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Stunning Gems: A Geometry Math Enrichment Lesson

Here is a math lesson that went so well that it would be a crime if no one replicated it. I’m sharing the details in hope that more students can benefit from the learning available. 

The lesson was for fourth graders who had been taught points, lines, angles, and triangles. 

Copy of Instructions

My aim was to create a lesson that students could experience independently. I wanted them to follow my directions without me walking them through the steps. For this reason, I tried to be as explicit as possible when describing what to do. At the same time, I had to balance this with simplicity, so that it did not scare students from the project. 

This was a math enrichment project. It was created to deepen the understanding of students who had proven proficient. The last thing I wanted was for this to be extra work. I had hoped it would be challenging and fun. It was. 

Although I’d hoped students would be able to follow the steps independently, most needed some guidance; at least in the beginning. Below, is an example of what I did when helping students successfully complete this math enrichment project. 

One way that you could adapt the lesson would be to provide the first two lines; Pre-drawn for students. I didn’t. I just dropped off the instructions and told students to use a blank sheet of paper. In addition to blank paper, students will need protractors and rulers with centimeters on them. 

Introduction

After a student read the introduction to the assignment, I paused the project to discuss what a “Stunning Gem” might be. I explained to the students that raw gem stones are found underground, and when they are discovered, they look like sea glass; dull. They were all stunned. I told them that it wasn’t until after a specialist; a jeweler; cut the rare stones and polished their sides that they shone beautifully. 

“The angles have to be made just right, with straight sides, so that light can bounce off of the translucent walls of the gem like a prism.” This set the stage for measuring accurately and drawing straight lines on their papers. 

Getting Started

Each student had a blank piece of paper, a pencil, and a medium sized protractor that had a centimeters ruler in the center of it, along with his/her own set of the instructions. I had a different student read each step out loud. We went step by step, with me modeling the work. 

At the very beginning I showed students how to draw the first line. I had them make a dot about one third of the way up from the bottom of their paper, right in the center. “Label that point A,” I told them. “Use your ruler to draw your 7cm line like this.” I drew my first line diagonally, at about a 45 degree angle toward the right side of the paper. 

The paper with the instructions on it does not specify any of this. I didn’t want to type so many steps that students were completely turned off. I figured they could just figure out what worked best. Make mistakes; fix them; “#ProductiveStruggle” is what I’ve been telling everyone this year. “It’s good for you! Builds character.”

After the first line is drawn, we label the end opposite point A, “B.” Next, we are instructed to “Draw an eight centimeter line twenty degrees from line segment AB.” Here, I show them that they ought to draw this new line closer to the center of the paper. I also tell students to draw their lines lightly. “Don’t press down too hard. You will most likely have to do some erasing. If you make dark, heavy lines, your gem will be less stunning with a bunch of erasures all over it.” 

I won’t go through every line of instruction here. You can see what happens when you read the steps. Suffice to say, there is a lot of protractor practice and line-drawing-procission involved. You definitely want students to help one another and be open to mistakes. Luckily, these are not actual diamonds.

Finished Product

It takes several minutes to complete the drawing, but it’s worth it. There are plenty of “Oooohs” and “Aaaaahs” when the final lines are drawn. 

My favorite part of this project is that the drawing is a tool for answering some geometric questions and even playing a game. Ahead of time, I typed the questions into a Google form, along with multiple-choice answers, so that the form will provide a score immediately upon completion. At the bottom of students’ “Stunning Gem” instruction paper is a code for the Google classroom where the assignment can be found. This was the first project of the year, so students had yet to access this new classroom. 

There were two assignments in the new Math Enrichment Google classroom. The first contained a Google doc with the same instructions students had just used to draw their stunning gem. This assignment asked students to take a photo of their drawing so that I could see how well they did. The Google doc was available for them to make another one, or in case they lost their original set of instructions while still making the gem drawing independently. 

The second assignment had the Google form in it. This form asks questions like “Which triangle is an isoceles triangle?” And, “Which triangle is scalene?” In addition to these identification questions, I made the drawing a game in the last few questions by asking students to look for the total number of triangles that can be formed with the lines. 

Finally, I set up the Google form so that students can see their peers’ answers and edit their own, once they have submitted their work. This is math enrichment, and my aim is to enrich; Not assess. They loved getting perfect scores… Albeit, eventually;)

Origin of Lesson

The idea of having fourth graders who were learning about lines, angles, and two dimensional shapes draw pictures of diamonds came from a lesson I did with third graders. They were writing diamonte poems, and I taught them how to draw a diamond to accompany their poetry. 

I took this concept of drawing a diamond, and tried writing instructions for sharing the steps with out an illustration. The product of pupils producing stunning gems from nothing but words was even more rewarding than I had expected. They were genuinely wowed by what they made. It felt like they had created something spectacular from nothing but lines and angles! 

Lessons Learned

First of all, a confession: I forgot to include a couple of steps in my first attempt at this lesson! A whole group of students was completely mystified, if not aggravated by the project when a substitute teacher tried to lead them through the drawing exercise without me. When I saw what they did and spoke with them, I figured out what was missing. There are lines that must connect the points on the top of the gem that I never mentioned. I fixed the instructions and pushed out the additional steps in an announcement via the Google classroom stream. 

Additionally, this is not a “Whole-class-project.” This is designed to enrich the math understanding of students who are already good at using a protractor and ruler. When there are too many students to check between steps, others get bored, misbehave, go ahead, mess up their drawings, and the lesson is less successful. 

Give yourself and your students plenty of time. This will take a good 45 minutes to do. You can have them come back to it, but it’s best to wrap up all of the drawing at once. Also, have them take the photo and put it in the Google classroom assignment while in your presence. They rarely do this on their own. 

Teaching Ratio

In order to help students understand a concept teachers often have to present it multiple ways. The more challenging the topic is to understand, the more methods might have to be used. I found this to be the case the other day when sharing the idea of ratio with my third graders.

I had presented my Quidditch quandary to the class; “Can a team win a game of quidditch without catching the snitch?” We had explored the ratio of two to one. This had gone fine. My third graders could easily understand the idea of a team being twice as good as another.

I paused at this point and introduced the term ratio. I explained that a ratio described the relationship between two quantities. I made sure that they understood quantity to mean the amount of something; In this case the number of goals a team has in relation to the number of goals another team has. So far, so good.

We worked through several examples of ratios on the board.

They did okay when we changed the ratio from 2:1 to 3:1. Now, one team was three times as good as the other. Things got hairy, though, when we changed the ratio to 3:2.

At first, I tried to just use numbers to show what was happening. “If you used a number that was divisible by three, it’s easier,” I began. “You could put that number on the left side of the colon.” I wrote a twelve on the board. “Then, break it up into thirds. Put two of the thirds on the right,” I told them. “What number goes into twelve three times?”

“Four.”

“Right.” I wrote eight on the right side of the colon… “12:8.” This may as well have been Portuguese to my third graders!

As it turned out, a couple of my students spoke Portuguese fluently! Not literally.

