The Box Problem Updated

Phew - finished correcting those Box Problems (see post from Sept 9th 2015). They were great. Really. But so time consuming to grade. And I'm efficient - been doing this for 25+ years. I have a system where I sketch out what each section should have and points for each part. Then I do the first item in the rubric for every student. Then the second item for every student. Then the third. And so on. This way I am consistent in what I am scoring and what comments I am making. Also I kept track of good responses to each part so I could share what a good response looks like. This all took about 6 hours. For about 45 projects in all. There goes my Saturday. (I did get in a hike and a little grocery shopping).

Of course when I correct any project/performance task I make notes of things to tweak for next year (or typos to repair). So what I posted on the 9th has been updated here.

One thing I added was a "Perseverance Score" on the rubric. Our math department is trying to integrate all the math practice standards into our work but one in particular is MP1:

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

In particular my rubric states the following:

1)     Perseverence  - (math practice standard 1) Demonstrates ability to make sense of problems and persevere in solving them.

This standard states that students should be able to do the following:  Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They are able to transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"

                     In this first project students will be allowed one clarifying question but more                  questions and more direct levels of help will result in deductions on this section.

These honors students don't like this very much. They want you to tell them that they are doing something correctly. Overall though I only had a few students come see me before this was due. Everyone was in good shape and seemed pretty well prepared for this project. The scores were good. Not too many As. Lots of Bs. Because there is a tweaky little thing in the graph that the students think is in the problem situation domain but really isn't.

And students just aren't comfortable with "being suspicious" of weird behavior on a graph. They just assume since it is in quadratic I it must be part of the problem situation domain. But a few were curious and investigated and really understood that weird little bit.  I LOVE it.

My mantra this year with my precalculus student is to BE CURIOUS. Investigate weird stuff. And we've been looking at functions that do weird things (especially rational functions and piecewise right now). So that little section in this project emphasizes that.

When I hand these projects back, I'm going to go over each part (briefly!) and show them a "good" response. There were lots of good responses for each part. But I chose one and photocopied it (deleting anything that showed the identity of the student who did that response) and will show it to the class with my document camera. I'll do this for each part but then also post a pdf document of all the parts together on my web page that students can look it over. And consider their responses with respect to what I deem as good responses. (I don't want to post it here, just in case sometime in the future a students stumbles upon it and uses it to do their project. You never know. If a teacher is interested in this composite document, just send me your school email address and I'll send it off to you).

Hopefully this will help the students with their next project - Minimizing the cost of a soda can.

http://www.debbiewaggoner.com/math-practice-standards.html

Algebra One Sequences - Honeycomb Performance task

We start the school year with a short little unit on sequences. Not the usual longer approach you see in Algebra 2 or PreCalculus with sequences & series. Just dabbling in arithmetic and geometric sequences. A little bit with Fibonacci and Pascal's triangle. And maybe some "other" types (like triangular numbers). It's a nice way to start the year, looking at patterns and making sense of them.

I've had a few posts on this unit already. Some specifics on my ISN foldables and some expanded thoughts on developing the rules for arithmetic and geometric sequences.

This post is to share a little more information on how we end the unit. We do the usual test with study guide prep. But every unit in Algebra 1 in our school has a performance task. This one is called the Honeycomb Performance Task.

The Basic Task:

You are an engineer for Plasticore Corporation who makes custom tables of varying sizes for banquet halls. You have been asked to design and manufacture round banquet tables with layered honeycomb cores. Each honeycomb in the core has a height of one foot.  The size of the honeycombs do not change, as the size of the table increases the number of honeycomb layers increases.  

a)  Your team at Plasticore has been asked to build a table that measures 15 feet in diameter. Each individual honeycomb cell costs $0.25 to manufacture, including material and labor costs. What is the total cost to manufacture the honeycomb core for your table?

b)  Plasticore just got an order from the Aquaturf for 80 tables that measure 15’ in diameter. What is the total cost for the honeycomb cores used to build the 80 tables?

c)  Since The Plasticore Corporation is located in the state of Connecticut, Aquaturf has to pay 6.35% sales tax on the total cost calculated in part (b). Also Plasticore charges a shipping and handling delivery fee of 10% of the pre-tax cost.  Calculate those two extra fees and find the final total cost of the Aquaturf order of eighty 15 feet diameter honeycomb core tables.

Preparation before the task:

Before students are given the performance task we do a little preparation (yes of  course there are all the lessons of this unit but also...) with a honeycomb exploration. 

