In plain English: MP6 is the Common Core Standard for Mathematical Practice about precision. Students communicate precisely, use clear definitions, state what their symbols mean and use the equal sign correctly, give units and label axes, calculate accurately and efficiently, and round answers to a precision that fits the context. By high school they examine claims using definitions. It applies in every grade 6-12 course.
Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
Common Core State Standards for Mathematics · Domain: Standards for Mathematical Practice (MP) · Practice group: Overarching habits of mind of a productive mathematical thinker Also written as CCSS.MATH.PRACTICE.MP6 · Official standard
MP6 is about precision in communication as well as in calculation. Proficient students "try to communicate precisely to others": they use clear definitions, state the meaning of the symbols they choose, use the equal sign "consistently and appropriately," specify units of measure and label axes. They calculate accurately and efficiently, and they express answers with a degree of precision that fits the problem: neither too rough nor falsely exact. By high school they examine claims and make explicit use of definitions.
This lesson gives each of these habits its own short task, drawn from grade 6-12 arithmetic, expressions, graphs, geometry and probability. Each task is labeled with a typical course.
Learning Objectives
By the end of this lesson, students will be able to:
Use the equal sign only between quantities that are equal, and state what each symbol means
Give every answer its unit and label graph axes with quantities and units
Write and use precise definitions to decide whether a claim is true
Calculate accurately and choose an efficient method
Round answers to a precision that fits the measurement and the context
Prior Knowledge Required
Students should already be comfortable with:
Operations with multi-digit decimals 6.NS.B.3
Scientific notation 8.EE.A.4
Area and circumference of a circle 7.G.B.4
Graphing a relationship on the coordinate plane 6.EE.C.9
Project three student answers. Pairs find what is imprecise in each one.
Warm-Up Prompt
"(1) To find the cost of 4 notebooks at $2.75 and a $6 pen, Ari wrote 4 × 2.75 = 11 + 6 = 17. (2) Bea wrote that the area of a 6 cm by 4 cm card is 24 cm. (3) Cal wrote that the answer to a speed problem is 12. What is imprecise in each answer? How would you fix it?"
Discuss. (1) The chain says 4 × 2.75 = 17, which is false; write 4 × 2.75 = 11 and 11 + 6 = 17, or 4 × 2.75 + 6 = 17. (2) Area is in square centimeters: 24 cm². (3) 12 what? Twelve meters per second and twelve miles per hour are very different speeds. Tell students that each fix is part of MP6.
Direct Instruction20 minutes
Teach five habits of precision, each with an example. Ask students to say which habit each example shows.
The equal sign means "is the same number as." Never use it to mean "and then."
Symbols and units: say what each letter stands for, with its unit, and give every answer a unit.
Definitions: when a claim depends on a word, state the definition and use it.
Calculation: choose an efficient method and check that it is accurate.
Rounding: report as many digits as the measurements and the context justify.
Using the equal sign correctly (Grade 6)
Write the work for "3 movie tickets at $12.25 and a $9 snack" without a run-on equal sign.
Equation: 3 × 12.25 + 9 = 36.75 + 9 = 45.75, so the total is $45.75. Every expression in the chain equals $45.75.
Units and sensible precision (Grade 7)
The diameter of a round garden is measured as 3.2 m, to the nearest 0.1 m. Find its area.
Equation: A = π(1.6)² ≈ 8.0424771 on the calculator. Because the diameter could be anywhere from 3.15 to 3.25 m, report about 8.0 m² (Diagram 2).
Making a definition explicit (Geometry)
Is a square a trapezoid?
Equation: It depends on the definition. If a trapezoid has "at least one pair of parallel sides," a square is a trapezoid. If it has "exactly one pair," a square is not. Say which definition you use.
Labeling axes (Grade 8)
A graph is a straight line from (0, 0) to (4, 20) with no labels.
Equation: Label it: time (hours) across, distance from start (km) up. Now the slope has a meaning: 20 ÷ 4 = 5 km per hour (Diagram 1).
