LESSONS
CORRESPONDENCE STUDY PROGRAM PAGE 15
ACTIVITY 1:
1.a) On graph paper, draw three right triangles
with legs that have the following lengths:
i) 3 and 4
ii) 8 and 15
iii) 5 and 12
b) Using the edge of another sheet of the same
graph paper, find the length of the hypotenuse
of each of these triangles, and record the data
into the first three columns of a table like the
one above:
c) Complete the chart.
d) Describe any patterns that you see.
e) What statement might you make concerning
the lengths of the three sides of a right
triangle?
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)i 3 4
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)vi t v w
222
LESSON 1Lesson 1 introduces the concept ofirrational numbers. Students will havelearned about square roots of numbersbut will now extend and deepen theirunderstanding of irrational numbers, howthey fit into the real number family, andhow they are used in applications.
Learning OutcomesUpon completion of lesson 1 students will beexpected to:
9A1 solve problems involving square root andprincipal square root
9A3 demonstrate an understanding of the meaningand uses of irrational numbers
9A4 demonstrate an understanding of the inter-relationships of subsets of real numbers
9A5 compare and order real numbers9B1 model, solve, and create problems involving
real numbers9B6 determine the reasonableness of results in
problem situations involving square roots,rational numbers, and numbers written inscientific notation
9C1 represent patterns and relationships in avariety of formats and use theserepresentations to predict and justify unknownvalues
MATHEMATICS 9
PAGE 16 CORRESPONDENCE STUDY PROGRAM
2. Howie Hiker wants to walk from point A to B
on the edges of a square grassy field with
walking paths as indicated in the diagram.
a) What possible paths could he take
(without retracing any path)?
b) What is the length of the longest path?
c) What is the length of the shortest path?
d) Are your answers to the questions above
exact or approximate numbers? Explain.
3. Stana lives in the downtown area of the city
where the houses are very close together. She
wants to clean the window on the second
floor, but she must use a ladder. Her ladder is
5 m long. The window sill is 3.5 m from the
ground, and on the side of the house nearest
the next house. The house next door is only
2 m from Stana’s house.
a) If she wants to place her ladder at the
height of the window sill, how far away
from the house will the foot of the ladder
have to be?
b) If she places the foot of the ladder as far
away as the house next door allows it to
be, how far up the wall will the ladder
reach?
c) What do you recommend Stana do?
d) Let us reflect on your solutions to the
above questions.
i) How did you decide how to record
your answer to questions a) and b)?
What other possible answers could you
have given to those questions? (Just
name a few.)
ii) Did you need to use a calculator to get
the answers to questions a) and b)?
iii) Does your answer to question c)
require a number?
iv) Billie’s answer to b) was m. Do
you think that is a suitable answer?
v) Martha’s answer was “about 5 m.” Do
you think that should be marked
correct? Explain.
vi) Explain why you used the
Pythagorean theorem to get your
answers in a) and b).
4. Can a circular table top with diameter 2.7
metres long fit through a doorway 2.5 metres
high and one metre wide? Explain.
T B
A E1000 m
LESSONS
CORRESPONDENCE STUDY PROGRAM PAGE 17
5. The boys at Pioneer Camp want to store a flag
pole in the storage room for the winter. The
storage room has the following dimensions:
12 units long, by 9 units wide, by 8 units high.
To the nearest tenth of a unit, what is the
longest flag pole that can be put in that room?
ACTIVITY 2:
6.a) In the middle of a sheet of graph paper
construct accurately a right triangle ABC,
with the right angle at ∠C, and the two legs,
each a unit long (a unit is two squares on the
graph paper).
b) Using the side AB as a leg, construct a
right triangle ABD with the right angle at
∠A, and leg AD = 1 unit.
c) Using the side BD as a leg, construct a
right triangle BDE with the right angle at
∠D, and leg DE = 1 unit.
d) Using the side BE as a leg, construct a
right triangle BEF with the right angle at
∠E, and leg EF = 1 unit.
e) Continue this process at least 8 more times.
f ) Calculate the length of each hypotenuse,
expressing your answer each time as an
exact answer (using a square root ( )
sign). What patterns do you notice?
