To introduce the crowd estimation problem play the video below up to the 1 minute 30 second mark.
Estimating Crowd Size
Thousands of people are on the Las Vegas Strip and in downtown Las Vegas each year to celebrate the birth of the new year. A massive fireworks show begins at midnight. According to the Las Vegas Convention and Visitors Authority, more than 318,000 visitors were in Las Vegas for New Year's Eve in 2019. This is the number of people that visited Las Vegas on New Year’s Eve but not the number of people that are out on the street of the Las Vegas Strip. Las Vegas Boulevard is closed for car traffic on New Year’s Eve so people can walk on the Boulevard.
Is it possible to get an accurate count of the number of people on the street of the Las Vegas Strip on New Year’s Eve? What do you need to do this?
Let’s look at just the crowd on Las Vegas Boulevard in front of the Bellagio.
-What is your estimate of the crowd in front of the Bellagio?
-What is your estimate for the lowest possible number of people?
- What is your estimate for the highest possible number of people?
Problem statement: Consider just the area from Flamingo Road to the Cosmopolitan. Determine a mathematical estimate of the number of people on Las Vegas Boulevard on New Year’s Eve.
Follow up question: The Las Vegas strip is about 4.2 miles long. How many people are on the complete strip on New Year’s Eve?
New Year’s Eve on the Las Vegas Strip is an exciting time and attracts large crowds. In this activity students estimate the number of people that are on Las Vegas Boulevard on New Year’s Eve. This is an engaging and relevant activity with multiple possible solutions and methods. Students can use a variety of mathematics in their solutions including area, measurement, proportional thinking, conversions, estimation, and mathematical operations. Students can also use the Internet as a resource to research information. Given the current COVID-19 pandemic teachers can decide whether students will solve the problem based on past crowds on New Year’s Eve in Las Vegas or with social distancing guidelines in place.
Elementary students may measure off a grid on the floor and see how many people could fit in the grid. They could then use the measure distance feature on google maps to calculate the area on the Las Vegas Strip. Students could divide the area of the strip by the area of the grid they used on the floor. This would provide the number of same-sized grids the students used that would fit on the strip. Students could use this number to multiply by the number of people they fit in their grid.
At the middle school level students may make use of the scale on the provided map to help calculate the area on the strip. They might use the Internet to help determine how many people fit into a certain square footage. They could then determine an estimate of the number of people on the strip.
While using similar methods as above high school students may consider more factors in their estimate. They may adjust their estimate so there is space for people to walk along the strip and then space for people who are staying in one place. They might also consider that there are barriers, trees, and a median where people could not stand. High school students may also come up with a formula for people per square foot based on the type of event or the density of a crowd. The follow-up question would require students to convert measurement units to solve. Limiting the number of people allowed to walk on the street would allow for social distancing to be followed. This would adjust the crowd estimate by just considering the maximum number of people who could safely be on the boulevard at one time.
After teachers have facilitated a discussion of students’ ideas, teachers can also share information about the Jacobs Crowd Formula. Herbert Jacobs, who was a journalism professor at the University of California, Berkeley in the 1960s came up with general guidelines for estimating crowds. He determined that generally a light crowd has one person per 10 square feet, a dense crowd has one person per 4.5 square feet, and a mosh pit like crowed has 1 person per 2.5 square feet. Jacobs became interested in investigating crowd estimation after looking out his office window and noticing crowds of students on the plaza below. The plaza’s concrete was poured in a grid so he used the grid and the number of students in each grid to begin to thinking through a way to estimate crowd size. Students can be asked what are the limitations of this method and the methods that they came up with. Students can also be asked in what situations estimating crowd size is used.
Battleship is a game of guessing, strategy, and logical thought. Students use their mathematical knowledge to set up their ships; then play against another person to be the first to sink their opponent's ships. Below are instructions for one version of the game Battleship.
Setting up your ships
To start, set up your grid so that it will be consistent for all players. Set the x-axis and y-axis to go from -15 to 15.
You will get three ships. The ships will be inputted as linear equations into desmos in the form of y = mx + b. For example y = 4x – 2. To determine the length of your ships you will need to either set the domain (x-values) or range (y-values) with the domain or range being 2 units, 3 units, and 4 unit intervals. (See example below).
Playing the game
You will alternate turns until all of your opponent's ships are sunk. Your torpedoes to sink your opponent's ships will come in the form of circles. If any part of the circle or inside of the circle touches a ship, that ship is sunk. The torpedoes will have different radiuses based on a roll of the dice. You will roll the dice before every turn.
If you roll a:
1- radius of 1
2 - radius of 2
3- radius of 3
4 – radius of 4
5 – radius of 5
6- radius of 6
You get to pick the center of your circle which can be any point in the grid. For the equation of a circle, the center of the circle is (h , k) and the radius is r.