Accurate and efficient calculation (Grade 8)
Compute (4.5 × 10³)(2 × 10⁻⁵) and 0.25 × 368.
Equation: Multiply the numbers and add the exponents: 9 × 10⁻² = 0.09. For 0.25 × 368, divide by 4: 92. Both methods are fast and exact.
After Example 2, show Diagram 2 and ask: "Why not report 8.04 m²?" (The last digit would claim a precision the measurement does not have; the true area could be 7.8 or 8.3.) After Example 3, ask students for another word whose definition changes an answer (for example, whether 0 counts as a natural number in different books).
Guided Practice15 minutes
Pairs rewrite three imprecise statements so that they are precise and correct.
(Grade 8) "The answer to the system is x = 5." Rewrite: give both values and say what they mean, for example "(x, y) = (5, 2), so the two lines meet at the point (5, 2)."
(Grade 7-8) "The slope of the cost graph is 3." Rewrite with units: "The cost increases by $3 for each hour of rental."
(Grade 7) "Approval rose from 40% to 60%, so it went up 20%." Rewrite: "It went up 20 percentage points, which is a 50% increase, since 20 is 50% of 40."
Ask: "In item 3, which word made the original statement imprecise?" ("Percent" was used for two different things.)
Independent Practice15 minutes
Students report each result with sensible precision and a unit, and write one sentence explaining their choice.
A spreadsheet gives the mean height of 23 students as 158.347826 cm. (About 158.3 cm, since heights are measured to the nearest 0.1 cm at best.)
A unit price is computed as $0.18666 per ounce. (About $0.19 per ounce, or 18.7 cents per ounce when comparing close prices.)
A recipe scaled by a calculator needs 1.3333 cups of milk. (1 1/3 cups, since measuring cups come in thirds.)
A trip takes 2.75 hours. (2 hours 45 minutes, which is easier to plan with.)
Closure5-10 minutes
Exit ticket:
Fix this work: 20 - 4 = 16 ÷ 2 = 8.
Give a precise definition of a square that someone could use to check a shape.
A runner's time is recorded as 12.3 seconds. Should a report give 12.300 seconds? Why or why not?
Differentiation Strategies
For Struggling Students
Give a precision checklist: equal signs, units, labels, definition, rounding
Have students read every equal sign aloud as "is the same as" while checking a chain
Provide a word bank of units and of precise terms (percent, percentage point, perimeter, area)
For Advanced Students
Ask for the range of possible volumes of a cube whose edge is measured as 5.0 cm to the nearest 0.1 cm, and a sensible report
Ask students to find two textbooks or websites that define a word (such as trapezoid or natural number) differently, and explain the effect on one problem
Ask students to write a short "style guide" for precise math writing for younger students
Assessment Guidance
What to Look For
Look for equal signs that join only equal quantities, answers with units, labeled axes and titles on every graph, definitions stated when a claim depends on them, and rounding that fits the measurement. Strong students explain why they rounded as they did and catch imprecise language such as "it went up 20%" for a change in percentage points. Watch for run-on equal signs, square units missing from areas, calculator displays copied in full, and arguments about shapes that never say which definition is used.
02
Classroom Activities
3 Activities
1
Definition Debate
20 minGroups of 3-4
Groups write precise definitions of four words, then test them on borderline examples. A good definition lets two people who disagree settle the question by checking it.
The Words and the Test Cases (with answers for the teacher)
Odd number: an integer with a remainder of 1 when divided by 2. Test: Is -7 odd? (Yes, -7 = 2(-4) + 1.) Is 0 odd? (No, 0 = 2 × 0 with no remainder, so 0 is even.)
Polygon: a closed figure made of line segments that meet only at their endpoints. Test: Is a circle a polygon? (No, it has no segments.)
Parallelogram: a quadrilateral with two pairs of parallel sides. Test: Is a rectangle a parallelogram? (Yes, its opposite sides are parallel.) Is a trapezoid with only one pair of parallel sides a parallelogram? (No.)
