Category Archives: linear functions

I Still Have a Question About…

Standard

We did not get through all I intended today to allow some students who wanted to watch the inauguration that opportunity.  But we did address a couple of more questions from the 2-minute reflection students had completed.  You can look back to the previous post to see the original task.

wp-1485006906618.jpg

We addressed the two blue questions in the after lunch class.  Why can’t you multiply the numbers by each other?  Well, lets see.  Again, as I did with another class, I asked them to add two numbers that would give us 18.  We graphed our responses, then graphed the equations x+y=18.  And likewise, give me two numbers that would multiply to give us 18.  We graphed our responses along with xy=18.

When we added the equation to the product set, students were caught off guard with what they saw.  WHY is there a graph in the third quadrant?  Will that red curve ever cross the y-axis?  Doesn’t it eventually get to the axis?  Again, just attempting to address their question, by looking at a couple of horizontal translations and introducing them to that boundary line called an asymptote, led to even more wonderings.  Which is what math class should be about.  As long as they were on task, I continued to go with their questions.  Only 3 students were not interested, who would likely have been off task no matter what I chose to do, so I made the decision to keep going with the majority’s curiosities.

Another student asked about our statement “x cannot be in the denominator” but yet when we find rate of change with a table of values, we compare y over x. Hmmmm.  Good question.  So I gave a table of values, asked the student to talk me through finding rate of change.  When we wrote our ratios, what values did we use?  Not the actual y and x values, but the change in y over the change in x.

The entire class really reminded me that we can say something with an intent, but what our students hear is something else…how important communication is, how important is it that we allow some time to process and clarify their misunderstandings.

Finally, we addressed the question, we’ve been told x’s exponent must be 1 in the linear function.  We’ve seen greater than 1, but if it is less than 1, can it still be linear?  Let’s see.  Go to y=, type in x and choose an exponent less than 1.  What do you see.  Share with your neighbors.  So, how would you respond to this question, students?

wp-1485007603679.png

To me, this was one of the most productive two days I’ve had in this class.  Students were engaged because we were addressing their questions.  I’m not sure I actually answered their questions, but I provided them with some examples that allowed them to answer their own questions.

Advertisements

Identifying Linear Functions

Standard

Linear Functions Organizer this does not include arithmetic sequences, which was earlier in the year, but I can refer back to our work with them to activate prior knowledge for this unit.  The next unit will be linear regression which will include correlation, describing scatterplots, finding regression equation with technology, using the equation to predict and finally introduction to residuals.

Students started with a pre-quiz similar to the one below.

id-linear-functions-pre-quiz

Identify Linear Functions This is a booklet with a Frayer Model for our notes, a variety of math relations to identify as linear or not and a 2-minute reflection grid on the back.  Prior to beginning our notes, I gave them 1 minute to jot down anything they thought they knew about linear functions.  Then we pair-shared before sharing with the entire class.  Then we took our notes. (as a follow up the next day, I gave them 2 minutes to jot down all they could remember about linear functions as a small retrieval practice).

wp-1484862594406.jpg

Our next task was created by cutting apart these relations and posting them around the room with a chart that asked if they agreed or disagreed with the example being a linear function.  Students received stickers to place on the chart as they visited each station.

wp-1484862653898.jpg

I was fairly accurate in which ones I thought we’d have to use for discussion, but a couple really surprised me.  These are the 4 we discussed following the carousel activity.

wp-1484862420975.jpg

I. y = 2x was the one I was not expecting.  When I asked if someone would share their thinking, one student said they thought x was an exponent.  Another shared they did see “the b” for y-intercept.  We looked at a table of values and graph to agree, and show the y-intercept was at the origin and indeed y = 2x was linear.

The other I failed to snap a picture of was graph K, a vertical line.  Yes, it’s linear, but not a function…two students got that one correct in this particular class.

Using the 2-minute reflection grid as our exit slip to see students thinking about the lesson, I was excited about some of their “I still have a question about…”

wp-1484862425668.jpg

On the reflection grid, if they have no questions, nothing is confusing, I ask them to give me a caution…something to be careful or / watch for.  Several of these questions encompass multiple students.  Some of them I only needed to clarify what was said.  Its pretty clear I was not communicating very well on a few of the.  I hear my “expert blind spot” showing up…”Of course squared is not linear, we learned it was quadratic in our functions unit!”  But so many students on the pre-quiz used vertical line test as their reasoning for linear…we had some side conversations about this misconception…that it shows functions, but does not prove if its linear.

