In plain English: MP7 is the Common Core Standard for Mathematical Practice about structure. Students look closely for patterns and structure: they split numbers to use the distributive property, read x² + 9x + 14 as (x + 2)(x + 7), see 5 - 3(x - y)² as 5 minus a non-negative amount, draw auxiliary lines in figures, and step back to change perspective. It applies in every grade 6-12 math course.
Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Common Core State Standards for Mathematics · Domain: Standards for Mathematical Practice (MP) · Practice group: Seeing structure and generalizing Also written as CCSS.MATH.PRACTICE.MP7 · Official standard
MP7 asks students to "look closely to discern a pattern or structure" and to use what they see. The official text gives a sequence of examples: seeing 7 × 8 as 7 × 5 + 7 × 3 before the distributive property is formal; seeing, in x² + 9x + 14, "the 14 as 2 × 7 and the 9 as 2 + 7"; and seeing 5 - 3(x - y)² as "5 minus a positive number times a square," so its value cannot be more than 5. In geometry, students recognize the significance of an existing line and draw an auxiliary line (an extra line added to a figure) to solve a problem. They also step back for an overview and shift perspective, and they treat complicated expressions as single objects.
The tasks come from grade 6-12 arithmetic, expressions and equations, quadratics, exponents and geometry, and each is labeled with a typical course.
Learning Objectives
By the end of this lesson, students will be able to:
Split numbers and expressions to use the distributive property for mental calculation
Factor quadratic expressions by seeing the constant as a product and the middle coefficient as a sum
Treat an expression such as (x - y)² or (x - 3) as a single object to find a maximum, a minimum or a solution
Draw an auxiliary line in a figure to find an angle, a length or an area
Step back and change perspective, for example by rewriting powers with a common exponent
Prior Knowledge Required
Students should already be comfortable with:
Applying properties of operations to write equivalent expressions 6.EE.A.3
Angles formed when parallel lines are cut by a transversal 8.G.A.5
Properties of integer exponents 8.EE.A.1
For the Algebra I tasks: multiplying binomials and factoring HSA.SSE.A.2
Run a short number talk. Students compute mentally, then share strategies.
Warm-Up Prompt
"Suppose you forgot that 7 × 8 = 56. How could you get it from facts you know? Now compute 99 × 14 and 25 × 48 in your head. What did you notice about the numbers that made them easy?"
Record strategies. For 7 × 8: 7 × 5 + 7 × 3 = 35 + 21 = 56, which is the official MP7 example of the distributive property. For 99 × 14: 100 × 14 - 14 = 1,386. For 25 × 48: 25 × 4 × 12 = 100 × 12 = 1,200. Name the habit: before computing, look at the structure of the numbers.
Direct Instruction20 minutes
Model five ways of using structure. For each, ask students what they noticed before any calculation.
Split a number into friendly parts to use the distributive property.
Read an expression's parts: which numbers multiply to the constant, and which add to the middle coefficient?
Treat a group as one object: (x - y)², (x - 3), x², and ask what that object can be.
Add a line to a figure when the given lines do not show the relationship.
Step back: rewrite or redraw the problem so that its structure is visible.
The distributive property in arithmetic (Grade 6)
Use 7 × 8 = 7 × 5 + 7 × 3 as a model to compute 6 × 47.
Equation: 14 = 2 × 7 and 9 = 2 + 7, so x² + 9x + 14 = (x + 2)(x + 7) (Diagram 1). For x² - 5x - 24: -24 = -8 × 3 and -5 = -8 + 3, so (x - 8)(x + 3).
An expression as an object (Algebra I)
Show that 5 - 3(x - y)² is never more than 5. Then find the largest value of 10 - (2t - 6)².
Equation: (x - y)² is a square, so it is 0 or positive, and 5 minus 3 times it is at most 5, equal to 5 when x = y. Likewise 10 - (2t - 6)² is at most 10, reached at t = 3.
