I noticed that my students, while using the quadratic formula, typically had trouble in one of two places. They either find the wrong value for b^2 - 4ac, or they have trouble simplifying the radical.

I decided to try having my first-timers evaluate the formula from the inside-out. Maybe if we put our focus on the most-likely-to-mess-up parts of the formula minus the noise of the whole thing, then we would be more successful and consistently correct.

Here's what I mean:

The solution looks like this: First, we find b^2 - 4ac.

Next, we simplify the radical. My students are awesome at this when it is an isolated operation, but often struggle when its part of a larger problem.

Then we put it all together.

At this point you may be done, or maybe you need to reduce.

Time will tell how much of a difference this approach makes (if any), but I have noticed a few other benefits:

We don't have a learning target for the discriminant in our curriculum, but if we did this would be an easy lead in. That number we found first? It has a name, and it can tell you what kind of solution you are going to have before you do anything else.

Also, I kind of like how the solution seems more efficient . . . it feels less noisy and crowded compared to writing out all the parts again and again as you simplify the answer.

# square root of negative one teach math

Just some things this high school math teacher thinks about

## Tuesday, October 3, 2017

## Thursday, September 14, 2017

### How to Drive to Sonic

Recently I noticed my students having a few misconceptions about the different strategies for solving quadratic equations:

Sometimes, when you take the factoring route, you find that your equation can't be factored. When this happens, its like a road closed sign on the path to your solution. The solution is still there, but you'll have to take a different route to get to it.

To follow up this discussion, I asked students to be in charge of road signs for these three paths. Your job is to put up some signs instructing people on which road to take. What do the signs say? Here is what one group created:

My favorite quadratic formula sign said "when you don't want to think about which road to take". They're not wrong.

For a follow up activity, this card sort would be perfect.

A student assumed that a non-factorable quadratic had no solution.

Several were surprised when quadratic formula and completing the square yielded the same solution.

I came up with this little analogy to help clear things up, and it ended up working pretty well.

I started out by talking about how students might drive to Sonic, a favorite fast food place that is 10 miles away from our tiny town. Students named several paths and we talked about their pros and cons. The interstate is fast, but it does not provide a direct path to Sonic. Most students would choose to take Old 40 highway, but it is well known for road construction and can be blocked for days/weeks/months at a time. Someone even pointed out that there is a dirt path through a corn field that many of us have used to bypass the road construction. Perfect.

Solving a quadratic equation is very much the same.

I started out by talking about how students might drive to Sonic, a favorite fast food place that is 10 miles away from our tiny town. Students named several paths and we talked about their pros and cons. The interstate is fast, but it does not provide a direct path to Sonic. Most students would choose to take Old 40 highway, but it is well known for road construction and can be blocked for days/weeks/months at a time. Someone even pointed out that there is a dirt path through a corn field that many of us have used to bypass the road construction. Perfect.

Solving a quadratic equation is very much the same.

On the left you have your quadratic equation. You want to get over to the solution, on the right. Factoring, complete the square, and quadratic formula give you three potential paths to get there.

Sometimes, when you take the factoring route, you find that your equation can't be factored. When this happens, its like a road closed sign on the path to your solution. The solution is still there, but you'll have to take a different route to get to it.

Other times, when you're completing the square, you run into a bunch of fractions. We aren't afraid of fractions here, but they can turn an otherwise straight path to the solution into a long and winding road. If we're looking for the most efficient way to get there, we want to avoid this situation.

Then there's the quadratic formula. It may not always be the most efficient, but it always works. It is particularly useful for avoiding the road block and the winding road.

My favorite quadratic formula sign said "when you don't want to think about which road to take". They're not wrong.

For a follow up activity, this card sort would be perfect.

## Tuesday, August 29, 2017

### Visionary

I always think its silly when people start a blog post by talking about how long its been since they've blogged, but . . . It's been over a year and that's probably worth mentioning. If any dear readers are still here, thanks for not giving up on me.

I'm starting the third week of a brand new school year. There have been some growing pains, but I will talk about those another day. I want to talk about one of our back-to-school inservice meetings. Our principals shared about the book Soup by Jon Gordon. At one point, they would describe a characteristic of a good teacher, and ask our groups of four to name one of our colleagues who most represents that quality.

One characteristic that was mentioned was "visionary". We were asked to name a colleague who is always evolving and learning new things, striving to get better. I was touched that many of my colleagues named me. This is the kind of teacher that I want to be and that I try to be, but I haven't felt particularly visionary in the past year(s). Even this year is off to a rough and chaotic start.

