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!

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. 

Give Your Teaching Life by Using Real-Life Situations to Teach

It’s math enrichment time again. The pacing guide has the fifth graders learning to add and subtract fractions. Our curriculum has an enrichment lesson that students can work on independently. It’s a grid that has empty spaces that need to be filled in order to help every row and column add up to the same sum. I like these types of exercises because they are puzzles and make the practice feel like a game

This is the enrichment activity provided by iReady. It’s a good tool for independent practice, but I wanted to provide a “Math Experience.”

I view my job as a math enrichment educator as deepening the understanding of already mastered math skills. Puzzles and games are a great way to make the skills easier to access, faster to use, and more accurate. They are excellent tools for regular ed teachers to provide for those students who have demonstrated competency in a math concept. This style of enrichment activity is good for keeping the advanced students busy while the teacher catches everyone else up to speed. For my part, I aim at helping students see the math from a different perspective, though. 

I cut all of the tiles for the ceiling before attaching them.

It makes a grid of mixed numbers!

Diamonds are beautiful, rare, and expensive gems used to make jewelry, but did you know that they are also the hardest substance on Earth? In fact, the fragments that are cut away from a diamond when it is shaped for a piece of jewelry were used to make saws and drill-bits that can cut through rock (2023). I’ve had the experience recently of using a diamond-studded saw blade to cut through ceramic tile and marble to make a new shower in my house. Just like adding and subtracting fractions can be used to complete a pretty grid, they are also very useful in everyday life. I hoped to make my students’ understanding of fractions more rich (enriched) by having them use their skills to help me measure tile precisely. 

This was one of the most difficult home remodeling projects I’ve tackled. I should use it to teach my students, I thought to myself. So I did.

Welcome to math-enrichment, real-world experience-mode. This project was full of fractions–really! I was excited to share it with my students. 

The first thing I did was think about the different ways I had to work with fractions in order to make the ceramic tiles fit my shower walls. I came up with a math problem that involved quite a lot of fraction work. This will be good, I thought, but perhaps there’s a simpler one to begin with. Even though it’s not the first part of tiling, I decided to have my students measure the window sill, a piece of white marble that had to be exact because the sides of the frame were already tiled. 

I put a photo of my window, minus the sill, into a Google jamboard. The photo shows my tape measure. “I took this picture, so that I would remember the exact measurement when I was at the granite store ordering my marble. I don’t want the gap to be too big on either side of the marble, or it will look weird,” I tell my students. “What I want is for the gap to be the same size as all of the other gaps between tile in the shower, ⅛ of an inch.” After explaining the parameters of the problem, my students have a goal: Make the marble look nice. It will look nice if the measurement is just right. “The space we have to fill is exactly 28 ½ inches wide,” I informed my students. “What width will the marble be cut in order to leave ⅛ of an inch on either side? Go!”

Anyone with experience measuring with rulers and tape measures for cutting materials will understand that you can simply back up your finger or the material on the tool to find the answer. Fifth graders have not had this experience, yet. Also, we were sitting in a classroom, instead of holding a tape measure up to the empty window frame.

After letting my students wrestle with the fractions for a few minutes with a couple of them finding success, I showed the class how to look at the problem, by holding up a yardstick. I placed my finger at the 28 ½ inch mark. I slid my finger over a fraction (⅛) of an inch and asked them how much I just “cut” off of the marble. They saw that it shrank by ⅛ of an inch. “Now, we need to take ⅛ of an inch off of the other side, but do we move our finger ⅛ of an inch from the beginning of the yardstick?” I paused to let them imagine what would happen if we did that. “No, we include the subtraction of that ⅛ of an inch right here,” and I slid my finger over another fraction (⅛) of an inch. “Now, how wide is our marble?” 

Some of my students had written 28 and 2/8 on their papers. When we looked at the yardstick together, we discussed that it is better to say 28 and ¼ of an inch… “Not just because you are supposed to reduce your fractions! But, because one is easier and faster to count than two; plain and simple.” This set us up for our next problem, the one I had come up with originally, but put off until after having practiced some measuring, first. 

Time for some tiling!

“Now, we need to cut some ceramic tile to fit my shower wall,” I pitched. “Remember, we want to keep gaps of ⅛ of an inch between everything; between the tile, between the corner of the wall and the tile, between the tile and the metal edging; everything.” (This isn’t just easy for students to forget. It is easy for someone measuring for actually tiling a shower to forget. I should know! I can’t tell you how many times I had to re-measure or recalculate because I forgot to account for the grout!) 

“Okay, so our wall is exactly 30 and ¾ inches wide, from the corner of the shower to the metal edging. I’ve already tiled the window wall,” I explained. “This 30 and ¾ inches is the space that we need to fill with tile and grout.” I showed them a tile and told them that it is two feet by one foot, rectangular. There was a photo of my shower wall in the Jamboard, too. I wanted them to visualize the final product. While I had written the entire problem into a Google jamboard, I shared the problem with them orally, also. It was easy for me to communicate, because I just did this! It was fresh on my brain. 

“We will be alternating the tile,” I told them. “We begin at the bottom of the shower wall and work our way up. The first row will begin with a whole piece of tile placed in the corner. The next row will begin with half of a tile starting at the corner, and so on.”

