MP8: Look For and Express Regularity in Repeated Reasoning
In plain English: MP8 is the Common Core Standard for Mathematical Practice about repeated reasoning. Students notice when a calculation repeats and turn it into a general method or formula: repeating remainders become a repeating decimal, repeated slope checks become the equation of a line, and a pattern of cancelling terms becomes the sum of a geometric series. It applies in every grade 6-12 course.
Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x² + x + 1), and (x - 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Common Core State Standards for Mathematics · Domain: Standards for Mathematical Practice (MP) · Practice group: Seeing structure and generalizing Also written as CCSS.MATH.PRACTICE.MP8 · Official standard
MP8 asks students to "notice if calculations are repeated, and look both for general methods and for shortcuts." When the same steps happen again and again, the repetition itself can be written down once, as a rule, an equation or a formula. The official text gives three examples, all used in this lesson: noticing that the long division 25 ÷ 11 repeats, so the decimal repeats; noticing that every check of whether a point is on the line through (1, 2) with slope 3 computes the same ratio, which gives the equation (y - 2)/(x - 1) = 3; and noticing how terms cancel in (x - 1)(x + 1), (x - 1)(x² + x + 1) and (x - 1)(x³ + x² + x + 1), which leads to the formula for the sum of a geometric series.
While they generalize, proficient students "maintain oversight of the process, while attending to the details," and they "continually evaluate the reasonableness of their intermediate results." The tasks come from grade 6-12 number, expressions, linear functions, sequences and finance, each labeled with a typical course.
Learning Objectives
By the end of this lesson, students will be able to:
Notice when a calculation repeats, and explain why a fraction's decimal must repeat
Turn a repeated calculation into a general rule, equation or formula and use it
Find a shortcut from a pattern and explain why it works
Keep track of the whole process while checking each step, and judge whether intermediate results are reasonable
Prior Knowledge Required
Students should already be comfortable with:
Long division and converting fractions to decimals 7.NS.A.2
Students do the long division 25 ÷ 11 by hand, writing every remainder, for six decimal places.
Warm-Up Prompt
"Divide 25 by 11 by hand and write the remainder after each step. What do you notice? Without dividing any further, what is the 100th digit after the decimal point? How do you know?"
Students find the remainders 3, 8, 3, 8, ... and the digits 2, 7, 2, 7, ... after the decimal point. Since the odd places are 2 and the even places are 7, the 100th digit is 7. Ask: "Why can't the division go on forever without repeating?" (Dividing by 11 leaves a remainder from 0 to 10, so within 11 steps some remainder must come back, and from then on the steps repeat.) Show Diagram 1.
Direct Instruction20 minutes
Model five cases of repeated reasoning. In each one, do the calculation a few times aloud, then ask: "What stays the same each time? Can we write it once?"
Do it a few times, carefully, and write every step.
Notice what repeats: the same operation, the same remainder, the same cancellation.
Write the general method once, with a variable for the part that changes.
Check the general method on a case you already did, and on a new one.
Watch the details and the big picture: is each intermediate result reasonable?
A repeating decimal (Grade 7)
Divide 25 by 11. Then use the same idea for 5 ÷ 7.
Equation: Remainders 3, 8, 3, 8 give 25/11 = 2.272727... For 5 ÷ 7 the remainders are 5, 1, 3, 2, 6, 4, then 5 again, so 5/7 = 0.714285714285..., a block of 6 repeating digits.
Repeated slope checks become an equation (Grade 8)
Is each point on the line through (1, 2) with slope 3? Test (3, 8), (4, 11) and (5, 13).
Equation: (8 - 2)/(3 - 1) = 3, yes; (11 - 2)/(4 - 1) = 3, yes; (13 - 2)/(5 - 1) = 11/4, no. The same check for any (x, y) is (y - 2)/(x - 1) = 3, so y = 3x - 1 (Diagram 2).
Cancelling terms and a geometric series (Algebra II)
Expand (x - 1)(x + 1), (x - 1)(x² + x + 1) and (x - 1)(x³ + x² + x + 1).
Equation: The results are x² - 1, x³ - 1 and x⁴ - 1: every middle term cancels. In general 1 + x + x² + ... + xⁿ⁻¹ = (xⁿ - 1)/(x - 1). For example, 1 + 2 + 4 + ... + 2⁹ = 2¹⁰ - 1 = 1,023.
