Download Unit Two Reasoning, Justification And Proof 18 Hours

UNIT TWO
REASONING, JUSTIFICATION
AND PROOF
18 HOURS
Math 521B
Revised Dec 20, 00
34
By the end of grade 11,
students will be expected
to:
E5 use inductive and
deductive reasoning
when observing
patterns, developing
properties and making
conjectures
Elaborations - Instructional Strategies/Suggestions
Introduction (6.1)
According to the 2000 NCTM Standards document, reasoning and proof
are “not special activities reserved for special times or topics in the
curriculum, but should be a natural part of classroom discussions.
Critiquing arguments and discussing conjectures are delicate matters;
plausible guesses should be discussed even if they turn out to be
wrong.”
Examples of reasoning are:
< algebraic or symbolic reasoning
< statistical reasoning
< probabilistic reasoning
< spatial reasoning
Reasoning is a process of thinking about a mathematical (or other)
question.
Justification is a rationale or argument for some mathematical
proposition. This can be inductive reasoning.
Proof is a justification that is logically valid, based on initial
assumptions, definitions and previously proved results. This is a more
formal argument and is usually a deductive level of reasoning.
There are two types of reasoning that we will examine in this unit:
< inductive
< deductive
Inductive Reasoning
Inductive reasoning is the process of recognizing patterns and making
generalizations or conjectures from the observations.
Scientific formulae are usually arrived at inductively.
Invite students to read and discuss the Triangle Sum Theorem and the
Exterior Angle Theorem on p.339.
Students could try to draw a number of triangles, measure the angles and
notice the pattern that the sum is always 1800.
Students could then extend one of the sides of these triangles in one
direction to form an exterior angle. Measuring the exterior angle and the
two remote interior angles in each triangle should support the Exterior
Angle Theorem.
Note to Teachers: Students will be expected to keep a Journal which
includes definitions, theorems, postulates and corollaries in a separate
notebook which may be used in the unit exam. This journal can continue
in the next unit as well.
35
Worthwhile Tasks for Instruction and/or Assessment
Inductive Reasoning (6.1)
Group Activity
Study the given equations and complete the conjecture started
below that holds for all the equations.
3 + 7 = 10
!1 + 5 = 4
9 + 13 = 22
7 + 11 = 18
Conjecture: The sum of two odd numbers is always an...
Suggested Resources
Inductive Reasoning
Math Power 11 p.340 #13,1-9,11,12,
15,16,21
Note: # 9 in the list above can be
done using a Reflect-View
Journal
Write a conjecture about combining two numbers using a
mathematical operation. Show a number of examples
supporting your conjecture.
Performance/Manipulative
Draw any quadrilateral. Determine the midpoints of each side
using a Reflect-View. Connect these midpoints. What kind of
quadrilateral seems to result? Either repeat your experiment
several times or compare your result with other members in
your group. What is your group’s conjecture? What kind of
reasoning has lead to your conclusion?
Group Discussion
Here are the first few rows of Pascal’s triangle. The sum of
the numbers is shown to the right. According to the pattern,
make a conjecture about the sum of the numbers in the 20th
row. In any row in general.
1
1 1
1 2 1
1 3 3 1
1 4 6 41
1 5 10 10 5 1
1 6 15 20 15 6 1
1
2
4
8
16
32
64
Pencil/Paper
a) Consider the number patterns to the right and verify that
they are correct.
b) What are the next two lines in the number pattern?
c) What will be the sum of the first ten positive odd integers?
d) Make a conjecture about the sum in any row.
36
1 = 12
1 + 3 = 22
1 + 3 + 5 = 32
1 + 3 + 5 + 7 = 42
1 + 3 + 5 + 7 + 9 = 52
By the end of grade 11,
students will be expected to:
Elaborations - Instructional Strategies/Suggestions
Inductive Reasoning (cont’d)
Testing the veracity of a statement with numerical or specific
examples is inductive reasoning.
E5 use inductive and
deductive reasoning
when observing
patterns, developing
properties and making
conjectures
Note to Teachers: Show a diagram like the one below to demonstrate
the concept of an exterior angle.
E6 use deductive reasoning,
construct logical
arguments and be able to
determine, when given a
logical argument, if it is
valid
E27 write proofs using
various axiomatic
systems
E28 demonstrate an understanding that reasoning
and proof are essential
and powerful parts of
mathematics
p1, p2 and p3 are the angles of the triangle
p4 and p6 are the exterior angles.
p5 is not classed as an angle exterior to the triangle. We will see in
section 6.3 that it is called an angle opposite to p3.