I teach gifted students. That is how I can present problems like these to 8 year olds and expect them to get them. Sometimes, like the present lesson, I have to do some extra teaching. But, for some gifted students, the math comes naturally. It is like a language to them. This is truly remarkable to witness. I captured one of my third grade student’s ratio-realizing moment on video. He used the numbers like a master painter might transition from one color to another. I was so impressed!

He had come up with 48:32 completely on his own. I wanted him to explain where these numbers came from. With a little prompting and help filling in the gaps, he and I recorded his thought process in the video I posted to X.

“I tried 45. Then I got into fifteen because three times fifteen equals 45. Then, I found out that was a tie,” the student breathlessly begins. He started to explain adding three to 45, but I interrupted him.

“How was that a tie?” I prompted.

“To do that,” the student recalled, “You get thirty, which is three hundred,” writing the numbers on the dry erase board as he spoke. “Then you add the snitch, which is 150.”

At this point in the video, I (unfortunately) talk over the student’s explanation. Before I began rolling the video recording, the student had raced through his explanation in his excitement to share his finding the correct answer. I wanted to help him clarify how he new 45 was not the correct answer, before moving on to the right number. With us talking at the same time, the audio is a little cumbersome, but I just kept the feed rolling. “So, show me 48,” I said, giving my student a thumbs up.

The student did not articulate audibly everything he had done, but he showed, through writing on the board, what numbers had been used. He had added three to 45, bringing Team A’s number of goals up to 48. Because 45 divided by three is 15, he knew a number three larger would have a quotient only one more when divided by three. In other words 45/3=15… Raise 45+3 to 48, and 48/3=16.

He instinctually got the relationship (the ratio) that caused each rise by three of the dividend to increase the quotient by one.

It was barely a step for this student to double the 16 to make 32. He then added the value of the Golden Snitch (150) to 320, which is how many points 32 goals would equal.

As the student rewrote his addition on the board, other students watched on. They noticed that the math communicator whom I was video recording had accidentally written “16 X 32” on the board. Someone began to point this out, commenting aloud, “Why’d you write sixteen times thirty-two?”

You can hear me tell this observer, “He’s thinking faster than he can write.” I didn’t want my scholar to lose his train of thought. Some students can be heard confirming that they see how 48 would work. At one point the most beautiful “Ah ha” moment can be viewed when a student realizes how the numbers fall into place unlocking the combination to the problem.

I was very proud of this third grader. It thrilled me to capture his “math talk” on video. Not every student understood the concept of ratio this easily, however. For this student, the numbers and ideas just fell into place. For others, the concept was clunky and the numbers were far from lining up neatly.

I tried guiding them through the same math I had already worked through with my fourth graders. They did fine with the computation, but the third graders were lost when it came to understanding the relationship between the two sides of the ratio. Fractions, multiplication, and division are all relatively new concepts for these students. Even though some of them have been multiplying numbers for years, understanding the concept is not long lived.

In addition to math, we explored the spelling of ratio, after one of my students mentioned that it was a word that did not fit a pattern. “There are other words that do what ratio does,” I mentioned. We brainstormed a bunch.

The students who understood how ratios worked wanted to do more math. They itched to prove themselves masters of arithmetic the way our video star had done. I gave them the new ratio of five to four (5:4), and they jumped on it.

At this point, there were some students who understandably did not know what to do with the five or the four. This was when I took the idea of division and simplified it into forming equal groups to show the relationship between the two sides of the ratio.

“Let’s start off with an easy number,” I suggested. “How about we have Team A score twenty goals.” I wrote a twenty under “Team A” on the board. “If Team A scored twenty goals, and the ratio is five to four, Team B will score more or less?” I figured we could start small.

“Less,” a couple of kids offered.

“Right. How much less might seem tricky to figure out.” The looks on faces told me that they agreed.

An idea occurred to me that I wanted to try. I drew a line of five circles on the board under the number five. I drew four circles under the number four. “What number would you multiply by five in order to make twenty?” I asked. When my students told me four, I wrote the number four inside each of the five circles under the five.

“Remember, ratio means relationship between quantities. That means what we have over here…” I pointed to the five circles with fours inside them, “We must have over here.” I then wrote four inside the four circles under four. “The five fours equals twenty. How much is four fours?” (I know it’s a lot of fours. It feels funny writing four so many times. I contemplated using a different number, but these worked well for [ha ha] my students;)

When they told me that it equaled 16, I wrote that under the four circles. Then, I erased the contents of the circles. I wrote a six in every single circle. “Five sixes equals what?” I wrote a thirty under the twenty. “Four sixes equals what?” Twenty-four got written adjacent the thirty.

An idea occurred to me that I wanted to try. I drew a line of five circles on the board under the number five. I drew four circles under the number four. “What number would you multiply by five in order to make twenty?” I asked. When my students told me four, I wrote the number four inside each of the five circles under the five.

“Remember, ratio means relationship between quantities. That means what we have over here…” I pointed to the five circles with fours inside them, “We must have over here.” I then wrote four inside the four circles under four. “The five fours equals twenty. How much is four fours?” (I know it’s a lot of fours. It feels funny writing four so many times. I contemplated using a different number, but these worked well for [ha ha] my students;)

When they told me that it equaled 16, I wrote that under the four circles. Then, I erased the contents of the circles. I wrote a six in every single circle. “Five sixes equals what?” I wrote a thirty under the twenty. “Four sixes equals what?” Twenty-four got written adjacent the thirty.

One of the students who was working independently had found the answer. When they announced it, we used it to work backward. “What number goes into 80 five times?” With a touch of division we figured out the answer, and I wrote 16 in each circle. “If you have 80 on this side, what number will you have on the other side?” Sixteen times four gives you 64.

To drive home the concept of ratio, I used several other numbers, ending with 500 to 400. “It doesn’t matter how big or how small the quantities,” I explained. “When they are related using the ratio five to four (5:4), they will reflect it by being divisible by five on this side, and four on this side,” I said pointing to the referenced space on the board. “Ratios are easier to understand and work with when we use the smaller numbers, so we reduce both sides to the lowest quotient, using the same divisor. What divisor would we use to reduce 500 to 400?”

While my lesson ended there, here are some ideas for exploring ratios. Compare the land mass between states, countries, counties, or continents. Contrast populations of people or animals.

You could get really scientific with it by exploring the natural ratio between predators and prey. How does nature balance the numbers between the two? Why don’t the predators eat all of the prey? What happens when the ratio becomes unbalanced, and there are too many herbivores? Research the deer population. Find out who is in charge of deciding the number of deer hunters are allowed to kill per season. How do they decide? What would happen if there were many more people getting hunting licenses and more deer than expected disappeared?

Photo by Lisa Fotios on Pexels.com

Investigate invasive species. What causes something to be considered invasive?

Finally, and perhaps more tame, research the ratio of ingredients in dirt. Some plants require more sandy soils. What is the relationship (ratio) between humus, sand, clay, and other materials in your land? This would introduce ratios with multiple numbers. Students could see that when one number goes up, they all do. Double the dirt, and every variable doubles. That is ratio. 