Honeycombs are a network of hexagons that bees create for their hives. They are a wonderful natural shape because they efficiently use material for the amount of area they create and they are strong. This task has you creating a network of honeycombs.

Work with your group to create the following honeycomb stages on hexagonal grid paper. You should have 5 different color pencils, markers or highlighters.  Let’s see how a honeycomb shape can be built starting with a single hexagon and building out.

PART ONE:

1A)  The center of your honeycomb is the black hexagon in the center of your grid paper.  For stage one, choose a colored pencil and color all the hexagons that touch the black hexagon.  How many hexagons did you color for stage 1?  Enter this number into the table below in the “Number of Hexagons Added” column. 

1B)  For stage two, choose a different pencil and color all the hexagons that touch a stage 1 hexagon.  You should now have two “rings” of hexagons colored.  How many hexagons did you color for stage 2?  Again, enter this number into your table in the “Number of Hexagons Added” column. 

1C)  Now choose a different color and color all the hexagons around this first layer you did. Enter this number into the table for stage 3. Continue this coloring method & recording for stages 4 & 5. 

Students then create a visual pattern with colors that they investigate using what they know about sequences. Everything is detailed in the document honeycomb patterns.

Once they've got all that (they do this in small groups and compare results, tweaking as they go). They are ready for the performance task.

We talk through the scenario. And I show them a model of a honeycomb core "table" that I made (with corrugated cardboard, an xacto knife and glue gun).

This model has an accompanying circular "cover" so they can see how the table has the core in it to stabilize and strengthen the table.

Then they are on their own to complete all the parts of this task. We do focus on the mathematical standards of "Make sense of problems and persevere in solving them", "model with mathematics" and "look for and make use of structure". Especially the perseverance piece. So there is no more classtime for this task, they are on their own and the rubric has a 4 point score for perseverance. (we've had a problem with students "not trying" so this motivates them to really use all the tools available to them and persevere!). 

All documents are found here. 

Car Comparison Performance Task - Exponential Functions

http://mrec.tv/car/car-comparison.html

Here's a performance task that's a great way to tie in what students know about exponential functions.

Big Idea

Students are given the following task:

Determine if there is a significant difference in value over time for two types of cars by determining and examining their approximate rates of depreciation and using that and other factors to determine which car would be a better value for your needs. 

Setting the Stage

First it's a good idea to brainstorm how students might compare two cars they are interesting in purchasing to determine which is the better deal. Is it enough to compare purchase price? Are there other factors they should be considering? Get a good conversation going, have students list things they should consider on their papers and then share in whole group on the board.

Students have learned about car depreciation in a general way. If they are still uncertain as to what that is you can search on youtube for a possible video to show them. I have students read this article on depreciation. I tell students they are going to compare two types of cars to see which is the better value. They will use depreciation rate as one way of comparing and then choose one other characteristic. We'll have time in class to collect data on the "Kelley Blue Book" site.

Then usually have students then read the article for HW and also read over the performance task details. They are also to think about two types of cars they might want to compare. They might want to compare SUVs and mini-vans. Or two-door sports cars vs 4 door sedans. Or american made vs foreign made. There are a lot of possibilities and encourage them to discuss with their parents about what they think. Students should come in with a general thesis question next class. They do not have to collect data yet or even choose specific makes of cars yet. We'll do that the next class. I also tell students they can work in pairs (this is the ONLY PT they can work together on. I hate having students work together on a graded assignment but every year students beg me to let them do this. And every year there are students who end up doing most of the work for their group who come to me to complain. Sigh. But I figure one PT for them to do together will appease them and it will be less grading for me. I do a big stern talk about being sure you carry your weight if you work in pairs, etc).

Collecting Data

The next class students come in all ready to collect data and start this PT - so we have a computer lab day. Students will collect data on the internet. I have to plan ahead to do this in our school, we don't have many full class sets of computers and I have to be sure to sign up a few weeks ahead of time. Warning - if you have students use ipads to access the site the "tablet" site is a bit different from the general webpage. So be sure you are familiar with what that looks like and how to navigate.

HUGE DISCLAIMER
This can be a little confusing. We really cannot collect true data for calculating depreciation because we can't buy a car and then record its value each year for 5 years. We have about an hour to collect our data. So you have to explain this very clearly and carefully to students so they understand how we are sort of fudging our data but that its still valuable to us even though we aren't truly tracking the value of a single car over time.

What are doing is collecting data on two cars that have been manufactured for at least five years. Students will choose two specific cars, go to Kelley Blue Book (http://www.kbb.com/)  and find the new price of their first car. I strongly recommend that you work through this yourself on the site before you have students do this. My performance task directions are pretty specific but they may redesign their site so things change (and tablet or phone mobile sites look different).