A shortcut from a pattern (Grade 6-7)
Compute 23 × 11, 45 × 11 and 72 × 11, then look for a shortcut.
Equation: 253, 495, 792: the outer digits stay and the middle digit is their sum. With a carry, 68 × 11 = 748. Why: 11(10a + b) = 100a + 10(a + b) + b.
Oversight and reasonableness (Algebra I)
$750 is invested at 6% per year, compounded yearly. Find the value after 1, 2, 3 and 10 years.
Equation: Each year multiplies by 1.06: $795, $842.70, $893.26, so after n years 750(1.06)ⁿ. After 10 years about $1,343.14. Reasonable: simple interest would give $1,200, and compounding adds a little more.
After Example 2, show Diagram 2 and ask what changes from check to check (the point) and what stays the same (the point (1, 2) and the slope 3). After Example 3, ask students to predict (x - 1)(x⁵ + x⁴ + x³ + x² + x + 1) before expanding it.
Guided Practice15 minutes
Pairs work on a seating pattern, writing each count before looking for the rule.
Task (Grade 7-8): Hexagon-shaped tables are pushed together in a row. One table seats 6 people, two tables seat 10 and three tables seat 14.
Draw 4 and 5 tables and count the seats. (18 and 22.)
What repeats? Each new table adds 4 seats: it has 6 sides, and 2 of them are joined to neighbors.
Write the general rule: seats = 4n + 2 for n tables. Explain the 2 (the two end sides).
Check: 4(3) + 2 = 14, which matches. Use the rule for 30 tables: 122 seats.
Ask a pair that wrote 6n - 2(n - 1) to show that it equals 4n + 2.
Independent Practice15 minutes
Students repeat a calculation, keep track of the intermediate results, and judge whether each one is reasonable.
(Grade 7) An $80 sweater is marked down 25%, and then the sale price is marked down another 20%. Repeat the multiplication step by step: 80 × 0.75 = 60, then 60 × 0.8 = 48. Is the total discount 45%? (No: $48 is 60% of $80, so the total discount is 40%.)
(Algebra II) Add 1/3 + 1/6 + 1/12 + 1/24 + ..., writing each partial sum: 1/3, 1/2, 7/12, 5/8, ... Each time, compare with 2/3. (Every partial sum is less than 2/3, and the gap halves each time, so the sums approach 2/3.)
Students write one sentence about the moment when an intermediate result told them whether they were on the right track.
Closure5-10 minutes
Exit ticket:
Without dividing, explain why 2/13 must have a repeating decimal.
Write one equation that checks whether a point (x, y) is on the line through (0, 4) with slope 2.
Name one shortcut you found today and the reason it works.
Differentiation Strategies
For Struggling Students
Use a table with columns "case," "calculation" and "result" so the repeated step is visible line by line
Build the seating pattern with tiles or counters before writing the rule
Color the terms that cancel when expanding (x - 1)(x² + x + 1)
For Advanced Students
Ask students to prove that the decimal of any fraction p/q repeats with a block of at most q - 1 digits
Ask for the sum 1 + r + r² + ... + rⁿ⁻¹ when r = 1/2, and what happens as n grows
Ask students to find and explain the shortcut for squaring numbers that end in 5
Assessment Guidance
What to Look For
Look for students who do a calculation two or three times on purpose, say what stays the same, and then write the general rule with a variable. Strong students check their rule against a case they already computed and explain why the shortcut works, not only that it does. They also stop to test intermediate results ("a sum of positive terms cannot be smaller than its last term"). Watch for rules guessed from one case, off-by-one errors such as 7 + 2n for a sequence that starts at 7, and repeated calculations that are never generalized.
02
Classroom Activities
3 Activities
1
Repeating Decimals Lab
20 minPairs
Pairs divide families of fractions by hand, notice the repeating remainders, and find shortcuts that predict the decimals without dividing.
The Fraction Families
Elevenths: 1/11, 2/11, 3/11
Ninths: 1/9, 2/9, 4/9
Ninety-ninths: 12/99, 47/99
Sevenths: 1/7 and 3/7
Answer Key for the Teacher
1/11 = 0.0909..., 2/11 = 0.1818..., 3/11 = 0.2727...: k/11 repeats the two digits of 9k.
1/9 = 0.111..., 2/9 = 0.222..., 4/9 = 0.444...: k/9 repeats the digit k.
12/99 = 0.1212..., 47/99 = 0.4747...: a two-digit number over 99 repeats those two digits.
