3) to be TRUE, then by its own admission, it
must be FALSE. If we assume it to be FALSE, then its opposite must be TRUE, which means
that the statement must be TRUE. Such a contradictory statement is called a Paradox. In this
case the contradiction was caused by the statement was making a judgement call about
itself. Such statements are said to be “self-referential” and they often lead to paradoxes.
In Boolean Logic, each individual proposition is given a short name. Propositions are
labelled, usually with letters like P, Q etc. so that they can be quickly identified.
o
Each proposition has a truth value of TRUE or FALSE, depending on the context.
TRUE is often represented as 1, while FALSE as 0.
o
Simple propositions can be connected together using AND, OR and NOT, to form
more complex arguments.
o
We can use propositions to make equations as we do in algebra. These are called
Boolean equations. For example, for two propositions P and Q, the equation
P=Q
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means that proposition P is TRUE whenever Q is TRUE, and P is FALSE whenever Q is
FALSE.
The symbols AND, OR and NOT are called Boolean operators. Fascinatingly, these three little
symbols can help us produce even the most amazingly complicated arguments or computer
programmes, and to check their validity.
Section 3- How do we use Boolean Logic?

Explain the practicalities of Boole’s work (Maths; Electrical Engineering;
Computer Science).

Ask the students to think of all the appliances that use electronic circuits
for example phones, laptops, cars, kitchen appliances etc. Link to gaming
and modern communication.

Encourage the students to provide examples of the use of Boolean logic.
Boole has become known as the forefather of the information age as Boolean logic
surrounds us every day, even though mostly unseen. Indeed, Boolean logic forms the basis
for computation in our modern computer systems and communication technology devices.
This is because you can represent any computer program, or any electronic circuit, using a
system of Boolean equations.
Basically, computers are made of electronic circuits. The only type of information they can
work with is a bit, that is a 1 or a 0, or equivalently TRUE or FALSE:
Diagram from:
http://www.petit
oiseau.org/author
/evan/ ).
The current is flowing,
so the light bulb is on:
TRUE
1
FALSE
0
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The building blocks for complex circuits are known as logic gates AND, OR, NOT:
This circuit represents the AND gate, and can be
written in Boolean logic as
A AND B.
The +6V point is the starting point of the current, and
0V is the destination. To reach the light bulb, the
current must pass through both A and B.
This circuit represents the OR gate, and can be written
in Boolean logic as A OR B:
To reach the light bulb, the current could pass through
A, or through B, or through both.
Click here to see the AND, OR gates in action:
http://www.technologystudent.com/elec1/dig2.htm
Exercise: Ask the students to imagine how a NOT (A) gate would look. For example, it could
be like a switch or revolving door which opens the circuit A when closing NOT(A) and viceversa.
Large sets of electronic circuits on one small plate are known as chips or microchips.
Computers consist of many chips placed on electronic boards called printed circuit boards.
The use of Boolean Logic in Web searches
The term is "Boolean," is often encountered when doing searches on the Web. In Boolean
searches, an "AND" operator between two words or other values (for example, "pear AND
apple") means one is searching for documents containing both of the words or values, not
just one of them. An "OR" operator between two words or other values (for example, "pear
OR apple") means one is searching for documents containing either of the words. For a
more comprehensive lesson in Boolean searches please refer to the search button on
www.georgeboole.com
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The use of Boolean Logic in Computer Games



Ask the students what computer/app games they play.
Get them to try to identify the AND, OR and IF…THEN operators in the games.
Candy Crush Saga below will show how Boolean operators play a central role to
the games
Candy Crush Saga has quickly become a popular (and addictive!) game. Behind it Boolean
operators are used in order to ensure that the game works by the rules.
To score points you must swap two pieces of candy to get three candies of the same colour
lined up. When this happens, the lined-up candies get crushed and more candies fall into
their places so that the game can continue.
The candy colours are placed in a grid like this:
The computer records
this as follows:
A1= ,b
, A2= y
y
, B2= r
B1=
……….
The game is based on a computer programme which uses the “AND” operator to decide
whether points have been scored or not. In the example above, the programme checks:
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B2= r
AND
TRUE
B3= r
AND
TRUE
Crush B2, B3, B4
Score 300 points.
