MP3: Construct Viable Arguments and Critique the Reasoning of Others
In plain English: MP3 is the Common Core Standard for Mathematical Practice about arguments. Students build a logical chain from definitions, assumptions and known results, test conjectures with cases and counterexamples, reason from data in context, state where an argument applies, and find and explain the flaw in a weak argument. It applies in every grade 6-12 math course.
Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Common Core State Standards for Mathematics · Domain: Standards for Mathematical Practice (MP) · Practice group: Reasoning and explaining Also written as CCSS.MATH.PRACTICE.MP3 · Official standard
MP3 asks students to build arguments and to judge the arguments of others. A viable argument is one that could convince a careful listener: it uses stated assumptions, definitions and previously established results, and links them in a logical chain. Students make conjectures (claims they think are true but have not yet shown), analyze cases, use counterexamples (single examples that show a claim is false), reason from data in its context, and learn to state the domain to which an argument applies. When they read someone else's argument, they decide whether it makes sense, find and explain any flaw, and ask useful questions.
The tasks in this lesson come from several grade 6-12 domains: number properties, equations and inequalities, geometry and statistics. Each one is labeled with a typical course, so a teacher can keep the tasks that fit the class.
Learning Objectives
By the end of this lesson, students will be able to:
Write an argument that uses definitions or known results and a logical chain of statements
Prove a claim by breaking it into cases, and disprove a claim with a counterexample
State the domain to which a claim or argument applies
Make and critique plausible arguments from data, taking into account where the data came from
Find and explain the flaw in an argument, compare two arguments, and ask useful questions about them
Prior Knowledge Required
Students should already be comfortable with:
Even and odd numbers and writing them as 2k and 2k + 1
Random samples and inferences about a population 7.SP.A.2
The Pythagorean Theorem and its converse 8.G.B.6
Angles formed by parallel lines and a transversal 8.G.A.5
Write three statements on the board. Students decide whether each one is always, sometimes or never true and write one reason for each.
Warm-Up Prompt
"(1) The sum of two odd numbers is even. (2) A number squared is greater than the number. (3) The product of two negative numbers is negative. Always, sometimes or never? Be ready to convince someone who disagrees."
Take votes, then ask for arguments. For (1), push past examples to a reason: (2a + 1) + (2b + 1) = 2(a + b + 1). For (2), a student who tested only 3 and 10 may say "always"; ask for a number between 0 and 1. With 0.5, 0.5² = 0.25 is less than 0.5, so the statement is only sometimes true. It is true for numbers greater than 1 and for negative numbers. For (3), the answer is never. Name the moves: a conjecture, a counterexample, a domain.
Direct Instruction20 minutes
Use Diagram 1 to name the parts of an argument. Then model five short arguments. Ask students to label each line: definition, assumption, known result or logical step.
State the claim clearly, including which numbers or shapes it is about.
Name what you start from: definitions, assumptions and results already established.
Build the chain: each statement follows from earlier ones. Break into cases when one chain cannot cover everything.
Test the claim: look for a counterexample. One is enough to disprove an "always" claim.
State the domain: say for which numbers or shapes the argument works.
Definitions and a logical chain (Grade 7-8)
Claim: the sum of any three consecutive integers is divisible by 3.
Equation: Let the integers be n, n + 1 and n + 2. Their sum is 3n + 3 = 3(n + 1). Since n + 1 is an integer, the sum is 3 times an integer, so it is divisible by 3 (Diagram 1).
A counterexample and a domain (Algebra I)
Claim: if a² > b², then a > b.
Equation: Counterexample: a = -5, b = 2 gives 25 > 4, but -5 < 2. The claim is true when a and b are both 0 or more, because squaring keeps the order of such numbers.
Breaking into cases (Algebra I)
Show that n² + n is even for every integer n.
Equation: Case 1, n even: n = 2k gives 4k² + 2k = 2(2k² + k). Case 2, n odd: n = 2k + 1 gives 4k² + 6k + 2 = 2(2k² + 3k + 1). Both cases are even, and every integer is even or odd.
Reasoning from data in context (Grade 6-7 statistics)
On five Fridays, the cafeteria sold 112, 98, 120, 105 and 115 pizza slices. A student argues: "The mean is 110, so order 110 slices every Friday."
Equation: The mean is 550 ÷ 5 = 110, but sales passed 110 on 3 of the 5 Fridays, so the cafeteria would run out on those days. A plausible argument asks what the order is for: if running out is worse than leftovers, order closer to 120.