1/7 = 0.142857... and 3/7 = 0.428571...: the same six digits in a different starting place, because the remainders go through the same cycle.
Procedure
Divide the first fraction of each family by hand and list the remainders
Predict the next fraction's decimal before dividing, then check
Write each shortcut as a rule and explain it with the remainders
Discussion Questions
Why can a division by 7 never produce more than 6 different remainders before repeating?
Why do 1/7 and 3/7 use the same six digits?
Use your rules to predict 8/11 and 35/99 without dividing.
2
Slope Detectives
20 minGroups of 3
Groups test several points against a line given by a point and a slope, notice that every test is the same calculation, and write that calculation once as an equation.
The Line and the Points (Grade 8)
The line passes through (2, -1) and has slope 1/2. Test the points (6, 1), (8, 2), (5, 0) and (10, 3).
Answer Key for the Teacher
(6, 1): (1 + 1)/(6 - 2) = 2/4 = 1/2, on the line. (8, 2): 3/6 = 1/2, on the line. (10, 3): 4/8 = 1/2, on the line.
(5, 0): 1/3, not on the line.
General check: (y + 1)/(x - 2) = 1/2, which rearranges to y = x/2 - 2.
Procedure
Each student tests one or two points and writes the full calculation
The group lines up the calculations and circles what is the same in all of them
Replace the tested point by (x, y) to write one equation, then solve it for y
Repeat with the line through (-1, 3) with slope -2, and compare the two general equations
Discussion Questions
What part of the calculation changed from point to point?
Why does the general equation fail at x = 2? What does the point-slope form y + 1 = (1/2)(x - 2) fix?
How does this connect to y - y₁ = m(x - x₁)?
Modification for Distance Learning
Groups use a shared spreadsheet: one column per point, one row per step of the calculation, so the repeated steps line up.
3
The Cancelling Chain
20 minPairs
Pairs expand a chain of products, notice how terms cancel, write the general result, and use it to add geometric series quickly. This is the third official example of MP8.
Procedure (Algebra II)
Expand (x - 1)(x + 1), (x - 1)(x² + x + 1), (x - 1)(x³ + x² + x + 1) and (x - 1)(x⁴ + x³ + x² + x + 1), writing every term before combining
Describe which terms cancel, and why only two terms survive
Write the general result for (x - 1)(xⁿ⁻¹ + ... + x + 1)
Use it to find 1 + 4 + 16 + 64 + ... + 4⁶
Answer Key for the Teacher
The products are x² - 1, x³ - 1, x⁴ - 1 and x⁵ - 1. Each term of x times the long factor cancels with a term of -1 times it, except the highest and the lowest.
General result: (x - 1)(xⁿ⁻¹ + ... + x + 1) = xⁿ - 1, so 1 + x + ... + xⁿ⁻¹ = (xⁿ - 1)/(x - 1) when x ≠ 1.
1 + 4 + ... + 4⁶ = (4⁷ - 1)/3 = 16,383/3 = 5,461. Reasonableness: the last term, 4,096, is about three-quarters of the total, as it should be when each term is 4 times the one before.
Discussion Questions
Why must x ≠ 1 in the general formula? What is the sum when x = 1?
How did writing every term before combining help you see the pattern?
What would change for (x + 1)(x² - x + 1)?
Challenge Variation
Use the formula with x = 1/2 to find 1 + 1/2 + 1/4 + ... + 1/2⁹, and explain what happens to the sum as more terms are added.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Why 25 ÷ 11 Gives a Repeating Decimal
The official MP8 example. Dividing 25 by 11 gives 2 with remainder 3. Each later step multiplies the remainder by 10 and divides by 11 again. When the remainder 3 appears a second time, the calculation is the same as before, so the digits 2 and 7 repeat forever: 25/11 = 2.272727...
Diagram 2: Repeated Slope Checks Become an Equation
The line through (1, 2) with slope 3, drawn to scale. To test a point (x, y), a student computes rise over run from (1, 2): (y - 2)/(x - 1). The points (3, 8) and (4, 11) give 3, so they are on the line; (5, 13) gives 11/4, so it is not. The calculation is the same every time, which is why it can be written once as the equation (y - 2)/(x - 1) = 3.
04
Homework Assignment
~30 min
MP8 Homework: From Repeated Steps to General Rules
Directions: For each problem, do the calculation for a few cases first and show them. Then state the general rule or shortcut, explain why it works, and check that each intermediate result is reasonable.
