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

day.

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.

No.Wonder. 

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.