An auxiliary line (Grade 8 / Geometry)
Lines ℓ and m are parallel. A bent path meets ℓ at 40° and m at 35°. Find the angle at the bend P.
Equation: Draw a line through P parallel to ℓ and m. It splits the angle into alternate interior angles of 40° and 35°, so the angle at P is 75° (Diagram 2).
Shifting perspective (Grade 8 / Algebra I)
Which is larger, 2¹⁰⁰ or 3⁶⁰?
Equation: Rewrite with a common exponent: 2¹⁰⁰ = (2⁵)²⁰ = 32²⁰ and 3⁶⁰ = (3³)²⁰ = 27²⁰. Since 32 > 27, 2¹⁰⁰ is larger, with no calculator needed.
After Example 2, show Diagram 1 and ask students to point to the 2 × 7 and the 2 + 7 in the picture. After Example 4, ask: "Why parallel to ℓ and m? What would a line in another direction give us?" (No known angle relationships.)
Guided Practice15 minutes
Pairs use structure on four items before calculating, then compare strategies with another pair.
(Algebra I) Solve (x - 3)² + 4(x - 3) - 5 = 0 by calling u = x - 3: u² + 4u - 5 = (u + 5)(u - 1) = 0, so u = 1 or u = -5, and x = 4 or x = -2.
(Algebra I) Is (n + 1)² - (n - 1)² always a multiple of 4? See it as a difference of squares: (2n)(2) = 4n. Yes.
Ask: "In item 3, why is it faster not to expand?" (The repeated (x - 3) is one object, so the equation is an ordinary quadratic in u.)
Independent Practice15 minutes
Students solve two geometry problems by drawing an auxiliary line or by stepping back.
(Grade 7) A square has side 10 cm. A second square is drawn inside it with its corners at the midpoints of the big square's sides. What is the area of the inner square? (Join the midpoints of opposite sides. These two segments cut the big square into 4 small squares, and each side of the inner square cuts one small square in half, making 8 congruent triangles. The inner square holds 4 of them, so its area is half of 100, or 50 cm².)
(Geometry) Find the sum of the interior angles of a hexagon. (Draw the 3 diagonals from one vertex: they make 4 triangles, so the sum is 4 × 180° = 720°.)
Students who finish write the general rule for an n-sided polygon and explain it with the same auxiliary lines.
Closure5-10 minutes
Exit ticket:
Factor x² + 8x + 15 and say what you noticed first.
Without graphing, what is the smallest value of 2 + (x + 1)²?
Describe one moment today when an extra line or a new point of view made a problem easier.
Differentiation Strategies
For Struggling Students
Use algebra tiles or grid paper for area models of products such as (x + 2)(x + 7)
Give a factor-pair table: list the pairs that multiply to the constant, then circle the pair with the right sum
Highlight repeated groups in an expression with one color before working with them
For Advanced Students
Ask students to factor x⁴ - 13x² + 36 completely and explain the structure they used
Ask for a proof, with an auxiliary line, that an exterior angle of a triangle equals the sum of the two remote interior angles
Ask which is larger, 5³⁰ or 3⁴⁵, and to invent a similar comparison for a partner
Assessment Guidance
What to Look For
Look for students who pause to look at an expression or figure before computing, and who can say what they noticed ("14 is 2 × 7," "this is a square, so it cannot be negative," "I need a parallel line here"). Strong students treat repeated groups as single objects, choose auxiliary lines that create known angle or triangle relationships, and rewrite problems to make their structure visible. Watch for students who expand everything by habit, who pick factor pairs with the right product but the wrong sum, and who draw extra lines that create no new relationships.
02
Classroom Activities
3 Activities
1
Structure Sort
20 minGroups of 3-4
Groups sort 8 expression cards by their structure, not by their look: difference of squares, perfect square, or a trinomial whose constant is a product and middle coefficient a sum. They factor each card and explain the structure that told them how.