Describe.
g) If you continued this process 20 more
times, what figure is being formed by the
1-unit long legs?
7. Mr. MacKinnon’s math class wanted to survey
the student body at the school to “find” the
most attractive rectangle. Below are some
examples that the student body voted for,
or against.
The winning rectangle looks like this:Use the BLM on page 19:
MATHEMATICS 9
PAGE 18 CORRESPONDENCE STUDY PROGRAM
a) By measuring, or using a compass,
construct a square AEFD with E on AB,
and F on DC.
b) Bisect DF, and label the midpoint M.
c) With centre M, an arc drawn through E
should also pass through C. This confirms
that the rectangle ABCD is a Golden
Rectangle, and that EBCF is another.
d) Turn the rectangle ABCD so that BC is at
the top. Using the same process that you
used earlier, make the square BGHE.
Now, GCFH is another Golden Rectangle.
e) Rotate the rectangle ABCD so that CD is
on the top, and make the square CIJG.
This make the rectangle IFHJ another
Golden Rectangle.
f ) Continue this process making a square
FKLI, which creates the Golden Rectangle
KHJL, a couple more times.
g) Measure carefully to two decimal places,
and calculate to four decimal places the
ratios: BC to DC, FC to BC, FH to FC,
JH to FH, LJ to JH, and so on. What
do you notice? This ratio is called the
Golden Ratio.
h) Using a compass, with centre F, draw arc
DE. With centre H, draw arc GE. With
centre J, draw arc IG. With centre L,
draw arc KI, and so on. The resulting
spiral is called the Golden Spiral.
8. The Divine Proportion was derived by Luca
Pacioli in the 15th century. It is found by
dividing a segment into parts so that the
length of the smaller part is to the length of
the larger part as the length of the larger part is
to the length of the entire segment.
a) Draw a segment that is 10 cm long and
divide it into two parts. Label the longer
part x. The segments are in the Divine
Proportion if the following is true:
Solving for x gives: x2 = 1 - x or
x2 + x - 1 = 0.
The result being: x = . Is this
value exact or approximate? Explain.
b) Using a calculator, determine a value for x
to 4 decimal places. Is this value exact or
approximate? Explain. How does it
compare to your answer in #7.g)?
c) On page 21 is the front facade of the
ancient temple Parthenon. Notice the
rectangle drawn around the front of the
building.
LESSONS
CORRESPONDENCE STUDY PROGRAM PAGE 19
A DCB
20 c
m
12.36 cm
MATHEMATICS 9
PAGE 20 CORRESPONDENCE STUDY PROGRAM
AC0
B
D
LESSONS
CORRESPONDENCE STUDY PROGRAM PAGE 21
Do you think the ancient Greeks knew
about the Golden Ratio? Explain.
9.a) Construct a triangle ABC with ∠A = 36°,
∠B = 72°, and ∠C = 72°. Show that the
ratio BC is to AB is the Golden Ratio.
b) Construct a regular pentagon. Measure
the side length of the pentagon, then
draw and measure one of the diagonals.
What is the ratio of the side length to
the diagonal length?
c) The Golden Ratio is closely connected to
the Fibonacci Sequence and to the
construction of a regular pentagon. The
Fibonacci Sequence is a list of numbers,
each of which is the sum of the previous
two. They begin like this ... 1, 1, 2, 3, 5,
8, 13, 21, ...
i) Write the next 7 terms.
