the radical rational…

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

Function Families & Why’d It Do That?

We began our week in Algebra I with Function Families.

This old task… here are the New link to files.

We eventually end up here as a wrap up. Students come to the board and share their sorts.


The following day we summarize their findings on a foldable…descriptions of the equations and graph shapes from their groups.  The inside of the foldable contains an example of each type of function, table of values and a graph.

I began with quadratic because I see the most mistakes here. Students will use their calculators and jot a number down without pausing to ask if it’s reasonable.  We had 10, -8 and -27 for the first table value.  Hmmm? How’d they get those?  I actually used an entire set of wrong calculations and graphed, then asked, Is that what you expected it to look like?  No. So we need to check our work and find the mistake.


We completed the first table and they were asked to write about what they noticed in the numbers. And we shared.

Next, we looked at the first differences. They wrote about their noticing again.  “Oh,” a girl says.  “That let’s me see what’s happening in the graph!”

And we finished with the second differences.

I went to the absolute value next.


One student claimed, it’s doing the same thing as the first but with different numbers. Another student disagreed because the numbers were constant and not changing like the first.  But the directions were the same.  I explained that different operations would cause the graphs to look differently and we were creating a guide to help us sort through the patterns and learn to recognize them.

In both cases, I heard students mention reflection, symmetry, matched – up referring to numbers in table, not the graph.

We continued with linear and the exponential. 



I began with 4^1 on this table and asked, can I write this 4*1 and it’s still 4?  Yes.  So, 4^2 would be 1*4*4 and 4^3 1*4*4*4.

Which means 4^0 would be 1* (zero 4s)…or just 1.

We had done simple function inverses prior to fall break.  I had used the -1 exponent to represent inverse.  So our discussion went back to 4^-1.  Student ask, “well, if exponents are repeated multiplication, would an inverse exponent be dividing?”  And we continue with that discussion. 

We ended the day with some reflection on our learning.  They were asked to tell which 2 functions were most alike and why.  Which 2 functions were most different and why.  Very eye opening to read some of their thoughts.

At the end of one class, a couple of students we still discussing something.  He shared, “I was wondering what I’d get if I graphed y=x^-1” and he showed me the graph.  Why does it graph that way he asked.  Why does it graph that, I asked him back.

His group mate shared, well, I graphed y=x^-2 and instead of reflecting into the 3rd quadrant, it’s like it reflected across the y-axis.  Why did it do that?  I replied, why do you think it did that? 

I told them both, that was my goal…to let them start asking their own questions…and to keep pondering their graphs, we would talk more about them next week.  It was a good way to end the week.


Pondering My Students with Different Experiences than Myself

Yesterday I made a trek to visit my college roommate in Louisville.  At least every other year since she was in Law School, we spend the day together at St. James Court Art Fair/Craft Show.  It truly is a day I always look forward to.

On the drive up, I listened to John Antonetti ‘ s 17000 Classroom Visits book.  At some point, Ruby Payne was referenced.  I have never read her materials, but this statement caught my attention and resurfaced multiple times throughout the


Considering three groups of people, those living in poverty, the middle class and wealthy and how they relate to society.  “For those facing poverty, the emphasis is on the present.  Decisions are generally short term and based upon emotions or basic survival.”

A tweet from @veganmathbeagle


and this post added to my thoughts.

At lunch the other day, one of our amazing paraprofessionals and I discussed the challenges of working with so many of our students.  Her comment, “they just can’t see toward the future, they don’t believe in themselves or their ability to change the path they’re on…”

As I shared some of these thoughts with my friend, who has worked in the family courts system longer than I’ve been in the classroom, I had the realization…I really have no clue. 

I grew up priveleged.  And I am grateful for my parents who instilled an attitude of hard work,  living within our means. We never had what I considered an excess, there were times I didn’t get to participate in trips or other activities at school but we always had what we needed.  I never went without food, clothing, a home, love or both of my parents.  I know was abundantly blessed. 

I remember one (yes, 1) argument between my parents when I was in 3rd grade and it stuck with me- for life.  I cannot fathom a home life where arguments are unending, a parent leaves or worse, being removed from my home.

My friend who has worked with many children through the years challenged me to close my eyes and think… someone enters your home, says you have 10 minutes to gather your things…what do you grab to take with you?  Then you’re taken to a home, with people you don’t know, they don’t know you, and you are expected to adjust to a knew set of rules.  What if you change schools-a new set of students, classes, teachers, expectations…how do you feel? Scared. Angry. Traumatized.


She shared something she heard a judge say to a parent once… my paraphrase: your decisions are keeping your children distracted from learning opportunities. You are taking away their future-an education is how they can change their future. 

So many of my students face things unbeknownst to me. 

