Yesterday, I had the privilege of attending a one-day conference hosted by Curriculum Associates, the company that produces the lessons that I use to teach my students math. During a whole-group general session between breakouts a few different leaders from the company got on stage to share some ideas. One of them was Kenneth Tan. He was in charge of speaking about some new ways of interpreting the data that diagnostic assessments provide.
He did a nice job making meaning from graphs and charts. One of the things that Kenneth shared caused me to remember a lesson I’d taught my 1st grade math enrichment class the day before.
Keynote speaker Glendaliz Almonte shares in Grand Ballroom Hilton Philadelphia at Penn’s Landing.
He had an image appear on the huge screen at the front of the room. It was a grid with around 12 or 16 seemingly random words in boxes. The audience was asked to try to remember as many words as possible. I figured there was a catch, and I took a couple of seconds to glance over the entire grid, getting a feel for the words. Were there any connections that could be helpful?
Just as I realized that the collection of words contained not only nouns and verbs, but adjectives and articles as well as prepositions, the image disappeared! I tried to quickly string together any words I’d remembered, forming a sentence, no matter how silly.
The speaker probably knew his audience was smart enough to think of this trick and, either to limit their success or to save time, switched slides from the grid of random words to one containing the sentence that I’d tried to piece together. Kenneth Tan remarked that data is only as helpful as it is meaningful. I liked the analogy.
The idea of connecting words made me remember my first grade math enrichment lesson from Monday. In that lesson, I had students join single-digit numbers to grow a double-digit number from the left side of the image larger and larger, until it eventually equaled the double-digit number shown on the right. This lesson was straight out of the Ready Math Teacher Toolbox (Lesson 20).
With the image imported into a Jamboard, students were able to trace over the provided lines with color. Each color was a different student’s work.
Typing out the process makes the lesson seem more complicated than it was. As you can see from the image, Ready Math had numbers in boxes. Some were double-digit, and some were single-digits. The double-digit numbers flanked the single-digit numbers. Students had to leap frog across the boxes with numbers to get from one double-digit number to the other, and the trip should create a balanced equation. Ready Math had an example that made the task plain as day.
The activity was an instant hit! Among other things, we discussed the relationship between the numbers. The 17 needed a nine to get it to 26. There were several ways to make nine, using the numbers that were available. We talked about combinations of numbers that could not work, and why, as well. One student (the blue line and numbers) wanted to go from 17 to nine, and then visit eight. Either they wanted to try something different, or they had recognized that eight plus nine equals 17. I let them try it, but they realized that the path would not “land them on” 26. They would over shoot their goal.
The Ready Math enrichment assignment had a few diagrams with varying numbers. I had different students come up to the Google Jamboard to draw paths and write equations that demonstrated getting from one double-digit number to the other.
I showed the first graders that every equation for a diagram had some parts that were the same. The first number, the double-digit number, was always the same. And, the sum, the double-digit number on the other side of the equal sign is the same for each. It is the middle addend(s) that change. To illustrate this fact, I drew two boxes in the middle of the equation. These symbolized the boxes from the diagram that held single-digit numbers. When we had finished working through the numbers that were available, we came up with some others that weren’t shown.
Then things got really interesting. The first graders felt bothered when one of the diagrams had numbers that were not being used. The starting number was 88, and we had to get to 95, a difference of seven. One of my students drew a line from the 88 to an eight. That would put the running total up to 96, one past the goal of 95. Rather than tell the student that he was wrong, I asked if there was anything that he could do to “Balance the Equation.”
With a little help from his friends, the first grader decided to change the operation from addition to subtraction between the single-digit numbers. In this way, he was able to incorporate the last unused number, a one. Sure, one had to be “taken away” in order to complete the algorithm accurately, but at least he was valued worthy of a place in our equation!
More than adding and subtracting, this was a lesson in making connections and building relationships between numbers. Finally, finding balance between the two sides of the equal sign is not just algebra. It is a life skill.
“Where did the 850 cans come from?” I was in the middle of sharing the iReady enrichment lesson (14) with my fourth graders when one of them asked me this question.
Have you ever had a student ask a question in order to postpone learning? If you’re a teacher, then that’s a silly question. Of course!
This is one of the few things that I remember from my elementary and middle school days. It was a thrilling challenge to try to come up with just the right topic or question that could throw the teacher off track.
We would hope and pray for a story. Then, we would artfully flatter and ask questions that would lead our pedagogue down the rabbit hole of memories, further and further… away from the lesson at hand.
Fast forward forty years. Today’s students still play the same tricks on their teachers! This past week I was engaging some fourth graders in math enrichment, when one of them tried steering me off task. Little did they know, that I practice Pedagogical Aikido.
Redirecting Energy
Aikido is a form of martial arts that is known for using an opponent’s energy (ki) against them. Masters of this study practice redirection.
Although I have not formally studied Aikido, I love its principles and attempt to use the philosophy of redirecting thought and energy within the walls of my classroom as much as possible.
For example, the other day when my student asked about the origin of the 850 cans in our math problem, I allowed the student to think that he had derailed the lesson. I told him that this was an excellent question. “850 cans is a lot of cans. Where would a school get that many cans for a fundraiser?”
The martial art Aikido uses a triangle to teach the redirection of energy. There are three components that work together to use an opponent’s attack against them, saving your energy and neutralizing the situation. It all starts with Balance, known as tachi waza (Aloia, 2020).
