Lesson Title: The Galton Board and Science
Grade Level:
9th – 12th grade
Content Objectives: Students will be able to build models of atoms that create foam board.
Language Objectives: Students will be able to identify the different types of atoms in the models they
built.
Vocabulary:
- Molecule
- Polystryrene
- Solid
- Liquid
- Gas
Next Generation State Standard(s):
HS-LS1-6
HS-LS1-1
Materials:
- Galton Board: Math in Motion
- Molecular Model Kits
- If you don’t have a kit, use construction paper
o White construction paper (hydrogen)
o Black construction paper (carbon)
o Glue
o Scissors
Preparation:
- Printed copies of the Galton Board rubric
- Gather materials
- Load Youtube videos:
o https://www.youtube.com/watch?v=6YDHBFVIvIs&t=26s
o https://www.youtube.com/watch?v=TYya2uPFvzo
- Molecular Model Student Kit: https://www.amazon.com/Duluth-Labs-Organic-Chemistry-
Molecular/dp/B016ZAWAIS
Lesson:
- Introduce the Galton Board by passing out the Galton Board: Math in Motion product.
- Provide students with time to use this product and be ready to discuss the following questions:
o What are you observing when you flip the Galton Board and the beads fall into the bin?
1
Answer Examples: There is an equal chance of the bead bouncing left or right; the
beads tend to fill towards the middle.
o What mathematical topics are connected to the Galton Board?
Answer Examples: Bell curve (normal distribution); probability; Pascal’s Triangle;
Binomial coefficients; Powers of 2; Powers of 11; Square Numbers; Prime
Numbers; Fibonacci Sequence; Sum of Squares. For this unit focus will be: Normal
distribution; Fibonacci sequence; Triangular numbers; Binomial coefficients.
o What careers connect to these mathematical topics?
Market Research Analyst; Meteorologist; Statistician; Operations Research
Analyst; Financial Analyst; Ecological/Environmental Statistics; Educational Testing
and Measurement; Environmental Health Sciences; Government Service.
https://www.ma.utexas.edu/users/mks/statund/careers.html will provide you
with further information on each of these careers.
- Watch Galton Board YouTube video to reinforce ideas just discussed:
o https://www.youtube.com/watch?v=6YDHBFVIvIs&t=26s
- Purpose of learning the basics of the Galton Board is for students to create their own board and
develop their understanding further of Law of Numbers.
- If you’d like, pass out the rubric for the Galton Board that will provide a structured vision for the
board. If you would like to challenge students, have students develop expectations.
- Students will develop their Galton Board using Foam board. Watch YouTube video to show the
science behind the foam board.
- Youtube video: How is expanded polystryrene produced?
https://www.youtube.com/watch?v=TYya2uPFvzo
- This process helps to develop your foam boards that are covered with a white paper and used for
posters or presentations.
- Pass out molecular models for students to create a 3D molecule. If you don’t have a set you can
order them or cut circles in different colors (2D model).
o Molecular Model Student Kit: https://www.amazon.com/Duluth-Labs-Organic-Chemistry-
Molecular/dp/B016ZAWAIS
o Colored Paper:
Carbon – black paper
Hydrogen – white paper
- Molecule Structures:
o Styrene (C8H8)
o Styrene is created from natural gas formed by a reaction between Ethylene (C2H4) and
Benzene (C6H6).
- Provide students with these molecule structures and have them try to build the molecules on their
own and discuss how they believe the molecule is structured.
2
Styrene Ethylene
: Hydrogen : Hydrogen
: Carbon : Carbon
Benzene
: Hydrogen
: Carbon
- Groups can present their work to strengthen their soft skills.
Check for Understanding:
3D molecules of chemicals
3
Lesson Title: Galton Board and Mathematics
Grade Level:
9th -12th Grade
Content Objectives: Students will be able to use Pascal’s Triangle to identify mathematical concepts.
Language Objectives: Students will be able to summarize the mathematical concepts they identified.
