## Integrating Sound with Math

Time to rethink my integration of science with math. My attempts to connect proportions of the human body with measurement went down in flames in my entry last year, so I’m focusing on Systems, Order, and Organization related to sound this time.

I know sound, math, and science are all suuuuuuper tight. What I don’t know is how to adequately organize my sound unit so it includes great inquiry-based investigations. My guiding framework is an annnnncient curriculum from the National Science Resources Center (published when I was in junior high) that has such profound extension activities as the one featured below:

Ugh. Not helpful. It’s worth noting that there are a whopping two math extension activities in this entire unit.

The wise and enthusiasticÂ Katie Weichert gave me some great ideas to chew on and think about. I wish I saw her more often. But in her absence, I had to get a move on.

So I started trolling the Internet.

This Aztec music lesson seems compelling.

I’m also interested in harmonics, but I don’t know how to build this into a full lesson. My students already use harmonic series as a procedure to line up from music class, so I wouldn’t need to go over the basic musical idea of third and fifth intervals.

THIS could be useful. It appears to be a sound generator. Could I have kids compose a song using fractions and then convert them to their frequencies? Speaking of composing music…

I imagine I could show snippets fromÂ Donald in Mathmagic LandÂ and have students generate questions from that? Yesssssss, I could totally do that… That way the learning would be authentic and related to the curriculum we already have in place!

My only concern remains starting with a video. I want to make sure I’m looking for an introduction that inspires perplexity, not just engagement. After the 27-minute video was released in 1959, Walt Disney admitted:

“The cartoon is a good medium to stimulate interest. We have recently explained mathematics in a film and in that way excited public interest in this very important subject.”

(emphasis is my own) Now in looking at moving from merely interest to investigation…… I suppose that recording student questions will take care of that fear, right? Then having their questions shape the following lessons?

Hmmmmm. Of course, there are a wealth of videos available on sound and math, but much of the information is so complex that I can’t figure out how to simplify it.

I’m also interested in looking at the materials used in instrument strings and the number of strings included in different instruments. How do the number of notes an instrument is capable of producing related to its system? Can systems be different sizes? Is a larger system necessarily “better” or more “complete?”

Anyway. Let’s see how this goes.

## Rigorous Math Every Day

The open-ended math from the Wall Street Journal a week or so ago was pretty rad. But lessons like those are admittedly woefully rare in my classroom. It’s a huge shame, right? Learning like that shouldn’t just be a once-a-month or even (eep) once-a-semester event.

So I started pondering why doesn’t math look like this in our classroom every day. I needed to keep myself real. Here’s what I came up with:

I’ve purposely chosen those phrases because I think we teachers sometimes use them as ultra-self-deprecating or unproductive language and the conversation just stops there. But I want to explain why these really are often valid concerns (or at least, valid-feeling concerns) and then focus on how I’m personally working to move past them.

Perhaps you’ve already heard me rail against people who say “I’m just not a math person” and seen me express frustration that the idea “math is sooo hard” is a bunch of bunk. That said, I’m still thoroughly unconfident in my own math abilities. I was mortified when I transposed two numbers in our soccer math. I freaked out when Mr. Brown informed me I HAVE BEEN DOING ORDER OF OPERATIONS TOTALLY WRONG. So it’s fair to say that when I deviate from our district frameworks, it’s a little stressful.

I’m moving past this excuse by being willing to really lean on my secondary-level colleagues. I love collaborating, but I don’t particularly love admitting that I need help. So this is a huge area of growth for me. Also, taking the leap to put detailed lessons online has given me a chance for feedback from folks from across the nation, like from my favorite ladies in the Midwest.

I was euphoric when our class completed its project last week. I was also exhausted. I can get sucked into manic cycles really easily. Although spinning my way into a cycle can be absolutelyÂ exhilarating. I need to be honest with my body and realize that it’s notÂ healthy for extended periods of time.

“The management is hard.” That’s what people tell me when I share our latest project. I agree, but not in the way they intended. Teachers often mean, “I’m going to have children stringing stuffed monkeys from the room if I open the lesson to exploration.” I share with my kids the explanation from Twyla Tharp’s book The Creative Habit that in order for creativity to take place, it happens within a system of order. A dance studio is essentially a bare floor and mirrors. An artist can’t create a masterpiece if she can’t dig out the right paints in her chaotic mess. And we can’t have deep, meaningful conversations about math in our lives if we’re not already solid in our class expectations.