“How many students does our school have?” I asked the class.
I could have squashed the student’s inquiry, telling him something like, “I don’t know where the number of cans came from. It’s hypothetical. Let’s just move on!” Or, “It came from Curriculum Associates, the authors of our math program. Don’t ask silly questions.”
If I had done that, I would have disrespected the student. A dismissive teacher or one who blocks the question head on is too hard, too strong; the lesson too one-sided. By allowing for the question in the first place, and then entertaining it, I had my center of gravity low to the ground. My metaphorical feet were spread wide apart and knees bent. The question didn’t topple my lesson. I was balanced.
In answering my question, the students were surprisingly accurate. Our school has around 700 students. “How many cans would we have if each student brought in one can?” I prompted. That was easy. “But, not every student will bring in a can… And, some will bring in more than one.” The easy back and forth of these simple concepts established a flexible, down to earth ease of thinking. It also revealed the problem. We don’t know where the 850 cans came from.
Next, it was time to Break Balance. This is the second part of the redirecting-energy triangle. “The opposite of balance is imbalance, or kuzushi. To break an opponent’s balance, one must first redirect their energy to one’s own advantage” (Aloia, 2020)
I shouldn’t be surprised, but I was very impressed, nonetheless, at how quickly my students figured out how many classrooms our school had. It was the advanced fourth grade math students receiving enrichment, after all!
I had begun the imbalance kuzushi by getting the class to come up with the total number of classes in the building. After figuring out that our school has five classrooms per grade and our school teaches six grades, if you include kindergarten, we discovered that there are 30 classes represented.
“Let’s say that our school collected 850 cans. How many cans would each class bring in?” The students had no clue where to start.
Antonio Aloia (2020) explains that kuzushi has two arms. The physical off-balancing of an attacker, parrying the opponent’s strike and redirecting the momentum of the assault, coupled with a strike of their own is what one normally thinks of when imagining Aikido. Um, of course there isn’t any literal physical contact with students, let alone “attacks,” but presenting this new problem of dividing up the number of cans by the number of classrooms was a cogitational assault of sorts.
The other arm of kuzushi is a psychological off-balancing. This is where a martial artist would “Distract a would-be opponent by bringing their attention to something else, be it an object on a building or something farther away and behind the opponent” (Aloia, 2020). Pedagogically, this happened when I changed the student’s original question from “where” to “how”: “Where did the cans come from?” turned into “How could a school come up with so many cans?”
While the martial art of Judo involves throws, Aikido keeps your opponent tight and controlled. Perhaps counter-intuitively, it is concerned with the well-being of the attacker. So, rather than toss my students aside to flounder with the problem of dividing 850 by 30 on their own, I guided them through the process of figuring out the answer.
I asked them how many cans there would be if every class brought in 10 each; 300. “Okay, maybe that was the first week of the fundraiser. If each class brought in another ten cans during the second week, how many cans would the school have collected?” We were up to 600 cans. They were starting to catch on.
One of the students used Google to divide 850 by 30. Rather than scold him, I asked him if it were possible for any of the classrooms to bring in .333333 of cans. This was a silly question. “What happens with the remainder from the division answer?” I asked. They didn’t know. “For our purposes, we will assume that the students from every classroom brought in 28 cans. The teachers brought in the rest.” My students were okay with this explanation.
The third side of Aikido’s redirecting energy triangle permeates everything. It is ki or energy. Don’t think of it as power or force, though. Ki is more like momentum.
“How big are our classrooms? How many students are there in a classroom?” I got several answers on this. We decided to use the number 20. “Let’s say that a quarter of the students don’t bring in any cans. If the rest are responsible for bringing in 28 cans, how many brought in two and how many brought in one?” My students just looked at me. I told them to try and figure it out on their own, and then I’d show them.
One student crushed it, and I had her show the class what she did. Then I modeled drawing a picture to solve the problem.
After all of this, I told my students, “Now that we have collected all of these cans, we need to put them in something to bring them to the food pantry that we are donating them to.”
“If Dylan went out and bought a bunch of boxes… Thank you Dylan! (Dylan is all smiles at this point; He may or may not have been the person to ask the question that started all of this;) And, if Dylan’s boxes are all the same size, holding six cans each, how many boxes would Dylan have to get?” I let them wrestle with that a little while.
When I was prepared to let them demonstrate their math on the board, I turned to the slide that had the original question on it. They reread the word problem as I decided on who would come forward to share their work first. A few students groaned and some others called out. “That’s the problem we just did!”
“Yeah?” I feigned ignorance.
I used someone else’s name when I told the story about getting bigger boxes; Ones that held 8, instead of 6 cans. “How many of those boxes were purchased?”
As it turns out, we never got to fully explore the last question, but a couple of students tried solving it in their heads. I had completely Aikido-ed them! Lol.
Redirecting energy can be an even more effective motivator than a cool lesson. Take their energy, spin it around, and use it against them. Students will feel like they’re in charge of their own learning, and in a way, they are!
Who wants to see a picture of Mr. Weimann with hair?
EVERYONE, apparently. I used this hook to get students to pay attention to my pitch of wallpaper hanging. They hung on my every word.
This was a math enrichment lesson, in which I presented a real world problem that my fourth grade advanced math students would struggle through, using many math concepts that they already know. Knowing what to do with the numbers is sometimes more than half of the problem.
