# Not all Relationships are Proportional – Making sense of y intercepts not equal to zero

### Warning – There are so many chances to over-teach in this set of lessons.  Stay focused on what you are trying to accomplish.Your main goal here is to introduce students to situations where the relationship is not proportional.  Live in the context of these situations and help students think about what that y- intercept represents in that situation.

Maybe this wasn’t a problem for you, but for many of the teachers using this curriculum for the first time, the Ant and Ladybug problem from Lesson 3.1 felt awkward. Lots of us found it uncomfortable that time was on the y axis.  In our experience, time is usually the independent variable, recorded along the x axis. We had years of experience with time measured along the x axis. Some of us had told previous classes of students that time was ALWAYS along the x axis. So when we ran into this problem that challenged what we had always seen, it was hard to adjust our thinking. We have mostly worked through those issues now, but my point here is that it felt awkward, because it was not what we had seen before, and we had to spend a little extra time thinking about and making sense of the situation.

Your students are about to have that same experience. They had no trouble with the Ant and the Ladybug, because they didn’t have the weight of previous experience to get in their way.  But non proportional relationships, lines that don’t go through (0,0)? That just feels wrong. Just like us, students need to spend a little time in a context to help them make sense of this unfamiliar situation. Lessons 5-7 are meant to let them do that

# Lesson 5 – Stacking Cups

My advice is to run activity 5.2 with the book closed.  I love this set of slides made by Morgan Stipe.1 We’ll look at a few to help you imagine the conversation you are leading. (You’ll have to imagine the animations).

This first slide starts as “low floor” as you can get.  Everyone should be able to engage with counting the cups and agreeing on the measure indicated by the ruler.  Then everyone is invited to think to themselves:

“If 6 cups is 15 cm, how tall will 12 cups be?”

The baiting to use proportional reasoning is strong, and students will almost all say 30cm. The

The moment of “Wait, what??” when it doesn’t work out as they expect is the setting of the hook.  Now they are invested in figuring out what is going on. After a little individual think time student should work in partners or groups to try to figure out what is going on, and then create an estimate for number of cups in a stack that is 50 cm high. Groups may benefit from having 3 cups to manipulate as they think about the situation.

This slide shares the final answer and serves to focus us on the purpose of this activity in the lesson:

The next activity brings back the term rate of change that they learned in lesson 3 and connects it graphically to the slope of the line. This one I had students work in their workbooks. To close the lesson they learn the term linear relationship can be used to describe any relationship between two quantities with a constant rate of change.

No equation writing yet. Just “hmm, it’s possible to have a relationship that is not proportional. Some don’t go through (0,0), but they still have a constant rate of change, which is the slope of the line.”

# Lesson 6- More Linear Relationships: Slopes, Vertical Intercepts, and Graphs

The warm up gives you a chance to continue to talk about rates of change. Use the opportunity to connect the idea that some patterns they notice are growing at a constant rate of change and others are not.

Red (+1, +1, +1) constant rate of change = 1

Blue( +3, +3, +3) constant rate of change = 3

Yellow ( +1, +3, +5) not a constant rate of change.  If we graph this it will not be linear.

The bulk of lesson 6 is a set of situations that students are supposed to match to graphs.  For every teacher who says there is not enough practice, here it is – a day of practice.

Because students have a fairly strong understanding of slope from a graph and a tenuous understanding of the meaning of the vertical intercept, matching is more easily done focusing on slope.  After they are matched, have a discussion of what the vertical intercept means in this situation. Morgan Stipe’s slides are again a wonderful resource.

At the end of the slide deck, Morgan has some great slides to consolidate and bring the learning together. (Keep imagining those animations. Answers appear after questions).

# Lesson 7 – Representations of Linear Relationships

Activity 7.2 gives one more opportunity for students to think about a situation that is not a proportional relationship and make sense of both the slope and the vertical intercept (aka y-intercept). You have the choice of running the activity as a lab or as a whole class demo that you lead.  7th grade teachers will tell you, labs are memorable experiences you can refer back to for the whole year. ( Drink mix anyone?2) But running it as a lab will take about twice the time and require gathering equipment.

The digital version linked in the curriculum is great (pictured below).

Activity 7.3  practices finding slope from 2 given points, and generalizes the slope formula in question 2 and 3. It also refers back to writing equations of lines as we did at the end of unit 2, using this slope formula.

If you are condensing or combining lessons for block days, these two activities could easily be done on separate days.

Coming next in Lesson 8: Connecting all of this (along with unit 1 learning about translating lines) to express equations of linear relationships as y = mx+b, where m is the slope and b is the vertical intercept.

1 Slides by Morgan Stipe are available for each lesson. The are linked on the teacher page under Community Created Resources.

2 Mystery Mixture lab from Grade 7 Unit 2:

## From deep in my archives – this is a lesson I used long ago to motive simplest radical form. It was one of my early ventures with Math Practice 7 – Look for and make use of structure.

So you are about to teach special right triangles, and to do that you want students to understand simplest radical form. The problem with simplest radical form from your students’ perspective is that it is a needless complication that does not add any meaning to the situation.  7.071 is much easier to understand as a quantity than 5√2. But we know that the patterns are easier to see if they use simplest radical form.

So . . .

What if you started with trying to get them to notice the pattern in the sides of a 45-45-90 triangle using decimals?  Here’s how I envision a possible intro lesson going.

1. Give them two angles in each triangle and have them notice they are similar. Then give them one side on each triangle and have them find other two sides (isosceles, pythag).

Here are some possible triangles you could use, and the answers they would get in decimal form:

It will be handy if the Pythagorean theorem work for these is on board to come back to, so you could have students put up this work.

1. Teacher says: “Using the patterns you see, can you predict the length of the hypotenuse for a 45-45-90 triangle with leg =7? No pencils allowed – predict in head.” Take guesses . . . .too high, too low, just right

Now use math to check who was closest.  Answer: 9.899

Students may use similar triangles to set up proportions and solve between triangles if they have worked with those recently. That is fine.

1. Tell students: “In this next unit we will be learning how to use the consistent ratio between two sides of the one triangle (for instance hypotenuse to leg in the ones we have been looking at) to find all the angles and all the sides of any triangles similar to it. So what is the ratio in this case?” Have students find.

1. Teacher talk: So decimal is consistent but not catchy. Could we have used it?

7(1.414) =9.898

Close. Off due to rounding.

Perhaps play with including more decimals to increase accuracy.

Would be nice if there was a way to be more exact without having to write out hundreds of decimals.

1. Tell students: “There is another way to express radicals to make pattern easier to see and use.” Go back to pythag work on board and break down the square roots. I would suggest for this moment that you use the greatest square method so they can understand what you are doing.

√50= √25∙2=√25∙√2=5√2

The point now is not to teach them how, but just to let them see helps see the pattern, so that they might see some value in learning how.

triangle 1:  5, 5, 5√2         triangle 2:  3, 3, 3√2         triangle 3:  4, 4, 4√2

1. Teacher: “Anyone want to guess the hypotenuse for a 45-45-90 triangle with leg 7 now? No pencil!    Answer:  7√2 !

How accurate is that?”

(Show them how to use calculator to check.) Pretty darn accurate!

“Can anyone guess the decimal  for   √2  ?”

1. Transition . . . “Before we look for more patterns, let me teach you how to rewrite square roots to help you see the patterns.” Or if you’re on a roll and feel experimental, cut a equilateral triangle in half to make a 30-60-90 and see if they can find the 30-60-90 ratio themselves in radical form. They may end up “inventing” how to break down simple square roots all by themselves.