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?

1gel 2gel topyh )1gel( )2gel( )topyh(

)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.