I give students a data collection worksheet to record their data.  They will use tables that look like this:

So the new price goes in for the y-value of "New". Then they look up the price of the same make & model car for the previous year and that goes for the y-value of a 1 year old car (x = 1) and so on with the used car prices. They will use the x values noted in the table when inputting data in their calculator NOT the actual year of the car.

They do this for both cars.

Warning - all sorts of things can happen when students are collecting data. They can find that one of their cars was not manufactured for a given year or that the value is lower for a newer year than an older year. Weird stuff can happen. So you have to think on your feet and decide what they should. Sometimes they have to start over and find another car model.

I know some teachers in my school limit students to a dozen different cars to choose from that the teacher researches ahead of time to be sure the data makes sense for those cars. This is a good idea. I have advanced students so I like them to learn about the quirks and limitations of working with the internet. They are pretty good-natured about having to start over so it has worked out fine for me.

Pulling it All Together

Now students have collected data and they have information about two cars. They need to create a nice table of all their data and get pictures of each of their cars. They should have something like this:

Then they have to apply the algebra they've learned, collect some more information about their cars and pull it all together to make a comparison. See the performance task details to see very specifically what they are doing.

The next class I like to talk briefly about what they do next. They need to have some sort of presentation of their results. They can turn it in as a powerpoint, as a booklet or as a poster. They do not have to present in class. Sometimes I get a pair of students who wants to present their powerpoint and I give them the opportunity as long as they can do it in ten minutes or less. But generally students want to just turn it in. It's a great idea for them to present their results but I'm always am working with time constraints - so much to cover, so little time. Anyhow, I show students this PowerPoint that gives a "sample presentation". I won't post this on line for students to refer to because I want them to have things in their own words but I like them to have a good idea of what makes a good and complete presentation. I don't give students any more class time to complete this task. They have about a week to turn in their final work. 

How I grade this

I have a teacher rubric with details on how I grade this. Feel free to contact me for this and I will send it out to you.

Unit 5 Scatter Plots, Correlation, and Line of Best Fit

Right after mid term exams we cover this mini-unit. The lessons take about a week (remember we meet every day for 82 minute blocks) then we work on our performance task. We use one class block to get started on this and then students get about a week to complete it.

Students worked with scatter plots in eighth grade, so they know about creating scatter plots and correlation. They might not have done much with lines of best fit and probably didn't learn how to do it on the graphing calculator yet. We do them both "by hand" and using the graphing calculator (TI-84 or TI-83 also works).

PreAssessment

Here's a "preassessment" kind of thing that works well in determining understanding and having students think about what we will be doing next. I started doing these in a multiple choice format so I could "plicker" to quickly assess how students are doing. If you have checked out https://www.plickers.com/ it's great!  You have to have a smart phone of some type (android or iphone) and the students hold up cards to display their answers and you read the answers by scanning across the room with your phone. It works really well and gives you a quick real time summary of what students know.

The Lessons

Correlation:

We start with what they understand correlation to mean. Then we fill in our left hand learning page about correlation.

Then students sort cards that indicate possible relationships. They decide if they have positive, negative or no correlation. Here are the "cards", caution students can take a long time cutting things out, you can hand this out the night before and have them cut them out for class (use envelope on inside back cover to hold the cut out cards).

I have them glue in a few cards of each. I think next year I'll make this into a foldable - just fold an 8.5x11 sheet of paper in half and create a little booklet. Front cover could be positive correlation, inside negative correlation and no correlation. It will give us more space. At bottom of each can have "sample correlation sentence" corresponding to one of the examples on that page. 

Scatter Plots by Hand:

Then we do work with making scatter plots by hand. Here are some materials I use with students: NBA players and Full Analysis example. And a foldable to summarize the steps. 

 We also talk about interpolation & extrapolation. Here's a foldable to illustrate this:

We fill in definitions of interpolation and extrapolation (apparently I didn't do that, this was before I started using a document camera). Next year I'll add a place to also write a correlation sentence for this example plus what type of correlation is shown.

Correlation and Causation:

The tricky thing about correlation is that most people assume the correlation implies causation, wrong!  This is a common misconception and something that warrants a bit of discussion. Here is an introduction worksheet that I have students work together on. *updated in 2016 I used a powerpoint to lead the discussion on this topic. All materials in the link at the at of this post.

