# Series: Engaging All Students in Common Core Math

Math.Practice.MP5

Common core State Standards

• Math:  Math
• Practice:  Mathematical Practice Standards
• MP5:  Use appropriate tools strategically.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, \"Does this make sense?\" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

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Math.Practice.MP4

Common core State Standards

• Math:  Math
• Practice:  Mathematical Practice Standards
• MP4:  Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

How Tall is the Flagpole?
Lesson Objective: Use indirect measurement to find the height of the school flagpole
Grades 9-12 / Math / Geometry
Math.Practice.MP5 | Math.Practice.MP4

#### Thought starters

1. How does Mr. Pack assess and further his students' knowledge of similarity criteria?
2. Why did Mr. Pack have students measure in meters?
3. How does Mr. Pack monitor and respond to students' learning needs?
Great lesson. I love how it was hands on. I also liked the fact that Chuck asked the students why they thought their answers were wrong instead of just telling them they were wrong.
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This is definitely an activity I will be using next year. I love the fact that this is so hands on. It is such good practice to use meter sticks and other tools. I love the enthusiasm of the students.
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wow. great lesson :)
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I LOVE when a teacher (especially math or science) can connect the learning to the REAL WORLD! That instantly makes kids more engaged when they're doing hands on projects with the understanding that, "This is how it can help you one day." Fantastic! :D
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Very good activity to teach application of similar triangles. Thanks a lot.
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#### Transcripts

• How Tall is the Flagpole Transcript

Chuck Pack [in class]: How tall do you think our flagpole is?

Student: Between 35 and

How Tall is the Flagpole Transcript

Chuck Pack [in class]: How tall do you think our flagpole is?

Student: Between 35 and 50 feet.

Chuck Pack: These are 9th grade students in geometry suing similar triangles to solve real-world problems. The objective today for the kids is to determine the height of the flagpole. The other objective is to get them to experience some mathematics in the real world and to model the world they live in - the world they see every day.

Chuck Pack [in class]: Alright. So, if I'm gonna use trigonometry, what device would I need? The mirror! Now, what could I do with a mirror?

Student: We can use the mirror to measure the angle of the reflection.

Chuck Pack [in class]: That sounds good. Now, why would the angle of the reflection be helpful to me?

Student: Because it's the same going in as it is going out.

Chuck Pack [in class]: Same angle going in as it is going out. Where is this angle if it's an angle we're talking about

Student: Eye level.

Chuck Pack [in class]: Oh, I've got to measure from my eye.

Chuck Pack: We've done some work with setting up proportions, working with similar triangles, I knew that it was not a strong standard for them because we've done benchmarks that have shown me that now all of them are able to say why two triangles are similar and why an additional piece of information would be required to prove they're similar. I saw that come out when I asked them, initially, how do I prove the triangle's similar?

Chuck Pack [in class]: What's required to have two similar triangles? What criteria do we --

Student: You have to have at least signs that are the same.

Chuck Pack [in class]: We have some proportional sides or now we've got one angle...

Student: Oh, you've got one angle, so two sides in between.

Chuck Pack: We saw Taylor trying to give us something but I really had to coach them into angle-angle similarity which I was wanting them to use. I know it's still weak and I'll have to keep coming back to that.

Chuck Pack [in class]: I only have one angle pair right now.

Student: You have a 90 degree angle that the --

Chuck Pack [in class]: Oh, the 90 degree angle right here and the 90 degree angle right there. Right? These triangles would have to be similar using angle-angle similarity and then, I don't know this height. Correct? Can I measure these other heights?

Student: Yeah.

Chuck Pack [in class]: Can we get any of those right here in the classroom?

Student: Yes. Our height.

Chuck Pack [in class]: Would I want to measure my height from the top of my head?

Chuck Pack [in class]: Oh, from my eyes. Ok. Lend me some tools, right? What we're gonna do is ---

Chuck Pack: Mathematical Practice 5 is about the strategic use of tools. When you say "mathematical tools" you think, ok, calculator. Or computer. There are tools that students can use that are not necessarily technology. There are some old technologies that are perfectly applicable in the modern day.

