The new school year has begun!!

While Common Core Geometry is consuming my time, a quick share:

**In creating a floor plan for their “dream home,” not a single student included a laundry room. **

I found that amusing.

Archive | Geometry RSS feed for this section
## Amusing

20
Aug
## Length, Area, and Volume

17
May
## Toblerone Challenge –> Can Top Challenge

28
Apr
## Play-Doh Solids

28
Apr
## Raffle Tickets to Teach Perseverence

8
Apr
## Area of a REGULAR Polygon

8
Apr
## Similar polygons- Using ratios

14
Jan
## Triangle Congruence Flashcards

1
Nov
## Tangent, sine, cosine FLASHCARDS

29
Feb
## The Golden Ratio

26
Jan

The new school year has begun!!

While Common Core Geometry is consuming my time, a quick share:

**In creating a floor plan for their “dream home,” not a single student included a laundry room. **

I found that amusing.

**Did you know**… that if you DOUBLE up the size of a chair by doubling up its dimensions in every direction, you will need FOUR times as much paint to paint that chair and that it will weight EIGHT times as much?

**Did you know**… that if a gorilla was to really proportionally increase to be the size of King Kong, that his guts would just explode and gush out of him from the sheer weight of itself? (The same concept prevents an elephant from ever getting a c-section!)

**Did you** **know***… *that if an ant was really the size of the ants in “Them!”, not only would they not be able to stand up on their legs, they would also suffocate immediately?

**Did you know**… that if you proportionally double up in size, you’ll only be able to jump the same height as you did when you were normal size? Likewise, if you got 100 times *smaller, *you will also be able to jump the same height as you did when you were normal size! (So how long *should *it have taken for the kids in “Honey, I Shrank the Kids” to get home?)* *

Interesting, right? All these facts take into consideration that very first **“Did you know…,”** that changing a linear dimension changes the area and volume *differently*. Intuitively we all have some understanding of this. I mean, imagine making an itty-bitty model of a wee little ant out of Play-Doh. How easily will the ant stand on its tiny legs? Now can you imagine building a 2-foot model of the same ant? You’re going to have some difficulty making it stand on the same proportional legs.

So I found it all the more interesting when I went online to search for great lessons on this concept and found NONE. I had never been able to teach this topic to very much detail due to our crazy pacing plans and, frankly, due to my hazy knowledge of the science behind it. However, this year, I had a couple of advantages that allowed me to actually try to plan a lesson- *time*, of course, and an awesome student teacher who is ridiculously skilled in the sciences!

Anyways, Day 1, I started with a lesson on the basic concept* and followed it up with a follow-up to an exit ticket from the area chapter. It was pretty interesting to see how many students didn’t make the connection to the worksheet that was previously given. I’d say it was about 50/50. Half who figured it out and half who simply multiplied the 52 square foot tiles by 12* inches per foot*… (I posted this student’s work because she had *asked me* if she could simply multiply by 12 or if she had to convert all the numbers on the page first and re-work it. Like a good teacher should, I said, “Exactly! Test it!”)

The next day was the fun day though. I had almost an entire block period to now talk about the implications of same concept in reality.

Again, looking online, I found almost nothing fun, nothing very deep, and nothing very useful to me, *except these notes… *which I devoured. And from these, we came up with this lesson…

__________________________________________________________________________________________

FACT: If you *double* up the **dimensions** in every direction, then the **area **multiplies by ____four____________ and the **volume** multiplies by ______eight________.

QUESTION: If you have a chandelier barely hanging from a rope and you doubled up the size, *including the rope that it hangs on*, will the chandelier still hang or will the rope snap?

Why or why not? (Make your argument here. You will not be graded based upon whether your answer is right or wrong, but on how well you make your case.)

By this point, most students understand that although the dimensions are doubled, the weight of the chandelier has now become 8 times heavier. However, italicizing the part about the rope has them also consider the fact that the rope has also gotten 8 times heavier proportionally. A very true fact. I have them in groups and ask them to talk about it and then write out their argument, emphasizing that they will NOT be graded on correctness.