Part 1: Notice and Generalize (Problems 1-3)
Divide 4 by 7 by long division until a remainder repeats. Write the remainders and the decimal. Without dividing further, find the 50th digit after the decimal point, and explain how you know.
Decide whether the points (4, 7), (6, 10) and (9, 15) are on the line through (2, 4) with slope 3/2. Then write one equation that does the check for any point (x, y), and solve it for y.
Compute 15², 25², 35² and 45². Describe a shortcut for squaring a number that ends in 5, use it to find 85², and explain why it works by expanding (10n + 5)².
Part 2: Formulas, Oversight and Reasonableness (Problems 4-6)
Use the pattern (x - 1)(x⁵ + x⁴ + x³ + x² + x + 1) = x⁶ - 1 to find 1 + 5 + 25 + 125 + 625 + 3,125 without adding term by term. Check that your answer is reasonable.
A student computes 1.5% monthly interest on $2,000 for 3 months and writes: month 1, $2,030.00; month 2, $2,060.45; month 3, $2,391.36. Check each intermediate result for reasonableness, find the error, and correct it. Then write the balance after n months as a formula.
Square tables are pushed together in a row. One table seats 4, two tables seat 6, and three tables seat 8. Explain what repeats, write a rule for n tables, and find the number of seats for 25 tables.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Repeated Cases
Several cases computed and shown
One case shown
No cases
General Rule
A correct general rule with a variable, explained
A correct rule without an explanation
No rule or an incorrect one
Details and Oversight
Every step correct and the rule checked on a known case
A minor error or no check
Many errors
Reasonableness
Intermediate and final results judged against estimates
Some judgment
None
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Each question asks you to notice a repeated calculation, express it as a general rule, or judge whether intermediate results are reasonable. 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
Dividing 5 by 11 gives the remainders 5, 6, 5, 6, ... and the digits 4, 5, 4, 5, ... after the decimal point. What is 5/11 as a decimal?
Answer: B
50 ÷ 11 = 4 remainder 6, and 60 ÷ 11 = 5 remainder 5, so the digits 4 and 5 repeat: 5/11 = 0.4545... Choice A writes the remainders as the digits. Choice D repeats only the 5, which would need the remainder 6 to repeat by itself.
Question 2 of 20 · Multiple Choice
Why must the decimal for a fraction with denominator 7 start repeating within 6 digits?
Answer: D
Dividing by 7 leaves a remainder from 0 to 6. If it is 0, the decimal ends; otherwise, there are only 6 possible remainders, so within 6 steps one repeats, and then the steps repeat. Choice C is false: 1/3 repeats after 1 digit. Choice A is true but is not the reason.
Question 3 of 20 · Multiple Choice
A student checks points on the line through (2, 5) with slope 4 by computing rise over run from (2, 5) each time. Which equation expresses this repeated check?
Answer: B
Rise over run from (2, 5) to (x, y) is (y - 5)/(x - 2), and every point on the line gives the slope 4. Choice A swaps the coordinates of the point. Choice C is run over rise. Choice D uses 5 as the y-intercept, but the line crosses the y-axis at -3.
Question 4 of 20 · Multiple Choice
Which point is on the line through (1, 2) with slope 3?
Answer: D
Check (y - 2)/(x - 1): for (0, -1), (-1 - 2)/(0 - 1) = 3, so it is on the line. For (2, 6) the ratio is 4, for (3, 9) it is 7/2, and for (6, 15) it is 13/5.
Question 5 of 20 · Multiple Choice
Use the cancelling pattern to find (x - 1)(x⁶ + x⁵ + x⁴ + x³ + x² + x + 1).
Answer: C
Every middle term cancels, leaving the highest power, x⁷, minus 1. Choice A uses the highest power already in the long factor, one too low. Choice B has the wrong sign. Choice D drops the constant term of the long factor.
Question 6 of 20 · Multiple Choice
What is 1 + 2 + 4 + 8 + ... + 2¹¹?
Answer: B
By the pattern, 1 + 2 + ... + 2¹¹ = (2¹² - 1)/(2 - 1) = 4,095. Choice A forgets to subtract 1. Choice C stops at 2¹⁰. Choice D adds one term too many.
Question 7 of 20 · Multiple Choice
Using the shortcut for multiplying a two-digit number by 11, what is 54 × 11?