Prime number: an integer greater than 1 whose only positive divisors are 1 and itself. Test: Is 2 prime? (Yes.) Is 51 prime? (No, 51 = 3 × 17.)
Procedure
Each group writes its own definition of each word before seeing the test cases
Groups apply their definitions to the test cases and revise any definition that gives a wrong or unclear answer
Groups compare definitions with another group and agree on the most precise version
Discussion Questions
Which test case broke your first definition?
Why does the definition of prime say "greater than 1"?
Is "a parallelogram is a slanted box shape" a definition? What is missing?
2
Label It Right
20 minPairs
Pairs receive three unlabeled graphs and two sample solutions. They write a situation for each graph and label it completely, and they correct the equal-sign errors in the solutions.
The Materials
Graph A: a line from (0, 50) down to (10, 0).
Graph B: a curve that doubles every unit, starting at (0, 3).
Graph C: a horizontal segment at height 40 from x = 0 to x = 6.
Solution 1: "12 × 3 = 36 - 6 = 30" for "12 boxes of 3 pens, minus 6 broken pens."
Solution 2: "x + 5 = 12 = x = 7" for solving x + 5 = 12.
Answer Key for the Teacher
Sample labels: Graph A, time (minutes) and water left in a tank (liters), draining 5 L per minute; Graph B, time (hours) and number of bacteria (thousands); Graph C, time (hours) and speed (km per hour) at a steady 40.
Solution 1: 12 × 3 - 6 = 36 - 6 = 30 pens.
Solution 2: x + 5 = 12, so x = 12 - 5 = 7. The original chain claims that 12 = x.
Procedure
For each graph, give a title, both axis labels with units, and a scale
Trade with another pair, who must state what the slope or the starting value means using only your labels
Rewrite both solutions so that every equal sign is true
Modification for Distance Learning
Share the graphs as images in a slide deck; pairs add text boxes for labels and type the corrected solutions in the speaker notes.
3
Measure, Compute, Report
20 minGroups of 3
Groups measure a desk top with a ruler marked in millimeters, compute its perimeter and area, and decide how precisely to report each result. Then they compare reports across groups.
Sample Data (invented)
One group measured a desk as 76.2 cm by 61.0 cm, each to the nearest 0.1 cm.
Answer Key for the Teacher
Perimeter: 2(76.2 + 61.0) = 274.4 cm.
Area: 76.2 × 61.0 = 4,648.2 cm² on the calculator. The sides could be up to 0.05 cm off, so the area is between about 4,641 and 4,655 cm²; report about 4,650 cm².
Procedure
Each student measures both sides once; the group records all three measurements and their mean
Compute the perimeter and area with the mean measurements
Decide how many digits to report and write one sentence explaining the choice
Discussion Questions
Why did different groups get slightly different areas for the same desk?
Why is the area's last digit less trustworthy than the perimeter's?
When would a carpenter need more precision than this?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Labeling Axes to Show the Quantities
The same line drawn twice, to scale. On the left, nobody can tell what it shows: a walk, a savings account or a filling tank. On the right, the title, axis labels, units and scale make the correspondence with the quantities clear, so the slope can be read as a speed: 5 km per hour.
Diagram 2: How Precise Should an Answer Be?
A diameter measured as 3.2 m to the nearest 0.1 m is really between 3.15 m and 3.25 m, so the area is between about 7.79 m² and 8.30 m² (blue band, to scale). The calculator's 8.0424771 suggests a precision the measurement does not have; "about 8.0 m²" is an honest report.
04
Homework Assignment
~30 min
MP6 Homework: Say It Precisely
Directions: Use the equal sign only between equal quantities, give every answer a unit, label any graph fully, and explain your rounding. When a question depends on a word, state the definition you use.
