Tag: education

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. 

Balderdash: Gamify Vocabulary

We all know that selective word-choice can enrich writing (Academy, 2021). It will lend clarity to a story, deepen emotional attachment to a character, and broaden the understanding of a topic. Contrastingly, misused vocabulary can harm the message of a text. And, weak words will water down its substance. Therefore, building a thorough understanding of a wide array of words will prepare students for increasingly effective communication. As it turns out, I have a game that will make your learners crave vocabulary-building. 

Would you like your students to beg you to learn new robust vocabulary? Try Balderdash.

I call the game “Balderdash.” (There is a board game version, but I’ve never used it. From the description, it seems to contain the gist of what I present here.) The name is unique enough to spark interest and be memorable. It also lends itself to the core of the game; Playing with unknown words. 

I’ve used Balderdash to introduce vocabulary, deepen background knowledge, and explore literature concepts for years.

This is how I introduce the game to my students. I start off by explaining that this is a game of definitions. 

“How many times have you been reading an entertaining story, and really enjoying yourself; Then you come across a word that trips you up? All of a sudden, you don’t know what is happening. Why are the characters acting so weird? What did you miss? They aren’t always big words, but misinterpreting the meaning of a word can turn a heretofore simple tale on its head (irony in italics;). 

“In this game, you will be presented with a word that you probably don’t know. That is okay. You aren’t supposed to know what the word means. If you DO know what it means, or if you think you know what it means, that is okay, too. But, don’t tell anyone. You can earn points by writing down a definition that is really close to accurate.”

Here’s how it works

Sometimes, the game originates organically. In the middle of conversation with my students, I might use a word that they don’t know. Rather than simply telling the Polite Pirates what it means, we break out Balderdash to have fun learning its definition.
  1. Have a list of your students handy.
  2. Hand out index cards or sticky notes. Tell students to keep the papers UNFOLDED. Also, do not write on the cards until instructed. (They must look the same.)
  3. Have your students write their names on the top. Everyone should use pencil, and don’t do anything to your card to make it appear unique. (You will understand why in a minute.)
  4. Next, come up with a word that students will not know. You could begin with “balderdash.” This would make the game that much more memorable! You could use a vocabulary word from a list of words you want the class to learn. You could even flip through the dictionary, looking for tough words.
  5. Write the word on the board, so that everyone spells it correctly. You may want to write down some phonetic tips. I will sometimes explain what part of speech it is; perhaps even a hint. (i.e. This is a noun; and although this game might be named “Balderdash,” that isn’t what it means.)
  6. Tell the students to make up a definition for the word. Explain to them that their peers are going to vote on which definition sounds most likely to be true or accurate. 
  7. While the students are writing their definitions, you look up the true definition of the word. (I don’t recommend relying on your own interpretation. Even if you do understand the word, it is best to deliver the scholarly definition first.) You will need to put the definition into kid-friendly language. Your definition will need to match the ones that students hand in.
  8. Walk around and collect everyone’s index card. I recommend using a bucket or top hat for this. 
  9. Read through the definitions to yourself, making sure that you understand what they say and are able to read them fluidly. You don’t want to supply any “tells” that one is NOT the accurate definition. If there are any that are similar to the actual definition, provide that student a point and remove the definition from the pile. (Reading 2 of the same definitions would let them know they are the true definition.)
  10. Next, place them back into the hat or bucket. Pull one at a time and read it. Do this once through without any voting. 
  11. Then, repeat the process, but after each reading, have students raise their hands if they think it is the correct definition. You can only vote once, and you can’t vote for your own definition. 
  12. Give the pupil who produced the phony definition a point for each vote. Whoever votes for the true definition also gets a point. 
  13. After each round, share who earned the most points and the real definition.

In the same way a serious athlete might take creatine supplements to boost muscle-building ability in the body during intense exercise (Creatine, 2021), using games to increase enthusiasm for definition development can motivate kids to grow their vocabularies. Rather than get fatigued when faced with unknown words in texts, your students will view these as opportunities for growth. 