Answer: A
Keep the outer digits 5 and 4 and put their sum, 9, in the middle: 594. Check: 54 × 10 + 54 = 594. Choice B puts the sum at the end. Choice D reverses the digits.
Question 8 of 20 · Multiple Choice
A student computes 1/(1·2) = 1/2, then 1/(1·2) + 1/(2·3) = 2/3, then adds 1/(3·4) to get 3/4, then 1/(4·5) to get 4/5. What general result does the regularity suggest for the sum of n such terms?
Answer: C
The sums are 1/2, 2/3, 3/4, 4/5: the numerator is the number of terms and the denominator is one more, so the sum of n terms is n/(n + 1). The reason is that each term 1/(k(k + 1)) equals 1/k - 1/(k + 1), so the middle parts cancel. Choice B gives 0 for one term.
Question 9 of 20 · Multiple Choice
A student finds the value of $1,000 growing 3% per year. After year 4 the intermediate result is $1,425.76. Is it reasonable?
Answer: D
1,000 × 1.03⁴ ≈ $1,125.51, a bit more than the $1,120 that simple interest gives. $1,425.76 is far too large, so an earlier step went wrong. Choice C, $1,012, is only 1.2% growth, far too small for 4 years at 3%.
Question 10 of 20 · Multiple Choice
A student expands (x - 1)(x³ + x² + x + 1). Which habit best shows "maintaining oversight of the process, while attending to the details"?
Answer: B
Writing every term keeps the details right, and checking that the result is x⁴ - 1, of degree 4, keeps the big picture in view. Choices A, C and D skip details where sign errors hide.
Question 11 of 20 · Multiple Choice
What fraction is equal to 0.363636...?
Answer: A
A two-digit block repeating forever is that block over 99: 36/99 = 4/11. Choice D is the terminating decimal 0.36, and choice B, 36/90, is 0.4. Choice C is 0.2727...
Question 12 of 20 · Multiple Choice
9 × 1 + 2 = 11, 9 × 12 + 3 = 111, 9 × 123 + 4 = 1,111. Following the pattern, what is 9 × 1,234 + 5?
Answer: C
Each step adds a digit to the first factor, adds 1 to the number added, and adds a 1 to the result. With four digits, 1,234, the result has five 1s: 11,111. Check: 9 × 1,234 = 11,106, and 11,106 + 5 = 11,111. Choice A has one 1 too few, and choice D ends with the 5 that is added instead of continuing the 1s.
Question 13 of 20 · Multiple Choice
A student approximates √5 by repeating x(next) = (x + 5/x)/2, starting from x = 2. She gets 2.25 and then 2.2361. Which statement best evaluates these intermediate results?
Answer: A
√5 is between 2 and 3 (since 4 < 5 < 9), and squaring 2.2361 gives about 5.0001, so the results are close and getting closer. Choice D is false: 2² = 4 is less than 5. Choice B expects whole numbers from a method that approaches an irrational number.
Question 14 of 20 · Multiple Choice
A student computes 1² - 0² = 1, 2² - 1² = 3, 3² - 2² = 5 and 4² - 3² = 7. Which general rule expresses the regularity?
Answer: D
The differences are the odd numbers 1, 3, 5, 7, and expanding confirms it: n² - (n² - 2n + 1) = 2n - 1. Choice B is the difference between consecutive results, not the result itself. Choice C fits n = 2 only.
Question 15 of 20 · Short Answer
Use long division to write 7/12 as a decimal. Which part repeats, and how do the remainders show it?
70 ÷ 12 = 5 remainder 10; 100 ÷ 12 = 8 remainder 4; 40 ÷ 12 = 3 remainder 4; 40 ÷ 12 = 3 remainder 4 again. So 7/12 = 0.58333..., with the 3 repeating, because the remainder 4 keeps coming back.
Question 16 of 20 · Short Answer
Write an equation for the line through (3, -2) with slope -2 by generalizing the slope check. Solve it for y and test the point (5, -6).
The check for any point is (y + 2)/(x - 3) = -2, so y + 2 = -2(x - 3) and y = -2x + 4. Test (5, -6): -2(5) + 4 = -6, so the point is on the line.
Question 17 of 20 · Short Answer
Use the cancelling pattern to find 1 + 3 + 9 + 27 + 81 + 243 + 729 without adding term by term. Check that your answer is reasonable.
The sum is 1 + 3 + ... + 3⁶ = (3⁷ - 1)/(3 - 1) = (2,187 - 1)/2 = 1,093. Reasonable: the last term, 729, is about two-thirds of 1,093, as it should be when each term is 3 times the one before.