Comparing two arguments (Grade 8 geometry)
Why do the angles of a triangle add to 180°? Argument A: "I measured three triangles and got 179°, 181° and 180°." Argument B: the parallel-line proof in Diagram 2.
Equation: Argument A gives evidence for some triangles, with measuring error. Argument B uses known results about parallel lines and works for every triangle, so it is the more effective argument.
After Example 2, point out that a counterexample does not end the conversation: it often shows where a claim does hold. After Example 4, ask: "What else would you want to know about these Fridays?" (Was one of them a holiday? Do sales change with the menu?) That is reasoning about the context the data came from.
Guided Practice15 minutes
Project three student arguments, one at a time. Pairs decide whether each is correct or flawed, explain any flaw, and write one useful question for the author.
"3/4 is greater than 2/3, and 4 is greater than 3, so a fraction with a bigger denominator is bigger." (Flawed: 1/2 is greater than 1/3 although 3 is the bigger denominator. Useful question: "Does your rule work when the numerators are the same?")
"x² = 25, so x = 5." (Flawed: x = -5 also works, since (-5)² = 25. Useful question: "Is there another number whose square is 25?")
"The diagonals of every rectangle are equal. A square is a rectangle. So the diagonals of a square are equal." (Correct: it uses a definition, that a square is a rectangle with four equal sides, and a known result about rectangles.)
Ask the class to respond to the author of argument 1 in a sentence that is respectful and specific, for example "Your rule worked for your example, but here is a case where it does not."
Independent Practice15 minutes
Students work alone on the domain of an argument. For each claim, they find a counterexample and then state a domain where the claim is true.
"Dividing a number by another number makes it smaller." (Counterexample: 6 ÷ 0.5 = 12. True for a positive number divided by a number greater than 1.)
"x²/x = x." (Counterexample: x = 0, where x²/x is undefined. True for every x ≠ 0.)
"A shape with four equal sides has four equal angles." (Counterexample: a rhombus that is not a square. True for rhombuses with one right angle, which are squares.)
Students compare with a partner and decide whose domains are the largest correct ones.
Closure5-10 minutes
Exit ticket:
Give a counterexample to "Every multiple of 4 is a multiple of 8."
A classmate says, "I checked five examples, so my rule is always true." Write a useful question you would ask.
What is the difference between evidence and a proof?
Differentiation Strategies
For Struggling Students
Give sentence frames: "I claim that ___ because ___. This is true for ___." and "I disagree because ___. For example, ___."
Let students test claims with a table of values before writing a general argument
Use pictures for parity arguments: an odd number as pairs of dots with one dot left over
For Advanced Students
Ask for a proof by cases that the square of any integer leaves a remainder of 0 or 1 when divided by 3
Ask students to write a claim that is true for every case they test but false in general, and to give the counterexample
Ask students to critique a statistical claim from a news article, focusing on how the data were collected
Assessment Guidance
What to Look For
Look for arguments that name what they rely on ("by the definition of odd...", "since the lines are parallel...") and that cover every case, not just the cases tested. Strong students know that one counterexample disproves an "always" claim but that many examples do not prove one, and they state a domain when a claim is only sometimes true. In critiques, listen for a specific flaw and a respectful question. Watch for students who treat agreement among a few examples as a proof, or who reject an argument without saying what is wrong with it.
02
Classroom Activities
3 Activities
1
Always, Sometimes or Never
20 minGroups of 3-4
Groups sort 8 statement cards into Always, Sometimes and Never. For every "Sometimes" card they give one example and one counterexample and state where the statement is true; for every "Always" or "Never" card they write a general argument.
The 8 Cards (with answers for the teacher)
Card 1: The product of an even number and any integer is even. (Always: 2k × m = 2(km).)
Card 2: A rectangle with a larger perimeter has a larger area. (Sometimes: a 1 by 7 rectangle has perimeter 16 and area 7, and a 3 by 3 square has perimeter 12 and area 9.)
Card 3: The mean of a data set is one of its values. (Sometimes: the mean of 2, 4, 9 is 5.)
Card 4: Adding the same number to the top and bottom of a fraction leaves its value unchanged. (Sometimes: for a nonzero number added, true only when the fraction equals 1; 1/2 becomes 2/3 when 1 is added to both.)
Card 5: A quadrilateral with four equal sides is a square. (Sometimes: a rhombus without right angles is a counterexample.)