In the examples on p.339-340 we see numerical examples used to
make conjectures about some observed situation. The examples come
from:
GCO E: spatial sense
GCO B: operation sense
GCO C: patterns and relationships
37
Worthwhile Tasks for Instruction and/or Assessment
Inductive Reasoning (cont’d)
Performance
Count the number of regions resulting when 2, 3, 4 and 5
points are connected and complete the table.Make a
conjecture about the pattern suggested in the figures below.
Then use the conjecture about the number of regions formed
when 6 points are connected. Finally, draw a circle, place 6
points on the circumference join them and count the regions
formed.
Suggested Resources
Inductive Reasoning (cont’d)
Solution
Activity
Complete the following calculations:
1×8+1=
12 × 8 + 2 =
123 × 8 + 3 =
1234 × 8 + 4 =
Make a conjecture about the pattern observed.
Use the conjecture to predict the answers to:
12345 × 8 + 5 =
123456 × 8 + 6 =
123456789 × 8 + 9 =
Use a calculator to check your predictions.
Solution
1×8+1=9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
98765
987654
987654321
This is the end of this pattern.
38
By the end of grade 11,
students will be expected
to:
Elaborations - Instructional Strategies/Suggestions
Analysing Conjectures (6.2)
In the previous section we observed patterns and made conjectures. Here
we will analyse conjectures by using specific examples. If we can find
even one counter-example that does not support the original conjecture
then the conjecture has to be revised or discarded altogether.
Invite students to read the first two paragraphs on p.343.
E5 use inductive and
deductive reasoning
when observing
patterns, developing
properties and making
conjectures
In science, the scientific method is used to make advances in scientific
knowledge. The steps involved in the method are:
Observation - some event or pattern is observed causing a person to
think about it.
Hypothesis(Conjecture) - a conjecture is made about the observations
E29 develop and evaluate
mathematical
arguments and proofs
Experimentation - the conjecture is tested by searching for a counterexample that will disagree with the conjecture
After many examples have been tested and a counter-example has not
been found, then the conjecture still hasn’t been proved conclusively but
it can be considered a:
Theory - if it is trying to explain why some event happens.
Law - if it merely describes what is taking place. Many times a
formula is used to aid in the description of the law.
Students should be able to see that what they are essentially doing is
using the Scientific Method.
39
Worthwhile Tasks for Instruction and/or Assessment
Analysing Conjectures (6.2)
Pencil/Paper
A number raised to a positive integral power yields a larger
number.
a) give two examples that support this conjecture.
b) give a counter-example that shows the conjecture is false.
Suggested Resources
Analysing Conjectures
Math Power 11 p.345 #1-3,5,7-10,15,
16
Pencil/Paper
Provide a counter-example to prove that each conjecture
below is false:
a) if x > 0, then x < x
b) if the diagonals of a quadrilateral are perpendicular, then
the quadrilateral is a square.
Research
Conjecture: If you live in a country bordering Cameroon to
the north, then you live in the Central African Republic.
Give two counter-examples that show the conjecture is false.
Journal
What is the purpose of finding counter-examples to
conjectures?
Written Assignment
Complete each of the following statements.
23 × 64 =
26 × 93 =
41 × 28 =
32 × 46 =
62 × 39 =
14 × 82 =
69 × 64 =
96 × 46 =
84 × 36 =
48 × 63 =
Make a conjecture based on the pattern above. Try to find two
counter-examples to the conjecture.
Group Activity/Open-ended Question
Make a conjecture on anything you have observed in the
world around you. Bring the conjecture to your group and find
two examples that support the conjecture and two counterexamples to the conjecture.
40
Solution
It seems that one can conjecture that
the product of a pair of two digit
numbers is the same as the product of
the numbers formed by reversing
their digits.
Counter-examples:
51 × 48 and 15 × 84
34 × 49 and 43 × 94
By the end of grade 11,
students will be expected
to:
E5 use inductive and
deductive reasoning
when observing
patterns, developing
properties and making
conjectures
E6 use deductive
reasoning,
construct logical
arguments and be able
to determine, when
given a logical
argument, if it is valid
E27 write proofs using
various axiomatic
systems
E28 demonstrate an understanding that
reasoning and proof
are essential and
powerful parts of
mathematics
Elaborations - Instructional Strategies/Suggestions
Deductive Reasoning (6.3)
So far in the unit students have been using inductive reasoning by
observing patterns, making conjectures and analysing the conjectures by
trying to find a counter-example. This is the creative, investigative openended form of reasoning that people use in their every-day lives.