I then take a vote. The vote for all 3 of my classes was close to 50/50. I say awesome, and before I give away anything else, I have one person from each camp make their case. I then take out a piece of rope to begin my “Consider…” lecture:

*Consider… a really heavy object hanging from this piece of rope. I want to make it stronger and so I double up the length, like so, because I know that this will increase the VOLUME by twice as much. Will that make the rope stronger? [Class agrees, no way!] So then, how do I make the rope stronger?? … exactly! I need to double up the rope by making it thicker. So what does that mean? What does the strength of the rope depend on? The volume? (no) The thickness (yes!). And how do we measure thickness?? … the cross-sectional AREA!*

*So will the rope snap of the chandelier snap or not?! Discuss…*

So yes, the rope definitely snaps. The cross-sectional AREA only increased four-fold (which is what the strength of the rope depends on) while the weight increased eight-fold. In this case, two chords of the thicker rope will hold the larger chandelier.

__________________________________________________________________________________________

FACT: The height that a body can jump vertically is directly proportional to the ratio of *muscle mass* to the *total mass* of the body.

, where C is a constant (enough).

QUESTION: If a human being who can jump 1.5 feet is doubled up in size (pretending like it’s actually possible), how high will the new larger human be able to jump? Explain.

This was the section that my student teacher had helped me out the most in. He came back w/ an equation that we boiled down and simplified to the one above (apologies to any physicist who might get offended by this). I helped to explain what this equation meant and how it made sense. I made sure they all knew that working out actually made your muscles *heavier, *which makes total sense in this equation. I had them discuss and then take a vote.

Their logic behind this one was so adorable. Most of them completely ignored the equation and went straight to guesses. “If twice as tall, muscles get bigger so they’re stronger… they can jump higher.” But the winner of all arguments…

*He jumps shorter! Like Mario! When Mario grows, he can’t jump as high! *

The group cracked up and then voted, yes, he jumps less than 1.5 feet… like Mario.

Final answer? This person will jump the same height of 1.5 feet. His/her muscle weight will definitely increase eight-fold, but so will the person’s total weight, canceling each other out.

QUESTION: Now reverse the idea. If the same human being is shrunk to half the size, how high can the new small human jump?

Likewise, the smaller human can jump 1.5 feet.

(So how long would it have taken the kids in “Honey, I Shrunk the Kids” to go home?!)

It would’ve taken the same amount of jumps as when they were full size! This was an awesome discovery after reading that one professor’s notes.

__________________________________________________________________________________________

FACT: Ants don’t have lungs. Air is received through “blind tubes” that cover the surface of the ants’ bodies where air enters through.

QUESTION: If an ant was to be blown up as big as the ants in the movie, “Them!”

(pretending the physical structure* *is possible), what can you conclude about the

air intake of the giant ant?

We didn’t have time in class to go over these, but these should be easier after all that we’ve learned. I should’ve given it to them for homework…

FACT: With larger animals came the development of blood circulation and lungs as a more efficient way of oxygen distribution. Lungs increase the surface area available for oxygen absorption by the blood.

QUESTION: If a human was blown up twice as big (again pretending like the physical structure is possible), how do the lungs have to increase along with the body in order to support the oxygen level needed to survive?

__________________________________________________________________________________________

Anyways, if there are any other lessons/facts out there that you think are good for this concept, let me know! I didn’t have this much fun lecturing in awhile!

*Some of the problems from that worksheet were either directly taken or modified from the NCTM Illuminations site.

Here was another fun one.

The Word file here: Toblerone Challenge

A few logistics:

- They got to work either alone or in partners. Winner gets ONE Toblerone bar, so if partnering up, they must share.
- 20 minutes was all that they had. After the Play-doh activity the previous day, they understood that the winner would be the BEST net for this Toblerone box in those 20 minutes. No extension of time.
- Only one sheet of cardstock was actually passed out to the groups. All other materials were available to them for pickup, if they needed- rulers, scissors, compasses, protractors. They had to build everything until they needed the gluing, pretending like I was the one who will be gluing (there was no glue involved).
- Collect all nets immediately and put them aside to determine winner when they are doing classwork.
- At the end of class, when announcing the winner, I took apart the real Toblerone box under the doc cam to see how the actual engineers did it. I then wrapped the chocolate bar w/ the winning net and awarded the winners their chocolate.
- Hw was to find the lateral area, base area, and total surface area of the box.

