Linear Equations – Plotting and Predicting Measurements

Given someone’s height, I wanted my students to see how well they could predict other measurements (like head circumference, arm span, foot length, and hand length).  They used measurement data that they had collected (their very own measurements), and then used scatter plots and lines of best fit for their predictions.

We are learning about graphing, including functions and linear equations.  They plotted ordered pairs the first day, and then learned about functions.  Now it was time to investigate linear equations.  This is an amazing part of mathematics not only because it is so interesting, but also because it is useful!

Plotting the Data

Plotting the Data

I have been told that students grow like weeds in middle school.  Since the beginning of the school year, my students have been recording their measurements every couple of months.  They measure their height, arm span, head circumference, foot length, and hand length in inches.  I had them fill out a Google Docs Survey which then organizes their data into a Google Docs Spreadsheet for us.

In class I had the students measure each other again so that recent measurement data could be included in our data set.  For homework, they were to enter their measurements onto the survey and measure each member of their families height in inches.  At the start of class I gave every student a piece of graph paper, a ruler, a worksheet, and a computer.  I then assigned each group a data set to analyze (head, arm span, foot, or hand).  We were making all predictions based on height (the x-axis), so everyone kept the height column and one more data set (the y-axis).  They duplicated the GDoc data sheet and deleted the columns of data that they were not assigned.

In order to determine the scale for the x-axis and y-axis, they calculated the range for their data sets.  This led to a great discussion about how this range would only make a good scale if we were just measuring middle school students.  But, we also wanted to be able to predict the measurements of people of other ages and sizes.  So, we expanded our range to find a more optimal scale that would cover all of the people that we had measured the night before.  We then discussed scale and counted the grid lines on our graph paper to determine how to number our axes.

After our axes were labeled and numbered it was time to plot points.  I let them plot for about 10-15 minutes, which was not near enough time for all of our data!  This needs to be a two day activity next year to get the most out of it.  I explained the line of best fit, and showed them a picture of a scatter plot with a line of best fit drawn on it (from Excel).  I had them draw what they thought was a line of best fit through their data points, and instructed them to draw their line across the entire paper so that they could make predictions for various heights.  **  When drawing their lines, be careful to instruct them not to cross the x-axis, as then their line will intersect the y-axis below 0.  People cannot have negative measurements! **

To show them how to make their predictions, I had them fill in the height data that they had collected the night before from members of their family.  They plotted their family members height on the x-axis and labeled them.  I illustrated how to go up to the line of best fit, then go over to the y-axis to predict their family member’s other measurement.  They recorded this and for homework tonight they will find the actual measurements of their family to compare.

Enter Mystery Guests!

This was our favorite part!  I arranged for 4 – 6 mystery guests to come in at this time.  I wanted to measure people of varying heights and ages.  Our mystery guests included students (1st graders, a 3rd grader, a 7th grader, some seniors) and teachers from our school.  We measured their heights and then predicted their other measurements using the students lines of best fit.  While the students worked on their predictions, I had the mystery guests measure each other to find their actual measurements.  The guests then walked around the room to see how close the students predictions were.

The mystery guest portion was not only the most fun, but also the most informative piece of the lesson.  We were able to discuss why some measurements were more accurate than others.  They discovered that a big factor overall was that our data came only from middle school aged students.  Thus, our data wasn’t as accurate of a predictor for people that weren’t “middle school size”.  Also, they determined that some factors were much better predictors overall, such as arm span.

For homework, they were to go home and measure their family members actual measurements to see how close their predictions were.  I also had the students write a reflection on our class wiki.

We had so much fun with this activity but it was a real crunch to get it done in 50 minutes!  I will definitely do this again next year, but I will take two full class periods.  I would like to go deeper into numbering the axes, spend more time plotting the points, and have more time at the end to analyze the results.  Next year we will analyze our data, spend more time plotting points, and then work on our families measurements the first day.  Our mystery guests will open day two!  From reading the students reflections, it was clear that they really enjoyed this activity too!  Win-win!

Solving for Y with Cups and Kisses!

My students beautifully tackled two step equations with ease, showing all of their steps along the way and solving to perfection!  So, when it was time to graph, I decided that we would spend a day solving for y before learning all of the different forms of the equation of a line.  Sometimes as a teacher, you assume a new concept will be easy for your students.  However, my students are treading through the new waters of Algebra, and what seems obvious to me is not always obvious to them.  After I spent hours making a fun “Solving for Y” Jeopardy game so that would could spend an hour solving equations for y with glee, we instead spent an hour of misery.  The kids were tortured by the problems that they thought were “way too hard” simply because they did not understand what they were supposed to be doing.

My mistake was that I assumed since they could solve for x in two-step equations so beautifully, that throwing a y in there would be no problem.  What I missed was that unlike students I teach in high school, these students had never solved equations with two variables before and simply did not know what to do with them.  Can you add them, subtract them, put them together?  Solve for y?  THAT is not an answer!  We are used to seeing x=12 as an answer, not y = 2x – 5.  Ohhhh….. Finally my light bulb went on.

I felt so bad.  I wasted a day of learning (for them, as I had learned a GREAT deal).  I was desperate for them to see how easy it was to solve for y, and let them get lots of practice at solving for y at the same time.  But, I had already lost a day, and some of their trust.  They did NOT like this “solving for y”.  I had messed up.

So, what do you do when you mess up and want to make it right?  Well, give them candy of course!  ; )  Thanks to an amazing brainstorming session with the wonderful, brilliant teachers on Twitter a new plan was devised.  We would solve for y in the style of “Hands On Equations” with small Solo cups, Hershey kisses, and integer counters.

I used two colors of Kisses (one for +x, one for -x), solo cups for y (sorry, no -y’s), yellow and red integer circles for the numbers, and my “Hands On Equations” equation mats.  I made a worksheet (at end) and modeled the equations with my new best friend, Elmo.  I had them model EACH equation, and write exactly what they modeled (in equation solving form) on their worksheet.  THEN, we moved on to solving without cups and kisses.

**  The BEST part of this activity was by far the cups!  **

When I had 3y = 6x + 3, I asked them, if 3 cups equals 6 Kisses and 3 chips, then how much does EACH cup (or y) have?  By using cups they could actually put 2 Kisses and 1 chip in each cup.  An unexpected plus of the cups was that the students did not once forget to divide BOTH the coefficient of x AND the constant by the coefficient of y!

Solving for Y with Kisses and Cups

Another highlight of the activity was the last problem.  It came down to 4y = x + 4.  I was so excited to see the students realize that each cup got 1/4 of a Kiss!  They discovered that they had to split the Kiss into fourths to put an equal amount of Kiss into each cup.  So, the answer was y = (1/4)x + 1.   My HS students in the past have always had trouble extracting the slope when they come up with y = x/4 +1.  I am really hoping Kisses help them greatly with this!  We will see this week as we delve into finding slope!  : )

My favorite part of the whole day was hearing kids say, “Wow, this is so easy!”.  Their favorite part was eating the 6 Kisses I allotted per student at the end of class.

Solving For Y with Cups and Kisses Activity Sheet