Card 6: The sum of two prime numbers is even. (Sometimes: 3 + 5 = 8, but 2 + 3 = 5.)
Card 7: Subtracting a negative number makes a value smaller. (Never: subtracting -4 is adding 4.)
Card 8: An angle and its supplement are both acute. (Never: two angles under 90° add to less than 180°.)
Procedure
Each student reads a card aloud and makes a conjecture before the group discusses it
The group may move a card only after someone gives a reason, not just a vote
For each Sometimes card, write the domain where it is true, if you can describe one
Discussion Questions
Seven of the cards are sometimes or never true. Which one surprised your group most?
Why is one counterexample enough to move a card out of Always, while ten examples are not enough to put it there?
For Card 6, which prime causes all the trouble, and why?
2
Critique Gallery
20 minGroups of 3, rotating
Four posters around the room each show a flawed argument. Groups rotate every 4 minutes, write the flaw on a sticky note, and add one useful question they would ask the author.
The Four Posters (with answers for the teacher)
Poster A (Algebra I): "Let a = b. Then a² = ab, so a² - b² = ab - b², so (a + b)(a - b) = b(a - b). Dividing by a - b gives a + b = b, so 2b = b and 2 = 1." (Flaw: since a = b, a - b = 0, and dividing by zero is not allowed.)
Poster B (Grade 7 percents): "An $80 pair of sneakers goes up 25%, then down 25%, so it is back to $80." (Flaw: the 25% decrease is taken from $100, so the price ends at $75.)
Poster C (Grade 7 statistics): "We asked the 30 members of the robotics club, and 80% want the school to buy more laptops. So 80% of the school agrees." (Flaw: the sample is not random, and robotics club members may care more about laptops than other students.)
Poster D (Grade 8 geometry): "These two triangles both have angles of 40°, 60° and 80°, so they are congruent." (Flaw: equal angles show the triangles are similar; one can be a scaled copy of the other.)
Procedure
At each poster, first decide whether the conclusion is false or only the reasoning
Write the flaw in one sentence and a useful question to the author on a second note
At the last poster, groups read the earlier notes and star the clearest explanation
Discussion Questions
On which poster is the conclusion obviously false? Why is it still worth finding the exact flaw?
What makes a question useful rather than just critical?
How could the robotics club change its survey to support its claim better?
Modification for Distance Learning
Post each argument on its own slide in a shared deck; groups add comments in rotating breakout rooms.
3
Which Argument Is Better?
20 minPairs, then whole class
Pairs compare two plausible arguments for the same claim, decide which is more effective and why, and then respond to a classmate who disagrees with the claim.
Argument A: 1/3 = 0.333..., and 3 × 1/3 = 1, so 3 × 0.333... = 0.999... must equal 1.
Argument B: Let x = 0.999... Then 10x = 9.999... Subtracting, 10x - x = 9, so 9x = 9 and x = 1.
Procedure
Partners each explain one argument to the other, naming the known results it uses
Together, write which argument is more convincing and what each one assumes
Respond in writing to this classmate: "0.999... is always a tiny bit less than 1, so they cannot be equal." Ask a useful question in your response
Notes for the Teacher
Argument A relies on accepting that 1/3 = 0.333... exactly. Argument B relies on subtracting two infinite decimals digit by digit. A good response to the classmate asks, "If they are different, what number lies between 0.999... and 1?" There is none: the gap 1 - 0.999... would have to be smaller than 0.1, 0.01, 0.001 and every other positive number, so the gap is 0.
Challenge Variation
Use Argument B's method to write 0.272727... as a fraction, and explain why the method needs 100x instead of 10x.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Parts of a Viable Argument
A viable argument states a claim, names the definitions, assumptions or known results it uses, and links them with a chain of statements to a conclusion that holds in every case. A single counterexample is enough to show that an "always" claim is false, and it often points to the domain where the claim does hold.
Diagram 2: A Proof That the Angles of a Triangle Add to 180°
Argument B from Example 5. Through C, draw the line parallel to AB. The angle marked a at C and angle A are alternate interior angles, so they are equal; the same holds for the two angles marked b. The three angles at C form a straight angle, 180°, so a + b + c = 180° for every triangle, not just the ones that were measured.
04
Homework Assignment
~30 min
MP3 Homework: Build, Test and Critique Arguments
Directions: Write complete arguments in sentences and symbols. Say what definitions or known results you use. When you critique an argument, name the flaw and write one useful question for its author.