Immediately after they turned in their net and before any kind of clean-up, there was a 5-minute bonus challenge.

I had the actual can in my hands w/ the actual label on it. The label was only taped down on one side so I could show how it wraps around the can and opens up to be a rectangle. I pretty much laid it all out save handing the can to them. I think I still only had about 2 groups per class understand how to find the perfect circle to go on top of the can. Most of them just eyeballed it. Ha!

Winners of that challenge got a mystery can that I had pulled the label off of last year for last year’s lesson. At least the expiration date was on there to prove it wasn’t expired yet! They just wouldn’t know what it contained until they actually opened it. Heh.

(The lessons from this chapter are not coming out in the proper order, but hopefully they can still be useful independently…)

What better way to gauge how well your students understand solids than by PLAY-DOH! This was tons of fun.

Each team got one tub of Play-Doh, some fishing string (to cut), and a plastic knife. The Play-Doh had to rotate hands every challenge. The challenges were on Powerpoint, one challenge per slide.

First slide, person #1: in 3 minutes, build your best… **tetrahedron. **GO!

At the end of the 3 minutes, everyone has to throw their hands in the air and wave ’em around like you just don’t care!… or just hold them there. I collect all the CORRECT tetrahedrons and then judge the BEST ones for 1st, 2nd, and 3rd.

The challenges/slides go something like this:

- build your best tetrahedron
- octahedron
- dodecahedron
- model of what the following net will build:

- Regular pentagonal prism
- Polyhedron whose cross-section is a triangle
- model of what the following net will build:

You can add or subtract what you like to the slides and to the time allotted for building.

To consider for next year: Add a drawing section and have them next DRAW THE NETS to a given solid.

Again, back to this area chapter. Once the basics of finding area is taught, it’s a great chapter to get them to start thinking of different ways to solve a problem. But how do we get our little trained monkeys to do so?! Especially when they are so used to teachers teaching them *exactly* how to do a problem and then having them mimic our process? I don’t know about you, but I can’t stand hearing “but you didn’t teach us how to do this!” when there’s really nothing new to have taught!

Here’s one solution: Raffle tickets!

I used this for group work on one very difficult problem that required multiple steps. For example, for this chapter, ONE problem that looked similar to the following examples. (This came after finding the area of a regular polygon given the apothem and a side length.)

**Find the area of a regular octagon with side length 6 cm.****Find the area of a regular octagon with radius 20 in**

Every group had these hints written under their one problem:

*Hints:*

*Did you try drawing a picture?**What do you***need**in order to answer the main question?*Can you draw any other parts that might be useful?**Did you try it multiple ways? (Try to re-draw, rotate the picture, split the picture, etc.)**Did you try using our extremely useful RIGHT triangles?**Did you double check your work? (Make sure you did not make any assumptions, that your answer makes sense, used units, maybe even try it a different method, etc.)**REMEMBER: You already have all the information you need in order to solve this problem!*

**Raffle Rules:**

- Every team receives
**5**raffle tickets to start. - Up to
**3**bonus raffle tickets will be awarded for the correct answer with proper and clear work. **One**bonus ticket can be earned for working well together.- Each
*question*that the teacher answers or*hint*that the teacher gives will**cost 1**raffle ticket. - Answer check is free only when the problem is finished. If correct, receive award. If incorrect, it will be made clear that there is a mistake. Each group can either try to find their own mistake without losing any tickets, OR teacher can point out where the mistake is at a
**cost**of**one**raffle ticket.

**20 minutes** were put up on the timer to solve just one of these problems on their own as a group of 3 or 4. In my classroom, each group also had their own large white board to work on.