An athlete takes creatine supplements to prepare their body for doing a little bit extra in each exercise, making the workout that much more beneficial. Balderdash is a reason for collecting large, complex, unfamiliar words, and tucking their meanings into memory banks for future use. Plus, it’s a lot of fun! game

Sources

Academy, E. (2021, November 12). Word Choice in Academic Writing: Tips to Avoid Common Problems. Enago Academy. https://www.enago.com/academy/word-choice-in-academic-writing-tips-to-avoid-common-problems/ 

Balderdash Board Game – the Game Of Twisting Truths. (n.d.). Mattel Shop. https://shop.mattel.com/products/balderdash-cfx43 

Creatine. (2021, February 9). Mayo Clinic. https://www.mayoclinic.org/drugs-supplements-creatine/art-20347591 

This 4th grader got to wear the “Balderdash hat” for our photo as prize for collecting the most tally marks.

Show Your (Math) Work (Part 2)

Algebraic notation in chess shows each move that a player makes. Ranks (rows) are notated with numbers, while files (columns) are designated a letter. Thus each of the 64 squares on a chess board has a name, the letter first, followed by the number. For example, the most popular opening in a game of chess is to place a pawn on the square e4. This square is located within the “e” file and on the fourth rank.

Chessgames.com provides the algebraic notation, along with some commentary, for “The Game of the Century.”

In order to show what happened during a chess move players write down not only the square that a piece was moved to, but also what piece was moved and whether something was captured, checked, or checkmated. (There are a couple of other symbols, but these are the major ones.) Abbreviations are used in order to save time. The moves are recorded so that the game can be analyzed and studied afterward. Most chess apps online and on phones will create the notation for you.

The algebraic notation of a chess game looks a lot like code. That’s because it IS! It is a string of commands. If you know how to read it, you could recreate a chess game, move by move. You could plug the code into a computer, as chessgames.com did for “The Game of the Century” between 13 year old Bobby Fischer and International Chess Master Donald Byrne originally played during the Rosenwald Memorial Tournament in New York City, October 17, 1956. 

For several years I have been teaching elementary-age kids (7-11) how to play chess. I’ve run a chess club that meets after school. While I always share information about how to read algebraic notation, I don’t require them to use it during play. I want them to learn the game and have fun. 

As students get better and begin playing in tournaments, however, they will need to learn to use notation. During competitive play, they will be required to write down the code for each move on a piece of paper. Next to their move, they will also write down their opponent’s moves. In this way, they will be able to “view” the game after its completion. Serious players do this to evaluate each move. Which ones were better than others? Where did I or my opponent falter? What was the critical move? What could I have done differently? 

When solving a math problem, it can be valuable to write down some notes. Why? Sometimes you do this to keep short term memory space available. If there are a lot of numbers and a lot of computation is going on, there might be too much to remember in your head. Along the route of figuring out the answer, you write down what you have so far. 

Another reason for taking notes can be showing the steps you took to solve the problem. This might help someone else see a path from beginning to end of the math problem, demonstrating how the answer was achieved. They could “replay” the action of solving the problem by reading the steps taken.

In school, this last “reason” is usually proposed by teachers who want to “see” how you did your work. They say, “Show me your work,” so often that it becomes grating. And, writing down each step of computation can become tedious for a student who just knows the answer. Back in January (2024) I wrote a blog (Part 1) about students communicating their math work creatively. Learning to write down the steps involved in solving a math problem has value, however.

Here are a couple of ideas.

First, instruct students to treat each math problem as though it were a rock-climbing problem. Ashima Shiraishi is a champion of solving problems. She wrote a book about it. “How to Solve a Problem: The Rise (and Falls) of a Rock Climbing Champion” tells the story of 13 year old Ashima figuring out a map that will guide her to the top of Golden Shadow, a V14 boulder in Rocklands, South Africa. The story is an excellent metaphor for not giving up. It also shows how Ashima uses imagery to map out a doable path up the face of a difficult climb. She assigns symbols to the holds and footings that she uses. As she ascends, time after time, for it takes many tries to successfully conquer a difficult problem, she creates a story in her head, using the beneficial symbols (holds), so that she can find her way. The book, illustrated by Yao Xiao, shows (on paper) what Ashima does in her head. 