When I was in college (the first time;), I began painting to pay my way. Back then, it was just the outside of homes (exterior painting). Upon graduating, I conducted an informal internship with a wallpaper hanger. He taught me all about interior painting; which is VERY different from slapping paint on siding; and he trained me to hang wallpaper.
This pic is from 20 years ago.
I was 22 years old, had long, curly, brown hair, and according to my students who did end up earning the privilege of viewing my old photos, quite tan (nearly all of them commented on this fact;). I worked with the professional wallpaper hanger for a little less than a year, hanging all kinds of paper in all kinds of homes.
About half of his work came from a restoration company that did insurance projects. These homes had suffered water, smoke, or fire damage.
Unlike a company that specializes in a niche of upper-middle class single family repaints, this work brought us into a wide range of residences. I worked in downtown Philadelphia and on the Main Line, a very wealthy area–This place has both Ferrari and Lamborghini dealerships, among others!
This was one of the stranger bathrooms I papered. I had to paper inside that skylight.
We rehung thick, tough paper on basement ceilings; and metallic, mirror-backed papers in tiny bathrooms. It wasn’t unheard of for me to have to remove several layers of paper in older homes, in order to get to a clean plaster surface.
It was fun to see so many different types of homes, meet a variety of peoples, and travel all over the area. I learned a lot about problem-solving because every job was completely different, and you didn’t get paid until it was done, and done well.
Eventually, I struck out on my own. I got married in 2000, and my wife Sonia and I started our own painting company called “Excellent Painting.”
We used feathers to produce the veins on this marbleized pillar.
We ordered lawn signs and door hangers, spread the word, and pioneered a full-service painting/wallpaper business that did it all! We even dabbled in faux finishes, popular at the time. Faux finish is when you use tricks and artistic techniques to make pillars and walls look like marble, stone, or wood.
Because we were the only company that could do all three, we got a lot of business, especially in the new developments that were popping up in suburban areas near where we lived. The market was ripe, and we were busy.
This is a photo of one of our neighborhoods. We would literally move our equipment from one house to the next, working our way around the loop. Having the interior of your home decorated by Matt and Sonia Weimann was a must.
There are many components to running a successful business. You have to market the company, spreading the word, so that people know to call you. Then you need to return phone calls and perform estimates. Those estimates need to be cheap enough that people will hire you, but expensive enough to cover the costs of supplies and provide a livable income.
Eventually, you have to deliver on your promises. You have to be able to successfully perform the work that you estimated. Be prepared to follow up on complaints, questions, and problems. If you don’t, an infection of bad publicity will spread around the neighborhood, and you can write that group of homes off! Do things well, and reap the rewards of moving from house to house, cutting down on marketing costs and time.
This is an advertisement from a small local paper that was mailed to many homes in our area. It was an awesome investment!
Many of the skills and lessons that I learned through entrepreneurship are tapped in my teaching. Some days the professional world mixes with pedagogical practice more than others. My favorite place to bring my background to life is in math lessons.
The other day, I treated my fourth grade math enrichment class to a treat of problem-solving that had them working hard, thinking hard, and learning hard. The challenge was to help me come up with a price for wallpapering a couple of bathrooms.
I shared a photo of a bathroom that I papered several years ago. Many opinions about the bathroom were unsolicitedly shared by my students. In order to get them motivated, I asked, “Who wants to see a picture of Mr. Weimann (that’s me;) with hair?”
Instant quiet. Hands involuntarily went to mouths to clamp lips shut.
“Listen up.” I proceeded to explain that you don’t just glue wallpaper to walls. You have to perform all kinds of preparatory work.
There’s taking down old paper, because many times the rooms that you are going to hang wallpaper in were rooms that were already wallpapered. This is a lot of work. You may not know this, but wallpaper is two layers. There is the thin vinyl layer that sometimes peels off in large flexible sheets, but other times has to be slowly, painstakingly, curse-your-existence-ly chiseled off of the walls in tiny, bitesize pieces; As, you can probably tell, I have had too many experiences like this! Then there is the backing, which more closely resembles actual paper. This, you wet and scrape off rather nicely. There are all kinds of tricks to removing wallpaper more efficiently, but suffice to say, you better estimate plenty of time for this part of your project!
Sometimes I used a steamer. Other times you can just spray warm ammonia water on the wallpaper.More often than not, you just chisel away at the top layer.
You aren’t done preparing the walls for wallpaper when all of the old paper is removed, however. Before you move on, you have to clean off all of the old glue! If you don’t, your new paper may not adhere properly. You do not want your new wallpaper to fall off the walls or bubble up. This process is not only messy. It can be deceiving. The glue is transparent, so that it does not discolor the wallpaper. Cleaning it off of the walls requires diligent and frequent assessments. Run your fingers over a cleaned wall. If you feel anything slippery or slimy, it is probably glue that still needs to be scrubbed off.
People don’t hang wallpaper the way they used to. My students will probably never practice this skill. But, I shared all of this information with them, so that they could see that there is much more to crafts like this than what meets the eye.
Once my students had been prepped with the horrors of wallpaper work, I brought out some numbers. I kept it simple with easy fractions to work with.
We would charge one whole dollar per square foot for actually hanging up the wallpaper, ½ a dollar for taking down the old wallpaper (Once in a while rooms don’t have wallpaper to remove. Plus, you want to make this a separate price so that customers appreciate all of your hard work, and it doesn’t seem overpriced.), and ¼ of a dollar per square foot for prep work. (This would include cleaning the walls, caulking cracks, and spackling holes and other imperfections that the paper won’t hide.) These prices are from when I was first starting out. I actually did this kind of thing, regularly.