In all fairness, I've had a lot of life happen outside of the classroom. Our now two-year-old son has had multiple surgeries to repair his cleft lip and palate. Then last spring our family grew by one more through what can only be described as a surprise adoption. The rare and special nature of how our family has grown is not lost on us, and we are thankful.

Being parents of three (particularly the two littles) is exhausting. Some days are all about survival. I'm going to cut myself some slack for my absence from blogging, but still . . .

I've been thinking about how this is the place where my vision began . . . The blogosphere is where I began to be more adventurous as a teacher . . . trying new things . . . writing about them . . . learning from others. I thought if I came back here it might trigger what I've been missing.

So here I am . . . inspired by my colleagues' perspective of me, inspired by a gentle nudge from @druinok a few months ago, and inspired by watching TMC17 from afar. I could spend a few more hours re-reading and editing this post, but instead I'll #pushsend.

I'm starting the third week of a brand new school year. There have been some growing pains, but I will talk about those another day. I want to talk about one of our back-to-school inservice meetings. Our principals shared about the book Soup by Jon Gordon. At one point, they would describe a characteristic of a good teacher, and ask our groups of four to name one of our colleagues who most represents that quality.

One characteristic that was mentioned was "visionary". We were asked to name a colleague who is always evolving and learning new things, striving to get better. I was touched that many of my colleagues named me. This is the kind of teacher that I want to be and that I try to be, but I haven't felt particularly visionary in the past year(s). Even this year is off to a rough and chaotic start.

In all fairness, I've had a lot of life happen outside of the classroom. Our now two-year-old son has had multiple surgeries to repair his cleft lip and palate. Then last spring our family grew by one more through what can only be described as a surprise adoption. The rare and special nature of how our family has grown is not lost on us, and we are thankful.

Being parents of three (particularly the two littles) is exhausting. Some days are all about survival. I'm going to cut myself some slack for my absence from blogging, but still . . .

I've been thinking about how this is the place where my vision began . . . The blogosphere is where I began to be more adventurous as a teacher . . . trying new things . . . writing about them . . . learning from others. I thought if I came back here it might trigger what I've been missing.

So here I am . . . inspired by my colleagues' perspective of me, inspired by a gentle nudge from @druinok a few months ago, and inspired by watching TMC17 from afar. I could spend a few more hours re-reading and editing this post, but instead I'll #pushsend.

## Wednesday, July 27, 2016

### Best TMC Ever!

I think twitter math camps are like children or students . . . you're not supposed to have a favorite. But if I HAD to pick a favorite, it would be TMC16. The reason is largely because there was so much usefulness in the sessions I attended. I came home excited for the upcoming new year and ready to implement what I've learned.

Here are a few things I hope you will hear more about this year . . .

1. Debate and Discussion in math class! My HOPE for this year . . . my #1TMCthing . . . is to put structure to the discussions that happen in my math class. I've had my students seated in groups of four for years. Years! And yet I've never felt like the conversations they are having have reached the productive level that I desire. Chris and Mattie did a great job of showing us how to structure discussion so that it focuses on answers AND reasons. There is always a "because". If you would like more detail, I recommend you read this post by Sam (specifically "Talk in the Math Classroom" near the end). He did a beautiful job summarizing our morning sessions.

2. Make it Stick! Unlike every other session, this one stressed me out. It exposed some weaknesses in the way that I teach that I really want to address. The problem is that the correction involves more of a fundamental shift than just a little tweak here or there. And I'm really ready for a year when "change everything" is not on my to-do list. So I decided to commit to making two baby steps: First of all, I'm reading the book and participating in the chat on twitter. This one is already underway. Yay me! Secondly, I'm going to begin spiraling practice of content through warm-ups. It's a start.

That's it. That's my 10%. Well, that and a few other takeaways . . .

*Getting Triggy: Kristen shared a great collection of trig activities. I heart trig, and I'm excited to add her goodies to my toolbox . . . and to make some of my old favorites BETTER.

*Box Method: My math department has just decided to begin using the box for all things polynomial, so I was excited to see how Anna presents these to her students. I love how the box pulls together multiplication, factoring, and division in a way that will hopefully lead to better conceptual understanding.

*New Desmos Stuff: Sadly, I did not get to attend the Desmos pre-conference. But card sorts and marble slides oh my! I'm as excited as everyone else to bring the latest in Des-awesome to my classroom.