Vocabulary:
- Pascal’s Triangle
- Binomial Coefficients
- Fibonacci’s numbers
Common Core State Standard(s):
- CCSS.MATH.CONTENT.HSA.APR.C.5 - CCSS.MATH.CONTENT.HSA.SSE.A.1.A
- CCSS.MATH.CONTENT.HSA.SSE.A.2 - CCSS.MATH.CONTENT.HSA.SSE.A.1.B
Materials:
- Pascal’s Triangle half sheet
Preparation:
- Print and cut the Pascal’s Triangle half sheet
- Load YouTube video
o https://www.youtube.com/watch?v=XMriWTvPXHI&t=3s
Part 1: Pascal’s Triangle Lesson
- Discuss with students what they already know about Pascal’s Triangle and what they want to
know about Pascal’s Triangle.
- Next, introduce Pascal’s Triangle by watching: The mathematical secrets of Pascal’s Triangle –
Wajdi Mohamed Ratemi.
- https://www.youtube.com/watch?v=XMriWTvPXHI&t=3s
- The Pascal Triangle will help students when building their Galton Board
- Pass out interactive notebook, half sheet of Pascal’s Triangle. Students fill out the triangle with the
required numbers. See below. Do the students remember how they can fill out the triangle?
- Teacher answers are below.
Start with 1 at the top. Remember 1+0= 1, hence why there is a 1 below and with
the box next to it.
Add the pair of two boxes to get the answer for the box below. Example 1 + 1 =
2.
4
- Have students turn to a partner and discuss what patterns or mathematical concepts they can find
with the triangle. Discuss this as a class when complete.
- Students will need different colors to locate mathematical concepts on the pascal triangle.
o See figure – Fibonacci’s numbers
o Red – Triangular Numbers
o Binomial Coefficients: For this, students will write their answers in line with the row.
o Teacher answers are below.
(x + y)0=1
1
(x + y)1= x + y
1
(x + y)2= x2 + 2xy + y2
1+1
(x + y)3= x3 + 3x2y + 3xy2 + y3
1+2
1+1+3 (x + y)4= x4 + 4x3y + 6x2y2 + 4xy3 + y4
1+4+3 (x + y)5= x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5
1+1+5+6 (x + y)6= x6 + 6x5y + 15x4y2 + 20x3y3 + 6xy5 + y6
Fibonacci’s Numbers Binomial Coefficients
- Students summarize their findings, once complete.
Check for Understanding:
- Pascal Triangle interactive notebook (Fibonacci’s numbers, triangular numbers, and binomial
coefficients)
5
Lesson Title: Galton Board and Technology
Grade Level:
9th -12th Grade
Content Objectives: Students will be able to predict, observe, and analyze data using the Galton Board.
Language Objectives: Students will be able to use technology to support them when analyzing data.
Vocabulary:
- Mean - Symmetry
- Bell Curve - Standard Deviation
- Total Area - Normal Distribution
Common Core State Standard(s):
CCSS.MATH.CONTENT.HSS.ID.A.4
CCSS.MATH.CONTENT.HSS.IC.B.6
CCSS.MATH.CONTENT.HSS.ID.A.2
CCSS.MATH.CONTENT.HSS.ID.A.3
RST.11-12.8
ISTE Standard(s):
2.DIGITALCITIZEN.B
5.COMPUTATIONALTHINK.B
7.GLOBALCOLLABORATOR.C
Materials:
- Galton Board: Math in Motion
- Path Predictions worksheet
- Smartphone or video camera
- Colored pencils or markers
- Bell Curve interactive notebook page
Preparation:
- Make copies of Path Predictions
- Make copies of Bell Curve interactive notebook page
- Load Youtube videos
o https://www.youtube.com/watch?v=gV1gUdFvI54
o https://www.youtube.com/watch?v=c11d3vVM5v8
- May need to locate video cameras, if students do not have Smartphones
Lesson:
6
Part 1: Path Predictions
- Pass out Galton Board: Math in Motion.
- Pass out Path Predictions sheet (can use for high school).
- Students will need their Smartphone or video camera to record and slow down a recording.
- Before students begin their observations, they will discuss predictions first.
o Do students believe they can predict the path of the beads? Why or why not?
o Is it nearly impossible to predict the beads paths? Why or why not?