Next, I shared a simple floor plan with my students. Some of them recognized the blueprint for what it was. I showed them the illustration of doors and asked them what the rectangle representing a closet was. We discussed what was happening in the picture for a minute. And then, I told them that our customer wants to wallpaper the two bathrooms.
The image was presented on an interactive Google Jamboard, so I could write on the board. I used a bright blue to rewrite the dimensions of the bathrooms in question. I told them that the ceilings were 8 feet high. When I turned around, I was met with incredulous faces. They had no idea what to do!
This was perfect, because it provided me an opportunity to teach. I told them that a good strategy when dealing with a difficult problem is to draw pictures. I drew four rectangles beside the image of the floor plan on the Jamboard. With a little prompting I got my fourth graders to figure out that we would need to find the square footage of the wall space that would be covered with wallpaper in order to create prices. “What are these four rectangles?” I asked.
I’ll confess that the first time I did this lesson (I have an AM and a PM 4th grade math enrichment class) I tried drawing a three-dimensional image of the bathroom on the side of the floor plan, and this ended up being too difficult to understand. I ended up pulling each wall from my drawing out and making individual rectangle representations.
In my second attempt, I cut straight to the 4 walls. This was less confusing.
Either way, it required some spacial thinking to understand what to do with the numbers.
This was my first (AM) lesson.
Once it was established that my picture of 4 rectangles were in fact the walls, we labeled the dimensions: Each one was eight feet high, and two were one length, while the other two were a different length.
In order to figure out the square footage of all of the walls, you solve the area of each, and add them together. This reads simple enough, but my students had never had to do anything like this before!
It took a little convincing to show them why we needed the square footage at all. I used the dimensions of our classroom and kept it to only one surface; the floor.
“How big do you think our room is?” I prompted. After a couple of guesses, I told them that I thought it was about 30 by 15. I then paced out the floor and found it to be 11 paces by 6 paces. “If we calculate each pace to be worth 3 linear feet, what are the actual dimensions of the room?” Wait for it. My students knew to multiply 3 times 11 and 6, respectively.
“Are 30 and 15 easier to work with?”
We used the dimensions of our classroom floor to practice. (The -100 was subtracting the cost of supplies.)
“They are compatible numbers,” Evie answered. I beamed with pride. We’ve been talking about using helpful numbers to do mental math a lot.
“That’s right; Much easier to work with. And, what is 30 times 15?” Blank stares. “What is three times 15?” Now we were thinking! When we figured out the answer to that, I wrote “15 X 3 X 10” on the Jamboard. “Thirty feels big, but pull it apart. It’s just 3 tens.”
“So we know that the floor of the classroom is about 450 square feet. How much would it cost to wallpaper the floor… Of course no one would do this, but what would it cost?” Everyone quickly understood that it would cost $450, since each square foot would cost one dollar.
Understanding #Math concepts well enough to be able to explain to peers, and then providing opportunities for #MathTalk
“Now, let’s say that some maniac had already wallpapered the floor before we were asked to. What?! Crazy, I know, but what if… We would have to remove that old paper before we can apply any new paper. How much will that cost? It’s fifty cents per square foot,” I remind them. It doesn’t take long to figure out half of 450.
“And, finally, there will be some prep work before actually wallpapering. That costs twenty-five cents per square foot.”
In order to illustrate this concept, I drew a square on a new, fresh Jamboard slide. I labeled it $1. Then I drew another square, the same size as the first, and drew a line down the middle. I labeled each half $.50. Before going any farther, my math enrichment students knew to halve the half.
While a couple of kids knew what to do with the twenty five from $225, some didn’t. I told them to break apart the number. “Don’t think of it as 225. What is half of two hundred?” I wrote 100 on the board when they said it aloud. “What’s half of 25? How do you know that it’s 12 and a ½?”
Evie-to-the-rescue-again explained that 24 divided by 2 is 12. The one extra that it takes to make it 25 can be split in half. “Nice,” I encouraged.
“So, how much would it cost to do the whole project on the floor of the classroom, provided some nut wanted to wallpaper a floor?” Staring students looked stunned. “You have the price for applying the wallpaper,” I said, circling the $450. “You have the price for removing the old wallpaper.” At that point, I circled the $225. “And, we just figured out that it would cost 112 and ½ of a dollar to prep the surface.”
“Do we add it all up?” a hesitant student asked.
“That’s right,” I said, trying to sound proud and hoping to steer my students toward feeling like they can do this. “Now, time to figure out what this bathroom would cost!”
I let them struggle a little before helping. They needed a bit of guidance. But, we figured out the square footage of all four walls. They did pretty good finding out what it would cost. And, the second bathroom was a little easier.
A funny experience happened at the very end of the afternoon lesson. This PM group of fourth graders originally entered the room with the announcement that they already knew the answer to the problem.
A couple of them had walked in on me finishing up my morning math enrichment lesson. They confessed that they already knew the final answer to be $952, “or something.”
I told them that I was going to change all of the numbers, so it will be different. They thought that sounded reasonable;)
I had intended to make the ceilings 9 feet tall for the PM class, but the concepts were difficult enough to understand that I didn’t feel like it was all that necessary. But, by the very end of the afternoon lesson, after I told the students to take the prices of the two bathrooms and add them together so that we can tell the customer how much it will cost…
“Wait,” they exclaimed. “You didn’t change the numbers!” Huge smiles stretched across their faces. They loved being tricked.