The 8 Cards (with answers for the teacher)
Card 1: x² - 49 = (x - 7)(x + 7), a difference of squares.
Card 8: x² + x - 30 = (x + 6)(x - 5), since -30 = 6 × -5 and 1 = 6 + (-5).
Procedure
Sort first, before factoring anything, and write one sentence per card on the structure you see
Factor each card and check by multiplying back
Find two cards that look different but have the same structure
Discussion Questions
Which cards looked hard but became easy once you saw a square?
How did seeing x + 1 as one object help on Card 5?
Why does Card 6 have a factor that does not break down further?
2
Auxiliary Line Challenges
20 minPairs
Pairs solve three geometry problems that are hard with the given lines and easy with one extra line. They record which line they added and why it helps.
The Three Problems (with answers for the teacher)
Problem A (Grade 8): Two parallel lines; a bent path meets the top line at 30° and the bottom line at 50°. Find the bend angle. (A line through the bend, parallel to both, gives 30° + 50° = 80°.)
Problem B (Grade 7): A trapezoid has parallel bases 6 cm and 10 cm and height 4 cm. Find its area without the trapezoid formula. (A diagonal makes two triangles: (1/2)(6)(4) + (1/2)(10)(4) = 12 + 20 = 32 cm².)
Problem C (Grade 8): An isosceles triangle has legs 13 cm and base 10 cm. Find its area. (The altitude to the base bisects it: √(13² - 5²) = 12 cm, so the area is (1/2)(10)(12) = 60 cm².)
Procedure
Try each problem for two minutes with only the given lines
Then add one line, mark the new relationships (parallel, perpendicular, equal) and solve
Explain to another pair why your line was the right one to add
Discussion Questions
What do the three helpful lines have in common?
In Problem C, why does the altitude land at the midpoint of the base?
Did any pair use a different auxiliary line in Problem B? Did it also work?
Modification for Distance Learning
Share the figures in dynamic geometry software; pairs add their line and paste a screenshot with labels.
3
Highest and Lowest Without a Graph
20 minGroups of 3
Groups find the largest or smallest value of four expressions by seeing their structure, as in the official example 5 - 3(x - y)², then check one of them with a table or a graph.
The Expressions (with answers for the teacher)
7 + (x - 2)²: the square is at least 0, so the smallest value is 7, at x = 2.
12 - 4(x + 1)²: the largest value is 12, at x = -1.
1/(x² + 1): the denominator is at least 1, so the largest value is 1, at x = 0.
(x² - 4)² + 3: the smallest value is 3, at x = 2 and x = -2.
Procedure
For each expression, circle the part that cannot be negative and decide whether it is added or subtracted
Write the largest or smallest value and where it happens, with a one-sentence reason
Check one expression with a table of values from x = -3 to 3
Challenge Variation
Write an expression whose largest value is 6, reached only at x = -4, and one with no largest value but a smallest value of -2.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Seeing the Structure of x² + 9x + 14
The official MP7 example as an area model. A rectangle with sides x + 2 and x + 7 splits into four parts: x², two strips 7x and 2x, and a corner of 2 × 7 = 14. Reading the expression this way, students "see the 14 as 2 × 7 and the 9 as 2 + 7," which is exactly what factoring needs. The strip widths are drawn to scale; x is drawn as 10 units.
Diagram 2: An Auxiliary Line Reveals the Angle
Lines ℓ and m are parallel, and a bent path meets them at angles of 40° and 35°. Drawing the dashed auxiliary line through the bend P, parallel to both, splits the angle at P into two parts. Each part is an alternate interior angle equal to one of the given angles, so the angle at P is 40° + 35° = 75°. The figure is drawn to scale.
04
Homework Assignment
~30 min
MP7 Homework: Find the Structure, Then Use It
Directions: Before you compute, write one sentence about the structure you notice. Then use it. Check each answer another way.