Ratio, Quidditch, and Stimming 

My second grade gifted students are very excited about reading Harry Potter. A couple boys talk about the books often. I’ve had to stop them before they give away any of the plot and spoil the literary experience for those who haven’t read as much. 

Caught being classy; These 2nd graders are finishing up “Who Would Win” books comparing insects.

The other day one of my students was pacing around the room conducting a monologue about quidditch. Everyone was working on wrapping up an end of the year project when this student took a break to stim. 

Over the years I’ve had a few students who stim. It took me a while to understand what this was, and even longer before I was familiar with the term. Stimming is an abbreviation for stimulation. It is when a person uses sound, motion, touch, or other stimulating sensations to soothe the spirit. A person may sing to themselves, wave their hands, repeatedly run their fingers over a piece of fabric, or pace the floor. It usually involves repetition.

The National Autistic Society suggests four reasons a person may stim. They could be using their stim to self-regulate the amount of sensory input they are receiving. By making their own noise, they are blocking out other sounds. Flapping one’s hands or pacing the floor puts the stimming person in charge of what their mind thinks about. It can be a way of dealing with stress and anxiety. Sometimes, a stim can just be enjoyable. Finally, there are times when people who stim do it in order to produce sensory input. Maybe their mind needs something to do, so they stim. 

Stimming can sometimes be distracting for other students in the classroom. It is best for everyone to understand that this is a completely natural and necessary thing for some people. 

I’ve grown accustomed to my 2nd grade stimmer. When he started talking about quidditch to no one in particular, I just let him pace and say his soliloquy. He was going over the rules for the game, and something he said got me thinking. He mentioned the golden snitch, and said that catching it would win you the game. I didn’t mean to be contrary, but I interrupted his stimming to ask, “Do you automatically win if you catch the snitch?” 

He paused his pacing and thought for a moment before explaining, “It is worth 150 points, so whoever catches it will most likely win.” That was a very good answer, but I saw an opportunity for a mini math lesson, that ended up turning into an awesome math lesson!

Another student chimed in, “Catching the snitch ends the match.” 

“But does it guarantee a win?” I challenged. 

The group thought about it for a second. The quidditch expert conceded that, “No. Catching the golden snitch doesn’t mean you win.”

In order to stretch their thinking and prove my point, I suggested they figure out how many scores the other team would have to make in order to win against a team that caught the snitch. Hardly a second passed before the quidditch expert told me the answer; sixteen. 

With each score producing ten points, sixteen would mean that the team without the snitch would have ten more points (160) than the snitch-catching team. “But, it is pretty unlikely that one team would score sixteen goals while the other didn’t earn even one,” I suggested. “What if the team that didn’t catch the snitch was twice as good. For every goal the snitch-catchers earned, the other team scored two. Then, how many goals would have to be scored in order to win without catching the snitch?”

I drew a T chart on the whiteboard. I wrote “Team A” on one side of the T and “Team B” on the other. At the bottom of  Team B’s column I wrote 150 and labeled it, “Golden Snitch.” I then wrote a two directly under Team A and a one under Team B. I explained that Team A scored two goals while Team B only scored one. “How many points does each team have?”

When the second graders told me that Team A had twenty points while Team B had ten, I commended them for multiplying the number of scores by ten, the value of each goal. Only, that was not accurate. I pointed to the “150” at the bottom of Team B’s column. “Team B actually has 160 points,” I told them. 

I wrote another “two” under Team A and an additional “one” under Team B. “Now how many points does each team have?” At this point it was forty to 170. We kept doing this until Team A’s score crept closer and closer to Team B’s. It took a few minutes to calculate, but my students were riveted to the math. Once Team A earned enough points to overcome the snitch catchers, I restated my original query: “How many goals did Team A need to score inorder to beat the team that caught the snitch?” 

They skip-counted to find the answer. I showed them that they could have multiplied the number of twos by two and get the same answer. I also explained to them that the idea of a team being twice as good as another is a ratio. I wrote 2:1 above the T chart. “This means Two to One,” I told them. “For every two goals that this team scored…” I pointed to Team A, “The other earned only one goal.” I left it at that. Our time was up, and they had consumed enough new terms and problem-solving for the morning. 

As you can see, there was a lot of math going on! Among other things, we figured out the difference between the teams to see how far we had to go for Team B to overtake Team A. The challenge was to find out the smallest number of goals necessary to win without getting the snitch.

I was so pleased with the spur of the moment lesson that I decided to reuse it when my fourth graders joined me. I wrote the question, along with some quidditch facts, on a Google Jamboard. A couple of fourth graders engaged in drawing on the picture of Harry and writing some random thoughts. I’ve found that letting students do this allows them the freedom to think creatively. This could very well be a form of stimming. 

I guided their work by asking the question verbally. I retaught the concept of ratio, which I had introduced earlier in the year. The fourth graders were much faster at figuring out the answer. They immediately guessed thirty goals might push Team A over the edge of victory. I wrote “30” under Team A, and asked how many goals Team B would earn with a ratio of two to one. They accurately told me fifteen. We multiplied both numbers by ten and added the snitch to Team B’s score: Tie. 

My students deduced we needed one more score for Team A to have more points than Team B. I told the class that Team B, having a ratio of two to one, could very well score one more goal before catching the snitch. “Just to be safe, we ought to say that Team A should score 32 goals to secure victory over a team that catches the Golden Snitch first when working with a ratio of two to one.” 

At this point one of my fourth graders realized that the answer was staring at them from the whiteboard on the other side of the interactive screen on the wall! “Yup,” I conceded. “I worked out this answer with my second graders earlier in the day. We figured it out a different way.” The fourth graders didn’t feel tricked, but just to be sure, I added, “You guys were really fast. It took my second graders and I a while to figure it out.” 

“Now, let’s have some fun with ratios,” I told them, as I wrote “3:1” on the board. “What if Team A was three times as successful at scoring quaffles than Team B, but Team B catches the snitch? Now, how many goals need to be scored by Team A to win?” 

We started off with thirty goals because it is a nice easy multiple of three and close to our previous answer. If Team A scored thirty goals, Team B would have scored only ten. After multiplying both scores by ten and adding the snitch, we found that Team A didn’t have to score that many goals:

30 * 10 > 10 * 10 + 150 With a difference of fifty (300 – 250 = 50), we figured Team A could have not scored a couple of goals and still have won, even if Team B caught the snitch. We tried again. With 27 goals, Team A would have 270 points and Team B would have scored only 9 goals (using the three to one ratio). Nine goals, at ten points a piece, plus the snitch would mean Team B earned 240 points (9 * 10 + 150). This was closer, but can we do better? Does Team A have to score that many points, or could it be less? Sticking with multiples of three, we gave Team A 24 goals. This would mean Team B would score eight. After calculating the math (24 * 10 = 240 and 8 * 10 + 150 = 230) we saw the difference between the two scores shrink to within one goal.

