Category Archives: Algebra I

I Still Have a Question About…

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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.

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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?

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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.

Identifying Linear Functions

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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.

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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).

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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.

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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.

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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…”

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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…

Stacking Cups… part 2 #MtbosBlogsplosion #myfavorite

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I like big cups, I cannot lie.

We stacked cups in the first few days of school…

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I’ve been stacking cups since…uh.  I think my first NCTM Navigating Through…  book was around 2002 or so.  Its been a while.  I have vivid memories of discussions in classes from room 125.  Yep.  It’s been while.  Long before there were songs about Solo cups.  My guess, a few of my sets of cups may be that old.

They’re a cheap resource.  Find a buddy or two, each buy some different sizes, split them up and you’ve got some varied sets of cups.  Hmmmm. What all can you do with cups?

I.  This past week, I began by displaying a single cup and asking students to generate as many questions as they can about said cup.  Set the timer.

II.  Turn to your groups and share your questions.  Then say whether it was mathematical in nature or not.  Each group shares out 1 question with the whole class.  Then if anyone had a question they wanted to share that had not been included.

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Yes, we actually looked at the etymology of cup…wondering where the name originated.

III.  a.  I went with “Why am I stacking cups?” as my transition to the task.  You guys are engineers today.  Packaging designers, specifically.  Design a box to ship a stack of 50 cups.  They needed tools, so I gave each group 4 – 7 cups (did I mention some of these cups may actually be older than some students?), each group with a different size/brand of cup and a measuring device.  Set the timer 5-7 minutes depending on class.

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III b.  As I monitor their work, I usually here a few moving in the wrong direction.  I pause the timer and their discussions…attention at the board:

I need some help.  One group has a stack of 5 cups measuring 14 cm, and their height for a stack of 50 cups would be 140 cm.  Do you agree or disagree with their response?  Turn to your group and discuss.  Set the timer.

I have some varied responses usually.  When I get to someone who disagrees, I ask how tall they think the box should be and they come to the board to explain their reasoning.

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III. c. Yes, believe.  You will sometimes have a class where no one disagrees with the 140 cm response.  Have them to create a table of values to record their measures for 1 cup, 2 cups, 3 cups, etc.  Set timer.  Usually during this time you will hear the a-ha’s.  Bring the class back together to discuss / share their thinking.  Modeling how the cups would be stacked.

Okay, so moving on now.

IV.  Once we feel fairly confident in our expressions. I ask them to find the height of a stack of ____ cups for their group.

V.  Well, what if I had a box that was 80 cm tall, what is the largest amount of cups could I ship in that box?

VI.  At that point, we share our expressions we’ve created for each type of cup.  I put all cups on display and ask groups if they can match the cup with its expression for  total height (cm).

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This leads to some light bulb moments for a few students.  They can now see how different parts of the expression represents different physical parts of the cup.  I always thought it would be fun to list the expressions on cards and they have to match to the cups and play the Race Game from The Price is Right.

VII.  For other practice, we use the expressions:

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  • simplify expression
  • find the total height of 50 cups
  • how many cups to make a stack of 80 cm?

VIII.  Closer choices

  • What’s one take-a-way from today’s task?
  • Something I learned… realized… or was reminded of…
  • How are the expressions alike?  different?
  • Which two expressions are most alike?  Explain.  Which two are most different? Explain.

IX.   Systems

Next, have students compare their cup stack to another groups stack of cups.  When will the two stacks be equal heights?  Just using my groups’ expressions above, they get at least 6 practice problems.  You can leave it as an open task – students can choose tables of values, creating equations to solve or even solve graphically.  The key component is to ensure they interpret their solutions (x, y) = (cups, stack height) within the context of the scenario.

A Light Bulb Moment #MTBoS30

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Here’s a shameful post – one of those things I knew it happened, and wanted to believe I knew WHY it happened, but in reality…I was struggling.  Until yesterday…  in after school, tutoring a student for upcoming comprehensive final.

I know how manipulating an equation will transform the graph of the function.  I can predict it quite efficiently.  I know and my students even recognize that y=a (x-h)^2 +k will translate left / right… “opposite” of what the sign / operation is in the parentheses with the h.  But why?

So one day, as I heard myself describing the transformation to a student, I thought…that sounds so stupid.  I mean, hello.  No wonder it doesn’t stick.  It makes no sense (what I had just said).  In my mind, I heard Pam, the student, asking why do we change the sign of the h, but not the k?  Why does the h move opposite, but the k translates direction of the operation?

I started really making more sense to myself when I focused on function transformations in Algebra 2 and transformations for plane figures in Geometry the past couple of years.  But I was definitely not satisfied with what I was saying.  I believe our work with equations of circles related to slope and pythagorean theorem is what started chipping away my lack of true understanding.  Because I began to explore, ask questions.  I was curious.

When I started having students create tables of values, seeing how the values changed with each transformation helped, but not to the level I’d like.

So, here it is folks… when we’re looking at the y=a(x-h)^2 + k…the h is actually NOT the x-coordinate of the vertex.  The h is the transformation back to zero (origin).  Can we look at it that way?  Does that even make sense?  The x-value is where we moved from the origin.  The h will return us back to the origin.  I know its not where I need to be yet.  But I’m open to listening to other’s ideas here.  I’m not satisfied with “it moves just opposite of what we think.”

