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.
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.
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.
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.
“I don’t know how I got it; I just know that this is the answer,” a frustrated student defends himself against the inquisition of an even more frustrated teacher who wants him to “SHOW YOUR WORK!”
You should have seen the students’ eyes bulge when I told them I was going to give them candy! LOL They were happy to gobble up the math, though.
But, what if he actually doesn’t know where the number came from? We don’t ask the toaster to “Show us how it heats up our bread.” When was the last time you insisted that the mechanic “Show you HOW they fixed your car”? (They always try to explain it to us, and I’m like, “Does it work? How much does it cost? I got stuff to do.” Ha ha;)
I recently had a math enrichment lesson with second graders where I told them what they didn’t know they did with a couple of mental math problems. We were working on comparing three-digit numbers. I had printed pictures of snacks that had prices on them. Teams of students were first asked to arrange the snacks in order from least to greatest price. Then I asked the class to compare the cost of three items to the cost of two others. The students didn’t have paper or anything to write on.
Please pardon my penmanship;)
After I received some successful answers, I asked the teams, “What did you do in order to produce those answers?” I got a variety of responses. Most teams told me the names of the operations. “We added the three numbers together, and then subtracted…”
One group explained what they did to complete the operations, and I was very impressed. While students were sharing, I took some notes on the board. I clarified what the group was communicating by drawing circles around numbers and pulling out concepts.
“You began by adding 65 cents to 55 cents,” I reiterated. Nods of heads confirmed the accuracy of my statement. What happens in a creative mathematician’s head is a little different from what one would do on paper, however, and I wanted to pull this out. These students hadn’t used an algorithm.
Here’s a post that shows 3rd graders communicating the use of compatible numbers to multiply.
“Fifty!” the group called out. We have been identifying compatible numbers, so they already knew to look for something more manageable.
“That’s right. And, in order to get to fifty, you have to adjust these a little.” I circled the 65 and wrote 15 on the side. Then I circled only the 5 from the ones of 55, and I wrote that near the 15.
If a student had paper in front of them, they might line up 65 and 55. Then they’d add the fives from the ones’ column and regroup with a “one” above the tens column… But, do we grown ups do this in the grocery store when we are comparing one item with another? No, we use mental math. We develop creative tricks that we may not even realize we use!
My aim is to unlock this mathematical creativity early in life. A secondary goal is to help students be able to communicate it.
“After adding the two 50s together, what did you do?” Everyone can see that there is still a 15 and a 5 written on the board. I wrote the sum before anyone called out, answering the rhetorical statement myself. “Now, you need to add this $1.20 to 99 cents. That sounds hard,” I teased, knowing that they’d already smashed that algorithm in their minds.
Letting students work in teams allows them more than just Social Emotional Learning (SEL). They help one another remember and recall sums and differences.
When I told them about using 100 instead of 99, several students silently shouted, “That’s what I did!” No one is going to carry a one from the tens to the hundreds column of a mentally constructed algorithm. And, we don’t always have paper. AND, do you really want to teach your students to be dependent on paper?!
Now, think about it, reader. Students are using subtraction in order to add numbers together. What 8 year old is going to be able to explain this abstract use of arithmetic in writing on a test or assessment?
Here, I’m having the group of 2nd graders “play” with numbers by lining their teams up in order of least to greatest, having constructed the largest number possible with the loose number cards I’d given everyone in each team. Get-up-and-move-around-math.
And, we (myself included) expect them to “Show their work!” I’m happy if they know what they are doing and get the correct answer. I’m nearly 50, and I only just learned how to show MY own work! LOL
What I found myself doing in the past was asking students who had performed mental gymnastics to achieve a remarkable mathematical feat to write down the steps they took. In other words, if you added up three numbers (65 + 55 + 99), and then subtracted a fourth from that sum, write it all down…
Even if you can’t describe the exact process of creating the sum or exactly what you did to subtract. Just tell me what you did with the numbers. I, like every other math teacher in the world, wanted to see more than just an answer!
I think that having students use mental math, and then having them explain what they did VERBALLY is helpful in sharing the mechanics of the creative math. It’s easier to verbalize than it is to write. I bet there are books written about this. (If you know of any, please share. Thank you.)
Didn't she do a great job. I love how every kid does it differently. One girl wrote on the screen, while she explained. As a T, I am thrilled to witness these Ss talking about their #ReadyMath. I can write comments and give #StandardsBasedScores in @Flipgrid #EastPennPROUDpic.twitter.com/Kux2rJbZTm
A tool I’ve enjoyed having students use to verbally communicate their creative math skills is Flip (formally known as Flipgrid). Kids can make videos of themselves talking about the math. They can also write on their screens to show what they did while talking about it. If they did the math on paper, they can take a photo of their work to include in their video. Finally, they can watch each other’s videos, get ideas for future creative math projects, and leave encouraging replies to each other. The platform is easy to navigate and teacher-friendly for leaving feedback and assessment info.
