Data Collection Labs to Model Functions

Data Collection Labs to Model Functions

I've posted on this topic before ... but today I'm adding some resources! I hope this is helpful to those of you who teach a variety of functions!

Using lab explorations to introduce functions in Algebra provides engagement as well as data collection analysis activities. Students use everyday materials to collect data. The data describes specific functions. Here are a collection of "lab" activities I've tried in the past few years plus others that have been shared online!

Linear (and Quadratic) Function: Pass the Ball

All the information needed for this activity is in this blog post. Students pass a ball to one person, timing the event. Then to two people, three people, etc. This is a very easy lab to set up.

Another activity is the message/whisper chain.  Students will enjoy this lab!

The Wave Lab is also a great introductory lab suitable for linear functions!

Pass the Book is similar to the previous two!

If you teach Algebra 2 and want to review both linear and quadratic functions in a single activity ... consider this looking through a tube idea!

Quadratic Function: Stacking Starbursts and Kangaroo Conundrum

Both of these activities are simple, table top activities that result in quadratic patterns. I provide instructions in this blog post.

The Water Flow Lab looks fascinating ... 

And EVERYONE loves a catapult!  Check out this data collection opportunity!

Square Root Functions: Inclined Plane Data Collection

A copy of the instructions can be found here. Students roll a marble on an inclined plane - varying the distance of the roll, and measuring the time it takes to reach 0.

You can model the movement of a metronome here using a bottle and spring!

Here's another version of a pendulum lab!

PhET has a pendulum virtual lab worth investigating if you don't have materials!

Exponential Functions: M 'n M Data Collection

This is popular for obvious reasons - students love to eat the m 'n ms after the experiment is complete. We do both parts ... exponential growth and decay. The handouts with instructions can be found here and here.

Paper folding also works for exponential functions ... and requires little preparation or materials!

Mathy Cathy explains how she uses the Sierpinski Triangle to model exponential functions!

Rational Functions: Spaghetti Cantilevers

Instructions are online.  Students bundle 2, 3, 4, 5, 6 pieces of spaghetti and hang a weight from the end without breaking the spaghetti.  They collect the data and analyze it.  The rational function is the model for the cantilever. 

MORE IDEAS!

Several labs are shared in this conference handout!

Also check out this Twitter thread where folks shared ideas for labs!

#MTBoSBlaugust: Exploratory Labs for Functions

Using lab explorations to introduce functions in Algebra provides engagement as well as data collection analysis activities.  Students use everyday materials to collect data.  The data describes specific functions.  Here are a collection of "lab" activities I've tried in the past few years:

Linear (and Quadratic) Function:  Pass the Ball  

All the information needed for this activity is in this blog post.  Students pass a ball to one person, timing the event.  Then to two people, three people, etc.  This is a very easy lab to set up.

Quadratic Function: Stacking Starbursts and Kangaroo Conundrum

Both of these activities are simple, table top activities that result in quadratic patterns.  I provide instructions in this blog post.

Square Root Functions:  Inclined Plane Data Collection  

A copy of the instructions can be found here.  Students roll a marble on an inclined plane - varying the distance of the roll, and measuring the time it takes to reach 0.

Exponential Functions:  M 'n M Data Collection

This is popular for obvious reasons - students love to eat the m 'n ms after the experiment is complete.  We do both parts ... exponential growth and decay.  I wrote a short blog post - without any real helpful information here and also here.  The handouts with instructions can be found here and here.

Rational Functions:  Spaghetti Cantilevers

Instructions are online.  Students bundle 2, 3, 4, 5, 6 pieces of spaghetti and hang a weight from the end without breaking the spaghetti.  They collect the data and analyze it.  The rational function is the model for the cantilever.  I write a blog post about this lab here.  And took a few pictures:

#MTBoSBlaugust: Parent Function Activities

Reviewing functions is often the first unit in Algebra 2 and/or Pre-Cal.  In Algebra 2 course, we discussed all the parent functions before delving into each one in detail.  It helped to set up the year, gave us opportunity to review key concepts about functions and to teach notation.

Here are a collection of activities from past years ...