B4= r
TRUE
The game will only allow you to swap two candies if this leads immediately to a candy crush
and score. For example, in the table below, you can quickly see that swapping C3 and D3 is a
good idea, while trying any other swap will just make the candies bounce back.
The game is not as smart as you; how can it decide whether to
allow C3 and D3 to swap, or whether to bounce them back?
Answer:
The game’s in-built programme allows it to check if a line-up
of 3
p -s can be formed by changing C3 into a
p
If 3
b -s will be formed by changing D3 into a
b
, OR
Using the OR operator, the programme checks if any of these will occur when swapping C3
with D3:
3
p
-s at C2:C3:C4 OR C3:C4:C5 OR C1:C2:C3 OR A3:B3:C3
FALSE
3
b
Up-coming Crush
OR
TRUE
Allow Swap
-s at D2:D3:D4 OR D3:D4:D5 OR D1:D2:D3
TRUE
The operator OR is used in many other ways in Candy Crush. One of the biggest problems
people have in Candy Crush is running out of lives. You start with 5 lives but when you run
out the game has a number of different ways by which you can regain lives and carry on
your candy crushing crusade…
OR
[1] Candy Crush gives you another life for every 30 mins you play (Takes too long. You HATE
this)
[2] Buy more lives from Candy Crush (the app creators LOVE this)
[3] Ask Friends on Facebook ( Your Facebook Friends HATE this)
[4] The easy, quick, free option – change your settings for unlimited lives (app creators HATE
this).
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Programming Challenge:
For the mystery Candy Crush table shown here, we
want to write a small programme helping the computer
to decide whether to allow a swap between A1 and B1.
Right at this moment, we can’t see the colours in the
boxes, but the computer has them recorded in its
memory, and it can check if two boxes have the same
colour. For example, if the boxes C3 and D4 have the
same colour, then you get that C3=D4 is TRUE.
For example, if the colour in B1 would line up with A2
and A3, then we should allow a swap of B1 with A1.
There are other colour line-ups that might be made
possible by the same swap. Can you fill in the two
empty boxes below?
B1=A2=A3
Up-coming Crush
OR
TRUE
Allow Swap
Solution:
If the the colour in A1 will be moved to B1, then it can line up horizontally with B2 and B3, or
vertically with C1 and D1. To see if this will work out well after the swap, you can check now
B1=A2=A3
A1=B2=B3
Up-coming Crush
OR
TRUE
Allow Swap
A1=C1=D1
If you’ve never coded before, give yourself a pat on the back: you’ve just done your first bit
of computer programming. It’s very useful too, as it can be used by the computer NO
MATTER WHAT THE COLOURS IN THE TABLE ARE.
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MineCraft:
Minecraft is a construction
game that makes heavy use of
Boolean logic in both the game
play and behind the scenes in
the inner workings of the game
itself. The game has many
modes… such as the creative
mode which allows you to build
for hours, if that’s what you’re
into … or the survival mode,
which can include designing and
constructing a fortress to keep
you safe from cuboid zombies (Not for the faint hearted!).
The game is known for its stand-out graphics… it is characterised by squares and jagged
edges. There are lots of materials available to build with, but the most interesting is
probably the Redstone block. Redstone can transmit Power, which is a lot like electricity: it
can travel through circuits and activate devices, like say a lamp or an elevator.
Here is a simple circuit: The lever is the INPUT; if you right-click on
the lever to switch it ON, you generate current through the
Redstone wire, making it glow red. This is the OUTPUT. It can
activate a device – in this case a piston.
INPUTS and OUTPUTS can only be in one of two forms: ON or OFF.
State
ON
OFF
Symbols used for the state in Boolean logic:
1
TRUE
T
0
FALSE
F
A Logic Gate is a simple circuit which receives one or more INPUTS in the form
TRUE/FALSE and gives a desired OUTPUT in the form TRUE/FALSE. A computer is collection
of millions of logic gates connected together. If you want to make a computer, you have to
know how to make logic gates.
Here is a simple logic gate made of some Redstone wire and a
Torch. As a rule, torches always generate current as OUTPUT,
except when current flows into them as INPUT :
INPUT: current into
torch
TRUE
FALSE
OUTPUT: current from
torch
FALSE
TRUE
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Because it turns TRUE into FALSE and FALSE into TRUE, a circuit made of a torch is called a
NOT gate, or inverter.
