Math

Engineering Math ~ Spinning our Way through Counting

Grade K-2, Math

Where do you see math in the following picture?

This chart came from a noticing I had that during a unit focusing on the concepts of design that the children in EY5 were very interested in spinning tops that year- Beyblades to be exact. These were the tops where you release the button and the top goes off and has a battle with someone else’s top to see which one will win and keep spinning. The problem was, the children kept bringing their Beyblades into school and naturally they would get lost or broken. So instead of banning them completely, I wondered . . . how can we make this at school so everyone can participate and where does math fit in with this activity?

The children discovered that if you stack 2 gears on top of one another, its makes a spinner that resembled how Beyblades work. When given an appropriate flick, the gears would turn rapidly around on the tabletop. It was fun to see how quickly the gear-tops could spin, but the children were discouraged because they didn’t actually know who’s top was going around more. So we thought, how can we count how many times the spinner goes around more accurately?

This is how the idea of using our iPads emerged!

How it worked is the children used the iPads to take a recording as the top was spinning using the slow-motion feature on the iPad. After the spinner came to a halt, they rewatched the spin in slow motion carefully counting every time the spinner went around. Sometimes the children needed to watch the video more than once to get a better idea of the exact number.

Once confident in the number they got, the children wrote down how many spins they got on the chart next to their name!

The children were so interested in this project, that they wanted to do it again and again. They tested different types of spinners, adding more or less gears to see if this sped up or slowed down the spin. They also did more than one spin to see if they could beat their record. The chart was the last step in this process to make our thinking visible, showcasing our hard work and to give the children a little more practice writing the big numbers they were determining when they counted each spin.

It was a memorable because the children took it into their own hands, and could go through the steps themselves. Additionally, coming from their own interests, the children were motivated to continue to add to the project and create a station that lasted in our learning environment!

Finding the Numbers in Football

Grade K-2, Math, Who We Are

Numbers are all around us; we just need to look for them. So often, we choose to ignore the numbers right in front of us—the numbers that are part of our everyday routines and the daily desires of our students. Or maybe we don’t choose to ignore them. Maybe we are so conditioned to a particular curriculum or methodology that we bypass the logic of everyday numbers entirely. We don’t mean to; it is just something that happens. But what if “page 24, section b, scenario number 8” was not what happened tomorrow? What if the only thing that happened was answering a single question . . . How did Ronaldo get so good at football?

Quick! Look carefully, there’s footballs flying at you. Left, right look everywhere. How many footballs do you see? A quick flash thinking exercise as a warm-up, a crash course in subitizing that keeps them alert for the game. The only preparation? Copy and pasting some footballs onto some slides, or even having the kids cut out some footballs themselves and gluing them onto index cards to challenge their friends. Tell the kids its a way to warm-up for the match. Ronaldo stays alert with his mind and his body to think quick during a game.

Time to warm up! Now that our minds are warmed up, it’s time to do the same with our bodies. Heading out to the track for some exercises builds both our strength and our rote counting skills. One by one, the children pick an exercise they want to lead and choose the count for how many repetitions we are going to do together. Most of the children count by ones, but some choose to challenge their teammates with skip-counting. As they go down with each push-up, you can hear them chant: “5, 10, 15, 20…”

Ronaldo always warms up before he plays.

Okay ready to play! But wait . . . we need Jerseys. Ronaldo is number 7, but what number do you want to be? The children design their own Jersey and write their favorite number on the back. Can you identify all the numbers from the Jerseys of your friends? What numbers are missing? We write down a list of the numbers we are missing in case more friends want to join us to play later.

Finally, we are ready to play, but there is a problem… there are 8 students. How many students should we have on each team? We use our paper players as concrete manipulatives, moving them around to figure out what even teams look like and practicing our problem-solving skills. “Even” means that the teams have the same number of players. It looks like there will be 4 players on each team, since 4 + 4 = 8. Let’s double-check with a part-part-whole model to ensure our teams have two equal parts!

It’s time to play, we just need a way to keep score. The children decide they can use a white board to write a tally each time one of the teams gets a goal. At the end, we will know how many goals each team scored. One child volunteers to be the official score-keeper, proudly recording each mark.

Great job, teams! The game is done. But before we can go to snack, we need to compare the numbers to see which team won. Twelve is larger than 4, but by how much? Let’s use the “count on” strategy to figure it out together: 5, 6, 7, 8, 9, 10, 11, 12. That’s 8 more than 4! Can anyone tell me how many goals were scored altogether? Let’s start tomorrow by calculating the total number of goals. I wonder if fewer or more goals will be scored the next time you play.