1) I Notice/I Wonder

I borrowed from G. Waddell's first of the year activity to get students thinking about the various functions.  I described my work here.

2) Parent Function Booklet is a tool students can use all year.  

Students create a page about each function, create a cover, and staple it together.

3) Odd One Out 

It is a "which one doesn't belong" activity that has multiple possible solutions.  It's a great way to get students discussing key attributes of functions.

4) Parent Function Analysis promotes math discourse.  

Students have 10 statements that may describe one or more parent functions.  They identify which functions each statement describes.

5) Synthesis Questions

1. Order the parent functions we are studying from least to greatest by the rate at which f(x) increases as x increases for x > 1.  Explain your thoughts.
2. Use the set of points {(-1,-1), (0,0), (1, 1)} to answer each question.
1. What parent function best describes the set of points?
2. If the points (-2,8) and (2, 8) were added, what parent function would best describe the set?
3. If the point (1, 1) were replaced with (1, -1) what parent function would best describe the set?
4. If the point (-1, -1) were replaced with (4, 2) what parent function would best describe the set?
5. Select 1 or 2 points to change or to add to create a different function than those already described.  Explain your selection and the parent function that would best describe the set.
3. Create “Who Am I” Riddles for 5 of the parent functions.  For each riddle use 4 to 6 clues.  Here is an example:  My graph is continuous.  My graph has an intercept at (0,0).  My domain is the set of all nonnegative real numbers.  My range is the set of all nonnegative real numbers.  The shape of my graph is sometimes referred to as an eyebrow.  What parent function am I?

 6)  Making Connections is about connecting concepts using vocabulary.

Do you have parent function activities (or families of functions)?  Share your link in the comments!

#70Days Function Fun

I found two  activities in CPM materials online  ... can't wait to use them! Love finding treasures here and there!  (I find it interesting to see what other textbooks offer in the way of activities and questions. Our adoption is a standard book (Holt) and we don't use it much.)

Your teacher will give you a set of four function machines. Your team’s job is to get a specific output by putting those machines in a particular order so that one machine’s output becomes the next machine’s input. As you work, discuss what you know about the kind of output each function produces to help you arrange the machines in an appropriate order. The four functions are reprinted below.

1. In what order should you stack the machines so that when 6 is dropped into the first machine, and all four machines have had their effect, the last machine’s output is 11?
2. What order will result in a final output of 131,065 when the first input is 64?

There was another activity - a scavenger hunt of sorts to use when identifying key vocabulary for relations and functions.  The document linked above has links to the materials needed. The gist of the ideas is that there are 10 relations/functions posted around the room in various formats.  The class is in teams.  Each team receives a clue (there are four different sets of clues).  As a team decides on the answer to a given clue, they return to the teacher to defend their answer and if successful to receive the next clue.  In the end there is one function that is the "treasure."

#70Days Students Asking Questions!

Day 7

Devotional thought from Practice Resurrection:  If we calculate the nature of the world by what we can manage or explain, we end up living in a very small world.

This is the context in which Peterson begins his exposition of Ephesians 1:3 - 14.  In the Greek Ephesians 1:3 - 14 is one long sentence, about 200 words, and noted as "one of the most splendidly Jewish passages of praise and prayer in the New Testament ... a prayer of blessing to the one God for his mighty acts in creation and redemption."

Today I am meditating on Peterson's paraphrase of the first few verses of this "song of praise" ... How blessed is God! And what a blessing he is! He’s the Father of our Master, Jesus Christ, and takes us to the high places of blessing in him. Long before he laid down earth’s foundations, he had us in mind, had settled on us as the focus of his love, to be made whole and holy by his love. Long, long ago he decided to adopt us into his family through Jesus Christ. (What pleasure he took in planning this!) He wanted us to enter into the celebration of his lavish gift-giving by the hand of his beloved Son. (from The Message, Eph. 1:3 - 6)

This morning I have been re-reading the chat we had last night about getting students to ask questions.  Here is a LINK to the article that we read.  And here is a COPY of our chat!

