2nd Maths Term 3 Unit 2: A Guide to Patterns in Numbers

2nd Maths: Term 3 Unit 2 - Patterns

Patterns in Numbers

Keywords: Rule, Arrange, Order

Recall

Observe and complete the patterns in numbers.

Travel Through

Roll and Scroll game

Two students can do the activity.

Make a number board with numbers from 1 to 37 as shown in the picture.

The first player throws a dice and a number appears on the face of the dice say 2.

The other player places a coin on the number 2 and skips to number position in twos such as 4,6,8,10 and so on till the last number. The first player records each of the number positions.

Now, the second player gets the chance to throw the dice. When the number 2 appears again the player tells all the number positions jumped orally. If he gets a number 3, then the first player moves a coin by skip counting in threes.

The second player record each of the number positions jumped. The above procedure is continued till the players are familiarised with the number patterns starting from 2, 3, 4, 5 and 6.

Learn

Find the rule and complete the patterns.

Activity

Count and write the number of ice sticks and complete the patterns using the rules.

Practice

Find the rule and write them in each of the pattern given below.

Pleasure Time

1) Write the number patterns for the given instructions.

(i) The last number is 21. The first number is 13. Write the rule.

13 (+2) 15 (+2) 17 (+2) 19 (+2) 21

(ii) The pattern begins with 39. The next number is 3 less than the previous number.

39(−3), 36(−3), 33(−3), 30(−3) 27

(iii) The middle number in the pattern is 45. The number previous to it is 40.

35 (+5), 40 (+5), 45 (+5), 50 (+5), 55

2) Draw sticks to continue the pattern.

Try This

Draw the missing black keys in a pattern.

Atom and Atomic Mass: A Detailed Explanation

Atom and Atomic Mass

ATOM AND ATOMIC MASS

As you know, anything that has mass and occupies space is called matter. Atoms are the building blocks of matter. Since matter has mass, it must be due to its atoms. According to the modern atomic theory, an atom contains subatomic particles such as protons, neutrons and electrons. Protons and neutrons have considerable mass, but electrons don't have such a considerable mass. Thus, the mass of an atom is mainly contributed by its protons and neutrons and hence the sum of the number of protons and neutrons of an atom is called its mass number.

Individual atoms are very small and it is difficult to measure their masses. You can measure the mass of macroscopic materials in gram or kilogram. The mass of an atom is measured in atomic mass unit (amu).

Atomic mass unit is one-twelfth of the mass of a carbon- 12 atom; an isotope of carbon, which contains 6 protons and 6 neutrons.

(Note: The symbol ‘amu’ is no longer used in the modern system and instead, it uses the symbol ‘u’ to denote unified atomic mass. The mass of a proton or neutron is approximately 1 amu).

1. Relative Atomic Mass (RAM)

As an atom is very small, its absolute mass cannot be determined directly. The early pioneers of chemistry used to measure the atomic mass of an atom relative to an atom of another element. They measured the masses of equal number of atoms of two or more elements at a time, to determine their relative masses. They established one element as a standard, gave it an arbitrary value of atomic mass and using this value they measured the relative mass of other elements. The mass obtained by this way is called relative atomic mass. In the beginning, the mass of hydrogen atom was chosen as a standard and masses of other atoms were compared with it, because of the existence of isotopic character of hydrogen (₁H¹, ₁H², ₁H³). Later hydrogen atom was replaced by oxygen atom as the standard. Now, the stable isotope of carbon (C-12) with atomic mass 12 is used as the standard for measuring the relative atomic mass of an element.

Relative atomic mass of an element is the ratio between the average mass of its isotopes 1/12^th to part of the mass of a carbon-12 atom. It is denoted as A_r. It is otherwise called “Standard Atomic Weight”.

Relative Atomic Mass

Modern methods of determination of atomic mass by Mass Spectrometry uses C-12 as standard. For most of the elements, the relative atomic mass is very closer to a whole number and it is rounded off to a whole number, to make calculations easier. Table 7.1 lists some of the elements of periodic table and their A_r values.

2. Average Atomic Mass (AAM)

How can one measure the atomic mass of an element? It is somewhat more complicated because most of the naturally occuring elements exist as a mixture of isotopes, each of which has its own mass. Thus, it is essential to consider this isotopic mixture while calculating the atomic mass of an element.

The average atomic mass of an element is the weighted average of the masses of its naturally occurring isotopes.

But, the abundance of isotopes of each element may differ. So, the abundancy of all these isotopes are taken into consideration while calculating the atomic mass. Then, what do we mean by a weighted average? Let us consider an element which exists as a mixture of 50% of an isotope having a mass of 9 amu, and 50% of another isotope having a mass of 10 amu. Then, its average atomic mass is calculated by the following equation:

Average atomic mass = (Mass of 1st isotope × % abundance of 1st isotope) + (Mass of 2nd isotope × % abundance of 2nd isotope)

Thus, for the given element the average atomic mass = (9 × 50/100) + (10 x 50/100) = 4.5 + 5 = 9.5 amu

(Note: In the calculations involving percentages, you need to convert percentage abundance into fractional abundance. For example, 50 percent is converted into 50/100 or 0.50 as shown in the a foresaid calculation.)

