**Explanations **

**Q1. Which of the following best describes 234 apples?**

**Explanation:**For K – 10, it’s fair to assume that numbers always represent quantities (but now there are numbers without representing any quantities and quantities not representable by any number). Numbers, such as 234 apples, are written in a particular numeral (which varies by the natural languages), e.g., 234 is the quantity written in the English language. In turn, numerals are made of digits, which is the decimal system are 0 – 9.

**Q2. Can you visualise how we get the quotient of ? In other words, can you find the quotient without any arithmetical operations?**

**Explanation:**

Have you ever read, or heard, a deductive/universal definition of division? We define division through examples and even that way the partition definition is given importance over the quotation definition. In quotation division one asks how many parts there are; in the partition, division one asks what the size of each part is. For example, in the division a/b, we read it as ‘how many b’s in a’. To get the quotient of 8/3 divide 1/3, we need to visualize ‘how many 1/3 in 8/3. And this is similar to visualizing ‘how many footballs in a box of 8 footballs’.

**Q3. Visualise what physical situation the expression 2 x (3 + 5) may represent.**

**Explanation:**

Even multiplication suffers from the same deficiency – there is hardly any universal definition of multiplication, e.g., what would a x b imply, always? What does 2 x 3 represent beyond the numeral 6? Is it any different from 3 x 2?

The default mode of teaching multiplication is to treat the multiplier and multiplicand as factors, wherein multiplier and multiplicand doesn’t matter. And that’s why 3 x 2 and 2 x 3 are the same things in ‘every sense’. But they aren’t!

To generalise, when we write the mathematical expression for the operation of adding a number ‘c’, ‘a’ times, we write:

Result = ac… and not … Result =ca.

When the multiplicand and the multiplier aren’t specified explicitly, we take the quantity written first- ‘a’, to be the multiplier and the quantity written second -‘c’, to be the multiplicand.

In 2 x (3 + 5), 2 is the multiplier. The multiplier gives the ‘number of repeats’, of the ‘desired quantity of something’.

And 3 + 5 must denote the multiplicand. The multiplicand is the thing that is being the focus of quantity/measurement

Also, 3 + 5 must denote the sum of two like quantities (Since the operation of addition is only valid for like quantities).

Therefore, the expression 2 x (3 + 5) can only signify ‘2 sets of 3 apples and 5 apples’

** **

**Q4. Comment on the expression, 7 + 3 = ?**

**Explanation:**

English literature knows that we ‘can’t add apples and oranges’, in broader terms, it implies that we can’t add, unlike things!

Thus, 7 and 3 must represent like quantities (because 7 and 3 are numerals for the number that represents something); so, if 7 is assumed to be 7 apples and 3 is assumed to be 3 oranges then we can’t add the two numbers.

** **

**Q5. ****Visualise, in most simple terms, why a kid may write 2 + 2 = 22?**

**Explanation:**

To be honest, 2 + 2 = 22 is the most ‘hurtful mistake’ a child can make; 2 + 2 being 3, 5, or even 6 can be explained as being unable to recall right number names, associating the right number names to quantities etc. But seeing 2 + 2 = 22, implies that the child has no sense of the quantity that the numerals 2 and 22 must represent! It’s a clear case of poor foundation of the numbers system – too early abstraction of math.

**Q6. ****Which of the following is not an example(s) of commutativity (i.e., commutative law, as used in math)?**

**Explanation:**

The simplest way to understand this question is to revisit how we name things; specifically, what must be observed in something to call it to commutative. In a way, what must be observed in math to call it commutative, and state the observed fact in the form of a statement that’s always valid (under given conditions, if any).

So, let’s start with the literal meaning of the word commutative. It literally means ‘exchange’. Thus, it implies situations where the exchange of the way things are doesn’t change the nature of the thing. For example, if you travel to a place by running half the distance and the walking the rest, then your travel is commutative (if the speed of running and walking don’t change), i.e., you can exchange the order of running and walking and yet reach the place in the same time of travel.

Expectedly, in math, commutative law is the name given to operations of numbers such that exchanging the numbers in the operations doesn’t change the result of the operations. For example, 2 + 3 = 3 + 2.

**Q7. ****Visualise as a fraction**

**Explanation:**

To understand a fraction, written as a/b, we need to understand what’re ‘a’ and ‘b’. ‘a’, the numerator, is the number of relevant part of the whole we are talking about, and ‘1/b’ is the size of each equal part of ‘any whole’ (i.e., any 1 thing – sun, earth, apple, cake, Rs 100), i.e., ‘b’ is the number of parts we have made of the one whole. Thus, ¾ implies taking 3 parts of 1/4, i.e., ¾ as a fraction is better read as 3 x ¼.

Similarly, 8/3 as a fraction is read as 8 x 1/3, thus implying 8 parts of 1/3. This can be best visualised by taking any one thing and taking eight parts of one–third each of the thing.

**Q8. ****Visualise a fraction equivalent of a decimal number having 3 decimal places.**

**Explanation:**

Natural, whole, Integer, fraction, decimal are various ways of quantifying things; just as whole and integers are used to represent ‘whole quantities’, fractions and decimals are used to represent ‘less than whole, or part’ quantities (and whole also).

We have two ways to represent part quantities – fraction and decimal – because in same situation it’s better to use fraction way of expressing the numbers, and in many situations, it’s better to use decimal way of expressing the numbers. Of course, we can change a number written as fraction into decimal form, and vice versa (because both can represent any ‘part numbers’).

The way we interchange fractions and decimals is as follows –

- The whole (if any) need not change in the interchange, because fractions and decimals differ in the way parts are written
- The part numbers (i.e., less than 1) are written such that the denominator is always an exponent of 10 (e.g., 10, 100, 1000) – thus we convert the actual denominator of the fraction to the closest exponent of 10 and the numerator changes accordingly
- Decimal place of a decimal number is the exponent of the fractional part of the decimal number

**Q9. Visualise the number of cm in 1000 m.**

**Explanation:**

This must be visualized as a division, i.e., what’s the measurement of 1000 m (i.e., measurement is in units of 1 m) if measured in a new unit – 1 cm. And we know what’s the exact division statement of the above situation is –

1000m/1cm, because in the division a/b, ‘b’ is the size of the new way ‘a’ is to be re-organized.

**Q10. What’s the first root of 25?**

**Explanation:**

The number itself. In mathematics, an n^{th} root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power *n* yields *x*:

r^{n} = x,

where n is the degree of the root. A root of degree two is called a square root. A root of degree three is called a cube root. Similarly a root of degree one is called the first root.

On account of this definition, the first root of any number would be the number itself.