This unit explores fractions as an extension of the whole number system. Students learn the meaning of the numerator (top number) and denominator (bottom number) in fractions symbols. They also learn to express equal shares as fractions and to order related fractions.
Fractions arise from the need to divide. That division may involve equal sharing or measuring. Many equal sharing situations can be solved without needing fractions. For example, 1/3 of 15 or 15 ÷ 3 can be accomplished by putting five objects in each of the three shares. However, other equal divisions of sets and objects require partitioning ones, e.g. 1/3 of 16 or 1/3 of a pie. Measurements in which the units do not fit into a space a whole number of times demand the use of fractions of that unit. For example, if a length of 13 cubes is measured with a unit of 4 cubes, 13 ÷ 4 = 3 ¼ units fit.
In this unit students learn about fractions as numbers which is a measurement idea. ‘Fractions as measures’ is arguably the most important of the five sub-constructs of rational number (Kieren, 1994) since it identifies fractions as numbers, and is the basis of the number line.
Fractions are symbols in two parts, the numerator and denominator (Lamon, 2007). In the fraction 3/4, three is the numerator and 4 is the denominator. The numerator, 3, is the number of parts being counted, and the denominator, 4, gives the size of those parts. Quarters are of a size that four of them make one (whole).
Describing the relationship between different lengths might involve whole numbers, e.g. twice as long, but also involve fractions, e.g. half as long. Fractions express an amount that is in reference to a whole. With measurement that whole, or one, is fixed. For example, consider the relationship between the brown and orange Cuisenaire rods. If the orange rod is defined as one (an arbitrary decision) then what number is assigned to the brown rod?
Some equal partitioning of the one is needed to create unit fractions with one as the numerator. For the size of the brown rod to be named accurately those unit fractions need to fit into it exactly. We could choose to divide the orange rod into tenths (white rods) or fifths (red rods), either would work. By aligning the unit fractions, we can see that the brown rod is eight tenths or four fifths of the orange rod.
Note that eight tenths and four fifths are equivalent fractions, and the equality can be written as 8/10 = 4/5. These fractions are different names for the same quantity and share the same point on a number line. The idea that any given point on the number line has an infinite number of fraction names, is a significant change from what occurs with whole numbers.
This unit can be differentiated by varying the scaffolding provided and altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
Tasks can be varied in many ways including:
The context of equal shares can be adapted to recognise diversity and student interests to encourage engagement. Support students to identify and explore other situations in their lives where equal sharing occurs. For example, sharing kai at home or sharing cards to play a game (the pack of cards represents one whole). Art and design often provide situations where shapes need to be equally partitioned.
Note that most of the fractions in these teachers’ notes are displayed as typed text, meaning that the vinculum (the line between the numerator and the denominator) is diagonal rather than horizontal. When writing fractions for students it is recommended that you use a horizontal vinculum.
In this session students learn to equally partition lengths and name the parts they create as fractions. Use Copymaster One to make a set of laminated pizza pieces. Three of each page is sufficient to make a useful whole class set.
In this session students learn to iterate unit fractions to form non-unit fractions, like three quarters and two thirds. They also learn to order fractions with the same denominator, like one quarter, two quarters, three quarters, and four quarters.
Expect students to be systematic in the way they find equivalent fractions.
How can we be sure we have found all the equivalent fractions?
One approach might be to start with fractions involving halves, e.g. 1/2 = 2/4 = 3/6 = 4/8 = 6/12 and 2/2 = 3/3 = 4/4 = 6/6 = 8/8 = 12/12. Eliminate all the cards involved. Move to thirds, then quarters, then sixths, etc. eliminating cards each time.
Extension:
Look at the ‘non-equivalent’ cards that are left (1/8, 3/8, 5/8, 7/8, 1/12, 5/12, 7/12, 11/12).
Why was it hard to find equivalent fractions for these fractions?
What fractions, that are not in the cards, would be equivalent to these fractions?
Extra hard challenge:
Find some fractions in your cards that are different by one twelfth.
How many can you find?
There are several pairs of fractions that have a difference of one twelfth, including one third and one quarter, five twelfths and one half, two thirds and three quarters, two thirds and seven twelfths or nine twelfths.
In this session students use their sets of fraction circles to compare fractions. They learn to distinguish easy comparison situations (same denominators or same numerators) from more difficult situations where both numerators and denominators are different.
In this session students transfer their learning with a circular regions model to develop a length model of fractions. The length model allows them to create a mental number line for simple fractions, and to realise fractions as an extension of the natural number system.
In this session the representation of a number line for fractions is developed.
Dear family and whānau,
This week we learned about fractions as numbers. We explored the meaning of the top and bottom numbers in a fraction. Ask your child about the meaning of the numbers in a fraction like three quarters, or seven eighths.
We used two types of material to explore fractions, circles, and rods. Challenge your child to divide circles up into equal parts and to draw a number line showing the size of fractions.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/fractions-numbers at 8:32pm on the 26th February 2024