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The Other Side of Zero - The Token Model

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Integers consist of positive numbers, negative numbers, and zero. Numbers to the left of zero on a number line are negative, and numbers to the right are positive.

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The Token Model uses two types of counters: a positive token representing +1+1 and a negative token representing −1-1.

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Zero Pair Rule: When one positive token and one negative token are paired together, they cancel each other out to equal zero. Mathematically, this is expressed as (+1)+(−1)=0(+1) + (-1) = 0.

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Addition with tokens: To add integers, combine the sets of tokens. Form as many zero pairs as possible and remove them. The remaining tokens represent the sum.

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Subtraction with tokens: To subtract, remove the specified number of tokens from a set. If there are not enough tokens to remove, add the required number of 'Zero Pairs' (which does not change the value) and then perform the subtraction.

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Additive Inverse: Every integer aa has an opposite −a-a such that their sum is zero. For example, the additive inverse of 55 is −5-5.

📐Formulae

a+(−a)=0a + (-a) = 0

a−b=a+(−b)a - b = a + (-b)

Positive+Negative=Difference in magnitude with the sign of the larger absolute value\text{Positive} + \text{Negative} = \text{Difference in magnitude with the sign of the larger absolute value}

+5+(−3)+2\begin{array}{r} +5 \\ + (-3) \\ \hline +2 \end{array}

💡Examples

Problem 1:

Find the sum of 33 and −5-5 using the token model.

Solution:

3+(−5)=−23 + (-5) = -2

Explanation:

Start with 33 positive tokens (+)(+) and 55 negative tokens (−)(-). Pair 33 positive tokens with 33 negative tokens to form 33 zero pairs. After removing the zero pairs, you are left with 22 negative tokens. Therefore, the answer is −2-2.

Problem 2:

Subtract 22 from −3-3 using tokens.

Solution:

−3−2=−5-3 - 2 = -5

Explanation:

Start with 33 negative tokens. You need to remove 22 positive tokens, but there are none. Add 22 zero pairs (two ++ and two −-). Now you have 55 negative tokens and 22 positive tokens. Remove the 22 positive tokens. You are left with 55 negative tokens, so the result is −5-5.

Problem 3:

Subtract −4-4 from −1-1.

Solution:

−1−(−4)=3-1 - (-4) = 3

Explanation:

Start with 11 negative token. You need to remove 44 negative tokens, but you only have 11. Add 33 zero pairs to provide enough negative tokens. Now you have 44 negative tokens and 33 positive tokens. Remove all 44 negative tokens. You are left with 33 positive tokens, so the result is 33.

Problem 4:

Perform the following vertical calculation: Subtract 1515 from 4040.

Solution:

40−1525\begin{array}{r} 40 \\ - 15 \\ \hline 25 \end{array}

Explanation:

In vertical subtraction, we subtract the units digit first (10−5=510 - 5 = 5 after borrowing) and then the tens digit (3−1=23 - 1 = 2).