Many times students (and adults too) know something but do not connect or apply that knowledge in a new context. In this lesson students use their knowledge of the definition of "percent" to write a percent as a fraction. Often when I ask students to write a percent as a fraction in simplest form they get stuck even though they know the definition and they know how to simplify a fraction. This lesson points out to them what they already know. I specifically tell them they are not going to learn anything new today, because they already know how to do this.
Students will also see that when they are solving percent proportions they are finding the percent of a number. Although they can find how many to expect out of any given total for a certain percent, they don't know that they are finding the percent of that total.
The warm up Warm up percent fraction equivalence.docx asks students to write fractions as percents and percents as fractions in simplest form. Students simplify, scale up, or do a combination of both to convert given fractions into percents. I flip the proportion around and ask the students how they might solve this one? For example, 12/15= ?/100 becomes 80/100= ?/15. Many students will recognize that it is the same problem and will intuitively know the answer, but will have difficulty figuring out how to solve it mathematically. It's the same process, but it sometimes throws them. I ask them how they solved it before I switched it (simplified, then scaled up) and can they do this one the same way? They get very excited to see that they do already have the skills, they just needed to try it.
I also want to show them that when they have solved this second proportion they have found 80% of 15. Some of them intuitively know this because they are equal ratios and both simplify to 4/5ths, but most don't make this connection. I like to show them the "box diagrams" as a visual scaffold box diagram to show percents.docx and show them that when we find 80% of something we are finding 4 - fifths of it. I don't spend a lot of time on this here. It is only a preview so they will feel more comfortable with it when we begin using them.
Students work on individual white boards, but have access to their math family groups. They all show their boards at once so that I can give corrective feedback if necessary and no one can opt out.
I start with the format they are more comfortable with, scaling to the percent (14/35=?/100). The second one is exactly like the first one only switched around(40/100 = n/35). One mistake I expect to see is writing 14 % instead of just 14. I don't say "no" or "wrong", but instead say "so, you found 14 out of 35, right?" (yes) "does percent mean out of 35?" (no, it means out of 100) "then is 14 out of 35 going to equal 14%?" I try to give feedback in such a way as to help them figure out their mistake themselves. Only after students have heard this a few times I might just say "lose the percent sign" and ask someone else to explain why. After this correction has been made I would follow up with "what is the percent?" so they can see that it was the fraction they started with. If it didn't confuse them earlier during the warm up I may also ask them what 40% of 35 is. (14) and I may model with the box diagram as discussed in the warm up, but again only if it didn't completely confuse them earlier.
I have them practice one at a time on similar problems asking similar follow up questions as needed.
75/100=m/24 20/100=c/30 12/40=x/100 35/100=x/60
At the end I ask them to compare and contrast the last two and discuss how they are similar and how different. The main thing I want them to come away with is that one is calculating the percent and the other gives the percent and calculates the percentage of another total.
Although the skills are not new this feels new to them and they may have been confused by the box diagram, so I want them to have some time to get started on homework homework fraction percent equivalence.docx. This makes them feel more confident and gives them access to help or confirmation. This is a good way to address homework completion problems.