I took the opportunity to point out the consistency in the scores changing. Team A’s score decreased by thirty with each new calculation, while Team B’s score only went down by ten, but every single time. We explored the idea that 30:10 is the same thing as 3:1. I taught the fourth graders that ratios, like fractions, can be simplified by dividing both sides by the same number. “Ratios explain the relationship between two quantities,” I told them. “The smaller the number the easier to understand how they are related. You don’t say 500 to 100. Five to one is just as accurate and easier to understand.”

Just for fun, I gave my group of fourth graders one more scenario; A more realistic one. I told them that the teams were more evenly matched. I made the ratio three to two (3:2). For every three goals Team A scored, Team B scored two. “Using this new ratio, what is the fewest number of goals Team A would have to score in order to win a match against a team that caught the snitch?”

Because we were still working with thirds, the problem was manageable. It didn’t take my students long to figure out that they could work with numbers that were divisible by three, double the third, and multiply both numbers by ten. We took one of our scenarios from the previous problem, 240 for Team A and 80 for Team B, which represented the three to one ratio, and adjusted Team B’s score to represent the new ratio of three to two. I just erased the 80 and wrote the equivalent of two 80s or 160.  

When we found that this new ratio caused Team B to pull away from Team A, students jumped at Team A earning much larger scores. Someone threw out the number 100. I asked them, “What would a third of 100 be?” When no one answered, I restated the question, “What three numbers could be put together to make 100?” Still nothing. I simply told them that it would be 66. I reminded them that a third of one is 0.33. They all sighed with remembrance. They knew that! “When you add some zeros…” I wrote one, and then followed it up with putting a couple zeros behind it. “You move the decimal over. And, since we are talking about a ratio of three to two…” I wrote 66 on the other side of the T chart. After adding 150 for the Golden Snitch, we noticed that the difference was still pretty large.

I showed my students that it was even easier if they began with scores that were divisible by three. From 100 we tried 90. That was simple, but the difference was still too great. Together, we guessed that 81 would be a multiple of three since it is divisible by nine, but how could we figure out the exact number that could go into 81 three times? A tiny bit of long division algorithm did the trick! We did this a couple of times and noticed a pattern. Each time we lowered the dividend by nine, the quotient dropped by three. A really remarkable recognition happened as we figured out the third difference over on our T chart, too. The differences between the two scores were showing signs of a pattern; Each time we shrunk the number of goals that Team A scored by nine, Team B’s final score would drop by exactly 60 points. It went from 810 to 750 to 690 to 630, consecutively! Wow!

The recognition of patterns within the math brought my mind back to what had started this fantastic exploratory math lesson; Stimming. There is something soothing about knowing what number is next. Being able to rely on the consistency of repetition can be comforting. Tapping, singing, pacing, and even skip-counting are all ways to occupy part of the brain, so that other parts may be freed to think. How much stimming a person needs is different for every individual, just like managing it in the classroom environment will differ, but finding just the right ratio can be magical.

Teaching 2nd Graders How to Draw Conclusions From Data

I was teaching some advanced second graders an enrichment math lesson the other day when I learned something. I often like to mix and combine skills, so that kids can see how math is really used, as well as make it fun. At the end of this particular lesson, I used the data we had collected throughout our time together to summarize what had happened. As I attempted to make sense of the numbers, I found myself making conclusions or at least forming a hypothesis that could be tested. When I pointed this out to my second graders, I thought to myself, “Wow, this is a pretty good lesson I’m learning right now.” I was thankful that I hadn’t stopped at only having my students do the original lesson’s math. 

The lesson involved making estimates and then measuring actual length to the nearest inch. The first thing I did was model. I took a wooden block out of a box; my “Box o’ blocks”!  After standing it up on the table, I asked, “How tall do you think this is?” I received some wild guesses from my second graders. Someone thought it might be a foot. Another student said two inches. 

I picked up the block and measured its length with a little mini ruler. I showed the students where the block ended on the ruler. They eventually settled on the idea that the block stopped between the five and the six. One of the students suggested that it was five and a half. 

After praising this smarty, I asked them if it was okay to measure the block laying down, because I was holding it flat in my hand. We were supposed to be measuring its “height.” Their spatial reasoning skills were sound, and we all agreed that we were measuring the length of the same side, no matter which direction it was facing. 

After teaching estimating and measuring and before breaking the group up into teams, I explained the directions. Each team would get some random blocks. They were to work together to build a tall tower. It had to be free standing; No holding it. Every block should be used. I would give them two minutes to construct the tower. When the timer goes off, the teams will form an estimate of how many inches tall the tower is. Once the team has decided on a number and communicated it to Mr. Weimann (me), they get a yardstick to measure the actual height. 

I would be keeping track of our estimates and accurate measurements. The idea was to try to get your estimate as close as possible. 

Next, it was time to form teams. I just had kids who were sitting near each other form teams to make it go faster. I dumped random blocks in front of each group and told them to get started. After the first two minute timer sounded, I stopped everyone. I had the groups come up with estimates of how tall their towers were. As each team shared their estimate, I had them provide a name for their team, as well. Clowning around, I purposely misspelled the names they gave me. That had them laughing. 

After I wrote the estimate under a team’s name, I handed them the yardstick. Watching them estimate the height was fascinating. One group had a girl who used two fingers squished together to climb the tower with the members counting as she jumped. They figured her fingers constituted about an inch. They were very accurate. Another group had a student using his arm, presumably thinking it was a foot long. After a round or two I reminded groups that the medium sized block was already measured. We found it to be exactly five and a half inches tall. They could use that in their estimates. I don’t think any of them did, but we can revisit that. 

As it turned out, we only had time for three rounds. After collecting all of the wooden blocks, I went over the chart that I’d made. I had second graders figure out the difference between each estimate and measurement. I wrote that data on the board in a different color. We then added all of the differences from each team together to total them on the bottom of each column. Although two teams had the same total, three inches , one of them had never supplied any data for one of the rounds; Their tower kept tumbling, and they were never able to estimate or measure it. 

We had more than one type of winner.

The “BeeKays” began rejoicing for having the largest total, presumably the winners, but other second graders squashed their victory dance with unwelcome information. Like golf, the total that is the smallest was winner. It took a little convincing, but I explained that the goal was to get the estimate as close as possible to the actual measurement. The smaller the difference, the better the estimate. “Look, the very first round had a team whose measurement ended up being exactly the same as the actual measurement! The difference between the two was zero. This zero was the winner of that round.”

Next, was the very cool part for me. Here is where I joined my students in learning, albeit through teaching. I had asked the students what data was. They eventually settled on the synonym, “information.” I added to this the word “useful” and proceeded to show them how we could use the data to draw conclusions. The team that had thought they won because the sum of all of their differences was the greatest, eleven and a half, did actually win something. They were the winners of which team improved the most. 

“What do you notice when you compare the differences of each tower they built?” I asked the group. “They got smaller and smaller. The first tower was estimated to be eight inches shorter than it actually was. The group over corrected a little on the second tower, estimating it to be a little taller than it was, but only by three inches. The final tower was within half of an inch of its estimate!” I pointed to each difference on the chart as I explained its meaning. “The estimates got more and more accurate, as the BeeKay team practiced.” I let that sink in. 