Chuck Pack [in class]: What are the tools that we're gonna need? What do you suggest we use? Maybe some meter sticks, maybe a tape measure. A mirror. We need a flagpole - we've got one of those, fortunately. I think we might need a calculator.

Chuck Pack [in class]: On your paper, let's get some god measurement for our height. I want you to work in pairs. One of you needs to designate themselves the person who's gonna come and get some meter sticks. Go ahead, take a moment, get your eyes off of the ground.

Chuck Pack: As I begin to move around the room, I notice that many of them had a meter stick on the floor and a meter stick upside-down.

Chuck Pack [in class]: Flip over like this.

Chuck Pack: And I'm trying to guide them to say, "you're being silly." You realize you've got that upside down! To make them think about, "well, does that make sense?" I think I see a 45. is that a 45? I really doubt that's true. Many of them were getting something around 40cm. Really? You think you're that tall, right? That's how far it is? "Oh, no!" It was that plus these others. And almost by every group, I had to do that.

Chuck Pack [in class]: That make sense? While we're finishing up, on that paper, do you see where it says include a sketch? You could be doing that right now, can't ya?

Chuck Pack: When you have your activity sheet, you can allow space there for your students to do work and they'll take a clue from you based on the amount of space you had. It's just a quick sketch that we can use to get an idea of what measurements are needed.

Chuck Pack [in class]: So, when you go outside, what do you need to take with you? Well, you need your mirror. And you need your measuring tape. Do you need the meter sticks?

Student: No.

Chuck Pack [in class]: We don't need the meter sticks any more. We leave those here. Once you're satisfied, you can go outside.

Chuck Pack [in class]: Now the hard part. Can you find it. Can you find the top of the pole? You see it? Once one of you sees it, the other needs to measure.

Chuck Pack: They were able to find the pole, and that's probably because I talked to them before about using mirrors and if you're gonna see the object you have to be away from the object and looking in the direction of the object. They knew to be cognizant of where they're standing and where they see the pole.

Student: Meters.

Chuck Pack [in class]: We'll do the math inside once we've gotten our measurements.

Student: So 9.6 to the mirror to the pole. And then 3 meters to the --- from me to the mirror. And then I am 156cm tall.

Student: You put the height of the flagpole over the eye height of the person and then you set that equal to the distance of the mirror to the flagpole over the distance from the person to the mirror. And then Algebraically work it out.

Student: And then you would cross-multiply then it gives you to that and then you divide there to give you 872.

Student: You've got this over this and you divide but how do you get the x?

Chuck Pack: He was wrestling with how to solve that proportion. It wasn't immediate for him to know, "I need to cross-multiply and then divide." I had to coach him to here's a variable and you have to isolate it. What do you need to do?

Chuck Pack [in class]: How would you solve that proportion?

Student: Swap 'em out?

Chuck Pack [in class]: That's gonna put your x on the bottom, right? So you have x, and you want x by itself.

Student: So you multiply it by...

Chuck Pack: He came up with it on his own that you need to multiply to do that.

Chuck Pack [in class]: Each meter is how many centimeters? It's 100? So you add 6 of those. How many hundreds is that?

Student: I didn't have six, I had centimeters, and thirty, so...

Chuck Pack [in class]: Well, try this, let's measure it out again. It's probably not 30cm, that's about this much.

Chuck Pack: 30cm is 30cm from that last meter mark. So, what was your last meter mark? Cause you know it's not this far from the meter to the pole. So, there were some "ah-ha" kind of moments to say, "hmm. It's interesting what our students do and don't come to the classroom already knowing that they really should know.

Student: It was 3 meters plus 30cm. Instead of just 30cm.

Student: Ok, alright. So you just forgot the meters.

Student: Yep.

Student: That makes sense. I was wondering why 68 was bigger than 30.

Chuck Pack [in class]: Ok, we can go back inside now.

Chuck Pack: Once we were back inside, I needed to give a few minutes for them to do the calculation.

Student: So, for the equation, you cross-multiply 700 times 168 and then it will be equal to 150cm.

Student: 1... 784.