I often see students erase their math when they have tried a set of steps that did not accurately solve a problem. “Don’t do that,” I tell them. “You could very well make the same mistake again. It’s helpful to see what didn’t work, as well as the correct answer. I (and your math teachers) want to see the journey you took, including the fruitless paths.” 

Second, teachers ought to present problems that are worthy of their students’ struggle. Ashima Shiraishi has an amazing story of being the youngest person to successfully climb very challenging problems. At the age of 14, she was the first woman (and second person ever) in the world to climb a V15 boulder problem. She has crushed youth competitions, winning several years in a row. When she and illustrator Yao Xaio chose a problem to use for the basis of producing a picture book for kids, they decided on one that was very challenging. Ashima fell many times, having to start over at the beginning. The text shares her process of imaging each place that her fingers or toes would hold as something specific. One was the dot at the end of a question mark. Another was the crook of her dad’s elbow. 

What if Ashima used a simple boulder to illustrate her problem-solving skills? One of the most important themes of the short book would be lost; The idea of developing and practicing persistence couldn’t very well be shared if there were no falls. 

Create fewer math problems that are more challenging. Cause students to have to struggle to conquer each one. If they are required to make memories, it will mean more. If you want your students to “show their work,” give them a problem so complicated that they need to write down notes to get all the way through it. 

The math problem should be a challenging chess opponent in a tournament. Your students should have to show each “move,” as they progress through the solution of the problem. 

There will be practice problems. These are like the rock-climbing walls that Ashima climbs to stay in shape and hone her skills. She will climb the same wall over and over. She will challenge herself to complete the climb faster with each ascent. A story may be imagined for the short climb, and it could be repeated in her head a hundred times, giving the climb life and familiarity. 

This is like memorizing an opening for chess. Increasingly good chess players will learn a variety of initial moves that great players have developed over the years. They will memorize these opening moves through many steps, so that they can anticipate and prepare for potential rebuttals. 

In conclusion, if you want your students to show their work, give them something worthy of writing down the steps it took to complete. Teach them how to make the sharing of their math into a story. In the same way that each move in an important chess game is vital to its end, have students think about what they are doing and share the most important parts of the process. You could even have them try to identify the most crucial part(s) of the problem-solving process. With each boulder of a math problem you give them that challenges their minds and they successfully solve, they should develop a hunger for tackling more and more formidable ones in the future. 

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. 

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. 

Paper Football Field Goal Line Plot Lesson

Driving question: What is the perfect length of a paper football field? We are talking, one that allows some paper footballs to score goals, but not every “kick.”

Goal: Students will create and use a line plot to categorize data in a way that makes it easy to interpret. They will analyze the data to determine the best measurement for flicking a paper football accurately.

Prep: I folded a paper football out of an ordinary,letter-sized piece of paper (8 ½ by 11 inches). You fold it the same way you fold an American Flag. Have one pre-folded, but this could be part of the lesson, if you have time. (I didn’t have X.) 

I placed two tables end to end, creating a lengthy runway for measuring. Before students arrived, I taped rulers to the table top the entire length of the two tables, about 3 inches away from the center. I put pieces of tape at each foot so that it would be faster and easier to locate the increment. 

Lesson: I told the students the object of the lesson was to determine the “goldilocks length” of a paper football field for this group of students. Another group may be better or worse at flicking the paper football. We are going to collect data that will help us tailor our “field” to our group. 

“We don’t want the field goal too close, or every single flick will score a point. We also don’t want the field goal too far away. Then no one will score! There will be a window where some will score, but some won’t. We will use data to find that sweet spot. And, we will use a line plot to help us read the data.” 

The first thing we did was figure out the width of the field goal, so that we could finish constructing our mock field. I had each student form right angles with their thumbs and index fingers. Then, touching thumb-tips, they placed their finger field goals on the measuring tape (ruler) I had already taped to the tables. As students shared the measurements of their finger field goals, I wrote them on the dry erase board. We had 6, 6, 5, 5 inches. 