After pointing out that the data shows some improvement in the other two groups, it isn’t as consistent as the middle group’s. As I time ended, I taught my second graders that we just analyzed data and developed conclusions based on the information we collected during our lesson. “What might we expect to happen if we built some more towers and continued this exercise of estimating and measuring?” I asked. 

Hands shot up all over the room. “They would get closer and closer,” someone shared. 

Putting it into mathematical terminology, I restated, “That’s right. The differences between the estimates and the measurements should get smaller and smaller as you get better at estimating.” 

Teachers use data all of the time to measure how students are doing. Do you ever show students how the data works? Give it a try. 

Making Math Connections: 1st Grade Double-Digit Addition

Snargg and Plory, iReady mascots

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.

Developing Real-World Math Problems: Adding & Subtracting Mixed Numbers

During an interview for a podcast with Curriculum Associates the other day I was asked how I use real world scenarios to enrich math lessons. I had explained to the interviewer that teaching is a second career for me. My experience of entrepreneurship as a residential custom painting contractor helps me introduce loads of business expertise in my math lessons. 

Blog about using tile cutting in a math lesson
Blog about using wallpaper hanging in a math lesson

The interviewer was looking to provide practical solutions for teachers to use. I took two seconds to imagine I was sitting in front of my computer (as I am right now;) and tried to remember the steps of making my lessons. 

The first thing I do is find the lesson in the i-Ready toolbox, and look at the “Extend Learning” assignment. I don’t usually use the i-Ready assignment verbatim, just in case the regular education teacher wants to assign it. I use it as a guide for my enrichment lesson. 

i-Ready provides paper lessons that can be assigned virtually or printed out.

For instance, this week my fourth graders were learning about adding and subtracting mixed numbers (Lesson 21). The extended lesson shares a story about a couple of kids filling a fish tank. Some mixed numbers are used, and kids are asked to do calculations that would require them to add and subtract the mixed numbers. 

Here’s a GIF I made showing our classroom 75 gallon fish tank. I made the stand that it is sitting on out of 4 X 4s.

I actually have a 75 gallon fish tank in my classroom, so this story could very well be perfect. However, I just wasn’t feeling the mixed number connection. There is no way that three friends would have three different buckets that all hold different mixed numbers of water with a fraction containing the same denominator. It felt too implausible. 

Regular Ed teachers could still use this paper assignment about students using mixed numbers to fill a fish tank.

I sat at my computer and thought, Where do I encounter mixed numbers? In addition to having run a successful business, I’m also a “Do it yourself-er.” I enjoy building things. Making things with my own hands and tools is satisfying to me. I made the stand that my fish tank sits on. I finished my own basement, complete with bathroom and laundry room. In short, I have come across plenty of mixed numbers! Developing an enrichment math lesson that uses mixed numbers will require me to make the work of adding and subtracting the mixed numbers both doable and easy enough for the fourth graders to understand. That is my challenge.

When I say that I like to use my hands to build things, I don’t mean paper airplanes. Check out this blog about my giant wooden sunken pirate ship classroom decoration.

The morning that I came up with “Fix a Bench” my first thought was to have my students figure out how many boards would fit on a small deck surface. Each board could be a mixed number in width. This would be similar to the fish tank assignment from i-Ready. Kids would just add them up to fill the space.

As I began researching and looking for pictures online to jazz up my presentation, I remembered that lumber is full of mixed numbers. The most common building material, the two by four, is NOT really two inches by four inches. I learned this ages ago when I expected several adjacent two by fours to equal a nice even round number. It was some wacky measurement, and I took a closer look at the dimension of the studs (two by fours) I had purchased. I was incredulous, thinking I’d been ripped off! 

Thinking this might be a fun fact to share with my students, I decided to have them explore having to use various sized pieces of lumber to make something. The fourth graders love mysteries, and I would wait until the very end to explain why two by fours are called that when they actually aren’t those dimensions. 

My lesson was originally “Build a Bench.” When I began planning what my students would actually do, however, I figured out that it would be easier to teach and explain if I had them only choose lumber to place onto an already existing frame of a bench. Thus “Fix a Bench” was born. 

The next part of developing a good real-world lesson is to create a “Sell.” You must come up with a pitch to draw the students in. “Today we’re going to fix a bench” isn’t good enough. Instead, I told my students that “It’s your parents’ anniversary (or birthday for single-parent families), and you want to give them something, but you have no idea  what! They have a special bench that they like to sit on, but the wood is rotting. You get the idea that you will fix this bench for them as a gift. Because you don’t have enough money to buy the wood to do the work, your parents agree to get it for you. Your labor and thoughtfulness is the present. In exchange for your parents footing the bill, you have to tell them exactly how much the lumber will cost.”

This little story makes sense to the students. Even if they don’t have a bench in their backyards or don’t have a backyard at all, they can imagine doing this kind of thing. Also, it gives them some good ideas of how to come up with presents for their parents that won’t cost them anything more than creativity and thoughtfulness. 

“How much does the wood cost?” the students instantly want to know. 

“Before I tell you the costs, I am going to need a helper… This person has to have very good penmanship. I will know that they can write very neatly by how well they listen to the explanation of the project…” I share this with a very stern look in my eye, as I scan the room for anyone not paying close enough attention. Every student straightens their body and widens their eyes. I proceed to share the dimensions of the bench frame.

I got the size of the bench by measuring one of the chairs in my room. The back was approximately 16 inches tall, and the seat was 14 inches deep. Instead of supplying these simple numbers, I turned them into mixed numbers. Sixteen inches turned into 1 ⅓ feet, and 14 inches transformed into 1 ⅙ feet. In order to narrow the focus of calculations, I made the bench exactly eight feet wide. This way, there wouldn’t be any trimming of the ends of the boards. Just choose eight foot long pieces. 

Now, it was time to show the students the materials available to them. I had found a list of lumber online that showed the names of the wood with the actual dimensions next to them. With this image on the screen in front of the class, I showed students how a two by four is actually 1 ½ inches by 3 ½ inches. A two by six, another common board measurement, is really 1 ½ inches by 5 ½ inches. And, a two by eight board is 1 ½” by 7 ¼”! 

The class needed a little guidance to get started with this lesson. I guided them through drawing a diagram of the important parts of their bench. We labeled the back, the seat, “And don’t forget about the single board that goes on top!” I told them. I had them figure out how many inches the mixed numbers would translate to. “Now, we have to fill these spaces (16 and 14 inches, respectively) with lumber,” I told them. “It would be easy if two by fours were actually two inches by four inches, but they aren’t! See if you can figure out how to make sixteen inches of surface using these mixed numbers.” I circled the widths of the “two bys” from the image. I had told them that we would only use those, because they need to be thick enough to hold a human’s weight. 

