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 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?

Understand the Question Commutative and Associative Properties: Addition of integers is commutative () and associative (). This means changing the order or grouping of terms in an addition expression does not change the final sum. Token Model Concept: Positive integers are represented by positive tokens (e.g., yellow). Negative integers are represented by negative tokens (e.g., red). Adding integers means pooling all tokens together. One positive token and one negative token cancel each other out to form a zero pair. Because the total pool of positive and negative tokens remains identical no matter which order they are collected or grouped in, the remaining uncancelled tokens—and hence the final sum—will always be the same. Step 1 · Evaluate an Expression in Different Orders Consider a 4-term expression: . Method 1: Add from left to right Method 2: Group the middle terms first Method 3: Group positive and negative numbers together All three methods yield the same value: . Step 2 · Explain Using the Token Model In the Token Model: Positive integers are represented by yellow tokens (). Negative integers are represented by red tokens (). Combining yellow token and red token forms a zero pair (). For the expression : yellow tokens red tokens yellow tokens red tokens Putting all tokens together: Total yellow tokens: Total red tokens: Cancelling out zero pairs: Since red tokens remain, the value is . Conclusion: The total count of yellow tokens () and red tokens () in the collection does not change regardless of the order in which we add or group them. Hence, the final sum is always the same. Answer > Yes. Adding terms in any order gives the same value because the total number of positive and negative tokens collected remains constant, resulting in the same net value after cancelling zero pairs. Common Mistakes Dropping Negative Signs: Dropping or misplacing the negative sign when rearranging terms (e.g., writing instead of ). Forgetting Zero Pairs: Forgetting that equal numbers of positive and negative tokens neutralize each other () rather than simply being subtracted without sign tracking. Confusing Addition with Subtraction: Assuming that changing order in subtraction works the same way; changing order without preserving the term's sign only applies directly to addition (, but ).

Arithmetic Expressions | A

Question 4

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?

Solution

Understand the Question

Commutative and Associative Properties: Addition of integers is commutative ( $a + b = b + a$ ) and associative ( $(a + b) + c = a + (b + c)$ ). This means changing the order or grouping of terms in an addition expression does not change the final sum.
Token Model Concept:
- Positive integers are represented by positive tokens (e.g., yellow).
- Negative integers are represented by negative tokens (e.g., red).
- Adding integers means pooling all tokens together.
- One positive token and one negative token cancel each other out to form a zero pair.
Because the total pool of positive and negative tokens remains identical no matter which order they are collected or grouped in, the remaining uncancelled tokens—and hence the final sum—will always be the same.

Step 1 · Evaluate an Expression in Different Orders

Consider a 4-term expression: $5 + (-3) + 2 + (-6)$ . Diagram 1

Method 1: Add from left to right

\begin{aligned} 5 + (-3) + 2 + (-6) &= (5 + (-3)) + 2 + (-6) \\ &= 2 + 2 + (-6) \\ &= 4 + (-6) \\ &= -2 \end{aligned}

Method 2: Group the middle terms first

\begin{aligned} 5 + (-3) + 2 + (-6) &= 5 + ((-3) + 2) + (-6) \\ &= 5 + (-1) + (-6) \\ &= 4 + (-6) \\ &= -2 \end{aligned}

Method 3: Group positive and negative numbers together

\begin{aligned} 5 + (-3) + 2 + (-6) &= (5 + 2) + ((-3) + (-6)) \\ &= 7 + (-9) \\ &= -2 \end{aligned}

All three methods yield the same value: $-2$ .

Step 2 · Explain Using the Token Model

In the Token Model:

Positive integers are represented by yellow tokens ( $+1$ ).
Negative integers are represented by red tokens ( $-1$ ).
Combining $1$ yellow token and $1$ red token forms a zero pair ( $1 + (-1) = 0$ ).

For the expression $5 + (-3) + 2 + (-6)$ :

$5 \rightarrow 5$ yellow tokens
$-3 \rightarrow 3$ red tokens
$2 \rightarrow 2$ yellow tokens
$-6 \rightarrow 6$ red tokens

Putting all tokens together:

Total yellow tokens: $5 + 2 = 7$
Total red tokens: $3 + 6 = 9$

Cancelling out $7$ zero pairs: $\text{Remaining red tokens} = 9 - 7 = 2$

Since $2$ red tokens remain, the value is $-2$ .

Conclusion: The total count of yellow tokens ( $7$ ) and red tokens ( $9$ ) in the collection does not change regardless of the order in which we add or group them. Hence, the final sum is always the same.

Answer

Yes. Adding terms in any order gives the same value because the total number of positive and negative tokens collected remains constant, resulting in the same net value after cancelling zero pairs.

Common Mistakes

Dropping Negative Signs: Dropping or misplacing the negative sign when rearranging terms (e.g., writing $5 + 3$ instead of $5 + (-3)$ ).
Forgetting Zero Pairs: Forgetting that equal numbers of positive and negative tokens neutralize each other ( $1 + (-1) = 0$ ) rather than simply being subtracted without sign tracking.
Confusing Addition with Subtraction: Assuming that changing order in subtraction works the same way; changing order without preserving the term's sign only applies directly to addition ( $a - b \neq b - a$ , but $a + (-b) = (-b) + a$ ).

Question 4

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