Question 18 of 20 · Short Answer
A student writes a rule for the sequence 5, 8, 11, 14, ... as aₙ = 5 + 3n and says the 10th term is 35. Check the rule on a term you know, find the error, and give the correct 10th term.
Check n = 1: 5 + 3(1) = 8, but the first term is 5, so the rule is off by one step. The correct rule is aₙ = 5 + 3(n - 1), and the 10th term is 5 + 3 × 9 = 32.
Question 19 of 20 · Short Answer
Figure 1 of a dot pattern has 5 dots, Figure 2 has 9 and Figure 3 has 13. Each figure adds the same number of dots. Write a rule for Figure n and find the number of dots in Figure 20.
Each figure adds 4 dots, so Figure n has 5 + 4(n - 1) = 4n + 1 dots. Check: 4(3) + 1 = 13. Figure 20 has 4(20) + 1 = 81 dots.
Question 20 of 20 · Short Answer
Use (x - 1)(x⁵ + x⁴ + x³ + x² + x + 1) = x⁶ - 1 with x = 10 to explain why 999,999 ÷ 9 = 111,111.
With x = 10: (10 - 1)(100,000 + 10,000 + 1,000 + 100 + 10 + 1) = 10⁶ - 1, that is, 9 × 111,111 = 999,999. So 999,999 ÷ 9 = 111,111. The cancelling pattern explains the familiar fact.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does MP8 mean in math?
MP8 is the Standard for Mathematical Practice "Look for and express regularity in repeated reasoning." Students notice when a calculation repeats and write the repetition once, as a general method, shortcut, equation or formula, while keeping track of details and checking that results are reasonable.
What are the official examples in MP8?
The text gives three: noticing that dividing 25 by 11 repeats the same calculations, so the decimal repeats; abstracting the equation (y - 2)/(x - 1) = 3 from repeated checks of points on a line with slope 3; and noticing how terms cancel in (x - 1)(x + 1), (x - 1)(x² + x + 1) and (x - 1)(x³ + x² + x + 1), which leads to the sum of a geometric series.
How is MP8 different from MP7?
MP7 is about the structure of one object, such as seeing x² + 9x + 14 as (x + 2)(x + 7). MP8 starts from a process repeated several times and asks what stays the same, so the repeated steps can be written once. Many tasks use both.
Why do fractions always give terminating or repeating decimals?
In long division by q, every remainder is a whole number from 0 to q - 1. If a remainder is 0, the decimal ends. Otherwise some remainder must come back, and from then on the same steps repeat, which is exactly the regularity MP8 describes.
Is finding a pattern from a few cases the same as proving it?
No. Repeated cases suggest a rule, and MP8 asks students to express it, but the rule still needs a reason: expanding (10n + 5)², or showing that the terms cancel, explains why it always works. That reasoning connects MP8 to MP3.
What does "maintain oversight of the process" mean?
It means keeping the goal and the overall plan in view while doing the detailed steps. When expanding a long product, a student writes every term (the details) but also checks that only the highest and lowest terms should survive (the oversight).
How can middle school students practice MP8?
Long division of fractions such as 4/7, mental shortcuts such as multiplying by 11, and growing patterns of tiles or tables are good grade 6-8 tasks. In grade 8, repeated slope checks lead to the equation of a line.
How does MP8 appear in high school?
Students derive the formula for the sum of a geometric series, write explicit and recursive rules for sequences, and build formulas such as 750(1.06)ⁿ from repeated percent growth. Each one turns repeated steps into a single expression.
How can I assess MP8?
Ask students to show a few cases, state the general rule with a variable, explain why it works, and check it against a known case. The homework rubric above has a row for each, plus one for judging intermediate results.
Which content standards are good for practicing MP8?
Good fits include converting fractions to decimals by long division (7.NS.A.2), repeating decimal expansions (8.NS.A.1), deriving y = mx + b from slope (8.EE.B.6), sequences written recursively and explicitly (HSF.BF.A.2) and the sum of a finite geometric series (HSA.SSE.B.4).
07
Related Standards
5 standards
These content standards are good places to practice MP8: earlier standards to start with, standards at the same level, and later courses where the practice keeps paying off.
Before this lesson
7.NS.A.2Prerequisite
Multiply and divide rational numbers, including converting fractions to decimals