“Look at that. You ended up needing to do all of the work, anyway.” We all laughed as they exited the room.
John Burger, second grade teacher extraordinaire (2012)
When I first began teaching, I had a mentor who was amazing. He taught second grade, and it was a calling more than a career. His name was John Burger. Rather than do Social Emotional Learning (SEL) lessons, he was SEL. Everything he taught had emotional and social lessons woven throughout it. Like myself, education was a second career for John. He had been an engineer before becoming a primary school teacher. More than the money, he was doing this because he believed in it.
In addition to John’s unique way of teaching, he used some teaching tools that I liked so much that I adapted them into my own repertoire. One was readers theaters. I have shared a few blogs about those in the past. Another was mental math games.
The mental math games were designed to make math fun and exciting. They are a great tool to use on the fly, because students don’t use paper or pencils. They should do it all in their head. Some of the games require critical thinking. Some rely on short-term memory and problem-solving. Others practice rote memory math facts.
One of my favorites comes with a story. John liked baseball, and each year, when he introduced this game to his second graders, he would tell them, “When a baseball is hit really hard, and instead of it going up into the air, it is a line drive, straight over the grass… If a worm were to stick his head up out of his hole just at that moment, it would get its head burned by that cruising missile of a baseball! This is called worm burning.” There would be all kinds of gasps as kids pictured a worm being scalped by a wizzing baseball. Then, in his soft, understanding style of sharing, John would explain how the game works. “I’m going to say a bunch of numbers, and I will tell you what to do with them in between. You have to try your best to keep up. I’ll go slow in the beginning, but then I will begin to speed things up. When I stop, you tell me the answer that’s in your head.”
“You have to keep the answer at the front of your brain.” Sometimes I lose it and have to stop.
The mental math game is fast. The teacher will use single digit numbers and a variety of operations, keeping track of the answer until the worm burner has run its course. When the teacher stops saying numbers and operations, students have to raise their hands with whatever answer they are left with. I have my students show me their answers with fingers; They raise the number of fingers that they think is the answer. The teacher (or student; I’ll have kids try it when they get good at it) who is sharing the math must keep the final answer under 10.
I often teach students this game at the beginning of the year and use it during whole-group bathroom breaks or times I need to keep students quiet. We get really excited when we know the answer and/or get it right, so it is hard to be completely silent. I remind students that they ought to only raise the accurate number of fingers. I praise students who do this well.
You change up the difficulty of the digits and speed by which you say them to adjust so that more students can participate. The students who are more fluent with their facts are affirmed as math whizzes, and that’s just the way it is. “Good for them. The rest of you can study and memorize your facts just the same.” I have witnessed students work on learning their facts and gradually move up the ranks in Worm Burning, until they became competitive with the best of the burners.
I usually try to do a few that everyone and anyone can get. You can weave in a couple of tricks, like multiplying the whole thing by zero. Then everyone gets in on the answer. It gets everyone to at least pay attention and listen. Also, the Polite Pirates perk right up when they hear me say, “subtract 99 or 98” because they know the going answer before that was probably 100, and we are back to only 1 or 2, respectfully. I’ve had lost souls jump back into the game at that point. It’s fun to see them grab ahold of confidence as they celebrate success.
Sometimes, but not always, I will go back and walk the class through the Worm Burn. I’ll demonstrate keeping the answer right there at the front of your mind. Often, the Worm Burn is so fast or long that I can’t remember all of the steps. The successful students are usually proud to help me remember, though.
Later on in the year I will introduce larger numbers that can be tricky. For example, I will have students multiply 25 by 4. “I don’t know that!” they’ll cry out.
“How many quarters are in a dollar?” I’ll ask them.
“Oh…” They get it, and then I will do a bunch of worm burners incorporating twenty-fives.
Another number I’ll throw in at some point is fifteen. “Three fifteens is the same thing as three quarters past the hour. How many minutes is that?” I’ll explain after stumping my students. Sometimes I’ll use alternative words like “dozen” or “double that” to keep things interesting.
By the end of third grade I would be throwing fractions into the mix. It’s a great way to cement the understanding of denominators dividing numerators. I will get the Worm Burn to the number 24, and then say, “What is a third of that?” Or, maybe I’ll start off with “Three fifths of fifteen (9), plus three quarters of four (3), divided by six…” and so on. Pause just enough so some students can get it, but not so much that others blurt out the answer. And, don’t make it so hard that no one gets it!
Gifted
This year I have transitioned from being strictly a third grade teacher to the gifted support teacher for kindergarten through fifth grade of my school. I was sharing the game of Worm Burning with my third grade gifted students, when I saw an opportunity to bring the math to the next level… and then some.
I never write the Worm Burn on the board. But I foresaw a unique teaching opportunity here.
I told my third graders about a trick that I often use in order to keep the numbers straight and maintain a going answer in my mind; I will use the answer in the next operation. For example, “Two times three, plus six…” I added the six to reinforce in my own mind that the product of two and three was six. I never write the Worm Burn on the board, but in going back to show this trick further, I wrote out the sequence of operations from a previous Worm Burn. This introduced the idea of squaring a number, which then lead to teaching exponents.