*Warm-ups: I've decided to go the spiraled-practice route for my warm-ups this year, but thank you to Jessica and Lisa for a really nice list of resources for great problems that can really be used for more than the beginning of class. When it's time for trig, I want to try a counting circle and clothesline for counting/ordering in radians!

*The Fun Math Game with The Lame Name

*Regrets: This is almost verbatim from last year's TMC post . . . I was lucky enough to travel to TMC with my entire math department (all four of us) for the third year in a row. I love sharing this experience with them, and coming home with some shared vision for our school. I get how rare it is to have that kind of collaboration at home. But once again I really missed out on connecting with the other attendees.

*Lunch: Following the Make it Stick session, I asked a few others if they would like to have lunch and continue our conversation. It was a GREAT time talking with some people who I really admire, and I learned so much it was like a whole other session. It took some stepping out of my comfort zone to make this happen, but I'm so glad I did. If I'm lucky enough to attend TMC again next year, more of this is on my list of goals FOR SURE. I don't want to come home from another TMC regretting that I didn't seize the opportunity to speak in person with those I admire and learn from throughout the year.

## Thursday, July 21, 2016

### The Post Before THE POST

I just got back from TMC16. It was amazing, but I couldn't bring myself to write a recap without writing this post first. This year has been hard, and my personal story was so intense that it changed the lens through which I see all other things…

Disclaimer: This post is long and it's not about math. I'd love for you to read but it's okay if you don't. I wrote it for me.

Last fall I posted about the birth of our son and his unexpected diagnosis of cleft lip and palate, but the challenges didn't stop there.

When Silas was five days old, a routine eye exam showed that there was a structural problem on the inside of his eyes called a coloboma. Doctors told us it was too soon to know how severely it would affect his vision, but they made sure we understood that there was no surgery to correct it. My husband and I were heartbroken as we thought about what his future would be like without sight.

There were other challenges, too. Silas cried. A lot. He struggled with eating and was eventually hospitalized with severe reflux. He needed a feeding tube to help him eat without pain. He had two surgeries to repair his cleft lip, get ear tubes, and repair a hernia. He had countless doctor appointments and tests, four ER visits, and another hospitalization. I spent every day terrified of what the future held for him.

When he was two months old I went back to my classroom, but I went through so many days on autopilot. I was numb. And weary. And I wasn't sure that I should even be there at all.

It has been the hardest year of my life. And it has changed me.

This year I learned to wake up every day knowing that, even when the unknown is way more than I can handle, I always have what I need to get through this day.

I learned that fears and worries are only good for one thing … robbing us of the beautiful thing that is happening at the same time.

I learned to rejoice, not because of our circumstances, but in spite of them. There is always something to be thankful for.

One day this song came on the radio: Tell your heart to beat again . . . every word applies to me, but I'll just share the chorus here:

Tell your heart to beat again

Close your eyes and breathe it in

Let the shadows fall away

Step into the light of grace

Yesterday's a closing door

You don't live there anymore

Say goodbye to where you've been

And tell your heart to beat again

Your heart to beat again

Beat again

And so I made an effort to jump start the pieces of my heart that had stopped beating. I reconnected with friends . . . I put on my running shoes and trained up for a half marathon . . . I even wrote a blog post about something that was happening in my classroom. It felt foreign, but good.

Along the way I also made peace with Silas's differences. I realized that I spent so much time worrying if he was going to be okay, when there was never any question that he was going to be okay. Silas is okay because his life is a gift from God and because he will always, ALWAYS, be deeply loved. No diagnosis will ever change that.

Meanwhile, my husband landed his dream job as a building principal in our home town. His new schedule would allow him to be home with our kids during twitter math camp. Could I go this year? SHOULD I go this year? He said that I should. I think he knew that reconnecting with my teacher self would be another small step towards my healing.

I'm so glad I did.

This week my teacher heart started to beat again.

P.S. I can't close without letting you know that there is so much good news for this happy boy! He is almost 11 months old and he amazes us every day! We conquered reflux and he now eats entirely on his own. He healed beautifully from lip repair surgery. We have learned that he does have low vision in his left eye, but with both eyes together his visual acuity is in the typical range for his age. His ophthalmologist says he will definitely be a visual learner! We are so thankful for the good news and for how far he has come.

## Friday, February 26, 2016

### It's Not About the Points

While many math teachers have stopped giving points for practice/homework, I confess that I'm over here still giving points. This semester, I decided to at least challenge my own attachment to points . . . particularly the thought that students will not do an assignment if there are not points attached to it.

