Common Core math: Fix or Replace

Recall that each real number has two square roots. 

The above statement is taken verbatim from page 260 of the Precalculus and Advanced Topics Module 1: Teacher Materials (see it here) as posted on the for engage^ny website.

That statement, as written, is false. The number 0, which is a real number, has one square root, 0 itself.

The error in this statement could be explained away as due to the occurrence of a root with multiplicity two, but I have to take exception to that.  This module is a 431 page document, and the term "multiplicity" is not used anywhere in that document. Since the document appears as a "cookbook" designed for someone who is otherwise unable to teach this topic, it would seem that such a detail should be addressed, or accounted for in some other way. A very simple way would be to rephrase the statement:

Recall that each non-zero real number has two square roots. 

Back to the module itself, take note that it is 431 pages long. 

Module 1: 431 pages

Module 2: 489 pages

Module 3: 398 pages

Module 4: 291 pages

Module 5: 254 pages

Thus the Precalculus and Advanced Topics modules total 1863 pages. That makes for a very long cookbook. With that many pages, I presume that not many people will read it cover-to-cover.

So who would read it? Could we presume, for a moment, that those in need of guidance are the ones most likely to pick it up? (Or should I say, "link it up"?) Does it really do a service to those in need if it contains easily fixable mistakes?

I notices this little error (be it of commission or omission) solely because I wanted to see how NYSED was going to deal with what I used to call "math 4", and I had a problem due to this as the link to what I wished to see was missing. (If the image below is unclear, the original is currently here):

Nowhere on this page was there a link to Pre-Calculus curriculum. The only links were to the side, and they were links to the 5 modules.

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Intermission: please NYSED, is it "Precalculus" or "Pre-calculus". I can go either way, bu please let me know your choice.

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So I began to look at the modules, and opened up Module 1.

While in Module 1 I looked at page 210. That page asked a question about a reflection across the line \[y =  - x\].   I would hope that by the time they get to math 4 the students would be aware enough to know that this reflection is the same as a reflection across \[y = x\] followed by a reflection through the origin.

A reflection across \[y = x\] is \[(a,b) \to (b,a)\] and a reflection through the origin merely negates the coordinates , so this rule should be simple to ascertain. The module does indeed connect complex numbers to the concepts of transformation of the plane.

Alas! This page ultimately uses the "new" topic to validate the "old" topic, using complex numbers to verify what was previously known about the composition of basic planar transformations. How about using previously known mathematics as a foundation for new mathematics? This module seems to do the reverse. (Hey teach, this new stuff is just a more complicated way of doing the old stuff!)

For those who are unaware, transformations of the plane are dealt with (well, I hope), in Module 1 from Geometry. See Topic C here.

And we wonder why students get turned off from math?

The plotting of complex numbers as points in the plane is commonly referred to as an Argand diagram, due to the work of amateur mathematician jean-Robert Argand. I searched this module for a reference to Argand, but found none. That is like discussing relativity without mentioning Einstein. (the module soes mention video games (See Lessons 22 and 23 in the module.)

Since this Precalculus module does involve square roots of complex numbers, I would feel remiss in not commenting on another statement I found  within  engage^ny , 

There are other numbers on the number line between the integers.  They are called square roots.

This statement is from  page 28 of Grade 8 Module 7 Lesson 2 . The second sentence in this quote is downright misleading. (Every number is a square root of some number!) Unless a dissertation on rationals, irrationals, and reals is to be had, the sentence should just be omitted.

We have to do better.

The more I dig into Common Core the more I wish it would be discarded.  Then maybe we could start it over and do it right.

Ellipses can be fun!!

This GeoGebra sketch shows a step-by-step creation of an ellipse.

I am a strong supported of GeoGebra in the classroom. Although it has a steep learning curve, its capabilities and utilities make that learning worth it. Keep in mind that you only have to create a file once. You can use it repeatedly, and modify and adapt it at will.

The complete file is at http://ggbtu.be/m2625435

Super Bowl Sunday is here! Time to be healthy!

I figured this would be a good day to share a picture that kind of describes the culture as we know it. The long diatribes I will leave to other people: all I wish to do is get people thinking. To start the thinking, I would ask a simple question.

What are the best items to place between the beer and the potato chips?

Can someone forward this to Stephen Colbert?

For the record, this photo was taken last March in a supermarket near Delray beach, Florida.

A Tusi Couple, and they are not even dating!

This is done in GeoGebra, but the setting is old.
Google "Tusi couple" and follow the evidence.
Explore.

 

Is this the best NYSED can do? (Turkey continued..)

The "commentary" following the problem referred to in How do you cook a turkey? (my blog post from yesterday) states the following (page 27 here):

This question measures A-CED.A because students must create an exponential
equation and use it to solve problems.

I beg to differ: the exponential equation is not created by the student, but handed to them and credited to Newton.

Once it is recognized that this problem does NOT address the part of the common core standards it claims to, its reason for existence in this setting is gone.

This problem is connected to "Mathematical Practice(s) 1 and 4 (see page 27 here).  Mathematics practice 1 is stated here (italics are mine):

MP 1 - 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. 

This problem expects students to jump in and attempt a solution using the equation they are given.

Mathematical practice 4 is here:

MP 4 - Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

This problem expects, almost requires, that the student do no modelling whatsoever. The modelling has already been done by Isaac Newton.

Come on NYSED, fix this.

How do you cook a Turkey?

The question below is taken from the New York State Common Core Sample Questions: Regents Examination in Algebra II (Common Core) Fall 2015. The complete file can be found  here.

This question makes reference to Newton's Law of Heating. Unless you are familiar with this law, I suggest you follow this link.

Now my concern: what is this law doing in an Algebra II exam of any kind?

Here we are at exam time, and we throw at our students a concept they are unfamiliar with in such a way that we believe we are only asking them this:

a) Solve for k: \[100 = 325 + \left( {68 - 325} \right){e^{ - 2k}}\]

 b) Use your value for k to solve; \[T = 325 + \left( {68 - 325} \right){e^{ - 7k}}\]

Everything else in this question, other than the use of these two items,  involves the careful reading of a passage in the realm of thermodynamics, interpreting the contents carefully, and properly substituting values into s formulas handed out freely.  All three skills are worthy, but should be tested in a familiar realm. The writer of the question knows that "object" here refers to the turkey, but would that be obvious to the novice, reading such a scenario for the first time?

This question also includes some either dangerous or misleading (or even false) information. 

The most important is that the time needed to bring a refrigerated turkey to room temperature would be far too long, allowing for growth of salmonella among other hazards. The typical cook may wait for the surface of the turkey to feel close to room temperature, but never waits for the whole bird to warm up.

Second, Newton's law of Cooling (the proper name of the law) is based on the object having a uniform temperature. In any other situation the law provides nothing more than an approximation. As noted above, that is not the case when we cook a turkey. (Ever notice that the temperature can read differently when the thermometer is moved to a different location in the turkey?)

Thirdly, take note of this, a typical chart (from Foster Farms)

The turkey in this problem is cooked for 7 hours (who knows, maybe more?). This is either a huge turkey in an industrial oven, or someone who likes their turkey as little on the burnt side.

No matter how I slice it (no pun intended), this problem comes across to me as being totally out of place in an Algebra II exam. It amounts to a "cookbook" problem, (pun unintended) the likes of which have no place even near an end-of-the-year Regents exam in Algebra II.

GeoGebra as a Presentation Tool

I made this this morning just to be an example of how GeoGebra can be used to create dynamic presentations for teachers.
It is pretty simple to follow and requires no knowledge of GeoGebra. It can be downloaded from GeoGebratube here.
Enjoy!