Then I thought about how different Worm Burning was from using Order of Operations. And, out comes PEMDAS! We were already talking about exponents!
I started out with a simple Worm Burn, “One plus three, divided by two, times seven, minus four, divided by five, plus one…” The answer is three. I wrote the burn on a Google Jamboard and showed the sequence of math. Then I told my students that if I were to do this math properly, the answer would be completely different. They were intrigued. “What do you mean, properly?” they wondered. I wrote the acronym PEMDAS on the board.
When we followed the rules for order of operations, our answer was much more complicated. With the help of Siri, we were able to divide numbers that didn’t have obvious answers. How do you divide three by fourteen? Is that even possible? Well, if you have three boxes of cereal, can fourteen people have some? How much of all of the cereal would each person get? Ask Siri.
The final answer came out to -.5858, which was really weird. They were unfamiliar with decimal points, let alone negatives. It was an eye-opening adventure.
From Worm Burning to diving down a rabbit hole of increasingly complex math concepts, my gifted third graders were happy to transition to reading about everyone’s favorite vampire rabbit, “Bunnicula,” and take a break from arithmetic.
This is my journal entry when I first came up with the idea.
Math enrichment, fifth grade style…
This idea came to me several weeks ago, but I hadn’t had a chance to throw it to my fifth grade gifted students until today. The topic that I started with was volume. The fifth graders were learning the algorithm to solve for volume at the time.
I wanted to come up with a reason they would need to discover the dimensions of a 3-dimensional space. My background as a painting contractor came in handy. When I estimated the prices for painting ceilings and walls of rooms, I had to do tons of math. How could I bring that experience into the classroom?
I had the idea of working backwards. I would give them a large number, and they would have to figure out the dimensions of the space.
I decided to turn my fifth grade gifted students into interior decorators. They would need to figure out the measurements of floor space and wall surfaces.
In creating my math problem I tried out a variety of numbers, multiplying length times width times height, until it created a nice round number. I made the ceiling 8 feet high, and the room 13 by 15. That comes to 1560 cubic feet. Before settling on this number, I tried breaking it up various ways. You could do 12 X 10 X 13 for a higher ceiling. This was good, because I wanted there to be more than one correct answer.
I wrote the problem out and put it on the board for fifth graders to read prior to class beginning. After the announcements, I read the problem aloud to everyone. We practiced our Ready Math routine, the same four-step method I wrote about in a blog about 2nd graders writing their own word problems. First, I asked the fifth graders what the problem was about. Then we discussed what we were asked to find. Next, we identified the information necessary for solving the problem. As it turns out, the only number is 1560, but what does this number represent? And, don’t you know an algorithm that can help you interpret this number? “Yes!”
I wrote L X W X H = 1560 on the board, when the fifth graders said it. “So, we identified the topic of the word problem; We know what we have to find; What are you going to do with this number and algorithm?” I guided my fifth graders. “You could try making some predictions. Plug in numbers and see what you come up with,” I suggested when I saw that they needed a nudge.
Some students were still stuck, so I asked them what they thought the space looked like. “A cube,” someone suggested aloud. Dylan jumped on deciding the room was not a cube. He used his iPad to find the cubed root of 1560 to be 11.5977, and since one of the parameters was that the dimensions are whole numbers, this option was off the table.
“You can’t use your iPad,” a peer protested.
“I never said anything about not being able to use iPads or calculators,” I offered. Fingers feverishly fought to open devices. Not everyone, though. There were some students who chose to stick with paper and pencil.
A group of girls asked to use some rulers. When I asked why, they told me that they wanted to get some ideas. This seemed perfect to me. They realized that they needed some background knowledge. They began measuring the classroom.
A student came to me with the dimensions 40 X 39 = 1560. At this point I brought eveyone together for a teaching moment. “If the room were 40 feet by 39 feet, and the volume of the three-dimensional space were 1560 cubic feet, how high would the ceiling be?” They thought about it for a second. Some multiplied 39 times 40 and discovered that it equals 1560. “The ceiling would only be one foot from the floor! Two measurements would make it a two dimensional space. You need a third measurement to give it depth,” I explain. “What could you do with these numbers?”
“2 X 20 X 39,” several students say at the same time.
“So, now we have a two foot high room. This needs to be a space that normal adult humans can walk around and live in.”
“4 X 10 X 39”
“We are getting closer. Why don’t you do to the 39 what you were doing to the 40? Try breaking it up.”
After a while, a few students were beginning to figure out more reasonable dimensions for a living space. They thought that they were done when they came up with three numbers whose product produced 1560, but “Oh, no! You still have to do the interior decorating work. Now that you know the lengths and widths of the walls, you must figure out how much hardwood flooring you need to get. And then, you have to calculate how many double-rolls of wallpaper to order,” I remind them.
I shared the word problem with the class via a Jamboard, so that they could share their work. They could write right on the Jamboard, or take a picture of their papers, using Jamboard. The classroom was electric with mathematicians calculating, communicating, collaborating. I don’t know about interior decorating, but these students were making my room look and sound great!
It’s October, and the second graders in my school are learning the basics of solving word problems. The arithmetic is super simple single-digit algorithms. Likewise, the stories couching the numbers are unadorned with character development, setting, or plot. Time for some #MathEnrichment!
My idea was for the 2nd grade gifted students to write their own word problems. Before creating our own, I thought I’d model some. I wanted to provide some math that was challenging, but doable, albeit with my help. As it turns out, my math riddles had the second graders perplexed to the point of paralysis.