- Students will then write down their predictions onto their Path Predictions sheet.
- Discuss how, in pairs, one student will record the beads, while one student flips the Galton Board
to have the beads fall through its Pascal Triangle.
- When recording, students will need to record until all beads fall and you are able to see where
they fell.
- After students are done recording, they will slow down the video camera and observe how the
beads fall. They will pick out beads to trace their path onto their Path Predictions sheet.
- Have students put their Path Predictions sheet and Galton Board aside before moving on to Part 2.
Part 2: Normal Distribution
- Pass out Bell Curve interactive notebook page
- Begin by watching Stats: What is a “Standard Normal Distribution”?
- https://www.youtube.com/watch?v=c11d3vVM5v8
- The focus is there is an equal chance for the bead to fall left or right; this creates the bell-shaped
curve.
- Students will need their Path Predictions sheet and their Galton Board again.
- This time when they flip their board, they watch the beads collected below and create a bell-
shaped curve.
- Students draw the bell-shaped curve that was created and label the parts of the curve from the
video they watched. Each row in the curve can be shaded a different color to help see each row.
o Label:
Shade the Total Area. What number is this equal to?
- Where the symmetry?
Probability
- Once students have drawn and labeled their bell-shaped curve, you can enhance this lesson with
high schoolers implementing probability.
- Youtube video: Stats: Finding Probability Using a Normal Distribution Table
- Resources:
o Probability: Part 2 (Distributions) by Michael T. Schmidt
o http://www.thephysicsmill.com/2014/04/13/probability-part-2-distributions/
o Quincunx Explained by Math is Fun
o http://www.mathsisfun.com/data/quincunx-explained.html
Check for Understanding:
- Path Predictions
7
Lesson Title: Engineering Your Own Galton Board
Grade Level:
9th – 12th Grade
Content Objectives: Students will be able to build their own Galton Board model using the knowledge
they’ve obtain throughout this unit.
Language Objectives: Students will be able to explain the mathematical concepts that the Galton Board
reinforces.
Vocabulary:
- Mean - Symmetry
- Bell Curve - Standard Deviation
- Fibonacci’s numbers - Normal Distribution
- Binomial Coefficients
Next Generation State Standards(s):
HS-ETS1-2
ETS1.C: Optimizing the Design Solution
Common Core State Standards:
CCSS.MATH.CONTENT.HSS.ID.A.4
CCSS.MATH.CONTENT.HSS.IC.B.6
CCSS.MATH.CONTENT.HSS.ID.A.2
CCSS.MATH.CONTENT.HSS.ID.A.3
RST.11-12.8
ISTE Standard(s):
7.GLOBALCOLLABORATOR.C
4.INNOVATIVEDESIGNER.C
Materials:
- Foam Board (1/2” thick)
- Push Pins (Sphere shape)
- Marbles (large and small)
- Pencil
- Ruler
- Sharpie
- X-Acto knife
- The Engineering Design Process sheet
8
Preparation:
- This lesson is an example of how students can build a Galton Board. To challenge students, allow
them to plan their Galton Board and the materials they will need. There are further links provided
that show other ways Galton Boards have been created. If you have a woodworking shop, use
wood or if you have a 3D printer, use the printer.
- Pascal’s Marble Run by Karl Sims: http://www.karlsims.com/marbles/index.html
- Galton Board Made with Legos: https://www.youtube.com/watch?v=aGiKdJ2npc4
- Physlab: http://physlab.org/class-demo/galton-board/
- Quincunx Board: http://mathematicsprojects.blogspot.com/2007/06/quincunx-board.html
Lesson:
- If students have not been introduced to the Engineering Design Process, use this Youtube video
The Engineering Design Process: A Taco Party:
https://www.youtube.com/watch?v=MAhpfFt_mWM
- Pass out The Engineering Design Process sheet.
o Discuss the steps of the process and students can share examples of each step. You can
use the Taco Party idea to help students.
o Define: How can we develop a Galton Board to analyze numbers and probability?
o Brainstorm: How can your group develop a Galton Board?
o Select: What type of project will best create a Galton Board and help the group to obtain
and analyze data?
o Identify: What materials, jobs of groups member will you need for the project?
o Create: Groups create their Galton Board.
o Test: Groups test their Galton Board. Can they obtain and analyze data? Are they able to
find the mean and create a bell curve (from the data)?
o Improve: Were students able to develop a proper Galton Board, collect, and analyze data?