Part 1: Symbols, Definitions and Labels (Problems 1-3)
A student finds the share of each of 2 friends for 3 concert tickets at $12.50 each plus a $4 parking fee, and writes 3 × 12.50 = 37.50 + 4 = 41.50 ÷ 2 = 20.75. Explain what is wrong with the equal signs, and rewrite the work correctly.
Use the definition "a trapezoid is a quadrilateral with at least one pair of parallel sides" to decide whether each shape is a trapezoid: (a) a parallelogram; (b) a kite that is not a rhombus; (c) a rectangle. Then answer again using "exactly one pair of parallel sides."
A phone uses data at a steady rate. After 0, 5, 10, 15 and 20 days it has used 0, 1.5, 3.0, 4.5 and 6.0 GB. Draw a graph with a title, labeled axes, units and a scale. State the slope with its units.
Part 2: Calculation and Precision (Problems 4-6)
Compute each one exactly, using an efficient method, and name the method: (a) 25 × 36; (b) 1.5 × 4.8; (c) (6 × 10⁵)(3 × 10⁻²); (d) 7/8 - 3/4.
Report each answer with a sensible precision and explain your choice: (a) the mean of the six test scores 85, 91, 78, 88, 94 and 82; (b) the length of each piece when a 2.5 m board is cut into 3 equal pieces; (c) the price per gram of a 340 g jar of peanut butter that costs $3.79.
A student claims, "The probability of rolling a sum of 7 with two dice is 1/11, because there are 11 possible sums." Use a precise definition of probability for equally likely outcomes to examine the claim, and give the correct probability.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Equal Sign and Symbols
Every equal sign is true and every symbol is defined
One misuse or undefined symbol
Several misuses
Units and Labels
All answers have units; graphs have titles, labels and scales
Some units or labels missing
Most missing
Definitions
Definitions stated and used to decide each claim
Definitions stated but not used
No definitions
Accuracy and Rounding
Correct results, rounded sensibly with a reason
Correct but rounded poorly or without a reason
Incorrect results
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Each question asks you to use the equal sign, units, labels or definitions precisely, to calculate accurately, or to report an answer with sensible precision. 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 line uses the equal sign correctly?
Answer: D
Only choice D states a true equality: (5 + 3) × 2 is 16. Choice A claims 5 + 3 = 16 by chaining steps. Choice B claims 5 + 3 equals 2 × 8. Choice C breaks the order of operations: 5 + 3 × 2 is 11.
Question 2 of 20 · Multiple Choice
A student finds the area of a 9 cm by 5 cm rectangle and writes "45 cm." What is imprecise?
Answer: B
Area counts square centimeters, so the answer is 45 cm². Choice A gives the perimeter, 2(9 + 5) = 28 cm, a different quantity. Choice C adds a digit but keeps the wrong unit.
Question 3 of 20 · Multiple Choice
A square's side is measured as 12.6 cm, to the nearest 0.1 cm. The calculator gives the area as 158.76 cm². Which report is most appropriate?
Answer: C
The side could be from 12.55 to 12.65 cm, so the area is between about 157.5 and 160.0 cm². Reporting about 159 cm² is honest; the hundredths in choice A are not justified. Choice B drops the square unit. Choice D claims even more precision.
Question 4 of 20 · Multiple Choice
Which is a precise definition of a rectangle?
Answer: C
Four right angles is a property anyone can check, and it includes squares. Choice B also fits kites and many parallelograms that are not rectangles. Choice A excludes squares and fits many non-rectangles.
Question 5 of 20 · Multiple Choice
A graph of a car's value is a line from (0, 24) to (8, 4) with no labels. Which labels make it most precise?
Answer: A
The labels must name the quantities and units: years since the car was bought, and its value in thousands of dollars. Then the slope reads as a loss of $2,500 per year. Choice C names general ideas without units, so the scale cannot be read.
Question 6 of 20 · Multiple Choice
Which is the most efficient accurate way to compute 6.5 × 14 + 3.5 × 14?