And wow! did they work their butts off! It gave every student an incentive to fight through the problem. I had a couple of groups refuse any hints and tried to find the mistake themselves. I had a few groups *not* get the extra 3 tickets because they did not finish on time, but they still kept the first 5 so they still had a chance at the prizes at the end. It was awesome. At the end, they all wrote their group number on the back of their raffle tickets and we pulled one ticket from a shoe box and the whole group won a prize. Can’t wait to try this again soon.

The area chapter is so much fun! This is the chapter where I can easily say, *it’s not about the formula, *but coming up with the formulas. It’s not about the answer, but explaining how you found your answer. Making connections is easy to incorporate and there’s just so many activities that can be done!

To start my posts for this chapter, an activity that I love. This worksheet is to ** discover** the area formula for a regular polygon. I really did not want to introduce yet another formula when this one is so easy to just split into triangles. However, I also wanted them to know the formula the book uses because I remember last year a bunch of students thinking the ‘a’ in (1/2)ap representing

**Sequence**: After covering the area of parallelograms, triangles, rhombuses, kites, trapezoids. I had been mainly focusing on coming up w/ the formulas for a rhombus, kite, and trapezoid using the area formulas for parallelograms, triangles, and rectangles only.

**Time of activity**: 30 minutes?

**Introduce:** Vocab words, “apothem” and “radius” of a regular polygon. (Review “regular”).

**Execution:** Partner work.

**A snippet of what the activity looked like**:

It goes on for a few more questions, but my favorite part that I added is this one:

They did a pretty swell job of figuring out why, I just had to help them a bit in actually getting it into writing.

*Full activity here, on my very baby-bud of a Geometry site.*

*Note (to self): ADD problem w/ connection of A=(1/2)ap to the area of a circle!*

Good group work problems are really hard to come by. Especially in math when looking for problems with multiple pathways to solving them. Here is one that seems quite simple, but kids definitely will struggle nicely with.

I ran it for the first time last year and I wasn’t quite ready for the struggle that students would have. This year went a LOT better (which is why I’m ready to share). I gave them this activity after doing an activity on The Golden Ratio. I talked a little more in depth about setting up proportions and what that means better than I did last year too.

Anyways, the problem starts like this:

Before digital cameras became popular, 35 mm films used to take pictures in a width-length ratio of 2:3. Most standard picture frames were also created back then to be 4” x 6”. Why does this make sense? Explain.

Nowadays, however, digital cameras take pictures in a 3:4 ratio, creating a need to crop pictures before they are printed if wanting to fit them in a 4” x 6” frame.

Below is an example of a digital photo the way it prints when sent to a printer without cropping. What is the problem when trying to fit it in to a standard 4” x 6” frame?

What will the new dimensions be if the digital photo were to be enlarged to fit the longer length of 6 inches? Then how much will you have to crop off? Explain with words and drawings.

Larger picture frames are typically 5” x 7” and 8” x 10”. In the same way, if a digital photo were to be enlarged so that there is no gap in the frame, how much of the picture would need to be cropped in order to fit each of the frames?

It’s a bit prettier and roomier on the actual worksheet, but if you work it out, it does a nice little twist by the time they have to fit the digital print into an 8″ x 10″ frame so that it tests their “monkey status,” as I like to call it (basically, how robotic they can get in solving a problem procedurally without understanding the concept of what they are doing). The last two problems in the worksheet are also nice and challenging.

**A bit about the execution **

*Misconceptions* to understand in advance:

- Students think that the proportions are
*additive,*instead of*multiplicative.*(I wish I had an app/ applet to tackle that misconception! Anyone know of one?) - Students will try to double the 3:4 ratio so that the short side of the photo matches the long, 6″ side of the frame. Might want to have props ready to explain that one.
- The picture I provide where the cloud picture does not fit the frame will throw them off. They have a hard time understanding that a 3:4 photo can have a short length of 4 inches to start. I’ve considered changing this picture, but instead turned it into a great way to
*start*explaining the problem without giving too much away.