Last year, on day 1, we started with our first "I Notice, I Wonder" activity.  I had made a chart of the seven functions we would study.  Students wrote down their noticing and wondering.  You can read about the activity HERE.

This year I am thinking about extending that activity into asking questions.  The chart could be my QFocus. Students could work in small groups around large whiteboards.  That way we can easily share our thoughts. I envision this activity occurring on Day 2 of our school year - when we are introducing the parent functions.  It will be interesting to see what questions students develop and how we can apply their questions in our unit.  One of the ideas discussed last night in our chat was inviting students to choose a question or two to answer for homework.  That way students have ownership - of the questions and the homework.  Hopefully that will be a win-win for them! I'm wondering also if I can include one or more of their own questions on our assessments during the unit of study!

Just now I was thinking about how we could hang our charts around the room - the noticing, wondering, questions ... but if we use the group whiteboards it's harder to save their work.  Yes - I do take pictures but printing those out chart size isn't an option for me.  We need to capture a summary of our work on "anchor charts" that we can refer to throughout the unit!

Before the summer is over, I plan to study more about students asking questions.  At Right Question Institute (Twitter #QFT) there are free resources.  My reading/thinking/planning pile is growing!

Last ... join us in our Twitter Chats ... we read one article a week and discuss them at 8 pm central time on Wednesday evenings!  See Read Chat Reflect Learn and #EduRead on Twitter!

Who Am I ... just a bit more!

Yesterday I wrote a little about my personal life.  Today I'm sharing a bit of my professional history.  I call it my checkered past!

Taught in three states: Virginia, Tennessee, and Texas.
Worked in three levels:  Elementary, Middle and High Schools.
Worked in both public and private schools.
I've been a classroom teacher, a gifted education resource teacher, an assistant principal, and a principal.
Taught every subject at some level - even a PE class (scary if you know me!)
Taught algebra 1, algebra 2, and precal - never geometry!
Presented at conferences at the local, state, and national levels.
I've lost track ... I think this is my 33rd year in education.

My experience is wide and varied.  And while I might not recommend so much change in a person's life, I learned a LOT by moving, working in different districts, working with different people.

I originally trained to be an elementary teacher.  My first job was in a private school where I had the opportunity to teach many different subjects at the 8th grade level.  It was there that I discovered that I liked teaching math.  I had enjoyed math in my K - 12 education, earned a minor in math in college. Eventually I went back to school to add enough credits to earn certification in secondary math.


As I have been typing this post, the idea of changes have crossed my mind.  I remember when the first desktop computers, the Commodore 64, were added to the school in which I was teaching.  I volunteered to teach programming ... BASIC.  It was great fun.

I remember being introduced to the TI 81 graphing calculator!  That invention changed the way we taught math.  Now we could analyze complicated graphs without spending so much time graphing by hand!  It was amazing!

With the changes in technology there have been changes in curriculum.  In those early days of algebra we spent much of the year working on algebraic manipulation skills with some minor attention to classic word problems, number, motion, work, etc.  Now in my current curriculum we spend about half of our time analyzing graphs, identifying attributes of functions and applying that understanding to problem solving.  We spend much less time on the manipulation skills (maybe we need a bit more)!

In the early days of teaching I knew the mechanics of math and delivered that instruction well. I know more now about the applications of math and more about its beauty ... and I also realize I have so much more to learn!  Last year I wrote a bit about the beauty of math.

So ... there's a bit of my checkered past.  Yes, I'm "old" ... a veteran.  But I'm also thriving, passionate, eager to know more, share more, teach more!  I love this quote: "Life should not be a journey to the grave with the intention of arriving safely in a pretty and well-preserved body, but rather to skid in broadside in a cloud of smoke, thoroughly used up, totally worn out, and loudly proclaiming, "Wow! What a Ride!"" (H. S. Thompson)
#MTBoS30
 

Planning our Conics Unit ... Day 1

We are testing tomorrow on our Rational Functions unit. It will be interesting to see the results. My formative data indicates we could use more time on this unit. But our calendar says we must move on. I offered tutoring review several mornings last week, and again this morning, tomorrow morning.  Several students have taken advantage of the opportunity for extra instruction.  But those few ... you know who they are ... haven't visited ... yet!

