Fraction Representations
**The big idea contained in this one lesson could span from grade 3–8, easily. See also: Ontario fractions learning pathway for the recent research on fractions.
**The intent here is to give an idea for a “low floor, high ceiling” way to open up a fraction learning pathway to all kids.
Lesson #1
Topic: Representing fractions in a variety of ways
Big Idea: Fractions can represent parts of wholes, parts of sets, parts of measures, division, or ratios. A fraction is not meaningful without knowing what the whole is. (Marian Small- Making Math Meaningful)
See also: Paying Attention to Fractions on the importance of fraction representations.
Curriculum Connections: grade 4- read, represent, compare, and order whole numbers to 10 000, decimal numbers to tenths, and simple fractions, and represent money amounts to $100
Learning Goals: The student’s work will demonstrate knowledge of/ability to:
- Represent fractions in a variety of ways with manipulatives and thinking tools
- Explain their reasoning about how they divided up “the cross”
Representation and reasoning and proving are our big process expectations here.
(Possible) Success Criteria:
The student’s work demonstrates knowledge of/ability to
- Represent fractions as areas, sets, using a number line
- Reason through a problem involving an area representation of a fraction
- Explains their reasoning about “one quarter” of a shape.
Overall Expectations
grade 4- read, represent, compare, and order whole numbers to 10 000, decimal numbers to tenths,and simple fractions, and represent money amounts to $100
Differentiation for Lesson #1 will be made to:
-variety of manipulatives and thinking tools will be available
-students will reason both verbally and on paper
Materials and Resources
Cubes, counters, pattern blocks, number line templates, paper to draw area models, loose parts to make sets, cuisenaire rods
Instructional Strategies:
Activating Thinking: how many ways can your represent 1/2 ? Challenge kids to show as many ways as they can.
Prompting questions: is there another way you can show ½?
Can you show ½ of a set? (kids often neglect the set as a model for fractions).
Which of your representations is most interesting?
Gallery walk so kids can see as many representations as possible. Share ones you find interesting. Share as many unique examples as you can with the whole class.
Sample of what the work could look like:
Developing Thinking:
(source: common #MTBoS task-blogged about many times)
This is 5 squares. It’s “5ness” disrupts kids’ sense of “one quarter”. We do that on purpose to make them think!
Create a version of this that has 4 blank crosses Challenge kids to do it 4 ways.
Questions:
Can you make an interesting pattern that shows 1/4?
How do you know you have made ¼?
Which of your representations is the most interesting? Pick one to defend to your classmates.
Consolidating Thinking
Use your document camera, or post on whiteboard. Run a bansho, where you decide what order to have kids share. Alternatively, you could have every kid pick their best one and share.
Important teacher moves: make sure you are highlighting a variety of representations of 1/4. Capture kids’ reasoning.
Key question: How are you sure you have one quarter? Could you show one quarter in a different way?
