In this lesson I expect a common misunderstanding to surface. I expect several students to try to scale up all of the ratios into percents without regard for whether they are part to part or part to whole ratios. Some math family groups will realize something is wrong when they get more than one percent to represent the portion of spotted lady bugs. I may hear "there can't be 20% and 25% that are not spotted". I also expect other students to support their claims by showing how they scaled the ratios up correctly. Only when we clarify the terms and define what the numbers represent do they fully understand that a percent is a part to whole ratio. We need to clarify that 20% means that 20 lady bugs are not spotted out of a total of 100 lady bugs and that 25% means that 25 lady bugs are not spotted out of a total of 100 lady bugs. Both can't be right. This is when they begin to realize that scaling up the ratio 3/12 is comparing non spotted to spotted lady bugs and not non spotted out of the total.
In this warm up I draw 15 lady bugs. 12 are spotted and 3 are not. I ask students to write ratios and percents to represent the lady bug population in terms of spots vs. no spots. Having students count and come up with their own numbers makes them more likely to define their terms and show what each number is counting. This way they are more likely to chose the correct ratios (part to whole) to scale up into percents. In order to encourage defining like terms I circulate and share with the class organizational ideas I see students coming up with. I also encourage them individually to define the terms within the ratios they have written by asking them to explain to me what each of the numbers in the ratios represents. Having the picture also makes it more concrete for my ELL students to see the numbers represented visually, they can see the actual lady bugs and whether or not they have spots rather than just the abstract words referring to them.
Before we move on to the exploration I ask students to share the ratios and percents they found. I ask students to identify them as part to part or part to whole and I write them on the board. This way if students make a mistake by scaling up a part to part ratio into a percent we have something to refer to that will help them find their mistake by asking is a percent part to part or part to whole? "so, if we want to end up with the number of spotted lady bugs out of a total of 100, which ratio should we scale up?"
Students look at a new population of lady bugs, showing 8 with spots and 2 without. They are asked to decide if this population is proportional to the last. I look for a student who is defining his/her terms and show the class. I ask the class what this student is doing and why in order to encourage them to take the time to define their terms.
I do not specifically tell them how to test for proportionality, so if they have trouble getting started I suggest they start by writing some ratios for this set of lady bugs like they did with the last and see what they can do with that. Once they see that the ratios "match" when they are simplified or that they have the same percent they remember that this shows equivalence or proportionality. If they really had trouble recognizing this as sufficient evidence I may try to connect them back to the sentence frame that was introduced in an earlier lesson that introduced ratio (Which is Blackest?) "for every 5 total there are 4 spotted" and then circle the lady bugs in sets of 5 for each population of lady bugs to show the same ratio.
I don't expect most of them to need this level of scaffolding, but I would have it at the ready, especially for ELL students. I just photocopied the original pictures of the lady bugs and wrote the sentence frame on one of the copies to have it ready for them.
This lesson was meant to clear up the confusion that arose in an earlier lesson (Spots vs. Not) about which ratios can and can not be scaled up to a percent as well as emphasize the importance of defining our terms.
During the wrap up I ask students to explain why they think the two populations of lady bugs are or are not proportional to each other to surface the mistake. There are really two mistakes that can be made, but I deal with both of them in the same manner. Dealing with a potential mistake with percent.docx
It is possible that no one will still be making this mistake. Depending on how prevalent it was in the earlier lesson I may suggest it myself by asking what I might be thinking if I said the percent without spots is 25% instead of 20% and then ask them to explain to me why I am mistaken.