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?

Understand the Question In the Token Model, an integer is represented by a set of tokens (for example, negative numbers are represented by negative tokens). Adding two terms means combining their collections of tokens into one single group. The total number and nature of tokens collected in the group remain the same regardless of which group of tokens is placed first. Thus, holds true because grouping is independent of order. Step 1 · Evaluate Term 1 + Term 2 Let ( negative tokens) and ( negative tokens). Combining and : Step 2 · Evaluate Term 2 + Term 1 Now, reverse the order and combine and : Since both orders result in negative tokens (), swapping the terms does not change the value. Answer > Yes. Using the Token Model, combining negative tokens and negative tokens in any order always results in negative tokens (), proving that . Common Mistakes Order Confusion: Thinking that placing tokens in a different sequence alters the total count or sign of the resulting group. Sign Mixing: Confusing positive and negative tokens when combining terms of the same sign.

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

Understand the Question

In the Token Model, an integer is represented by a set of tokens (for example, negative numbers are represented by negative tokens).
Adding two terms means combining their collections of tokens into one single group.
The total number and nature of tokens collected in the group remain the same regardless of which group of tokens is placed first.
Thus, $\text{Term 1} + \text{Term 2} = \text{Term 2} + \text{Term 1}$ holds true because grouping is independent of order.

Step 1 · Evaluate Term 1 + Term 2

Let $\text{Term 1} = -3$ ( $3$ negative tokens) and $\text{Term 2} = -2$ ( $2$ negative tokens). Diagram 1

Combining $\text{Term 1}$ and $\text{Term 2}$ :

\begin{aligned} (-3) + (-2) &= \text{3 negative tokens} + \text{2 negative tokens} \\[0.6em] &= \text{5 negative tokens} = -5 \end{aligned}

Step 2 · Evaluate Term 2 + Term 1

Now, reverse the order and combine $\text{Term 2}$ and $\text{Term 1}$ : Diagram 2

\begin{aligned} (-2) + (-3) &= \text{2 negative tokens} + \text{3 negative tokens} \\[0.6em] &= \text{5 negative tokens} = -5 \end{aligned}

Since both orders result in $5$ negative tokens ( $-5$ ), swapping the terms does not change the value.

Answer

Yes. Using the Token Model, combining $3$ negative tokens and $2$ negative tokens in any order always results in $5$ negative tokens ( $-5$ ), proving that $\text{Term 1} + \text{Term 2} = \text{Term 2} + \text{Term 1}$ .

Common Mistakes

Order Confusion: Thinking that placing tokens in a different sequence alters the total count or sign of the resulting group.
Sign Mixing: Confusing positive and negative tokens when combining terms of the same sign.

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