L1-10 · Keep a model testable
A 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.
Level 1 · Grades 4–6 · about 40 minutes
Big idea. A 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.
Before this lesson. L1-03, L1-05, L1-09
Grown-ups. The guide for adults is at the end of this page.
What you will learn¶
I can find the chance that two dice add to 7 by listing all 36 pairs.
I can write a model card with a prediction and a result that would make me doubt the model.
I can compare my real counts with my prediction without changing the rule afterward.
Story: Testing the dice¶
Maya and Leo want to know how often two dice add up to 7. They list every way the dice can land. The first die can show 1 to 6, and so can the second. That makes 36 pairs, and with fair dice each pair is equally likely.
Six of the pairs add to 7. They are 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, and 6 and 1. So their model says the chance of a 7 is 6 out of 36, which is one sixth. In 36 rolls, they predict about 6 sevens.
Before rolling, they write a model card. “What would make us doubt the dice?” asks Leo. They agree on a rule: 0 or 1 sevens, or 12 or more. With fair dice, a result that far off happens only about 2 times in 100.
Then they roll 36 times and tally every sum. They get 3 sevens in 36 rolls. That is 3 out of 36, fewer than 6. Leo groans. “Is the model wrong?”
Maya points to the card. “We said we would doubt it at 1 or fewer. Three is lower than we guessed, but it is not past our line. We have no reason to doubt the dice yet.” They write what happened on the card, without changing the rule.
All 36 ways two dice can land, a run of 36 rolls, and the model card.
What the animation shows. A 6 by 6 grid shows every way two dice can land, and the six squares that add to 7 light up. The screen shows 6 out of 36, which equals one sixth, and a prediction of about 6 sevens. A bar chart then fills up during 36 rolls and ends at 3 sevens in 36 rolls. Last, a model card lists the question, assumption, prediction, doubt rule of 0 or 1 sevens, or 12 or more, and what happened.
Hands-on: My model card for two dice¶
About 20 minutes.
You need:
Two dice
A cup to shake them in (optional)
Paper and a pencil
A ruler for drawing a table
Steps:
Draw a 6 by 6 grid. Write 1 to 6 across the top for the first die and down the side for the second die.
Fill each square with the sum of its two numbers. Circle every square that shows 7. How many did you circle?
Write your model card. Question: how often do two dice add to 7? Assumption: the dice are fair.
Add your prediction to the card: about 6 sevens in 36 rolls.
Add your doubt rule before you roll: 0 or 1 sevens, or 12 or more.
Write the sums 2 to 12 down the side of a new page. Roll both dice 36 times and make a tally mark next to each sum.
Count your 7s. Write the result on the card under what happened.
Compare with your doubt rule. Do you have a reason to doubt the dice? Do not change the rule now.
Practice together¶
Using the 6 by 6 grid, how many pairs add up to 2? What is the chance of rolling a sum of 2?
Check your answer
Only one pair adds to 2: a 1 and a 1. So the chance is 1 out of 36. A sum of 7 is six times as likely.
Try it yourself¶
1. Sam says, “My lucky model says something will happen.” Sam will not say what or when. Is this a good model? Why or why not?
Check your answer
No. No test could ever prove it wrong, because almost anything counts as something happening. A good model makes a clear prediction that could turn out false.
2. Leo rolls two dice 36 times and gets 0 sevens. Using the rule on the card, what should he write under what happened?
Check your answer
Zero is inside the doubt rule of 0 or 1 sevens. He should write that he got 0 sevens and that he now doubts the model. Next he could check the dice and roll again.
Talk about it¶
Why should you write your doubt rule before you roll, not after?
Maya and Leo got 3 sevens, not 6. Why is that not enough to call the dice unfair?
If four friends each roll two dice 36 times, will they all get the same number of sevens? Why not?
Check yourself¶
1. Which of these is a testable prediction?
A. Dice are fun to roll.
B. In 36 rolls of two fair dice, we will get about 6 sevens.
C. Something interesting might happen someday.
Show the answer
B. In 36 rolls of two fair dice, we will get about 6 sevens.
This prediction gives a number that a test can check. If the count were very far from 6, the test would make us doubt the model.
2. Your doubt rule is 0 or 1 sevens, or 12 or more. You roll 36 times and get 9 sevens. What should you do?
A. Change the rule so 9 counts as too many
B. Throw out the rolls and start over
C. Write down 9 sevens and say there is no reason to doubt the dice yet
Show the answer
C. Write down 9 sevens and say there is no reason to doubt the dice yet
Nine is not past either line of your rule, so you keep the model for now. Changing the rule after you see the result would not be a fair test.
3. Remember Lesson 5. Why do scientists flip a coin to choose who goes in which group? (Review of L1-05)
A. So the groups always have exactly equal averages
B. So no person can steer the strong ones into one group
C. So fewer people are needed
Show the answer
B. So no person can steer the strong ones into one group
A coin does not take sides, so hidden differences balance out on average. It does not promise exactly equal groups every time.
Words to know¶
- model card
- A short record of a model: the question, what you assume, what you predict, what would make you doubt it, and what happened.
- prediction
- What you expect to happen before you test, based on your model.
- assumption
- Something you take to be true so your model can work, like the dice being fair.
For grown-ups: about this lesson
Why it matters. The Common Core asks students to build a probability model, here 36 equally likely ordered pairs, and to compare its predictions with observed frequencies, explaining possible reasons for a mismatch (7.SP.7) NGA Center and CCSSO, 2010. OpenStax defines equally likely outcomes and relative frequency, and states the law of large numbers: the relative frequency tends toward the theoretical probability as the number of trials grows, so a short run of 36 rolls can still land some way off Barbara Illowsky and Susan Dean, 2023. The NGSS practices ask children to make predictions, plan investigations, and use evidence to judge an explanation NGSS Lead States, 2013. The doubt rule of 0 or 1 sevens, or 12 or more, is a two-sided cutoff: with fair dice, the binomial chance of a count that far off in 36 rolls is about 0.023. The roll sequence comes from the course’s seeded random number generator (seed 1), so the 3 sevens in the story, the video and this guide match.
Watch for these mix-ups.
If the count is not exactly 6, the model is wrong. (Counts vary; 3 sevens in 36 rolls is not unusual for fair dice.)
A test that agrees with the model proves it is true. (It only means the test gave no reason to doubt it; a later test still could.)
It is fine to choose the doubt rule after seeing the results. (Choosing it afterward lets you excuse any result; set it in advance.)
A sum of 7 has one way to happen, 3 and 4. (Order matters on two dice: 3 then 4 and 4 then 3 are different pairs, so there are 6 ways.)
To make it easier. Use 18 rolls and predict about 3 sevens. Skip the doubt rule numbers and simply ask: would 0 sevens surprise you?
To make it harder. Pool the 36-roll tallies from several people. Plot how many sevens each person got, and see that most counts land between 3 and 9. Then write a model card for the sum of 2, which has a chance of 1 out of 36.
Limits. Passing a test does not prove a model is true, and one unusual result does not prove it false. About 2 times in 100, fair dice will break this doubt rule by chance alone.
On the computer. Grown-ups can see where this idea leads later in Lab: Chance and repeated trials, written for older learners.
Sources for this guide. NGA Center and CCSSO, 2010Barbara Illowsky and Susan Dean, 2023NGSS Lead States, 2013
Next: L2-01 · Name quantities with variables
- 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
- Barbara Illowsky and Susan Dean. (2023). 3.1 Terminology. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-1-terminology
- 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