I had taught my students how to average numbers earlier in the year. They were bouncing with the information, now. “It’s 5.5,” a girl offered. 

“How do you know?” I queried.

A boy suggested that it was right in the middle of the numbers. I affirmed this by circling the middle four and five. The girl who had provided the original answer shared what she did to get it, and what one should do to find the average of several numbers. “You add all of the numbers, and then divide by the number of numbers.” We discussed dividing 22 by four in order to review fractions and decimals, and to double-check our answer.

Next, we used mini (six inch) rulers to measure five and a half inches distance between the already taped down ruler and a new one. I had the students tape it down. Now, we had a runway that was the average field goal width, running about ten feet long. 

I demonstrated how to flick the paper football. Each student got three tries. If any of them were duds (didn’t fly), we conducted a retry. There were a few very short flicks, but all in all we collected some valuable data. 

This line plot is not great because the line is not accurate. There ought to be measurements that do not have Xs. Every 1/3 foot should be labeled.

About half of the flicks landed between the two rulers; within the field goal range. These measurements were written on the board in one color. The flicks that did not land between the rulers were recorded in a different color. All of the measurements were recorded to the nearest ⅓ of a foot, in order to use mixed numbers on our line plot. 

Once the line plot was finished, it was easy to see the window where the field goal ought to be erected. There was a collection of accurately-flicked colored Xs up to a point. Then the other color, the color of missed flicks began to move in. At a certain point there were no longer any accurate flicks. The brackish space containing both colors contained the available distances. 

Some students wanted to place the field goal at the first measurement that recorded a miss. I explained that, were we to place it there, nearly every flick would score a point. Even the misses that went far could pass between the goal posts before veering off to the side. I drew a picture illustrating what I meant. 

We drew lines at measurements that we thought the field goal would work best. Then we discussed pros and cons referencing the data.

One student wanted the goal posts erected right before the very last successful data point; The last one to land between the two rulers. I told him that “This would guarantee that only one person would get one point for one flick out of… How many did we do? That might be too frustrating, and not very fun.” 

We ended the lesson without deciding on the perfect distance. Basically, the thing to do was to use the data that we collected to try out some reasonable distances, and see which ones were more fun. The beauty of the paper football field goal game is that the field is so malleable. It is all about fun, and that’s what I hoped the line plot lesson would generate. If nothing else, it was memorable. 

Building Strategic Thinking with Dominoes

My 5th grade gifted class revisited the game of Dominoes last week. It took some review, but they enjoyed playing the game. I told them that one of the reasons I had taught them the game was because it is a classic that they could play with grandparents and other elderly people, bridging the gap between generations. The game has been in existence for over 900 years! 

In addition to the game being old, it also presents an opportunity to practice strategic thinking. In an effort to prove this to my 5th graders, I have begun dreaming up scenarios where a player might use analytic skills to make a counter-intuitive move that would benefit them in the long run. 

There are times during a game when you have more than one Bone (Domino) that you can play, but none of the plays will give you points. Sometimes, it does not matter which one you put down, but other times you can plan ahead. Much like you would in chess, you can set up future moves by arranging the Bones to meet your needs. Playing them in a particular order would benefit you more. 

I planned on showing my 5th graders what I meant by setting up scenarios of games and taking pictures. I have done that many times to teach the problem-solving aspect of Dominoes. 

Using photos as teaching tools works great on interactive devices.

Then I thought, Why not have my gifted students make up the puzzles themselves? I will give them the parameters, and they have to try to figure out how to show the need for strategic thinking through constructing an image of a hypothetical game. 

The puzzle would be an image showing Bones (Dominoes) already played, Bones available to a player (standing up so Pips or dots were showing), blank sides of the opponent’s Bones, and maybe a Boneyard (unused Dominoes). 

If you are a novice Dominoes player, some of this vocabulary might be new to you. Bones are the game pieces, named after what they were originally made out of; Ivory or elephant tusks (bones). The Boneyard is made up of the unused Bones lying face down. Face down means that the Pips or dots on the bones are not showing. All you can see is a blank Bone or the uniform design that is printed/carved on every one of the 28 Bones of the set. Bones often have something decorative on the side without Pips, so that players can identify the 0-0 Bone more easily. Every Bone has two numbers on it. There are two ends of the number side of a Bone. No two Bones have the same combination of numbers. Beginning at 0-0, the Bones go up to 6-6. 