In this section students will be encouraged to examine and discuss
conjectures that are logically consistent.
Deductive Reasoning, or logical reasoning, is a process of proving a
conjecture in general terms and not by using specific examples of the
conjecture.
There are three ways to write deductive proofs:
< two column proofs
< flow chart proofs
< paragraph proofs
These three methods will be illustrated in Unit 6. In this unit we will use
only the paragraph proof method.
A Greek, Thales, was one of the first mathematicians to make a number
of geometric conjectures and prove them deductively. One of his
students, Pythagoras, continued linking chains of logical reasoning.
Euclid, in his work, Elements, created a deductive system composed of
the following:
Undefined terms and postulates
In any deductive system there must be certain assumptions (undefined
terms) and ground-rules (postulates) to go by. These are simply accepted
as being true without proof. These assumptions differ from Euclidean to
non-Euclidean geometries. Examples of undefined terms are: point, line
and plane. Examples of postulates are: For any two points there is
exactly one line that contains them. Through a given point there is at
most one line parallel to a given line.
Definitions
A definition is a statement that clarifies the meaning of a word using
undefined terms or previously defined terms.
Theorems
If any conjecture can be proved deductively it is classed as a Theorem.
Corollaries
Any statement or theorem which follows directly from a previously
deduced theorem is a corollary.
41
Worthwhile Tasks for Instruction and/or Assessment
Deductive Reasoning (6.3)
Group Activity
Conjecture: The sum of four consecutive integers is
equivalent to the first and last integers added, then multiplied
by two.
a) defend the conjecture inductively by showing 2 examples.
b) prove the conjecture deductively.
Group Presentation
Have each group prove one of the following inductively,
showing three examples. Then they are to prove the
conjecture deductively. Each group will present their proofs to
the class.
< the sum of any 2 odd integers is even
< the product of any 2 consecutive integers is even
< the product of any 2 even integers is even
Pencil/Paper/Discussion
Prove the following inductively, showing three examples.
Then prove it deductively.
Conjecture: The sum of a two digit number and the number
formed by reversing its digits will always be divisible by 11.
Presentation
Find four consecutive integers whose sum is 44. Show
inductively and deductively that it is not possible.
Performance
Prove inductively(with 3 examples) and deductively.
Choose any number
×4
+ 10
÷2
!5
÷2
+3
Group Activity
Make a series of steps (like the previous problem) of five or
more steps that always results in a final answer of 6.
42
Suggested Resources
Deductive Reasoning
Math Power 11 p.349 # 1-4,8-10,14,
17
By the end of grade 11,
students will be expected
to:
Elaborations - Instructional Strategies/Suggestions
Deductive Reasoning (cont’d)
Initiate a discussion on the definitions for complementary and
supplementary angles. Supplementary angles can be visualized thus:
Theorem - If two angles are equal, then their supplements are equal.
E5 use inductive and
deductive reasoning
when observing
patterns, developing
properties and making
conjectures
In all 3 situations the sum = 1800
E13 explore constructions
for various geometric
configurations and
deduce proofs of their
validity
E27 write proofs using
various axiomatic
systems
Opposite Angles Theorem
When two lines intersect, the opposite angles are equal.
Students will be asked to prove the above conjecture and thus it will
become a theorem.
Note to Teachers: To ensure that students have a mental picture of
opposite angles, generating the opposite angle diagram as shown below
might be helpful. (unlike complementary or supplementary angles,
opposite angles must share a common vertex),
Step 2 extend each side of p1 in the
opposite direction
Step 1
Step 3 The angle formed by these new sides ( p2) is the angle opposite
p1.
Worthwhile Tasks for Instruction and/or Assessment
43
Suggested Resources
Deductive Reasoning (cont’d)
Deductive Reasoning (cont’d)
Journal
Write a definition for complementary and supplementary
angles. Draw three possible diagrams for each.
Performance
In the August 1973 issue of Scientific American the following
problem appeared. From an unused matchbook containing 20
matches, tear out from 1 to 9 matches and discard them. Count
the number of remaining matches. Add the two digits of this
number and tear out this many additional matches. Finally tear
out two more matches.
a) Try this three times. What is the result? (inductive
reasoning)
b) Prove that this will always work.(deductive reasoning)
Communication
Prove the following inductively(showing three examples) and
then prove it deductively:
The square of the sum of two positive integers is greater than
the sum of the squares of the same two integers.