Using only 3 ½ (the width of a two by four) won’t work for the back of the bench. My students figured out that four of these boards will get you to exactly fourteen inches of wooden surface. That leaves you with a two inch gap, and “We don’t want any spaces. Neither can we saw any boards to resize them. There aren’t any boards that are exactly two inches wide. Can you take away one of the two by fours, and find a different size board that fits nicely?” 

When my students take 3 ½ away from fourteen, they have 10 ½”. “What is the size of the space, now?” I ask this while pointing to a gap that I’ve illustrated on my drawing of the bench we are fixing. They figure out that the empty space is exactly 5 ½ inches wide. “Are there any 5 ½” wide boards that we can purchase?” Yes. The 2 by 6 is that width. 

Are we done? Definitely not! “You were all such hard workers and very good listeners that it is very difficult to decide who could be my writer,” I tell my class of math enrichment fourth graders. 

One of the students actually volunteered another, saying, “Nahum has really good hand-writing. You should have him write on the board.” 

“Are you nominating your friend?” I inquire. He admits it, and several students second the nomination, suggesting that Nahum really does have good handwriting. “Well, okay, then. Come on up here,” I extend the invitation and commend Nahum’s friend for being classy.

As Nahum prepares to write on the board, I open my laptop. I have pricing from a lumber yard ready to go. We now write down the amount of money each board will cost us. I have Nahum give the writing tool to other kids after he writes a couple of prices, so that more students get a chance to write on the board. We only supply the prices of the two-bys, because those are the only ones we are using. 

Students proceed to figure out the cost of 3 two by fours and 1 two by six. When they think that they are done with the project, I point out that we still have to figure out the seat of the bench. They happily begin problem-solving that challenge independently. It took a surprising amount of time for them to figure out that we had already answered the question of what boards could be used. The 4 two by fours that we had added up earlier had totalled exactly fourteen inches, which is the size of the seat! 

When they began adding up four prices of two by fours, I pointed out that we already knew how much three of them cost. “Why not just add the cost of one more to the first number?” I suggest. Grateful for the idea, they do this. 

Preempting the “I’m dones!” that were about to fill the room, I reminded them, “Don’t forget about that top board… The one that goes on the top of the back of the bench.” Happy groans and more pencil scratching ensued. 

Just when my students thought that they were finally done, and Mr. Weimann couldn’t come up with any more surprises, I told them, “It would be very classy if you figured out how much your parents would have to pay in sales tax.” Epic groaning accompanied smiles and students beginning to hunch over their iPads. I told them to use calculators and that our state sales tax was six percent. This was the icing on the cake. 

After a few seconds, I modeled for them, asking Siri, “What is six percent of thirty-eight dollars and fifty-six cents?” When she told me, I wrote it on the board for them. 

Because I had created Google Jamboards with all of the information preloaded on them, I was able to see each individual students’ work. I had waited until Nahum and partners had neatly written the prices into the slide with lumber details before I pushed the Jamboard out in a Google classroom assignment. I had the software “Make a copy for each student.” Students knew that, although they were allowed to work with partners and I helped them solve several parts of the problem on the board, they had to add their own version of the details, showing their work

Before students left my room I explained why two by fours are actually mixed numbers. The lumber is cut at exactly two inches by four inches, but when it dries, it shrinks. Of course the students wanted to know why lumberyards don’t correct for this or call the wood by another name. The young minds cried foul and felt tricked! I told them that it has been this way for a long time, it is easier to say “two by four” than “one and a half by three and a half,” and the price of the wood that you feel like you are being cheated out of goes into having to store it while it dries, before selling it. It isn’t like the mill cuts a two inch by four inch pied of wood for you to bring home, you build with it, and it shrinks on your home. That would be worse. In the same way that creating this lesson required several steps, when one wants to make and use a two by four, you measure the wood, cut the lumber, let it dry, measure it again, and then you can work with it.

The One-Room Schoolhouse Game

The other day my elementary school had an assembly. When it was over, a fourth grade class was left with no teacher. She had attended a meeting that was running over. Since I didn’t have a class at the moment, I decided to bring her students upstairs for her. As we waited in the fourth grade classroom for the teacher to return, I wondered what I would do to maintain a semblance of sanity. I decided on a game centered on behavior: The One-Room Schoolhouse Game.

When the teacher, accompanied by an instructional assistant (IA), entered the room, they couldn’t believe their eyes. Every student was sitting up ramrod-straight. Every single eye was trained on me. The IA verbally queried, “What is happening in here?” She was incredulous. I was in my element.

Educators are show(people). We are either putting on a performance to attract the attention of our students. Or, we are ringmasters, making sure the mayhem stays within the bounds of 42 foot diameters (the official size of a circus circle).

This is an image of the Claussville One-Room Schoolhouse, the same that my class visited on many field trips.

There are many ways to perform, but methods for maintaining order is more limited. Many years ago I came up with a unique way to do both at the same time. 

When the IA, who had seen the game in action, entered the faculty room at lunchtime, she practically burst with enthusiasm, announcing to the other teachers what she’d seen. Some of the staff members wanted in on this game of good behavior that students seem to enjoy. I explained the rules to them. Since then, I’ve had a couple of teachers ask me to share with their classes how the game works. I’m more than happy to oblige.

Then, I searched my blogs to see if I’d written about The One-Room Schoolhouse Game. Unbelievably, I hadn’t! So, here goes.

It began many years ago. I brought my 3rd grade class of “Polite Pirates” (what I called my group of students) on a field trip that visited several historical sites around our area, all in the same day. It was a fast-paced adventure of a field trip, running from one stop to another, practically assaulted with information. 

I made this video from photos and video of my 2016-17 class visiting Claussville.

We began the day at Trout Hall, the original house of the Allens, the family that founded Allentown, Pennsylvania. The building is staged to look like it had during the Revolutionary War era. A tour guide led us around the house and explained many interesting facts. 

Students get to ring the Liberty Bell replica.
Students explore the replica of a Conestoga wagon.

From there, we traveled only a few blocks away, where the Liberty Bell had been housed during a portion of the Revolutionary War, hiding it from the British. The Liberty Bell Museum has a life-size replica of the original bell, a slightly smaller than lifesize model of the wagon that would have carted the famous bell, along with several others from Philadelphia to our town. They have a nice mural and story display that they show. They also provide a favorite feature of every field trip; a gift shop! This is where I acquired my famous “One-Room Schoolhouse” bell. It is a mini replica of the Liberty Bell.  

This is my 2021-22 class in front of the Claussville One-Room Schoolhouse.

The final stop of the day was  an old school building that was preserved to look the way it had when it was used as a one-room schoolhouse as late as 1956. The Claussville Schoolhouse is where I learned the practices that what would turn into the rules of my One-Room Schoolhouse Game. 

A tour guide walked us through what it would be like to go to school one hundred years ago. Every grade, from first through eighth was taught in that one room! Because of this, the school master had to be very strict. 

I explain all of this nowadays when I introduce The One-Room Schoolhouse Game. Believe it or not, me acting like a big meanie is one of the more fun aspects of the game! Students love tempting me to pretend to yell at them. Suffice to say, this is far from my typical conduct. 