In the meanwhile, I am pulling together our next unit.  I could just do one worksheet after another as that is what is in our school's files.  But I'm trying to mix in technology that will enhance student learning, hands-on activities that will promote critical thinking, and problem solving - applications that make sense.

Here is my plan so far ...

On test day, the homework will be an invitation to read background information on conics, to tweet something they learned, to try to find 3 interesting facts about conic sections and to begin exploring the equations in Desmos. Already some of my sophmores are whining ... "What?!? We are going to tweet math stuff?  I'll have to create a math Twitter account!"  Oh my!

Then on Day 1 of the unit ... students will have some background IF they do they homework.  As students come in, I'll have their tweets on the screen ... and invite them to work in teams to create "top ten" lists of interesting facts they learned about conics in their homework.

Next, I'm using Cindy Johnson's conic cards.  Students will sort cards ... carefully looking for patterns, key attributes to sort the cards.  In this first round of sorting, students will work with partners, and will try to sort one whole deck into four piles - and explain why.  The extension would be to organize those four piles into sets of 3 cards representing specific conic sections.

To follow up with the sorting, I'll use Discovery Education 10 minute video from the series, Math Factor: Physical Properties of Conic Sections.  The video will provide students with a clear picture of how the conic sections are sliced from a double-napped cone.  I thought about having students slice the sections from play dough but decided against it.

We'll use the work students did with sorting cards and the video to start filling out a graphic organizer about the four sections.  The graphic organizer doesn't have to have all the details on this day - we will revisit it daily to add notes.

Last we will spend a few minutes in direct instruction reteaching completing the square.  Since we have spent time on this skill, I'm hoping that a short review will be enough for students to work with conic equations.

For homework, students will practice using completing the square, and I'm asking them to find an example of a conic section in their home, capture a pic, and tweet it!

On Day 2, we explore circles.

Rational Functions ... how do you teach them?

I thought I'd write about how we structure our rational functions unit.  I'm curious what others do with it.

(Background ... we are Texas - a non common core state; we have 90 minute classes every other day)

We split the unit into two parts.

Part 1
Day 1:  Data Collection: How is the resulting graph like others we have studied?  How is it different? (We start every function unit with a data collection day).  For rational functions we used the spaghetti cantilever activity.  Find it here.  Students work in teams, collect data, answer questions about independent, dependent variables, discrete vs continuous functions, and domain and range.  Then they  try to find a function that fits their data.

Day 2:  Transformations:  Transformations are a huge part of our curriculum and by this time of the year, students are very familiar with how "a," "h," and "k" affect functions.  So we start with the basic rational function in transformation form.  Students use Desmos to track transformations and describe them.  With this day we talk about asymptotes and how they are a major attribute of this function.  (Asymptotes were introduced in our exponential/logarithm unit).  Last we express the domain and range of functions!

Day 3 & 4:  We start this day with a review of factoring and how to use factoring to simplify algebraic fractions.  Then we being the work of dissecting the attributes of the rational functions based on the factored forms of functions.  We start with discontinuities - the vertical asymptotes and holes.  Then we discuss the horizontal asymptote; next we find the intercepts; and last we write the domain and range. 

Day 5: We review ... presenting the content in various ways.  Here is a sample.

Day 6:  We test ... this test is short and could be combined with a introductory activity to the second half of the unit.  I chose to give my students time to work on their Desmos Creative Art unit.

(This is where we are now)

Part 2
Day 1:  Students use their understanding of simplifying algebraic fractions to multiply and divide fractions.  This is a skill practice day.  We also solve simple rational equations - the factors cancel leaving simple linear equations to finalize.  Students play Tic Tac Toe after doing some routine work.

Day 2:  Students use rational functions in introductory problem solving.  This year I am using Illuminations, "Light It UP" unit.  Students will work in groups ... I may help them get started - use leading questions - but then I want them to grapple with the problems themselves as much as possible.  I will set up the experiment ... maybe in 2 - 3 stations so that students can do the experiment as well.

