Foundational Concepts: Integers and Fractions Review

TL;DR

Before we add rational numbers, let's quickly review integers (whole numbers and their opposites) and fractions (parts of a whole). Understanding how to work with these separately is key to combining them correctly. We'll refresh on their basic operations and properties.

1. The Mental Model

Think of integers as steps on a number line, going forwards or backwards from zero. Fractions represent dividing something into equal pieces and taking a certain number of those pieces.

2. The Core Material

What are Integers?

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Integers are all the whole numbers (0, 1, 2, 3, ...) and their negative counterparts (... -3, -2, -1). They don't include fractions or decimals.

Adding and Subtracting Integers

• Same Signs: Add the absolute values and keep the original sign.
  • Example: 3 + 5 = 8
  • Example: -3 + (-5) = -8 (which is the same as -3 - 5 = -8)
• Different Signs: Subtract the smaller absolute value from the larger absolute value. Keep the sign of the number with the larger absolute value.
  • Example: 7 + (-3) = 4 (7 is larger than 3, and 7 is positive)
  • Example: -7 + 3 = -4 (7 is larger than 3, and -7 is negative)
  • Example: 3 - 7 = 3 + (-7) = -4

Multiplying and Dividing Integers

• Same Signs: The result is always positive.
  • Example: 3 * 5 = 15
  • Example: -3 * -5 = 15
• Different Signs: The result is always negative.
  • Example: 3 * -5 = -15
  • Example: -3 * 5 = -15

What are Fractions?

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A fraction represents a part of a whole. It has two main parts: a numerator (the top number) and a denominator (the bottom number). The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have.

Equivalent Fractions

Different fractions can represent the same value. To find an equivalent fraction, you multiply or divide both the numerator and the denominator by the same non-zero number.
* Example: 1/2 is equivalent to 2/4 (multiply top and bottom by 2).
* Example: 6/9 is equivalent to 2/3 (divide top and bottom by 3).

Comparing Fractions

To compare fractions, they need a common denominator. Find the least common multiple (LCM) of the denominators, then convert each fraction to an equivalent fraction with that new denominator.
* Example: Compare 1/3 and 2/5.
* LCM of 3 and 5 is 15.
* 1/3 = (1*5)/(3*5) = 5/15
* 2/5 = (2*3)/(5*3) = 6/15
* Since 5/15 < 6/15, then 1/3 < 2/5.

Fraction Operations Recap

Let's visualize the fraction conversion process.

graph TD
    A["Start with Fraction 1 and Fraction 2"] --> B{Need Common Denominator?};
    B -- Yes --> C["Find LCM of Denominators"];
    C --> D["Convert Fraction 1 to Equivalent Fraction"];
    C --> E["Convert Fraction 2 to Equivalent Fraction"];
    D --> F["New Fraction 1 (Same Value)"];
    E --> G["New Fraction 2 (Same Value)"];
    F & G --> H["Perform Operation (Add, Subtract, Compare)"];
    H --> I["Simplify Result (if needed)"];
    I --> J["End"];
    B -- No --> H;

3. Worked Example

Let's work through an example combining integer and fraction concepts:
Evaluate: -5 + (1/4 - 2/3)

1. Work inside the parentheses first: 1/4 - 2/3

   • Find a common denominator for 4 and 3. The LCM is 12.
   • Convert 1/4: (1*3)/(4*3) = 3/12
   • Convert 2/3: (2*4)/(3*4) = 8/12
   • Now subtract: 3/12 - 8/12 = (3 - 8)/12 = -5/12

2. Now substitute back into the original expression: -5 + (-5/12)

   • This is the same as -5 - 5/12.
   • To combine an integer and a fraction, write the integer as a fraction with the same denominator.
   • -5 as a fraction with denominator 12 is -60/12 (since -60/12 = -5).
   • Now add: -60/12 + (-5/12) = (-60 - 5)/12 = -65/12

So, -5 + (1/4 - 2/3) = -65/12.

4. Key Takeaways

• Integers are whole numbers and their opposites, extending infinitely in both positive and negative directions.
• When adding integers with different signs, subtract the smaller absolute value from the larger one and use the sign of the larger.
• Fractions represent parts of a whole, with the numerator showing parts taken and the denominator showing total parts.
• You need a common denominator to add, subtract, or compare fractions effectively.
• Multiplying or dividing both the numerator and denominator by the same non-zero number creates an equivalent fraction.
• Remember the rules for multiplying/dividing integers: same signs give positive, different signs give negative.

Common Mistakes to Avoid:
- Forgetting to change the sign when subtracting a negative integer (e.g., 5 - (-3) becomes 5 + 3).
- Not finding a common denominator before adding or subtracting fractions.
- Incorrectly applying integer sign rules when multiplying or dividing.
- Confusing the numerator and denominator's roles.

5. Now Try It

Spend 15 minutes reviewing these concepts by solving the following:
1. Calculate: -12 + 7 - (-3)
2. Calculate: -4 * (-6) / (-2)
3. Compare using <, >, or =: 3/7 and 5/14
4. Calculate: 2/3 + 1/6 - 3/4
5. If you have 3 pizzas, and each is cut into 8 slices, and you eat 5 slices, what fraction of the total pizza did you eat?

Success looks like correctly solving all these problems and feeling confident about the integer and fraction rules.