Step 3: Curriculum Exploration
This step allows you to examine curriculum materials and fraction models. It provides fuel for thinking about items 2-5 of the unit plan for fractions that you have been working on.
|Researcher: Which fraction is smaller? 64 or 65
Chris: 64. Because 4 takes longer to get to 6 than 5 does.*
Like Chris, many students reason about fractions in ways that work in certain situations but not in others. Chris may have an image of a fraction as a quantity less than one and as a number of pieces in a whole with some missing. His reasoning would be helpful if he were comparing the size of 46 and 56, but leads him to the wrong conclusion in the example above.
Students’ images and understandings of fractions are shaped by their encounters with mathematics textbooks and materials and by real life experiences. What images of fractions do your students typically draw when asked to make sense of a fraction problem? The illustration below shows the fraction 3/4 shaded in eight different models.
Models of the fraction 34
As Tad Watanabe’s article “Initial Treatment of Fractions in Japanese Textbooks” notes, each of these models has different strengths and shortcomings for building student understanding, and there is not one single “best” model. Ultimately, students will need to understand that 34 represents several very different situations, such as 34 of a meter, 6 out of 8 squares colored on a checkerboard, and 3 girls in a group of 4 children. The resources here focus on the linear measurement context for fractions because it is often neglected in U.S. textbooks, although it is common in other countries and in research-based programs.
Explore the linear measurement model with a challenging hands-on activity for adults!
To find out how to prepare for the mystery strip activity, click hereCreate one strip of paper exactly 1 meter long and do not mark the strip in any way, see example below:
Then create the blue mystery strip by downloading this pdf file: mystery strip.
You will need to save the file. Before printing the strip, check your pdf print settings are as described in the mystery strip pdf.
- Without using standard measuring devices, express the length of the mystery piece (a fractional part of a meter) in meters. How might students solve the problem and what they might find challenging? What understandings of fractions would help students solve the problem?
Read over “Fractions” (pp. 57-64 of the Grade 3B Tokyo Shoseki textbook, 2006). It may be useful to know that 8 periods of 45 minutes are allocated for this unit, roughly one period per textbook page.
- Consider how this unit might help students:
- See non-unit fractions (fractions with a numerator other than 1) as accumulations of unit fractions (fractions with numerator of 1)?
- See the connection between whole numbers and fractions?
- Connect area, linear measurement, and number line models?
- Consider: How do the images and understandings of fractions that students might develop from the 3B textbook compare with what students might develop from your own curriculum’s introduction of fractions?
If you like this activity and want to do more of it, you can analyze the 4B Fractions Unit (pp.38-51) in the same fashion.
In sections 2-5 of your fraction unit plan (unit flow and rationale), capture any key learnings from examining the curriculum.
*Adapted from Cramer, K. & Wyberg, T. (2007). When getting the right answer is not always enough: Connecting how stuents order frations and estimate sums and differences. pp. 205-220. W. G. Martin & M. E. Strutchens (Eds.) The learning of mathematics. 69th Yearbook. Reston, VA:NCTM, p.215.