The example below shows two hits and one miss. It will likely take more turns to sink ships, but this is just an example. The equations for each circle are inputted into desmos in the fourth, fifth, and sixth, lines. You should keep track of your opponent's equations in one Desmos window and your equations in another Desmos window.
Click on the following link for other versions of Battleship that can be played with Desmos.
Stohlmann, M. (2017). Desmos battleship. The Australian Mathematics Teacher.
Mathematical modeling is garnering national and international focus due to the many benefits that it can provide to students including increased engagement, understanding through multiple representations, and discourse.
The Scale the Strat activity is connected to health science and an event held in Las Vegas sponsored by the American Lung Association, whose mission is to save lives by improving lung health. Each year at the Stratosphere Tower people climb up 108 floors of the tower consisting of 1,455 stairs to raise money for the American Lung Association. Shaun Stephens-Wale has the fastest time running up the stairs in 7 minutes and 3 seconds. At the start of this problem a video is shown about the event.
The question is then posed if Stephens-Wale would beat someone in an elevator to the top of the tower. Students realized that someone in an elevator should arrive at the top first.
The question for the problem is then posed. How many floors would someone in an elevator have to be stopped at in order to tie Stephens-Wale running up the stairs? This problem is connected to creating equations that describe numbers or relationships. In figuring out this problem students make different assumptions and approximations including if the elevator is being used by guests of the casino or staff, the speed of the elevator, and the time it takes for people to get on or off the elevator at each stop. Students have also used the Internet to help develop their solution.
An example solution involves making an assumption that the elevator takes 1.5 seconds to go up each floor and approximately 17 seconds of wait time for each floor it stops at. This takes into account people getting on and off the elevator. The equation would be the following then:
423 = 108 (1.5) + 17x which when solved gives about 15 floors with stops. In this activity the importance of choices and assumptions are highlighted for mathematical modeling.
Stohlmann, M. (2020). STEM integration for high school mathematics teachers. Journal of Research in STEM Education, 6(1), 52-63.
Science, Technology, Engineering, and Mathematics (STEM) education has received increased interest in the past decade. Integrated STEM education can provide students with relevant and meaningful experiences that develop STEM knowledge and 21st century competencies. The world is increasingly becoming more reliant on STEM knowledge which makes quality integrated STEM education imperative for students.
There are five main tenets for assessing or developing integrated STEM curriculum.
One of the main concerns that has been raised in regards to integrated STEM education is the need for further curriculum development where the STEM disciplines are integrated in a meaningful way. At the elementary education level there is great potential for this to occur.
I analyzed and described integrated STEM curriculum from articles in two elementary education journals: Science and Children and Teaching Children Mathematics. Both of these journals are published by leading organizations in mathematics and science education. Science and Children is published by the National Science Teaching Association (NSTA). Teaching Children Mathematics is published by the National Council of Teachers of Mathematics (NCTM). I identified 85 articles that included integrated STEM curriculum. Click below for a full description of curriculum. The table below that has a description a few of the articles.
Use dice or playing cards (1-9; Ace =1) and grid paper.
Take turns rolling two dice or picking two playing cards. The two numbers that are picked are used for the width and length dimensions of a rectangle. The rectangle can be placed anywhere in the grid paper as long as it does not overlap with another rectangle. Play continues until one player does not have a place for their rectangle. The picture below shows the start of a game.
Each player should have a 3 x 3 grid. In each box a number from 2 to 12 is placed. The same number can be used more than once. The goal of the game is to be the first person to have all of your boxes crossed off. Once all players have their boxes filled in, two dice are rolled. The two numbers are summed and all players check their boxes to see if the number appears in a box. Players may only cross one number off for each roll. For example, if a player had the number 6 in three boxes and a total of 6 is rolled, the player can only cross off one six. The picture below shows the start of a game with a sample board. The two dice continue to be rolled until one player has all numbers crossed off.
After playing the game multiple times, ask students what totals are more likely to be rolled. The image below has the number of ways each total can be rolled and the probability of each sum being rolled.
Small, Large, or Target
This game can be played by having players try to get the smallest number, the largest number, or be closest to a target number. This should be decided at the start of the game along with what operation will be used (addition, subtraction, multiplication, or division). The number of boxes used in this game can be varied with 2, 4, 6, or 8 boxes. 10 playing cards are needed for these games (2-9 and Ace =1 and Jack = 0). Let one player pick one of the 10 cards while the cards are face down. After the card is selected each player needs to decide what box to place the number in. Once a number is placed in a box it cannot be changed. The selected card is then removed from the pile and another card is selected. Play continues until all boxes are filled with a number. Players then solve the problem and the winner is determined. The winner is based on what was selected at the start of the game. Either trying for the smallest number, the largest number, or being closest to a target number.
