The whole point of this lesson is for students to learn to use the context of the problem to help them make sense of the numbers and persevere in solving the problem. The key for the teacher is to ask students to clarify what the numbers represent, count, and compare. I use the sentence frame "for every....there are..." that was introduced in an earlier lesson (which is blackest?) to help students visualize pattern. Students are still struggling with multiplicative thinking that is necessary for understanding ratio and proportion. This lesson continues to support and strengthen additive thinking in order to develop and strengthen multiplicative thinking. Spending time having students examine multiple strategies is useful here.
In this warm up warm up percent to ratio.docx students are told that a survey shows that teens prefer drinking energy drinks to water by a ratio of 3 to 2. They are asked how many total teens could have been surveyed. First I ask students to define what the numbers represent (what they are counting or comparing) and we write out the sentence frame "for every 3 teens who prefer drinking energy drinks there are 2 that prefer water". Referring back to this context helps them test the possible totals that could have been surveyed.
Drawing a model on the board may help ELL students participate better in the discussion. I may draw EEEWW EEEWW to show E (students who prefer energy drinks) and W (students who prefer water). When I circulate to groups, especially those with ELL students, I suggest they copy down this model so they can show where the numbers are coming from. They may circle or count to show that there may be 5 total, then 10 total. Once this pattern starts to emerge suggesting a table may also be helpful for ELL students. When developing academic language it is always helpful to have a visual model to reference when hearing and speaking the new language.
Going over this warm up may surface different ways of thinking about the problem. I also expect them to make some mistakes if they ignore the context and don't use it to make sense of the problem. I record all suggestions for possible totals on the board and ask why they make sense, or if they make sense. I expect them to suggest 5, and would go over that one first, because the likely explanation would be a breakdown of how many prefer energy drinks and how many prefer water, which I can model easily with a table. This makes it easier for students to notice and correct mistakes. As students notice the pattern as multiples of 5 they may suggest some really big possible totals like 1675 and I would ask the class to discuss in their math family groups if this is possible. This is a good way to get them to engage in argumentation and also helps my language learners to practice and listen to explanations.
When we come to the total of 100 I would ask them what percent of the students prefer energy drinks and water, just to remind them of this special ratio.
I ask them also to discuss how they can figure out the number of student preferences for any given total. I pay particular attention to having them explain similarities between additive and multiplicative methods in order to build and strengthen both. It is really important not to shut down additive thinkers because it may inhibit their transition to multiplicative thinking.
This exploration asks students to figure out a question Changing percent to ratio.docx similar to the warm up, except that the survey results are given as a percentage rather than a ratio. Students are asked how many teens might have been surveyed if 80% preferred Company A's video games.
Students first need to figure out what the 80% represents, counts, or compares. As soon as a student mentions that 80% means 80 out of 100 I ask the group how they know this is true (the definition of percent) and then point out that 80% is actually two numbers, which makes sense since it is a comparison. When students explain that the 80 is the number of teens who prefer Company A's video games and the 100 is the total number of teens surveyed I model this in a table like the warm up problem. One error students may make is thinking that the 100 refers to the teens who prefer another company. If this is the case I ask them to briefly discuss it in their math family groups and hope that someone comes up with the percent being "out of" 100.
We can then go over the possible numbers of teens who could have been surveyed. I record the responses in the table and ask how many prefer company A. If they suggest it, I also include a row for the other company. If they only suggest multiples of 100 I might ask if a total of 50 is possible. (It is always important to write the number in the table when I ask for students who don't speak English. I also put a question mark in the table to show what we are looking for) What about 15 or 35? I might ask what is the smallest total, but I want to first give them the chance to come up with it on their own. Once we've gotten down to the lowest total possible I may scale back up to 15 or 35, which they may not have thought possible before.
At each step it is important to ask students to verify it in their groups to allow for some quick peer instruction for those students who need clarification or primary language support.
Students work individually on white boards, but have access to their math family group for help. At the count of three students hold up their answers on white boards at the same time. This way I can see everyone at once, give corrective feedback, and ensure that no one is opting out. When just a couple of students mess up I work with them on the next problem. When the whole class messes up I explain to the group and give them a similar problem to practice on.
I have them practice solving proportions by simplifying and scaling up:
7/20 = ?/100 3/5 = ?/100 11/25 = ?/100 9/10 = ?/100
11/100 = ?% 11/50 = ?% 11/20 = ?% 3/10 = ?% 3/5 = ?%
I like to have the same numerator in my later examples because it helps to emphasize the scaling. When students first start learning about percent they know that the emphasis is on the denominator and a few of them forget to scale up the numerator. When this happens it is easier for them to see their mistake if you can ask if 11 out of 100 is really the same as 11 out of 20.