ii) Take the ratio of successive terms,
compute to five decimal places, then,
copy and complete the table:
1 1÷1 00000.1 8 43÷12
2 2÷1 00005.0 9
3 3÷2 66666.0 01
4 5÷3 00006.0 11
5 8÷5 21
6 31÷8 31
7 12÷31 41
Parthenon
MATHEMATICS 9
PAGE 22 CORRESPONDENCE STUDY PROGRAM
d) Complete this construction of a regularpentagon (use the BLM.2, page 20):
i) Bisect radius OA. Label the midpointM. Draw segment MB
ii) With a compass, using M as the centreand MB as the radius, swing an arcthrough radius OC. Label theintersection point P.
iii) With a compass, using B as the centreand BP as the radius, swing an arcthrough the circle, near C. Label theintersection point Q.
iv) Segment BQ is the side of an inscribedpentagon. Mark off the remainingfour sides around the circle using yourcompass. Draw a diagonal andcompute the ratio of a side to thediagonal. How does this valuecompare to the ratio in #9.b)?
ACTIVITY 3:
10.a) Using a calculator, write each of theequations to as many decimal places as yourcalculator allows.
i) 3 ÷ 5 =ii) 1 ÷ 4 =iii) 13 ÷ 25 =iv) 13 ÷ 50 =v) 5 ÷ 125 =
b) Describe the patterns that you see in the
above results. The results are called
terminating decimals. Why do you think
that is what they are called?
c) Create an equation like the equations in
10.a) that results in a three digit decimal
that terminates.
d) Using a calculator, write each of the
equations to as many decimal places as
your calculator allows.
i) 3 ÷ 9 =
ii) 1 ÷ 3 =
iii) 6 ÷ 9 =
iv) 13 ÷ 99 =
v) 15 ÷ 999 =
e) The above results are examples of
repeating decimals. How do the repeats in
iv) and v) differ from those in the first
three?
f ) Make up a question that repeats like those
in i), ii), and iii).
g) Make up a question that repeats like those
in iv), and v).
h) All the above are examples of rational
numbers. Some have digits that terminate.
Some have digits that repeat, and some
have periods of digits that repeat. Give
another example of a rational number with
a repeating period written as a fraction.
LESSONS
CORRESPONDENCE STUDY PROGRAM PAGE 23
i) In #10. d), part v), your answer should
look like this ... 0.015015015 if your
calculator allows for 9 digits in its display,
and truncates the rest. Some calculators
might display this number as ...
0.01501502, where instead of truncating
the number, the calculator rounds the last
displayable digit. What does your
calculator do?
j) Write the decimal displays for the
following: 1 ÷ 7 = ___, 2 ÷ 7 = ___,
3 ÷ 7 = ___, ... 6 ÷ 7 = ___. Look
carefully at the results and describe any
patterns that you see. Classify these as
what kind of rational number?
k) Are numbers that are divided by 13
displayed with repeating periods? Do they
behave like numbers divided by 7?
Explain.
l) Is the Golden Ratio an example of a
rational number? (Hint: change the
expression to a decimal).
m) When a number with a decimal does
not terminate or repeat it is called an
irrational number. (Rational numbers can
always be displayed as a fraction with an
integer numerator and an integer
denominator). State which of the
following are irrational...
i) ii) iii)
iv) v) vi)
n) Create an irrational number that is:
i) greater than 1 but less than 1.1,
ii) greater than 1.11 and less than 1.12.
o) In order to prove that a number is
irrational, you will need to review prime
factorization.
Example: 990 = 99 ⋅ 10
= 9 ⋅ 11 ⋅ 2 ⋅ 5= 2 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 11
So, 990 has a total of 5 prime factors
(Three is counted twice since it appears
twice in the 9.)
p) Start the factorization of 990 by writing
990 = 3 ⋅ 330. Do you get the same prime
factors?
q) Start the factorization of 990 a third way.
Do you get the same prime factors?
r) Each whole number (or integer) has only
one prime factorization. Find it for the
following ...
i) 12 ii) 345 iii) 6789
MATHEMATICS 9
PAGE 24 CORRESPONDENCE STUDY PROGRAM
s) Find the prime factorization for several
perfect square numbers. Try to find one
that has an odd number of prime factors.
