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Level 2 · Algebra, evidence, and causal maps

Ages 12 to 15 (grades 7 to 9), working mostly on their own

Level 2 icon Grades 7–9: Ages 12 to 15 (grades 7 to 9), working mostly on their own

The big question: When does a pattern in data tell us what causes what, and how would we check?

Learners write rules with variables and functions, read graphs with care, and calculate probabilities inside a group. They use natural frequencies to reverse a condition, see a comparison flip when groups are combined, and draw causal maps with chains, common causes, and colliders. They learn why setting a value is different from selecting records with that value. Finally they write a two-outcome chance as a vector and compare it with the state vectors used in quantum physics.

At a glance

Who it is forLearners aged about 12 to 15 who have finished Level 1 or are comfortable with fractions, simple probability, and reading tables. They can work through a lesson alone. A teacher, parent, or study partner can join the discussion questions and look over the project.
Lessons10 lessons of about 45 minutes, plus a 3-part final project
PaceTwo lessons a week takes about six weeks, including the final project. Each lesson splits well into two sittings: story, animation and activity first, practice and checks second.
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 activity and keep a record.

  4. Work through the guided example, then try two on your own.

  5. Discuss, then answer the three check questions. Commit to an answer before you open it.

  6. Teachers and parents: read the short guide at the end of the page.

What you will be able to do

I can...Practised in
O1I can name quantities with variables and units and write a rule as an equation.L2-01
O2I can use a function rule and tell a model’s output apart from a measured value.L2-02
O3I can read a scatter plot, describe its trend, and say what it cannot show.L2-03, project
O4I can calculate a conditional probability from a two-way table and test whether two events are independent.L2-04, project
O5I can reverse a condition with natural frequencies, taking the base rate into account.L2-05
O6I can show how a comparison can flip when groups of different sizes are combined.L2-06
O7I can read a causal map and tell observing apart from intervening.L2-07, L2-08, project
O8I can write a two-outcome chance as a vector, and state a claim with its evidence and limits.L2-09, L2-10, project

Lessons

LessonBig ideaHands-on
L2-01 · Name quantities with variablesA variable is a letter that stands for a quantity that can change, like a number of tickets or a cost in dollars. An equation states a rule that links variables. Some rules are set by people, like a price list. Others only describe a pattern in data, and a pattern is not the same as a law of cause.Three price plans on one grid
L2-02 · Functions map inputs to outputsA function is a rule that gives exactly one output for each input. When we use a function as a model of the real world, its output is a prediction, not a measurement. The gap between them is called the residual: observed minus predicted.Function machines and a coin tower
L2-03 · Read plots without inventing causesA scatter plot shows how two quantities vary together. You can describe its direction, form and strength, and spot points that do not fit. The plot alone cannot tell you what causes what, which way a cause runs, or what happens outside the range of the data.Plot it, then doubt it
L2-04 · Calculate conditional probabilityA conditional probability counts inside one group. The group you condition on becomes the whole, so its total goes on the bottom of the fraction. If knowing one thing does not change the chance of the other, the two events are independent.Doubles and big sums
L2-05 · Reverse a condition with Bayes’ ruleP(positive | condition) and P(condition | positive) are different numbers. To turn one into the other, count the people in an imagined group. When a condition is rare, most positive tests can be false alarms, even from a fairly good test.Grow a frequency tree
L2-06 · See a reversed comparisonOne option can win inside every group and still lose when the groups are added together. This happens when the options get very different mixes of easy and hard cases. Which comparison to trust depends on the causal story behind the numbers.Find the flip, then build your own
L2-07 · Read a causal graphA causal graph is a map of what we think causes what. Three small shapes appear again and again: a chain, a fork, and a collider. Looking only inside one group can hide a link in a chain or a fork, but it can create a fake link at a collider.Sort the stories, then play the admissions game
L2-08 · Observe or interveneWatching who chose something and assigning it to people answer different questions. When a common cause pushes both the choice and the result, the watched gap can be much bigger than the real effect. A coin that decides who gets what removes that push.Watch, then assign
L2-09 · Use vectors for two outcomesA probability vector (p0, p1) lists two chances that add to 1, so it sits on a line segment. A qubit state vector (a0, a1) lists two amplitudes whose squares add to 1, so it sits on a circle. Two different state vectors can give the same chances in one measurement and still be different states.Points on a line, arrows on a circle
L2-10 · Build an evidence recordA claim is only as strong as the evidence behind it. An evidence record writes down the claim, where it came from, the calculation, what was assumed, how uncertain it is, and who the data covered. It keeps what was tested apart from what was only assumed.Write your own evidence record

Final project: My evidence record

Over three sittings, the learner investigates a question with a two-way table, draws a causal map, and writes an evidence record that states what the data can and cannot show.