Written Assignment
A motorist drove 300 km at a speed of 100 km/h. On the
return trip the motorist was able to go at only 60 km/h. What
was the motorist’s average speed?
a) What is your intuitive answer?
b) Answer the following questions to deductively arrive at the
answer.
< How long did it take to drive there?
< How long did it take to drive back?
< What was the total time spent driving?
< What was the total distance travelled?
< What was the average speed?
44
By the end of grade 11,
students will be expected
to:
Elaborations - Instructional Strategies/Suggestions
Conditional Statements (6.5)
A statement is a sentence that is either true or false. It is not an opinion.
Ex. A quadrilateral is a polygon having exactly four sides.
E25 explore “converse”
and understand the use
of the phrases
“if...then” and “if and
only if” in definitions
and proofs
E26 construct logical
arguments, and be able
to determine the
validity of an argument
E28 demonstrate an understanding that reasoning
and proof are essential
and powerful parts of
mathematics
When a statement is written in “If...., then....” form, it is a conditional
statement.
Ex. If a polygon is a quadrilateral, then it has exactly four sides.
»
hypothesis
º »
conclusion
º
The hypothesis is what is assumed to be true.
The conclusion should follow logically from the hypothesis.
Symbolically, a conditional statement can be written as:
a º b ( read as: if a, then b)
Converse
If the hypothesis and conclusion are interchanged, the resulting
statement is the converse of the original statement. Symbolically it can
be written:(b ºa). Generally the converse does not have the same
meaning as the original statement.
Ex: Original: If an animal is a horse, then it has four legs.
Converse: If an animal has four legs, then it is a horse.
Bi-conditional statements
The statements below can be read both ways and still have the same
meaning.
original: If a polygon is a quadrilateral, then it has exactly four sides
converse: If a polygon has exactly four sides, then it is a quadrilateral.
They could have also have been written in bi-conditional form:
A polygon is a quadrilateral iff it has exactly four sides.
No, there is no spelling error in the above statement. It is read;
A polygon is a quadrilateral if and only if it has exactly four sides.
Definitions can be written in bi-conditional form. In fact this is used as a
test of a good definition in any field of study (not just mathematics). If a
statement’s converse is logically equivalent to the original then the
statement is a good definition.
Contrapositive
If the original statement (a º b) is rearranged into this form (not b º
not a) it is given the name contrapositive of the original statement and is
always logically equivalent to the original statement.
45
Worthwhile Tasks for Instruction and/or Assessment
Conditional Statements (6.5)
Pencil/Paper/Discussion
Write the following statement in conditional form;
“A triangle has at least two acute angles.
Suggested Resources
Conditional Statements
Math power 11 p.361 #1,2,4,5,7,9,10,
12,14-16,18,
20,21,23,26,
28,30
Pencil/Paper/Discussion
Write the converse and contrapostitive of the above
conditional statement. Determine the validity of these
statements.
Pencil/Paper/Discussion
Write each of the following definitions as a bi-conditional
statement;
a) A rhombus is an equilateral quadrilateral.
b) Supplementary angles are two angles whose sum is 1800.
c) Perpendicular lines are lines that intersect at right angles.
Technology
Use the TI-83 to test the truth of:
If 4x!3 > 7x ! 15, then x < 4.
2nd test 3: > will put the > symbol on
the screen
Technology
p.363 Green #1 #1-5, Green #2 #1-6
Note to Teachers: Read p.363 on
Boolean Algebra
With the standard window
zoom 6:standard
and press graph
press trace and cursor along to test the above conclusion
Journal
Try to create a good definition of some object or concept in
the real world. Check the validity of its converse to see if it is
a good definition.
46
To test the endpoint to see if it is part
of the solution set, press trace and
simply enter the x value of the
endpoint (here x = 4) and press enter
Notice that if y = 0 then the endpoint
is not part of the solution while if y =
1 then the endpoint is part of the
solution.
SCO: By the end of grade
11, students will be
expected to:
E6 use deductive reasoning
construct logical
arguments and be able
to determine, when
given a logical
argument, if it is valid
Elaborations - Instructional Strategies/Suggestions
Equality properties (p.384)
Challenge students to read and discuss the properties on p.384.