Day 3:  I have to get pumped up for this day ... we will add and subtract algebraic functions and simplify complex fractions.  My students are so squeamish about fractions.  I would really like to talk with their previous teachers about how fractions were handled in middle school, algebra 1, and geometry. This will be a skills practice day!  I'm thinking I will give every student a rational function on an index card.  They will partner up with another student - find the sum and difference.  Then I'll play a few seconds of music.  When the music stops, they  partner up again - find the sum and difference again.  We may do this 4 or 5 times.

Day 4:  We solve rational equations.  I haven't thought this far yet ... but I'd like to create some form of partner practice that is self-checking.

Day 5:  We will tackle the more traditional word problems that involve rational functions like rate of work problems.

Day 6: I will use this as a catch up day.  I know 90 minutes sounds like a long time but it's every other day ... and so often skills that I think we master in our 90 minutes need reviewing.

Day 7:  This will be our official test review day.

Day 8:  We will test!

This about 5 school weeks!  The only other unit that we spend this much time on is the quadratic unit.

So ... how do you address rational functions?  Do you have any activities that help students with adding and subtracting?  What are your favorite application problems for rationals?

Creative Art Project Introduced!

Yes, some students were ho-hum ... but others ...

... explored all the trig functions to see what shapes they would make
... imported a picture into Demos to see if he could outline it with equations
... discussed the virtues of a single drawing versus a scene
... asked if abstract drawings were acceptable
... tried a few basic functions and how to limit the domain and range

I wish now I hadn't given students 3 whole weeks to complete the project ... I can't wait to see their art!

So what is expected and why?

Our algebra 2 curriculum is structured around a series of parent functions.  We introduce the concept of functions at the beginning of the year.  And then we start marching through seven of them!  With linear functions we explore systems of equations.  Then we look at absolute value functions and how they are related to linear functions.  We end our first semester with an intensive study of quadratic functions ... first examining the graph, transformations, and using graphs to solve problems.  And then we solve quadratics and problem solve some more.

In our second semester we jump into radical equations first since they are inverses of quadratics ... and then we take a detour to study rational exponents.  Next up are exponential functions and logarithms.  And last we learn about rational functions.  That's where we are now.  We are finishing our first unit on rationals which is the graphing unit.  Then we will simplify, solve, and apply rational functions to word problems.

So this creative art project is planned with the purpose of reviewing these seven functions, their transformations, domain, and range.  Students must use at least five of the seven studied to create art.  Their artwork must have at least 12 equations total but as we discussed today, most will have many, many  more.

In the past I would have students create this work on paper.  BUT oh my, DESMOS to the rescue!  How much nicer to have the art online, equations clearly identified, easy to see what students did! Now the focus is on transformations, limiting domain and range ... not on their ability to graph the functions by hand.  The thinking process is different ... better from my perspective!

Here is a link to the handout I gave students.  I can't wait to share their work with you all!  Check back after April 28!

Excuses! 2048! Testing! Spring Fever!

I have neglected blogging lately!  It could be due to playing 2048 too much!  A moment of honesty ... my students were winning and I hadn't won yet.  I would get frustrated; make wrong moves.  So I had to keep playing to win at least once!

Second reason for not blogging is the crazy schedule for the past week and a half.  Two days of STAAR testing messed with several days of class.  Finally I think we are back on a regular schedule - all students should be ready for their unit test on attributes of rational functions on Tuesday.

Even with the above excuses and a heavy case of spring fever, I am excited about the Desmos creative art project I just assigned to students.  The students seemed excited too.  I can't wait to see their creations.  The purpose of the project is to review the seven parent functions we have studied, their transformations, as well as the necessary emphasis on domain and range.  Projects will be posted towards the end of April.

We are about to embark on solving rational equations and applying those skills to word problems.  I'm still working on lesson plans but I know we will use Illuminations problems from the activity Light It Up.

If you have good ideas about how to teach students who recoil from factions how to add and subtract algebraic ones, I'd love to hear your thoughts!

Working on Instructions for Project in Desmos

I'm working on instructions and a rubric for students' project in Desmos.  The purpose of the project is to review the seven parent functions we have studied and to give students an opportunity to demonstrate creativity with those functions.  A key skill will be limiting domain and range.