It wasn’t just the numbers. Somehow, I’d forgotten what Ready Math had taught me! There is a specific way to read word problems. Even the smartest of the smart; kids who can wrestle with and make sense of the math; won’t be able to decipher what is being asked of them if they aren’t taught how to comprehend what is going on in a math story. It is like an Olympic runner being dropped into the middle of a forest and expected to sprint to the finish line that she can’t even see.
I constantly tell my students that challenges are fun. These kiddos weren’t buying it. Challenges are only fun when there is some hope.
Like an idiot, I dragged my students through my word problems, doggedly showing them what the numbers were doing. They were good sports. When I let them write on the Google Jamboard, they perked up. In the end, they left my room with number hurricanes storming their cerebrals. I was left to pick up the mess of math misconceptions strewn about the streets of seeming failure.
That was last Friday. Over the weekend I remembered; There is more to solving a word problem than crunching numbers! The Ready Math curriculum instructs teachers to have the students use a 4 step approach to solving word problems. When you break the process down this way, it is much more manageable.
First, read through the word problem and decipher what the story is about. Don’t worry about the numbers. What is the topic? Are we talking about reading books or alien monsters that can control your actions with their minds?
Next, identify the important information. What are the tools you’ll need to fix this problem? Don’t be fooled into thinking that unnecessary numbers or information will be needed. Sometimes there are superfluous facts stirred into tricky math stories to trip you up!
Before doing any math, you must figure out what you are asked to find. (This step might come second. I can picture needing to know the end goal prior to identifying the important info.)
Finally, we begin doing some math. Show all of your work/thinking.
There’s a fifth or bonus step that I told my 2nd grade gifted students about, that has to do with communication. Just like we include publishing in the “Writing Process” and the “Scientific Method,” we are not done solving our word problem until we share the answer. Make sure to label the numbers with whatever unit of measurement or name of thing you are talking about!
The answer of this word problem doesn’t even have any numbers in it! But, you better show and be prepared to explain your work for full credit.
Tuesday morning, the day after Indigenous People/Columbus Day, we were back at it with more challenging word problems. When I first showed my students this Jamboard, I had the “sticky notes” layered on top of the word problem. As we read and discussed what each one meant, I moved them to the sides and shrunk them, so that they all fit on the right for reference. Only then, did we read this word problem.
My 2nd graders kept yelling out numbers, like there was a contest for who could solve the problem first. It was humorous to tell them that every single number that they would say, no matter what it was, would be wrong. This was perfect for drawing their attention to step 3. “What are you asked to find?”
“Are you supposed to provide a number as an answer?” I asked my students. This got them thinking. And, even after solving how many candies each kid had, they still needed to compare the numbers in order to really finish the problem. It wasn’t enough to just know how many each had.
This is only the first of many word problems to come for our 2nd grade gifted students.
Once we had successfully solved our Trick or Treat word problem, it was time to make up our own. Before getting creative, we decided on the numbers and operation. We would have the mathematicians subtract seven from twenty.
I wanted to include everyone’s ideas. That is why the math story has flowers named after a student’s pet bunny, a main character named “Kid Pineapple, robots, laser blasts, and lots of the word “stinky” in it. Ha ha. We had a figurative blast coming up with our story.
We look forward to making up and writing down many more for our other 2nd grade friends to solve… The Ready Math way.
I used to use a giant set of Dominoes to introduce the game to The Polite Pirates (my 3rd graders).
Playing the game of Dominoes is an excellent way to introduce and practice problem solving. I’ve used this game for several years in my 3rd grade classroom, and now I am introducing it to my gifted students.
This past week was the first week that I met with my elementary (K-5) gifted students. Obviously, the various grade levels were in different places when it came to math concepts. Second grade is working with word problems. Third grade will be tackling multiplication soon. Fourth grade is focused on geometry right now. And, 5th grade is preparing to use formulas to solve for volume.
Dominoes can be used to introduce multiplication, using cumulative property, strengthening mental math, not to mention strategic thinking.
Tuesday morning’s second grade gifted lesson began with my typical introduction to Dominoes. I told the students that each tile is called a “Bone” and the pile of unused tiles is the “Boneyard.” They learned that there are always 28 Bones in a game, and that every single Bone is different. Each one has two numbers on it, even the ones that look empty! “What number is on this side?” I asked holding up a Bone with a six on one side while the opposite was blank. They didn’t skip a beat in guessing “Zero.”
It took a little longer to explain adding the ends of each Line of Play. “You add up any number that is at the tip of an arm, no matter how long or short the line is,” I told them. We practiced some play. The adding wasn’t a problem. We discussed using the cumulative property to switch up the order of numbers, so that the mental math was easier. “Look for combinations that create ten,” I told them. “Rather than adding 4 + 5 + 4 +2 equals fifteen, combine the 4, 4, 2, first. Then you instantly know that the five makes fifteen.”
I let students draw and write on the Google Jamboard to show and explain their thinking.
“Points are only awarded when the sum is a multiple of five. The ends of all of the lines of play add up to ten. Is that a multiple of five?” They knew that ten was made up of two fives. “How many fives are in 15? 20? 50? 500?” Flawless computation… Multiplication, here we come!
“Whenever you earn points, you say, ‘Give me ten…’ or however many points you earned.” I learned this from a fun video that taught me how to play ages ago, and elementary students LOVE this aphorism.