My friend kind of giggled when I shared that I was older in school before I realized that Foster was not the students’ last name.  I’m sort of ashamed to admit that, but I simply did not know.

I now see that I have the privelege of instilling hope of a better future.  Its challenging sometimes to connect with certain students, helping them see there are adults who care and believe in them.  They have reasons to not trust adults.  But it also feels like a burden that takes its toll on me emotionally.

They have every right to feel what they feel, but I have failed in the sense that I want them “to get over it, move on and learn this math!”  We have expectations in our classrooms without considering their point of view.  They have young minds and young hearts and need a way to process their experiences. 

I sit wondering, how many of my colleagues were ever in Foster Care?  We are given a challenge that we have no experience with and expected to make a difference.

How would my attitude change if I sat and listened to a panel of students who have been or are in foster care?  I need to be aware of how they feel, what they think in order to provide the best opportunities of support and learning for them.


I do the best I know how- just this past week, a student described me as kindhearted.  I am not sorry for how I was raised-I am utmost grateful.  I want to understand better.  I welcome any suggestions of resources that would bridge this chasm between myself and my students with different experiences.

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Building Those Pathways #eduread

A big part of our Make It Stick discussion this summer was giving students opportunity to retrieve information.

Another aspect, I’ve battled with in using INBs was giving students all the information instead of letting them decide what’s helpful and create their own notes.  Very few ninth graders are skilled in creating their own notes – so how can I help them develop this skill?

So here’s what I came up with…


The workbooks have an intro to each lesson – a couple of paragraphs, highlighted vocabulary and usually some type of graphic or example.  Here’s what I did yesterday…

Step 1:  Read (2 minutes)

Step 2: Pair (1 minute) – share 1 thing you read.

Step 3:  Whole-Class (I scribe what they share) something you read that you think is notable – something you feel is important or that you want to remember, something that made you wonder.  I labeled each slide for each class so I could reference it today

Then they worked in their groups to complete a lesson and practice.

To begin class today, each student had their white boards.

Step 1: Brain dump (1-2 minutes depending on class) – list anything and everything they can recall from the past two days in class.

Step 2:  Give one, get one! (1 minute) – meet with 3 different people, look at their list and get one thing from them to add to your list.  Let them get one thing from your list.

Step 3:  Return to desks.  Pull up slide from yesterday’s share… and I asked, anything you wish to add?  And we added to our list from yesterday.

You can tell they are not asked to retrieve this way often enough.  It was hard for a few of them – only a handful of things were on their lists initially.

I loved the openness of the brain dump.  It helped me really see their focus/thinking.  One student listed things from 2 weeks ago, which made me think he had not fully processed this week’s work yet.  I had graphs, details from specific problems, a list of vocabulary – but everyone had something written down.

Give one – get one, I heard them saying:

“Oh yeah, I forgot about that!”  “Where did you get that idea?” “From the science lab bacteria problem.” “When the extra truck came in to fill the pool.”

I reminded them – like walking a path across the field to my granny’s house ( a story I told them during our 1st week) – we are trying to wear a path…  So what do you think will happen tomorrow when we do a similar routine?  I’m looking forward to their growing list of “all things functions.”  Everything they have listed is exactly what I would have had them copy down from my notes (minus a few details of course) – the big difference, its their thoughts and ideas, not mine.  I even had students list discrete and continuous! Go figure.

My intentions are to take their generated lists and allow them to sort / connect them (Making Thinking Visible) to create their own page of notes later in the unit.

It was a good day.

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Real Life Water Line

After some good old Dan Meyer Graphing Stories last week, we began our next phase of functions by predicting what our graphs (#s coops & water height) would look like for these containers:



And then we actually scooped water to see how close we were…


Walking around, listening to conversations…
“NO. It can’t go back down, you’re still adding water, so the height of water keeps increasing.”

“It might slow down or speed up, but it won’t decrease until we empty the vase.”

“So we need to reverse the steepness…where it’s steep, we need to flatten it out and where it’s flT, we need to make the graph steeper.”

I required them to complete a group graph to predict before I gave them their scoops to start data collection.

Several had to go back and finish details like labels and scales on graphs…a good reminding activity.

One student asked – do you do activities like this often?  It makes it (math class) fun.  

It was a good day.

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Recursive Models #8minreflection

When planning my first unit with sequences, I just assumed Recursive models would wait until Algebra 2.  Last week, my students took their first benchmark for the year and what do you know, but a question about  a recursive model.

F.IF.3  Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

I’m struggling to see the difference in these 2 standards.  As I look at examples, I feel they are much the same or at least co-exist within a problem.

As I watched students during their benchmark, I was aware of the recursive formula question.  When I looked all of my classes results, only 38% got it correct.  As I looked at their response distribution, most students picked an example that at least corresponded to the give sequence in some manner.  However, one class in particular, response distribution was 35%, 18%, 29%, 18% which says to me they are unsure of the notation.  The subscripts are throwing them off.  Something I need to help them make sense of.