If not, were in this project can it become improved?
o Share: Students present their Galton Board, their process of developing their Galton
Board, and findings to their classmates.
- Students move into their groups to begin discussions on developing their Galton Board.
- You can break down this lesson into days (example below):
o Day 1: Define, Identify, Brainstorm, and Select
Students may need a couple of days to obtain materials
Students draw their selected solution and identifying the materials needed to
create the solution.
Example: Students decide to use Foam board, push pins, and marbles to develop a
Galton Board.
o Day 2: Develop a prototype
Does the Galton Board properly move beads from the top to the bottom? Are
students able to find the total area, standard normal distribution, and symmetry?
o Day 3: Develop a prototype
In the prototype, students can also draw out Pascal’s Triangle onto their project.
Example: Students measure out equal squares (to create the Triangles) and make
a circle of were the push pin will be placed.
9
Example: Students create a box, or use a shoe box, and place their Pascal Triangle
in the box. This will collect the marbles that students drop to move through the
Galton Board.
o Day 4: Test the prototype
Example: Students test their Galton Board by placing the main board into the
collect box and drop large or small marbles one by one.
To collect data, one student will drop the marbles, one student will write down
what number column the marble dropped into.
Once data is collected, students should be able to find the mean and develop a
bell curve.
o Day 5: Improve the prototype
o Day 6: Share your solution with the class.
- When students have engineered their projects, they share with the class how they developed their
Galton Board and show the Galton Board in use.
Check for Understanding:
- Students develop a Galton Board using the Engineering Design Process
- In their presentations, students share how they used the Engineering Design Process and what
worked and didn’t work for their Galton Board.
- The Galton Board, students developed, helped them to collect and analyze data. High school
students were able to use the data to find standard normal distribution and symmetry.
10
PASCALS TRIANGLE HIGH SCHOOL LEVEL WORKSHEET
Fibonacci’s Numbers
Fill out Pascal’s Triangle. Next, color code the sequence numbers and, on the line, write the
equation to get to the sequence number.
Binomial Coefficients and Triangular Numbers
Fill out Pascal’s Triangle. Next, color the Triangular Numbers red and write down the Binomial
Coefficients onto the line.
Bell Curve
Complete the sheet when watching, Stats: What is a “Standard Normal Distribution”? (https://bit.ly/2Hw18Kl)
C
What are three key ideas, that come to mind, when seeing Normal Distribution?
1) Example drawing of a bell-shaped curve, including the
total area and symmetry locations.
2)
3)
What are two key ideas, that come to mind, when seeing Standard Normal Distribution?
1)
2)
Bell Curve
Complete the sheet when watching, Stats: What is a “Standard Normal Distribution”? (https://bit.ly/2Hw18Kl)
What are three key ideas, that come to mind, when seeing Normal Distribution?
1) Example drawing of a bell-shaped curve, including the
total area and symmetry locations.
2)
3)
What are two key ideas, that come to mind, when seeing Standard Normal Distribution?
1)
2)
NAME:_________________________
Path Predictions
Prediction of the beads movement:
Recording of the Beads
Pick out certain beads, when watching the recording, and color code their paths onto the
pascal triangle below.
Bell Curve
After flipping the Galton Board, draw the bell shaped curve that the beads created.
KEY
Path Predictions
Prediction of the beads movement:
Students predict how they believe the beads will fall when the Galton Board
is flipped. Can they predict the paths or is it impossible?
Recording of the Beads
Pick out certain beads, when watching the recording, and color code their paths onto the
pascal triangle below.
EXAMPLE
KEY
Bell Curve
After flipping the Galton Board, draw the bell shaped curve that the beads created.
EXAMPLE