I don't have the rubric yet ... it needs to be simple.
Here are the instructions for the project.  I would love your feedback!

Lesson Planning - Rational Functions continued

Day 1 on Rational Function is tomorrow.

Students will share their research on cantilevers.  That will take only a few minutes.  I want students to think briefly about the possible variables in designing a cantilever.

Then we will build our own spaghetti cantilevers.  Students will measure the deflection in centimeters as they hang a weight on the end of spaghetti.  The variable will be the number of spaghetti pieces.  We will start with just one piece, then a bundle of 2 pieces, then 3, 4, 5, up to 8 pieces of spaghetti.  If our data collection is clean we will get data that creates a rational function curve.  This activity will serve as the introduction to the function. 

As we analyze our data, I want to make a connection to the concept of reciprocals.  We will examine the linear parent function and what happens when we graph it's reciprocal.

Last, I've prepared a guide for students to examine the transformations - what happens to the parent function when a, h, or k are involved using desmos.  I'm hoping students will have time to start the exploration in class and will finish it for homework.

Micro-Project: Cantilevers!

This semester one of my goals is to give students opportunities to work on mini-projects - mostly one-page explorations of math outside of the typical curriculum.  I wrote about the one we did in February here.  (I wanted to share some of these but I didn't ask students to leave their last names off of the projects ... learning that if I want to post their work, I have to specify first names only).

Today I gave students a "micro" project ... smaller, shorter than even a mini project!


We start Rational Functions on Friday.  We are going to start our unit with a math lab to collect and analyze data.  The lab on Friday uses spaghetti to make a cantilever.  We'll measure the deflection as we increase the number of pieces of spaghetti.

I suspected that students might not know what a cantilever is.  So I asked them to find an interesting example of a cantilever.  I created template slides in Google Presentation.  Each student puts their fascinating find on a slide with the link to the picture and a short blurb about the cantilever.

After a short test today, students had time to begin their work.  Already nearly half of my students have created their slide.  Here are examples (I haven't proofed these) ...





A couple of the students as leaving today "made fun" of how easy this homework is.  I'm OK with that.  Now when we work with our spaghetti on Friday we will have some visual ideas of cantilevers and the variables involved in making them work!

Our next project ... using our seven parent functions to create a work of art in Desmos!

Lesson Planning Rational Functions

I participated tonight in the @alg2chat!  I often have plans to participate but too often get caught up in evening activities.  I'm glad I took the time tonight to visit ... definitely will do so next week as well!

In our chat teachers mentioned having just completed the Rationals unit and students having difficulty with it.  Since we begin Rational Functions on Friday, I was all ears!  It's been several years since I taught Algebra 2 so every unit is a new adventure for me.

I know as we get into solving rational equations that students will struggle with the algebraic manipulation.  We have a couple of weeks before we get to that part.  As if our custom with each parent function we start first with the graph ...

As the discussion progressed, Jen Silverman shared a couple of activities from a precalculus class.  One of them caught my eye.  The document was a discussion starter.  It had several graphs and equations with this question: Make as many connections between the equation and the graph as you can.

That was just the kind of open ended question I've been looking for as a way to start the thought process about rational functions.  The equations on her document were more difficult than the ones we will study.  So I simplified ... and scaled down to just one graph, one question, one discussion.

Here is my copy of the activity.

Lesson Planning ... spring break is over ...

Long breaks spoil me ... can't say I'm ready to go back to school ... I know once I get there I'll be fine :)

So this weekend as I was looking over my plans for Monday I was not satisfied with what I had.  The topic of the day is properties of logs.  I decided to visit math blogs to see what others had done with this topic.  I found many ideas.  Here are my plans for tomorrow:

Lesson Plan March 17

1) Opening Activity:  We may share spring break stories ... or I might show this 3 minute video on St Patrick's Day little known facts!  I also love this infographic - St. Patrick's Day By the Numbers!

2) Warm-Up with this powerpoint ... practicing just a little bit of mental math related to logs.