Pick one question, or make up your own:

Steps:

  1. Write the question, the variables, and their possible values.

  2. Collect or copy the data into a two-way table (at least 40 records).

  3. Calculate at least two conditional probabilities and say whether the events look independent.

  4. Draw a causal map with at least one other variable that could explain the pattern.

  5. Say what experiment, if any, would test the causal claim, and why it might not be possible.

  6. Write the evidence record: claim, source, assumptions, calculation, uncertainty, and scope.

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

PartGetting startedGot itCan explain it
Question and variablesI wrote a question.I named my variables and their values.I can say why my question is a statistical question.
CalculationsI made a table.I calculated conditional probabilities correctly.I can explain what each denominator is.
Causal mapI drew arrows.I drew a map with a possible common cause.I can say how the map changes what my numbers mean.
Evidence recordI stated a conclusion.I listed my assumptions and uncertainty.I can say what new data would change my mind.

Materials for the whole unit

How progress is checked

Each lesson ends with three check questions, one of which looks back at an earlier lesson. Each activity produces a record (a table, a graph, a calculation, or a causal map). The final project is an evidence record judged with a four-part rubric.

For grown-ups: how this unit is designed

Like the earlier levels, this unit was planned backward from its outcomes Jay McTighe and Grant Wiggins, n.d., and each lesson moves from recall, to a worked example, to practice, to discussion Merrill, 2002, with retrieval and spaced review built in Dunlosky et al., 2013. Each lesson has a short animation made with Manim Community Edition The Manim Community Developers, 2026, driven by the same numbers as the text. The probability ideas follow OpenStax Introductory Statistics Barbara Illowsky and Susan Dean, 2023Barbara Illowsky and Susan Dean, 2023Barbara Illowsky and Susan Dean, 2023, Bayes’ rule is taught with natural frequencies, which research shows people reason with more accurately Gigerenzer & Hoffrage, 1995, the reversal lesson draws on a classic admissions study Bickel et al., 1975, and the causal maps follow Pearl’s framework Pearl, 2009Bareinboim, 2025.

Ways to adapt it.

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

  • Learners can show what they know with a table, a graph, a diagram, a calculation, or a spoken explanation.

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

  • Optional notebooks let learners run the same calculations in Python.

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

StandardWhat it asksSource
CCSS 6.EE.6Use variables to represent numbers and write expressions for real-world problems.NGA Center and CCSSO, 2010
CCSS 8.F.1A function is a rule that assigns to each input exactly one output.NGA Center and CCSSO, 2010
CCSS 8.SP.1, 8.SP.4Construct and interpret scatter plots and two-way tables.NGA Center and CCSSO, 2010
CCSS 7.SP.8Find probabilities of compound events with tables, tree diagrams, and simulation.NGA Center and CCSSO, 2010
CCSS S-CP.3Conditional probability of A given B as P(A and B)/P(B), and independence.NGA Center and CCSSO, 2010
CCSS S-ID.9Distinguish between correlation and causation.NGA Center and CCSSO, 2010
NGSS 6-8 practicesAnalyze and interpret data; construct explanations supported by evidence.NGSS Lead States, 2013

Watch & listen

Bayes theorem, the geometry of changing beliefs. 3Blue1Brown · 15 min · 2019. 3Blue1Brown presents Bayes’ theorem as the geometry of changing beliefs.

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). 3.4 Contingency Tables. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-4-contingency-tables
  6. Barbara Illowsky and Susan Dean. (2023). 3.2 Independent and Mutually Exclusive Events. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-2-independent-and-mutually-exclusive-events
  7. Barbara Illowsky and Susan Dean. (2023). 3.5 Tree and Venn Diagrams. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-5-tree-and-venn-diagrams
  8. Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684–704. 10.1037/0033-295X.102.4.684
  9. Bickel, P. J., Hammel, E. A., & O’Connell, J. W. (1975). Sex Bias in Graduate Admissions: Data from Berkeley. Science, 187(4175), 398–404. 10.1126/science.187.4175.398
  10. Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3(none), 96–146. 10.1214/09-ss057
  11. Bareinboim, E. (2025). Causal Artificial Intelligence: A Roadmap for Building Causally Intelligent Systems. https://causalai-book.net/
  12. 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
  13. 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