Tag Archives: mathlesson

Pythagorean Theorem Lesson

5 Jan

I always think it would be instructive and so entertaining to have my students “discover” the Pythagorean Theorem rather than tell them about it. But I am always caught between a pacing schedule and a benchmark, and end up pushing ahead. This year, I took some baby steps towards a more proof-oriented, discovery based approach.

How I taught it:

  • Introduced the concept by drawing a 5, 12, 13 right triangle, and then drawing the squares off of each side. The students determined the area of each square: 25, 144, 169.    Then we did a 6, 8, 10 right triangle, and did the same thing. I asked the class if they noticed anything interesting about how those areas  relate to each other. The students figured out the connection.  I then moved to explain that this is the Pythagorean Theorem and it works for all right triangles in the world.   Now I revealed the formulas :

            a² + b² = c²          and            leg² + leg² = hypotenuse²


  • Next was my Pythagorean Theorem Rap:

Today we’re goin’ wrangle a right triangle,
we’re goin’ seal its fate if we concentrate,
we’re goin’ calculate the side lengths of a right triangle.

We’re going to let loose the hypotenuse,
we’re going to use a tool called Pythagorean’s rule.
It’s fair to share, so prepare to blare:

            leg² + leg² equals hypotenuse²

           leg² + leg² equals hypotenuse²

           leg² + leg² equals hypotenuse²

Now that’s the cool Pythagorean rule!

  • Next we apply the theorem to problems, emphasizing how to document your work. I created this little worksheet to get the labeling of the triangles down which I stress is the most important step.  Pythagorean graphic organizer.
  • This year I also offered extra credit if they created a poster, video, or powerpoint on a Pythagorean Theorem proof. I showed them a couple of examples:

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Here’s pictures of a couple of the many posters  turned in by students. Some creative work!

pyth-poster1     pyth-poster2

Estimating in Intervention class

8 Oct

I teach a math intervention class during 1st period. I’m loving this class because I stepped away from just doing more of the same (drills, reteaching, etc) to creating more of a culture of explaining your thinking, improving overall number sense, and tuning critical thinking skills. I spend some time with white boards, practicing their regular math class’ topics, and because they are all my students, I give them a practice quiz on Fridays before they get the real one in class (benefit of a 1st period intervention class). But mostly we are focusing on thinking and explaining your thinking!

Last week we started sharpening our estimation skills with Andrew Stadel’s estimation180 website. The kids (and me) are absolutely loving this activity. I got a couple of other teachers at my school doing it, and we are learning so much. The students get really excited trying to come up with an accurate estimate, and on articulating their reasoning (some strong opinions in this class). As a class we reach a consensus regarding the estimate and reasoning, but then it’s so satisfying to see the actual answer.  We also have fun reading the other classes  estimates and reasoning.  We even had another class come and visit my class (impromptu) because they wanted so desperately to understand how we came to our estimate (and they wanted to brag that they had the better estimate ;-).  Not only will we continue with this estimation activity, but my students want to give Mr. Stadel some suggestions for other estimation pictures. Thanks Mr. Stadel!

Lessons Learned on Data Analysis Plots

22 Sep

This week I explained line plots, stem & leaf plots, and box and whisker plots.

The good:

— Used the “lowest unique number” game as the data for creating a line plot, stem & leaf plot, and box & whisker plot. The kids love this game. They write on a piece of paper a number from 1 to <class-size>  that they think will be the lowest number that no one else will pick. Here’s the worksheet we used to create all the plots:  Lowest Unique Number Worksheet

— After the kids asked who uses these plots, and I was sort of wondering the same thing since I never really came across stem & leaf plots or box & whisker plots while I was in industry, I started searching. I found this cool stem & leaf plot (thanks wikipedia) of  a Japanese train schedule. I got a lot of miileage out of showing this image along with some scientific uses of box & whisker plots (google-images).

The not so good:

— Students are very quick to say “I don’t get it” ALOUD, when they haven’t even been exposed to a complete idea yet. I need to prepare them better for the discomfort of not understanding something while they are still gathering information about it.  One idea is to show them just pieces of a picture, and ask them if they know what the whole image looks like yet. No. However, are they confident that once all the obstructions are taken away that they will see and understand the entire image? Yes. This is similar to the process of learning most things that are interesting to learn.

Lessons Learned on 2 Step Equations

16 Sep

This past week I taught my 7th graders how to solve 2-step equations. But did they actually learn how? Just graded their quizzes, and the mean score was about 75%. Arghhh!

I’ve been teaching this topic for years, and this year I did a few things differently. I think a few ideas were  good, and others still need work.

The good:

  • I introduced 2-step equations and how to solve them with a demonstration.  I showed the students 3 cups. I told them that the cups hold exactly the same number of paper clips. The number of paper clips in those cups along with the 6 in my hand total 18 paper clips. How many paper clips in each cup? This was awesome; in each class, the student I called on to explain their thinking on solving the problem, solved the problems exactly how we normally solve 2-step equations: “First I subtracted 6 from 18 and got 12. Then I divided by 3 since there are 3 cups. So each cup has 4 paper clips.” Sweet. I then went and explained their thinking using the variable, c, for cup and so on…  Where did I get the idea for this demonstration? In the most unlikely place: their Holt texbook. Ha, who would have thunk?
  • After we concluded that a good way to solve an equation was to zero-out the constant term that is on the same side as the variable, and then to make the coefficient 1, we created a flow map (thinking map) for their notes. I think this is a great time to emphasize the vocabulary: constant, coefficeint, and term.
  • I had students come to the board and work out the homework problems twice this week. Students love it, and I get to see where they are having problems.

Things needing work:

  • Many students are having a difficult time with equations like 2m-3=-9.  They are not seeing the 3 as negative, so they are subtracting 3. Previous years I would have them immediately change the subtraction sign to adding the opposite. This year I was trying to get them to use inverse operations instead. Instead of subtracting 3, lets add 3.  I *thought* they were getting it, and I think they sort of did. But this is where most of the mistakes were made on the quiz. More practice with this on Monday!
  • I had my students check their work, by substituting their answer into the variable. We practiced this again and again. However, I am still seeing students work where they get the answer incorrect, and then they check their work, and they rush to believe that their check is true. I somehow want them to slow down and do the check as objectively as possible…yah right.  This is  driving their teacher crazy.  Maybe some multiple choice problems would help here.