“Give me ten!” one of my second graders beckoned. I put 10 on the board, and we played a couple more sets before moving on to our Self Portrait project. Even though I only have my gifted students for 40 minutes at a time, I found doing more than one mini lesson to be helpful in keeping their interest.
After school, I came up with a word problem for Wednesday using Dominoes. I typed it into a Jamboard. Even though I tried to word it in such a way that you could visualize the game in your mind, I went ahead and used Domino tiles (Bones;) to make a model of the hypothetical game on a table and took a photo. I imported the picture on the Jamboard slide with the word problem text.
I snapped this pic, so that I could show my students the thinking behind their word problem. I am hoping to inspire them to use their “Gifted Journals” to jot down ideas through drawing diagrams and writing notes just like this.
When Wednesday arrived I was excited to try out my word problem. The first group of students that I met with was 5th graders. Because I do car duty, I get to my room at the same time as my students. This morning I had placed a note on the open door, instructing them to try to work out the answer to this problem while they waited for me to show up. The word problem was presented on a giant Google Jamboard that I’d rolled in front of the entrance to the room.
The group was crowded around the screen when I got to my room. I enjoyed listening to their discussion of possible moves. From what I heard, a couple students were aware of the general concept of matching numbers. They didn’t quite understand adding all of the numbers from the end of each line of play, but that was okay.
When one of the students decided the trick to solving the problem was to simply take a new Bone from the Boneyard, I invited the 5th graders into the room. Before talking Dominoes, I praised their persistence and told them that I was proud of how hard they worked. I purposefully pointed out the failure to solve the problem as an example that not everything will come easily to them. It is my job to come up with challenges that stretch their thinking, and I intend to make not only their intellects but also grit grow.
My 5th grade gifted students experienced something new this morning: bewilderment.
As I explained the solution to the puzzle, I used it as a teaching tool. When I informed them on how the points work, they noticed that placing the Bone with the two on one side at the end of one of the lines of play would bring the total sum of all of the arms to 20, a multiple of five, and thereby an opportunity to earn points. “Good. But, there is a third and less obvious option,” I told them. “This one,” I said, pointing to the Double that had a six on both sides, “Is a Double. It is special. Not only can you play it with one six touching the line of play (also a six), but you can set it perpendicular to the line, thereby creating an extra arm.” I waited for someone in the group to notice the new sum.
It didn’t take long for a fifth grader to notice that all of the sixes add up to 30 points. “Turning the Double on its side, created a fifth number to add. Since all of the numbers were the same, you can’t help but have a multiple of five! It is literally five of the same number.”
I tried the word problem on my daughter Scarlet who had a brilliant reason for using the Double instead of the 2-6. In addition to the points, you would cause your opponent to have to take from the Boneyard, because chances are they won’t have a six. Limiting their options was very strategic thinking. I shared this strategy with my 4th graders who already knew the game well.
Next, it was 2nd grade’s turn. These kiddos had just learned how to play the game for the first time the day before. I used the word problem to reteach some of the ideas. More than focusing on problem-solving, I walked my second graders through the mechanics of writing a word problem. We discussed the details of Dominoes that I included in my little story. “Why did I mention multiples of five?” I prompted. “The goal is to earn points.”
I had toyed around with the idea of only having words. When I originally crafted the word problem, I didn’t have the Dominoes laid out on a table. I tried to provide just the right amount of text for students to be able to visualize all of the details necessary for solving the problem in their heads. At the last minute I decided to include a photo with the problem. This was only the third day I was teaching these students! Watch out, though. Word problems, here we come!
Pride flooded my person when fourth grade arrived in my room and crushed this problem within seconds. I’d taught all but two of these students how to play Dominoes last year. When I saw how quickly they solved the puzzle, I wished I’d hidden the photo.
I pointed out the verbiage at the end of the word problem. “Why is it okay that I didn’t say what numbers were on the Double in the word problem?” I asked them. I had left out this vital information on purpose, and I wanted to see if they could find the puzzle within the puzzle. I did this kind of trick with them all last year. They understood that it had to be two sixes because the word problem stated that “You have two bones that you can play.” If the double were any number other than six, you couldn’t play it.
Here’s a question I failed to float: “Is it possible that the player has more than two Bones?” And, this is where a photo is limiting. Because you can see only two Bones on the table, it is difficult to imagine there being more than two. But, yes, the parameters of the problem leave that detail open. You have two that are “able to play,” but you may have more. I’ll have to work that riddle into a future problem!
My last group of the day was third grade. Here, I used the word problem differently. After exploring the numbers and solving the puzzle, plus revisiting how to play the game, I drew their attention to the structure of the word problem. It was a story, providing characters (you and a friend), a setting (a game of Dominoes), and a conflict. “This word problem has a plot,” I pointed out. (Plot is the language arts concept that third graders are learning right now.)
The problem of which Bone to play is the climax of our story. The resolution is a mystery. You could use either Bone. Which one is better? What is the third, less obvious option? How will the story end?
And, this is the end of my blog about this amazing word problem. But, is it? Through the process of fleshing out the many lessons that coated these Bones, I have come up with some additional puzzles. Here is a taste: “What is the highest score that can be collected in one play of Dominoes? What would the Bones look like at the ends of each line of play?” I think I’ll provide manipulatives; Bones; for pupils to push around when solving this. Good luck!
The Captain of Class 