A focus on vocabulary in math is vital for students to be able to discuss mathematics with precision. In the meme above it is conveyed that the student thinks the word "cross" refers to a crossover dribble in basketball. There are often words that have an everyday meaning and then also an academic meaning. In math, there is important vocabulary that has a mathematical meaning that students need to develop understanding with. The table below has some examples.
Consistency in teaching math within a grade level and across grade levels is important. Consistent usage of vocabulary can help reduce cognitive load on students and help them make connections better. It also assists students in understanding mathematical questions that are posed and to discuss math clearly with other students.
Word walls are a great way to help students learn vocabulary. There are a variety of games that can be used with word walls. One example is below.
Bluff word wall game
Split the class into 2 teams. Ask a question of one team about the words on the wall. Students who would like to be considered to answer stand up. They can either KNOW the question or BLUFF. The teacher or someone from the other team chooses one student to answer. If they get it correct, the team gets points for everyone standing up (even if they were bluffing!). If the student does not get it correct then the team gets no points. The question moves to the next team OR you can tell the answer and give the next team a new question.
For most students, games are an everyday part of their lives. For example, between June 2018 and March 2019, 125 million new players registered to play the online video game Fortnite, putting the total number of players at nearly 250 million. A large portion of these players are school-aged kids. In addition, universities now offer full-ride scholarships for students good at video games with esports scholarships. Kids can play games for hours with little to no breaks and the time can go by quickly because they are so engaged. The question becomes why can’t learning in school foster this kind of engagement?
When students play games they persevere in problem solving, try new approaches, use all of their resources, and continue to develop their strategies when encountering setbacks of failures. These are all characteristics that will help students be successful in life and in the math classroom.
In the math classroom game-based learning should move beyond just incorporating drill and practice though. An example of a typical game includes students solving traditional, non-contextual practice problems in order to get more speed for a race car and attempts to take advantage of students’ interest in video games. However, in this type of game, students only receive feedback if the answers are correct or incorrect and do not receive support for improving their conceptual understanding. These types of games also emphasize that mathematics is about speed and focus on the memorization of ideas
The following are three productive examples of game-based learning.
An escape room is a game in which teams solve multiple puzzles using clues, hints, and strategy in order to figure out how to escape from a locked room. Setting up a mathematical escape room can be a great way for students to apply and practice mathematics they have learned. When students are engaged in a task with high intrinsic motivation it creates an ideal environment for learning. Students also learn valuable teamwork and communication skills as they learn from each other. See if you can figure out the following from one of the escape rooms I have developed.
Programming and robotics
There is a movement for more schools to require computer science courses for students. Current technologies for programming are becoming more user-friendly, which can make it more likely for mathematics teachers to feel comfortable integrating programming. Exposing students to this work is essential as applications software developers is the largest STEM occupation. The following images are some example games: golf and racing.
Desmos has a collection of activities with some of the activities being game-based.
One example game is mini golf. Students plot points in the coordinate plane in order to make a slide that will get marbles to land in a hole.
Stohlmann, M. (In press). Escape room math: Luna’s lines. Mathematics Teacher: Learning and Teaching PK-12.
Stohlmann, M. (2019). Integrated steM education through open-ended game based learning. Journal of Mathematics Education, 12(1), 16-30.
Stohlmann, M., & Kim, Y.R. (In review). Game-based learning: Robotics and escape rooms. The Australian Mathematics Education Journal.
Watch the following video and see how the math anxiety was overcome.
Nate, the fifth grader, dreaded these days at school. It was another math class where a chapter test was being handed back. Nate did not even want to look at his score as Mrs. Miller handed him his test. There at the top of his paper was a big fat grade of "F". Read the book to find out what Nate learns during an engineering design project that helps him to do well in mathematics.
In my book, The Natural?, Nate holds the assumption that some students are just naturally good at math and others cannot do it. This is a message that is too prevalent in schools and is something that students latch on to. This message is too often reinforced by parents who may say, "I never got math." It is vital to emphasize that all students are capable of doing well in mathematics. The table below has phrases to avoid and what to state instead. We want students to hold a growth mindset, that they can learn new things and improve, as they approach mathematics.
Positive math messages
Math like life takes effort
Math is not a spectator sport!
Have a growth mindset—you can always learn new things and improve.
Persevere- Find your grit.
Math is important for understanding life and graduating from college.
Stay positive and optimistic. Everyone can do well in mathematics!
Deliberate practice is purposeful and systematic. In mathematics it is important that students learn from their mistakes. They need to realize what they know and what they need to work on or understand better. The quality of the time spent practicing is more important than the quantity of time.
4 Components of deliberate practice
Stohlmann, M. (2019). The natural? Seattle, WA: KDP.
Duckworth, A. (2016). Grit. New York: Scribner.