So I set a lofty goal: 100% of students completing 100% of assignments. Whenever I had time to walk around the classroom, I carried a clip board and asked students to show me their completed assignment(s). Not done? No problem. What questions do you have? What can I do to help you finish up?** I did some repeated asking and follow-up and asking again. Over time, it got easier as students realized that not completing the assignment was not an option.

I also discovered that, more frequently than I expected, lack of completion was really due to lack of understanding. Many times it came disguised as laziness, disinterest, and the like . . . but really the student just didn't know how to do the math.

When an assignment was complete (and correct!), I recorded the points on my clip board. I used circles to indicate assignments that were late and highlights for assignments that took more than two weeks to collect. I ended up with a sheet for each class that looked like this:

Since everyone ended up with all of the points for all of the assignments, it really brings one big question to mind . . . What's the point of the points?!

I'm finally believing that students don't complete assignments because of points. Students complete assignments because of accountability. I would argue that there are forms of accountability that are more affective than points. I never had 100% completion when I was only assigning points.

Another unexpected outcome (which shouldn't have been surprising), is that assessment scores were higher as a result of "Operation 100%". In the past, I spent a lot of time orchestrating/scheduling remediation and re-assessments. Since students ended up understanding the content better on the front side of assessments, I spent significantly less time on that type of thing this semester.

**My school has some structures in place that helped tremendously with follow-up here. I assigned many students to our school's tutoring room. It is held during the school day and staffed by a few teachers and lots of National Honor Society students. For students who understand the content but were just dragging their feet on completion, I had the option to assign them to academic lunch.

So I set a lofty goal: 100% of students completing 100% of assignments. Whenever I had time to walk around the classroom, I carried a clip board and asked students to show me their completed assignment(s). Not done? No problem. What questions do you have? What can I do to help you finish up?** I did some repeated asking and follow-up and asking again. Over time, it got easier as students realized that not completing the assignment was not an option.

I also discovered that, more frequently than I expected, lack of completion was really due to lack of understanding. Many times it came disguised as laziness, disinterest, and the like . . . but really the student just didn't know how to do the math.

When an assignment was complete (and correct!), I recorded the points on my clip board. I used circles to indicate assignments that were late and highlights for assignments that took more than two weeks to collect. I ended up with a sheet for each class that looked like this:

Notice fewer circles/highlights during the second half of the quarter?!

Since everyone ended up with all of the points for all of the assignments, it really brings one big question to mind . . . What's the point of the points?!

I'm finally believing that students don't complete assignments because of points. Students complete assignments because of accountability. I would argue that there are forms of accountability that are more affective than points. I never had 100% completion when I was only assigning points.

Another unexpected outcome (which shouldn't have been surprising), is that assessment scores were higher as a result of "Operation 100%". In the past, I spent a lot of time orchestrating/scheduling remediation and re-assessments. Since students ended up understanding the content better on the front side of assessments, I spent significantly less time on that type of thing this semester.

**My school has some structures in place that helped tremendously with follow-up here. I assigned many students to our school's tutoring room. It is held during the school day and staffed by a few teachers and lots of National Honor Society students. For students who understand the content but were just dragging their feet on completion, I had the option to assign them to academic lunch.

## Friday, February 5, 2016

### A Way to Make Their Day

One of my nephews is in my Algebra 2 class this year. Today is his birthday, and I gave him an assessment. Happy Birthday, Kale.

I wanted to make the day special for him, so I wrote some notes and drew some pictures on the inside of his assessment. Then I did a little magic as I was passing them out to make sure the "special" assessment landed on his desk.

I loved the grin on his face as he worked his way through the assessment. That's when I realized that I have lots of other students who celebrate birthdays on assessment day, or who just need a little extra encouragement . . . This is going to have to be my new thing.

I wanted to make the day special for him, so I wrote some notes and drew some pictures on the inside of his assessment. Then I did a little magic as I was passing them out to make sure the "special" assessment landed on his desk.

He volunteers as a junior firefighter and carries a pager to alert him when help is needed. He's not supposed to leave school for fires, but oh how he would love to!

Kitty inspired by this post from Jonathan.

17!

I loved the grin on his face as he worked his way through the assessment. That's when I realized that I have lots of other students who celebrate birthdays on assessment day, or who just need a little extra encouragement . . . This is going to have to be my new thing.

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