Question 4

The sum of integers stays the same. This is true even if we change their order. It is also true if we group them differently.

Step 1 — Checking with an example

Let us take an expression with four terms. We will use both positive and negative integers. Let the expression be $5 + (-3) + 2 + (-6)$. We will add these numbers in different ways.

First way: Add from left to right.

$5 + (-3) + 2 + (-6)$

$= (5 + (-3)) + 2 + (-6)$

$= 2 + 2 + (-6)$

$= 4 + (-6)$

$\boxed{\mathbf{-2}}$

Second way: Group the middle terms first.

$5 + (-3) + 2 + (-6)$

$= 5 + ((-3) + 2) + (-6)$

$= 5 + (-1) + (-6)$

$= 4 + (-6)$

$\boxed{\mathbf{-2}}$

Third way: Rearrange terms and group them. We group positive numbers together. We group negative numbers together.

$5 + (-3) + 2 + (-6)$

$= (5 + 2) + ((-3) + (-6))$

$= 7 + (-9)$

$\boxed{\mathbf{-2}}$

All three ways give the same answer. So, adding terms in any order gives the same value.

Step 2 — Explaining with the Token Model

Let us use tokens to understand this. We use yellow tokens for positive integers. We use red tokens for negative integers. Adding integers means putting all tokens together. One yellow token and one red token cancel each other. They form a zero pair. The final tokens show the sum.

Let us use the expression $5 + (-3) + 2 + (-6)$. This expression means we have: Five yellow tokens for $5$. Three red tokens for $-3$. Two yellow tokens for $2$. Six red tokens for $-6$.

Let us put all these tokens together. We have a total of $5 + 2 = 7$ yellow tokens. We have a total of $3 + 6 = 9$ red tokens.

Now, we cancel out the zero pairs. Seven yellow tokens cancel out seven red tokens. We are left with $9 - 7 = 2$ red tokens. So, the final sum is -2.

The total number of yellow and red tokens does not change. It does not matter how we combine them. The total collection of tokens is always the same. So, the final tokens left will be the same. This is why the sum is always the same.

Answer

Yes.