Percents on a Linear Model

I've used linear models before to introduce percent problems.  As I work to focus more on conceptual understanding in my classroom, I amped up my use of this tool and I can really see a difference in my students depth of understanding.  Because it was my first year incorporating all of this, I was adding resources and discovering ideas as I went.  I am just putting all the ideas here.  You will have to decide which order makes sense for your instruction.

Here is the gist of the linear model in case you are not familiar.  It relies on students ability to partition a number line and use proportional reasoning.

Clothesline Math and Estimation 180

Before starting our work with percents we had been focusing on proportional relationships.  One of the activities we had done is a double clothesline task from Andrew Stadel about yogurt.  It helped that my students were familiar with the double clothesline before going into percents.  The linear model was not a new concept and we were able to focus on percents as the new information.

Here is the first double clothesline we did for our work with percents.  Because 31 is not an easy number to work with we were focused on estimates.  I liked that as an introduction because then students weren't focused on having the same answer as their classmates.  They focused on understanding the strategies.

Notice/Wonder works great too during the introduction phase because there are no wrong answers.  (So often students feel like they need to understand a new concept before we have even talked about it.  Where does that come from?  Is that in all grades or just middle school?)  Notice/Wonder helps students make connections with what they know and see the relationships in the numbers.  Here is a picture of that and you can see our double clothesline on the side.  

As part of my daily routines, we do an Estimation 180 task once a week.  I now have 2 permanent clotheslines at the front of my room (still deciding the best permanent spot).  One day I decided to throw our estimates up on the clothesline and calculate some percents.  This number line is based on this candy corn task.  Students always write down a low estimate, high estimate, and best estimate.  (There is a video of how to use a single clothesline with Estimation 180, using all 3 estimates.)  In my classroom, the best estimate is what we put on the top number line on white paper.  After we reveal the answer we put 100% on the bottom number line.  Then we have some fun.  For this particular picture we:

• Find 10%.  Shared strategies include: 
• Partition the number line:
• divide by 10 because if we partition the number line by 10 on the bottom to get 10% we can do the same on the top.  
• Use scale factor/multiplier for this proportional relationship
• divide by 10 because the fraction form of 10% is 1/10.
• Multiply by 0.1 because it is the decimal form of 10%
• Set up a proportion
• divide by 10 If __/10 = 893/100.  The we are multiplying by 10/10.  In the other direction we would divide by 10.
• Please note that cross products is not a strategy for my students at this point.  Until they learn to solve equations with fractions we cannot make mathematical sense of this strategy.  I do not allow my students to use strategies that don't make mathematical sense. 
• Find 60%.  Shared strategies include:
• The same as above only with 60%
• Multiplying the 10% amount by 6, taking advantage of the proportional relationship.
• Finding 50% by dividing by 2 and then adding 10%.  
• Find 150%.  This was our first time going beyond 100%.  We aren't working on this but I thought it would be good for students to see that the number line goes beyond 100.

The key to all of this is in the discussion.  You will notice that divide by 10 came up three different times.  It is important for students to understand why dividing by 10 works.  It is also important as the teacher to fully understand the students' thinking and which strategies they are using.  Some students may know that dividing by 10 works but cannot explain why.  The double number line helps them see the reasoning and the relationships.

At the end of the unit I realized that the questions students were still struggling with had the 100% value missing.  For example: if the sale price is $35 and I got 25% off what was the original price?  

With that in mind, I plan to do some Estimation 180 where, after their estimates, I give them a percent value like 10% or 20% and have them figure out what the answer is.  Then we can "reveal" through the Estimation website to see if they are correct.

Geogebra Interactive

Geogebra has a great interactive tool by Duane Habecker using the Singapore Bar Model.  This is a great way to let students explore with numbers.  Notice the numbers, proportions, table, etc as you increase and decrease the percent, the part, the whole.

I was learning EdPuzzle and created this video.  Since I was just learning it isn't the best, but it gives you some idea of the exploration that can be done with this tool. 

Open Middle

I love open middle problems because they allow the students to try different numbers and see what happens.  It is important for students to notice patterns and relationships instead of just being told what they are.  I created these differentiated problems to use after a formative assessment.  Based on where students were with their understanding of percents and the linear model I gave them a different page to work on.  Here are some pictures so you can see the range of problems.  (The last 3 come directly from www.openmiddle.com.  You can find them if you search percent.) 

These worked very well, but I needed something more for those that were still struggling to understand.  Here is what I created. (I did submit these to Open Middle so they may appear on that site at some point.) There are different questions that can be asked with the first few number lines.  You will see those in the linked document.

These are great visual representations, but some students needed something more concrete as they worked.  Enter the cubes.  With the Open Middle number line, I filled in the 100% value.  I kept it reasonable to use cubes and evenly divisible.  Students connected the cubes to represent the full number line.  Then they partitioned the cubes to represent the number of sections in the number line.  

What I loved about this is that it worked for so many levels of division and fraction understanding.  It was accessible to all of my students.  Some knew exactly where to separate the cubes.  Others broke them into random sized groups, knowing how many groups they needed, and stacked them up so they could see if they were the same height.  With the cubes matching the number line, students were able to fill in the rest of the linear model.  Then we just repeated with a new whole.  The blank boxes of the problem also allow for starting at 25% or 10% and having the students build the cubes to find 100%.  

Our textbook does a great job at the beginning of the school year of sprinkling number line problems in the homework.  Students have to fill in missing values on a partitioned number line and find various distances on the number line.  Using the cubes is something that I plan add next year to that skill for students who are struggling with those problems.  

In conclusion

So that is the long and the short of it.  At this time I do not have a recommendation of exactly how to implement these and in what order.  It was a lot of experimenting for me this year and most likely it will be different each year.  Hopefully you can find something useful here to engage your students and deepen their conceptual understanding of percents.  Please feel free to ask questions or share other activities with a linear model in the comment section.