In conclusion, while I always instinctually knew that forcing a kid to write down everything they did in their head could squash their creativity, I never knew how to bridge the gap between teacher and student; The chasm between the answer (what the student produces) and the process (what the teacher cares most about) before now. I’d tried varying techniques with varying results. My new thing is to verbally walk them through tricks I’d use to do mental math. Through this process, they recognize some of what they are already doing in their minds. They are learning how to communicate it. And, some students are learning creative ways to play with numbers.
How do you challenge yourself as a teacher? I’ve been working on making my pedagogy more powerful by pushing lessons forward with STEAM. Originally known as STEM (Science, Technology, Engineering, and Math), STEAM introduces the Artistic element to the closely related fields. When you are teaching a STEM subject, it is common to integrate more than one letter of the acronym (Hertz, 2016), but one of my modus operandi is to try to incorporate as many of the letters as possible into any given lesson. Powering my teaching with as much STEAM as possible is an art form to me.
I sometimes use art as a reward for completing work. “Once you finish solving the math, you may draw the aliens you encounter on your adventure!” I told enrichment students at the beginning of this math lesson.
Some lessons are more successful at incorporating all five letters of STEAM than others. A recent math lesson got my creative juices flowing, and turned into a beast of learning for my 4th grade gifted students. It all began with an enrichment lesson from Ready Math.
4th grade gifted students use art to get even more out of STEM.
Fourth graders had been reviewing place value as well as adding and subtracting large numbers. This is the perfect example of a math concept becoming boring for advanced students, those who have shown mastery of the subject matter. Once these kids conquer the use of algorithms, plugging in numbers to get sums and differences becomes mindless. Adding and subtracting is like riding a bike. Why make them pedal a stationary one when we could take it on a trip, instead? This is where I come in.
My first stop on developing math enrichment lessons is to check out the iReady Teacher Toolbox for ready-made resources that I may build upon. Curriculum Associates, the makers of Ready Math, has developed an enrichment assignment accompanying nearly every lesson. These are worksheets that can be printed for students to work on independently. You can push the assignment out digitally via Google classroom, too. The Ready Math team has provided a “Teacher Version” that contains answers; Even potential answers to open-ended questions, so that teachers know what to expect!
For this lesson, I took a screenshot of an iReady enrichment assignment, and used it as the background of a Google Jamboard. I like Jamboards because my students can draw and write on their iPads. They can create new blank slides, as well as duplicate the one with the instructions on it. It is easy for me to make a template, and then create a copy for each student through Google classroom.
This zany lesson, “Planning a Trip,” had students pretend to explore an imaginary planet, Zanyville. It introduced gigantic numbers and was very open-ended. I included some more instructions that I shared orally, when I initially rolled out the project. “You must tell me the number of miles from the beginning of the trip where each of the stops happen. Also, you have to provide the distance between the stops.” Students were instructed to use “sticky notes” in Jamboard to share this information. They showed their work on other slides.
The 4th graders worked hard and were engaged in making their maps. Only, the maps were grossly disproportionate. Some students made the third stop only “one mile” from the end of the trail, but placed it a fifth of the distance from it along the route they drew in their Jamboard. In actuality that distance would represent tens of thousands of miles! It is true that I did not instruct students to partition their maps with accuracy, but when I witnessed the misuse of proportion, I saw a fertile opportunity for learning. Enter, STEAM!
My vision for math enrichment is to dig into each concept, finding riches that deepen the understanding of my students.
The next time that I met with the 4th grade math enrichment students, I taught them proportionality. There weren’t any Ready Math lessons on this, but it fulfilled my mission for the enrichment group perfectly! My vision for enrichment is to deepen students’ understanding on math concepts. We do not rush ahead or necessarily “do more” math. I try to present novel ways of using the concepts that students are learning in the classroom. Sometimes, lessons include real-world problems, using the math that they are learning to perform a task they will most likely encounter some day. Other times, we may play a game that requires students to view the numbers in a unique way. In this instance, I wanted my enrichment students to learn about the relationship (ratio) between numbers.
I modeled partitioning a space on the whiteboard. With students’ input, I placed numbers in appropriate places on the makeshift number line. After a short lesson on this, I had students revisit their “Planning a Trip” lesson from the previous week. They were to make a number line and show where each stop would happen. Their Google Jamboards showed excellent progress in understanding this concept.