My next failure as a teacher saga…I don’t do a good job of helping students differentiate between linear functions and arithmetic sequences.  I’m starting to muddle an understanding.  At a moment in time, they are comfortable with each idea, but they continue to mix up when its a first term, n=1 OR an initial value n=0.  The best I can do for now, verify your equation works for the values…

Landmarks & Falling Objects #MTBoS30

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Algebra I Quadratic Equations

We have prom this weekend, FFA ( a lot of my students ) are gone to Mammoth Cave, so I thought this would be a valuable way to end the week with so many students out and distracted.

Dan Meyer’s 3 Act Falling Rocks

Dan Meyer’s 3-Act Dropping Glowsticks

So, our practice problems consisted of finding the heights of tall landmarks (student generated list) and solving to find the time it would take an object to hit the ground after being dropped from the top of that landmark.

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As we continue working with quadratic functions / equations next week, we’ll take a look at Coyote & Road Runner and the Catapult to introduce parts of a parabola. Post discussing these  files: Quadratic Files.

For more skills practice / workbook 356/357.

Just 1 Second / It Can Wait #MTBoS30

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As an introduction to Quadratics, we’ve been looking at speed and stopping distance in Algebra I.  The formula d = 0.045v^2 is a great way to start practicing solving quadratics by square roots.

We created a table of values for various speeds.  Asked the question, if you double your speed, will your stopping distance double?  If you triple your speed, will your stopping distance triple?  Hmmmm.  Good discussions.  Why/Why not?

For the speeds, 40, 50, 60 mph, which interval had the greatest rate of change?

Was Jake Speeding?


Jake had an accident.  The accident report states the skid was about 222 ft.  Was Jake speeding?  Provide evidence for your claim.  How fast was Jake driving?

Some compared the stopping distance to previously calculated for given speeds, then reasoned where to begin their own calculations.  Again, I am proud of their thinking – no trying to replicate something I had said or done.  I will take their reasoning any day over replicating my procedures and thinking.  There were students who solved the equation algebraically and some students picked up on their classmates procedure, once it was shared.  But it was their thinking, not mine.

Just 1 Second


If Jillian was driving at 58 mph.  How far would she travel in 1 second?  Hmmm.  One would think this was a simple conversion problem, but some students were stuck.  Okay, give me a value you believe is too low.  Too high. Pair share.  I wrote several values on the board and allowed students to confirm or dispute.  Finally, after some sharing, we agreed on 85ish feet.

We step out in the hall and student walks off 85 feet.  1 second.  That’s not much time.  That’s a pretty far distance.  When we return to the room, I play a PSA from ATT Close to Home, It Can Wait.

It Can Wait PSA

I’m using this as part of our Program Reviews (Integrating Arts/Humanities, Practical Living/Vocational Studies, Writing, Global Awareness). Students will include a math fact we’ve used this week in our intro to Quadratics to write a 20 second PSA or create a print ad for Distracted Driving… Just 1 Second.

 

 

Braking Distance and Speed #MTBoS30 Day 2

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8 minutes…here goes…

A quick retrieval from last week’s exponential growth / decay to begin our day in Algebra I. Four functions:

  • y=1200(1.13)^x
  • y=1200(2)^x
  • y=1200(.91)^x
  • y = 1200(2)^-x
  1.  60 seconds:  Tell how they’re alike / different.
  2. State which ones are increasing and decreasing without “doing any math / graphs.”
  3. Graph each, sketch, noting the end behavior…does it support your description?
  4. Let x=3, find the function value for each.  Is it greater or less than the initial value?

Texting Olympics!

Whaaaat?

That’s right, thanks to Heather Khon, we did a little sprinting and hurdles in class today.  

Sprint:  staying for help in math pick me up at 4

Hurdles:  Laker baseball has a winning season!!! woooo!  #lakerproud

Recorded our times and will use them tomorrow as we look at breaking distances more…adding on to our distances how far we’ll travel in just 1 second at various speeds, then leading up to a written PSA, print PSA, recorded 30 second PSA to support ATT’s campaign, Its worth the Wait.

Here’s where we began our new content – a look at some quadratic data.

These lessons are taken from A Visual Approach to Functions, 2002, Key Curriculum.

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  1. Notice / Wonder about the speeds and braking distances.
  2. Create a scatterplot, then run a quad reg because we don’t see a constant rate of change for linear and no common rate of multiplication for exponential.
  3. Some are amused when they see this crazy function pop up and just how well it models their data when graphed.

f(x) = .0975x^2 – 2.5x + 65.25 (its pretty close to this, but I don’t have it written down with me.)   What’s that, you want to know how far you’ll travel at 90 mph, Brooke?  Let’s see.

The lessons go on with varied graphs asking students which models given sentences.  Three tables of values are shared, asking them which is likely the data for their chosen graphs.  And finally, equations – which one matches each data set.

Again, open questions because students were not limited to a single strategy.  Some students chose to use x-values as inputs into the equations and find the one that modeled.  Another group used their data to create a scatter plot and/or regression equations, and others found the rates of change within the tables and compared to determine if linear/quadratic.

All in all, I was please to see such varied approaches – and  correct in their reasoning.  Looking forward to continuing these conversations tomorrow as an intro to quadratic functions.  Maximizing Area will follow braking distances.