Part 1: Build and Test Arguments (Problems 1-3)
Use the definitions of even and odd numbers (2j and 2k + 1 for integers j and k) to prove that the sum of an even number and an odd number is odd. Explain each step to a classmate who has only checked examples.
Prove by cases that the square of every integer leaves a remainder of 0 or 1 when divided by 4. (Hint: an integer is either 2k or 2k + 1.)
Give a counterexample to each claim, then name a domain on which the claim is true: (a) 1/x < x; (b) x³ ≥ x²; (c) (a + b)² = a² + b².
Part 2: Critique and Compare (Problems 4-6)
Dana solves 5x² = 20x by dividing both sides by 5x and gets x = 4. Explain the flaw, give the solution she missed, and write one useful question you would ask Dana.
A coffee shop records drinks sold on six rainy weekdays (142, 155, 149, 160, 151, 147) and six sunny weekdays (120, 131, 118, 126, 135, 122). The data are invented. The owner argues that rain increases sales. Compute both means, decide whether the argument is plausible, and name two facts about the context you would want to check.
Two students prove that the diagonals of a rectangle are equal. Argument A places the rectangle at (0, 0), (a, 0), (a, b) and (0, b) and computes both diagonal lengths. Argument B shows that the two triangles that share the bottom side of the rectangle, each containing one diagonal, are congruent by SAS. Carry out Argument A, then compare the two arguments: what does each one assume, and which would you use to convince a friend?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Logical Chain
Each step follows from definitions, assumptions or known results, and every case is covered
Mostly logical, with a gap or a missing case
Examples only, or no argument
Counterexamples and Domains
Each counterexample works and each domain is correct
Counterexamples work but domains are missing or wrong
Counterexamples do not work
Critique
The flaw is named exactly and a useful question is asked
A flaw is named but vaguely
No flaw identified
Communication
Clear sentences with correct symbols
Understandable with some unclear steps
Hard to follow
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Each question asks you to build an argument, test a claim, decide where a claim is true, or critique someone else's reasoning. Your score updates as you answer, and Reset quiz clears everything.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which number is a counterexample to the claim "Every prime number is odd"?
Answer: B
2 is prime (its only factors are 1 and 2) and it is even, so it disproves the claim. Choices A and C are odd but not prime, so they say nothing about primes. Choice D is prime and odd, so it agrees with the claim.
Question 2 of 20 · Multiple Choice
Which value of x is a counterexample to the claim "For every number x, 2x > x"?
Answer: C
For x = -4, 2x = -8, and -8 is less than -4, so the claim fails. Choices A, B and D are positive, and for every positive x, 2x is greater than x. The claim is true on the domain x > 0.
Question 3 of 20 · Multiple Choice
Which known result is needed to justify "If two angles of a triangle measure 50° and 60°, the third angle measures 70°"?
Answer: A
The argument is 180° - 50° - 60° = 70°, which uses the triangle angle sum. Choices B and C are true results, but nothing in the claim involves vertical angles or an isosceles triangle. Choice D is about angles around a point, not inside a triangle.
Question 4 of 20 · Multiple Choice
A student wants to prove that n² + 3n + 5 is odd for every integer n. Which split into cases is most useful?
Answer: D
Parity decides the answer: if n is even, n² and 3n are even; if n is odd, n² and 3n are odd and their sum is even. Either way n² + 3n is even, so adding 5 gives an odd number. Choices A, B and C split the integers in ways that do not control whether n² + 3n is even.
Question 5 of 20 · Multiple Choice
Lia argues: "The sum of two square numbers is never a square, since 1 + 4 = 5 and 4 + 9 = 13 are not squares." What is wrong with her argument?
Answer: C
Examples cannot establish a claim about all pairs, and one counterexample breaks it: 9 + 16 = 25 = 5². Choice B keeps the same flaw with one more example. Choice A is false: 5 and 13 are not squares.
Question 6 of 20 · Multiple Choice
A website asks its readers to vote, and 70% of those who vote want a four-day school week. A student says this shows that most parents in the city want one. What is the strongest critique?
Answer: A
The data came from volunteers on one website, not a random sample of parents, so the context limits the conclusion. Choice C rejects all sampling, which is too strong: a random sample can support an inference. Choice B argues about the size of the majority instead of where the data came from.
Question 7 of 20 · Multiple Choice
Which argument best shows that the product of any two odd numbers is odd?