Here is a gate which receives two inputs. It is enough for
lever 1 OR lever 2 to be ON, and current will flow. Hence
this is called the OR gate. We describe it by a truth table :
Lever 1 input
TRUE
TRUE
FALSE
FALSE
Lever 2 input
TRUE
FALSE
TRUE
FALSE
Output
TRUE
TRUE
TRUE
FALSE
The OR gate can be used in scenarios where you want to have options, for example if you
are creating a hidden door which you want to be able to access from each side of the wall.
Let’s take a look at the AND gate. It has two inputs: lever 1 and lever 2, and is supposed to
generate a current only when both lever 1 AND lever 2 are switched ON:
Here is how the AND gate works, in a truth table:
Lever 1 input
TRUE
TRUE
FALSE
FALSE
Lever 2 input
TRUE
FALSE
TRUE
FALSE
Output
TRUE
FALSE
FALSE
FALSE
The AND gate sounds simple enough,
but in Minecraft its construction is a
bit tricky: you place a torch after
each lever, connect them by two
wires coming together, and then add
in an extra torch:
The Key Redstone Torch will only
generate current if no current flows into it from the two torches next to the levers. Those
two torches can only be OFF if BOTH levers are ON. So only when BOTH INPUTS are ON, do
we get an OUTPUT of ON. That’s exactly what the AND gate should do.
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We have constructed:
(lever 1) AND (lever 2)
in a round-about way:
Key Redstone Torch
In other words:
NOT
(NOT
(lever1) OR NOT (lever2) )
Torch 1
Wires
meeting
Torch 2
(lever 1) AND (lever 2) = NOT (NOT (lever1) OR NOT (lever 2))
which is an equation in Boolean algebra. Later on you will play with such equations and
understand them better, but for now it suffices to say that writing logic rules in equation
form was George Boole’s great insight which made all of this possible. Indeed, using insights
from Boolean algebra, it can be shown that any type of logic gate in MineCraft can be
constructed simply by putting together several NOT gates and OR gates. Similarly, today’s
engineers use Boolean Algebra to make electronic circuits suited to ever more complicated
tasks.
Here are the main MineCraft logic gates. You may use truth tables to discover what each
does, and/or read more at the excellent MineCraft resources on the web.
Graphics used are taken mostly from http://www.minecraftforum.net/forums/minecraftdiscussion/redstone-discussion-and/350815-redstone-guide-v1-21
which also has great explanations. Another great source is
http://www.nerdparadise.com/tech/apps/minecraft/redstonelogic/
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Section 4 – Truth tables



Establish what a truth table is.
Explain the difference between the operators AND/OR.
Draw the truth table
What is a Truth Table?
As we said, in Boolean logic every proposition can be either TRUE or FALSE. This is twovalued logic. Truth tables are a useful tool for working out how different propositions
logically relate to each other.
Example: Truth table showing the relationship between P and NOT(P).
The first column in a truth table is the statement P; the second column is the statement
NOT(P ) (the opposite of P). The first row in the truth table shows that whenever P is TRUE,
then NOT(P) is FALSE. The second row shows that if P is FALSE, then NOT(P) is TRUE.
P
TRUE (T)
FALSE (F)
NOT (P)
FALSE (F)
TRUE (T)
NOT, AND, OR
Class game:
The purpose of this game is to exemplify how trying out different scenarios/contexts and
completing truth tables can help us decide how to choose correct opposites of complex
statements.
Choose two students, let’s call them Anna and Brian, and hand each of them an instruction
card as below. During the game, Anna and Brian will have their back turned away from the
class. At each prompt, they will turn around acting as told on the instruction card.
Instruction card:
Stage 1: Anna acts
Brian acts
.
Stage 2: Anna acts
Brian acts
.
Stage 3: Anna acts
Brian acts
.
Stage 4: Anna acts
Brian acts
.
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While Anna and Brian are preparing for stage 1, ask the rest of the class to contradict the
following statement using only the terms AND, OR and NOT:
Statement
Opposite statement?
Anna AND Brian are happy.
Then ask Anna and Brian to turn around for stage 1 and ask the class to fill in the truth
values in the table. For example, in the (extremely likely) occurrence that someone in the
class had suggested “Anna AND Brian are NOT happy”:
Statements
Stage:
Anna AND Brian are happy.
