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Level 1 · Chance, waves, and fair tests

Ages 9 to 12 (grades 4 to 6), with a grown-up or teacher nearby

Level 1 icon Grades 4–6: Ages 9 to 12 (grades 4 to 6), with a grown-up or teacher nearby

The big question: How do we describe chance, test fairly, and tell what waves can do that coins cannot?

Learners turn counts into fractions and probabilities, compare what a model predicts with what really happens, and use random assignment to make a fair comparison. Then they meet waves, which can add up or cancel. That idea, adding first and then squaring, is the first step toward quantum physics.

At a glance

Who it is forLearners aged about 9 to 12 who have finished Level 0 or can count, compare, and read simple tables. They can work through most of a lesson alone. A grown-up or teacher checks the record sheet and joins the talk questions.
Lessons10 lessons of about 40 minutes, plus a 3-part final project
PaceTwo lessons a week takes about six weeks, including the final project. Any lesson can be split into two sittings: story, animation and activity one day, practice and checks the next.
Total sittings13

How a lesson works

  1. Warm up: answer the look-back question from memory.

  2. Read the story and watch the animation.

  3. Do the hands-on activity and keep a record.

  4. Practice together, then try two on your own.

  5. Talk about it, then answer the three check questions. Say or write your answer before you open it.

  6. Grown-ups: read the short guide at the end of the page.

What you will be able to do

I can...Practised in
O1I can write a fraction from a count and use it to describe part of a group.L1-01
O2I can build a probability model and find the probability of an event and of its opposite.L1-02
O3I can tell a probability apart from a frequency, and explain why more trials help.L1-03
O4I can find a probability inside a group by using the right total.L1-04
O5I can explain how random assignment makes a comparison fair.L1-05, project
O6I can draw a common-cause diagram for two things that go up and down together.L1-06
O7I can describe a wave and explain how two waves can add up or cancel.L1-07, L1-08
O8I can explain why not knowing a hidden result is different from the two-path experiment with light, and write a model that a test could prove wrong.L1-09, L1-10, project

Lessons

LessonBig ideaHands-on
L1-01 · Fractions describe part of a wholeA fraction compares a part with the whole. The top number counts the part. The bottom number counts all the equal parts in the whole. Different fractions can name the same amount.Fold, shade, and group
L1-02 · Build a probability modelA probability model lists everything that could happen and how likely each one is. When every outcome is equally likely, the probability of something happening is the number of outcomes that count, out of all the outcomes. A probability goes from 0, which means impossible, to 1, which means certain.Three probability models
L1-03 · Repeat and recordA probability is what a model predicts. A frequency is what you actually count. With more trials, the frequency usually gets closer to the probability, but it rarely lands on it exactly.Coin flip record
L1-04 · Count within a conditionWhen a question says “of the squares” or “among the kids with glasses”, it picks out a smaller group. That group becomes the whole. The bottom number of your fraction is the size of that group, not the size of everything.Shape sort and two-way table
L1-05 · Assign a fair comparisonTo test two things fairly, let chance decide who gets which. A coin does not care who is strong or weak. So hidden differences between the groups balance out on average, but not perfectly every time.Coin groups or picked groups?
L1-06 · Look for a common causeTwo things can go up and down together even when neither one causes the other. Often a third thing causes both. Looking only at the pattern cannot tell you which way the causes run.Find the hidden cause
L1-07 · Track a waveA wave is a pattern of ups and downs that travels. Its amplitude is how high a crest rises above the middle line. Its wavelength is the distance from one crest to the next. The rope itself only moves up and down.Rope waves
L1-08 · Combine waves, then squareWhen two waves meet, their heights add at every point. In step, they make a bigger wave. Half a cycle apart and the same size, they cancel. For light, brightness goes with the amplitude times itself, so the order matters: add first, then square.Add two waves on paper
L1-09 · Unknown is not the same as quantumIn the two-path experiment with light, some spots stay dark because the amplitudes from the two paths cancel. Ordinary chances can only add up, but amplitudes can cancel. When a coin is hidden under a cup, you do not know how it landed. Not knowing which way a coin landed is different: the coin already landed one way.Hidden coins and paper arrows
L1-10 · Keep a model testableA good model makes a prediction that a test could prove wrong. Before you test, write down what result would make you doubt the model. Then test it and write down what really happened.My model card for two dice

Final project: My chance and cause investigation

Over three sittings, the learner runs a small experiment with random assignment, compares a model with real counts, and writes a model card.

Pick one question, or make up your own:

Steps:

  1. Pick a question. Write a prediction and the model behind it.

  2. Plan a fair test. Use a coin or a die to assign which choice each trial gets.

  3. Run at least 20 trials. Record every result in a table.

  4. Turn your counts into fractions, and compare them with your model.

  5. Make a graph of your results.

  6. Write a model card: question, assumptions, prediction, what would prove it wrong, and what happened.

How it is checked. Each part has three steps up. Celebrate every step.