The atomic masses of elements, given in the periodic table, are average atomic masses. Sometimes, the term atomic weight is used to mean average atomic mass. It is observed, from the periodic table that atomic masses of most of the elements are not whole numbers. For instance, the atomic mass of carbon given in the periodic table is 12.01 amu, not 12.00 amu. The reason is that while calculating the atomic mass of carbon, both of its natural isotopes such as carbon-12. and carbon- 13 are considered. The natural abundance of C-12 and C-13 are 98.90 % and 1.10 % respectively. The average of the atomic mass of carbon is calculated as follows:

Average atomic mass of carbon

= (12 × 98.9/100 ) + (13 × 1.1/100) = (12 × 0.989) + (13 × 0.011) = 11.868 + 0.143 = 12.011 amu

So it is important to understand that if it is mentioned that the atomic mass of carbon is 12 amu, it refers to the average atomic mass of the carbon isotopes, not the mass of the individual atoms of carbon.

Calculation of average atomic mass - Solved Examples

Example 1: Oxygen is the most abundant element in both the Earth’s crust and the human body. It exists as a mixture of three stable isotopes in nature as shown in Table 7.3:

Atomic Mass of Oxygen:

The atomic mass of oxygen = (15.9949 × 0.99757) + (16.9991 × 0.00038) + (17.9992 × 0.00205) = 15.999 amu.

Example 2: Boron naturally occurs as a mixture of boron-10 (5 protons + 5 neutrons) and boron-11 (5 protons + 6 neutrons) isotopes. The percentage abundance of B-10 is 20 and that of B-11 is 80. Then, the atomic mass of boron is calculated as follows:

Atomic Mass of Boron:

Atomic mass of boron = (10 × 20/100) + (11 × 80/100) = (10 × 0.20) + (11 × 0.80) = 2 + 8.8 = 10.8 amu

Atoms and Molecules: An Introduction to Modern Atomic Theory

Atoms and Molecules - Introduction

INTRODUCTION

You have learnt, in your lower classes that matter is around us everywhere. Matter is made of atoms. Curiously the idea of atom was first proposed by the Greek philosophers in the fifth century BC (BCE). But, their theory was more philosophical than scientific.

The first scientific theory of the atom was proposed by John Dalton. Few of the postulates of Dalton’s theory about an atom were found incorrect by the later on studies made by J.J. Thomson, Rutherford, Neils Bohr and Schrodinger. In the light of the result of the researches most of the limitations of the Dalton’s theory were removed and a new theory known as the modern atomic theory was put forward.

‘The main postulates of modern atomic theory’ are as follows:

• An atom is no longer indivisible (after the discovery of the electron, proton, and neutron).
• Atoms of the same element may have different atomic mass. (discovery of isotopes ³⁵₁₇Cl, ³⁷₁₇Cl).
• Atoms of different elements may have same atomic masses (discovery of Isobars ⁴⁰₁₈Ar, ⁴⁰₂₀Ca).
• Atoms of one element can be transmuted into atoms of other elements. In other words, atom is no longer indestructible (discovery of artificial transmutation).
• Atoms may not always combine in a simple whole number ratio (E.g. Glucose C₆H₁₂O₆ C:H:O = 6:12:6 or 1:2:1 and Sucrose C₁₂H₂₂O₁₁ C:H:O = 12:22:11).
• Atom is the smallest particle that takes part in a chemical reaction.
• The mass of an atom can be converted into energy (E = mc²).

The modern atomic theory is the basis for all the studies of chemical and physical processes that involve atoms. You have studied the most fundamental ideas about an atom in your lower classes. Let us discuss some more concepts about atoms in this lesson.

Concept Map: Atoms and Molecules

Empowered Women at Work: Reading Comprehension Exercise | Prose Analysis

Reading - Empowered Women Navigating The World | Prose

I. Read the data below and answer the following questions.

Choose the correct answer.

1. What is the data about?

   • a. women empowerment
   • b. women power
   • c. women at work
   • d. women at home

2. Identify the three jobs where the same percentage of women work.

   

   Ans: b

3. In which field of work is women’s involvement the second highest?

   • a. Logistics stores
   • b. Home maker
   • c. Medicine
   • d. Administration/Human resource

4. Percentage of women working in finance is the same as

   • a. Home maker
   • b. Information Technology
   • c. Technical Field
   • d. Administration/Human Resources

5. What is the difference between the percentage of women working in logistics and Medicine?

   • a. 8
   • b. 11
   • c. 13
   • d. 5