So, the management I’m talking about isn’t the student-secretly-reading-under-the-table-instead-of-doing-math business. And it’s not because issues like that don’t exist in our classroom — the aforementioned situation actually happened last week and was dealt with swiftly. I’m talking about the mental gymnastics I put myself through as I’m wandering about the classroom facilitating conversations. Although the brain only takes up 2% of our body weight, it uses 20% of our energy, according to Bill Bryson‘s A Short History of Nearly Everything.

So I’ve gotta keep myself mentally in shape. That means reading tons of books I love even when other teachers tease me. That means blowing off grading homework for a night to paint my nails. That means making time for my physical health and not necessarily devoting hours of lesson planning each day.

Not enough hours in the day. I’m, frankly, super-pissy when I hear teachers say this, and then five minutes later I’m nodding at the truth in it. Because yes, our job is impossible and yes, there are insane demands coming at us from all angles. But I feel like you can’t automatically default to complaining about time without carefully looking at how you currently do spend your time.

For me, this has meant a intentional devotion to super-quick transition times and an up-tick in the priority I make in keeping my room clean so I don’t have to scrounge for materials. Now, my goal is shifting to providing great math instruction by still letting me be a human.

Among neuronormative folks, the generalÂ consensusÂ is I’m an overachiever. *I* don’t feel that way, but apparently the speed with which my brain works and the resulting efficiency I have in completing mental tasks makes me one. When I think ofÂ overachieversÂ in my mind, I definitely don’t want to be someone who spends hours constructing the perfect math centers that can only be used for a week or two.Â I’m certainly not that extreme, but I admit I’m still working on this. Mainly because I get sucked into interesting information online and can’t pull myself out. But limiting myself to a half-hour of prep time before class begins seems to have been a good boundary to set.

I want a system, whether it just be an internal mental process or a procedure I can use in my classroom, to ensure that I’m pursuing great math with my kids but I’m not spending hours in the staff lounge or on the Internet to do it. I suppose a time-hog that others might forgo would be the time I spend documenting my process and further questions I have through blog posts here, but the writing-about-it part is just fun.

I could continue writing, I suppose. But I’m off to redo my nails. Because I’m only going to really be a good teacher if I know when it’s time to let go.

## Air Time

When you take time for yourself, good things follow. In this case, it was some REALLY AWESOME MATH.

Friday morning, I missed the bus (oops) and was able to drive to work at a legal speed.

I had an opportunity to drink some nummy nummy coconut mocha coffee and read my Wall Street Journal. And lookie what I found!

Iâ€™ve been thinking a lot about Dan Meyer commenting on how we spoon feed each step to a problem solving situation, and so today, I went out on a skinny limb and used this graphic to help us work on our measurement skills. I wasn’t sure where our work would take us, but we’re early on in the unit, so many of my students are still working to measure accurately using a ruler.

I showed them the graphic, and Samuel helped me pronounce all the playersâ€™ names. He was our resident expert. Then I opened the floor to mathematical questions.

Hereâ€™s what we brainstormed as our big questions.

Â The questions with green dots to the left are the ones we decided to pursue.

Then, people started asking more â€œnitty-grittyâ€ questions, which we identified as being the â€œquestions along the wayâ€ you had to answer to get to your big ideas. We kept this poster up as we worked. I stayed near my computer so I could capture students’ comments.

â€œYou need to know how big the field is,â€ Savanah spoke up. I handed her my iPad so she could find the field size. She paused. â€œDo I need to know like, how BIG it is or how long the sides are?â€ Â â€œI think youâ€™re asking me whether you need the area or the perimeter?â€ â€œYeahâ€¦ ohhhh, I need the length of the sides.â€ Hereâ€™s the information she found.

After checking another site to verify the accuracy of her information, we added the dimensions of the field to the poster. (Yes, I know I could have taken a screen shot of the iPad, and I did, but I couldn’t get the image sent to my computer. Hrmph.)

