This is a totally awesome activity, in more ways than one. I set it up slightly differently: First of all, I built the actual box with 5 padlock hasps that all have to be unlocked to get into the box, in which I put a bunch of cheap candy.

The class I started with was my 2 sections of Algebra 2 classes. These students have been giving me fits all year. We have an “Advanced Algebra 2” class in our school for students who are college-bound in a STEM career, so the students I have in my “regular” Algebra 2 are students who (at one time) professed an interest in college, but let’s say not STEM motivated. Some of them are not very motivated at all.

I generated worksheets for these students on solving systems of equations, 10 systems for each lock (50 total problems). KUTA Infinite Algebra 2 software made quick work of this. Students had to solve each system then add up all the answers to get the combination to one lock. I divided the class into 5 groups and gave each group a worksheet. The awesome thing was they (of course) didn’t get all the right answers at first, and had to go back and CHECK ALL OF THEIR ANSWERS to figure out which one was wrong. They initially wanted me to come around and check their work and tell them where they made a mistake. I have to admit that I have probably done way too much of this in the past. However, I made them work with each other to check answers. I specifically showed them how to have another person in their group do the problem in parallel so that they can compare answers.

They were 100% engaged. Even some of my normally least engaged kids:

Once a group finally got a lock open, I dispersed the first group out to the other groups to help check answers. Basically EVERY problem had to be done CORRECTLY before ANY kid got ANYTHING. I have never seen them work together like that before.

I was initially concerned that I had made this activity too hard. It seemed very difficult for the students to make sufficient progress, and they made many disappointing trips to the box with a wrong combination. However, they were super excited when they finally got their locks open. They jumped up and down and cheered and laughed. For SOLVING SYSTEMS OF EQUATIONS!

I really like the cooperative nature of this activity. I like that it is NOT a competition and that some students have to feel like they “lost.” Every kid got a small handful of candy. It’s winners all the way around.

This activity is adaptable for almost any content. All you need is problems with numeric answers that can yield a 3-digit number for a lock combination.

Down the road bonus: yesterday I gave them just a normal worksheet on graphing linear inequalities. The cooperative nature of the lock box activity carried over and they were working cooperatively on that worksheet as well! This activity seems to foster and teach cooperative work in a culture where that was not the norm.

My only concern with this activity is that I not overdo it. I think the effectiveness would wear off if I did it too often, but I want to do it often enough so that the cooperative work part is reinforced.

I have a set of worksheets ready to go for Geometry on triangle sums.

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Steven Leinwand

Be forewarned: this is not another angry screed against the Common Core State Standards. To the contrary, after 25 years of fighting for a coherent and teachable mathematics curriculum, I am proud to express my admiration and support of the K-8 Common Core mathematics standards. I am proud to acknowledge the end of the “math wars.” It is deeply disappointing, however, that the 9-12 mathematics standards just do not yet measure up, and even more disturbing, there do not appear to be any mechanisms for revision in place.

I really like the K-8 Common Core mathematics standards because there are *fewer* standards – 27 to 38 standards per grade where recently there were often 38 to 45 or more. I like them because there are clear examples embedded into the standards to clarify their meaning. I admire the fact that these standards replace the current absurdity of 20% equivalent allocations of time to number, measurement, geometry, statistics and algebra, with standards in which number and geometry now get strong emphasis early, whereas algebra and statistics get their appropriate due *after* quantity and shape concepts are largely established. I really like how the developmental progressions or learning trajectories have been wisely drawn from high-performing countries and present a mathematical coherence that has been sadly missing in the United States. And I am very pleased that computational procedures have, in many places, been delayed by a grade, thus providing time and room for building much stronger conceptual foundations for understanding *why* algorithms, like subtraction with regrouping or long division, work. In summary, the K-8 Common Core mathematics standards are truly internationally benchmarked; are accurately characterized as more coherent, clearer, deeper and fewer; and thus provide powerful guidance to teachers. As a result, the K-8 standards are simply fairer, more coherent, and more teachable.

