Page 5 - 新思维数学学生用书7 样章

How to use this book

How to use this book

In this book you will find lots of different features to help your learning.
1 Integers

Questions to find out what you

know already. Getting started
1
2 Expressions, formulae and equations
Put these numbers in order, from smallest to largest: 9, −7, 6, −5, 3, 0.
2 Find the multiples of 9 that are less than 50.
3 Find the factors of 15.
2.1 Constructing expressions
4 Work out 13 2 − 12 2 . Write your answer as a square number.
When you count objects, you use the positive whole numbers 1, 2, 3, 4, …
Whole numbers are the first numbers that humans invented.
In this section you will …
You can use these numbers for more than counting. Key words
For example, to measure temperature it is useful to have the number 0 (zero) and
•
use letters to represent numbers
What you will learn in the unit. negative whole numbers −1, −2, −3, … coefficient
You can put these numbers on a number line.
•
use the correct order of operations in algebraic expressions
1 Integers 1, 2, 3, 4, … are sometimes called positive numbers to distinguish them from the constant
•
write and use expressions.
negative numbers −1, −2, −3, −4, … expression
Positive and negative whole numbers together with zero are called integers.
1.1 Adding and subtracting integers equation
In this unit you will learn about integers and their properties.
In algebra you can use a letter to represent an unknown number.
An expression contains numbers and letters, but not an equals sign. equivalent
An equation contains numbers and letters and an equals sign. expression
In this section you will … Example: 5n + 4 is an expression. term
Key words
Important words to learn. integers unknown
5n + 4 = 19 is an equation.
•
add and subtract with positive and negative integers.
In the expression 5n + 4, there are two terms. 5n is one term. The other term variable
is 4. inverse
Integers are positive and negative whole numbers, together with zero. inverse operation
You can show integers on a number line. The letter n is called the variable because it can have different values. Tip
The coefficient of n is 5 because it is the number that multiplies the variable.
negative integers
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 In the equation 5n + 4 = 19, n is the unknown number, 5 is the coefficient of 5n + 4 means
number line
5 × n + 4. Use
Integers greater than zero are positive integers: 1, 2, 3, 4, … n, and the numbers 4 and 19 are constants. A constant may also be written the correct
positive integers
as a letter, such as π. π is the ratio of a circle’s circumference to its diameter.
Integers less than zero are negative integers: −1, −2, −3, −4, … It is approximately 3.14. order of
You can use a number line to help you to add integers. 2 Expressions, formulae and equations 7 operations.
You can use a letter to represent an unknown number to solve problems.
Tip
Step-by-step examples showing Example: Shown is a bag of sweets. You don’t know how many sweets are in Do the
Worked example 2.3 The ‘…’ (called multiplication
the bag.
how to solve a problem. Simplify each expression. an elipsis) shows before the
a 2x + 3x b 7y − 2y c 4p + 3q + 2p − q d 5t + 7 − 3t + 3 addition.
that the lists
Answer continue forever. You will learn
Worked example 1.1 a 2x + 3x = 5x 2x and 3x are like terms, so add them to get 5x. more about π
n– 3 sweets
b 7y − 2y = 5y n sweets 7y and 2y are like terms, so subtract to get 5y. later in your
Work out: c 4p + 3q + 2p – q = 6p + 2q 4p + 2p = 6p and 3q − q = 2q, but 6p and 2q studies.
n represents the unknown
Three sweets are taken out of the bag.
are not like terms so you cannot simplify any
a −4 + 6 b 8 + −3 c −3 + −5 number of sweets in the bag. Now there are n − 3 sweets left in the bag.
further.
5t − 3t = 2t and 7 + 3 = 10, but 2t and 10 are not
Answer d 5t + 7 − 3t + 3 = 2t + 10 Worked example 2.1
like terms so you cannot simplify any further.
a You can use a number line to help you. 2 Expressions, formulae and equations
Mathew is x years old. David is 4 years older than Mathew. Adam is 2 years younger than Mathew.
These questions will help you Kathryn is three times Mathew’s age. Ella is half Mathew’s age.

Start at −4. Move 6 to the right.
6
develop your skills of thinking Exercise 2.3 Write down an expression for each person’s age.
This is part of Bethan’s homework. Bethan has made a mistake in
–5 –4 –3 –2 –1 0 1 2 3 4 5 6 every answer. Explain what Bethan has done wrong. Work out the

−4 + 6 = 2
and working mathematically. 1 Erik has yellow, green and blue bricks. a b c
correct answers.

The length of a yellow brick is a.
b Start at 8. Move 3 to the left. You move to the left because it is −3. 32

The length of a green brick is b.
Question
The length of a blue brick is c.
Work out the total length of these arrangements of bricks.
–3 –2 –1 0 1 2 3 4 5 6 7 8 9 Give your answer in its simplest form.
Multiply out the brackets.
8 + −3 = 5 a a 4(x + 4) b b 2(6x − 3)
?
?
c Start at −3. Move 5 to the left. c 3(2 − 5x) d 6(2 − x)
c d
? Solution ?
–8 –7 –6 –5 –4 –3 –2 –1 0 1 e a 4(x ?+ 4) = 4x + 8 f b 2(6x − 3) = 12x – 3
?
−3 + −5 = −8 2 Match each expression, a to f, to its correctly simplified expression, Tip
c 3(2 − 5x) = 6 + 15x
d 6(2 − x) = 12 − 6x = 6x
i to vi.
The first one has been done for you: a and v. Remember that
4 8 a x + x i ii 7x x is the same as
1x.
b Look again at Question 6. What method did you use to answer this question?
x
5x + 4x
c x + 6x iii 4x
Do you think this was the best method or would you use a different method if
d 5x − 2x iv 9x
you had to answer the question again?
e 2x − x v 2x
f 8x − 4x vi 3x
When I
7 Arun looks at these four expressions. expand the brackets
42 in all of these expressions,
2(12x + 15) 3(10 + 8x) the answers are all the
same.
4(6x + 26) 6(5 + 4x)
Is Arun correct? Explain your answer
and show your working.
Think like a mathematician
8 Discuss with a partner the answers to these questions.
a When you expand the brackets in the expressions 3(4b + 5) and 3(5 + 4b),
do you get the same answer?
b When you expand the brackets in the expressions 2(5c − 1) and 2(1 − 5c),
do you get the same answer?
Give evidence to justify your answers.
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