I briefly introduced recursive models Friday, but we worked with them more today.  I had a student ask, so is this like a function of the term before it?  Hmmm.  Sort of.  Yes.  Across the board 3 of my classes are very strong in terminology and understanding functions.  So this was a connection.  I saw students eyes widen and they nodded at our discussion.  Alrighty then.  Let’s try another.  And there ya go, the connection was made, they were able to “see” the process within the model.

Student Reflection

I wrote 3 different models on the board.

1 minute, think to yourself:  How are they alike?  How are they different? Now, turn to a friend and share your thoughts.

1 minute, think to yourself:  How do I know which model is which? What do you see/look at to help you decide? Now turn to a friend and share your thoughts.

seq models

So many good things shared.  Its amazing how I can have 4 different ways of seeing something, but yet, each is beneficial somehow.  Some of their comments: Two of them have a1(first term), some have d, some have r.  But what I heard again and again – they all have (n-1) but the location is different.  What does that location of n-a tell us?  Once, its a factor for repeated addition, another its an exponent for repeated multiplication and the recursive, its a subscript for the term before.

So, their conclusion…the math is not hard.  Knowing what the notation means makes it difficult.

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Radical Rummy

I received this file about 5 years ago at KCTM in Bowling Green.  Kari from WKU shared it.  I apologize I cannot remember her last name to give credit.


She actually used it to play a card game style activity.  I copied sets onto different colored cardstock and laminated, I have enough sets we usually do groups of 3 people.

I do this activity along with Go Fish for simplifying radicals. 

There are four different forms of each value.  Students use calculators to match cards with same value.  We create a poster as a whole class.  Then notice and wonder. 

I like how students develop their own understanding of rational exponents, negative exponents and radical forms.  It’s a great intro activity.

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Complicated to Doable

This was the Visual Pattern for the bell ringer and some of their thoughts…


So we ended up with some form of these 3 equations in both classes today…


I got tired of writing step # and total blocks so I asked if we could shorten it a bit…so we defined n and x as those quantities!

And majority of students raised their hand saying those equations looked hard or confusing…

Until…we picked a step number.  Woe.  That’s doable.  We substituted 4 as our step # and all they saw was addition, subtraction, multiplication, parenthesis…  we just went from this complicated thing to something I even do without a calculator!

And the arithmetic showed each equation actually resulted in the same values.  

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Flexibility with Numbers

Is that part of numeracy?

Each week, I have used a different thinking structure/resource as my bell ringers.  On Monday, I model, but the rest of the week, students work within that structure.

My intention is build a tool box of sorts. 

Four 4s to review Order of operations.  Krypton and Math Dice would also work here. to help students see that several different ideas can be correct because of good reasoning.

We actually did visual patterns during class a couple of weeks ago, but this week, I am using non-linear examples.

Yesterday, my thoughts lingered from a discussion (several, actually) about this number pattern…


The top version was in a problem set and 90% of the students skipped it because “I’m not very good with fractions” (YET, I added every time).  I’m not talking my struggling learners, I’m talking…everybody.

It erks me, they didn’t even look at the problem to investigate what’s changing, what’s staying the same…they saw fraction and went on.

Do you immerse them in fractions to build that confidence to at least pause… and take a look?

Next week, we will start class with Counting Circles, the following week Counting Circles with some fractions involved somehow.  Or should I use the clothesline here, somehow? 

A long weekend will give me some time to think on these things.

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A Fun New T-Shirt

I was super excited about these new tees we ordered last week. 


Just in time for our first home football game, faculty/staff tailgating.  There was also a vintage blue with red font.   Yep. We were the cool kids at the game…well, except for our student section.

They wore all white and had a paint war an hour before the game on the backside of our track.  Always goofy fun seeing them covered head to toe in paint.  Uhh. No hugs, or even handshakes…an air – high five was about all I could take.

I’m proud of the work so many have been putting in to make a change in the culture of our building.  Hoping it will continue and carry through…overflowing to our classrooms and our students…creating a place they want to be…a place they can laugh and learn and create good, lifelong memories.

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Fitbit Fun


So here’s my fitbit screen this evening.  Hmmm.

I remember when I hit 1,000,000 and thought that was a big deal. Nah, not so much anymore.

I notice a lower average than I would like but I was pretty lazy this weekend.

I also notice my lifetime steps is not crazy far from 2,000,000. 

I wonder if I’ll hit that mark before the end of the year?

If not, how many steps per day would I need to average in order to get there by December 31?

If I averaged 10,000 steps a day, what day will I reach my new goal?

I wonder what questions my students might ask?

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