3) Discovery  Discussion of Properties of Logs using Kate's outline.

4) For practice I created 4 sets of problems.  In each set there are six problems to condense and six problems to expand.  I orchestrated the problems so that the condensing and expanding are the same problems.  Students will work the problems on the cards they choose.  Then students with the same colors will get together and check their work making sure their answers match the problems on the A and B cards.  We will do a second round, so students will swap colors with a classmate and repeat the process.

5) Last I have a typical routine worksheet for students to complete.

Students will take a short test on properties of logs in their next class.

We tried to finish our unit on Exponentials and Logs before spring break but didn't quite make it.  This is the last topic in the unit.  Rational (or Reciprocal) Functions are up next!

Spring Break

This week is Spring Break - and yes, it's almost over.  The tension creeps up my neck as I type those words!  Spring Break came a bit early this year, but it was very much needed ... by teachers and students!

So this week ...

• Shopping for furniture for our newly enclosed patio
• Creating a special gift for my niece who is getting married in April
• Discovering the joy (and trials) of Legos as my 4 year old grandson's eyes light up
• Reading - thanks to free books identified by BookBub
• Catching up on MTBoS bloggers and their classroom activities
• Gardening ... putting potatoes and tomatoes in the raised bed boxes
• Eating out ... finding 1/2 price appetizers at Jack Allen's Kitchen to be amazing

And yes, thinking about our next unit in Algebra 2 ... rational functions.  I'll write more about it in a separate post.  We'll start with creating spaghetti cantilevers to collect data and analyze.  Then we will head over to Desmos to observe rational functions on the graph and identify attributes/transformations.  The notes needed seem heavy ... I've been thinking about how to organize them for students.  And that's as far as I've gotten!

And now to relish the remaining hours of spring break ... 

Weaving short discovery activities in with direct instruction

Lisa wrote this blog post recently on using less direct instruction in math class.

The math community online often blog about the discovery activities, the explorations, the big activities.  But often we don't discuss clearly how direct instruction and those activities are intertwined.  We discuss even less the day to day details when we are working on the ordinary, the nitty gritty!  I had hoped my 180 blog would have more depth, be more reflective, even about the ordinary days but I find that really difficult. One of my goals is to learn how to reflect better on those ordinary days!

I structure my class typically one of two ways.  After our warm-up or knowledge check, I invite students to engage in a short discovery activity.  I follow that up with specific notes and then additional practice.  The other structure I use is starting with notes first and then group practice that has a self-checking mechanism built in.

Finding short discovery activities is a challenge.  I am learning to use Desmos as we explore each of the required functions in our curriculum.  Setting up an exploratory activity to identify transformations on Desmos is easy and engaging!

When introducing exponential functions, we started with a very short activity.  I gave students this problem:

Each spring, the nation’s top 64 college basketball teams are invited to play in the NCAA tournament. When a team loses, it is out of the tournament. Complete the following on your own.

1. How many teams are left in the tournament after the first round?
2. Create a table to show the number of rounds and the number of teams left at the end of each round. 
3. How many rounds of games must be played?
4. Graph the points from the table on the grid below. 
5. Examine the graph. Is the function linear? Quadratic? Other? Be ready to explain.
6. If the table was extended indefinitely, what would happen to the y-value?

After students worked on this activity for just a few minutes, I asked them to share their work with their table groups.  I asked students to specifically discuss #5.  Then I asked groups to share their description of the graphs they created.  Their descriptions were the lead-in to my direct instruction about basic attributes of the exponential function.

After basic notes, it was time to explore our work on Desmos.  I gave students this basic form:

Use https://www.desmos.com/calculator to explore exponential and log functions

What to enter in the calculator	Describe what happens in this column
Enter bx and use the slider “b”
Enter 2x-h and use the slider “h”
Enter 2x + k and use the slider “k”
Enter a * 2x and use the slider “a”

Use this activity to prepare to explain how a, h, and k affect the exponential function.  You can also view other bases; I used 2, but you could use any other number!  After completing the basic transformations, try creating equations with multiple transformation on your own.  Record your work.

