Category Archives: Right Triangles

Solving Right Triangles

Solving Right Triangles

One of the NCTM Principles to Actions Mathematics Teaching Practices is to build procedural fluency from conceptual understanding. We began conceptual understanding in our lesson on Trig Ratios. So we started our Solving Right Triangles putting-it-all-together lesson with some “Find the Error” (what’s wrong with the procedure) problems from our textbook.

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Students worked a few minutes alone before sharing their thoughts with their teams. What can you find right about the given work? What is wrong about it? How can you correct the given work?

We started our whole class conversation with #1. Students began to recognize how many options there were for correcting the given work. The Class Capture feature of TI-Nspire Navigator gave students the opportunity to see and compare each other’s calculations. They looked at and compared trig ratios for complementary angles D and F.

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We moved to #2. What error did you find?

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The triangle isn’t right. I don’t think we can use sine, cosine, and tangent for triangles that aren’t right.

Not yet … but eventually you’ll be able to.

And then BK said, “But I worked it out.”

If I haven’t learned anything else, I’ve at least learned to hear my students out. I hadn’t noticed what he’d done when students were working individually, so I didn’t know what we were about to get into. “How did you do that?”

I drew in an altitude from angle B and made two right triangles.

And so he did.

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He recognized half of an equilateral triangle, so he used 30-60-90 triangle relationships to get enough information to write a ratio for tangent of 55˚.

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This is what happens when students learn mathematics steeped in using the Standards for Mathematical Practice. Students practice make sense of problems and persevere in solving them. They practice look for and make use of structure. Even in what seems at first glance like simple, procedural problems.

And so the journey continues, with the practices slowly becoming habits of minds for my students’ learning, seeing glimpses of hope more often than not …

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Posted by on March 29, 2015 in Geometry, Right Triangles


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Trig Ratios

Trig Ratios

How do your students experience learning right triangle trigonometry? How do you introduce sine, cosine, and tangent ratios to them?

NCTM’s Principles to Actions includes build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. In what ways can technology help us help our students build procedural fluency from conceptual understanding?

Until I started using TI-Nspire Technology several years ago, right triangle trigonometry is one topic where I felt like I started and ended at procedural fluency. How do you get students to experience trig ratios?

I’ve been using the Geometry Nspired activity Trig Ratios ever since it was published. Over the last year, I also read posts from Mary Bourassa: Calculating Ratios and Jessica Murk: Building Trig Tables about learning experiences for making trigonometric ratios more meaningful for students. Here’s how this year’s lesson played out …

We first established a bit of a need for something called trig (when they finally get to learn about the sin, cos, and tan buttons on their calculator that they’ve not known how to use). I showed a diagram and asked how we could solve it. We reserved “trig” for something they couldn’t yet solve.

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We use TI-Nspire Navigator with our TI-Nspire handhelds, and so I can send Quick Polls to assess where students are. Sometimes Quick Polls aren’t actually so “quick”, but these were, along with letting students think about what we already know and uncovering a few misconceptions along the way (25 isn’t the same thing as 18√2).

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Next I asked each student to construct a right triangle with a 40˚ angle and measure the sides of the triangle.

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I sent a Quick Poll to collect their measurements.

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Then we looked at the TNS document for Trig Ratios. Students can take multiple actions on the diagram. I asked them to start by moving point B. What do you notice? We recorded their statements for our class notes.

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Then I asked them to click on the up and down arrows of the slider. What do you notice?

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What ratio of side lengths is used for the sine of an angle?

You all constructed a right triangle with a 40˚ angle and recorded the measurements. What’s true about all of your triangles?

  • The triangles are all similar because the angles are congruent.
  • The corresponding side lengths are proportional.
  • We know that sin(40˚) is always the same.
  • So the opposite leg over the hypotenuse will be the same?

Will it? We sent their data to a Lists & Spreadsheet page and calculated a fourth column, opp_leg/hyp. What do you notice?