Answer: A
Both products share the factor 14, so the distributive property lets you add 6.5 + 3.5 = 10 first: a one-step mental calculation. Choice D is also accurate but takes three steps. Choice B is accurate but slow. Choice C is fast but only an estimate, and a poor one.
Question 7 of 20 · Multiple Choice
A poll's approval rating rose from 40% to 50%. Which statement is precise and correct?
Answer: B
The difference of the two percents is 10 percentage points, and 10 is 25% of 40, so it is a 25% relative increase. Choice A is ambiguous: "10%" could mean either. Choice C mixes up the two measures.
Question 8 of 20 · Multiple Choice
Use the definition "a rhombus is a quadrilateral with four equal sides." The claim "Every square is a rhombus" is:
Answer: D
Using the stated definition, a square qualifies, since its four sides are equal. Choice A adds a condition that is not in the definition. Choice C assumes a shape cannot belong to two categories, but a square is both a rectangle and a rhombus.
Question 9 of 20 · Multiple Choice
For the equation d = 18t, which statement gives the meaning of the symbols precisely?
Answer: B
Precise symbol meanings include units, and they explain the constant: 18 kilometers per hour for a cyclist. Choice A names the quantities without units, so 18 could be meters per second. Choices C and D say nothing about the situation.
Question 10 of 20 · Multiple Choice
A spreadsheet averages five stopwatch times, each recorded to 0.01 s, and shows 52.3846. Which report is most appropriate?
Answer: D
The times were recorded to hundredths of a second, so the mean is reported to hundredths: 52.38 s. Choice A claims precision the stopwatch does not have. Choices B and C throw away precision that the data do have.
Question 11 of 20 · Multiple Choice
Which answer to "How fast does the snail move?" is precise?
Answer: C
A speed needs a unit of distance per unit of time. Choice A gives the number with no unit, so it could be 0.03 meters per second (60 times faster). Choice D gives no unit and no scale at all.
Question 12 of 20 · Multiple Choice
What is (3 × 10⁴)(4 × 10⁻⁶)?
Answer: D
Multiply 3 × 4 = 12 and add the exponents: 10⁴ × 10⁻⁶ = 10⁻². So the product is 12 × 10⁻² = 1.2 × 10⁻¹, or 0.12. Choice A multiplies the exponents. Choice B adds 3 and 4. Choice C subtracts the exponents, -6 - 4 = -10, instead of adding them.
Question 13 of 20 · Multiple Choice
Which definition of "even number" is best for writing a proof?
Answer: A
The form 2k turns "even" into algebra that can be used in a proof, and it works for negatives and 0. Choice B is true for integers but hard to use in algebra. Choice C lists examples, which is not a definition. Choice D fits every number: 7 = 3 + 4.
Question 14 of 20 · Multiple Choice
A scatter plot's axes are labeled "hours" and "score." Which change makes it most precise?
Answer: A
Precise labels say which hours and which score, with units, so the correspondence with the quantities is clear. Choice C may help the reader but does not fix what the axes measure. Choice B removes the meaning.
Question 15 of 20 · Short Answer
Write the work for this problem with every equal sign true: 5 shirts cost $8.40 each, a $5 coupon is subtracted, and then 8% sales tax is added.
Cost before tax: 5 × 8.40 - 5 = 42 - 5 = 37. With tax: 37 × 1.08 = 39.96. The total is $39.96. As one expression: (5 × 8.40 - 5) × 1.08 = 39.96.
Question 16 of 20 · Short Answer
A circular pond has a radius measured as 4.3 m, to the nearest 0.1 m. Compute its area and circumference, and report both with sensible precision and units.
Area: π(4.3)² ≈ 58.088 m², reported as about 58 m². Circumference: 2π(4.3) ≈ 27.018 m, reported as about 27 m. The radius is known only to 0.1 m (from 4.25 to 4.35 m), so more digits would be falsely precise.