Part 1: Structure in Numbers and Expressions (Problems 1-3)
Use the distributive property to compute mentally, and show how you split each number: (a) 8 × 97; (b) 15 × 24; (c) 6 × 7.5.
Factor by seeing the structure, and say what you saw: (a) x² + 13x + 36; (b) x² - 2x - 35; (c) 9x² - 25; (d) (x + 2)² - 9.
Without graphing, find the largest or smallest value of each expression and the value of x where it happens: (a) 8 - 2(x - 3)²; (b) (x + 4)² - 6; (c) 3 + 5(2x - 1)².
Part 2: Lines and Perspective (Problems 4-6)
Lines ℓ and m are parallel. A bent path meets ℓ at an angle of 48° and m at an angle of 31°, as in Diagram 2. Find the angle at the bend, and explain which auxiliary line you drew and why the angles are equal.
Solve each equation by treating a repeated expression as one object: (a) (2x + 1)² - 5(2x + 1) + 6 = 0; (b) x⁴ - 13x² + 36 = 0.
Which is larger, 5³⁰ or 3⁴⁵? Decide without a calculator by rewriting both with the same exponent, and explain your steps.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Noticing Structure
The structure is named before computing
Named for some problems
Not named
Using Structure
The structure is used to simplify the work
Correct answers by longer methods
No use of structure
Auxiliary Lines and Perspective
A helpful line or rewriting, with a reason
A line or rewriting without a reason
Missing
Accuracy and Checking
All answers correct and checked another way
One or two errors
Many errors
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Each question asks you to notice a structure in a number, an expression or a figure, and to use it. 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 expression equals 9 × 47 and is easy to compute mentally?
Answer: B
47 = 40 + 7, so by the distributive property 9 × 47 = 9 × 40 + 9 × 7 = 360 + 63 = 423. Choice A multiplies the parts instead of adding the products. Choice C adds instead of multiplying.
Question 2 of 20 · Multiple Choice
Which is the factored form of x² + 11x + 28?
Answer: D
28 = 4 × 7 and 11 = 4 + 7, so x² + 11x + 28 = (x + 4)(x + 7). Choice A has the right product, 28, but its sum is 16. Choice C has the right product but gives -11x. Choice B uses the coefficients themselves as the numbers.
Question 3 of 20 · Multiple Choice
If x² + bx + 30 = (x + 5)(x + 6), what is b?
Answer: B
The middle coefficient is the sum 5 + 6 = 11, and the constant is the product 5 × 6 = 30. Choice A confuses the sum with the product. Choice C subtracts. Choice D adds 30 and 5.
Question 4 of 20 · Multiple Choice
What is the largest value of 9 - 2(x + 3)²?
Answer: C
(x + 3)² is a square, so it is 0 or more, and subtracting 2 times it from 9 gives at most 9, reached when x = -3. Choice A is the x-value where the largest value occurs, not the value. Choice D doubles 9.
Question 5 of 20 · Multiple Choice
For which value of x does 4 + (x - 5)² have its smallest value?
Answer: A
The square (x - 5)² is smallest, 0, when x = 5, so the expression is smallest there, with value 4. Choice D gets the sign wrong. Choice B gives the smallest value instead of the x-value.
Question 6 of 20 · Multiple Choice
Lines ℓ and m are parallel. A bent path meets ℓ at 25° and m at 60°. Using an auxiliary line through the bend, what is the angle at the bend?
Answer: A
A line through the bend parallel to ℓ and m splits the angle into alternate interior angles of 25° and 60°, so the bend angle is 85°. Choice C subtracts the angles. Choice B is 180° - 85°, the supplement.
Question 7 of 20 · Multiple Choice
Which auxiliary lines make it easiest to find the area of a regular hexagon with side 4?