Take the numbers 6 and 8. 6 = 2 ⋅ 3, and
8 = 2 ⋅ 2 ⋅ 2 (look at that!, 8 is a perfect
cube).
6 has two prime factors, an even number
of prime factors, and 8 has three, an odd
number of prime factors. When we square
them, we get ...
62 = (2 ⋅ 3)2 = 22 ⋅ 32
82 = (23)2 = 26
i) Explain why any perfect square must
have an even number of prime factors.
ii) Explain why any number that is equal
to twice a perfect square must have an
odd number of prime factors.
t) If p and q were whole numbers or
integers, and you had , it would
follow that , so then ,
and then p2 = 2q2.
i) Explain why p2 must have an even
number of prime factors.
ii) Explain why 2q2 must have an odd
number of prime factors.
iii) Explain why p2 cannot equal 2q2.
iv) You can conclude that there can be no
whole numbers or integers p and q
such that , and therefore is
irrational.
v) Use the same method to show that
is irrational.
vi) Show why is not irrational.
ACTIVITY 4:
11. Write each number on a separate slip of paper.
-3, 5, 0.7, , ,
0.323 323 332 333 32..., 0, , -9,
3.2, , π, 768, ,
0.123 123 123..., -0.7, .
a) Place all the natural numbers in thecentre of the table.
b) Place all the whole numbers in the upperright corner of the table.
LESSONS
CORRESPONDENCE STUDY PROGRAM PAGE 25
c) Place all the integers in the upper leftcorner of the table.
d) Place all the rational numbers in the lowerright corner of the table.
e) Do any numbers remain? Put them in thelower left corner. What kind of numbersare they?
f ) Copy the numbers onto your lessonpages as you have them grouped.
g) Are there any numbers that might fitinto two different groups? Explain.
h) Are there any numbers that might fitinto three different groups? Explain.
i) On your lesson pages, write all thenumbers from smallest to largest downyour page.
j) Which Venn diagram best describes therelationship among natural, whole,integer, and rational numbers, if onecircle represents each set? Copy yourchoice and label each circle with anappropriate symbol: N, W, I, or Q (whereQ is rational).
using labels like N, W, I, Q, and and Q
(irrationals).
l) Copy and complete the table placingcheck marks in the appropriate cells forthe set to which each number belongs.Remember that it is possible for a numberto belong to more than one set.
m) Terms that describe numbers also describepeople. “I feel most like a(n) _________number”.
i) Copy the previous sentence and fill inthe blank with one of the followingwords: natural, whole, integer,rational, irrational.
ii) Provide 5 examples of numbers from
each set.
Q
I
N
W
QI
WN
rebmun N W I Q
)a 7-
)b 0
)c
)d 21
)e 65.0
)f
)g 3.0
)h
)i ..433343343.0
37
-78
Q
1. 2.
k) Make a Venn diagram with circles that
would represent the Real number family,
MATHEMATICS 9
PAGE 26 CORRESPONDENCE STUDY PROGRAM
iii) Write at least five sentences to explain
why you feel most like the type of
number you chose.
12. Respond to the following:
a) Show how to express the number 5 in the
form of .
b) Show how to express the number in
the form of .
c) Show how to express the number 4 in the
form of using a value of 3 for b.
d) Evaluate:
e) Are any of the numbers in d), above,
rational numbers? If so, which? If not,
why not.
f ) Johnny estimates square roots like this:
“the square root of 10 is a little bigger than
the square root of 9, which is three, so is
a little bigger than 3.” Use a similar
method to estimate the square roots of:
g) Which does not belong, and explain why
not:
h) Write the following numbers in order from
smallest to largest:
This is the end of Lesson 1Make sure you have completed all of the
assignment questions.SEND YOUR ANSWERS TO YOUR MARKER NOW.