The Plan: In order to demonstrate strategies for play, I am going to have my 5th graders come up with puzzles that point to weighted plays. In other words there will be better moves than others. People trying to solve the puzzles will have to analyze the potential moves. Which one is better and why? Puzzle-solvers will be required to explain the move they chose. 

This is from the beginning of the year.

The Work: Arrange Bones as though they had been played in a game. This means matching the ends of Bones; Six is connected to six, three to three, etc. There ought to be four lines of play that a player can connect a Bone to.

Each player has Bones left to play. One set of Bones is standing up, with the number of Pips showing. These are the Bones that the puzzle-solver has to work with. (Normally, when I am teaching Dominoes to students, I have them lay all of the Bones down, so that every student can see all of the Pips. This is so that every single play is a lesson on problem-solving. When one plays a real game, you do not show your Bones to your opponent.)

The Bones that the puzzle-solver has to work with (the ones showing Pips in the image) should have numbers that can be played. They contain the number that is present at the ends of the lines of play. One of the Bones that can be played would cause the sum of all four ends of the lines of play to add up to a multiple of five, which is how one acquires points in Dominoes. This would seem like the best choice to complete the puzzle. 

Because we want this to be a puzzle that causes Domino players to grow in their understanding of the game and not just an illustration modeling how to play, we aren’t going to make the correct answer to our puzzle be an obvious choice. A good head-scratcher will require a player to look beyond the obvious play. 

If four Bones with the same number have already been played, and the puzzle-solver has two of the remaining Bones with that same number, how likely is it that the opponent of the puzzle-solver has any Bones with that number? 

Here is your task: Make it so that playing the Bone that does NOT create a multiple of five is the better play. 

I’ll have to share this lesson with my 4th graders, as well!

How could this happen? If the opponent of the puzzle-solver is forced to draw a Bone from the Boneyard, rather than playing a Bone, not only will they not earn any points, but they will be growing the number of points that the puzzle-solver will get at the end of the round; The round that the puzzle-solver is now more likely to win because they have fewer Bones left than their opponent. 

At the end of each round the player who uses up all of their Bones first gets points from the Pips that are on their opponent’s remaining Bones. In order for the play that did not make a multiple of five in the first place (at the beginning of the puzzle-solving exercise) to be the better play, the final play must provide more points than the potential multiple of five. 

If the multiple of five would have been fifteen, and there is no way, given the Bones that are left, for the puzzle-solver’s opponent to have a total of Pips greater than fifteen (you always round up, so sixteen would go up to twenty), then not playing the multiple of five during play would not necessarily be a winning strategy. Typically, you would play the multiple of five, get the points, and hope for the best. This exercise is designed to show my 5th graders that if you plan ahead, the delay of point acquisition could very well bring a windfall of greater point tallies. Not only is this a good life lesson, but it can help them play the game better in the future. 

Now, if you want to try to figure out how to create a puzzle that fulfills these requirements on your own, without any help, go for it. You can return to this writing when/if you get stuck and need some guidance. The next section provides some helpful hints. 

If you aren’t sure where to start, or you have hit a mental block, check out these ideas.

Some Helpful Hints: 

Limit the available Bones. You can do this several different ways. One is to only give the puzzle-solver two bones to choose from. 

Another way to limit the potential outcomes is to make the lines of play long. Have most of the Bones from the set showing in the lines of play, so that the potential Bones of the opponent is narrowed to only a few possible numbers. The puzzle-solver can reverse-engineer the game to figure out what Bones are left to be played. It’s like “card-counting,” but legal;)

A very effective strategy for creating a doable puzzle is to limit the numbers in play. Idea: Make the ends of the line of play all the same number, and the puzzle-solver has the remaining Bones that contain that number. For example, there is a one at the end of all four lines of play. There are only seven Bones that have a one in them. If four of these are played, and the puzzle-solver has the remaining three, then the opponent cannot possibly play any of their Bones. 

