Category Archives: High School Math

Point-Slope

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I honestly have no clue how many times / classes / students I’ve taught Point-Slope as a rote formula to memorize, plug in values and simplify. ¬†No discussion, except that its derived from the slope formula.

But after some discussion / great convo on Twitter, I’ve arrived at the solution for now.

To introduce it simply as a vertical translation of a direct variation graph.

https://www.desmos.com/calculator/nb8le3gwnr?embed

Why do we add here, but not here…

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For some reason there were 6 kids in after school with me…all working on different topics and courses. Yikes. I had to be on my toes that day.  

One student in particular was working with polynomial operations.  Why do we add the exponents on these… but not these…?

Well.  I paused. Why do we?  That expert blind spot…I know that’s what we do, but why?  We aren’t actually adding exponents but that’s what appears to be happening. Help.

Hmm. Well.  Uhm.  Flashback to my daughter’s 2nd or 3rd grade math.

Sort of like place value.  The math doesn’t work because of the rule. Someone made up the rule as a short cut because that’s what happens when we do the math…

And this is what I began with…

Too often students see polynomial expressions as some obscure thing. Have we failed them by not encouraging them to see them as numbers?

Please share how you address this same question, why do we add exponents here, but not here?

 And saying that’s the rule is not acceptable. ūüėä

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…

Monster Trucks #teach180

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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.

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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.

Balance Benders

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I have good intentions of hitting the ground running add term Christmas Break.  A day out for a water line break and our first snowday baffles those plans. 

Today I had students finishing the MAP test, others doing wrong answer analysis on their midterms, others recising their NTI assignments and a group who had completed all.  The latter group, I circled up some desks, said move over here, bring a pencil and gave them packets from Balance Benders!

My instructions… these shapes represent quantities. See of you can figure them out.  There ended up being 6 students in that circle, but it was fine.  Good discussion, critiquing rhe reasoning of others, checking their own work.  Wow.  Excited discussion. 

I need to locate my level 3, these kids will be ready for it soon!

An Update on a Starfish 

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Heather shared The Starfish Story which will always be my favorite. ¬†Here’s ¬†a link to a day I needed that reminder…

But the post below…needs an update. ¬†Later that year, the student emailed me to let me know they had completed their credits and graduated. ¬†In the Spring 2016, their sibling let me know they moved to Alabama before they would leave for Basic Training in the Air Force last that year.

Just before midterms week last semester, I received a phone call – from the front office asking if I was available to meet with a visitor. ¬†This young man stopped by to see myself and a colleague. ¬†They were home for ¬†a while before transferring to their new assignment. ¬†Here’s what made me most proud. ¬†They were in fatigues (I apologize if that’s not the correct wording), standing proud. ¬†But when jROTC students walked by, shook his hand, saluted him and thanked him for his service. ¬†I couldn’t help but tear up because the respect this young person had always deserved was given to him. ¬†I was so proud for him – that these young minds saw him as a leader.

From August 2015…

Part of an email I received today… Thank you for the time you spent on me! You reinforced my belief to always challenge myself, one of the reasons I am currently enlisting in the Air Force. T…

Source: One Good Thing…