The Game

Now you know where the idea for the game originated, let me explain how it’s played. I introduce the game on the very first day of school. It contrasts the comfortable social/emotional learning (SEL) environment that I foster and points out how nice I actually am. 

Right off the bat, I talk about what school was like a hundred years ago. Because there weren’t any buses, schools only serviced local communities. Every kid walked to school. The building housed every grade, from first through 8th! A teacher was responsible for managing lessons for eight grades, all at the same time. And, there was only one teacher. They could not afford to have any disorder or misbehavior, so the teacher was very strict. 

“There were many rules for doing things. First of all, the students had to sit up straight.” As I say this, I morph into a very serious, demanding persona. Every student whom I have ever shared this game with has always straightened his/her posture at this point. “There is absolutely NO talking.” I look around the room, daring students to even think about speaking out loud without being called on. The students are loving this. They are on pins and needles, waiting to see who might get into trouble. I have transformed into a scowl-wearing, grumbly ogre of a teacher. 

Power of Role-Play

The tension mounts as I walk around the room. I might at this point explain some of the primitive punishments that would have been used during the days of the one-room schoolhouse. It’s not impossible for me to slap a ruler onto a desk to really amp up the mystery of the experience. (Of course, I make certain all of my students understand that corporal punishment is unacceptable in connection with education… but, that is how it was.)

Once I have my students good and primed, it’s time to share the rules of the game. When they are called upon, they are to stand, push in their chair, name the person or persons they are addressing, and make their statement in a complete sentence. I always make it clear that they are to “Refer to me as Mr. Weimann, or Captain Weimann if you’re classy, and you wants to be classy! If you are answering a question, and the information is pertinent for your classmates to pay attention to, you say, ‘Captain Weimann and fellow classmates…’ If there is another adult present, and your answer would benefit that grown up, you say, ‘Captain Weiman, Ms. so and so, peers and/or fellow classmates…’ You get the idea.” We practice this a bit. 

That’s pretty much it. I throw in some variations or additional rules throughout the year to keep the game interesting and fresh. One day I might go crazy about neatness, causing everyone to clean up their desks and the floor around their work spaces. Another day I could go nuts about hearing them breathing. They love it, the more extreme and just beyond grasp the parameters are. A really fun one is pretending to be the mean one-room schoolhouse master and prohibiting all smiling. They practically burst, but work so hard to meet the requirement! 

The trick is to do the game just enough, both in the number of times you play it and how long it lasts. It slows everything down; the making kids stand up, push in chairs, reanswering because they forgot to say who they were talking to or didn’t use a complete sentence. 

This is the tiny bell that turns the room into a “One-Room Schoolhouse.” So little: So powerful!

The students love the game, though. They usually beg me to ring the bell, for I explain that the tiny bell that I got on that very first field trip to the Claussville one-room schoolhouse has magical properties. Whenever I ring it; and only when Mr. Weimann rings it; we are magically transformed into a one-room schoolhouse. This, of course, is a huge part of the game. 

Sometimes, I will pretend to be unhappy with what is happening in the room, and “threaten” to ring the bell. The class will gasp in mock shock. If I do ring it, they all groan and moan, as if they were just strapped into straight jackets against their wills. …But, they quickly sit up straight, quiet down, and see who the big, mean school master from the past will pretend to pick on first. They are always disappointed when we return to the humdrum of contemporary time, and I turn back into fun, loving Mr. Weimann. 

The Claussville One-Room Schoolhouse Bell
What would it feel like to be a student back then?

Back to the Future

This game is like “The Floor is Lava,” in that no one gets hurt, it’s all just pretend, and it amps up the excitement of an otherwise typical walk-through-the-park day. It is perfect for jazzing things up right in the middle of mundane stuff. A little ring of the bell commands everyone’s attention to both behavior and their tasks at hand. 

I had no idea how different the 4th grade class that I was sharing the game with was acting, but their teacher and the accompanying instructional assistant recognized that something unique was going on. Rather than explain the game, myself, I allowed the teacher to call on students, who nearly burst with pride to show off their new skills. 

It took a couple of kids to get all of the rules out, but once I felt comfortable that the teacher understood what was going on, I was free to leave the past. On my way out I whispered, “I’d wait a couple of minutes before ringing that bell. Ride it out;)” 

You don’t want the game to go too long. Fatigue can set in. For the game to be fun, it has to be special, so don’t play it too often or too long. Let the students petition for it a few times, and then surprise them after returning from lunch or in the middle of a multiplication game. Use it for a transition to another lesson. Above all, have fun.

This video shows all three stops of the Polite Pirates’ Lehigh Valley, PA historical sites field trip (2018).

A Bookworm’s Quizzical Poem

I wrote A Bookworm’s Quizzical Poem several years ago. I shared it with my second grade gifted students yesterday when we came across a poem in the book we were reading (365 Penguins).

When I asked them how they knew the poem in the book was poetry, they told me that the letters were fancy. This made me laugh. It reminded me of a blog that I wrote about poetry being intimidating (License to Poetry). While poetry can be beautiful and help you feel fancy, I want to encourage you to guard against reserving it only for ivory towers.

Teaching 3rd grade helped melt my fear of poetry. With an audience of 8 and 9 year-olds, I not only wrote it, but felt free to share and interpret it with my students. I even began to search the Internet for poems that had to do with topics I was teaching in order to deepen my students’ experience during lessons!

As readers of Jack Prelutsky and Shel Silverstein can attest, poetry can be loads of fun to read. It can also be fun to write.

My second grade gifted students and I discussed and explored the rhyming words at the end of each line of the poem from our book. We counted the syllables from each line. The message of the poem was pretty obvious, but it lent to the idea that poetry can be about sharing emotions. Finally, my students and I used the last line to spark a lesson about apostrophes.

They knew that apostrophes show possession, like in the title of my poem, A Bookworm’s Quizzical Poem. And, they understood the concept of showing the missing letters in contractions, but I blew their minds when I told them about poetic license to slice off whole sections of words!

“Why would a poet need to shorten a word?” I prompted. They were quick to identify the trick of squeezing the correct number of syllables into lines. This is when I dug out my old poem.

Did you know that bookworms are real live insects. Back when the glues that bound books were more organic than chemically produced and the pages were pulpier than they are now, larvae would literally feast on the texts.

My poem plays on this idea of a worm eating books, both literally and figuratively. In line 15 the dad says that his son’s book “Was no more than food for my belly.” The next line compares reading a good book to eating something delicious, though.

These comments on the right of the poem are questions that I thought of for helping my students get more out of the text. I ended up typing them into a Google Form to use as a teaching tool. Feel free to use the poetry and form, if you wish.

I always enjoy pointing out the word “shooking” in line 36. It is so silly. It feels weird to say it out loud. After discussing its job of rhyming with “looking” from line 35, I explain to my students that this is a perfect example of me simply having fun with words.