Time to have our snack to fuel up for some recess and more football! Ronaldo would be back on the field before no time.

More Than, Lesson Than Activities

Grade K-2, Math
Pattern Blocks

Pattern blocks are a great way for young learners to explore the concept of More Than and Less Than because they can be sorted easily by color and by shape. Children can start using terms such as “square” and “hexagon” in their vocabulary when talking about math and still easily be able to categorize them without getting into the concepts of number of sides. Children can either make groups that show the concept of more or less or make puzzles for their friends to challenge them to determine which one has more and which one has less.

Creative Counters

What counters do you have in your classroom that kids are already familiar with and love using? For me, it’s these little teddy bears. For this activity, the children found a partner and each played the game “How Many?” by reaching into the bin to pull out one handful of bears. Each partner counted how many bears they had and then the team determined who had more and who had less. We introduced the greater than and less than symbols, but the kids can also verbally talk about how many they got.

Graphing Extension: Jo Boaler Lesson

Grade 3-5, Math

Take a look at the image on the left. What do you notice? You may recognize this image to be a graph. You may also know that this graph comes from a lesson written by inquiry mathematics advocate Jo Boaler. Boaler’s goal in her lesson, is for students to understand that graphs show comparisons. However, you’ll notice one difference between the image on the left and the image Boaler uses in her lesson plan. In this image, I have taken out the labels on the axis. My reasoning? Because I believed my students could figure that part out themselves.  

I asked the following set of questions to my 4th grade classes.

Sample Student Work
  • What can you tell me about this image? 
  • What would you label the axises? 
  • Where would you place a zebra? What about a dog?

Not only were my students able to correctly label the axises, but they also were able to infer where a dog and where a zebra would be appropriately placed on the graphs. Interestingly, each group had their own perspective as to where a “dog” and where a “zebra” would go providing opportunity for a unique classroom discussion on perspective. There can be different breeds of dogs, so why couldn’t this be placed differently!

Good inquiry mathematics stretches with the ideas of the children in the classroom. We took this one day exercise and turned it into a 3-day math project, as students wanted to create their own graphs similar to Jo Boaler’s animal one. My students researched topics such as flags of countries and weights of planets, and built their own graphs comparing and contrasting areas of interest. While starting off as just a math lesson, this project grew to incorporate geography, science, and social studies into math.

Some examples of my student’s final graphs. They wanted to leave the axis’s off to have other peers and community members guess what they were comparing on their graphs. Can you guess?

Tangram Fractions

Grade 3-5, Math

For this math lesson, I broke my students into small groups and allowed them time to play with tangrams to explore how the shapes fit together. Once they had about 5 minutes to play, I asked them some guiding questions to begin discovering fractions.

  • How many times does the small triangle fit into the larger triangle? 
  • How many times does it fit into the small square? 
  • If the small triangle fits into the small square 2 times, what fraction of a small square is a small triangle?

Of course these questions were basic for my 4th grade students, but I wanted them to determine that although all the shapes are different sizes, each shape can be measured using the smallest triangle. 

Next, I let my students play again! I gave my students 5 more minutes to play with the pieces but told them that by the end of the 5 minutes they must have some sort of picture created with the blocks. This picture should not have more than 10 pieces, and should include at least 3 different colors of pieces. I showed them this picture of a house as a simple picture I made using my requirements. 

Once each group of students agreed on a picture, I first went around and took a picture of each one. This is mostly because I didn’t want someone to bump the table and the kids to not remember what color piece went where. With the images documented, I had my students pull out their math notebooks to answer the following question: What fraction of your total picture is each color?

Answering was a challenge for many of my students. Even with the guiding questions at the beginning, I had students tell me that in the picture on the right blue is 3/10th of the total because 3 pieces were blue out of 10 total. But with some patience and some partner work, most of my students came to quickly understand this misconception and determine a strategy to accurately answer this journal prompt!

Area and Perimeter of Compound Shapes

Grade 3-5, Math

As an introduction to compound shapes I arranged pieces of artwork my students had completed into three different shapes on the board: an L, a snake, and a straight line. The artwork came directly from our unit on imagination, where students drew pictures to fill a nine frame square.The kids loved using their art as the base of math, connecting to the problem because their pieces were being used on the board. 

First, I asked my students: Does each compound shape have the same area? Answers varied and as a class we calculated that each shape does indeed have the same area.

Next, I asked my students: does each compound shape have the same perimeter? Again, answered varied and we calculated as a class that each shape had the same perimeter as well.

For our independent challenge I had students work out an answer to the following question:

Is there a way I can arrange these boxes to have the same area but a different perimeter? 