Anna AND Brian are NOT happy
F
1) Anna
, Brian
F
.
Both statements are FALSE at this stage. This means that “Anna AND Brian are NOT happy”,
cannot be the opposite of “Anna AND Brian are happy”. After some trial and error, the
class may find the correct opposite “Anna OR Brian is NOT happy”. You may now complete
stages 1-4:
Statements
Stages
Anna AND Brian are happy.
1) Anna
, Brian
.
2) Anna
, Brian
.
3) Anna
, Brian
.
4) Anna
, Brian
.
Anna OR Brian is NOT happy
F
T
T
F
F
T
F
T
Conclusion: “Anna OR Brian is NOT happy” is the correct opposite of “Anna AND Brian are
happy”, because whenever the first statement is T, the other is F and the other way around.
In other words, when negating a proposition, AND changes into OR.
Logical contradiction does not go all the way to the other extreme like in “Anna AND Brian
are NOT happy”, but rather finds a “middle way”. When people are fighting, they tend to
forget about the middle way, which is to say that logical reasoning flies right out of the
window.
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In Boolean logic, we denote propositions by symbols:
A: Anna is happy
happy.
B: Brian is happy.
A AND B: Anna and Brian are
Our exercise above can look more formally like this:
A
T
T
F
F
B
F
T
T
F
A AND B
F
T
F
F
NOT (A AND B)
T
F
T
T
NOT A
F
F
T
T
NOT B
T
F
F
T
(NOT A) OR (NOT B)
T
F
T
T
Where the arrows symbolize changing the truth values in one column to their opposites.
By comparing the red columns we have just proven the following Boolean equation:
NOT (A AND B) = (NOT A) OR (NOT B)
Other examples of Boolean equations are:
A OR A= A,
A OR (NOT A) = T = 1
A and (NOT A) = F = 0
(A AND B) OR (A AND C) = A AND (B OR C),
Boolean equations are immensely important in practice because they help simplify the
construction of electronic devices, for example by contracting unnecessarily long and
complicated combinations of electrical circuits to shorter ones.
IF…THEN
We now look at the conditional statement IF P, THEN Q. In symbolic form it is written as
P→Q .
Class Puzzle:
One day the famous scientist Dr. Doom makes a public announcement:
“IF there is an earthquake tomorrow, THEN this entire building will fall down.”
The next day, everybody talks about how Dr. Doom was so wrong. What happened in the
meanwhile?
Answer: There was an earthquake, AND the building DIDN’T fall down. Notice that if there
had been no earthquake Dr. Doom’s statement would have been true.
Conclusions: NOT ( IF P, THEN Q) is the same as P AND (NOT Q).
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Section 5 - Puzzles





The tables are an important part of the answering of the question. The students will use the
table to eliminate false statements through the operators AND/OR.
Organise the students to work in teams of 2 or 3.
Distribute puzzle sheets to the students.
Read the puzzle aloud to the students.
Explain the procedure of filling out the table using each of the statements provided.
Warm up Game- Time 10 minutes
ON, OFF and IN-BETWEEN…
You are standing outside of a room. The room
has no windows and the door is closed. Inside
the room is a light bulb. Outside of the room
there are three light switches. One of these
switches belongs to that light bulb. The light
bulb is not lit and the switches are in off state.
You can turn them on and off as many times as
you like but only enter the room once. How can
it be determined which of these switches is connected to the light bulb?
Solution: First turn ON the first switch and leave it for few minutes. Then turn OFF the first
switch and ON the second switch. Now enter the room. If the light bulb is lit, the second
switch must be connected to it. If it is not lit, it might the first or the third switch. Now touch
the light bulb, it is hot it will be the connected to the first switch. Now if it is cold, then it
must be the third one.
Boole2School Challenge 1 – Time 15 minutes
The First Sudoku Toilet Paper
Inspector Simon Mart of Scotland Yard was looking at the
interrogation statements of three well known criminals. It had
already been established that one of them had stolen the Very
First Roll of Sudoku Toilet Paper, which is, of course, an object of
immense historical value. It had also already been established
that of the three suspects, exactly one spoke the truth.
Inspector Simon Mart looked at their statements:
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Albert: I am innocent.
Bill: Charles stole it.
Charles: I am innocent.
Who stole the toilet paper?
