Students are introduced to the percent proportion as a pair of equivalent fractions with the same denominator. I do not expressly teach "cross multiplication", because eventually they will derive it on their own. The concept of equivalence gets lost in the algorithm and students don't retain it, because they don't understand why it works. When they use common denominators just to compare the numerators, eventually they will drop the denominators. In this lesson I want them to use scaling up or down in order to calculate the percent as a fraction "out of 100".
I have students work in their math family groups to figure out the problems to help them connect to prior learning. I want to encourage them to try out the ideas they come up with and not first ask me if it is the right thing to do. I will respond to such questions with "do you think that might help?", "how could you find out if it helps?", "why don't you try it and see what happens?". (mp1) I use contextual problems so they can view the calculations as tools for figuring something out rather than a process to follow. (mp)
This warm up warm up percent proportion intro.docx tells students the definition of percent by saying that it means "out of 100" or (similarly to the sentence frame they have become accustomed to since the "which is blackest?" lesson) "for every 100". It asks students to figure out what percent of students like math class best if one in every 5 say they prefer math class.
Students work together sharing their ideas first in small groups and then sharing with the class. It is important to let them share with small groups first to help them sort out and test their ideas. It also gives ELL students a chance to practice how to articulate their ideas in a less threatening setting.
As I circulate I encourage students to also express their ideas mathematically. I expect most students to multiply and scale up the ratio 1/5, but some may be adding repeatedly. Again, I ask these students to show their idea mathematically and then when we share it in class we can have a discussion about the similarities and differences between this method and the multiplicative method, one of which will obviously be comparing efficiency. warm up percent proportion intro multiplicative vs additive methods.docx
Students calculate the percent from a fraction in three more scenarios contextual percent practice.docx similar to the one in the warm up. While students are just practicing the skill I think it is still important to include a context so they view the calculation as a tool for figuring something out rather than a math exercise that has nothing to do with the world.
We go over each one with the whole class and have students explain how they did it. If there was a discussion earlier about both multiplicative and additive methods during the warm up I want to bring it up here as well. Although we want students to be thinking multiplicatively it is important to include a discussion of additive methods if they are being used by students in the class. The best way to develop multiplicative thinking is to build on, connect to, and strengthen their additive thinking. As I see that the multiplicative method is being used by more and more students I may ask why they chose this method over the other method, just to reinforce the efficiency. This is a good way to help them choose appropriate mathematical tools for solving problems. (mp)
Students work on individual white boards, but can work with their math family groups. On the count of three they all show their answers at once.
12/25 = ?% 7/20 = ?% 4/5 = ?% 3/4 = ?%
If students are having trouble I may have them practice on the intermediate step of finding the equivalent fraction "over 100" first.
If students are getting it really quickly I may have them work on some fractions that they have to simplify:
60/200 = ?% 120/300 = ?%
Having access to math family groups during white boards is really helpful for all the students, but especially so for ELL students. I try to make sure that ELL or Spanish only students are grouped with at least one bilingual student and I try to make sure all their group members are supportive. I keep a closer eye on these groups as well as groups with students who need extra support of any kind. I spend more time "training" these peers in talking to each other. I make sure they ask each other, "did you do it differently?", "what did you do?", etc. I have them put their work in the center so that everyone can see. I make sure they point to what they are talking about on their board or on some one else's. I also remind them to keep an eye on everyone's board and take notice when one of their math brothers and sisters is having trouble getting started.