As a high school math teacher, I taught a wide range of students from ESL Algebra 1 and regular Geometry to Pre-AP Algebra 2. The resounding similarity I saw between all of my students was that, for some reason, students cringe when they see rational numbers. They feel like rational numbers automatically make the problem “hard”. I was amazed that by high school, students were still struggling with something as simple as converting from a fraction to a percent. Perhaps this is because, as teachers, we sometimes teach our students shortcuts that leave out the logic behind the scenes of the procedures they learn.
The IES Practice Guide, which is supported by research evidence, recommends that teachers “'help students understand why procedures for computations with fractions make sense’ and ‘develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication…’ (Siegler et al., 2010).”
Some common shortcuts teachers use are changing the fraction to a decimal then multiplying by 100 or changing the fraction to a decimal then moving the decimal to the right twice.
Examples:
Neither method above leads students to a percentage as the final answer, unless the student “remembers” to tag it on at the end. Units are crucial when converting in any context. In order for students to understand why they must multiply by 100% rather than 100 when converting from fraction to percent, units must be used properly.
Instead, students should be taught to set up proportional relationships, including units, between the fraction and unknown out of 100%. It is important that students understand that when the units of the numerator and denominator are the same, they cancel and the fraction is unit-less.
Example:
25 students went on a field trip and 5 wore a hat. What percentage of the students wore a hat?
20% of the students wore a hat on the field trip.
Through the process above, students see why they are multiplying by 100% and why the units in their answer must be a percentage. Also, students can use number sense to reason that x must be a percentage between 5 and 100.
Summing It All Up
Fellow teachers: it’s not safe to assume that our students understand why they are doing a particular procedure, even if it is one they “should” have mastered several grade levels ago. If we take a little more time to illustrate examples with labeled units and explanation, we will hopefully catch any previous misconceptions our students have and steer them on the right path towards math success.
Now it’s your turn. Share with us common misconceptions, similar to what we described above, that you’ve found in your classroom!
Resources:
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/
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