Question 17 of 20 · Short Answer
A polygon is called regular if all of its sides are equal and all of its angles are equal. Use this definition to decide whether a rhombus with one 60° angle is a regular polygon. Explain.
A rhombus has four equal sides, so the first condition holds. Its angles are 60°, 120°, 60° and 120° (consecutive angles add to 180°), so they are not all equal. Both conditions are required, so it is not regular. The only regular quadrilateral is the square.
Question 18 of 20 · Short Answer
A weather station records the temperature every 2 hours for one day. Describe the title, the axis labels and the scales you would use for a graph of the data.
Title: "Temperature at the weather station, every 2 hours." Horizontal axis: time of day (hours), from 0 to 24 in steps of 2. Vertical axis: temperature (°C), with a scale that covers the day's lowest and highest readings, for example from 0 to 30 in steps of 5.
Question 19 of 20 · Short Answer
Compute exactly: 2/3 + 5/6 - 1/4.
Use the common denominator 12: 8/12 + 10/12 - 3/12 = 15/12 = 5/4, or 1 1/4.
Question 20 of 20 · Short Answer
A student describes the graph of y = 3x - 2 by saying "the graph goes up." Rewrite the description precisely.
The graph is a line with slope 3: each time x increases by 1, y increases by 3. It crosses the y-axis at (0, -2) and the x-axis at (2/3, 0).
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does MP6 mean in math?
MP6 is the Standard for Mathematical Practice "Attend to precision." Students communicate precisely: they use clear definitions, say what their symbols mean, use the equal sign correctly, give units and label axes, calculate accurately, and round answers to a sensible precision.
Is MP6 just about getting the right answer?
No. Accuracy is one part, but MP6 also covers precise language, symbols, units, labels and definitions. An answer of "24" to an area question is accurate but imprecise without "cm²."
Why does MP6 mention the equal sign?
Because the equal sign has one meaning: "is the same number as." Students often use it to mean "and the next step is," writing chains such as 4 × 2.75 = 11 + 6 = 17 that state false equalities. MP6 asks for the equal sign "consistently and appropriately."
How many decimal places should students give?
As many as the context and the measurements justify. A length measured to 0.1 cm does not support an area to 0.001 cm². Money is usually given to the cent, and counts of people are whole numbers. Students should say why they rounded as they did.
Does attending to precision mean using more digits?
Not always. Extra digits can be false precision: reporting 8.0424771 m² for a garden measured to 0.1 m claims a precision the measurement does not have. MP6 asks for "a degree of precision appropriate for the problem context."
Why do definitions matter so much in MP6?
Many disagreements in math are really about definitions. Whether a square is a trapezoid depends on whether a trapezoid has "at least one" or "exactly one" pair of parallel sides. By high school, MP6 asks students to "make explicit use of definitions" when they examine claims.
What should a precisely labeled graph include?
A title, a label on each axis naming the quantity and its unit, and a consistent scale. With those, a reader can tell what each point means and what the slope or other features represent, as in Diagram 1.
How can I assess MP6?
Include precision in every rubric: true equal signs, units, labels, stated definitions and sensible rounding. Short tasks such as "fix this work" or "report this calculator result sensibly" check MP6 directly.
Which content standards are good for practicing MP6?
Good fits include fluent decimal operations (6.NS.B.3), scientific notation and choosing units (8.EE.A.4), defining appropriate quantities (HSN.Q.A.2), choosing a level of accuracy (HSN.Q.A.3) and precise geometric definitions (HSG.CO.A.1).
Is MP6 the same as being neat?
No. Neat work helps, but precision is about meaning: a neat page with run-on equal signs or missing units is still imprecise, and a rough sketch with clear labels and a stated definition can be precise.
07
Related Standards
5 standards
These content standards are good places to practice MP6: earlier standards to start with, standards at the same level, and later courses where the practice keeps paying off.
Before this lesson
6.NS.B.3Prerequisite
Fluently add, subtract, multiply, and divide multi-digit decimals