The addition property can be read two ways:
a) If a = b, then a + c = b + c
or
If a = b and c = d, then a + c = b + d
The second statement above is used to solve Systems of Linear
Equations by elimination.
Initiate a discussion on the subtle differences in the logic between
substitution and transitive.
Substitution
If you have an equality a = b then because “a” equals “b” one can be
substituted for the other in any other statement containing a or b.
Ex: If a + b = c and b = d, then a + d = c
Transitive
Many times it is interchangeable with substitution but not always.
Notice the two statements are linked(ie. the 2nd part of the first sentence
is the same as the 1st part of the second sentence).
In logic symbolism it is written:
If a 6 b and b 6 c , then a 6 c
This pattern is called a syllogism. Patterns can be composed of more
than 2 statements, for example:
A good chess player thinks ahead a number of moves something like
this:
If I move to a, then my opponent will move to b.
If my opponent moves to b, then I will move to c.
If I move to c, then my opponent will move to d.
If my opponent moves to d, then I will move to e.
Notice the syllogistic pattern. A good chess player may have 3 or 4 of
these possible scenarios worked out for 5 or more steps in each scenario.
47
Worthwhile Tasks for Instruction and/or Assessment
Equality properties
Math Power 11 p.384 Green #1,2
Equality properties (p.384)
Pencil/Paper/Discussion
State the reason (using the equality properties) that will justify
each statement to be true.
a) If pA = 300 and pB = 300, then pA = pB.
b) If AB = CD, then AB ! XY = CD ! XY
c) If pA = pB and pB = pC, then pA = pC.
Journal
Explain the differences in the reasoning between the
substitution and transitive properties.
Communication
Communicate to the class your step-by-step solution which
includes justifying each step using the equality properties for
each of the following:
a) 2 x − 5 = 5 x + 4
b) 3( x − 2 ) = x + 2
c)
Suggested Resources
3x + 4
= −4
5
48
By the end of grade 11,
students will be expected
to:
Elaborations - Instructional Strategies/Suggestions
Congruent Triangles (p.364)
Naming an angle
Challenge students to give four ways to name the angle indicated;
Solution
E7 explore properties of
and make and test
conjectures about two
and three dimensional
figures
pABC
pB (if only 1 angle at vertex B)
pCBA
p1
Terminology for triangles
Invite students to discuss the terms for triangles:
< side opposite a certain angle
< included angle
< included side
ex.
side AC is the side opposite pB
pA is included between sides AB and AC
side BC is the side included between pB and pC
Definition of congruent triangles
Allow students time to brainstorm on the idea of congruent triangles.
Encourage students to come to a consensus on a good definition of
congruent triangles. (See top of p.364).
The order in which the vertices of congruent triangles are written in the
congruency statement determines the correspondence of the angles and
sides.
ªABC – ª LPR
49
pA = pL
pB = pP
pC = pR
BC = PR
AC = LR
AB = LP
Worthwhile Tasks for Instruction and/or Assessment
Congruent Triangles (p.364)
Pencil/Paper/Discussion
For
Congruent Triangles
Math Power 11 p.364 #1-4
p.365 #1-8
ªABC, state:
a) the side opposite pC
b) the angle included between sides BC and CA
c) the side included between pA and pB
Group Activity
ª
Suggested Resources
–ª
Given that PNT
XAL, list all pairs of congruent
sides and angles. (A sketch of these triangles may be helpful).
Journal
Write a good definition of congruent triangles.
50
By the end of grade 11,
students will be expected
to:
Elaborations - Instructional Strategies/Suggestions
Congruent Triangles
Students have a geometry set( or you could demonstrate this on the
blackboard).
E15 explore constructions
for various geometric
configurations and
apply them in problem
solving
Invite them to construct the three following triangles:
a) a triangle with sides 8 cm, 12 cm and 14 cm long
b) a triangle with sides 7 cm, 9 cm and an included angle of 500
c) a triangle with angles of 600 and 750 and an included side of 9 cm.
For each of the above constructions, all students should have
constructed the exact same triangle( they are congruent).
Inductively, the student groups should be able to conclude that each of
the above gives sufficient information to construct congruent triangles.
A student does not have to get all 6 pairs of sides and angles congruent
before deducing the triangles to be congruent.
