Say goodbye to hours of lesson planning and hello to easy-to-use, standards-aligned curriculum that makes math more approachable for students and less overwhelming for teachers
Say goodbye to hours of lesson planning and hello to easy-to-use, standards-aligned curriculum that makes math more approachable for students and less overwhelming for teachers
Teachers who use the curriculum share resources here. They range from reviews to projects to Smart Notebook Files for smart board use.
Take the polynomial challenge! Try to come up with as many equations as you can for the 20 polynomial graphs given in 40 minutes of time.
Here’s the worksheet:
And here are the links to the challenges. Each is a Desmos graph that you can experiment with to find the correct equation. Hyperlinks are also included on the worksheet, but you can get to each challenge here.
So, I recently had the pleasure of speaking at the NCAMS Winter Luncheon about issues with Common Core Algebra II. I shared work that I had been doing and, of course, had the ubiquitous Power Point presentation as well as other resources.
I promised the great Long Island teachers who were attending that I’d put up a post with many of the materials that I showed them. So, here it is. I’m going to try my best to remember all of the requests I had for shared materials.
NCAMS Winter Luncheon Power Point
Link to Statistical Simulators and GAISE Report
Link to Topics Not Tested on CC Alg II Regents
Those were the main links that folks asked me to include. I thought I would also give links to some of the Desmos graphs that are tied to CC Algebra II.
Unit Circle and the Sine Function
General Unit Circle with Inscribed Right Triangle
The Standard Form of a Parabola with Focus and Directrix
These last three are what I call “Fitting Games.” They are images of graphs you can give kids to try to fit them by experimentation. I’ll be coming out with many more of these because I think they are a great way to get kids to think about what equation models a graph using trial and error.
Sean Finity of Marion Central Schools sent us a great follow-up lesson to our Unit #6.Lesson #11 – Locus Definition of a Parabola. He introduces the “Standard Form of a Parabola”, i.e.
4p(y-k)=(x-h)^2
You may have never seen this formula before, or perhaps you have. It can be used to generate the formula for a parabola if you know it’s vertex (h,k) and the distance between the vertex and the focus (or equivalently the distance between the vertex and directrix). This is the parameter p.
This formula shows up in one of NYSED’s Common Core Algebra II nonsample-sample questions from last spring.
Sean has put together a great lesson on how to use this formula and shared it with us. Here’s his lesson (with a homework set):
U8D4 General Equation of a Parabola
I created a Desmos interactive graph that allows people to play around with the focus and the directrix and to see how they interact with this formula. Here it is:
The Standard Form of a Parabola – Desmos Activity
I will admit, I’m not the biggest fan of the Standard Form formula, but I really like Sean’s lesson and, given the likelihood of this topic on the exam, I think it is a good one to add in. Again, I can think of very little less “Common Core” than memorizing a formula that doesn’t really link back to the locus definition of a parabola (well, it does, but I don’t know how many kids will appreciate that fact). Notice that this formula is not in the standards nor is it on the NYSED approved formula sheet. But, it sure makes finding the formula of a parabola given its focus and directrix a lot easier.
Thanks Sean!!!
Last year Julie Merana-Spanarelli submitted tons of great stuff to us, mostly SMART Notebook files for Common Core Algebra I. She has made pretty much a file for every eMATHinstruction Common Core Algebra I lesson. We’ve posted the first three units earlier this year. Here are Units 4, 5, and 6. Thanks Julie!!!
Unit 4
CCAlg1-U4L1-Proportional Relationships
CCAlg1-U4L3-Nonproportional-Linear-Relationships Smart
CCAlg1-U4L4-More-Work-Graphing-Linear-Functions-Lines Smart
CCAlg1-U4L5-Writing-Equations-in-Slope-Intercept-Form Smart
CCAlg1-U4L6-Modeling-with-Linear-Functions Smart
CCAlg1-U4L7-More-Linear-Modeling Smart
CCAlg1-U4L8-Strange-Lines-Vertical-and-Horizontal Smart
CCAlg1-U4L9-Absolute-Value-and-Step-Functions Smart
CCAlg1-U4L10-The-Truth-About-Graphs Smart
CCAlg1-U4L11-Graphs-of-Linear-Inequalities Smart
CCAlg1-U4L12 Introduction to Sequences
CCAlg1-U4L13-Arithmetic-Sequences Smart
Unit 5
CCAlg1-U5L1-Solutions-to-Systems-and-Solving-by-Graphing Smart
CCAlg1-U5L2-Solving-Systems-by-Substitution Smart
CCAlg1-U5L3-Properties-of-Systems-and-Their-Solutions Smart
CCAlg1-U5L4-The-Elimination-Method Smart
CCAlg1-U5L5-Modeling-with-Systems-of-Equations Smart
CCAlg1-U5L6-Solving-Equations-Graphically Smart
CCAlg1-U5L7-Solving-Systems-of-Inequalities Smart
CCAlg1-U5L8-Modeling-with-Systems-of-Inequalities Smart
Unit 6
CCAlg1-U6L1-Simplifying-Expressions-Involving-Exponents Smart
CCAlg1-U6L2-Zero-and-Negative-Exponents Smart
CCAlg1-U6L3-Exponential-Growth Smart
CCAlg1-U6L4-Introduction-to-Exponential-Functions Smart
CCAlg1-U6L5-Percent-Review Smart
CCAlg1-U6L6-Percent-Increase-and-Percent-Decrease Smart
I’m going to be speaking in Rochester tomorrow about the use of statistical simulation in Common Core Algebra II. I think this is a deep topic that we are only beginning to understand. So far as I can tell, inferential statistics has never been taught in a standards system that clearly places simulation ahead of more formulaic approaches. Don’t get me wrong, I completely support the move for more experiential education in probability and statistics.