Chuck Pack [in class]: I think at this point, you've all set up your proportions nicely so that proportion should have been something like: Height of the flagpole to the - your eye-height should be like distance to the mirror from you, distance mirror flagpole. Right? Then, how did you solve the proportion? What do we do to solve the --

Student: Cross-multiply. That's a great idea, cross-multiply. Each of you has an answer to what you think it is, right? Alright. Do you think you're all agreeing? I seem to hear some different answers. Let's collect all of our data together. And maybe we can find an average. What'd you get, Dalton?

Student: 7.3 meters

Chuck Pack [in class]: 7.3 - Travis, what did you get?

Student: I got 7.425 meters

Chuck Pack [in class]: 7.425 - Caitlin, what did you get?

Student: I got 8.89m

Student: 19.4m

Chuck Pack: They did go to, "what's your answer? What's my answer? Let's compare." Am I right because I got this or am I way off, am I weird, am I normal. What am I? That's why I wanted to measure in meters cause I knew they weren't comfortable with them. Now, they'd be much more comfortable with the standard units and then, they could only know they were right if they started to compare and so I was baiting them, I was setting them up for, "let's do a dot plot." Which is something that they would have seen before."

Chuck Pack [in class]: Here's all of our data scattered about. Do you see anything special, unique, unusual?

Student: The one over on the right.

Chuck Pack [in class]: Oh, ok. You had a name for this point. What did you call that?

Student: Outlier.

Student: Not high.

Chuck Pack [in class]: It's shaken and that's because the other kids didn't get that. But what if you are the right answer, you know?

Student: That one is way too long but that one is way too short.

Chuck Pack: The fun part is, "well, is there a right answer?" Yes, there actually is. Do you have it? "I don't know."

Chuck Pack [in class]: So, I'm wondering - what are some possible sources of error we might have encountered in taking our measurements?

Student: We're at different distances from the flagpole so it changed ours pretty well and then we also didn't measure -- with the tape, we didn't measure from the middle of the mirror, we measured from the end of it and we weren't very precise with our measurements.

Chuck Pack [in class]: The precision of your measurements might have caused some problems.

Student: I thought we were measuring in meters when in fact we were measuring centimeters so I got that it was 150 meters tall which isn't right whatsoever. So then we figured out it was centimeters.

Chuck Pack [in class]: Based on this work that we've done as a class, what do you think is a good guess, a good guess-estimate of how tall the flagpole is in meters?

Student: 8 meters.

Chuck Pack [in class]: About 8 meters?

Chuck Pack: Once we saw 8 meters was a pretty decent estimate, they still don't have that feel for what is 8 meters. I provide it for them on the sheet -- a conversion.

Chuck Pack [in class]: How many feet is that?

Student: 27

Chuck Pack [in class]: Do you think the pole is buried in the ground somewhat?

Student: Yeah.

Chuck Pack [in class]: About how far do you think it is in the ground?

S; 6 feet.

Chuck Pack [in class]: So how long would that make the pole?

Student: Like 28 feet?

Chuck Pack [in class]: That's about what I would guess, something in that 30 foot range.

Chuck Pack: This activity was a real life application to use of similar triangles. It's the world they live in, the world they see every day with some mathematics that is available to them - right at their hands. It's not complicated.

Chuck Pack [in class]: Why might you need to be able to do this? What's the skill?

Student: What if loggers wanted to fell a tree? They might want to know how tall it is so they can tell how long it's gonna take for it to land and how - where it might hit? They wanna clear that area before it falls.

Student: Construction.

Chuck Pack [in class]: What might they do in construction.

Student: For roads - how far you have to make it to make it one lane, or how far you have to clear it out.

Chuck Pack [in class]: Our real world is the math always precise? Real data is messy so mathematics is about taking our messy world that we live in and trying to make sense of it. Trying to model the real world with some simple equations. And what you saw was it's a pretty simple ratio to set up. And a pretty simple proportion to solve. But collecting that data was a little bit messy, wasn't it? Yeah. That's real life.

END

#### School Details

Tahlequah High School
591 Pendleton Street
Tahlequah OK 74464
Population: 1206

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