Context: In an expression having two terms, swapping them does not change the value: Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Arithmetic Expressions | A

Question 1

Context: In an expression having two terms, swapping them does not change the value: $\text{Term 1} + \text{Term 2} = \text{Term 2} + \text{Term 1}$

Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Solution

The total number of tokens stays the same, no matter the order.

Step 1 — Representing the terms

Let us use tokens for our terms. A negative token is a red circle. Let Term 1 be -3. This means we have 3 negative tokens. Let Term 2 be -2. This means we have 2 negative tokens.

Step 2 — Adding Term 1 and Term 2

First, we take Term 1's tokens. We have 3 negative tokens. Then, we add Term 2's tokens. We add 2 negative tokens to them. Now, we count all the tokens. We have a total of 5 negative tokens. So, Term 1 + Term 2 equals -5.

$(-3) + (-2)$

$= \text{3 negative tokens} + \text{2 negative tokens}$

$\boxed{\text{5 negative tokens}}$

Diagram 1

Step 3 — Adding Term 2 and Term 1

Now, let us change the order. First, we take Term 2's tokens. We have 2 negative tokens. Then, we add Term 1's tokens. We add 3 negative tokens to them. Again, we count all the tokens. We have a total of 5 negative tokens. So, Term 2 + Term 1 equals -5.

$(-2) + (-3)$

$= \text{2 negative tokens} + \text{3 negative tokens}$

$\boxed{\text{5 negative tokens}}$

Diagram 2

Answer

Yes, using the Token Model, adding 3 negative tokens and 2 negative tokens in any order always results in 5 negative tokens.

More questions in A

Context: In an expression having two terms, swapping them does not change the value: $\text{Term 1} + \text{Term 2} = \text{Term 2} + \text{Term 1}$

Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Context: Let us consider the expression $(-7) + 10 + (-11)$ again. What happens when we change the order and add $-7$ and $-11$ first, and then add this sum to $10$ ? Will we get the same sum as before? We see that adding the terms of the expression $(-7) + 10 + (-11)$ in any order gives the same sum of $-8$ .

Q. Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also.

Context: Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than $3$ terms also.

Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

What happens to the value of an expression if we increase or decrease the value of one of its terms?

Some expressions are given in following three columns. In each column, one or more terms are changed from the first expression. Go through the example (in the first column) and fill the blanks, doing as little computation as possible.

Expression Engineer!

Using three 3's along with the four operations (addition, subtraction, multiplication, and division) and brackets as needed we can create several expressions. For example, $(3 + 3)/3 = 2$ , $3 + 3 - 3 = 3$ , $3 \times 3 + 3 = 12$ , and so on.

Using four 4's, create expressions to get all values from 1 to 20.
Using the numbers 1, 2, 3, 4, and 5 exactly once in any order get as many values as possible between $-10$ and $+10$ .
Using the numbers 0 to 9 exactly once in any order, make an expression with a value 100.
What other similar interesting questions can you ask?