Fractions can be hard for students to understand because they can represent many different things and can be represented in many different ways. Students bring prior personal knowledge to school and it is the teacher’s responsibility to build off of this knowledge. Teachers should understand their students’ prior knowledge in order to use and apply teaching methods that will be most fitting for their class (Flores & Klein, 2005). It is important to have students learn through context and content rather than memorization (Sharp et al. 2002). This is why we should allow them to learn through picture and concrete materials before introducing symbols (Ex. 3/4) (Sharp et al. 2002). The materials that we use with students should have the ability to be divided and subdivided unless we are specifically using them with set models (Flores & Klein, 2005). For example, brownies are a better choice than marbles because you can’t divide a marble (Flores & Klein, 2005). It is also important to show or use pre-cut fraction pieces for accuracy (Flores & Klein, 2005). Teachers will need to use different materials and presentations to see what works with their class (Flores & Klein, 2005); these ideas can be derived from students, teachers, research, own experiences, etc. I have some ideas for materials that could be used in the classroom in order to learn different types of fractions.
Area models involve sharing something so we need to cut it into smaller pieces (Karp et al. 2015). The most frequently used model are circular ‘pie’ pieces (Karp et al. 2015); however this is not always the most helpful as children find it difficult to divide a circle evenly. We should use rectangles to divide evenly at the beginning until students become more familiar. Rectangles can be used in many ways. For example, you could divide a rectangle evenly by folding paper, dividing a candy bar or cake, using a geoboard and elastics, etc. There are even tools and apps that can be used to replace the geoboard. Here is an example of a student using a geoboard app to find the relationship between 1/2 and 3/8.
The student in the video used the Geoboard App to compare the two fractions. On this application, you can change the size of the board, add a number grid, draw, use the calculator, and even change the board from a rectangle to a circle. The circle geoboard application would be useful in the classroom because it already divides the circle into even pieces that the students can use to further divide if they wish. Once students understand how to evenly divide a circle, they can begin to use circular materials such as pie, pizza, cake, tortillas, fraction circles, spinners, etc. Another material that could be used is pattern blocks. The students could find what fraction the different shapes represent if, for example, the hexagon was a whole. You could start with the hexagon as the whole and as the students become more familiar, you can try this with the other shapes being wholes as well. Here is a video that explains this concept further:
This can be done through tools and applications as well such as The Math Learning Centre Pattern Shape App and The Math Playground Pattern Block App.
Linear models involve the length of the whole divided into equal lengths (Karp et al. 2015). A fraction is seen as being a specific distance from zero to the whole. There are many materials that are helpful for students to understand linear fractions; two of these include number lines and cuisenaire rods. Number lines have been recognized as an essential model when teaching fractions mostly because students can then understand that fractions are numbers and can see where they are placed in regard to the whole (Karp et al. 2015). Number lines can be made in many ways including folding paper, writing on paper, using a wall, using applications, etc. Here is an application that I found that would be useful to use in the classroom: The Math Learning Center Number Line App. With this application, you can have a number line with predetermined number ticks or you can custom make your own number line. You’re able to make the ticks on the number line narrower or wider, draw on the application, hide the numbers on the line and fill them in yourself or uncover them, make jumps on the number line, use the calculator, etc. Cuisenaire rods are another linear model that can be used; they are ten rods that are measured according to the smallest rod (Karp et al. 2015). Each rod is coloured so students can quickly identify which rod is which (Karp et al. 2015). Here is an introductory video for cuisenaire rods:
Cuisenaire rods can be used in many ways. One example of this is to not always have rod ten represent the whole. If you want your students to look at 1/4’s and 1/8’s then you could select rod 8 as the whole. You could also make a whole by attaching two rods. For example, you could attach rod ten with rod two to have your students looking at twelfths (Karp et al. 2015). There are online tools and applications that can be used to explore cuisinaire rods such as The Math Playground Math Bars and The Math Toy Box Number Blocks. Some other ways to represent this model include rulers, linking cube trains, and fraction bars or strips.
Lastly, there are set models which involve a set of objects that make up a whole (Karp et al. 2015). For example, a set of ten cars could make one whole; one of these cars would be 1/10. This model is especially difficult for students to understand as they tend to think that the one object is already the whole (Karp et al. 2015). Students who have difficulties with set models can use a piece of yarn or string to tie around the objects so they can see it as a whole (Karp et al. 2015). You could make set models out of anything that the class has an interest in such as dinosaurs, skittles, two-coloured counters, bingo chips, foam shapes, pattern blocks, themselves, etc. Set models can be easily made in the classroom; although there are online tools and applications as well. Here is one that I have discovered called the Toy Factory which allows students to make their own toys in the toy factory and then determine how many are red, how many are monkeys, etc. Although this model is more difficult for students, it is important for them to see fractions in all three models (area, linear, and set) (Karp et al. 2015). Some materials are able to be used for all three models. One of these materials is LEGO; here is a video that shows some ways LEGO can be used in math and with fractions. LEGO can be used as an area model by covering the area of a larger LEGO piece with smaller LEGO pieces. LEGO can also be used as a linear model by creating a line with the LEGO pieces and measuring. Lastly, LEGO can be used as a set model by using many LEGO pieces to make a whole. This is a new idea for me and I cannot wait to use LEGO in the classroom.
References
Flores, A., & Klein, E. (2005). From students’ problem-solving strategies to connections in fractions. Teaching children mathematics, 452-457. Retrieved January 19, 2016.
Karp, Bay-Williams & Van de Walle. (2015). Developing fraction concepts. In Elementary and middle school mathematics: Teaching developmentally (9th ed., pp. 284-309). Toronto, ON: Pearson Education Canada Inc.
Sharp, J., Garofalo, J., & Adams, B. (2002). Children’s development of meaningful fraction algorithms: A kid’s cookies and a puppy’s pills. Making sense of fractions, rations, and proportions, 18-28. Retrieved January 12, 2016.
