Page 25 - 新思维数学教师用书9 试读样张

1 number And CALCuLAtion

Starter ideas They will see 2.645 751 311 or similar. Note that the
number of decimal places can vary with different
Getting started (10 minutes) calculators.
Resources: Getting started exercise at the start of Unit 1 Since 2 = 4 and 3 = 9 then, as you can see, 2 < 7 < 3.
2
2
in the Learner’s Book Ask ‘Is this a rational number?’, ‘Does the decimal
Description: Ask the learners to do the questions. After number eventually terminate?’, ‘Is there a repeating
a few minutes check the answers. Do this by asking a sequence of digits?’ The answer to each question is no.
learner to give the answer. Then ask them to explain The proof of this is too advanced for learners at this stage,
why. Use this to check that learners are familiar with but you can explain to them that the square root of any
the prior knowledge required for this unit. This includes positive integer that is not a square number (1, 4, 9, 16,
the concept of a rational number, square roots and cube …) will be similar to this. It has a decimal expansion that
roots, positive integer indices and the index rules for does not terminate and does not have a repeating pattern.
multiplication and division (positive indices only). Since it is not rational it is called an irrational number.
Other examples of irrational numbers are the cube roots
Main teaching idea of any number that is not a cube number. Ask learners to
decide whether the following six numbers are rational or
Irrational numbers (10 minutes)
irrational.
Learning intention: To understand that there are
3
.
numbers on the number line that are not rational numbers. 25; 250; 343 ; 3 81 ; 62 5 ; 625.
Resources: Calculators Learners should work in pairs. Check the answers after
a minute or two. After this activity, learners can start
Description: Ask ‘What does rational number mean?’ Exercise 1.1.
Agree on two points:
• You can write a rational number as a fraction. Answers: 25 = 5 rational; 250 = 15.811… irrational;
3
• The decimal expression will either terminate or have a 3 343 = 7 rational; 81 = 4.326… irrational;
.
repeated sequence of one or more digits. 62 5 = 7.905… irrational; 625. = 2.5 rational
Ask learners to use a calculator if necessary to find the Differentiation ideas: For more confident learners,
7
15
5
1
decimal form for 12 , 18 , 3 and 6 . ask them to find the squares of successive decimal
16 15 7 17
approximations to 7 = 2.645 751 3…
Answers: They will find:
5
12 = 12.3125 (this decimal terminates). 2.6 = 6.76; 2.65 = 7.0225; 2.646 = 7.001 316;
2
2
2
16
2
7 2.6458 = 7.000 257 64
18 = 18.466 666 6… (here the digit 6 repeats).
15 Beyond this the answers will be rounded because of the
1 limit of the calculator display.
3 = 3.142 857 14… (here there is a sequence of
7
6 repeating digits 142 857) Ask ‘What do you notice?’ They should see that the
answers get closer to 7, but the number of decimal digits
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6 = 6.882 352 941… (a calculator does not show increases by two each time. This makes it likely that the
17 decimal value of the square root will not terminate.
enough digits to see the repeating pattern. Explain that
there is in fact a pattern of 16 repeating digits and
15 · · Plenary idea
6 = 6.882 352 941 176 470 5 where the sequence from
17 Summary (5 minutes)
8 to 5 is repeated). Resources: None
Now ask learners to use a calculator to find 7. Description: Ask learners to draw a diagram to show
the relationship between integers, rational numbers and
irrational numbers. Can they do it?

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