The ideas that underlie the statistics standards, especially in CC Alg II, come directly from the GAISE Report (Guidelines for the Assessment and Instruction in Statistics Education). I have to say, for a report with such a laborious name (notice that assessment comes before instruction), it’s a really great read. And, yes, I know how weird that sounds. Here’s the report:
GAISE Report on K-12 Statistics Education
The report encourages the use of statistical simulation to help students understand the connection of probability to inferential statistics. Given the almost impossible task of assessing a student’s understanding of simulation or the use of it, we will likely have to teach the traditional statistical formulas. Keep in mind, these formulas are not mentioned in the standards nor in the NYSED clarifications of the CC Alg II standards. Yet, what else can they place on a standardized tests. Can’t you just wait for the mnemonic for margin of error = 2* standard deviation/square root of n? That cannot be a pleasant tune.
We’ve put together web based apps and TI-84 calculator programs to do the three major simulations suggested in the standards and in the GAISE Report. After the description of each simulation, I give a link to the web app and a link to the calculator code.
Sample Normal Distribution: In this simulation, the user specifies the mean and standard deviation for a normally distributed population. Random samples of any size are then generated a user specified number of times and a distribution of the sample means and sample standard deviations are produced. This simulation can be used to establish how likely a particular sample mean would be from a given population.
Sample Normal Distribution Web Based App
Sample Proportion Simulation: In this simulation, the user specifies a population proportion value ranging from 0 to 1 (population p). Random samples of any size are then generated a user specified number of times and the distribution of sample proportion (p) values is produced. This simulation can be used to establish how likely a sample p value (p hat) would be given a population with a different p.
Sample Proportion Simulation Web Based App
Sample Proportion Simulation Calculator Code
Difference of Sample Means: In this simulation, the user inputs data sets from two treatments. The data sets are scrambled to produced random groupings of outcomes a user specified number of times. For each random grouping, a difference of treatment means is calculated. A final distribution of these differences is produced. This simulation can be used to establish how likely an observed difference in treatment means would be by chance assignment alone. This is equivalent to determining how likely the observed difference is due to the treatments.
Difference of Sample Means Web Based App
Difference in Sample Means Calculator Code
I look forward to seeing how these statistics standards play out in the coming years, both in how we educate students and how we assess them. I hope that the powers that be clarify those standards soon. I’ve always felt that I can be most creative with my teaching when the rules of assessments are clearly laid out there.
We are entering a great stage of education where internet based technology, such as our web based simulation apps, will allow students to easily experiment with the crucial (but under appreciated) connection between probability and statistics. It is a rich connection that can come alive with simulation. I hope that we have contributed both tools and lessons to get us part of the way there. We know there is still great work to be done and look forward to contributing more lessons to this topic as the year goes on.
See you all in Rochester!
Kirk
So, I’ve been doing a lot of work lately on Desmos and one thing I thought I’d try was to create Desmos graphs that helped teachers show students Riemann sums and how they converge for large numbers of partitions. It took a bit, but I think I’ve created some nice Desmos graphs for right, left, and midpoint Riemann sums.
I’m embedding the code here for each. Just click on the graph and you will be taken to the Desmos graph corresponding to the particular type of Riemann sum. Feel free to change the function, the bounds, and (of course) the number of partition intervals.
Right Riemann Sum:
Left Riemann Sum:
Midpoint Riemann Sum:
So, I was driving around with my son Maxwell (Max) and we were discussing giant pumpkins. The whole family loves carving pumpkins each year and we had just seen a television show about the giant squash species. So, he posed the question:
“How big do you think the largest pumpkin in the world is?”