â€œBut whatâ€™s a yard?â€ â€œWho can answer that?â€ â€œItâ€™s three feet,â€ Ivy answered. â€œHow can you check to see if you agree?â€ â€œWell, I could look in my math book, but I remember what yard sticks last year look like, and I know there are three rulers.â€ (I knew we’d need to convert from yards to feet to inches so they’d be able to convert the lengths they measured on their papers into the actual lengths)

â€œWell, then you need to multiply by three to get the length â€“ 120 times three.â€ â€œWoah. Howâ€™re we going to do that?â€ â€œUse a known fact, 12 x 3.â€ â€œ36?â€ â€œYeah, 36.â€ â€œSo itâ€™s 360 feet.â€

They did the same for the other side. Then a group of students wanted to determine the linear distance the ball traveled for each player. I asked how many inches long their picture was, and Marcos stopped us all.

Marcos: WAIT. You blew up the picture from your newspaper article. So our picture isnâ€™t the same size as yours and the distances will be all different. (I photocopied the graphic at 121% so it’d be easier to read than my original copy of the newspaper.)

Me: Nice. That would be a problem if the image were STRETCHED like a rubber band and warped, but since it was enlarged to scale, weâ€™ll be okay AS LONG AS you donâ€™t let me use my original copy, okay?â€

Marcos: Okay. So the field is 11 inches long.

“You know, if they would have just included a map scale on this picture, we wouldn’t have to do ANY of this measurement.” “I guess that’s why Miz Houghton wants us to be able to use map scales in social studies.”

Then a few of us worked to create this poster.

We knew the field was 11 inches in our image, but we wanted to know how far just ONE inch would be because then we could find out how far Jone Samuelson’s 6-inch kick actually went. We also knew how long an actual field was, so we tried to find the relationship between the two.

Using a fact family (the triangle drawn above) helped us figure out the ratio. Or. What I initially THOUGHT was the ratio. DO YOU SEE MY GLARING ERROR??? I didnâ€™t notice until lunch. I neglected to convert the 240 feet into inches so the units matched. Drat. I frantically called AP Calculus teacher James Brown to make sure I didn’t make any further errors.

So after lunch we converted 240 feet into inches, THEN used the ratio and found out that one inch in our picture equalled approximately 33 feet.

Some students switched to using calculators for these larger computations, which gave us a chance to talk about how calculators represent 1/2, equivalent fractions (5/10), etc. Above, Alejandra calculated how many feet David Villa kicked the ball (5 inches, according to her measurements, making the kick 165 feet). I asked her about the “33 in. in a inch” she wrote, and she said, “Oh no no no, it’s not 33 INCHES or that would be like a mini soccer field.” So she was also looking at reasonableness of answers.

Another group wanted to know how far the balls would have gone if they were kicked on the moon. Again, I told them to ignore the parabolic motion and just look at linear distance. I know the physics of this arenâ€™t entirely correct, but I didnâ€™t think itÂ  hurt the integrity of the original problem situation.

Oh, actually! Selam originally asked how far the ball would go in SPACE, but Maya pointed out that if the they were in space, the player and ball would both push off each other and the ball would never land (AMAZING INSIGHT, RIGHT???). So we clarified that the ball would be kicked on the moon, where there was still a force acting on the ball, but a lesser force than what we’d find on Earth.

Adam went to the classroom library to find out what the gravity was on the moon. Hereâ€™s the passage he found, from the DK Eyewitness Book UNIVERSE.

Eayn: It says the gravity is one-sixths of Earth!

Me: So the gravity is 1/6 of the gravity on the Earth. So if we are converting from the moon, what would we have to do to the distance we calculated for the ball kicked on Earth?

Me: Where did you get three from?

Milena: Multiply it times five.

Me: Five? Where did you get that from?

Milena: If the moonâ€™s gravity is 1/6, then the rest of the fraction thatâ€™s left is 5/6.

Me: Ohhh, I think I see what youâ€™re picturing in your head. But think of the gravity on the Earth as being one whole, and the gravity on the moon being 1/6 of that whole. You’re not looking at the other 5/6ths.

Vy: Youâ€™d multiply it times six.

Me: Where did you get six from?

Vy: If itâ€™s dividing by six to get the pull on the moon, then youâ€™d multiply by six to show how much further the ball would go when it has a sixth of the gravity slowing it dowwn.

Me: So youâ€™re saying that fractions can be a way of dividing.

Vy: Yep. And then the opposite, er, inverse, is multiplying, so you times by 6.

