Slopes Aren’t Always Positive

Unit 3 lesson 9 today (11/9/19) in grade 8 was overall a success. The cool down was a bit brutal with students making slope triangles without lattice points and drawing a line through a point with a slope of -2 but not making horizontal correct, overall it was a success.

It starts out with one of my favorite WODB prompts which allowed me to dust off my sentence frame anchor chart ( I believe in seasonal charts so when we are using that routine it comes out of storage).

I’ve actually used this exact prompt at Back to School night at the beginning of the year with parents and at the end of the year’s Open House to model one of our routines. They talk about steepness and some remember slope. The most unique answers I’ve seen is s is the shortest segment.

At this point in the unit, according to the Course Guide, students are expected to be producing the language of slope by lesson 5. So, it is possible they may say line V has a negative, decreasing, downwards slope. According to the lesson narrative, they are not expected yet to say positive or negative slope.

The launch to the first activity is awesome. It provides the context of fare cards on a subway. My students were able to relate to this because I said, “raise your hand if you’ve ever ridden on BART?” So, students were able to offer how you buy a fare card and then when you put it into the machine it takes off money. I made sure they knew that in this problem it was a flat fee per ride, unlike BART and other systems where they take off more per ride based on how far you travel.

In the activity, students find out how much money is left on the card after 0, 1, 2, and x rides, graph the results, and answer how many rides Noah can go on before his card runs out and where do you see this on the graph.

In the synthesis, you ask students, “why does it make sense the slope of your graph is -2.5 rather than 2.5?” After discussing that the amount on the card decreases by $2.50 every ride, and the points are decreasing on the graph, you ask where we see when the card runs out. This brings up 16 rides. “What do we call that point on the graph?” Many students will remember “x-intercept.” Also in the synthesis, you bring back the graphic from the warm-up showing that two lines have a slope triangle with a vertical of 1 and a horizontal of 3, but they are not the same slope. One is negative 1/3 and the other is positive 1/3.

Since many students struggled with the expression with the variable, I made a point of asking students exactly how they calculated 1 ride, and wrote 40-2.50*1 and 40-2.50*2 to emphasize the structure of the x being the number of rides being multiplied by the price per ride. It’s important to increase access to this abstract thinking for all students to see this pattern and be able to reason about x rides being 40-2.5x as seen below:

We also got side tracked a little bit about mental math when students were getting 40-2.50 incorrect. I think some may have got 38.5, so I talked about how you can subtract 40-3 to get 37, and since 3 is .50 away from 2.50, you can add the .50 back on.

In the next activity, students are asked to notice and wonder about this graph:

Screen Shot 2020-03-03 at 10.00.41 PM

Then students describe what is happening with the graph, ensuring that the x axis is no longer number of rides but days passed. They also plot and label 3 points, write an equation, and write down what makes sense for the slope of the line.

Some students see that all the coordinates share a y coordinate of 20, so the equation is y=20. Many students did not see this. I encouraged students to make sense of the structure of the first equation, y=40-2.5x, and how it could relate to this new graph.

And this is where a very proud moment happened. A student of mine that is frequently tardy, disinterested, uses the restroom every day, gives little effort, and tests my patience a lot came up with equation below: y=20-0x with no help from myself or their peers. I was so proud that I wrote the person’s name on the board. I also took the chance to call their family with a positive phone call home to show that they showed me that when they give forth their best effort that they showed they can achieve.

Students realized that the slope must be zero because it’s not increasing or decreasing, so it could be y=20+0x also or even y=20 since anything times zero is zero.

There is an optional activity after this, and then a really awesome lesson synthesis. The instructions are “Ask students to pretend that their partner has been absent from class for a few days. Their job is to explain, verbally or in writing, how someone would figure out the slope of one of the graphed lines. Then, switch roles and listen to their partner explain how to figure out the slope of the other line.” This is really awesome to listen to. It’s amazing formative assessment to see students talk about what they learned using a graphic that’s not in the book, so no one has a head start on it. One line has a slope of 3/4 and the other is -1/2, so as the teacher we are able to walk around and eavesdrop during the synthesis to call on people to share with the class. Finally, the cool-down is calculating 2 slopes, interpreting what a slope of zero could mean, and drawing a slope of -2 through a given point. This last question was quite difficult for students.

This is one of my favorite lessons, and a memorable day in my classroom, way back in Unit 3. I like this lesson because the context is culturally relevant. Students can relate to it. A fare card, money, thinking when the money will run out, etc. Also, writing equations is a really important skill that they’ve worked on in 6th and 7th grade and will continue to do in 8th grade and beyond.

Author: mrjoyce180

6/8th grade math teacher using open up resources tweeting from@martinsean

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s