Unfortunately, none of these positive characteristics apply to the 9-12 Common Core mathematics standards. It is almost as if the developers ran out of time, energy and wisdom after producing the K-8 standards. The 9-12 standards are not internationally benchmarked; they provide no alternatives to the underperforming Algebra 1, Geometry, Algebra 2 sequence that exists virtually nowhere else in the world. Instead of organizing the 9-12 standards into far more sensible integrated Math 9, Math 10 and Math 11, we are given aggregated standards for Quantity, Algebra, Functions, Geometry, Modeling, and Statistics, with many standards so broad they cross course lines. Most debilitating is that we have lost the essential characteristics of *fewer and deeper,* as teachers are expected to squeeze 156 standards into three years of instruction. We make no progress and serve few students when teachers are asked to teach as many as 59 standards within a single course, resulting in more-of-the-same race through the curriculum with far too little mastery or understanding. While the coherent learning trajectories and progressions are so clearly delineated across grades within the K-8 domains, the complete absence of progressions at 9-12 severely undermines their power and quality. In summary, the 9-12 mathematics standards seem to be an afterthought and, to be blunt, they are essentially unteachable.

Then there is the mathematical content itself. To their credit, the standards emphasize functions, modeling and statistics, but they fail to remove any of the increasingly obsolete algebra to make room for these critical new emphases. To their credit, the standards move significant chunks of rational, teachable algebra to 8^{th} grade, but fail to remove much more than matrices and secants from what remains for 9-12. Why do students with no intention of a STEM career need to learn about imaginary and complex numbers? Why is the focus on polynomials still so strong when its role in non-calculus mathematics is so small?

Algebra 2 may correlate with college success, but the reality is that most of Algebra 2 content has very little to do with career readiness, workplace success or effective citizenship as pointed out so powerfully in a recent National Center of Education and the Economy report. When essentially 100% of 5^{th} graders move to 6^{th} grade and when essentially 100% of 8^{th} graders move to 9^{th} grade, it makes sense to have a common, undifferentiated curriculum for all students. But when high school feeds students into minimum wage jobs, the military, community colleges, technical schools, minimally competitive colleges and highly competitive universities, it is absurd to have a single set of mathematics standards. Differentiation of content in high schools is essential. Sure the existing 9-12 standards include 56 “+ standards” to denote what is only appropriate for “STEM intending students.” There were only 485,000 STEM majors out of 1.5 million college graduates in the class of 2007. This is from a pool of 2.8 million high school graduates in 2003, out of a cohort of 3.9 million people born in 1985. It is therefore time to ask why the Common Core still includes so much algebraic pre-calculus content that only 1 in 3 college graduates and only 1 in 5 high school graduates are ever likely to need.

To date, there is no governance structure for the Common Core. There are no mechanisms to revise or fine tune. It is time for the National Governors Association and the Council of Chief State School Officers to reclaim a leadership role or delegate a revision to a national panel of knowledgeable stakeholders. It is time to acknowledge that the 9-12 Common Core mathematics standards are not strong enough and do not adequately serve students, teachers or society. It is time to partner with Achieve, the National Council of Teachers of Mathematics, the original writing team, PARCC, Smarter Balanced and others to fix what needs to be fixed sooner rather than later so that all the positive aspects of the K-8 mathematics are not lost in the current morass of 9-12.

Steven Leinwand is a Principal Research Analyst at the American Institutes for Research in Washington, D.C. and a former president of the National Council of Supervisors of Mathematics.

]]>My principal related an anecdote about being an administrator in a school at one time and being contacted by a research firm regarding his 3rd grade literacy rate. They were trying to predict the future prison needs for the area, and were using literacy rates in 3rd grade as an indicator.

In 2011 the Annie E. Casey Foundation released a study that found that students who don’t read proficiently by the end of 3rd grade are four times more likely to drop out of high school than their peers.

Apparently there are rafts of research that show a strong correlation between literacy in the third grade and a host of other negative outcomes later in life.

It seems that we have tackled this problem as “If we can just get the 3rd graders to read well, everything else will take care of itself.” I have an issue with this approach. As any beginning statistics student will tell you, “CORRELATION is not CAUSATION.” We have assumed that illiteracy in the 3rd grade CAUSES of all of the negative things that come after, and that if we can just get them to read well by 3rd grade it will cure all their ails. There doesn’t seem to be much evidence that this is the case.

Now, I don’t mean to belittle the importance of literacy, or to imply that if we can improve literacy rates that things won’t get better for these students, because I think it probably will, but I think we need to proceed with caution here, and we also need to consider some other lurking variables (such as poverty, neglect, abuse, alcoholic parents, etc.) when talking about how to “fix” these children.