(It is perhaps worth noting that Vy has not voluntarily spoken in front of the class in the past year and two months)

Wow. So now that we knew how to find distances on Earth and on the moon, we plugged away, with at least three people needing to agree on their measurements to the nearest half-inch before we would post the results. (reviewing our estimation and rounding unit from earlier in the year)

As we approached second recess, we posted what we’d come up with so far.

We also reflected on what we’d learned over the course of the day, and on the math we used.

As you can see, we didn’t finish everything, so some students asked if they could finish the calculations during Math Daily Five. UM, YES OF COURSE.

What suggestions or modifications do you have to offer me and my students? Where can we take things from here? Other thoughts?

## Book of the Week: A is for America

Every Monday, I highlight a book from our school bookroom along with lesson plan suggestions. I hope you find this useful, and please leave a comment with any suggestions or additions!

A is for America, by Devin Scillian

I love pretty much anything published by Sleeping Bear Press, and the bazillions of alphabet books they’ve printed are, by and large, pretty wonderful. We have a mentor text copy of A is for America ready to go in the bookroom.

You can access an extensive activity guide for almost every Sleeping Bear Press book here.

The author of this book is also the nightly news anchor for Channel 4 in my beloved Detroit, and it’s pretty awesome to see a “celebrity” author who can write pretty darn well.

There is aÂ CAFE menuÂ included with this mentor text, and Iâ€™ve highlighted these as suggested lessons:

• Back up and reread.Â This is a pretty dense text. I actually intended to post this lesson two weeks ago, but since then, *I* as a teacher have had to back up and reread the book several times. In the past, I’ve used Sleeping Bear Press alphabet books over several days, reading two letters (and reviewing each of the previous letters using call-and-response). Often, we talk about backing up and rereading if the text is CONFUSING, so it could be important to talk about backing up and rereading if the text is just plain DENSE.

• Use dictionaries, thesauruses, and glossaries as tools.Â In my time away from teaching social studies, I forgot about the fabulous tool hidden in the back of our textbooks known as the Gazeteer. A “geographical dictionary,” isn’t that brilliant? I know I often tell students to not worry if they can’t pronounce a proper noun in text, but wouldn’t it be great to give each student a letter from the book and have them investigate each of the locations featured in their letter?

Please add any lessons or supplemental materials to the book bag so future teachers can utilize your good thinking!

Comments and constructive feedback are always welcomed. Please let me know if these lessons were useful in your class!

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## Book of the Week: The Runaway Dinner

Every Monday, I highlight a book from our school bookroom along with lesson plan suggestions. I hope you find this useful, and please leave a comment with any suggestions or additions!

The Runaway Dinner, by Allan Ahlberg

This is a great silly, nonsense book that reads like an extended version of “Hey Diddle Diddle” plus The Gingerbread Man.

Also, apparently I read this back in January 2011 and book talked it, whoops…

Allan Ahlberg has a bunch of other books, especially poetry books, that might be worthwhile to investigate.

There is aÂ CAFE menuÂ included with this mentor text, and Iâ€™ve highlighted these as suggested lessons:

• Infer and support with evidence.Â At the beginning of the story, and several places in the middle, the author insists the story is completely true. Ask students if they agree, and ask them why the narrator would have purposely,Â blatantlyÂ lied like he did.

• Reread text.Â A cumulative story like this has reread text kind of built into it. To infuse a lesson on author’s craft, talk with students about why the author may have chosen this device for the story. It’s not quite as sing-songy as “There Was an Old Woman,” so why does it still work?

• Ask someone to define the word for you.Â Items like ketchup, carrots, and french fries can’t be easily defined using a dictionary. In younger grades, consider using realia to support this lesson so students will be familiar with the dining utensils and foods they encounter as they read.

Please add any lessons or supplemental materials to the book bag so future teachers can utilize your good thinking!

Comments and constructive feedback are always welcomed. Please let me know if these lessons were useful in your class!

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## Book of the Week: Er-lang and the Suns

Every Monday, I highlight a book from our school bookroom along with lesson plan suggestions. I hope you find this useful, and please leave a comment with any suggestions or additions!

Er-lang and the Suns, Retold by Tony Guo and Euphine Cheung

Er-lang and the Suns: A Tale from China is a text from the SFA Roots program. There should be one master copy of the Roots lesson plans in the bookroom. There are check for understanding questions on post-its throughout at least one of the three teacher copies.

