This lesson is designed to help students understand the connections between the different representations of rational numbers. It follows the previous lesson (Percent Graffiti) in which I asked students to reflect on different representations. In this lesson I hope to clarify and strengthen some of those connections as well as reinforce the idea of equivalence in proportional relationships. Students shared in small groups many different ways of explaining and representing rational numbers and in today's lesson I hope to make those ideas visible to the whole group. I want everyone to understand that a value can be represented in several ways, specifically as a fraction, ratio, decimal, and percent. They will be using different models for finding the percent of a number which will require this background knowledge. For those students who are still unsure about proportion I really want to emphasize the idea of equivalence.
In this Warm up I ask students to draw a picture of the fraction 1/10 and describe it in words. I circulate and look for Different ways students are drawing the fraction and ask them to put them up on the board. As they describe what they did I call student attention to the idea that the denominator tells us how many equal parts the whole is divided into. It is important that this idea comes from students and not the teacher. I may ask them to describe their figure or "why is one part shaded?", "how many equal parts are there?", "how did you know to do that?", "why are there ten equal parts?", etc.
One misconception that is likely to come up and is worth exploring is the idea that 1/10 shows one shaded part and 10 un-shaded parts. It might come out when you ask "how many parts are not shaded?" This provides an opportunity to highlight the importance of defining our terms as well as reminding the class of the other types of ratios that could be used to describe this quantity. To get them to notice the discrepancy I might ask "are there really 10 parts un-shaded?" to which I would expect a student to either point out there are actually 9 un-shaded and 10 total parts. This might be a good time to review part to part vs. part to whole ratios and that only part to whole ratios can be scaled up to a percent.
Depending on the vocabulary students are using to describe the fraction in words you might be able to ask what other ways we could write that value to generate decimals and percents. If their vocabulary does not easily give rise to multiple representations you can ask them to read the fraction outloud and listen for "one tenth". I am really looking for all the different ways they can represent that value mathematically. (2/20, 0.1, 0.10, .1, 10%, etc.) I am hoping to use this list to generate a clarifying conversation about how equivalence relates to proportional relationships.
Once students have generated multiple representations for 1/10 I ask them to explain how they know these are all proportional to each other. Warm up follow up question.docx I also ask them how they could show that they are all proportional to each other. As I circulate I listen for ideas about place value, simplifying, scaling to common denominators, and graphing and ask students to share the idea with the class and show how they might use one of these ideas to prove it. I want to reinforce the idea that equivalence shows a proportional relationship.
This exploration asks students to show how they would find 10% of a number. Some students show their work on white boards and some on paper. On the screen I show them a box diagram as a possible model. I want to see if they can transfer their use with fractions to a percent. If they don't automatically relate 10% to the fraction they will have trouble with this model. Be ready to scaffold their fraction sense with fraction circles! One pitfall to this method is that you are finding 10% and dividing by 10 and students want to overgeneralize so that they would divide by 30 when finding 30%, etc. It is really important to relate the dividing of 10 to the denominator in 1/10. The box diagram helps reinforce this idea.
Some students will try to use the box model and others will use different methods. I make sure multiple methods are demonstrated. I really want to zero in on the "decimal slide" and focus on this mental method. If no one remembers it from the earlier lesson (The cup half full) I allow them to use any method they choose and I keep track of the results until they notice the pattern again. The ability to find 10% easily in their heads is crucial to building up other percents mentally in this and future lessons. It is really important that students really understand why it works to "slide" the decimal. Relating the movement to the place value chart is really helpful. I have bundles of straws ready to scaffold what happens in the place value chart when multiplying and dividing by 10.
Another pitfall is when student grab ahold of the "decimal slide" and try to use it as an algorithmic "trick" without making sense of the problem first. They may slide the decimal one space to the left when I ask them for 10%, but then slide it another space to the left when I ask them for 20%. I like to do two things when I see this. I like to ask if it makes sense that 20% would be less than 10% and I like to relate it to the box diagram and ask (if 20/100=2/10) how many of the tenths we are asked for in 10% and how many in 20%. This also might be a good time to bring back the "cup" diagram from the earlier lesson (The cup half full). Either model can be useful in emphasizing the idea of doubling or scaling from 10% in their heads.
I ask students to find 10% of any number any way they like. I may need to write "10% of ____" to clarify that I want them to choose their own number. As I circulate I look for multiple methods and ask students to go up and show it on the board. I really want to see box diagrams, decimal slide, and scaling ratios. For each method (one at a time) I ask students to explain what was done and why in their small math family groups and then I ask someone to explain for the class. I may ask how each method shows dividing by ten or how a student knew to divide the box into 10 equal parts. I want to end by circling the 1/10 (which I expect to find in the ratio scaling method: 10%=10/100=1/10=6/60) and ask how this tells us to divide by 10. The lack of fraction sense can really get in the way here and I want to hammer in that a denominator of 10 divides the whole into ten equal parts! If I had to use the fraction circles to scaffold earlier in the lesson I may use that to help emphasize this point.