Answer: B
Segments from the center to the vertices split a regular hexagon into 6 equilateral triangles with side 4, each with area (√3/4)(16) = 4√3, so the hexagon has area 24√3. Choice A also works but leaves harder trapezoids. Choice D adds a curve that gives no new lengths.
Question 8 of 20 · Multiple Choice
Which expression has the structure of a difference of squares?
Answer: D
(x + 3)² - 25 is (something)² - 5², so it factors as (x + 3 - 5)(x + 3 + 5) = (x - 2)(x + 8). Choices A and B are sums of squares, which do not factor over the real numbers. Choice C has a cube, not a square.
Question 9 of 20 · Multiple Choice
Which expression equals 39 × 41?
Answer: A
39 × 41 = (40 - 1)(40 + 1) = 40² - 1² = 1,599. Choice D has the wrong sign. Choice B subtracts 2 instead of 1². Choice C is 1,562.
Question 10 of 20 · Multiple Choice
Which is larger, 2⁶⁰ or 3⁴⁰?
Answer: D
With a common exponent, 2⁶⁰ = (2³)²⁰ = 8²⁰ and 3⁴⁰ = (3²)²⁰ = 9²⁰. Since 9 > 8, 3⁴⁰ is larger. Choice A looks only at the exponents. Choice B multiplies base and exponent, which has no meaning for powers.
Question 11 of 20 · Multiple Choice
Solve (x - 2)² = 49 by treating x - 2 as a single object.
Answer: C
If (x - 2)² = 49, then x - 2 = 7 or x - 2 = -7, so x = 9 or x = -5. Choice A forgets the negative root. Choice B adds 2 to 49 before taking any square root. Choice D adds and subtracts in the wrong places.
Question 12 of 20 · Multiple Choice
Which description gives the structure of 3(x + 2)² - 5?
Answer: A
The expression is 3 times the square of (x + 2), then 5 is subtracted, so it is never less than -5. Choice D ignores the square: expanded, the expression has degree 2. Choice B counts terms in a form that is not written.
Question 13 of 20 · Multiple Choice
Which auxiliary lines best help find the sum of the interior angles of a pentagon?
Answer: C
The diagonals from one vertex cut a pentagon into 3 triangles, whose angles together make up the pentagon's angles: 3 × 180° = 540°. Choice B forgets to subtract the 360° around the center point, which is not part of the pentagon's angles. Choice A is wrong about the shapes: one diagonal splits a pentagon into a triangle and a quadrilateral, not two triangles.
Question 14 of 20 · Multiple Choice
Which factorization of 16x⁴ - 1 shows its structure completely over the integers?
Answer: C
16x⁴ - 1 = (4x²)² - 1² = (4x² - 1)(4x² + 1), and 4x² - 1 is again a difference of squares: (2x - 1)(2x + 1). The factor 4x² + 1 does not factor over the integers. Choice A is equal to 16x⁴ - 1 but stops one step early. Choice B is not equal: it has middle terms such as -32x³. Choice D gives 6x², which does not cancel.
Question 15 of 20 · Short Answer
Show how 12 × 15 = 12 × 10 + 12 × 5, and then use the same structure to compute 12 × 115.
Factor x² - 3x - 40. Explain how the structure tells you which numbers to look for.
Look for two numbers whose product is -40 and whose sum is -3: -8 and 5. So x² - 3x - 40 = (x - 8)(x + 5). Check: (x - 8)(x + 5) = x² + 5x - 8x - 40 = x² - 3x - 40.
Question 17 of 20 · Short Answer
Explain why 7 - (2x - 1)² can never be greater than 7, and find the value of x for which it equals 7.
(2x - 1)² is a square, so it is 0 or positive. Subtracting it from 7 gives 7 or less. The value is exactly 7 when (2x - 1)² = 0, that is, when x = 1/2.
Question 18 of 20 · Short Answer
An isosceles triangle has a base of 16 cm and two legs of 10 cm. Draw the altitude to the base, and use it to find the height and the area.