The congruency postulates are:
a) SSS 3 pairs of congruent sides
b) SAS 2 pairs of congruent sides and 1 pair of congruent included
angles
c) ASA 2 pairs of congruent angles and 1 pair of congruent included
sides
51
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Congruent Triangles
Congruent Triangles
Activity
Write the reason why the triangles below are congruent:
a)
Math Power 11 p.382-3 Green #1,2
b)
Performance
Draw a diagram that illustrates two triangles that are
congruent due to:
a) the SSS postulate
b) the SAS postulate
c) the ASA postulate
d) the AAS postulate
e) the HL postulate
52
By the end of grade 11,
students will be expected
to:
Elaborations - Instructional Strategies/Suggestions
Direct and Indirect Proof (6.6)
Invite students to read p 366 to the beginning of example 1 on p367.
Direct proof is essentially deductive reasoning. See examples on p 369370.
E6 use deductive
reasoning, construct
logical arguments and
be able to determine,
when given a logical
argument, if it is valid
E28 demonstrate an understanding that reasoning
and proof are essential
and powerful parts of
mathematics
E29 develop and evaluate
mathematical
arguments and proofs
Indirect proof is known as “proof by contradiction”. Whatever we are
asked to prove, assume the opposite to be true.
Using the given information and the negation of what we are asked to
prove, reason until a contradiction is reached. We can then deduce that if
the opposite of what we were asked to prove is false then the other
possibility must be true.
See the explore exercise on p.366 and examples 1,2 and 7 on the
following pages for worked examples of indirect proofs.
Invite students to discuss the above direct and indirect examples.
Discussion should be encouraged on the following Theorems and
postulates:
< Parallel Lines Theorem ( and its converse)
< Parallel Line Postulate
< Triangle Sum Theorem
< Third angle Theorem
< Exterior Angle Theorem
< Isosceles Triangle Theorem (and its converse)
< AAS
–Postulate
Note: This might be a good time to do numerical problems on some of
the above theorems: (See Suggested Resources column)
53
Worthwhile Tasks for Instruction and/or Assessment
Direct and Indirect Proof (6.6)
Pencil/Paper
Using indirect proof, what is your first assumption in trying
to prove: “If a number is odd, then its square is odd”.
Group Activity
Use indirect proof to solve the following puzzle;
John, Harry and Mary are different heights. Who is the tallest
and who is the shortest if only one of the following statements
is true.
John is the tallest
Harry is not the tallest
Mary is not the shortest
Group Activity
Use indirect proof to solve the following. If only one of the
three people is telling the truth, how many friends does Joan
have?
John claims that Joan has at least 100 friends
Harry remarks that Joan certainly doesn’t have that many
Mary states that Joan must have at least one friend
Pencil/Paper
Use direct proof to prove that the difference between a two
digit number and the number formed by reversing the digits is
divisible by 9.
54
Suggested Resources
Direct and Indirect Proof
Math Power 11 p.372 #1-7,10,12,
16-18
Math Power 11 p.377 #29-31
Problem Solving Strategies
p.379 #1, 5, 7, 9, 10
Numerical Problems
Math Power 11 p.382 #1 Green
p.383 Mental Math #1-4(angle
measures)
Inductive Reasoning Worksheet
Pencil/Paper/Technology
Complete the following table and make a conjecture about a rule that describes these figurate numbers.
Predict the next two terms and draw diagrams describing them. Use regression analyse to determine the
rule or formula for each sequence of numbers.
a) Square numbers:
Solution
Quadratic sequence
tn = n 2
b) Rectangular numbers
Solution
Quadratic sequence
tn = n(n + 1)
c) Triangular numbers
Solution
Quadratic sequence
tn =
n ( n + 1)
2
Journal
When pool balls are racked on a pool table at the start of a game what shape do they make? What is the
shape of the holder? How many pool balls are there? Which of the above types of figurate numbers does
55
the number of pool balls fit into?
Journal/Research
Write a few paragraphs on the history and meaning of the term “figurate number”
Pencil/Paper/Technology
Complete the following table and make a conjecture about a rule that describes these figurate numbers.
Predict the next two terms and draw diagrams describing them. Use regression analyse to determine the
rule or formula for each sequence of numbers.
Journal
Can you speculate on the name
of the above figurate number pattern?
56
Manipulative
Shown on the following pages are the nets of the five Platonic Solids. Use these nets to build 3-D models
of each of the solids. Then use the models to complete the table. Express in writing the pattern you have
observed. Finally generate Euler’s Rule.
Platonic Solid
# Faces
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
57
# Vertices
# Edges