But, the puzzle has the puzzle-solver making the next play. How can the puzzle-solver cause their opponent to have to draw from the Boneyard? See if you can figure it out.

There are a couple of ways to solve this problem. One answer is to provide the puzzle-solver with a double. A double has the same number on both sides. When this is played at the end of a line of play, it keeps that number going! 

Another solution requires more work, and could therefore be trickier for the puzzle-solver to find. Make it so that all of the Bones that the puzzle-solver possesses have numbers on them that can’t be played. You have to position every bone that has any of the other numbers on them within the lines of play. No need to worry about your puzzle-solver using up their Bones because every one of theirs contains the same number as the ends of the lines of play. 

Stack the Pips. Create lines of play that have low numbers, thus ensuring that the Bones that the opponent possesses are more likely to have higher Pip counts. In this way, even if the puzzle-solver would make a fifteen or twenty with the false-solution-Bone (the one that would make a multiple of five and seems to be the better choice for the puzzle-solver to choose), the total Pips that the opponent would have must be greater than the multiple of five. This number work is truly statistical thinking. Out of all of the Bones still available, how likely is it for the opponent to have a high enough number of Pips for the counterintuitive play to benefit the puzzle-solver more? 

This puzzle would allow for the opponent to make a play or two before the puzzle-solver is out of Bones. My student would have to work through all of the possible outcomes to ensure that the puzzle-solver would come out on top. 

3rd graders learn to play Dominoes

Try it out, and make the puzzle fool-proof. When making the puzzle, turn all of the Bones over so that the Pips are showing. Create a model of lines of play. Give the puzzle-solver the Bones they will work with. Now, look at the Bones that the opponent could have. Adjust the lines of play, so that there is no possible way for the opponent to have a way of winning. You also have to double-check that there are only Bones that would cause the opponent to have more Pips than the false-solution. Then turn over the Bones that form the Boneyard, and stand up a couple that represent the unknown opponent’s Bones. 

Conclusion:

Normally, I will do a lesson like this, and then write a blog about it. This is different. I have used my writing to think through what I want to have my 5th graders do. 

My aim is to have them build their understanding of the game of Dominoes and learn statistical analysis through the process of constructing their own puzzles, rather than just solving mine. Hopefully it will be successful, and I can write a follow up blog about how wonderful it went… or the lessons I learned through its execution, pun intended;) 

If you try this idea or one like it, please share your results. I’d love to learn feedback and improve future teaching. 

Sources:

Marcus, M. (2020). How to Play Dominoes . Cool Math Games. https://www.coolmathgames.com/blog/how-to-play-dominoes

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.

Comparing Fractions Through Arch… itecture

Humans love to construct (Lorek, 2018). Is it in our instinct? Sarah Lorek (2018) contrasts the idea that beavers building a dam would be considered “natural” versus humans constructing a dam being “man-made.” The structures look different, and have slightly different purposes, but the reason for construction may not be as disparate as assumed. 

[I searched the web for articles that speak to the (my) hypothesis that constructing things might be a human natural instinct and came up short. This would most likely be a very difficult thing to prove, but I’d love to read about it. If anyone knows of any literature or can point me in the direction of a good source, please share.] 

From my own personal experience, I can say that I have witnessed students come to life when building blocks are made available, and some of my favorite activities involve construction. I have fond memories of sitting in the middle of a mess of Legos, Lincoln Logs, and Robotix when I was a kid. I spent hours building spaceships, cities, and robots. Now, I remodel my house (recent bathroom project), labor on landscaping, and generally enjoy working with my hands (famously failed pirate ship project).

I’ve written about the unique and exciting experience of putting building materials in front of my students. From Building Bridges, using Blocks to teach measurement, Adding Blocks, Purchasing Blocks to use in business plans, constructing Bridges as Object Lessons to teach SEL, and creative Playtime, my students are no strangers to building materials and hands-on lessons. Even during this past lesson, I heard students professing this to be the best Math Enrichment lesson, yet! 