As an author, you have the power and authority to write whatever you want. You can bend the rules. You could even break the rules… a little. If you break them too much, your text won’t make any sense. Just enough, and your writing is interesting, attention-getting, fun, and memorable.

The phoenix’s presence at the end of the poem symbolizes the healing of the relationship between the son and dad, as well as the renewal of ideas. The son will be the next dad. Generations continually move on. Phoenix tears, not that our poem’s bird cried, are supposed to have healing properties.

Also, according to “Ancient Origins” (2021), anyone standing near the pile of ash left when a phoenix burns up can’t help but tell the truth. Is this why the dad confessed to taking the young bookworm’s book? Perhaps it lent to the honesty by which the son told the dad all about his love of poetry and reading.

Whether you love my poem or not, I hope that you open your heart to poetry in general. I have thoroughly enjoyed reading and writing it since I have let go of fears and insecurities.

Here is a video of me reading my poem for my class. I made this during the pandemic when we were all stuck at home, but it also served to differentiate the lesson for students who might have struggled reading. The video is embedded in the Google form so kids can re-experience the poem right before answering the questions.

Buttery Batch of Math Cookies

This is the introduction that I used on my fifth graders.

In preparation for teaching a math enrichment lesson to my fifth graders, I looked at the iReady “Extension” activities in the Ready Math “Teacher Toolbox,” and I found a problem that I liked a lot. (iReady and Ready Math are products of Curriculum Associates. My district has been using it for several years, and I like it a lot.) This lesson (14) is all about using fractions to solve word problems.

Here’s an image of the worksheet that a teacher could photocopy or share via Google classroom. Because I have the luxury of actually teaching enrichment lessons, I decided to do some explaining before handing over the problem. Also, I opted to make a few tweaks, too. In my experience recipes usually call for specific measurements of butter, not a number of “sticks.” Therefore, I covered up the word sticks in the problem and wrote in “cups.”

This changed the outcome of the answer quite a lot. Now, students would not have enough butter to complete the recipe. They could access new sticks of butter, but if they did that, then solving the problem wouldn’t require wrestling with all of the fractions presented in the partial sticks. That’s when I imagined the real-life experience of baking cookies after having worked all day at making a big meal, like Thanksgiving.

Needing soft butter for recipes is a real thing. Also, who doesn’t love consolidating? We can clean up all of those partial sticks of butter and make cookies at the same time!

I shared what a typical day of cooking in preparation for a large Thanksgiving meal looks like at my home with my fifth grade students, setting the stage for having several fractions of sticks hanging around. With the instruction to use up the warm butter first, and then dip into the cold butter from the new package, I set my students loose to calculate how much butter would be left.

Many students jumped on adding up all of the fractions. They began figuring out compatible denominators, so that they could combine every partial stick and find out what they had in all. “But, do you have to do that?” I asked them. No one wanted to venture a guess.

“What are you asked to find?” I pressed.

“Two and half cups of butter,” someone accurately answered. Without saying anything, I drew two of the worst cups ever drawn on an interactive board, followed by half of a third. I made fun of my drawings, which everyone helped with, pitching in their own digs. Once that settled down, I pointed out the lines I’d drawn through the middle of each measuring cup.

“Why’d I do that?” Earlier, we had discussed that fact that one entire stick of butter was the equivalent of half a cup. The students understood better than they could put it into words, so I articulated the concept for them, “Each half of a cup was one stick of butter.”

Then, we looked back at the fractions. It was easy to see that 1 2/3 + 1/3 would be able to fill one whole measuring cup. That leaves us with three fractions with differing denominators. “Before working out a common denominator to add up all three, think about what you are trying to do,” I instructed. “What is your aim?”

I showed the students that 1/2 a stick of butter + two of the 3/4 would equal one whole. “That would take care of half of a measuring cup,” I told them. Also, I should mention that I crossed out halves of measuring cups, as we discovered combinations of partial sticks of butter that would fill them.

“If we used up two of the quarters to combine with the 1/2 a stick and create a whole stick, how many quarters are left?” One quarter. “And then, we have 5/8 of different stick left.”

They instantly got it. We were 1/8 short of a whole stick of butter. In the end we needed one whole cold stick of butter, plus 1/8 of an additional stick to add to all of our warm butter fragmented sticks to fill our two and a half measuring cups.

The Ready Math extension lesson (14) has a second question that I left as is. The catch is that my students used our additional left over cold butter (2 7/8 sticks) from my adapted first problem to solve it. I let them struggle with this one for a few minutes before I showed them the short cut of drawing pictures.

“You might think it childish to draw pictures,” I began. Fifth grade is the oldest grade in my school, so these were the seniors of the place. “…But, I find it easier to manage some problems when I sketch what is happening.” I had been watching them crunching numbers, making common denominators again, and subtracting fractions. Now, within a handful of seconds, I showed them how many quarters could be made from two sticks of butter! I pointed out the idea of labeling the quarters in order to keep track of my thinking. I wrote a B above each “batch” of cookies. Sure, I could just count the quarters, but when it came to the last stick, it will be important to identify what portions of butter will complete a batch.

As I divided the last rectangle into eighths, I asked, “What am I doing to this last stick of butter?”

Rather than answering my question, they were chomping at the bit to be the first to spew the solution to the problem. “Eleven and 1/8!” more than one fifth grader shouted at the same time.

“No, that’s incorrect,” I casually, but cautiously counseled. Rewording what they had yelled in order to make plain the problem with their answer, I said, “You cannot make 11 AND 1/8 batches.” The emphasis on the word “and” did the trick.

“You can make eleven batches, and you’ll have 1/8 of a stick left over,” a student corrected.

“Perfect,” I affirmed. “Drawing pictures might seem silly, but look at how simple it is to see the answer. We didn’t do any denominator work past doubling up the number of sections in the last stick. I hardly did any math, beyond simply counting!

“When you are taking standardized tests, you get scrap paper. Use it. Draw pictures. Illustrate word problems. Take the time to label parts of your illustrations. Make sure that you understand what you are being asked. What is your goal? What are you supposed to find? It’s not just a number. It is the solution to a problem. In real life, it is a key that will unlock a problem. Be a problem-solver; Not a human-calculator,” I told them.

Photo by Elliot Fais on Pexels.com

In conclusion, my aim is to turn these advanced math performers into problem-solvers. With this goal in mind, I try to make lessons that force students to use what they have learned in their regular math class in a way that is not only compatible with what they would find in the “real world,” but forces them to understand how to use the skills. I often allow my students to use calculators because the problems I prepare for them require more knowing what to do with the numbers than practicing running through algorithms. AI can learn how to crunch numbers, but will it be able to successfully manage a kitchen full of amateur chefs laughing, telling stories, and making meaningful memories, all the while measuring butter for cookies after already cooking and eating a Thanksgiving dinner?

To combat the threat of AI, don’t try to make humans better than machines. That just makes them more like machines. I say, grow the human-ness of students. This is getting pretty deep, so I’m going to go eat a buttery cookie while I chew on these ideas for a future blog;)

Wallpaper Based Learning: Math Enrichment

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. 

“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.