the radical rational…

in search of innovative ideas with a well-balanced approach for the math classroom

What if I Never Taught Factoring

While working in our polynomials unit, a page had some review problems.

Students where given equations similar to:

image

And asked to solve. 
I changed the directions… to graph the expression in Y= and record the x-intercepts.
Then look for connections between the intercepts and given equations. 

Some nice conversations took place. 

After a discussion/sharing, there was some confusion about signs being different from the expressions.  We talked about the location of the intercepts, sharing how we could create factors…which connected back to translations.

Some students shared how to tell if intercepts were on same side of origin or one positive, one negative.

Others shared – if they substituted the x-intercepts back into the equation, the result was zero.

And finally, someone shared how when they looked at the coordinates of x-intercepts (x,0) the y-value was =0. Bingo. Connecting to the functions zeroes.

So, I asked…what happens when we’re given this x(x^2+5x+6)=0?

Their thinking and reasoning can show through… I  just need to get out of the way.

1 Comment »

Combining Polynomials

This is like something that others have done often.  Maybe I have even used something similar at times in the past, but today a group of students had a big a-ha moment.

We have reviewed the basic polynomial operations and like robots they can do it well for the most part.

The last 10 minutes of class, I popped the graphs
f(x)=x+2 and g(x)=2x-1 on the board.
I pointed out specific points, asking for their sum and recorded on the board.

image

2+-1
1+3
3+4
8+8…
This graph only shows part of our discussion.

Next, I graphed a purple function f(x)+g(x).

“Holy shoot” was exactly what a student said. Yep, they edited the version just for me.  But there were several ah’s and oh’s. 

Thanks to Desmos…they could see the math they’d been doing on paper.

We combined f(x)+g(x) on paper to get 3x+1.  I graphed it and they were amazed it hit our points.

I know this isn’t so amazing for many, but the look on my students faces today…they got it.

I am looking forward to more…

1 Comment »

Cubing? #challenge #mtbos #cubinginclass

So my superintendent, @fordmich, shared a list of instructional strategies…

image

Great list, should tag it as a reminder for when I am planning? But cubing? What is it?  So I looked For the Teachers and  some samples.  Oh.  I can do that.

It reminded me of @druinok’s Function Dice and Function Operations activity…that inspired a popular pin  Polynomial Stations. ha, Pinterest even suggested a pin from my own blog the other day… 

I have also used foam dice to generate coordinates on several occasions.

So I am challenging anyone reading this post to create an activity (no matter your content area/grade level) using the idea of “cubing” as linked in the samples above and use it in your classroom before the end of February.  Blog your idea, share your experiences, reflect and make suggestions on ways to improve next time!  Either post a link in comments or tag me in a tweet. 

Afterall, teachers…we are our best resources!

Leave a comment »

Snow Days for Days 2015

image

8 inches of snow…can you determine approximate length of pencil?

Highs in the teens/20s  this week…assuming snowdays  for days in south central Kentucky.

Leave a comment »

First/Last Five #slowmathchat A1

I appreciate @mjfenton’s slow math chat, great little bits to make me pause and ponder daily. 

Years ago, I just @jenncrase’s Flashback Layout daily, 4 MC & 1 OE question.  Each day focused on a specific strand for review: Number/Computation, Geometry, Algebra, Functions, Prob/Stats. 

The past 2 years my bell ringers have evolved into a variety of thinking of math thinking.  I try to change our line-up every 6 weeks or so, just to help students experience a variety.

Friendly Class Starters I have used in our rotation:
Estimation 180
Number Talk Tuesday
Would You Rather Wednesday
Counting Circles
Graphing Stories
Visual Patterns
Three-Act Thursday
Test Prep Tuesday
Flashback Friday

This Semester we began with Make A Difference Mondays.  Students were given short articles (What Do You Stand For?) featuring teens who overcame obstacles to better themselves, their school or community.  Each week they have a quick prompt to respond to based on the article.  This is a great way to help students start the week on a positive note. 
Week 1 was on goal setting and the following day we created these hands-reaching for our best, on the fingers they listed short term and long term goals.  The palm contains the legacy they wish to leave behind.

image

This month, each day contains Flashbacks from Geometry and Algebra I in efforts to review common topics they may see on next month’s statewide ACT.  On Friday, they take a short review assessment to practice.

My last 5 minutes is an area for growth. I KNOW the importance of student reflection and wrap up.  However, my timing is off and I run out of time and students are sometimes rushed.

I like:
Post-it note quizzes
2 – minute reflection: 1 thing to remember, 1 question they still have, 1 aha moment, 1 improvement to make
On an index card…Watch-fors:  reminders of common mistakes to watch for
What was easiest part of lesson? Why? What was most difficult part of lesson? Why?
Odd Man Out – give 4 examples of what we are learning, they tell which one doesn’t belong and tell why.
Steph O’Reillys suggestion -students create and solve their own problem based on the day’s lesson.  She even includes some of them on upcoming assessments!