PartGetting startedGot itCan explain it
My question and modelI wrote a question.I wrote a question and a prediction from a model.I can say what my model assumes.
My fair testI compared two choices.I used a coin or die to assign the choices.I can say why random assignment makes it fair.
My recordI wrote some results.I recorded at least 20 trials and turned them into fractions.I can say why my fractions are not exactly what the model predicted.
My model cardI said what happened.I compared my results with my prediction.I named a result that would have proved my model wrong.

Materials for the whole unit

How progress is checked

Progress shows up three ways. Each lesson ends with three check questions. From Lesson 2 on, one of them looks back at an earlier lesson. Each activity produces a record (a table, a graph, or a diagram) that a grown-up can look at. The final project uses a four-part checklist.

For grown-ups: how this unit is designed

Like Level 0, this unit was planned backward from its outcomes Jay McTighe and Grant Wiggins, n.d., and each lesson moves from recall, to an example, to doing, to talking it through Merrill, 2002, with built-in retrieval and spaced review Dunlosky et al., 2013. New in Level 1, each lesson has a short animation made with Manim Community Edition, an open-source animation library forked from the tool Grant Sanderson created for his 3Blue1Brown videos The Manim Community Developers, 2026. The animations use the course’s own tested simulation code, so the numbers on screen match the text. The probability ideas follow OpenStax Introductory Statistics Barbara Illowsky and Susan Dean, 2023Barbara Illowsky and Susan Dean, 2023; the wave ideas follow OpenStax Physics Paul Peter Urone and Roger Hinrichs, 2020William Moebs, Samuel J. Ling, and Jeff Sanny, 2016.

Ways to adapt it.

  • Each story comes with an animation and a written description of what it shows.

  • Learners can show what they know in a table, a graph, a drawing, or by explaining aloud.

  • Each lesson has a way to make it easier and a way to make it harder.

  • Activities use cheap household things, and the animations play without sound.

Connections to school standards. These lessons connect to, but do not replace, the standards below.

StandardWhat it asksSource
NGSS 3-5 practicesPlan and conduct fair tests in which variables are controlled and the number of trials considered.NGSS Lead States, 2013
NGSS 4-PS4-1Develop a model of waves to describe patterns in terms of amplitude and wavelength.NGSS Lead States, 2013
CCSS 3.NF.1Understand a fraction 1/b as one part of a whole cut into b equal parts.NGA Center and CCSSO, 2010
CCSS 4.NF.1Explain why two fractions are equivalent, using visual fraction models.NGA Center and CCSSO, 2010
CCSS 6.SP.1Recognize a statistical question as one that expects variability in the data.NGA Center and CCSSO, 2010
CCSS 7.SP.5 to 7.SP.7Probability as a number from 0 to 1; approximate it from data; build and compare probability models.NGA Center and CCSSO, 2010

Watch & listen

Introduction to fractions. Khan Academy · 5 min · 2013. Khan Academy introduction to fractions.

More for every level on the Watch and listen page.

References
  1. Jay McTighe and Grant Wiggins. (n.d.). Improve Curriculum, Assessment, and Instruction Using the Understanding by Design Framework. ASCD white paper. https://jaymctighe.com/wp-content/uploads/2024/10/UbD-White-Paper.pdf
  2. Merrill, M. D. (2002). First principles of instruction. Educational Technology Research and Development, 50(3), 43–59. 10.1007/BF02505024
  3. Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving Students’ Learning With Effective Learning Techniques. Psychological Science in the Public Interest, 14(1), 4–58. 10.1177/1529100612453266
  4. The Manim Community Developers. (2026). Manim Community. GitHub repository README. https://raw.githubusercontent.com/ManimCommunity/manim/main/README.md
  5. Barbara Illowsky and Susan Dean. (2023). 3.1 Terminology. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-1-terminology
  6. Barbara Illowsky and Susan Dean. (2023). 1.4 Experimental Design and Ethics. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/1-4-experimental-design-and-ethics
  7. Paul Peter Urone and Roger Hinrichs. (2020). 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period. OpenStax, Physics. https://openstax.org/books/physics/pages/13-2-wave-properties-speed-amplitude-frequency-and-period
  8. William Moebs, Samuel J. Ling, and Jeff Sanny. (2016). 16.5 Interference of Waves. OpenStax, University Physics Volume 1. https://openstax.org/books/university-physics-volume-1/pages/16-5-interference-of-waves
  9. NGSS Lead States. (2013). Appendix F: Science and Engineering Practices in the NGSS. Next Generation Science Standards. https://www.nextgenscience.org/sites/default/files/Appendix%20F%20%20Science%20and%20Engineering%20Practices%20in%20the%20NGSS%20-%20FINAL%20060513.pdf
  10. NGSS Lead States. (2013). 4-PS4-1 Waves and Their Applications in Technologies for Information Transfer. Next Generation Science Standards. https://www.nextgenscience.org/pe/4-ps4-1-waves-and-their-applications-technologies-information-transfer
  11. NGA Center and CCSSO. (2010). Common Core State Standards for Mathematics. Common Core State Standards Initiative. https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf