the beginning: numbers (solucionario)€¦ · material aicle. 3º de eso: the beginning: numbers...

3Material AICLE. 3º de ESO: The beginning: numbers (Solucionario)

SOLUTIONS

Numbers… from the beginning

3) Examples:

a) When you add two whole numbers you ALWAYS obtain a whole number: TRUE.

b) When you divide two whole numbers you ALWAYS obtain a whole number: FALSE, 5:2=2.5, not a whole number.

c) When you subtract two whole numbers you ALWAYS obtain a whole number: FALSE, 5-9 is not a whole number.

d) When you multiply two whole numbers you ALWAYS obtain a whole number. TRUE.

4) b) The text is…

At the beginning people used whole numbers to count their goods and identify things, and they were able to add and multiply those numbers (obtaining other whole numbers). They could subtract when the minuend was higher than the subtrahend. But people created other kinds of subtraction to express debts and other situations with other kinds of subtraction: if the subtrahend is less than the minuend the result is NOT a whole number! These new numbers are called integers. The set of the integers contains the whole numbers, the zero and the opposite of every whole number (expressed by the sign “-” before it, like -6). So the result of a sum/multiplication/subtraction of two integers is another integer, that can be zero (0), a positive number (2, 6, 152…) or a negative number (-24, -85, -652…).

5) a) Match the two halves of the sentences.

First half of the sentence Second half of the sentence

With a rational number you can express by a fraction

You can represent a rational number a numerator and a denominator

A fraction has two parts: that represents an integer is 1

The denominator of the fraction the division of two integers

4 Material AICLE. 3º de ESO: The beginning: numbers (Solucionario)

Denominator: it means the number of equal parts in which we divide the whole. This case we divide the whole into 4 parts. Every part is ONE fourth, we use ordinal numbers.

Numerator: it indicates the number of parts we consider. This case the fraction means three fourths.

b) Fractions in English.

(1) Listen to your teacher and complete the text, the words you (and your partner) have to use are given below.

A fraction has two parts:

If the denominator is equal to two, the parts are not “seconds”, every part is a half. So ½ is one half, 5/2 means five halves, etc.

Another possibility, especially if the numbers are big, is to read the fraction using this formula: “numerator” over “numerator”, without using ordinals. So we can read this way: fifty over thirty.

6) Equivalent fractions.

a) The correct ones are:

7) Everything in order! Who eats more pizza??

a) Ordering fractions: how to distinguish the greater fraction?

• If two fractions have the same denominator… the greater one is the fraction with the greater numerator.

• If two fractions have the same numerator… the greater one is the fraction with less denominator.

• If the numerators and denominator are not the same, then… you have to change them by two equivalent fractions with the same denominator, and proceed as before.

50___30

/ /


b) Who should pay more??

8) The true sentences are:

You can do four mathematical operations with rational numbers obtaining another rational number. Some integers are whole numbers. Some rational numbers are integers.

9) Ordering words:

A fraction is called a proper fraction when the denominator is greater than the numerator.

A fraction is called an improper fraction when the denominator is less than the numerator.

When the numerator and denominator are equal, the fraction represents the unit (one).

Two fractions are equivalent when they represent the same number (the division they represent has the same result).

You can obtain equivalent fractions by multiplying/dividing the numerator and the denominator by the same number.

Mary Paul John Who had to pay more?

sandwich

Paul

pizza

Paul

cake

John


10) Remembering practical things!

b) Time to calculate: practice!

1)

2)

3)

4)

5)


11) Powers:

a) Text and operations:

A power is a mathematical expression used to represent a multiplication whose factors are equals. Example: 2x2x2x2 = 24

A power has 2 parts, called base and exponent.

The base means the repeated factor and the exponent means how many times the base is repeated. To read a power you use ordinal numbers. Example: you read 34 as three to the fourth power. If the exponent is 2 you say “squared” and if the exponent is 3 you say “cubed”. If the base is negative and the exponent is even, the result is positive. If the exponent is odd the result is negative. If the exponent is negative the power is equal to a fraction whose numerator is 1 and the denominator is the power with the exponent positive.

Calculate the value of the following powers.

1)

2)

3)

4)

5)

6)

7)

8)

9)

1) 125

2) -‐ -‐1024

3) -‐8000

4) 11390625

5) -‐6765021

6) 7225

7) 1/125

8) -‐1/257

2560000

1) 125

2) -‐ -‐1024

3) -‐8000

4) 11390625

5) -‐6765021

6) 7225

7) 1/125

8) -‐1/257

2560000


b) Properties of powers:

12) Decimal numbers and fractions:

a) For every fraction write the corresponding decimal number.

c) The final result of the investigation is:

• If the denominator can be divided only by 2 and/or 5, it is an exact number.

• If there is a factor different to 2 or 5, the decimal will be recurring.