Other things noteworthy:

- Order-wise, I had it mandatory that they finish pages 1 and 2.
- Pg 4 came next as optional for those finished early. **They must read page 3 in order to understand page 4!
- Pg 3 is a great example to do in getting them to use a variable to represent
*all*16:9 rectangles. I think it’s a great example for a lecture also because it reinforces the*multiplicative*idea. OR, it can be for your super SUPER kids. - Be ready to
*listen*to what they are thinking and encouraging their chosen route! Be open to being surprised by their outcomes!For example: as I was explaining the problem itself, one student yelled out the right answer by simply enlarging and shrinking the photo in his head.

3:4 became 6:8 or 9:12, for example, which became**4.5 : 6**, which was the ratio we needed. He doubled the ratio, tried tripling the ratio, then he halved that tripled ratio… I hope that makes sense.

Anyways, you can find the full packet here.

I’m starting to wish every worksheet was available in flashcard form! They TALK about the problems, they ASK questions, they rotate/point to the problems as they explain, they debate, they make up games on their own… and they get to practice a million times over w/out wasting a million paper!

So with this set, in the BOX widget to the left, they simply have to identify if they triangles are congruent by which postulate/theorem or if it cannot be proven, say it cannot be proven. I gave them the cards right after teaching them SSS, SAS, ASA, and AAS. I had pulled out all the ones they did not know yet beforehand. After I taught them HL and the Isosceles Triangle Theorem, I added those cards in and gave it to them again.

TO PRINT:

Print pages 1 and 2 back to back, 3 and 4 back to back, and 5 and 6 back to back. Pages 5/6, however, are double sets so you only need to print half as many as the other two pages.

I printed a nice cardstock class set for myself (one per pair). Students use this in class and if they need to draw on it to explain to each other, I give them patty paper so they don’t mess up my cards. I also printed out a bunch of extra copies (not cut) on regular colored paper for students to take home if they need more practice.

I threw this last card in today because after training them on how to identify AAS vs ASA, for example, they forgot what these were actually proving to start

!

Nothing all that brilliant in this post regarding the actual teaching of sine, cosine, and tangent… BUT, I do have a couple of resources that might be useful. If not brilliant, it was at least extremely effective. =)

I took the Prentice Hall California Geometry workbook, cut out and combined the tan, sin, and cos problems. I created flashcards with them (of 18 problems) and had them practice just setting them up with a partner. For example,

Then I gave them a worksheet with the exact same problems to actually finish. Be aware that we only did this after focusing on tangent alone for about 2 block days.

The template for the flashcards and the worksheet is in the Box widget.

- Print double-sided.
- The first 4 pages are the flashcard problems. Print them double-sided onto cardstock.
- I cut off 2 inches from the bottom and had them all about 3″ x3″.
- One set per partner. Teach them
*how*to use flashcards w/ partners. - Don’t give them the classwork until
*after*they have mastered the flashcards.

As for the trig introductory lesson itself, our Geometry team has been working on adapting the CPM intro of using slope, but it needs a WHOLE LOT of tweaking Especially since our students are not used to CPM, we can’t just throw it at them…

This is coming soon…

I’ve never done a lesson on it, though I’ve mentioned in classes before. Now I wish I could have done more!!

I started by asking them questions like,

- What makes a body “proportional”? Why aren’t babies proportional? Why aren’t Barbies proportional?
- What is the standard of being proportional? What am I comparing with?
- Why is the Vitruvian Man so famous? Why does DaVinci have a “code”? =P

By the time they were done measuring themselves and finding all these different ratios, I told them that over the years, people have discovered one ratio that keeps coming up in nature, even in the human body. This ratio can also be found in the Fibonacci sequence.

The Fibonacci sequence is a nice place to have them discover the Golden Ratio and compare their measurements to it because it doesn’t depend on the accuracy of their measurements.

If there was time, we then went on to measuring the face and talking about how the ideal “beautiful” face is said to be in the Golden Ratio, though this varies by greater amounts. Then we talked a bit about whether a standard can be set at all or not.

What I *wish *I did afterwards, however, is that I *really wish* I had them sketch out a Golden Body by setting up ratios equal to the Golden Ratio (proportions). That would have led quite nicely into my *next *lesson on similar polygons… which I will post possibly tomorrow…

For now, what I’ve got of the Golden Ratio is in the Box widget.