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Tag Archives: CCSS-M G-SRT.4

Pythagorean Theorem Proofs

CCSS-M G-SRT.B.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

We’ve been teaching our CCSS Geometry course for three years now, and this is the first year that we have been able to spend more than a little class time on proofs of the Pythagorean Theorem. (Our students are coming to us knowing more mathematics than three years ago. Our students are coming to us more willing to take risks and use the Standards for Mathematical Practice than three years ago. We are making progress just in time for our legislators to decide that collaborating with other states to write standards and assessments was a bad idea.)

We started with the Mathematics Assessment Project formative assessment lesson (FAL) on Proofs of the Pythagorean Theorem. This FAL is one that includes student work. Students focus on SMP3: construct a viable argument and critique the reasoning of others.

As students practice look for and make use of structure, I asked them to share what they noticed and wondered.

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Then we looked specifically at a diagram drawn to scale, and students noted what they knew to be true (and why).

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As we started to examine the student work proofs so that students could critique the proofs, SC asked to go back to the previous page. I wonder what will happen if we reflect the outer right triangles about their hypotenuses into the center square.

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What do you think will happen?

The triangles will make a square.

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I think I’ve said before that technology slows me down in the classroom. Students notice and wonder more than they did before, and the technology gives us the chance to see what happens so that we can make sense of why it happens mathematically. I am not the only expert in the room. The student who gets mathematics without technology is not the only expert in the room. Our use of technology increases our confidence and lifts all in the room to experts. And so the journey continues …

 
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Posted by on February 1, 2015 in Dilations, Geometry, Right Triangles

 

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The Side-Splitter Theorem

CCSS-M G-SRT.B.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

One of the theorems that we prove in our unit on Dilations is the Side-Splitter Theorem.

At the end of class one day, I checked to see how intuitive the Side-Splitter Theorem is.

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Not so intuitive, apparently.

(I didn’t show them the correct answer.)

But we started class the next day with our diagram and our learning goal, look for and make use of structure.

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What do you see that isn’t pictured?

We set the learning mode to individual. Students worked alone for a minute or two to see what they could see.

I monitored. And selected. And sequenced.

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And then we talked.

Which auxiliary lines help us determine a relationship between the segments?

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Our work with dilations and similarity has been mostly with triangles. Do you see any similar triangles? Why are they similar?

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We generalized our results. And then I sent back the original question.

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And so the journey to make sense out of what isn’t always intuitive continues …

 
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Posted by on January 21, 2015 in Dilations, Geometry

 

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Similarity Theorems

 

Similarity Theorems

Prove theorems involving similarity

G-SRT 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G-SRT 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

I have struggled some to know which additional theorems we should include when teaching G-SRT 4. So, at least for now, my best source for whether to teach something that we have traditionally included has been to ask my juniors and seniors who have recently taken a high stakes standardized test for college entry and scholarships. I get that my only reason to teach something shouldn’t be whether or not it is “on the test”, but I don’t want my students to be at a disadvantage when they go to take the test.

We started the Dilations unit by showing that two triangles are similar when there is a dilation and if needed, set of rigid motions, to map one figure onto another. In our lesson on Similarity Theorems, we discuss similarity criteria for triangles in terms of the traditional similarity theorems: AA~, SAS~, SSS~. We formatively assess that students can use the similarity theorems.

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Are the figures similar? If so, why?

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If the figures are similar, then what is the value of x?

Students always do well with the first triangle but not with the second. Asking the second uncovers student misconceptions.

We briefly explored 3 other theorems dealing with similar figures.

 

1. A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. What is the relationship between the midsegment and the third side of the triangle? We used part of the Geometry Nspired activity called Triangle Midsegments for this exploration.

This is an example of where technology makes me slow down. It would be so much faster for me to tell students that the midsegment is parallel to the third side of the triangle and that it is half the length of the third side of the triangle. But I have found that students remember the information longer when they make sense of it: from constructing the midsegment to observing what happens as the triangle changes.

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We formatively assess to be sure that students can use what they have observed.

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And then we prove the theorem.

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Where would you start?

What Standard for Mathematical Practice would be helpful?

 

2. The Side-Splitter Theorem.

Observe what happens.

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Formatively assess to be sure students can use what they observed.

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And then prove the theorem.

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Where would you start?

What Standard for Mathematical Practice would be helpful?

We used look for and make use of structure. The auxiliary line gives us something two cases of “a line parallel to one side of a triangle divides the other two proportionally”, and then we can use the parallel lines and transversal to show that the two triangles are similar to each other by AA~.

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3. The Angle Bisector Theorem.

Observe what happens.

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Formatively assess to be sure students can use what they observed.

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And then prove the theorem.

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Where would you start?

What Standard for Mathematical Practice would be helpful?

What auxiliary lines would be helpful?

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We actually didn’t get to completely prove The Angle Bisector Theorem…our time ran out. But I think that’s okay. We covered the Angle Bisector Theorem because my former students suggested it is helpful. And because we have taken care all year to observe what happens and prove most of what happens, my students realize that the result isn’t magic.

I started reading The Joy of X this morning. I love Steven Strogatz’ quote at the bottom of page 5: “Logic leaves us no choice. In that sense, math always involves both invention and discovery: we invent the concepts but discover their consequences. … in mathematics our freedom lies in the questions we ask – and in how we pursue them – but not in the answers awaiting us.”

And so the journey to ask more questions continues …

 
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Posted by on January 26, 2014 in Dilations, Geometry

 

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