Join with me in support of the Common Core

I have tried to stay out of the politics swirling around the standards and focus this blog on helping people who are trying to implement them. And, after this post, I will keep it that way here at Tools for the Common Core.

But I’ve decided it’s time take a stand against the swirling tide of insanity that threatens our work, so I’m starting a new blog called I Support the Common Core. It will provide resources, links to articles, rebuttals, and discussion to help those who are fighting the good fight. If you sign up you will be getting emails and calls for action from me and others. Tools for the Common Core will remain available for those of you who prefer a quieter life and just want to get on with your jobs.

The success of this effort depends on you. If only 10 brave souls sign up I will thank them and close down the effort. If 1,000 of you join then we can get something done (and I promise there will be jokes).

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Statement by CBMS presidents in support of the mathematics standards

I thought my readers might be interested in seeing this. The Conference Board on the Mathematical Sciences is an umbrella organization for the various societies in mathematics, applied mathematics, statistics, symbolic logic, and mathematics education.
Support Statement for CCSSMath

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Please post questions in the forums

If you have questions about the standards, please click on the Forums tab above and post them in the appropriate forum. There are forums for each K–8 domain and high school conceptual category, and a general forum for questions that do not fit in any of these.

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Lesson Plans to Accompany Published Tasks

This is a guest post by Morgan Saxby, a fifth grade teacher in Chesterfield County, Virginia, who works with Illustrative Mathematics.  Morgan has begun writing lesson plans to accompany published mathematics tasks.

A clear step after developing high-quality mathematical tasks is to develop accompanying lesson plans.  I wrote seven lesson plans to accompany published tasks, all of which I tested in my classroom.  My goal was to write lesson plans that guided students to the level of thinking required by both the standards and the practices.

One example is the lesson plan for the task What is a Trapezoid?, aligned to standard 5.G.B.4.  A student who is able to successfully complete the task not only knows the relevant content, but can also skillfully construct viable mathematical arguments (Practice 3).  The obvious question to teachers is, “How do we get students there?”  The lesson plan Plane Figure Court is one possible way.  In it, students serve as “lawyers,” charged with proving or disproving a particular mathematical statement.  For example, the statement, “A square is a rhombus” has a lawyer arguing that this is true, and a lawyer arguing that this is false.  I required that students create justifications, even if they knew their justification was wrong.  The other students (the jury) decided the case based on the mathematical arguments made, not on what they thought was correct.  My end goal here was to help students to recognize valid (e.g., a square is a rhombus because it has four congruent sides) and invalid mathematical arguments (e.g., a square is a rhombus, because if you turn it a little it looks like one).

The format for the lesson plans is consistent through each one.  The first section includes the objective(s), an overview, and the standards to which the lesson are aligned. The second section includes a detailed lesson plan, as well as suggestions for assessment and differentiation.  The third section includes commentary and relevant attachments, such as worksheets or diagrams.  Some lesson plans, like Cooking Time 1, include student work.

The initial seven lesson plans are listed below, and others will be added in the future.  Tasks with lesson plans will be tagged “Lesson Plan Included”, and are accessible under the “Resources” heading.

5.NF How Much Pie? / Cooking Time 1

5.NF How many servings of oatmeal? / Cooking Time 2

5.NF Making Cookies / Cooking Time 3

5.NF Salad Dressing / Cooking Time 4

5.OA Video Game Scores / The Order of Operations

5.MD Cari’s Aquarium / What is Volume?

5.G What is a Trapezoid? / Plane Figure Court

We’ve also been working on developing review criteria for lesson plans that develop Illustrative Mathematics tasks into full-blown lessons.  The criteria are available here.

If you’ve taken a look at the lesson plans and the criteria, we’d love to hear your feedback by September 1.

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Progression on the Number System (Grades 6–8) and Number (HS)

Here is the latest progression hot off the presses. It includes the Grades 6–8 Number System domain and the Number part of the High School Number and Quantity category (Quantity part coming soon). As usual, please comment at the forum, here.