Beginning and end are critical to setting the tone as well as reinforcing take a ways.  Looking forward to reading what others are doing!

Leave a comment »

First/Last Five #slowmathchat A1

I appreciate @mjfenton’s slow math chat, great little bits to make me pause and ponder daily. 

The past 2 years my bell ringers have evolved into meaningful time.  I try to change our line-up every 6 weeks or so, just to help students experience a variety.

Friendly Class Starters I have used in our rotation:
Estimation 180
Number Talk Tuesday
Would You Rather Wednesday
Counting Circles
Graphing Stories
Visual Patterns
Three-Act Thursday
Test Prep Tuesday
Flashback Friday

This Semester we began with Make A Difference Mondays.  Students were given short articles (What Do You Stand For?) featuring teens who overcame obstacles to better themselves, their school or community.  Each week they have a quick prompt to respond to based on the article.  This is a great way to help students start the week on a positive note. 
Week 1 was on goal setting and the following day we created these hands-reaching for our best, on the fingers they listed short term and long term goals.  The palm contains the legacy they wish to leave behind.

image

This month, each day contains Flashbacks from Geometry and Algebra I in efforts to review common topics they may see on next month’s statewide ACT.  On Friday, they take a short review assessment to practice.

My last 5 minutes is an area for growth. I KNOW the importance of student reflection and wrap up.  However, my timing is off and I run out of time and students are sometimes rushed.

I like:
Post-it note quizzes
2 – minute reflection: 1 thing to remember, 1 question they still have, 1 aha moment, 1 improvement to make
On an index card…Watch-fors:  reminders of common mistakes to watch for
What was easiest part of lesson? Why? What was most difficult part of lesson? Why?
Odd Man Out – give 4 examples of what we are learning, they tell which one doesn’t belong and tell why.
Steph O’Reillys suggestion -students create and solve their own problem based on the day’s lesson.  She even includes some of them on upcoming assessments!

Beginning and end are critical to setting the tone as well as reinforcing take a ways.  Looking forward to reading what others are doing!

Leave a comment »

Isn’t There a Formula?

Last week I offered some review each day on coordinate skills, as we have ACT coming up next month.

Rather than posting 2 points and the midpoint formula, I only wrote the word midpoint and 2 points, then asked for a point they knew was NOT the midpoint.

As they responded with wrong answers, I asked them to convince me (ode to @steveleinwand) -how did they KNOW it wasn’t the midpoint?  And their responses were varied from number sense to a graphical picture, always referring to betweeness, not exactly in the middle.  I cheered them on-that their reasoning here could help them eliminate wrong answers on ACT.

Someone asked, isn’t there a formula? Me: Yes.  Do you know the formula? …  well.  Does anyone KNOW the formula? …. silence…  finally, a student asked, Is it something like x1-x2 divided by 2? And y1-y2 divided by 2?

Hmmm. I wrote it down and said, let’s try it.  The result, students quickly said, no, that doesn’t work, explaining why.

My issue with giving a formula and working multiple problems…about 60-70% (my guesstimate) forget the formula, not because they can’t use it, but because they never move it from their working memory to long term, it’s not internalized.  They use it for an upcoming test, then trash it and move on to something else.

In both classes, students intuitively found the distance between the coordinates, divided by 2, then counted that many units from either endpoint.

As I asked my students why so many were not math fans, one student stated – because there’s only 1 way to do the problem and if you don’t get it, you’re doomed.

I said to her, you are exactly right. And you are exactly wrong. 

There’s never just 1 way to solve a problem.  But often times, students are only presented with 1 way.  So when they don’t get it, they give up. 

In the end we had multiple ways suggested to find the coordinates of the midpoint, and they were all correct. Similar to these…

image

It continues to make me sad that students think there is exactly one way to do every problem.   I am slowly trying to change their mindset.

6 Comments »

Students Making Sense of Quadratics

I realize some folks will bash me for sharing this from an Algebra 2 class, but based on benchmarks, most of my students have major gaps in quadratics. 

I began with reviewing multiplying 2 binomials on our whiteboards.  I shared the box/area model and several smiles celebrated because they “saw it” and were doing it correctly!

Last week, I pulled out a box of Algebra Tiles.  We literally explored building squares.  I wish I had taken pictures because some of their squares were like a grandmother’s beautiful quilt blocks.  I began tying it back to our box/area models -I’d rather think of it as leading (not forcing) their thinking – but they were quickly picking up the patterns. 

image

We then began looking at the algebraic equivalents, again, with a sketch along side allowing them to “see” the process.

Our next step was to find the missing value without tiles/picture models…and then I asked them to review their multuplying with 5 expressions alongside.

image

“What? You think they’re the same thing?!?” I asked,  “Prove it to me. Well, by-golly-jee. You are on to something!”

The following day in class, I made a HUGE ordeal of different ways to write zero.

image

I explained our next few minutes were a process. But we talked about it, step by step, completing the square, adding ‘that zero’ in our expression, the separating the trinomial and 2 constants.  Rewriting our trinomial as a binomial squared.

