Let's tackle this project step by step!

15. a) Three Proper Fractions

A proper fraction is a fraction where the numerator (top number) is less than the denominator (bottom number).

1. Fraction: $\frac{1}{4}$

   • Rectangle Representation: A rectangle divided into 4 equal parts, with 1 part shaded.

2. Fraction: $\frac{3}{5}$

   • Rectangle Representation: A rectangle divided into 5 equal parts, with 3 parts shaded.

3. Fraction: $\frac{2}{3}$

   • Rectangle Representation: A rectangle divided into 3 equal parts, with 2 parts shaded.

15. b) Three Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

1. Fraction: $\frac{5}{4}$

   • Rectangle Representation: A rectangle divided into 4 equal parts, with 5 parts shaded (this will require shading 1 full rectangle and 1 additional part in a second rectangle).
   • Mixed Number: $1\frac{1}{4}$

2. Fraction: $\frac{7}{3}$

   • Rectangle Representation: A rectangle divided into 3 equal parts, with 7 parts shaded (this will require shading 2 full rectangles and 1 additional part in a third rectangle).
   • Mixed Number: $2\frac{1}{3}$

3. Fraction: $\frac{9}{5}$

   • Rectangle Representation: A rectangle divided into 5 equal parts, with 9 parts shaded (this will require shading 1 full rectangle and 4 additional parts in a second rectangle).
   • Mixed Number: $1\frac{4}{5}$

15. c) Three Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction.

1. Mixed Number: $2\frac{2}{3}$

   • Rectangle Representation: Two full rectangles fully shaded, plus another rectangle divided into 3 equal parts with 2 parts shaded.
   • Proper Fraction: $\frac{8}{3}$

2. Mixed Number: $1\frac{3}{4}$

   • Rectangle Representation: One full rectangle fully shaded, plus another rectangle divided into 4 equal parts with 3 parts shaded.
   • Proper Fraction: $\frac{7}{4}$

3. Mixed Number: $3\frac{1}{2}$

   • Rectangle Representation: Three full rectangles fully shaded, plus another rectangle divided into 2 equal parts with 1 part shaded.
   • Proper Fraction: $\frac{7}{2}$

Would you like a visual representation of these fractions and mixed numbers? If yes, I can create the shaded rectangles for each one.

Questions for Further Exploration:

1. How do you determine if a fraction is proper or improper?
2. Why is it useful to convert improper fractions to mixed numbers?
3. Can you explain the relationship between mixed numbers and improper fractions?
4. What are the steps to convert a mixed number to an improper fraction?
5. How can proper fractions be represented as decimals?
6. What are some real-life examples where you would encounter improper fractions?
7. Can you explain how to add two mixed numbers together?
8. How do you subtract mixed numbers?

Tip:

When converting a mixed number to an improper fraction, multiply the whole number by the denominator, then add the numerator. Place this result over the original denominator!