Answer: B
Choice B uses the definition of odd and algebra that works for all integers a and b. Choice A is evidence from three cases, not a proof. Choice C restates the claim. Choice D is a plausible idea but leaves the key step, that the last digit of the product is odd, without a reason.
Question 8 of 20 · Multiple Choice
A student checks that √(x²) = x for x = 2, 5 and 10 and claims it is true. On which domain is the claim actually true?
Answer: D
The square root symbol means the non-negative root, so √(x²) = |x|, which equals x exactly when x ≥ 0. For x = -3, √9 = 3, not -3. Choice A ignores negatives, and choice C picks the wrong side: for x < 0, √(x²) = -x.
Question 9 of 20 · Multiple Choice
A student is solving 3(x - 4) = 2x + 1. Step 1 is 3x - 12 = 2x + 1 (distributive property). Which is a correct next step with a correct reason?
Answer: B
Subtracting 2x from both sides keeps the equation true and gives x - 12 = 1, so x = 13. Choice A should add 12, not subtract it. Choice D gives the right equation with a false reason, and a viable argument needs both. Choice C adds 2x on the left but takes 2x away on the right, so the two sides no longer stay equal.
Question 10 of 20 · Multiple Choice
Ben claims, "The diagonals of a parallelogram are always perpendicular." Which question would best help him test his claim?
Answer: A
A rectangle that is not a square is a parallelogram whose diagonals are not perpendicular, so this question leads Ben to a counterexample. Choices B and C ask about his process but do not test the claim. Choice D has an obvious answer that does not bear on the diagonals.
Question 11 of 20 · Multiple Choice
Which response best justifies that a triangle with sides 7, 24 and 25 is a right triangle?
Answer: A
49 + 576 = 625 = 25², and the converse of the Pythagorean Theorem says a triangle whose sides satisfy a² + b² = c² is a right triangle. Choice B only shows the triangle exists. Choice C is false: the longest side is opposite the largest angle, which need not be a right angle.
Question 12 of 20 · Multiple Choice
Kim writes: "Since -2 < 1, multiplying both sides by -3 gives 6 < -3." What is the flaw?
Answer: B
When both sides of an inequality are multiplied by a negative number, the order reverses: 6 > -3, which is true. Choice C makes a sign error, since (-2)(-3) = 6. Choice D is false: multiplying by a positive number keeps the order.
Question 13 of 20 · Multiple Choice
A student surveys 25 randomly chosen students at her school, and 15 of them prefer a later start time. Which conclusion is most reasonable?
Answer: D
15 out of 25 is 60%, and a random sample supports an estimate for the school, with some variation. Choice A claims too much precision. Choice B goes beyond the population that was sampled. Choice C rejects a random sample, which is the standard way to make such an inference.
Question 14 of 20 · Multiple Choice
Ana says 0.25 × 0.4 = 0.1 because 25 × 4 = 100 and the factors have 3 decimal places in all, giving 0.100. Ben says 0.25 is 1/4, and 1/4 of 0.4 is 0.1. Which statement is true?
Answer: C
Both arguments are correct and reach 0.1. Ana's rule follows from place value (hundredths times tenths gives thousandths), and Ben uses the equivalence 0.25 = 1/4. Choice D drops the decimal places; 25 × 4 = 100 is the product of the whole numbers, not of the decimals.
Question 15 of 20 · Short Answer
Prove that the sum of any four consecutive integers is even.
Let the integers be n, n + 1, n + 2 and n + 3. Their sum is 4n + 6 = 2(2n + 3). Since 2n + 3 is an integer, the sum is 2 times an integer, which is the definition of even. The argument works for every integer n, including negatives.
Question 16 of 20 · Short Answer
Use cases to show that |x| ≥ x for every real number x.
Case 1, x ≥ 0: then |x| = x, so |x| ≥ x is true. Case 2, x < 0: then |x| = -x, which is positive, and x is negative, so |x| > x. Every real number falls into one of the two cases, so |x| ≥ x for all x.
Question 17 of 20 · Short Answer
Give a counterexample to the claim "Every quadrilateral with perpendicular diagonals is a square." Then name a smaller set of quadrilaterals for which the claim is true.
Counterexample: a rhombus that is not a square, or a kite, has perpendicular diagonals but is not a square. A smaller domain: for rectangles, the claim is true, since a rectangle with perpendicular diagonals has four equal sides, so it is a square.