I spent a lot of years (and a lot of money) taking fish oil supplements trying to get my HDL (the good) cholesterol up where it was supposed to be. Much research was done that showed that people with higher levels of HDL have a much lower incidence of heart disease and heart attacks, so everyone was gleefully taking supplements trying to jack up their HDL so as to reduce their cardiac risk. Recently a study was published that followed people like me that had, shall we say, artificially elevated their HDL’s, and it turns out that people who elevated their HDL with supplements had the *same rate of heart disease and heart attacks** that people with lower levels of HDL did.* In other words, elevating your HDL with vitamins did *nothing* to improve your health.

You see, what is likely is that both higher levels of HDL and the corresponding rates of heart disease are determined by some other factor that we have not yet discovered or identified, and both high HDL and low rates of heart disease are *symptoms* of some other, larger, syndrome. Simply treating one of the symptoms does not cure the disease.

I’m afraid that what’s going to end up happening is that we’re going to put all our eggs into this literacy basket and its going to turn out that both illiteracy in the 3rd grade and the host of other negative things that come after are just *symptoms* of a larger syndrome (probably poverty), and simply addressing one of the symptoms is not going to turn out to be the panacea that we hope it will. I think that children who are illiterate in the 3rd grade probably have by this time been imbued with a number of qualities and characteristics that are going to leave them at risk for all the negative things that happen afterwards.

I think that this problem, like many of our other problems in public education, is going to require that we as a society tackle the thorny and complex problem of poverty in our country.

]]>Students are paired up. Each pair gets a snack bag of supplies. Each snack bag contains:

- two red sticks about 9 cm long
- two blue sticks about 7 cm long
- two black sticks about 12 cm long
- several (at least four) white (or some other color) sticks that are full length (app. 15 cm)
- two 40 degree angles
- two 80 degree angles
- two 60 degree angles

Divide the materials between the two students so that each student has one of each item (two of the whites).

Students explore the congruence shortcuts by trying to build different triangles given certain criteria. For example, if you’re exploring SSS, each student should use a black, blue and red stick to build any triangle they can. They then compare triangles to decide if they are congruent or if they are different, or if it’s even possible to create triangles that are different given that the three sides are the same:

There’s few rules: the colored sticks must be straight and connect at the ends. The white sticks are “magic” and can connect anywhere (not necessarily the endpoints). Students use the white ones to form sides of any length.

Since the resulting triangles are always congruent, SSS is enough to prove triangle congruence.

Another example: SAS. Each student must use a blue stick, a red stick, and the 60 degree angle between them (each triangle has SAS in common). Then see if it is possible to build different triangles:

Since only one triangle results from this initial setup, SAS is enough to prove triangle congruence.

SSA (or ASS) is the tricky one, of course. For this setup, tell students to use the red stick, the blue stick, and the 40 degree angle. Tell them the 40 degree angle must be at the end of the red stick, but not between the blue and the red sticks. Some students should be able to come up with the two different triangles possible with this scenario:

Which means that SSA doesn’t work as a congruence theorem (SSA is not sufficient to prove triangle congruence)

All of the triangle congruence possibilities can be explored with this kit. Have fun!

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When are we going to learn that privatizing public education creates schools focused on profit not schools focused on student learning?

]]>Here’s my first one from high school Geometry:

G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent, when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

I find standards like this frustrating. The basic standard itself is fairly vague: Prove theorems about lines and angles. Well heck, there’s a lot of those. The standard goes on to list as examples three particular theorems which are covered by most Geometry courses. Is this meant to be an exhaustive list? Are any other “theorems about lines and angles” going to be fair game?

What exactly are students going to be expected to do? Literally just prove these three theorems? If two angles are vertical angles then they are congruent? Or will this be buried as a step in a more involved problem?

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The second was by Grant Wiggins, and talked about the mistake of breaking things down into too-small pieces so that each piece becomes meaningless and useless. It is located here.

These two articles seem to be at odds with each other, and I’m trying to reconcile the differences between the two. Any thoughts?

]]>It’s funny. Until you start to think about it and realize how many important qualities and characteristics we say we value, but are ** not on the test**. If it’s not on the test, then it doesn’t get taught, or at least doesn’t get taught well, with fidelity.

My Algebra 1 classes have been working on a Sierpinski Triangle. Here’s the result so far. I think I’ll keep it and see if I can get the next iteration done next year!

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