Ok. Why in the world would anyone want to do this?  I told them we were finding hidden information.

As they arrived at this form (x+4)^2 – 9, I paused, reminding them to think back on our function transformations before Christmas break.  How would this function y= (x+4)^2 – 9 move on our graph from this one y=x^2?  Quiet. “Move left 4 and down 9!” Someone exclaimed.  Really? Are you sure? We graphed the two and yes, it did just that.  So what does this tell me about my parabola?  They didn’t say vertex. Or minimum.  They said it shows us how the graph was transformed. 

I will take that.

I then asked them to move left 4 and down 9 from the origin.  What have you found? The lowest point.  The vertex. The minimum. All their responses, not my statements.

We set our expression equal to zero and solved the equation, using our inverse operations.  They made the connections with the x-value of the vertex being the “center line” of the parabola.  They realized the +- 5 were steps in either direction from the center line.

image

I most appreciated the questions they asked on #3, 4 and 7.  Several chose #7 thinking it was shorter, thus less work. Snafoo. No middle term. What happens? 

I suggested they look at it from a transformations point of view.  Someone shared-It doesn’t slide left or right, only down.  Another student said-well, that’s the easiest equation to solve! (Yep.)

Why did #4 bother some? The middle term had an odd coefficient.  But once they shared their thinking, ok. Got that one too!

#3 was what we math folks recognize as perfect square trinomial.  But for the students, it was an a-ha.  Again, using the transformations context, we moved right 5, but not up or down.

L: But I thought all quadratics intersected x-axis twice?  I asked – did this one? No.

What about y=x^2 + 3?  It moves up 3. Ok. How many times did it intersect the x-axis? It doesn’t.   Hmmm.

A student who is rarely engaged then asked, if you can make a parabola that doesn’t intersect the x-axis, can you find one that doesn’t intersect either axis?  Me: Can we? What would it look like? S: Noooo. As its going up, increasing, it would be increasing outward, too!  More discussion, between them. Me not included. I was smiling.

And their questions were what drove our lesson today.   And I was so excited, telling them their questions make me think! And when they’re asking questions, their brains are processing the information – making it their own.

It was a good day.

2 Comments »

Musical Chairs #eduread

Thanks to #eduread chat last night with @druinok and @algebrasfriend – a simple, no prep idea to get students out of their seats…Musical Chairs… something @druinok shared she had done in some of her classes last week. 

image

Today, after a mini quiz, we had less than 15 minutes left.  Students picked up a worksheet (given a scenario, they were asked to match a graph).  I chimed up some good ol’ Bluegrass from The Boxcars and Dan Tyminski.  Basically they wandered around the room, music stopped and they grabbed a seat, paired up with someone and discussed scenario #1. 

When everyone had a response marked, I started the music up again.  They smiled as they mixed around the room – some grumbled at my music choice.  Music stopped, they grabbed a chair, paired with a new person and discussed #2.  There were disagreements-(yeah!)-some nice math talk going on.  We continued right up until the end of class.

After lunch, the next class came in asking if they would get to play musical chairs, too. Cha-ching.

Thanks to my #mtbos peeps for adding a bit of joy to my classroom today!

Leave a comment »

A Struggle with Function Inverses

So as we were doing this today – simple enough using inverse operations to find the inverse function…

But as we did this one f(x) = (x-5)^2 to this f^-1(x) = sqrt(x) + 5

Student question… because you had x-5 in parenthesis, won’t the +5 also be inside the radical?

I’m just curious how anyone else responds to this.

We picked #’s and evaluated the expression, modeling transformation on a number line…but referring to inverses as a way to return where we started.  

f(8)=(8-5)^2 = 9

Simple enough…working backwards… sqrt(9)=3+5=8…back where we started x=8.

But what happens if x=3…

f(3)=(3-5)^2=(-2)^2=4

Working backwards…sqrt(4) = 2+5 = 7…not where we started at x=3.

Is this where you talk about restricting the domain in order for the inverse function to be defined?

Please don’t judge, I’ve taught straight up procedures in the past, even focusing on mostly linear inverses.  But I want my student to understand what they’re doing and why and be able to expand their understanding to more complex scenarios on their own.

Listening to them discuss this initial question, made me realize how Building Functions and Transformations has huge impact on foundational understanding.

The other realization today…very few of my students are comfortable and fluent in equivalent expressions. 

For example…
f(x)= 6-2x some asked if they could rewrite -2x+6? Sure!

But then when discussing their results, we saw several equivalents…but they did not recognize, rather argued others were incorrect.

image

We picked a value for x, then tested to see if they were equivalent.  A few were a bit perplexed they were the same value. 

This is one of those assumptions I’ve made but realize we need to address/refresh.

Would a matching, always/sometimes/never be sufficient?

3 Comments »

Follow

Get every new post delivered to your Inbox.

Join 1,255 other followers