From Enrichment to Gifted Instruction
This is where the lesson took a turn from enriching math understanding to present problem-solving and higher-order thinking through gifted instruction. In order to illustrate the disproportionality of the original maps, I’d have the gifted 4th graders use one of their own to display an accurate account of the journey.
How would we do this? I would lift the trail from the map. The trail would become a rope that could rise up off of the map. This would allow my gifted students to account for extra mileage and figure out how someone may have covered the distance of tens of thousands of miles, but look, from only glancing at a two dimensional map, as though they had only hiked a handful of miles. My plan was for the fourth graders to learn about topography.
I gave my students a rope that was about 50 feet long. Their initial task was to tie pieces of yarn onto the rope symbolizing stops on the trip. They used what they had learned from the previous math enrichment lesson to do this.
“If the entire trail was 498,013 miles (this was on the planet Zanyville, remember), and the first stop was only 35K miles from the start, you couldn’t mark it halfway along the journey!” I reminded them.
We brought the rope into the hallway, so that we would have plenty of room to #STEAM. The first thing my students did was section the rope into five equal parts. It being a 50 foot rope, each of the five sections was ten feet. As it turns out, the tiles in the hall were one foot wide, so each one symbolized ten thousand miles, nicely. Once we had some reference points, we figured out where each stop would be and tied a piece of yarn to represent the place.
I then took one of the disproportionate maps that one of the gifted students had drawn, and I enlarged it onto poster paper that covered a large portion of the classroom floor. Next, we placed the rope showing our well-proportioned stops onto the map. We taped the pieces of yarn to the stops drawn on the map. There were huge portions of extra rope between some of the stops, and it was too short in other places.
I asked my students to “Imagine that this rope isn’t a rope, but a journey. Someone actually did travel these miles, and they traveled along the route that is represented on this map,” I told them. “How could this be?” Their first idea was that the “journey” coiled around and around; As in, the traveler had actually walked in circles. “There is a loop drawn on the map,” I pointed out. “You should assume that the line drawn on the map is the exact path the person traveled.” I wanted them to discover the concept of elevation on their own.
“The journey (they were very keen on calling the rope ‘journey’ and yelled at me when I mistook it for ‘rope’;) goes up!” several students shouted in unison. Feigning surprise, I encouraged my students to show it. I told them to make it happen. “Elevate the rope to show traveling up off of the paper.”
Watch the time-lapse of my 4th grade gifted students creating a model of their journey.
The students started grabbing everything within reach to raise their “journey” off of the paper. I gave them a few parameters: They could use anything in the room, other than my personal things. It had to follow the line of the map. The journey has to stand independently. “It can’t need you to hold it.” I heard the students mentioning how much fun this was several times, as they stacked Rubik’s cubes, built structures out of KEVA planks, and draped rope Journey over the 4’ high faux pilings I’d made for another project.
After finishing, I asked “How could we show these mountains on a two-dimensional map?” This is where topography enters the scene. Before exploring the topic, I had the gifted students brainstorm ways to display three dimensions on a flat piece of paper. They came up with drawing small images, like triangles symbolizing mountains. Information in a key was paramount. I showed them some maps that had varying colors used to illustrate elevation. They liked that idea. Finally, I introduced contour lines. These are drawn around the mountains, and show the incline of the slopes.
Once I did my best to explain how contour lines worked, I showed my students a video (above) made by HikingGuy.com. In the film, “Hiking Guy” takes a two-dimensional, topographical map and imports it into Google Earth Pro. He overlays the topographical map onto the exact geographical area that it represents. With Google Earth in 3D mode, the Hiking Guy swivels the view, so that you can see the depth of the mountains and valleys. We were in awe of the effects. I could have told my students that a contour interval is the amount of elevation between two contour lines on a given map, but seeing it displayed via three-dimensional modeling drove the concept home.
Next, it was time to collect some data to use in our map-making. We used yardsticks to measure how tall our mountains were. (I would have used centimeters as the measuring increment, in order to keep with everything scientific using the metric system, but since the Ready Math assignment had started the project off with “miles” on Zanyville, we kept to the standard system.) Before clearing away all of the materials elevating our journey, we marked the beginnings and ends of each mountain along the route on our map. In this way we would know the edges of the bottoms of our mountains for drawing contour lines.
Prior to drawing our contour lines, we needed to figure out what our contour interval would be. The first thing we did was figure out the height of each mountain, according to tens of thousands of miles. If a mountain is only nine inches high, as was Mount Rubik’s (we named all of our mountains), and a foot (12 inches) represents ten thousand miles, what is the elevation of the summit? We figured out that nine inches is 3/4 of a foot, so we concluded that our nine inch mountain was 3/4 of ten thousand; or 7.5 thousand miles high. The taller mountains required us figure out how many feet fit within the total number of inches. Some rounding was used, and we came up with some valuable summit information that was transferred to the paper map.