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Of course their ratios aren’t exactly the same, but that’s another good discussion. They are close. And students noticed that one entry has the opposite leg and adjacent leg switched because the leg opposite 40˚ is shorter than the leg opposite 50˚.

We didn’t spend long looking at the TNS pages for tangent and cosine … students were well on their way to understanding a trig ratio conceptually. They just needed to establish which side lengths to use for cosine and which to use for tangent.

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There’s a reason that #AskDontTell has been running through my mind as I have conversations with my students and reflect on them. Jill Gough wrote a post using that hashtag over two years ago: Circle Investigation – #AskDontTell.

What #AskDontTell opportunities can you provide your students this week?

[Cross-posted at T3 Learns]


Posted by on March 12, 2015 in Geometry, Right Triangles


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Practicing Formative Assessment: Hopewell Triangles

This year’s Mathematics Assessment Project Hopewell Triangles task on similarity and right triangles played out differently than previous years.

2013 – Right Triangle

2013 – Similarity

2014 – Misconceptions

In general, students didn’t have as many misconceptions as the prior year.

As students were working by themselves, I did see one misconception that I was sure to bring out in our whole class discussion.

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(Look at ∆D above.)

On another trip around the room, I saw this on his handheld, which helped him correct his own mistake for ∆D.

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I particularly enjoy seeing different ways that students explain why the triangles are similar (#3) and why or why not the triangle is a right triangle.

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I sent a TNS document with questions to collect student responses for some of the questions. When I collected it after students worked individually, we had a 70% success rate. (I’ve changed the Student Name Format to Student ID to keep student names concealed.)

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At that point, we changed to Team mode, and students talked with each other about their work and I told them that they could change answers in their TNS document as they discussed their work. Students are making it a practice to not mark an answer unless they have an explanation to go with the answer. When I collected their work the second time, I knew that our whole class discussion needed to start with the third question. (I also knew who needed to come in during zero block for extra support.)

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Also of note is that the first collection of question 3 had these results:

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But the final collected had these results:

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Why is ∆1~∆A?

Why isn’t ∆1 similar to ∆F or ∆E?

We finished the discussion by discussing a misconception that students had last year. Anna thinks that ∆2 is a 30-60-90 triangle. Do you agree? Why or why not?

And so the journey continues, using formative assessment to make instructional adjustments to meet the needs of the students who are currently in my care …

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Posted by on February 28, 2015 in Dilations, Geometry, Right Triangles


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30-60-90 Triangles

In our 30-60-90 triangle lesson, we first find out what students already know about 30-60-90 triangles. We deliberately let them practice look for and make use of structure. Some students compose the triangle into an equilateral triangle and note that one side is double the length of the other (we eventually attended to precision to note that the hypotenuse is double the length of the shorter leg).

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Some students rotated the triangle 180˚ about the midpoint of the hypotenuse; others rotated it 180˚ about the midpoint of the longer leg.

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Other students decomposed the triangle by drawing the altitude to the hypotenuse and noted that they formed two additional 30-60-90 triangles.

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Our lesson (partially from the Geometry Nspired activity Special Right Triangles) provided the opportunity for students to look for and express regularity in repeated reasoning using the equilateral triangle. What changes and what stays the same as you grab and move point B?

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Recording side lengths in a table also provided the opportunity for students to look for and express regularity in repeated reasoning. But after our 45-45-90 lesson the day before, I thought it would be okay to skip that step and move to #3.

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As soon as we recorded some of the student responses to #3, I realized I had made a mistake. They weren’t all getting that the longer leg is the shorter leg times √3. I had tried to rush to the result instead of giving students the time to notice when calculations are repeated, to evaluate the reasonableness of intermediate results, and to look for general methods and shortcuts.

We took a step back. They all used the Pythagorean Theorem to determine the altitude for an equilateral triangle with a side length of 10, and then in the little time remaining we used our technology to confirm the result and help us generalize the relationship between the side lengths of a 30-60-90 triangle.

