A Visual Model for Percents

Over the last few weeks, I have been working on percents with my sixth graders.  With the move to the Common Core, some slight tweaks have taken place in what parts of percent I have to teach. I have been focusing on this standard

CCSS.Math.Content.6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent

The prerequisite skill for this is to be able to fluently move between writing numbers as fractions, decimals and percents.  My students did quite well with this and after some initial instruction and modeling with magnetic base 10 pieces (The magnetic part makes this a new favorite!), decimal squares and fraction/decimal tiles they showed good levels of proficiency based on a quick formative assessment I gave.



We finished up this skill with a QR code scavenger hunt that had students rewriting numbers as fractions, decimals and percents.

When we moved onto finding the percent of a number, I felt like kids were doing so well.  A week later when we started working on problems where they are given a part and a percent and had to find the whole, we stumbled into a few roadblocks.  It is amazing how similar these problems can sound.

For example:
What is 60% of 300?  VS 300 is 60% of what number

After a LOT of discussion and MANY examples, I felt like my students had this.  I gave another formative assessment and I had a few kids who showed me that they really didn't get it.

I was able to pull them out for a 40 minutes booster group on Friday and tried to figure out where I went wrong.  While I am trying to explain the big ideas and model them again with various models that we had used in class I discovered a new model that I could use for percents.

I turned away from the group for a moment to grab my water bottle and inspiration struck when I saw my 100 bead strings hanging on the wall.  Turns out, they make a great model for percents whether you are given the part or the whole.  Check this out!

First we worked on the connection between 10 percent of a number and 1 percent of a number

I expect my students to be fluent in finding 10 percent because of course they are fluent with dividing by 10.  This bead string shows another representation of why finding 10 percent of a number is just like dividing by 10. 

Here we are illustrating that once you know 10%, dividing by 10 will give you 1%

Here the students are asked to find 30% of 700.  They use what they know about 10% and can see how that relates to 30%

We could see this would help us visualize the percent of a number but what about finding the whole when given a percent?

I presented them with this problem  

They took 20 beads and said that is worth 30.  They know this part and need to find out the whole.

One student suggested that since we knew 20% was 30, 10% would be 15.  Then we needed to multiply 15 by 10 to get the whole.  

Another student suggested we need to multiply 30 by 5 because we know 20 beads (percent) is worth 30 and there are 5 groups of 20 in 100.  Both these ways helped the other kids in the group think more flexible about percent problems.  

Of course, I do not want my sixth graders staying at this concrete stage forever but I think this model will help them keep the ideas of finding percent of a number and finding the whole given a part straight.  I also like how it is a model they can go back to in their heads even long after the bead string is gone.  I plan on spending another day with them connecting their ideas to the bead string and gradually trying to transition them from using the concrete model to a mental model.

What models do you use with students when you work on percent problems?  Please respond in the comments below!