Some of the questions, I allowed other students explain their reasoning to help clarify their understanding.

I know I shouldn’t have favorites, but in this list…

Why can’t you multiply the numbers by each other?  We tried it.  Add 2 numbers that will make 18.  Create table of values, find rate of change, graph it.  Yep, that’s linear!  Multiply 2 numbers that will result in 18.  We created a table of values of their answers, found the rate of change and graphed them.  No, that’s not linear!

If an exponent is less than 1, can it be linear?  We will try it tomorrow as our bell ringer.  But I look forward to exploring their questions more!

I told them how excited I was about their questions and posted them on our “THINKING is not driven by answers, but by QUESTIONS” board.  One student had the biggest smile and as she said, Look!  I’m so proud, my question is on the board!  Something so simple, yet, my hopes are that it will encourage her to ask more questions.

One student asked me, but isn’t it disrespectful to ask questions and interrupt the lesson?  Nooooooo.  I love when you ask purposeful, curious questions you wonder about!  Finally, a break-through to get them to start asking and wondering more…

Monster Trucks #teach180

Standard

I grabbed a Monster Truck from my box and asked students to brainstorm all the questions they could ask about the truck.  Pair-Share their questions, then choose one to share with the entire class.  Anyone have any other questions not on the list, you’d like to add?  Every time, someone wonders how fast it will go.

wp-1484257600950.jpg

I’ve use Chris Shore’s lesson Monster Cars for several years in Algebra I, but this was the first year I used a video to hook students as I introduced the lesson.  What does this have to do with Math Class?!?

I can’t wait to continue the lesson next week!  Its grabs their attention a bit more than a typical textbook lesson.

Cool Shoes #onegoodthing

Standard

I struggle to find the balance between just the math and in a context.  That’s why I love modeling with math.  I have for years, tried to provide a context – a hook to grab students’ attention, and reference back to when working with “just the math.”

This week, we’ve looked at multiple ways of writing equations of lines – when given varied information:  Slope and y-intercept, slope and a point, given 2 points, given a point and parallel to a given line.  But its been very generic.  Just the math.  Some loved it. Some hated it.   But none saw a purpose or reason for it.

Anyway, today, we collected data.  Created scatter plots.  We drew a trend lines.  Chose a couple of points…to write the equation for our line of best fit.  Why did I choose to do “just the math” prior to using the lesson hook?  Not sure.  I’m still battling which should come first.  But today, I wish I had used the task first.  Provided a need for the math.

coolshoes

 

It was in a class that we used Cool Shoes from Chris Shore’s big blue book Math Projects Journal (one of my favorites for years!)…that my day was made.  The task uses height to predict shoe size.  In our discussion, I mentioned how online shopping sometimes allows you to click a sizing chart.  A student all of a sudden exclaims – “Thank you!”

Me, “Okay.  for what?”

The student explains – “Finally, I see a purpose for all of this stuff!  A real, purposeful use.  Somewhere this can actually be used, be helpful.  How real people can use these math skills to do something.”  Smiling. Smiling. Smiling.

I went on to share – That’s what math really is…looking for patterns, modeling those patterns and using our models to predict, make connections, etc.  Why do they weigh us when we go to the doctor?  How do they know how much meds to prescribe when we are sick?  What happens when someone has a cancerous tumor?  How do they measure it?  How does the oncologist decide how to treat it?  How do insurance companies set rates?  How do businesses make projections for upcoming projects?

One small glimpse.  A student saw a purpose.  The student smiled in math class (finally).  I smiled.  It was a good day.

 

Linear Equations Card Match #made4math

Standard

Let me first say – I did NOT create this set of cards.  I received them in a session at KCM about 3 years ago.  Kudos to whomever they belong. 

 I was looking for resources to use during my RTI and ran across a box I had used in the past. 

LinearEquationsMatch – the file of the cards.

You can do several different sorts with them.  POINTS-SLOPE, POINTS-EQUATION, GRAPHS-EQUATIONS, etc.

2013-01-14 11.08.59

I have each complete set on different colors of cardstock, so I can have several sets out at once, but none of them get shuffled.