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Modeling Progression, Take 2

After talking to some teachers at PCMI on Tuesday and hearing from my fellow standards writer Jason Zimba I decided to do a quick fix on the modeling progression. The previous version ventured into territory that has been discussed on this blog: the different possible meanings of the word “model.” I decided this could be confusing, so edited it down so that it now sticks to the meaning of the word as used in the standards. The new version is here.

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Progressions preface and introduction, and updates to Algebra and Functions

A a couple of things today. First, a a draft of the front matter for the Progressions, including an introduction explaining the sources of evidence, organization, and terminology for the standards. It also lists the members of the work team that produced the Progressions, who have been sadly unacknowledged until now. I would like in particular to call attention to the work of our editor, Cathy Kessel, who is also an occasional contributor to this blog.

Second, thanks to work of Al Cuoco, we have updated versions of the Algebra and Functions Progressions.

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Draft of Modeling Progression

Isn’t summer wonderful? This has been sitting on my desk for a while, waiting to be typeset. Some teachers at PCMI this summer needed it for their c-TaP projects, so I finally got to it. As always, this is still only a draft. Please leave comments in the appropriate forum.

[Edited 4 July 2013. Please go here for the latest draft.]

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The three Rs in MP8. And the E. And the L.

Standard for Mathematical Practice number 8 is probably the hardest for people to wrap their heads around:

MP8. Look for and express regularity in repeated reasoning.

There are too many words in there: regularity, repeated, reasoning. I’ve seen a lot of people latching onto one or two of these. If it’s regular, it’s MP8! If it’s repeated, it’s MP8! If it’s both regular and repeated, it must really be MP8!! One thing that is fairly regular and repeated is generating coordinate pairs from an equation in two variables. So there are lots of fake MP8 lessons out there about generating points from a linear equation in two variables to draw the graph of the equation, a straight line. The more points, the better—it’s more repeated that way. And regular.

But that word reasoning is also important. There’s precious little reasoning involved in generating coordinate pairs from an equation. But if we turn the question around, there’s lots of reasoning. Instead of going from an equation to a line, let’s go from a line to an equation. Consider a line through two points in the coordinate plane, say (2,1) and (5,3). How do I tell if some randomly chosen third point, say (20,15), is on this line or not? Given any two points on a line in the coordinate plane, I can construct a right triangle with vertical and horizontal legs, using the line to form the hypotenuse, as shown here.

Why_a_line_is_straight

It is a wonderful geometric fact that all of these triangles are similar. (Exercise: prove this!) So, if (20,15) is on my line, then the triangle formed by (20,15) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:

$$
\frac{15-1}{20-2} = \frac{3-1}{5-2}?
$$

Oops. Not true. So (20,15) is not on the line. Let’s try (20,13) instead. If (20,13) is on the line, then the triangle formed by (20,13) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:

$$
\frac{13-1}{20-2} = \frac{3-1}{5-2}?
$$

Yes! Both sides are equal to $\frac23$. And in fact, to confirm, the reasoning works the other way: if the ratios are equivalent, then the triangles are similar, then the base angles are the same, so the hypotenuses of these two triangles are on the same line. (Exercise: prove all this, too!)

So we have a way of testing whether points lie on the same line. (This is Al Cuoco’s point tester; google it.)

After testing a lot of points, we look for some regularity in our repeated reasoning. Every one of our calculations looks the same. We can express the regularity by a general statement: to test whether a point $(x,y)$ is on the line, we check whether

$$
\frac{y-1}{x-2} = \frac{3-1}{5-2}.
$$

By our reasoning, every point on the line satisfies this equation, and no point off the line satisfies it. We have discovered the equation for the line by expressing regularity in our repeated reasoning.

All the words in MP8 are important: reasoning, repeated, regularity, and also express and look for. See this post by Dev Sinha for more discussion.

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New Common Core Discussion from ETS

Check out this new document from ETS, also linked on the tools page, that describes the Common Core Math and English standards through tasks.

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