This is an origin story covering how the Earth finally got reprieve from its seven suns that shone nonstop. There are plenty of other origin stories to compare and contrast with. As always, pre-read these texts before sharing them with students, as they are appropriate for different ages.

The end of the book contains a brief history of China and the Han people.

As mentioned earlier, there are three copies of this book if you want to use them as a grade-level team mentor text.

There is aÂ CAFE menuÂ included with this mentor text, and Iâ€™ve highlighted these as suggested lessons:

• Make a picture or mental image.Â At the end of the book, there’s a brief passage that talks about how the illustrations were designed to match the tone of the story. Ask students to pick and sketch 5-7 of the most important images that they think are critical to telling the story. To take this a step further, then have them write a brief caption for each picture. Huzzah! They’ve now also used the strategy of…
• Retell the story.Â See above.
• Compare and contrast within and between text.Â See above for plenty of other origin stories. Perhaps students could select their favorite and document the similarities and differences with Er-lang and the Suns.

Please add any lessons or supplemental materials to the book bag so future teachers can utilize your good thinking!

Comments and constructive feedback are always welcomed. Please let me know if these lessons were useful in your class!

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## Book of the Week: Red-Eyed Tree Frog

Every Monday, I highlight a book from our school bookroom along with lesson plan suggestions. I hope you find this useful, and please leave a comment with any suggestions or additions!

Nic Bishop is a brilliant photographer. Joy Cowley does a nice job of using pretty basic text to create a quick narrative of a tree frog’s day. There aren’t any text features, but there is a “Did You Know?” section in the back.

There’s an !OFFICIAL! FWPS lesson plan around main idea and details included in the bag. It focuses on activating prior knowledge.

I love everything Anita Silvey does, and you should definitely check out her Red-Eyed Tree Frog essay.

Additionally, there’s some extra content available on Houghton Mifflin’s website.

There is aÂ CAFE menuÂ included with this mentor text, and Iâ€™ve highlighted these as suggested lessons:

• Use main idea and supporting details to determine importance.Â As mentioned above, there’s a pre-designed FWPS lesson plan for this in the book bag. You might also talk about how nonfiction books are sometimes intended to be read out of linear order — for example, reading the Did You Know section at the end of the book first won’t spoil the story like it would if a fiction book were being read.

• Cross checkingâ€¦ Do the pictures and/or words look right? Does it sound right? Does it make sense?Â Because the comprehension lesson is about activating prior knowledge, this might be an opportunity to explain a time when cross-checking might NOT work. If a student has never seen or heard the word “katydid,” for example, no matter how many times they look at the text and picture, it won’t magically make sense.
• Flip the sounds.Â There’s a point where the frog is stalked by a “hungry boa snake.” If students pronounced the word correctly on the first try, ask how they knew they didn’t need to try flipping the sound first. Explain that as they become better readers with more strategies, the slower, more cumbersome strategies like flipping the sound won’t be as critical for them on a regular basis.

Please add any lessons or supplemental materials to the book bag so future teachers can utilize your good thinking!

Comments and constructive feedback are always welcomed. Please let me know if these lessons were useful in your class!

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## Book of the Week: Annie and the Wild Animals

Every Monday, I highlight a book from our school bookroom along with lesson plan suggestions. I hope you find this useful, and please leave a comment with any suggestions or additions!

Before you get started on anything Jan Brett related, you’ve got to stop whatever you’re doing and go straight to visit Mrs. Eltrich or Mrs. Burn. They’ve put together a pretty fabulous Jan Brett author’s study that might be useful.This book has post-its with open-ended questions attached to several pages to use during reading.

This book was originally paired with Caldecott-winning book The Big Snow, but that text hasn’t been added to the mentor text library as of this posting.

You can see Annie and the Wild Animals read aloud here:

I’m pretty impressed with the literature guide here. I honestly don’t know that there’s much I can add beyond that!

There is aÂ CAFE menuÂ included with this mentor text, and Iâ€™ve highlighted these as suggested lessons:

• Compare and contrast within and between text. Spoiler alert! Annie’s cat has kittens. In the past, before Bob Barker’s daily reminders to spay and neuter our four-legged friends, this text might have been a great one to make predictions and confirm them at the end. Older students can discuss how the book would be different now that it’s nearly thirty years after it’s been written.
• Infer and support with evidence. This strategy could be used regardless of whether students predicted Taffy would have kittens or not. If few or no students are familiar with the signs of a cat about to have kittens, it’s a great opportunity for a discussion of how difficult it is to infer if you don’t have much prior knowledge and how important it is to have heightened awareness of the world around us. If students DO pick up on the signs of Taffy’s pending delivery, proceed with a regular inference lesson.