Students loved this challenge as it provided them with an opportunity to disprove the belief that shapes with the same area must have the same perimeter. They ultimately discovered that if they moved these four nine-frame boxes into a square that the area remained 36 square units, but the perimeter switched from being 30 units to being only 24 units around. 

Whiteboard Angles

Grade 3-5, Math

This was a fun center idea for students to determine Obtuse, Acute, and Right Angles. The kids taped up some old tables outside in the common space to create intersections of different sized angles. Next, in small groups the class took turns labeling each angle with it’s corresponding name based on it’s size.

Labeling Tables

A few days later, once we introduced angle degrees and measurement using protractors, the kids were again able to use these tables in small groups, this time to measure how big each angle was. The tape provided an extra challenge, as the kids figured out they needed to measure from the inner tape line

Math Talks

Grade 3-5, Grade K-2, Math

I believe Math Talks are one of the best methods to both challenge advanced students and support struggling students. In a math talk, students are given a problem, or question, to which there may be one or multiple answers, but there are many strategies of which this answer(s) could be found.

Steps to a Successful Math Talk

  1. Have students work individually on the problem or question introduced– I tell my students, if you have figured out one way to solve the problem, do another. Find as many strategies or solutions as you can.
  2. Walk around and jot down what methods students are using– I keep a running list of strategies that both work and do not work with names next to them so I have a plan for who to invite up to share their work.
  3. Come back together as a class to share– I call on the students who’s names I wrote down and tell them which strategy I would like them to talk about.
  4. Students teach each other– I used to write down the answers for my kids and have them talk through their thinking as I write, but I’ve found that once my class feels comfortable with each other, the students can write and explain their own strategies while using their presenting skills.
  5. Allow the class to ask follow-up questions once a student shares– I also ask clarifying questions, especially if the student arrived at an incorrect solution so that as a class we can discuss the misconception. At the end of each share, I ask the class, who else used this strategy? This is important as it allows students who were not asked to share to be recognized for their work.

We use Math Talks quite frequently in my 4th grade classrooms. I found the Math Talk from which these pictures are taken to be a great pre-assessment for my students in terms of addition. Most of my students were able to use the standard algorithm for addition to solve the question 39+84=. Many of my students chose to use some type of visual representation for the problem, either a number line, or an organizer with breaking apart the numbers. One of my advanced students was even able to recognize that 39 and 84 are both divisible by 13 and solved this problem by first dividing each by 13 then multiplying 13 by the total he divided by.

M&M Estimation

Grade 3-5, Math
Inquiring about how to use the scale

Who doesn’t like mixing food with learning? Today we investigated how many M&M’s we thought were in one bag using a scale to weigh in grams. As a whole class, we inquired into how the scale works, playing with it’s functions. Next, I drew popsicle sticks to determine groups of four students to work together to reach a conclusion of how many candies there were. The group got to pick either a crispy bag of m&m’s or a regular bag of m&ms to experiment with. The rules were simple: use any strategy you want to figure out how many m&m’s are in the bag. Yes, you can open the bag and use the m&m’s but no, you cannot count the m&m’s one by one, you must somehow use the scale.

Hint: Look at the back to figure out what the total weight of the package is.

Working in small groups to organize our thinking
Weighing the M&M’s in small amounts

The results for this lesson were varied. Many students got frustrated since when they weighed only one m&m the scale weighed 0kg but when they weighed two m&m’s it sometimes weighed 2kg. Regardless, the students had fun, used teamwork, developed skills on how to use a scale, and were creative in their approaches to get reasonable results whether or not they were totally accurate.

Pizza Three-Act-Task

Grade 3-5, Math
Pizza Shop Menu

During a weekend trip to Seoul, I went to a pizza shop in Itaewon selling a wide variety of types of pizza. My friend and I debated over whether we would pizza multiple slices or a whole pizza, and ultimately decided that we would go for slices to try more kinds. Good thing we did since each slice was huge, and 3 slices made up half a pizza! This got me thinking . . . mathematically, this wouldn’t make sense for the pizza shop to even sell whole pizzas at all! This is how I got my idea for a lesson I taught the next day at school.

Act I: What do you notice in this picture? What do you wonder about what you are looking at?

Students discussed that this must be a pizza shop and how the numbers represented the prices. They determined that triangles must mean prices for single slices and circles must mean prices for full pizzas.

Act 2: How many slices do you think the pizza shop should sell in their whole pizzas?

We discussed what they remembered from a previous unit on businesses and how businesses normally price items in order to make a profit.

Act 3: This pizza shop sells 6 slices per pizza. How does this compare to your answer? Do you think the pizza shop should change the number of slices per pizza? Why or why not?