Well, that made me think of many math problems, from the obvious relationship between the diameter or circumference and the weight. to estimating how many giant pumpkins would fill the Arlington Auditorium (a truly huge structure). I settled on just trying to predict this year’s world record pumpkin weight by looking at past trends. Here’s the problem in both MS Word and Pdf:
How Much Will the Largest Pumpkin Weigh
How Much Will the Largest Pumpkin Weigh
I give data for the last six years of world record pumpkin weights, which progressively are getting larger. Students then need to mathematically model the problem to predict the world record weight this year (getting near 2,500 pounds!).
I like this problem because the data is messy, yet approachable by students in both Algebra I and Algebra II. It is a real problem that has no clear correct answer as that has yet to happen. Clearly, students can use a regression approach and both linear and exponential are pretty good (r-values in the .95 range). But they could also just look at the average increase in weight per year.
I love extending this problem in both courses by collecting predictions by the entire class and then doing the statistics on them. Really neat to then finally see what the record is this year and how close the predictions are.
The whole problem lends itself two two of the primary Mathematical Practices from the Common Core:
Mathematical Practice #4: Model with Mathematics
Mathematical Practices #5: Use Appropriate Tools Strategically
Have a great Halloween season! Enjoy the fall before the darkness of winter spreads over us (hey, what can I say? not a fan).
So, Chris Desmond from Nottingham High School in Syracuse, where I did my formative student teaching many, many years ago, has done an analysis with his fellow teachers of all of the questions that have been asked so far on the Common Core Algebra I exam given in New York state.
Their analysis comes in the form of an Excel spreadsheet where they’ve analyzed the percent of questions that are tied to a given CC Standard. What is really, really amazing is that they also include snips of all of the questions themselves. This is really amazing work and would be great for any school to analyze at this point.
The Excel file is large because of all of the graphics that are embedded in it. Here is is:
Well, we’ve gotten a wonderful end of summer present from our friend Julie Merana-Spanarelli from Central Islip CSD.
She sent us SMART Notebook files for all of the eMATHinstruction lessons for Common Core Algebra I. I’m only posting Units 1 through 3 here, but will post more as the school year progresses. Thanks Julie for this great beginning of the year contribution.
CCAlg1-U1L1-Rates-Patterns-and-Problem-Solving Smart
CCAlg1-U1L2-Variables-and-Expressions Smart
CCAlg1-U1L3-The-Commutative-and-Associative-Properties Smart
CCAlg1-U1L4-The-Distributive-Property Smart
CCAlg1-U1L5-Equivalent-Expressions Smart
CCAlg1-U1L6-Seeing-Structure-in-Expressions Smart
CCAlg1-U1L7-Exponents-as-Repeated-Multiplication Smart
CCAlg1-U1L8-More-Complex-Equivalency Smart
CCAlg1-U1L9-More-Structure-Work Smart
CCAlg1-U1L10-Translating-English-to-Algebra Smart
CCAlg1-U1L11-Algebraic-Puzzles Smart
CCAlg1-U2L1-Equations-and-Their-Solutions Smart
CCAlg1-U2L2-Seeing-Structure-to-Solve-Equations Smart
CCAlg1-U2L3-A-Linear-Equation-Solving-Review Smart
CCAlg1-U2L4-Justifying-Steps-in-Solving-an-Equation Smart
CCAlg1-U2L5-Linear-Word-Problems Smart
CCAlg1-U2L6-More-Linear-Equations-and-Consecutive-Integers-Games Smart
CCAlg1-U2L7-Solving-Linear-Equations-with-Unspecified-Constants Smart
CCAlg1-U2L8-Inequalities Smart
CCAlg1-U2L9-Solving-Linear-Inequalities Smart
CCAlg1-U2L10-Compound-Inequalities Smart
CCAlg1-U2L11-More-Work-with-Compound-Inequalities Smart
CCAlg1-U2L12-Interval-Notation Smart
CCAlg1-U2L13-Modeling-with-Inequalities Smart
CCAlg1-U3L1-Introduction-to-Functions Smart
CCAlg1-U3L2-Function-Notation Smart
CCAlg1-U3L3-Graphs-of-Functions Smart
CCAlg1-U3L4-Graphical-Features Smart
CCAlg1-U3L5-Exploring-Functions-Using-the-Graphing-Calculator Smart
So, we’ve already published a map showing which units and lessons all of the CC PARCC EOY standards show up in. But, until recently, I hadn’t done the mapping in the “other direction” as I like to think about it.
In other words, I hadn’t given a table of contents that showed what standards were hit in each lesson. Well, I’ve got that done now and am presenting it here for your use as we approach the beginning of the school year.
Common Core Algebra II Table of Contents with Standards
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