Line Week - Day 4

Day 4 of  Line Week; today we practiced identifying slope and intercept from graphs!  First a side note, I usually test each Friday - I consider this to have a couple of benefits - one it allows me to pull lots of review problems all year long not allowing any topic to lose mastery (cause if you don't use it you'll lose it), two it gives me needed chair time.  When I teach I am up with the class, their doing, I am guiding, discussing, sometimes showing; Friday tests allow me to plan, grade and handle whatever is pressing (today it was ordering MS math team t-shirts). 

So I broke with tradition a little and shortened Algebra's test so we could work on identifying slope and y-intercept as a class.  We plotted a few lines, discussed slope and reinforced the last few days (then a "half-test").  Day 5 is Monday - wrap up day -- I like that it will happen after a weekend, it lets us review and discuss the concepts again.

The project over the next week is create two real world problems and graphs, one representing a positive slope situation and one with a negative slope.  I would expect problems like "John had $20 in his wallet and did Frankie's paper route for $5 per day, how much money does John have after five days?"  with a graph. It is a start (also all my projects are done until correct - so if it takes multiple revisions cause a student does have understanding they will work at it til they get an acceptable "product"  - like a workplace).    The real challenge is to take the next step, perhaps Line week 2, where we take data and decide if it is linear (something to add to Geometry?).  

So not a bad week, next week we will practice and I will assign "line homework" next week.  I never assign homework of the concept we are learning -- homework is practice of "known" things.  So even during "Line Week" we did not graph lines outside of class.  They practiced factoring....

Til Day 5.....

Line Week - Day 3 -- Identifying slope & y-intercept

So today was day 3 of "Line Week."  (insert proper theme music here)  We were in the computer lab again using our self-made excel sheets to graph different lines and look for relationships (see day 2 post).  We tried a Jeopardy game where I graphed a line and we identified the slope & y-intercept.  It worked okay with me graphing on the computer and projecting for the class, but did not work well when the students broke into groups of 2 and 3 to do with themselves.  We worked through it, but it is not something I will do again.

We then discussed a graph about money and days!  I graphed the line y = 3x + 12 in the first quadrant, with the x-axis being days and the y-axis being money.  They quickly saw the y-intercept of $12 and the end total at day 7 of $33.  I then asked how much money I was paid --- and the shout of $33 with quickly yelled! Then I got to say "Nope, why?"   That was awesome!  We worked our way thru the problem - starting with the slope of $3 per day, or the fraction $3/day.  That led to the discussion that $12 was the amount of money I had prior to the seven days, leading to the answer of $21 for 7 days.  This was a great entry into the mx and b having to be like terms to add discussion.

I asked for units on m, x and b --- without much effort the class was all able to explain the units of m-$/day, x - days and b - $.  I then worked thru mx -- $/day times days -- which is $!  The same as b!   That then lead to the  discussion of what is y = mx + b.

y is the vertical point, m is RISE/RUN (from our day 1 work with the stairs), x is RUN and b is a RISE.

So y = mx + b is RISE = RISE/RUN*RUN + RISE, where the RUN cancels, so RISE = RISE + RISE!  Like terms.  This lead into a couple minutes of talking about how we should think of m.  Again big ideas and fairly good returns.

As always - when the really good learning started it seemed the class ended, but will pick up with Day 4 tomorrow and Day 5 on Monday!  A weekend will allow for some good review of the previous week on Monday.....

Line Week -- Day 2 - Graphing slope and y-intercept in Excel

So today was day 2 of Line week and it went okay, but not as well as the first day.  We spent our class period creating a spreadsheet in microsoft excel that had 2 lines where we could easily adjust the slope and y-intercept of each line.  I was hoping we could do that in 20-25 minutes, it took about 30 minutes, which left only 15 minutes for plugging in different values (here is the file).

We started with shifting one curve over the other (using y-intercept), making one curve steeper (using slope), one student quickly discovered parallel lines, and so on.  We were able to play around and some students started noticing how the y-intercept effect the "vertical position" of the line (yippie!).  When I asked how they could guarantee the lines crossed in our viewing screen they figured out the idea of using positive slope in one equation and negative slope in the other.   (Tomorrow we see if they figure out using the same y-intercept!)

The graph started to lead to good discussion and reinforced the slope lesson, but we were not able to finish with why mx and b are like terms, which I hoped to do per my previous post.  We will need to pick up tomorrow with placing units on our equations to help give it a context!

So it did not go as well as hoped, but we still created an excel spreadsheet that can now be used to solve 2 equations for 2 unknowns where the ordered pair are integers between -6 and 6!  And we will continue tomorrow with line week - using our spreadsheet to find more relationships and make/test hypothesis.  Then we will move to looking at plotted lines and figuring out formulas -- I am thinking a Jeopardy format - I plot the line on screen, they answer the equation in the form of a question - "What is y = 3x - 2?"

Overall another good day!

Line Week! Day one - Slope!

So today was the first day of "line week."  We started by breaking into teams and sending them out to staircases through-out the school to measure how 'steep' they are/were.  I did this without any direct instruction on how to measure steepness, it was the first thing we did in class.

I only allowed about 15 minutes and each team had to post their solution prior to the end of the time.  The answers were all over the board and we spent time discussing and revising but we concluded that rise and run were the critical components!  That allowed us to move into lines and graphing stairs of a steepness ratio of 1 - which we then defined as slope.

Then  we were able to graph stairs of 1:1 rise to run, 2:3 rise to run, and 3:1 rise to run, etc.  Then I graphed slope of 1 and another line of slope -1 and we discussed if they were the same.  We then made some analysis of the lines with the idea that the x-axis was class periods and the y-axis being the money they had in their possession (to receive, positive slope or pay, negative slope).  That worked really well!  That ended the period and we are now moving into the computer lab to use units to tie slope to y = mx + b using excel.  My goal is to have the students see that by multiplying mx you get the same units as b.....  And Why that allows us to add (like terms!).


Day one was a success....  (playing the theme from Jaws did not hurt either)

Line Week! Slope, slope, slope....

Slope & lines, lines & slope.  It is  truly one of the most important concepts in Algebra - if truly understood many things are easy, if simply memorized it leads to many forgetful problems.  So with that thought in mind I am setting up "Line Week" -- like "Shark Week" - except cooler and many more attacks resulting in gruesome death (students love the joke...).

So I made the joke(s) last week, and now I sit here Sunday morning trying to create the lesson that really results in students understanding of slope, intercepts, etc.  I teach 8-12th grade - so these students have seen (and forgotten) lines, they have been using x,y tables to plot points and create lines -- but with no real world understanding of slope etc.  {You may think what kind of person promises "line week" without a plan...  - well I do!  Lines are important and it is time for lines!}

I have spent the last 2 hours combing the internet for something (and if you pictured Mel Brooke's Space Balls combing the desert you're my kind of person) and have really not found anything amazing.  I have founded some fun songs - that I may use once they understand.  And some good worksheets, I may use those once they can do at home alone (in a month) for practice.  But nothing that really ties Rise over Run to y =mx + b....

My plan is to use a stair step example that I read about on I Want to Teach Forever, but then I want to tie slope and the slope-intercept equation.  It is units that will do it!  Perhaps it will be dollars per day for slope, thus using x as days - mx will become dollars...  b is dollars initial and thus with some practice the two will make sense.  That is the tie, seeing that mx and b are the same units (thus like terms).  Then zero slope is no money, and undefined slope is winning the lottery.... or being robbed.....

Hopefully there will be another post of something amazing, or at least acceptable...  Cause from combing the internet there seems to be very little that teaches the concept - just simply the procedure.