The lessons compiled here (see links in the bottom of the page) were developed as part of the Ph.D. research projects of Leonardo Barichello and Rita Santos Guimarães. The material was not conceived for publication, but as part of our research projects. However, due to the interest of the teachers that took part on our research to use the lessons again, we decided to review, organize and share them.

All the material were conceived for low achieving groups of secondary students in a British school, but I do not see major reasons for them not to work with different groups. The lessons were enacted at least three times (by different teachers) and changes were made based on the experience.

Although the material was not professionally reviewed, we expect it to be free of mistakes (none of us is native speaker) and we hope it is useful for any teacher interested in the approach described below. Feel free to use and reproduce it fully or partially. But if you do so, we would like to ask you to contact us (barichello@gmail.com and guimaraes.rita@gmail.com) or leave a comment below telling us about the experience. This may help us to improve the lessons in the future!

This compilation contains 12 lessons divided into 3 packs covering the following topics: a) definition of fractions, b) equivalent fractions, c) comparison of fractions, and d) addition and subtraction of fractions.

Each lesson is made of a one-page document with comments for the teacher, a starter (usually a shorter task that revisits topics from previous lessons) and the main tasks. Sometimes an extension sheet is also included.

• Zip file with all the lessons in pdf format: here
• Zip file with all the lessons in odt format (LibreOffice): here
• Zip file with extra tasks (pdf): here
• Zip file with images for the cut-outs (Inkscape): here

### Rationale

The starting point of these lessons is the idea of building all the knowledge regarding fractions on a solid and common basis, instead of teaching it as a set of unconnected procedures and definitions.

The choice of visual representations as the basis is informed by scientific literature showing that visual models and visual skills are of utmost importance in the learning and doing of mathematics (Rivera, 2011). Also, there are evidences suggesting that learning based on visual representations can be specially beneficial for students with low prior knowledge (Gates, 2015).

It is important to clarify that we do not intend to cover multiple representations with these lessons. In fact, we focused on one model (the rectangular area model) in order to deepen the understanding of the model, allowing the students to get familiar with its properties and actions and, then, use it as a basis to build their knowledge of fractions. The rectangular area model is being used as a grounding metaphor (Lakoff and Núñez, 2000) for fractions.

As we expect the students to construct their knowledge, it is important to avoid: a) demonstrating how to solve the questions before letting them have a go, and b) synthesizing what was done in a lesson as a rigid procedure. Although some lessons may seem procedure-oriented, the approaches shown should not be presented as rigid procedures, but as systematizations of what is being done by the students.

Also, it is part of our assumptions that doing math with visual representations are a valid way of doing math. Therefore, the lessons will not cover explicitly the symbolic procedures. However, this can be done by the teacher by complementing these lessons with new lessons or with extra tasks.

### Materials

The lessons use cut-outs (see details in the intro file) and short animations (available on Youtube). Both are intrinsically embedded into the lessons.

### References

Gates, P. (2015). Social Class and the Visual in Mathematics. In S. Mukhopadhyay and B. Greer (eds) Proceedings of the 8th International Mathematics Education and Society Conference, Portland State University, Oregon, Vol 2.
Lakoff, George, and Rafael Núñez. (2000) Where Mathematics Come from: How the Embodied Mind Brings Mathematics into Being. Basic books.
Rivera, F. D. (2011). Toward a visually-oriented school mathematics curriculum: Research, theory, practice, and issues. Springer Science & Business Media.