“Guess what,” I exclaimed. “I have a treat for you: The tile that you thought (I thought;) was two feet wide isn’t! Like everything else in life, it is a little short; a quarter of an inch short to be precise. It’s your job to figure out exactly how wide to cut the tile that will fill the gap that is left within the 30 ¾ inches space. Don’t forget about the ⅛ inch grout between everything! Go.” 

Student mouths hung slack-jawed. I used the Jamboard to demonstrate the math. I drew lines that represented the sides of the shower wall. “This is the corner,” I told them, pointing to the line on the right. “And, this is the metal edging that is the end of our tiling.” I pointed at the line on the left when I said this. “Now, how wide is the space between the two lines?” I prompted. 

It took some waiting, but finally one of the students ventured, “Thirty inches?” 

“Thirty and what?” I reminded them.

“Three quarters of an inch,” they completed. I told my students that this was the width of my tub (30 ¾ inches).

“Okay,” and I wrote the complete distance in the space between the two lines. “Now, let’s begin by allocating ⅛ of an inch over here.” I drew a little circle next to the line that represented the corner. “What comes after the ⅛ inch of grout?” Pause. Wait. Patiently persist in waiting. “Look back at the problem.”

“A tile,” someone says out loud.

“Yes,” I affirm the correct answer. “We put a whole tile on the wall. How much distance does that take up?” I went ahead and wrote on the Jamboard while they were thinking: 24” – ¼ of an inch.

“23 and ¾ of an inch,” a student answered before I was done writing. Pleased, I put the length into the designated space in our row of tile. 

“Now what?” I asked.

“We figure out how wide the remaining space is,” a student said in a half statement/half question.

“Yes, but…” I wanted to provide positivity, but needed to help them toward accuracy. “Don’t forget about the ⅛ of an inch on either side of the remaining tile. You have to leave space (⅛”) between the two pieces of tile and also a space (⅛”) over here.” I drew tiny circles and labeled them on the Jamboard. “Use all of these measurements to figure out where I will cut my tile to place on the wall,” I told them. “Be careful!” I warned. “I don’t want to waste any tile. I have just enough.”

There were a few ways to figure out the answer. You could add up all of the fractions. And then, subtract the mixed number from 30 ¾ inches. You could also subtract each individual “space” away from 30 ¾ inches to see what is left. This method is like sliding your finger down the yardstick. After letting the students try the math on their own, we discussed. 

If you thought that the previous problems were difficult, buckle up. “Now, I have a real treat for you,” I told my students. “That first whole tile; The almost two feet wide tile; We have to cut a hole in it. The spout for my tub has to go through it.” Groans. “That’s right! I want the pipe, which is exactly one inch in diameter, to be in the center of my wall. What does that mean?”

“Halfway,” someone says. 

“Yes.” I let that sink in. “How far from the corner is the pipe?”

One of my fraction experts quickly tells me that it ought to be placed 15 ⅜ inches from the corner. I was pretty impressed, and had the student explain how he had converted ¾ into 6/8 in order to divide the fraction in half. 

“That’s right, but that is the center of the pipe,” I tell them. After explaining the term diameter, I point out that half of an inch will be on one side of 15 and ⅜ and half an inch on the other. “Where do we cut the hole for the pipe? …Don’t mess up! We don’t want to waste a great big whole tile!!”

After they fight with fractions for a minute, I remind them of the ⅛ inch of grout that precedes the beginning of the whole tile. Many groans follow this reminder. 

The whole lesson ends with me warning the students that, “After all of that measuring and math, you better be sure to cut the tile on the correct side of the line you draw, because your diamond blade will shave 1/16 of an inch off of whatever you’re cutting. In other words, you could figure out that the hole for your pipe has to be cut 14 ⅞ from the edge. Mark that measurement on your tile. And then, when you go to make the cut, slice the tile on the wrong side of the mark, which would make the distance from the hole to the edge of the tile 1/16 of an inch short. What would that be?” I posed the question, but didn’t actually expect them to figure it out. They knew I was messing with them, and they all gathered their things to leave. 

“Wait! What about our next row of tile?” I laughed as they left shaking their heads. 

Tony Wagner, the author of “Creating Innovators: The making of young people who will change the world” (2012), describes the practices of some highly effective schools. One of them is Olin College, a small engineering school that is pioneering hands-on learning.

“In classes at Olin, the primary goal is not the acquisition of knowledge. The goal is to develop a set of skills–or, in Jon Stolk’s terms, competencies–by solving a problem, creating a product, or generating a new understanding. Knowledge is important, but it is acquired on an “as needed” basis. It is a means to an end. Traditional academics often criticize this approach for being too utilitarian and lacking an appreciation of learning for its own sake, but the evidence is that Olin students are very well prepared for graduate school and better prepared for work, with managers who have been surveyed by the college reporting that Olin students who’ve just graduated act as if they’ve had three to five years of experience. Learning research shows that students understand and retain much more of what they learn when they have studied and used the knowledge in an applied context.” (Wagner, p. 175)

“When will we ever use this in real life?” is a question many math students will utter when trudging through seemingly pointless pedagogy. Infuse life into your teaching by showing how the lessons are used in real-life.  

Sources:

Development History of the Notched Rim Lapidary Diamond Blade. Barranca Diamond. (2023). https://www.barrancadiamond.com/home/history.html#:~:text=Richard%20Felker%2C%20a%20pharmacist%20develops,natural%20and%20manmade%20stone%20products.  

Wagner T. & Compton R. A. (2012). Creating innovators : the making of young people who will change the world (1st Scribner hardcover). Scribner.