Task 6-2: creating math entities

This is an extension of my system of equations exercise posted on July 21st.  These are 3 math tasks with a potential assessment method.

Create a system of two equations that has one solution.  Show what it looks like on a coordinate plane.  I would use an objective, formative assessment.

Create a system of two equations that have no solution. Show what it looks like on a coordinate plane.  I would use a subjective, formative assessment since I am looking for a higher level of thought and would value a correct answer as well as a good effort.

Create a system of three equations.  You do not need to solve, but show several graphs of what a solution might look like.  Talk about what the graph may tell you.  For example, is there one solution for all 3 lines based on your graph?  I would use a constructive response assessment.

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Math Tasks, Bloom’s, and Assessments

The following are tasks around quadratic equations that are tied to Bloom’s levels of thinking.  Included with each level is a potential way to assess the task.  Definitions of these assessments are in my August 12th entry at this blog.

Draw 2x^2+4x+8 = 0 on a coordinate plane.
Formative and objective assessments.

Review what happens when 2x^2 becomes -2x^2.
Formative and objective

Find the zeros of these parabolas.  Are they the same?
Formative constructed response

List some real life examples that parabolas could be used to implement.
Subjective and constructive response

Summarize how a quadratic equation is different from an exponential function.
Authentic and performance assessment

List 10 questions relating to quadratic equations that could be used on our next test.
Self-assessment and peer-assessment

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Internet Usage Data Visualization Lesson Plan

First, here is an interesting site that had a lot of information that could be used for data visualization.  Each lesson is meant to show how color can be used on a map to illustrate differences between countries.


This particular site shows internet users by country:


The lesson plan, as is, is a good plan for younger students – it involves color coding countries according to the amount of Internet users.  In order to be more appropriate for a high school algebra class, I would have the students derive percentages and produce other ways to show the data (e.g. pie charts, bar graphs, etc.).

It could also lead into interesting discussions as to why some countries have more users than others?  Is there any scenario where the data could be misleading?  Can some users be counted more than once?  Could there be better way or data to understand internet usage by country?

For higher grades in high school, students could look at multiple data sets and determine if there are any correlations.  For example, does GDP have any correlation to internet usage?  Look at all the data sets at the www.indexmundi.com site and make a conjecture between two of the data sets.  Then use a scatter plot and regression analysis to see if there is truly a correlation.

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Types of Assessment & Use

Formative assessments are on-going assessments, reviews, and observations in a classroom. Teachers use formative assessment to improve instructional methods and student feedback throughout the teaching and learning process.

Summative assessments are typically used to evaluate the effectiveness of instructional programs and services at the end of an academic year or at a pre-determined time. The goal of summative assessments is to make a judgment of student competency after an instructional phase is complete.

From http://fcit.usf.edu/assessment/basic/basica.html

I use quizzes and exit slips to gauge if students are understanding the material.  If many students do not know the material, I will reteach it.  If it is a small number, I will use independent time with these students to catch them up.

Assessment (either summative or formative) can be objective or subjective. Objective assessment is a form of questioning which has a single correct answer. Subjective assessment is a form of questioning which may have more than one current answer (or more than one way of expressing the correct answer).

From http://www.academickids.com/encyclopedia/index.php/Assessment

I use objective for more rote type of learning.  So, in Algebra, I would use it for addition, subtraction, multiplication, and division of integers.  I use subjective questions when helping students understand concepts.  For example, why do you think the slopes of parallel lines are the same?

I use self-assessment to help students correct and learn from their mistakes.  I have not used peer assessment often since it sometimes leads to public humiliation.  The only time I have used computer assessment is for city and state-mandated benchmarks.

With selected response assessment items, the answer is visible, and the student needs only to recognize it.  With constructed response assessments (also referred to as subjective assessments), the answer is not visible — the student must recall or construct it.

From http://fcit.usf.edu/assessment/constructed/construct.html

I rarely use multiple choice type tests since it allows guessing and makes cheating a bit easier.  I like to see the students’ work since it may provide some insight as to where the student is struggling.  Constructed responses are much tougher to grade, but can be tailored to higher levels of thought (e.g. analyze or evaluate …).

The term performance assessment (PA) is typically used to refer to a class of assessments that is based on observation and judgment.  It has been argued that performance measures offer a potential advantage of increased validity over other forms of testing that rely on indirect indicators of a desired competence or proficiency.

From http://www.answers.com/topic/performance-assessment

I have used PA (indirectly) during some guided practice activities.  I may put a step or two for student s to follow on the board, and then walk around and observe how the students solve the problem.  I have also used this when doing some role-playing activities to help students learn classroom procedures.

Assessment is authentic when we directly examine student performance on worthy intellectual tasks. Traditional assessment, by contract, relies on indirect or proxy ‘items’–efficient, simplistic substitutes from which we think valid inferences can be made about the student’s performance at those valued challenges.