The altitude from the top vertex meets the base at its midpoint, making two right triangles with legs 8 cm and h and hypotenuse 10 cm. h = √(10² - 8²) = 6 cm, and the area is (1/2)(16)(6) = 48 cm².
Question 19 of 20 · Short Answer
Solve x⁴ - 10x² + 9 = 0 by seeing it as a quadratic in x².
Let u = x². Then u² - 10u + 9 = (u - 1)(u - 9) = 0, so u = 1 or u = 9. Since u = x², x = ±1 or x = ±3, four solutions.
Question 20 of 20 · Short Answer
Without expanding everything, decide whether (x + 3)(x - 3) - (x² - 9) equals 0 for every x. Explain using structure.
(x + 3)(x - 3) has the structure (a + b)(a - b), which always equals a² - b² = x² - 9. Subtracting x² - 9 from itself gives 0 for every x.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does MP7 mean in math?
MP7 is the Standard for Mathematical Practice "Look for and make use of structure." Students notice how numbers, expressions and figures are built, and they use that structure: splitting numbers, factoring, treating a group as one object, drawing an auxiliary line or changing their point of view.
What are the official examples in MP7?
The text mentions seeing 7 × 8 as 7 × 5 + 7 × 3, seeing in x² + 9x + 14 "the 14 as 2 × 7 and the 9 as 2 + 7," drawing an auxiliary line in a geometric figure, and seeing 5 - 3(x - y)² as "5 minus a positive number times a square," so its value cannot be more than 5.
Is MP7 the same as factoring?
Factoring is one important use of structure, but MP7 is broader. It includes mental arithmetic with the distributive property, finding maximum and minimum values, comparing powers, and adding lines to figures in geometry.
What is an auxiliary line?
An auxiliary line is an extra line drawn in a figure to reveal a relationship, such as a line parallel to two given lines, an altitude in an isosceles triangle, or a diagonal that splits a polygon into triangles. A good auxiliary line creates angles or triangles you already know facts about.
How is MP7 different from MP8?
MP7 is about seeing structure in a single object: an expression, a number or a figure. MP8 is about noticing that a calculation repeats, and turning that regularity into a general method or formula. Both are about patterns, but MP8 starts from repeated reasoning.
What does "see complicated things as single objects" mean?
It means treating a group such as (x - 3) or x² as one unit. In (x - 3)² + 4(x - 3) - 5 = 0, calling u = x - 3 turns the equation into u² + 4u - 5 = 0. In 5 - 3(x - y)², seeing (x - y)² as "a square" shows at once that it cannot be negative.
How can middle school students practice MP7?
Number talks are a good start: 99 × 14 as 100 × 14 - 14, or 25 × 48 as 100 × 12. Grade 7 students can rewrite expressions to see a problem differently, and grade 8 students can draw auxiliary parallel lines to find angles.
How do I assess MP7?
Ask students to write what they notice before computing, and give credit for using structure, not only for the answer. Questions such as "Without graphing, what is the smallest value of...?" or "Which line would you add?" test structure directly.
Which content standards are good for practicing MP7?
Good fits include generating equivalent expressions (6.EE.A.3), rewriting expressions to shed light on a problem (7.EE.A.2), angles with parallel lines (8.G.A.5), using structure to rewrite expressions (HSA.SSE.A.2) and choosing equivalent forms to reveal properties (HSA.SSE.B.3).
Why does MP7 matter beyond algebra class?
Seeing structure saves work and prevents errors. A student who sees that 5 - 3(x - y)² is at most 5 needs no graph, and one who sees a shared factor can simplify a formula before using it. The same habit helps when reading data tables, code or any complicated system.
07
Related Standards
5 standards
These content standards are good places to practice MP7: earlier standards to start with, standards at the same level, and later courses where the practice keeps paying off.
Before this lesson
6.EE.A.3Prerequisite
Apply the properties of operations to generate equivalent expressions
