MAE3270 Module 3 - Lecture 4: Fractions and Division

Chapter 1: Misconceptions

Equivalent Fractions

  • Some students think that these 3 examples are equivalent

    • The first example is a half, the next one is two thirds, and the next one is 3 quarters

    • Visual representation shows that they are not equivalent in size

    • Students need to be able to visualize and understand equivalent fractions

  • Students have misconceptions about calculating equivalent fractions

    • Some think they can add or subtract a number to the numerator and denominator

    • Actually, they need to choose a number to multiply or divide both numerator and denominator by

    • Students also think they can cross out the same numbers in the numerator and denominator, but it's not correct

    • They need to understand that they are dividing by a certain amount

Simplifying Fractions

  • Example of simplifying fractions: 20/30

    • Students need to select a common factor that divides equally into both 20 and 30

    • The common factor is 10, so 20/10 = 2 and 30/10 = 3, giving an answer of 2/3

  • Example of simplifying fractions: 25/35

    • Common factor is 5, so 25/5 = 5 and 35/5 = 7, giving an answer of 5/7

Adding Fractions

Another misconception is adding fractions

  • Students think they can just add the numerators and denominators

  • Actually, they need to ensure that the denominators are the same

  • Example: 3/6 + 2/6 = 5/6

Working with Remainders

  • Sometimes it's not possible to divide amounts equally and there will be remainders

  • Remainders can be written as a whole number, fraction, or decimal depending on the situation

Chapter 2: Division to Make Fractions

  • Sharing lollies among children

    • Each child gets 5 and 2 thirds lollies

    • Remaining lolly needs to be cut into 3 equal sized pieces

    • Each child gets 2 of those sections

  • Dividing children into cars

    • 17 children and 3 cars

    • Cannot have 2 thirds of a child

    • Options: 6 children in each car, 6 in 2 cars and 5 in the other, or consider how many will actually fit and if an extra car is needed

Chapter 3: Prime Numbers

  • Representing prime numbers as arrays

    • Prime numbers can only be represented as a single array (row or column)

    • Rows and columns must be equal

    • Example: Number 7 can be represented as a row or column with 7 squares

  • Prime numbers and composite numbers

    • Prime numbers have exactly two factors: 1 and itself

    • Composite numbers have more than 2 factors

    • Composite numbers can be written as the product of prime numbers

  • Using factor trees to find prime factors

    • Start with the composite number at the top

    • Select two numbers that multiply to give the composite number

    • Continue breaking down the numbers until prime factors are obtained

Chapter 5: Conclusion

  • Product of prime factors

    • Write down the prime factors next to each other separated by multiplication

    • Example: The product of prime factors for the number 48 is 2 times 2 times 2 times 2 times 3

  • Real numbers and fractions

    • Fractions and division are interchangeable

    • Fractions represent quantities that may or may not be whole

    • Proper fractions are less than 1 whole, improper fractions are greater than 1 whole, and mixed numerals consist of a whole number and a fraction

    • Utilize various models and representations when teaching fractions

    • The part-whole concept underpins fractions