MathG8 Week4 Nov27-28

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Math grade 8 on Nov. 27 & 28, 2010

1

Notice

The notes are the same as before, but the questions in class and homework are different with

before. Please review the notes and do the questions in class and homework.

Counting and patterns 1

1. Counting

Counting is the action of finding the number of elements of nature numbers.

The traditional way of counting consists of continually increasing a (mental or spoken) counter

by a unit for every element of the nature numbers, in some order, while marking (or displacing)

those elements to avoid visiting the same element more than once, until no unmarked elements

are left; if the counter was set to one after the first object, the value after visiting the final

object gives the desired number of elements.

Counting sometimes involves numbers other than one.

For example, when counting money, counting out change, when "counting by twos" (2, 4, 6, 8,

10, 12) or when "counting by fives" (5, 10, 15, 20, 25).

2. Counting using tally marks.

Tally marks are a quick way of keeping track of numbers in

groups of five.

One vertical line is made for each of the first four numbers;

the fifth number is represented by a diagonal line across the

revious four.

3. Number Patterns

A sequenceis a pattern of numbers that are formed in accordance with a definite rule.

We can often describe number patterns in more than one way. To illustrate this, consider the

following sequence of numbers {1, 3, 5, 7, 9, }.

Clearly, the first term of this number pattern is 1; and the terms after the first term are obtained

by adding 2 to the previous term. We can also describe this number pattern as a set of odd

numbers.

Alternatively, We can represent a number pattern by using special symbols. For the number

pattern under consideration, let tnrepresent the nth term.

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2

By trial and error, we find that:

By observation, we notice that we can describe this number pattern by the rule

where tn is nth term of the sequence.

4. Formula and Tables

A table of values can be generated from the rule

as shown below.

5. Finding the Algebraic Rule

We use algebra to study rules that describe the behavior of everyday things, for example, the

behavior of the height of a ball when it is thrown upward or the amount outstanding for a loan

after a number of regular repayments.

By finding a pattern in observed values (i.e. measurements), we are often able to discover a

rule that allows us to make accurate predictions.

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5

Type 1st 2nd 3rd 4th 5th 6th

Triangular

Value 1 3 6 10 15 21

Square

Value 1 4 9 16 25 36

Pentagonal

8. Points on a Circle

Image Points Segments Triangles Quadrilaterals Pentagons Hexagons Heptagons

1

2 1

3 3 1

4 6 4 1

5 10 10 5 1

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Question in class

Counting

1. One needs 4221 digits to number the pages of a book. How many pages does this bookcontain?

2. How many animals do I have if all but two are dogs, all but two are cats and all but two are

hamsters?

3. How many times is the digit 3 used in writing all of the integers from 1 to 100?

4. What is the digit in the ones' place in the expansion of 347?

5. In the game of 6**9, a person chooses 6 different integers from 1 to 9. If the order in whichthe numbers are chosen is not important, how many choices are possible?

6. In how many ways can 7 different colours be arranged in a row if the first colour is always

red, the last colour is always blue and the middle colour is always green?

7. How many integers between 1 and 1000 (inclusive) do not contain the digits 8 or 9?

8. There are 15 Blue Jays and 14 Orioles perched in 3 trees. Each tree has at least 4 Blue Jays

and 2 Orioles. If no tree has more Orioles than Blue Jays, then what is the largest number of

birds that can be in one tree?

9. If our days were divided into 10 hours (with no need for a.m. and p.m.), the new hours into

100 minutes and the new minutes into 100 seconds, what new time would it be at 6 p.m.?

Patterns

10. What is the next number in the list1 1 1 2 5

, , , , ,1, ?6 3 2 3 6

11. What is the next term in the sequence 2 4 8 1 6 3 2, , , , ,3 9 2 7 8 1 2 4 3

?

12. How many rectangular pieces of wood of size 20 cm by 45 cm can be cut from a plywood

sheet of 120 cm by 240 cm?

13. What is the next term in the sequence 1, 3, 5, 11, 21, 43 ... ?

14. I have 6 different books, 3 with red covers and 3 with blue covers. In how many different

ways can I arrange these books on a shelf so that no two books of the same colour are next to

each other?

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Homework

Basic problems

Continue each pattern with the next two numbers.

1. 19, 28, 37, 46, 55, 64, 73, __ , __ 2. 89, 81, 73, 65, 57, 49, 41, __ , __

3. 63, 58, 53, 48, 43, 38, 33, __ , __ 4. 29, 35, 41, 47, 53, 59, 65, __ , __

5. 8, 17, 26, 35, 44, 53, 62, __ , __ 6. 5, 8, 11, 14, 17, 20, 23, __ , __

7. 84, 78, 72, 66, 60, 54, 48, __ , __ 8. 53, 56, 59, 62, 65, 68, 71, __ , __

9. 2, 11, 20, 29, 38, 47, 56, __ , __ 10. 36, 40, 44, 48, 52, 56, 60, __ , __

Use the clue to fill in the missing digit.

1. The number 40 is divisible by 3. 2. The number 72 is divisible by 18.

3. The number 85 is divisible by 9. 4. The number 2 6 is divisible by 4.

5. The number 4 0 is divisible by 2. 6. The number 501 is divisible by 7.

7. The number 38 is divisible by 19.

8. The number 25 is divisible by 15.

9. The number 9 is divisible by 16. 10. The number 30 is divisible by 5.

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7. A sequence of numbers is defined as follows: The first two terms are 1 and 2. Each

succeeding term is 3 times the previous term minus the term before that. What is the 5th term

in the sequence?

8. The integers are arranged in five columns as follows. If this pattern is continued, in which

column will the number 98 be found?

9. John has just purchased five 12-foot planks from which he will cut a total of twenty 34-inch

boards for a ramp. If we disregard the thickness of the cut, how many total inches of board will

be left over when John is done cutting? (Reminder: 1 foot = 12 inches.)

10. Pick a four-digit number and write it down. Now rearrange the digits to form a second four-

digit number. Subtract the smaller number from the larger number and divide the result by 9.

What is the remainder?

11. Just for today, the symbol (N)signifies the value obtained by alternately adding and

subtracting, from least to greatest, each of the positive factors ofN, not includingN. For

example:+ (20)= +1 2 + 4 5 +10 = 8. Find the value of+ ( 36).

MathG8 Week4 Nov27-28

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