Purpose

Students have been introduced to fractions in their regular math classes. I was planning to enrich a lesson about comparing fractions. Our math curriculum had a nice worksheet that would have students create quilts with limited colored squares, and then compare the fraction of each color. Students can still do that lesson, but independently in the classroom. I created a lesson that was hands-on and interactive.

Students would use colorful connecting blocks to construct an arch. I had a picture of the arch in Washington Square, NYC on the board when they entered the room. We discussed the idea of arches for a few seconds before I shared the parameters of the project. 

Arches are symbolic, letting in light and allowing people to walk through walls. They are old; developed during the second century B.C. They are also timeless, in that they are still used today. They have even been known to hold magical properties! Vampires must receive permission or an invitation before entering someone’s home (through an arch), and the arch makes an appearance in Harry Potter as a passageway between this life and the beyond.

A memory that popped into my head that will definitely date me is that of the movie “Ernest Goes to Camp” (1987). The image on the cover of the movie shows the arch that Ernest is working on at the very beginning of the movie, surrounded by several scenes from the flick.

Parameters

As per a recent lesson that I learned about setting tight parameters, I constructed very strict limits on student creations. Students were allowed exactly 24 cubes. I chose this number because we have been learning and playing Math 24 a lot recently, and because the number 24 is divisible by many numbers. The lesson could include reducing fractions, making common denominators, and more through starting with 24 for a denominator.

The blocks could only be red, green, blue, and/or yellow. At least three different colors were to be used. This ensured that there would be some comparisons between fractions. 

Students could work independently or with a partner, but no groups larger than two. Each student, whether working with a partner or alone, would be responsible for putting information about their project into a Google classroom assignment. 

Activity

Once the assignment was explained, I set them loose. They were busy bees, buzzing around the blocks, loving the building. Creativity bubbled in the classroom. There was a purposefully leaning arch, decorative arches, symmetrical aches, one was made as short as possible while still fulfilling the definition of “arch,” and one even mimicked the pointed style of the Gothic Arch. So impressive! 

I stopped the students when some of the first arches were being completed, so that I could instruct them on what to do next. I had a couple of girls hold up their arch for an example. We took a photo of it, which is what I wanted everyone to do. This way, I could assess the accuracy of the numbers. It demonstrated to students that they must provide evidence.

I model taking a photo of the photo I just took after teaching what to do.

Next, we counted the number of different colored blocks. The girls had made their arch symmetrical and used the same number of colors for each of the three colors. (This is something I fixed the next time I taught the lesson, that afternoon; “None of the colors can be the same number.”) 

I showed the students how to write the fractions next to the color of blocks that I had provided for them in the software that they were manipulating. Then, we wrote an equal sign between the two fractions. The way I got the girls to include “greater than” and “less than” in their project was by combining colors. “The fraction of cool color blocks was greater than the fraction of warm color blocks.” 

Time was allotted for producing fractions and making comparisons on iPads. In order to do even more comparing of fractions, I then had students take pictures of a neighboring team’s arch. They then imported that photo into their Google Jamboard project so that both their arch and their neighbor’s arch were side by side on a slide. Now, they got to compare the fraction of red blocks from their arch with the fraction of the other arch that was constructed with red blocks. This exercise involved talking about the math, sharing out, and self assessing. 

In the end, students enjoyed not only comparing fractions, but constructing them, building knowledge, and cementing learning into a fun and memorable experience. 

Sources

Azzarito, A. (2021, August 19). From Architectural to Artistic, Arches Are Trending. SemiStories. https://semistories.semihandmade.com/design-history-arches/

Lorek, S. (2018). Ancient Architecture and the Human Need to Construct. Trimble Construction. https://constructible.trimble.com/construction-industry/ancient-architecture-and-the-human-need-to-construct  

Sinclair, L. (2014, December 19). The History of Architecture in Eleven Arches. The Architectural Review. https://www.architectural-review.com/essays/the-history-of-architecture-in-eleven-arches

The Top 10 Construction Toys of All Time. Michigan Construction. (2017, December). https://blog.michiganconstruction.com/the-top-10-construction-toys-of-all-time