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And so the journey continues … learning more each day that providing students deliberate learning episodes steeped in using the Math Practices is much more effective than having them haphazardly figure out the math.


Posted by on February 7, 2015 in Angles & Triangles, Geometry, Right Triangles


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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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Half of a Square

Sketch a smaller square inside the given square so that the area of the smaller square is half the area of the larger square. Write an explanation to convince another that your new, smaller square is truly half the area of the original square.

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I’m sure that I didn’t write this question, but I don’t remember from where it came.

What would your students get right?

What misconceptions might they have?

I was surprised at some of my students’ misconceptions (and miscalculations) and explanations.

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Measuring errors.

Calculation errors. Lots of “simple” calculation errors: Half of 36 is 16. Half of 25 is 13.25. Half of 5 is 3.5.
More measuring errors.

Square vs. rectangle errors.

And someone getting at the immeasurability of irrational numbers.

The test was summative. But even though the test was summative in my gradebook, I can still use the responses to inform the learning experiences that I give my students in the future. And so I will, as the journey continues …


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Team Sorting – Right Triangles & Circles

We’ve been sorting students into teams on the first day of each unit this year. After the team sort for Dilations, we were motivated to again make the cards challenging and interesting.

We went to the cafeteria so that students would have plenty of room to find their teams. I told them that we have 6 teams and 30 students.

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I overhead the following:

  • I’ve just got two sides.
  • Am I supposed to calculate the side or the angle?
  • I’m noticing we both have 60-30 triangles.
  • You would have to find the hypotenuse, and you would have to find the leg.
  • Do you have 33 degrees?
  • We think we have a team: two sides and an x.
  • Who has an x, an angle, and a hypotenuse?
  • Do you have the included leg or the opposite leg?
  • Is everybody here today?

Eventually, all teams sorted themselves.

From the beginning, students began to notice when we can use the Pythagorean Theorem and when we can’t.

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Students began to notice the difference between the opposite leg of the acute angle of a right triangle and the included leg.

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Students began to notice when we were looking for the angle of a right triangle instead of one of the sides.

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Students began to notice special right triangles.

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Yesterday, we sorted into teams for our circles unit.

Students began to notice the difference between central angles and inscribed angles (even though they don’t know the names of them yet).

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Students began to notice the difference between diagrams with two chords and two secants (even though they don’t know the definition of a chord versus a secant).

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Students began to notice the difference between diagrams with tangents only and diagrams with a secant and a tangent.

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Just in case you want to use or edit, you can find all of our team sorting cards here.

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Posted by on January 28, 2015 in Circles, Geometry, Right Triangles


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Inscribed & Circumscribed Right Triangles

Several students noted in their reflection on our circles unit that the performance assessment tasks helped them “combine everything that we learned to find the correct answers to challenging problems”.

We started with a diagram from the Mathematics Assessment Project formative assessment lesson. Inscribing and Circumscribing Right Triangles

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and asked students to

1. Figure out the radii of the circumscribed circle for a right triangle with sides 5 units, 12 units and 13 units. Show and justify every step of your reasoning.

2. Use mathematics to explain carefully how you can figure out the radii of the circumscribed circle of a right triangle with sides of any length: a, b and c (where c is the hypotenuse).

The second task provided a bit more structure. Circles in Triangles comes from a Mathematics Assessment Project apprentice task.

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Students were given the following:

This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units, 4 units, and 5 units long.

1. Prove that triangles AOX and AOY are congruent.

2. What can you say about the measures of the line segments CX and CZ?

3. Find r, the radius of the circle. Explain your work clearly and show all your calculations.

I wonder what would have happened if we had asked students to determine the radius of the inscribed circle without as many auxiliary lines already given in the figure? Instead, what if we had used the same figure as the first task?

Another teacher taught this lesson to my students. She saved their work for me because even with the auxiliary lines drawn, students made sense of the structure (and particularly the expressions they wrote for the segment lengths) in different ways.