Example Property

32 ·∙ 52 = ( 3 ·∙ 5 )2 = 152

32 : 52 = (3:5)2 an ·∙ bn = (a .b)n

an : bn = (a:b)n

32 ·∙ 35 = 32+5 = 37

39 : 35 = 39 -‐ 4 = 35

an ·∙ am = an+m

an : am = an-‐m

= 73x5 = 715 (an)m = an·∙m

50= 1 ; 51 = 5 a0=1; a1=a

FRACTIONS DECIMAL NUMBERS

0.5, 0.625, 1.4, 0.024

0.9, 0.15, 0.18

2.1666…, 0.208333…, 1.91666…

2.476190476190…, 2.222…, 0.051948051948…, 0.769230769230


Numbers… going further

1) Telling a story. a) Look:

Pythagoras was an ancient Greek philosopher and mathematician. He was born on the island of Samos (570 BC.). He travelled widely when he was young and he went to Egypt and others places searching for more and more new things to learn.

They studied numbers deeply, doing interesting classifications, discovering important things about them. Their investigations are very important even today. You surely remember the famous Pythagoras theorem (he was not the one who discovered it but he studied it deeply). The theorem is applied to right-angled triangles:

So C is a number that multiplied by itself equals 2. There is NO rational number like this mysterious C! The theorem was certainly proved as true, and then… what

happened?

But there was a little problem. The numbers they adored were whole and rational numbers (divisions of whole numbers). They did NOT believe in other different kinds.

Around 530 BC he moved to Croton (Greece) and founded a religious sect. His followers were convinced that “numbers are everything”. They believed that numbers represented magical things, it was a religion.

The problem was this: if you apply the Pythagoras Theorem for a=1 and b=1, you obtain

It seemed “gods” (numbers) were playing with the faith of Pythagoras’ followers. There was a great crisis. The monster had appeared: no rational numbers were there!! They called them “alogos” (“no sense” in Latin).

Now you are going to know them. Are you ready??

c) Questions:

o Who was Pythagoras? Where was he born?o Where did he go when he was young?o Where and when did he move to?o What did he find?o What did he and his followers believe?o Which important theorem did they study? What did it establish?o What was the problem? How did it affect them?o What can you say about the new numbers?

a2+b2=c2

The sum of the areas of

the two squares on

the legs (a and b)

equals the area of the square on

the hypotenuse

(c).

12+12=c2 1+1=c2 2=c2


2) Rational or not?

b) True or false? Why?

• With integers you can divide numbers. It’s true because you obtain a rational number.

• With rational numbers you can do every mathematical operation. It’s false because you cannot do the squared root of a negative number.

• To divide numbers properly you need irrational numbers. It’s true because you cannot divide some integers obtaining another integer (you obtain a rational number).

• Whole numbers and integers are irrational numbers. It’s false because they are not decimal numbers and can be expressed as fractions.

• Some numbers can be rational and irrational. It’s false because a number has infinite non repeated decimal figures or not.

• Irrational numbers have always infinite decimal figures. It’s true because if not they are exact decimals, so rational numbers.

• Rational numbers never have infinite decimal figures. It’s false because recurring decimals do.

• Rational and irrational numbers have infinite decimal figures. It’s false because there are rational numbers without decimal figures (integers).

Interesting “mistakes”

2) c) Complete chart (including relative errors):

Real value Approximation Absolute error Relative error

1h 5m 10s 1h 5m 7s 3s 3/3910=0.000767…

23.56/23.48 cm 23.52 cm 0.04 cm 0.00169…/0.00170…

1456.76 km 1457.02/1456.5 km 0.26 km 0.000178…

30º 5’ 2’’ 30º 5’ 2’’ 0.00001846…

50.20501 g 50.30 g 0.09499 g 0.00189…


5) c) Practice to learn well!

Time to be radical

1) a) Remember...

• The squared root of 9 is 3 because 32 is equal to 9.

• The squared root of 100 is 10 because 102 = 100.

• The squared root of 144 is 12 because 122=144.

b) The squared roots are respectively: x, 9, 14, x, 126.

Sometimes we cannot calculate the squared root because there is no squared root of a negative number.

2) It is like this:

The cubed root of 125 is 5 because 53 is equal to 125The fourth root of 16 is 2 because 24 is equal to 16The fifth root of 243 is 3 because 35 is equal to 243

The sixth root of 1 is 1 because 16 is equal to 1The seventh root of 128 is 2 because 27 is equal to 128

3) a) Picture:

b)

Index 3

Radicand 8

Radical 3√8

Root 2

Real number Round to the… Approximation

215.574 units 216

85.74 tenths 85.7

9801.8656 thousands 9801.866

=2

radicalindex

radicant = root


4) a) Calculating

b) It is…

5) Aproximately…

6) Chart:

8) Solutions: see http://www.vitutor.com/di/re/r_e.html

a) 3 b) 3.464 c) .

Radical Power

Impossible

-3

2

5

36

-15

-6

impossible

the beginning: numbers (solucionario)€¦ · material aicle. 3º de eso: the beginning: numbers...

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