After exploring the effects of a, h, and k, and discussing their results with their partners, I pulled the class together again to emphasize a few key points in the form of direct instruction.  Students took notes.

We ended the class with questions something like these:

1)      Given the function 2^(x+4)-3, write the new equation if this equation were translated up 4 units, left 1 unit and vertically compressed by 2/3.

2)      Given the function (1/4)^(x-2) + 5, write the new equation if this equation were translated left 3 units and down 1 unit.

3)      Given the function -3^{(x -7)} – 2, write the new equation if this equation were reflected across the x- axis, translated right 3 units and up 3 units.

On the next day we started with notes, learning to differentiate between exponential growth and exponential decay.  I demonstrated how to solve problem situations involving exponential functions using graphs.  The notes were short/sweet.  I then gave students problems to solve as partners. 

We used a similar format to introduce logarithms.  Instead of a problem situation to create the first graph as in the March Madness example, we used the concept of inverses to invite students to "discover" the basic attributes of the logarithm function.  The process was similar - next we went to Desmos to consider the transformations.  And last we worked on problem solving using graphs and tables.

Weaving in and out of short discovery activities, following up with notes and practice seems to be effective.  I do have some students who want me to give them notes straight up without having to grapple with the math first.  They were quite clear about that in my first semester reflection piece.  And I understand that they feel tentative and are fearful of making mistakes.  It is for that very reason I don't give up on providing short discovery activities.

Mini-Project and Lesson Planning

This week I want to get back into my blogging routine.

I gave a unit test last Wednesday and Thursday on the attributes of exponential and logarithm functions.  Friday and Monday are are given to the district benchmark test.

Then TUESDAY .... we just into solving exponential and logarithm functions.  I am working on last minute plans for the unit!

So with the long testing sequence, I decided to assign a mini-project.  I borrowed ideas from Fawn and from Sam to create a very similar mini-project assignment.






This is a very specific homework assignment ... requiring students to explore some math beyond our typical curriculum.

Already a few students have sent their work via email.  A couple of them worked on a post from Math Munch and explored the app called Silk.  Oh what fun!  Because students talked about it, I had to go to Silk ... I created a flower and loved it!

Tuesday is coming fast ... I know I'm starting with Mathalicious, iPod dPreciation!  From there ... hmmm ...

The "boring" bit

I read Dan Meyer's post on Teaching the Boring Bits this week.  Our topic beginning Friday is rational exponents.  We will spend just 2 class days on the skills of simplifying and solving with fraction exponents.  This is in preparation for delving into the exponential and logarithmic functions.

The focus on manipulating algebraic expressions - skill only - has the potential to fall under that category some call the boring bits.  I decided to flip the two lessons ... partly because of the non-interesting factor ... a way to shake up the classroom.  That alone of course isn't enough necessarily to make the task any more interesting.  (I also work in a 1:1 environment and look for ideas and structure to use the technology we have more fully).

So I re-read Dan's post.  I have a couple of thoughts.

I can engineer an argumentative discussion around the question, "Is 0 to the 0 power equal to 1?"

And/Or ... in the stations I am setting up, I can structure my feedback to be just enough at just the right time so that they "grapple" with challenging problems.

I'm looking for a few challenging problems that will stretch my students' thinking in this skill-based unit.  Unit starts tomorrow ... still working out the kinks!

Flipping the classroom

In the midst of our study of functions we will take a two-day detour to learn how to manipulate rational exponents.

I decided that this 2-day unit would be the perfect time to try flipping the classroom.  The unit is skill-based, and I want students to have as much class time to practice as possible.  The notes are simple to give via video.

The preparation for the unit is huge.  I'm spending a lot of time preparing the stations we will use in class as well as planning the examples I want to put in the videos.

The outline of the stations looks like this:

Day 1 ... simplifying expressions with rational exponents

• Manga High 
• Card Sort
• Search 'n Shade practice

Day 2 ... solving equations with rational exponents

• Edmodo challenge problems (EOC style)
• Error Analysis
• Solving Practice

I'll share the bits and pieces soon.  I left most of the work today on my school computer.  The unit starts Friday ... so I need to step up the preparation!