Authentic assessment is a good indicator whether a student truly understands the concepts and application.  Non-authentic assessments are not as descriptive.  For example, a person may just be regurgitating information from memory or just simply may have guessed correctly.

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Math Manipulatives (Info from Twitter session)

Tuesday August 9 @ 3:30pm

What are “brilliant” activities with manipulatives?
Followed #mathchat using TweetDeck

Wasn’t thrilled about using Twitter for a forum, but decided to give it another try using TweetDeck.  Read on to see how it went …

I actually did my first tweet explaining that a manipulative is something the student can hold and “play” with.  Was immediately told that I forgot the “mathchat” hashtag.

Was asked if online was manipulative.  I replied only if student was controlling cursor.  Someone else agreed, stating that manipulative implies a hands-on experience.

Still figuring out how to reply to a tweet using tweetdeck.  Right now toggling back and forth between twitter and tweetdeck.

This link: http://mathchatarchive.wikispaces.com/004+Manipulatives has a lot of good suggestions for manipulatives.  Noticed that are esteemed professor is on this list with using finger multiplication and binary finger counting!

Gave coordinate plane on floor example using tape and students as “points”.
Cassyt said he does same thing on floor but does it to the song “Tootsie Roll”.

When asked for my “must have” manipulative, I responded graphing calculators for algebra.  Chat then got interesting – someone pointed out that there is some resistance to using them.  I thought resistance was due to calc doing work and student not understanding underlying causes.  Participant said it was also due to assessment issues.  Tough to assess whole class.

Someone talked about using m&ms and wine gums for probability manipulatives.  Wine is apparently a descriptor for the colors of the candy, not alcohol content! http://www.aquarterof.co.uk/maynards-wine-gums-p-109.html

Someone asked me to check out this cool calculus map:
Pretty neat visualization of what I would call a product roadmap for learning math.

All in all, I had a much better experience using this technology (Twitter) the second time around.  I still have a lot more to learn, but this was definitely a good step forward.  Plus, I got a site with a bunch of interesting links for math manipulatives!

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Statistics Lesson Plan (Percent Body Fat)

Lesson Objective for 9th Grade Algebra 1 Class:  Use scatter plots and lines of best fit to see if simple body measurements can determine someone’s body fat percentage.
Use these techniques to predict the body fat percentage of ourselves and others (males).

Agenda:  I will show you how to use Fathom* to investigate whether neck size is a good determination for weight.
You will then use Fathom to determine if stomach/abdomen circumference is a good way to determine body fat percentage AND predict what your body fat percentage is.
Homework will be to take measurements of male relatives and predict their body fat percentages.


Do Now/Guided Practice/Homework document

Powerpoint Presentation introducing lesson

*for those without Fathom, data is available here


to be imported into Excel to create the necessary scatter plot diagrams.

I can email you my Fathom file if you would like.

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Data Visualization Tools

Here are 5 data visualization tools I thought were pretty neat and have applicability to teaching:

1.  http://www.wikimindmap.org/viewmap.php?wiki=en.wikipedia.org&topic=algebra.  This site creates of mindmap of information contained in wikipedia.  This particular example shows one of algebra and can be used to show students just what algebra is.

2.  http://www.antaeus-data.com/concepts/about.html.  Antaeus is client based software that helps to look at data with several variables.  To me, this software is like scatter plots on steroids.  This product is beefy but also light enough to be used for high school students when teacher correlation.  It also does some neat stuff like superimposing data onto maps so that the information is much more readable.

3.  http://www.visualthesaurus.com/trialover/.  This was pretty cool and a great literarcy tool.  Type in a word and you get a visual representation of meanings with the possibility of exploring like words and meanings.  I typed in algebra and spent several minutes going off into pure mathematics and other related subjects!  Note:  there was a trial version of this software but I believe there is a subscription cost for continued use.  In a similar vein, http://www.lexipedia.com looked pretty awesome.

4.  Although I looked for predominantly math oriented visualization tools, I had to suggest a few that try to visualize a person’s personality.  http://www.psfk.com/2010/05/a-data-visualization-of-your-personality.html and http://www.brainpickings.org/index.php/2009/08/10/mbti-map/.  I think these sites are not only fun but would help teachers learn about their students and also help students learn about other students.

5.  Fathom from Key Curriculum Press.  This is not free, but many schools have site licenses.  If not, a teacher can obtain one for about $40.  This is a tool that does a ton of stuff.  It’s strength is in graphing data and helping with regression techniques.  You can import data or create your own data (e.g. flipping a coin or pulling a card from a deck).  In addition, you can do a lot of graphical representation with a coordinate plane.  For example, you can create a parabola with sliders (similar to Geogebra) and show students graphically what happens to the graph when the coefficients of the graph change.  For trigonometry classes you can create the random walk on a coordinate plane and show exactly where a person would “wander” to.

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