• Ample easy reading. If students have read this book (perhaps with Mrs. Eltrich or Mrs. Burn! :)), remind them that in a book as complex and detailed as Annie and the Wild Animals, there’s plenty to return to and explore, particularly if they first discovered the book a year or two ago.

• Ask someone to define the word for you. Mrs. Eltrich has already printed out vocabulary cards for several challenging or uncommon words in the text. Talk with students about how if you know a word is particularly unusual and you don’t anticipate many will know it, you choose to give them the word ahead of time.

Please add any lessons or supplemental materials to the book bag so future teachers can utilize your good thinking!

Comments and constructive feedback are always welcomed. Please let me know if these lessons were useful in your class!

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## Book of the Week: I Wonder… About the Sky

Every Monday, I highlight a book from our school bookroom along with lesson plan suggestions. I hope you find this useful, and please leave a comment with any suggestions or additions!

I Wonder… About the Sky, by Enid Field

This is an older book, and it seems like it’s out of print, but our bookroom has a copy, so let’s go with it. I think it can anchor a couple of pretty critical thinking skills. Consider pairing it with one of these resources:

Wonderopolis. The name pretty much speaks for itself.

I Wonder Why… New series on our local NPR station.

There is aÂ CAFE menuÂ included with this mentor text, and Iâ€™ve highlighted these as suggested lessons:

• Ask questions throughout the reading process.Â This is pretty self-explanatory. The entire book starts with “I wonder…” and then some element about the sky or weather. For example:Â
• Predict what will happen, use text to confirm.Â I notice that often when I make KWL charts with my students, we neglect to follow up on them. (whoops) Consider copying a few of the pages, posting them around the classroom (or the hallway — the photographs are pretty neat), then letting students add answers or new learning as they find them.

Please add any lessons or supplemental materials to the book bag so future teachers can utilize your good thinking!

Comments and constructive feedback are always welcomed. Please let me know if these lessons were useful in your class!

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## Book of the Week: If You Are a Hunter of Fossils

Every Monday, I highlight a book from our school bookroom along with lesson plan suggestions. I hope you find this useful, and please leave a comment with any extra ideas or additions!

Well, this book set off quite a bit of pondering on my part. Originally published in 1980, much has happened in the world of dinosaurs since this was created. So at what point does a nonfiction book do more harm than good in a library? You may have seen me tackle this question earlier this month.

I attempted to find a recording of the Reading Rainbow episode this was featured on. Instead:

You can SEE what I searched for! Did Reading Rainbow seriously Rickroll me???

Anyway. There are a billion and a half dinosaur links and lessons and related books I could suggest to you. So I’ll just share two of my favorites.

Sue at The Field Museum in Chicago. Sue, in my mind, is, for kids, the reason museums were made. The whole reason students clamor to go on a field trip. Sue is a beautiful, huge, wonderful, magnificentÂ specimen that every scientifically-inclined human on Earth should go see. I haven’t met her yet. But I will.

Dinosaur National Monument. I visited the monument in 2004 reporting on the Colorado River Trips program. When you wake up in the morning, you wonder how anyone got any dinosaur-digging done because the whole area is so breathtaking.

Also, if you’d like to search for fossils in Washington State, here’s how. I priced out a Wildwood trip to Stonerose Interpretive Center, but it’s obviously out of our price range. Like, way way way out of our price range. Boo.

Anyway. Baylor’s story is a lovely piece of poetry-ish prose that celebrates the spirit of exploration and discovery. The art is OK, and it hold up well to more than thirty years of age. So I think it will stay in our library for now â€” it’s still absolutely relevant. (Although some new research points to Pteranodon being more bird-like than bat-like)

There is aÂ CAFE menuÂ included with this mentor text, and Iâ€™ve highlighted these as suggested lessons:

Please add any lessons or supplemental materials to the book bag so future teachers can utilize your good thinking!

Comments and constructive feedback are always welcomed. Please let me know if these lessons were useful in your class!

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