Then we summarize with an ISN foldable. (but I put the examples from the foldable on the board before handing it out for some discussion first).

with cute little saying I found on pinterest:

Common misconception - students think a relationship is either a correlation or a causation. They don't see that something is either a correlation OR a correlation and causation. It's a bit tricky and abstract for them. 

Outliers & Influential Observations:

Next topic is Outliers and Influential Observations. I've started to do both because not all "distant data points" are outliers. If they follow the general trend line they are influential observations. I didn't actually know that and I think it's important to know. Here's a classwork discussion worksheet. And here is my ISN foldable for summarizing. *again in 2016 I used a powerpoint to lead the discussion on this. files all available at the end of the post.

Scatter Plots on the Graphing Calculator:

Now we bring in some technology. Students learn how to create scatter plots on their calculators. Not all my students have the same operating system so my foldable has instructions for both. I like to have students use the [VARS] menu to insert the linear regression equation directly into Y1 and calculating the linear regression looks a bit different depending upon the operating system. Here is their instruction foldable. I expect advanced students to learn this and know how to do it without instructions on assessments. (they need to get very fluent with graphing calculator).

I also like the "diagnostics" to be "on" so they can examine the regression coefficient to evaluate the strength of the correlation. Here is the foldable for that.

Data Sets:

Here are some data sets that I then have students work with and here is the answer key. 

Performance Task:

Our summative assessment for this unit is a performance task that has students compare fat and carbohydrate content of snack foods. It's a fun activity with some good analysis. *previously I had a question that introduced an item that would be an outlier. After correcting my 2016 performance tasks I tweaked my write up a bit more (in addition to fixing typos on the rubric) so that we have a weird item in our data to start (steak!) and let students discuss what that does in the write up. All update files below.

All my files for this unit can be found HERE. I did update some of my ISN inserts in 2016. They are not hugely different and the window for blogging about them is gone so I'll probably blog about them in 2017! But all the documents can be found in my box file posted above.

Sequences Algebra One

I am used to teaching sequences and series in Algebra 2 & Pre-Calculus. But four years ago when we were given the task of aligning our Algebra 1 curriculum with the Common Core - lo & behold the first unit was "Patterns" which really was all about sequences - arithmetic, geometric and other sequences (Fibonacci, etc). And they wanted us to use function notation rather than sequence notation and students were supposed to naturally discover the formulas for a term in an arithmetic sequence and geometric sequence, ha! We gave it our best shot. After teaching this unit for three years I see now that the Common Core no longer has this as their first unit and that sequences are blended into other units. We have fine-tuned it over the last 3 years and aren't going to toss it just yet. Connecticut is making their own "Algebra for All Common Core" curriculum, so we'll just wait and see what they want us to do (it will include an end of year exam for all students in the state taking Algebra 1).

It's actually kind of fun to start with patterns, it gets students thinking mathematically. I have students do some exploring with patterns in worksheet they get for homework on the first day of school. You can find that here.

Then we develop the concepts & vocabulary. And "come up with" the arithmetic formula using what they already know about the slope intercept form of linear equations (y = mx + b) from there they see they have to find the "0 term" for the equation/formula. Geometric is harder to develop, so we kind of just give it to them. We revisit both sequences when we do linear and exponential functions. Overall it's a short unit with a cool little performance task at the end.

Foldables for Sequences

I use three sets of foldables for my ISN. Remember my page numbers are all wrong, I'm going to be better about the left side even & right side odd this coming school year. Also these are my first attempts at foldables and I'm not crazy about how everything lined up. I got better at that as the year progressed. The heading for each is a link to access that foldable via box.com.

INTRODUCTION TO SEQUENCES

ARITHMETIC SEQUENCES

GEOMETRIC SEQUENCES

Those are all Left Side Learning Pages (3 in all for this unit). The Right Side Reflect pages for opposite arithmetic & geometric are:

I don't have the actual examples - just notes to myself as to what to have students do.

And we do some cool stuff with the recursive rule and our calculators (TI-83 or TI-84). You can set them to generate a sequence recursively just by kind of following the rule. Type in the first term, [enter] then type in [+] "the common difference" [enter], You will see the first & second terms on the calculator screen. Then just keep typing in [enter] to generate more terms (the 3rd, 4th, etc). Not especially efficient if you want the 50th term but it's cool to do. And we talk about this is how computers generate terms. But they can do it super fast.

And if you are interested in my performance task, it's kind of cool. I made a honeycomb model out of cardboard for students to see what the core of the table looks like. Of course that is at school so I can't post a picture yet! Honeycomb Performance Task with the exploratory task we do the day before the PT and honeycomb graph paper.