Page 6 - 新思维数学学生用书9 样章

7 Shapes and measurements
10 a
11 Here is a decimal: 5.020 020 002 000 020 000 020 000 002… of π’ means
These investigations, to be carried
out with a partner or in a group,
will help develop skills of thinking
and working mathematically.
In this exercise, you have looked at the properties of rational and
irrational numbers.
a Are the following statements true or false? 2 words.
Questions to help you think
about how you learn.
b Here is a calculator answer: 3.646 153 846
Summary checklist
This is what you have
learned in the unit.
7 Shapes and measurements
Questions that cover what you
have learned in the unit.
At the end of several units,
there is a project for you to
carry out, using what you
have learned. You might make
something or solve a problem.
158
14 The circumference of a circular lawn is 32.56 m.
Work out the area of the lawn.
Give your answer correct to the nearest square metre.
15 This is part of Dirk’s classwork. 1.1 Irrational numbers
Question
Use a calculator to find
Work out the circumference and the area of a circle
i ( 21+ )( 2 −1 ) ii ( 31 )( 31 How to use this book
+
×
+
−
×
).
−
)( 41
×
) iii ( 41
Tip
with diameter 18 cm.
b Continue the pattern of the multiplications in part a.
Write your answers in terms of π.
‘Write your
c Generalise the results to find ( N +1 )( N −1 ) where N is a positive integer.
×
Answer
d Check your generalisation with further examples. answers in terms
C = πd = π × 18 = 18π cm
7.1 Circumference and area of a circle
2
2
2
Arun says: A = πr = π × 9 = π × 81 = 81π cm you can leave
the answers
Think like a mathematician as 18π cm and
a
Critique this method of writing the answers. Does it have any
There is a regular
6 pattern: one zero, then 81π cm . You do
Work with a partner to answer this question.
advantages or disadvantages?
2
b So far in this unit you have used the formula A = πr 2 .
two zeros, then three ions. Write your answers in terms of π. not have to work
Answer these quest
zeros, and so on. This is out the value
In questions 1, 4a and 4b, you found the area when you
i
Work out the circumference of a circle with
were given the radius.
a rational number.
diameter 25 mm.
In questions 2, 4c and 4d, you found the area when you of 18 × π and
a Is Arun correct? Give a reason for your answer. Tip 81 × π and write a
ii Work out the area of a circle with radius 12 mm.
were given the diameter.
rounded answer.
iii Work out the circumference of a circle with
Can you write a formula to work out the area which uses
When you write
b Compare your answer with a partner’s. Do you agree? If not, who is correct?
d (diameter) instead of r (radius)? an algebraic
radius 22.5 cm.
iv Work out the area of a circle with diameter 40 cm.
Write your formula in its simplest form.
formula, try to
c
The diagram shows a semicircle.
Test your formula on questions 4c and 4d. Does it work?
use letters not
Show that
Compare your formula with other pairs in the class.
the area of the semicircle is 72π m
i
i For the rest of the questions in this exercise, use the π button on your calculator. 24m
The sum of two integers is always an integer.
ii the perimeter of the semicircle is (12π + 24) m.
ii 7 The sum of two rational numbers is always a rational number.
Work out i the area and ii the circumference of each circle.
The sum of two irrational numbers is always an irrational
iii Look back at this exercise.
Give your answers correct to one decimal place (1 d.p.).
number.
a
b
a How confident do you feel in your understanding of this section?
diameter = 32 mm
radius = 5.6 cm
8
b What can you do to increase your level of confidence?
This is part of Pria’s homework.
The answer is rounded to 9 decimal places.
Can you decide whether the number is rational or irrational?
Question
Summary checklist
Work out the area of this semicircle.
Answer
I can use the formulae for the circumference and area of a circle.
Area of semicircle = half of area of circle
I can use square numbers and cube numbers to estimate square roots and
11.85cm
Estimate: d ≈ 12, so r ≈ 6 and π ≈ 3
cube roots. 1 1
A ≈ × 3 × 6 2 = × 3 × 36 = 54 cm 2
2
2
I can say whether the square root or the cube root of a positive integer is
Accurate: r = 11.85 ÷ 2 = 5.925
rational or irrational.
1
A = × π × 5.925 2 ≈ 55.14 cm 2
2
144
Check: 55.14 cm 2 is close to 54 cm 2 ✓
Check your progress
Use Pria’s method to work out an estimate of the area and calculate the accurate area of a
semicircle with
1 Work out the circumference of these circles. Use the π button on your calculator.
b
a
radius = 6.2 cm
radius = 14.85 m
Give your answers correct to two decimal places (2 d.p.). d diameter = 19.28 m. 11
c
diameter = 14.7 cm
2 a Work out the area of these circles. Use the π button on your calculator.
b
radius = 3.4 m
diameter = 12.5 cm
Round your answers correct to 2 d.p.
Give your answers correct to three significant figures (3 s.f.).
a diameter = 12.5 cm b radius = 3.4 m
3 Work out the area of this compound shape. 141
Use the π button on your calculator.
Give your answer correct to one decimal place (1 d.p.).
Project 1 Cutting tablecloths
8 cm
Project 1
6 cm
4 Work out the shaded area in this diagram.
Cutting tablecloths
Use the π button on your calculator.
Imagine a square piece of cloth 1 metre by 1 metre that could be altered to make a
Give your answer correct to two significant figures (2 s.f.).
tablecloth for a rectangular table.
You could cut off a strip 20% of the way along the square, rotate it, and attach it to
10 cm
the other edge to make a rectangle. There would be a little bit of cloth left over!
25 cm
5 a Write these masses in order of size, starting with the smallest.
5 tonnes 5 milligrams 5 nanograms
5 grams 5 micrograms 5 kilograms
b Underneath each mass in part a, write the mass using the correct letters for the
units, not words. For example, underneath 5 grams you write 5 g.
• The purple square is the original tablecloth.
• The blue rectangle is the new tablecloth.
• The red piece shows the cloth that is left over.
Look at the diagram.
• What percentage of the original cloth has been used to make the new tablecloth?
Instead of cutting off a 20% strip, you could cut a 10% strip, or a 15% strip, or a
different percentage.
Choose some percentages to try. For each example, think about the following
questions:
• What percentage of the original tablecloth is used to make the new tablecloth?
• What percentage of the original tablecloth is wasted?
• Is there a quick way to work out the percentage of cloth used and wasted, if you 5
know what percentage strip was cut off?
Then answer these questions.
• To make a rectangular tablecloth in this way, with an area of 75% of the original
cloth, what percentage strip would you need to cut off?
• To make a rectangular tablecloth in this way, with an area of 50% of the original
cloth, what percentage strip would you need to cut off?