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Then they were asked to determine the radius of the inscribed circle for a 5, 12, 13 right triangle, and then they were asked to generalize their results.

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This diagram shows a circle that just touches the sides of a right triangle whose sides are 5 units, 12 units, and 13 units long.

4. Draw construction lines as in the previous task, and find the radius of the circle in this 5, 12, 13 right triangle. Explain your work and show your calculations.

5. Use mathematics to explain carefully how you can figure out the radii of the inscribed circle of a right triangle with sides of any length: a, b and c (where c is the hypotenuse).


I wonder how technology fits in with tasks like these?
We had a skeleton of a diagram prepared for students who wanted to use it.

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Students had to measure themselves for the construction to be helpful in making sense of the mathematics. I find that the technology can be helpful for those who don’t know where to start. What do you notice when the triangle is right?

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Can it help them reason abstractly and quantitatively, starting with the quantitative and building to generalization? Can it help them make sense of problems and persevere in solving them, when they don’t know what else to do on paper?

We also had a skeleton of a diagram prepared for the inscribed circle for those who wanted to use it.

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What auxiliary lines (or segments) would you construct and measure for the inscribed circle?

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The technology can help them make a conjecture about the length of the radius, and then they can go back to the mathematics to help them understand why.

And so the journey continues … Maybe some year I will be brave enough to start our Circles unit with this task and let the mathematics unfold in the context of the task as it is needed.


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Seven Circles … Again

We tried Seven Circles I from Illustrative Mathematics a few weeks ago.

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At the end of class one day, I showed students the diagram and what question they might explore with it. I collected their responses using an Open Response Quick Poll and have shown the results below.

What does this figure have to do with geometry?            1

if we connected each top vertex of the triangles, will it make a hexagon?         1

whats the area of all the circles       1

Why are the circles in this shape?  1

what are the circles forming?          2

what is the area of all of the circles            1

why are we looking at circles?         1

are the spaces between the circles triangles?       1

what are the seven circles forming?           1

do all the circles have the same diameter?            1

do all the circles have the same diameter  1

Are the circles’ diameters the same?          1

Can the measures of the triangles that can be drawn through circles be calculated quickly  1

why are all the circles touching?     1

Why are the circles in that certain arrangement?            1

Can you find the area for that?       1

Is there a way to solve non 90° triangles? (With sin, cos, tan, or the other trig functions)      1

Can the circles be mapped onto each other with a rigid motion?           1

when you look at the image what do you see?      1

are the 6 figures that look like triangles in the gaps of the circles considered triangles since their sides arent straight    1

what are the circles for        1

are all of the circles congruent to each other?       1

What is the significance of the circular pattern?   1

How can you find the measurement of each circle           1

What are the triangular looking spaces in between the triangles called?          1

Some students were interested in the space between the circles. Other students wondered whether the circles were congruent. The task is given below.

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My students felt like it was pretty obvious that this could work with 7 congruent circles. I gave them different sized coins so that they could play. What if the circle in the middle is not congruent to the others? Will this work for 6 congruent circles? Or 8 congruent circles?

After students played for a few minutes, I sent them a TNS document that a friend made to explore this task. I used Class Capture to watch while students used the technology to make sense of the necessary and sufficient conditions for 6 circles and 7 circles in the given arrangement. Who had something interesting to discuss with the whole class?

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Many students saw the regular pentagon or regular hexagon with vertices at the centers of the outside circles and used that to make sense of the mathematics. While I was watching them, I was trying to figure out how we should proceed as a class. We started with Claire’s work. What do we know?

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We saw a dilation. We saw central angles of a regular pentagon. We saw isosceles triangles, which we bisected to make right triangles. We saw an opportunity to use right triangle trigonometry. We looked for and made use of structure. We reasoned abstractly and quantitatively.

And before the bell rang, we looked back at the picture with 7 circles and recognized that the 30-60-90 triangles require that the radius of the center circle equal the radius of the outer circles.

