Page 5 - 新思维数学学生用书8 样章

5 Angles and constructions
How to use this book
The sum of the angles in a triangle is 180°. Can you remember the
first time you were shown this?
How to use this book
You may have measured the angles and added them.
You may have cut a triangle out of paper and folded it. 2 Expressions, formulae and equations
This does not prove that the sum of the angles of any triangle is
180°. It only shows that it is true for the triangles you have drawn A formula is a set of instructions for working something out.
It is a rule written using letters or words.
and that it is a reasonable conclusion. The plural of formula is formulae.
A proof is a logical argument in which a reason is given for each People use formulae in everyday life to work out all sorts of things.
6
Collecting data
An employer may use a formula to work out how much to pay the people who work for them. For
In this book you will find lots of different features to help your learning.
step.
Over 2000 years ago the Greek mathematician Euclid wrote a book example, they could use the formula P = R × H, where P is the pay, R is the amount paid per hour and
H is the number of hours worked.
called The Elements. He used logical arguments to prove many Doctors may use a formula to assess a person’s health. For example, they could use a formula to find
Questions to find out what you the person’s body mass index (BMI).
facts in geometry and arithmetic. His book was the most successful
mass
textbook ever written. It is still in print today. This formula is: BMI = height 2 , where the person’s mass is measured in kilograms and their height
know already. Getting started is measured in metres.
Euclid started by defining basic things such as a point and a straight line. He also made a set of
If a person’s BMI is too high or too low, the doctor may ask them to lose or put on weight, to make
1
statements which he thought everyone could agree with. These were called axioms.
Give an example of
them healthier.
An example of one of his axioms is: a discrete data b continuous data c categorical data.
2
Things that are equal to the same thing are equal to one another. You want to choose a sample of 3 boys and 3 girls from your class.

Describe three different ways to do this.
2.1 Constructing expressions
3
a
From this simple starting point, he proved many complicated results.Write one advantage of a large sample size.
In this unit you will look at several proofs. b Write one disadvantage of a large sample size.
4 You want to find the number of brothers and sisters of a group of children.
In this section you will …
Describe two different ways you could collect this data. Key words
use letters to represent numbers
•
What you will learn in the unit. In some country, elections are held every two years to choose members for the Congress. This is one coefficient
5.1 Parallel lines
9 Sequences and functions
use the correct order of operations in algebraic
•
constant
of the groups of people who run the country.
expressions
4
Copy these sequences and fill in the missing terms.
Each province chooses representatives. The number they can choose depends on how many people equivalent
•
use words or letters to represent a situation.
a
3
3
b
6
1
live in that province. The more people live in a province, the more representatives they can choose. It expression
1
1
2, 4 ,
,
, 22 ,
, 8 ,
5, 8 , 11 ,
,13, 15
In this section you will … is therefore very important to keep accurate records of the number of people living in each province. linear expression
Key words
5
5
7
7
7
5
c
1
3
1
1
1
d
You can write an algebraic expression by using a letter to represent an ,
, 68 ,
, 23
, 24,
100, 89 ,
25, 24 ,
, 47
,
Important words to learn. To maintain accurate records, a census is held every 10 years. A census is a way of collecting data. It term
4
2
2
2
2
•
use geometric vocabulary for equal angles formed
alternate angles
unknown number.
e
f
, 24, 23.6,
,
,
,
, 22.4
, 8.9, 9.2,
,
8,
when lines intersect. is a questionnaire that must be filled in by every household in that country. unknown
In the expression 3n + 8 there are two terms. 3n is one term. The other
corresponding
There was a census in 2010. The questionnaire had only 10 questions, which people were able to
term is 8.
answer in about 10 minutes. The results of the 2010 census showed that there were 300 000 000 variable
Think like a mathematician
angles
The letter n is called the variable, because it can have different values.
This diagram shows two straight lines. people living in that country.
How can you answer these questions without working out more of the terms
5
The coefficient of n is 3, because it is the number that multiplies the
geometric
Angles a and c are equal. They are called a b in the sequences?
variable.
transversal
vertically opposite angles. d c a In the sequence 0.4, 0.8, 1.2, 1.6, 2, 2.4, …, what is the first term
The number 8 is called a constant.
13.1 Calculating probabilities
vertically
Example:
Angles b and d are equal. They are also b greater than 10? 1 1
Is 45 a term in the sequence 5, 7 , 10, 12 , 15, ...?
Step-by-step examples showing Let n represent a mystery number. 2 2
opposite angles
vertically opposite angles.
You write the number that is 5 more than the mystery number as n + 5
Is 5 a term in the sequence 30, 26 , 23 , 20, ...?
1
c
Vertically opposite angles are equal. Worked example 13.1a 2 3 1 3
3
or 5 + n.
how to solve a problem. The pro bability that it will be sunny tomorrow is 40%.
Discuss your answers.
Angles a and b are not equal (unless they are both 90°).
The probability it will not rain tomorrow is 95%.
You write the number that is three times the mystery number
They add up to 180° because they are angles on a straight line. Find the probability that tomorrow
as 3 × n or simply 3n.
6
Write the first three terms of each of these sequences.
a will not be sunny

The first one has been started for you.
b it will rain. 28
a
first term is 8
Answer term-to-term rule is: multiply by 2 then subtract 5
104 a P(not sunny) = 1 − P(sunny) = 100% − 40% = 60%
first term = 8
124 b P(rain) = 1 − P(not rain) = 100% − 95% = 5%
second term = 8 × 2 − 5 = 16 − 5 = 11
third term = 11 × 2 − 5 = − 5 =
Worked example 13.1b

Two unbiased 6-sided dice are thrown.
b
first term is 15, term-to-term rule is: subtract 9 then multiply by 3
These questions will help you Find the probability of getting
c
first term is 12, term-to-term rule is: divide by 2 then add 5
a the same number on both dice
7
The first three terms of a sequence are 8, 10, 14, …
develop your skills of thinking b a total of 6 Which of these cards, A, B or C, shows the correct term-to-term rule?
a
c a total of 9 or more.
and working mathematically. Answer A multiply by 3 then subtract 14
a The diagram shows all possible 6 × × × × × ×
outcomes. B divide by 2 then add 6 × × × × × ×
5
There are 36 outcomes 4 × × × × × ×
altogether. C subtract 3 then multiply by 2
The loop shows the outcomes Second dice 3 × × × × × ×
b
Which is the first term in this sequence greater than 50?
with the same number: (1, 1), 2 × × × × × ×
(2, 2) and so on. 1 × × × × × ×
200 There are 6 of them. 6 1 2 3 4 5 6
The probability is which is First dice
36
equivalent to 1 6
4
275