The purpose of this lesson is twofold. One goal is for students to discover and understand why we slide the decimal two spaces to the right when writing a decimal as a percent. The other goal is to address various number sense gaps with fractions and place value. This lesson relies on prior knowledge and when connecting to prior knowledge that is weak or faulty there is a wide range of gaps. It is important to listen to student thinking and questions. Questions & observations can be a clue to how the student understands correctly or not and can cue teacher to the type of information needed to bridge the gap. It is also really important to listen carefully to student explanations as they begin to develop and strengthen their understanding. Too often we listen "for" certain things and end up shutting down their thinking before we really know what they meant. Students often don't express their mathematical understanding exactly as they intend on their first try, especially ELL students. Instead we really need to listen and probe for further explanation until we really understand what their model is. Pay attention to diagrams. Some of the most elegant models I have found come from diagrams drawn by ELL students. If I hadn't stopped and asked them to explain their "doodle" I would have missed it!
The first problem in the warm up Warm up perfracimals 1.docx asks them to find 80% of 50. Some may use the box diagram which I will use to emphasize the idea of the fraction as division. The denominator tells us how many equal parts to divide the whole 50 into. Some students will simplify the fraction 80/100 and scale it up to 40/50, which I will use to address the idea of equivalence. Many students with weak fraction sense struggle with equivalent fractions. Warm up perfracimals 1 notes.docx Some students might even solve this problem by finding 50% of 80 instead. This was something that we noticed in a previous lesson (Is that a coincidence?) Seeing multiple methods helps students make connections that develop their number & operation sense and help them make sense of the math.
The second problem 3/12=n/40 is the one that sets up the calculator exploration in this lesson. As a whole class we solve the fraction problems by dividing with the calculators. This helps remind them how to correctly use calculators and helps to scaffold for some common misunderstandings. Warm up perfracimals 1 notes.docx After calculating the decimal value of the fractions I ask what percent the fractions represent. Some students may scale the fraction up to a percent, but I expect most to use the decimal place value. To reinforce this later method I ask them to read the decimal number with its place value name. The place value chart remains on my board during the next several lessons.
The last question asks students to place the decimal point in $12. This helps to avoid the common misunderstanding that whole numbers don't have decimals (if they don't see it then it is not there). It is important to get this idea out there before the calculator exploration, because I don't want them getting stuck and not being able to "see" the pattern.
This is a quick exploration that gives students a brief introduction to dividing fractions to find it's decimal value. Using the calculator helps avoid the knowledge gaps in long division for now and allows students to focus on the connections between fraction notation and place value. Students unfamiliar with using calculators may enter the division backwards. It is a good idea to have some fraction circles or some other concrete manipulative to address the question of why the answer is different when they switch the numbers around.
I give them three or four fractions one at a time 3/4, 3/75, 4/5, and 12/16. I ask them to read each fraction as a division problem, which I record on the board, and ask them to do the division on the calculator. I record the answer and ask what else they know about that decimal number (for example 0.75). Some students may tell me it can also be written as 75% or as 75/100. If they don't give me both I might ask "how could you write that as a percent?", or "what other fraction would give us the same value?" I chose these first three fractions because they represent a two digit decimal ending in the hundredths place, a one digit decimal ending in the tenths, and a one digit decimal ending in the hundredths place. The last two get mixed up frequently and I want them to notice the difference right away. I expect there to be disagreement about them in the groups and I want them to use the place value chart as evidence to convince their peers. This is a good way to incorporate argumentation and peer instruction.
This is really another exploration in which students are using the place value of decimal numbers to help them determine the equivalent percent. During the calculator exploration they found that by reading the decimal number with it's place value name they could easily write the percent. Each student works on an individual white board, but has access to their math family groups for peer instruction. Everyone raises their white boards up at the same time so I can give corrective feedback when needed and no one can opt out. Students are asked to turn the decimal numbers (one at a time) into the equivalent percent, which I then record on the board.
.12, .45, .06, .60, .6, .7
For each one I remind them to read the decimal with its place value name to help them. I expect to generate some argument with .6 as several of them may write 6%. I find that asking them for .06 earlier helps many of them catch their own error, because they recognize the amount as a previous answer and they can see that there is something different about the decimal. When a student catches their own mistake they want to catch other's as well and so begins the peer instruction. I expect students to use a variety of methods to help convince their peers. Some may use the place value and some may scale up a fraction. Either way students are using mathematical evidence to and arguing about the math!
Once all of these are recorded on the board I ask students to look for a pattern. I tell them to look for how every single decimal number changed when it became a percent. If students have trouble finding the pattern I tell them that I notice there is a decimal in the front of all the decimal numbers and ask where it went? This usually gets them to notice that the decimal point moves two spaces to the right. This is something they may have seen in a previous class, but did not remember. It is so important for them to make sense of this pattern and understand the reasoning behind why it works so that they have a way to remember it. Students have really great ways of explaining why it makes sense and it is really important to listen for how they understand it. "It uncovers the hundredths place", "It makes it a whole number out of 100", "It shows that it is out of 100 with a percent sign instead of being two spaces behind a decimal". Instead of explaining it to them I just model their ways the best I can on the board.
I ask students why it makes sense that moving the decimal two spaces to the right would show us the percent. Many of the groups have already had this discussion, but now I want to share their thinking with the class. I may ask certain students to share if they had a really nice way of explaining it and they don't volunteer. There are so many things students have to remember in math, not to mention all of their other subjects. I find it is better to understand than to remember because it is more permanent.