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We only touched the surface of what we can learn from this task. Last year, we didn’t even do that. Last year, I shared the task with students during their performance assessment lesson, but we spent all of our time on Hopewell Triangles. This year, we got to it, but I know that our exploration could have been better. We began to answer what are the necessary and sufficient conditions for 6 circles. And in the process, we came across an argument for why the 7 circles must be congruent. But we didn’t really solve the conditions for 6 circles.

I wanted to write about this as a reminder that we are all learning. In this journey, I am finding good tasks out there to try with my students. And I am more confident about some than about others. Even though I don’t know exactly how the tasks should play out in the classroom, I am going to keep trying them. And I’m not going to throw out the task just because we didn’t get as deep into the mathematics as I wish we had. I will try again next year as the journey continues …

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Posted by on February 24, 2014 in Circles, Geometry, Right Triangles


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Unit 7: Right Triangles – Student Reflections

I have just read through student reflections for our unit on Right Triangles.

Unit 7 – Right Triangles

Level 1: I can use the Pythagorean Theorem to solve right triangles in applied problems. G-SRT 8

Level 2: I can solve special right triangles. G-SRT 6

Level 3: I can use trigonometric ratios to solve right triangles. G-SRT 6, G-SRT 7

Level 4: I can use trigonometric ratios and special right triangle ratios to solve right triangles in applied problems. G-SRT 8

Similarity: G-SRT

Define trigonometric ratios and solve problems involving right triangles

G-SRT 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT 7. Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.


7A – 45-45-90 Triangles

7B – 30-60-90 Triangles

7C – Trigonometric Ratios

7D – Solving Right Triangles

7E – Performance Assessment (Hopewell Triangles)

7F – Mastering Right Triangles

7G – Assessing Right Triangles

Most students felt like the Math Practices they used most in this unit were Construct viable arguments and critique the reasoning of others and Attend to precision.

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

  • Trigonometry is a completely new subject for me, so no, none of the lessons were repeats to me. I had no idea about anything except for the Pythagorean Theorem honestly.
  • A student from Iran: yes, I leaned part 7A,7B and 7D in middle school, but the difference was that in middle school we just taught how to use Pythagorean theorem and that the side of triangle opposite to 30 angle is half of h, … which were formula but i didn’t know about sin and cos.
  • The only part of this lesson that I knew how to do before was the Pythagorean Theorem.

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit? If so, how?

  • Trigonometric ratios and solving right triangles were the most helpful lessons because we learned the most new information in these lessons. What we learned in these lessons also applies to all triangles, not just a special type of right triangle.
  • The self-check bell ringer that we did a few days ago was very helpful to me. When I know that one of my answers are wrong, I persevere in solving it correctly.
  • When we finally put it all together and we were solving right triangles really helped me to finally grasp the whole idea. I had slowly learned and built a solid base and the solving right triangles lesson put it all together for me.
  • I believe that the Trigonometric Ratios helped me the most because they allow me to solve a triangle with a different set of information.

Our Trigonometric Ratios lesson came from Geometry Nspired. I’ll write a post about it some day.

What have you learned during this unit?

  • I have learned how to use sine, cosine, and tangent, and also the purpose of these formulas. Before I was taught what the formulas were used for, I was always curious as to what they meant on the calculator. Now not only do I know, but I also know how to work problems using sine, cosine, and tangent with accurate results.
  • I have learned that there are “special” right triangles, and that there are quicker ways to solve these triangles. I also learned about trigonometry, which is something completely new to me. It was definitely new, and a little hard since it was the first time I saw it.
  • I have learned how to solve right triangles, how to do trigonometry, and how to think about the problem before I try to draw it out.

What I have learned during this unit?

I have learned that the “I can” statements are really important for students. We reviewed them daily so that they could see where we were and where we were going. We are slowly making progress towards having students recognize what they are learning (I can …) and how they are learning it (Standards for Mathematical Practice). And so the journey continues …


Posted by on February 17, 2014 in Geometry, Right Triangles, Student Reflection


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