Question 18 of 20 · Short Answer
Max solves (x - 3)(x + 2) = 6 by writing x - 3 = 6 or x + 2 = 6, so x = 9 or x = 4. Explain the flaw and solve the equation correctly. Then write one useful question you could ask Max.
The zero product property works only when the product is 0; a product of 6 does not force a factor to be 6. Correct: x² - x - 6 = 6, so x² - x - 12 = 0, (x - 4)(x + 3) = 0, and x = 4 or x = -3. Max's x = 9 fails: 6 × 11 = 66. His x = 4 works only by luck. A useful question: "Can you check x = 9 in the original equation?"
Question 19 of 20 · Short Answer
In an invented study, 40 randomly chosen tomato plants got a new fertilizer and had a mean height of 64 cm after six weeks; 40 other plants without it had a mean height of 58 cm. Write a plausible argument about the fertilizer and one question about the context you would ask before accepting it.
A plausible argument: the fertilized plants averaged 6 cm taller, and since the plants were randomly assigned, the fertilizer may have caused some of the difference. Questions about context: Were both groups grown with the same light and water? How much do heights vary within each group? Would another random split give a similar difference?
Question 20 of 20 · Short Answer
Two arguments claim that the angles of a quadrilateral add to 360°. Argument A: "I measured three quadrilaterals." Argument B: "A diagonal splits the quadrilateral into two triangles, each with angles adding to 180°." Which is more effective, and for which quadrilaterals does Argument B need extra care?
Argument B is more effective: it uses a known result and works for every quadrilateral, while A only gives evidence. B needs care for a quadrilateral with an angle greater than 180° (a non-convex one): one diagonal lies outside it, so you must choose the diagonal that lies inside. Then 180° + 180° = 360°.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does MP3 mean in math?
MP3 is the Standard for Mathematical Practice "Construct viable arguments and critique the reasoning of others." Students explain why something is true using definitions, assumptions and known results, test claims with cases and counterexamples, and judge whether other people's arguments make sense.
What is a viable argument?
A viable argument is one that could convince a careful listener. It states its claim, says what it starts from, and links each step to earlier ones. It does not have to be a formal two-column proof; in grade 6 it might be a clear explanation with a diagram.
Why are examples not enough to prove a claim?
Examples show that a claim works in the cases checked, but an "always" claim is about every case. The claim "n² + n + 41 is prime" works for n = 0 to 39 and fails at n = 40, where it equals 41². A general argument covers the cases nobody checked.
What is a counterexample?
A counterexample is a single case that makes a general claim false. The number 2 is a counterexample to "every prime is odd." One counterexample is enough to disprove an "always" claim, and it often shows where the claim does hold.
How do I teach students to critique without being negative?
Teach a routine: first say what the argument gets right, then name the exact step that fails, then ask a question that helps the author fix it. Sentence frames such as "Your rule works for ___, but what about ___?" keep the focus on the mathematics, not the person.
Does MP3 apply to statistics, or only to proofs?
Both. The text of MP3 says students "reason inductively about data, making plausible arguments that take into account the context from which the data arose." In statistics, a good argument considers how the data were collected, how much they vary, and what population they represent.
What does "determine domains to which an argument applies" mean?
It means stating for which numbers, shapes or situations a claim is true. "Squaring makes a number bigger" is false in general but true for numbers greater than 1 and for negative numbers. Naming that domain turns a false claim into a true one.
How can I assess MP3?
Ask for written arguments and critiques, not only answers. A rubric with rows for the logical chain, counterexamples and domains, critique, and communication works well, as in the homework above. Short prompts such as "Always, sometimes or never? Convince me" are quick checks.
Which content standards are good for practicing MP3?
Good fits include inferences from random samples (7.SP.A.2), explaining a proof of the Pythagorean Theorem (8.G.B.6), explaining each step in solving an equation (HSA.REI.A.1), proving theorems about triangles (HSG.CO.C.10) and evaluating reports based on data (HSS.IC.B.6).
Is MP3 only for high school geometry proofs?
No. MP3 applies in every grade. Middle school students justify answers with diagrams, number properties and data; high school students also write formal proofs in algebra and geometry. The text notes that students "at all grades" can read others' arguments and ask useful questions.
07
Related Standards
5 standards
These content standards are good places to practice MP3: earlier standards to start with, standards at the same level, and later courses where the practice keeps paying off.
Before this lesson
7.SP.A.2Prerequisite
Use data from a random sample to draw inferences about a population