Now for our contour interval. How much elevation should each space between contour lines represent? We want the lines to be meaningful, but too many would make the map cumbersome to produce as well as read. We took each of our summits and found a number that was doable.
STEAMing Up Your STEM
Believe it or not, including art with the traditionally scientific fields of STEM has been mildly controversial. “The focus of STEM is developing rigorous math and science skills through engineering. How can you focus on other subjects (such as art) without losing the mission of STEM or watering down its primary purpose?” (Jolly, 2014). People who think this way are trying to isolate the left side of the brain (Pietrangelo, 2022). They imagine that opening STEM up to the arts would allow right-side brain activity to infiltrate and weaken STEM, but Jolly (2014) points out that this is viewing it all backwards. Art is already used in engineering, product design, creative math, and out of the box science. Strengthening our artistic use of STEM will make all four subjects so much more powerful.
Slow cookers like crock pots use lower heats to cook food over longer periods of time, deepening flavors and breaking down meat muscles, so that food becomes soft and tender (2022). A pressure cooker, on the other hand, cooks meals much faster. This mechanism has a seal that helps maintain the pressure within it. Steam builds up and pushes into the meats and other ingredients, breaking down the foods to make them tender, as well as infusing the flavors. You may not have tons of time to teach a STEM lesson. Use art to make the learning more memorable.
Saunas are small rooms that are heated up to 160 Fahrenheit. The health benefits of sitting in one of these hot beds for a short amount of time include clearing the pores of your skin, relaxing muscles, and burning calories, as well as increasing blood circulation (Hussain & Cohen, 2018). Add some water to the hot rocks in these wood-paneled spaces, and you have yourself a steam room. A steam room can help clear ones sinuses, loosen joints, and repair broken skin tissue (Johnson, 2023). [One thing to watch out for in a steam room is dehydration. It may seem ironic to dehydrate when surrounded by so much water, but the heat will cause moisture to leave ones body. Make sure to drink plenty of water and keep your time limited.] Using art in your lessons can add health benefits to your teaching. Students who favor their creative right side brain work might remember the lesson more. The artistic element could clean out misconceptions for visually-oriented students.
The difference between STEM and STEAM is as subtle as sauna vs steam room and slow cooker vs pressure cooker. I’d say that teachers are most likely using some art in their STEM lessons… I can also imagine a STEM teacher feeling pressure to utilize art which could lessen the lesson. Does titling teaching “STEAM” allow for art, promote its use, elevate art to STEM prominence, or simply point out the fact that it was there all along?
I like to think of art as the glue that holds the Science, Technology, Engineering, and Math together. I’m surprised we could see out the windows; There was so much STEAM happening in our classroom!
Enriching the Enrichment
A lesson that I began in the middle of this long mapping of a “Journey” project involved slope measurement. I thought that figuring out the value of the angles of the mountains would be useful. As it wasn’t helpful in completing the drawing of our map, I dropped it after spending only a little time on it. The information was very interesting, however, and I can foresee using it in on a slightly different angle of the lesson; Switchbacks.
Our Journey would not have gone straight up the sides of the mountains. A switchback is when a trail travels at an incline, but also more parallel than perpendicular to the mountain’s rise. After a short distance, the trail will “switch” directions and zig zag up the side of the incline. This means that the hiker will walk a much longer distance in order to get to the top of the incline, but it is an easier hike. What does this mean for our Zanyville trek? The journey, being its given length, will not go up as high; This will lower the peak or summit of each mountain. But, by how much?
Students can explore steilhangs, learning more about earth’s geography. Furthermore, they could research the differences between Earth and other planet’s geographies.
I had students conduct a scientific examination through dropping small plastic cubes on top of one another to form a miniature naturally formed mountain. They were not allowed to touch the mountain with their hands. What was the angle of elevation? We placed a ruler parallel to the side of the slope and used a protractor to measure the angle of elevation. It was about 15 degrees.
It will require some advanced geometry to recalculate the distance from the base of the mountain to the summit, given new measurements of switch back angles and lengths between directional changes. I might incorporate these elements to the lesson when I do it next year. It’s time to wrap this up.
Finally, I would like to see my students graph the elevation of the trip. Between zero miles and 498,013 miles, where did the trail rise and fall, and how much?
I use a running app that uses GPS to track my progress. It calls out my splits at five minute intervals. It also provides charts showing variations in my speed, cadence, and elevation. I want my students to create an elevation chart similar to the one from my running app. It ought to display where the traveler would begin and end hiking up/down mountains, as well as how high the peeks were.
The Captain of Class 