Hint: Consider assuming the guilt of a suspect and then evaluating the truth of all
statements based on this assumption. Try filling out this table with TRUE or FALSE.
Charles is Guilty
Bill is Guilty
Albert is Guilty
Albert: I am innocent.
Bill: Charles stole it.
Charles: I am innocent.
Solution:
Suppose Bill speaks the truth. Then Charles stole it, and lies. But both Albert and Bill would
be speaking the truth, which contradicts the statement that only one person speaks the
truth.
So, Charles is innocent, and Bill must be lying.
Charles is Guilty
Albert: I am innocent
TRUE
Bill: Charles stole it
TRUE
Charles: I am innocent
FALSE
Bill is Guilty
Albert is Guilty
That leaves only Albert and Bill as suspects.
If Bill were the thief, both Albert and Charles would be speaking the truth, again an
impossibility, so Bill must be innocent.
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Charles is Guilty
Bill is Guilty
Albert: I am innocent
TRUE
TRUE
Bill: Charles stole it
TRUE
FALSE
Charles: I am innocent
FALSE
TRUE
Albert is Guilty
If Albert were the thief, Albert and Bill would both be lying, while only Charles speaks the
truth. This means that Albert must be guilty.
Charles is Guilty
Bill is Guilty
Albert is Guilty
Albert: I am innocent
TRUE
TRUE
FALSE
Bill: Charles stole it
TRUE
FALSE
FALSE
Charles: I am innocent
FALSE
TRUE
TRUE
Boole2School Challenge 2 – Time 15 minutes
Einstein puzzle with houses
There are four bungalows in our cul-de-sac, numbered 1 through 4 from left to right. They
are made from these materials: straw, wood, brick and glass.
Mrs Scott's bungalow is somewhere to the left of the wooden one and the third one along is
brick. Mrs Umbrella owns a straw bungalow and Mr Tinsley does not live at either end, but
lives somewhere to the right of the glass bungalow. Mr Wilshaw lives in the fourth
bungalow, and the first bungalow is not made from straw.
Each person lives in a different house made of a different material from all the others. Who
lives where, and what is each person’s bungalow made from?
Solution:
Label the statements a to g.
a)
b)
c)
d)
Mrs Scott's bungalow is somewhere to the left of the wooden one.
The third one along is brick.
Mrs Umbrella owns a straw bungalow.
Mr Tinsley does not live at either end.
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e) Mr Tinsley lives somewhere to the right of the glass bungalow.
f) Mr Wilshaw lives in the fourth bungalow.
g) The first bungalow is not made from straw.
Fill in the information as follows:
House
#1
#2
Name
Mr Tinsley (d)
Built of
wood (a), straw(g)
#3
#4
Mr Wilshaw (f)
brick (b)
glass (e)
Indeed, (a) tells us that the wooden bungalow is not the leftmost, just like (e) tells us that
glass is not the rightmost.
We haven’t used (c), which tells us that Mrs. Umbrella and straw must occupy the same
column together. The only one available is column #2.
House
#1
#2
Name
Mr Tinsley (d)
Mrs Umbrella
Built of
wood (a),
Straw
#3
#4
Mr Wilshaw
Brick
glass
straw(g)
Therefore by (d) Mr Tinsley must live in #3.
House
#1
#2
#3
#4
Name
Mr Tinsley (d)
Mrs Umbrella
Mr Tinsley
Mr Wilshaw
Built of
wood (a),
straw(g)
Straw
Brick
glass
Which leaves Mrs Scott, living in #1, the glass bungalow. On the other hand, in column #1,
the only material available is glass, hence #4 must be wooden.
House
#1
#2
#3
#4
Name
Mrs Scott
Mrs Umbrella
Mr Tinsley
Mr Wilshaw
Built of
Glass
Straw
Brick
Wood
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Section 6 – Follow up
Have you and your students enjoyed this class?
This lesson plan was put together by Maths Circles Ireland with help from Boole2School project
support officer Kathy Bunney. We found many excellent online resources, in particular this one:
https://justpuzzles.wordpress.com/category/logic/
Maths circles aim to help students develop their problem solving skills in an environment that isn't
driven by any goal other than the enjoyment of investigation and discovery.
To find more lesson plans for junior cycle students, designed in a similar style, on various
mathematical themes, click here.
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