This lesson illustrates the importance of building self confidence and resilience in our students. It threatens to be the perfect storm with several gaps in prior knowledge colliding. We have to let students know that struggling is okay and we will be their to support them. Teachers will be unable to bridge any gaps we can't see. Students have to feel safe enough to expose their deficiencies. Be ready to scaffold and intervene at several different points to address weak or faulty fraction sense and decimal sense. When students feel they don't understand something they tend to over rely on trying to remember an algorithm. They try random things rather than really try to make sense of the context. It is really important to be ready to scaffold or intervene and meet students at their level.
Warm up cup half full.docx This warm up tells students that If I bought 100 cupcakes the bakery will give me 10 more for free. Then I ask if I bought 20 cupcakes would that be enough for the class? I also provide them a box diagram to use for modeling the problem. I expect many students will write the percent as a fraction, simplify it and then scale it up to a denominator of 20. If no one uses the box diagram I will ask them to help me model it's use, because it helps them determine that dividing by 10 is a valid way to get 10% of a number. I am a little afraid that students will think you can find the percent just by dividing by the percent we want, since, in this one case it works. To help avoid this problem I make sure to start with 10/100 and ask if there is any way to avoid having to divide the box up into 100 pieces. By the time we simplify the fraction there is enough separation between "10%" and "dividing by 10" to avoid the confusion.
I am also sure that I ask how many cupcakes we end up with, because I can expect some students to have forgotten to add the 2 (10%) to the 20. Asking them what I had to do to get those 2 for free is usually enough to get them to notice their mistake. When they say 22 is not enough for the class I ask if it would be enough if the bakery offered me 20% instead? This sets them up for the mental math strategy we will be using today.
Before we go over the problem, however, we have to do a little front loading. Most of my students have weak decimal sense and many do not know where to place a decimal in a whole number. This deficiency will get in their way of seeing the pattern of sliding the decimal. I have some numbers written on the board (9,426; 15.50; 12; 0.45; 3). As students finish up their warm up I ask them to come up and show where to place decimal points in the numbers. When we go over this I reiterate that the decimal marks the end of the whole part of a number.
I tell them there is a shortcut that allows them to find 10% of any number in their heads without doing any division, but in order to find it we need to do some division and look for a pattern. One more day of doing the division and they never have to do it again...unless they forget the pattern. I ask them to think up several different types of numbers for us to divide by 10 and I record them on the board. To encourage them to try different types I have to say things like "good, we have an even number..." which will hopefully prompt an odd one, "we have a small number", "several whole numbers", "decimal number ending in the tenths place", etc. When they offer up several numbers that are even or small I remind them that we need to find a pattern that works for all the different types of numbers, so we don't want all of them to small and even. A good variety might look like: 20, 4, 505, 0.11, 66, 72.5, 14, 101. I like to have about 8 or 9 so that each member of the group does 2 or three problems. I also remind them to help each other and double check each others work, because if we make a mistake we won't see the pattern.
I tell them to look for a pattern that would help them find 10% of any number without having to do the division. This usually needs some scaffolding, because many of the students have not placed the decimal in every answer and suggest that maybe seeing the numbers side by side would help. I write two columns on the board: total and 10%. I write each total and ask them what 10% is.
I ask for a show of hands who thinks they see the pattern and I tell them to tell their groups what they think. Then I will pick up another color marker and say it might help to put in the decimals. The different color helps them see it, but I don't want them to know I did it on purpose. Most of the students will have heard the idea from another student if they didn't notice it themselves and now I give them the chance to test it out. I erase the numbers and put up new numbers (43, 8, 12.5, 0.25, 0.7) one at a time and call on volunteers to give me 10%. This can go really fast and students who get it can't help but explain it to their groups in whatever language they want. Because they have their work in front of them they can point it out more easily to each other, which is particularly helpful to ELL students. In one video you can see how one boy is trying to use hand motions to demonstrate what another student is trying to explain. I told him afterwards how helpful this was.
I start by asking students to find 10% of several numbers. Students work together on individual white boards and show their answers at the same time. I start with a number that they could easily divide by 10 in their heads, so that I can remind them what to do when they forget the shortcut. The shortcut is not valuable if they don't have a way of retrieving it when they forget.
10% of 40 10% of 300
So, as I circulate and find students who are having trouble getting started I can just ask what is 1/10th of 40? I can rely on their math families to provide the shortcut.
Then I move on to numbers that already have a decimal placed in them. (29.4, 3.25, 0.75) Some of my students are still grappling with where the decimal point goes in a whole number and I don't want this to get in the way in the earliest stages. Then I move on to whole numbers. (17, 3)
This is when I introduce my cup diagram, which I draw on the board and fill to the top with "water". I point out that when it is 100% full it has 80 ml. of water in it. I draw a water line near the bottom and label it 10% on one side and ask how many ml of water it would be if it were only 10% full. (8) Since this is a different way of asking for 10% of 80 some students may need me to ask "we are finding 10% of what total?" If they don't know then I expect one of their group members to be able to help, but I think the word "total" will help.
Now I can ask how many ml the cup would have when it is 20%, 30%, 40%, 50% full. They can see they just multiply the 10%. I ask for multiple methods of explaining so students can see the connection between repeated adding and multiplication. Many of my students are still working on multiplicative thinking. Once they get to 50% someone will likely come up with just halving the total.
The second cup has a 25 ml. capacity. This way they can practice finding 10%, 20%, etc. with decimal numbers.
Even if they struggle with this I point out that they are able to do more mental math percents than they thought they could at the beginning of the lesson. I know several of them did not believe they could find 10% of any number mentally, let alone 20% and more. When we struggle with confronting weak and faulty prior knowledge it is important to boost confidence at the same time to encourage perseverance.