MP1: Make Sense of Problems and Persevere in Solving Them
In plain English: MP1 is the first Common Core Standard for Mathematical Practice. It asks students to make sense of a problem before solving it: find the givens, constraints and goal, plan a pathway, try simpler or special cases, monitor progress and change course, and check the answer by a different method. It applies in every grade 6-12 math course.
Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
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.MP1 · Official standard
MP1 describes what proficient problem solvers do before, during and after they solve: they explain the problem to themselves, look for an entry point (a first step that gets the work moving), analyze the givens (what is known), the constraints (the limits an answer must respect) and the goal, and make a conjecture about what the answer will look like. While they work, they try simpler or special cases, monitor their progress and change course when a plan is not working. At the end they check the answer by a different method and ask, "Does this make sense?"
This lesson teaches the practice directly, using short tasks from several grade 6-12 domains: ratios, expressions and equations, functions, geometry and statistics. Each task is labeled with a typical course, so a teacher can keep the tasks that fit the class. The practice is the same in every course; only the content changes.
Learning Objectives
By the end of this lesson, students will be able to:
Restate a problem in their own words and name its givens, constraints and goal before calculating
Make a conjecture about the form of an answer and choose a solution pathway
Use simpler forms and special cases to find a pattern that solves the original problem
Notice when a plan or a graphing window is not working and change course
Check an answer with a different method, compare two approaches and decide whether the result makes sense
Prior Knowledge Required
Students should already be comfortable with the content of the tasks they will use:
Ratio and rate reasoning 6.RP.A.3
Solving multi-step problems with rational numbers 7.EE.B.3
Tables, graphs and equations of linear functions 8.F.B.4
For the Algebra I tasks: graphing quadratic functions and completing the square HSA.SSE.B.3
Project the prompt. Students may not calculate for the first three minutes. They write what they notice and what they wonder on sticky notes, then share with a partner.
Warm-Up Prompt
"The student council is ordering T-shirts. Company A charges a $45 setup fee plus $6 per shirt. Company B has no setup fee and charges $9 per shirt. Before you calculate anything: what question could be asked here? Which company would you guess is cheaper for 10 shirts? For 30 shirts? What kind of answer do you expect: one number, a range, or a rule?"
Collect conjectures. Many students guess that Company B is cheaper for small orders and Company A for large ones, so the answer to "Which is cheaper?" is a rule with a break-even point (the number of shirts where both cost the same). Only then let students test the conjecture: 6n + 45 = 9n gives n = 15, so Company B is cheaper below 15 shirts and Company A above. Point out what happened: the class made sense of the problem and predicted the form of the answer before computing.
Direct Instruction20 minutes
Introduce the cycle in Diagram 1, then model each habit with one worked example. Think aloud: say what you notice, what you try first and why you change course.
Make sense: restate the problem, then list the givens, the constraints and the goal.
Plan: make a conjecture about the answer (its size, its units, its form) and pick a pathway.
Simplify: if the problem is big, solve a smaller or special case first and look for a pattern.
Monitor: every few minutes ask, "Is this getting me closer?" If not, change course.
Check: confirm the answer with a different method and ask, "Does this make sense?"
Givens, constraints, goal (Grade 6-7 ratios)
A class of 28 students makes trail mix with 3 cups of granola for every 2 cups of raisins. Each student gets 1/2 cup of mix. How many cups of granola are needed?
Equation: Goal: cups of granola. Entry point: the total mix is 28 × 1/2 = 14 cups. Granola is 3 of every 5 cups, so 3/5 × 14 = 8.4 cups (and 5.6 cups of raisins).
Special cases and a pattern (Grade 8 / Algebra I)
In a league, each of 12 teams plays every other team once. How many games are played? Start with 2, 3, 4 and 5 teams.
Equation: Games: 1, 3, 6, 10. Each of n teams plays n - 1 others, and each game is counted twice, so games = n(n - 1)/2. For 12 teams: 12 × 11/2 = 66 games.
Change course: transform the expression (Algebra I)
A toy rocket is launched from a 1 m platform. Its height is h = -5t² + 30t + 1 meters after t seconds. In the standard window the graph runs off the screen.
Equation: Complete the square: h = -5(t - 3)² + 46. The top is 46 m at t = 3 s, so use the window t from 0 to 7 and h from 0 to 50 (Diagram 2).
Check by a different method (Grade 6-7 statistics)
Maya's first four test scores are 78, 85, 91 and 82. What score on the fifth test gives a mean of 85?
Equation: Method 1: 5 × 85 = 425 and 78 + 85 + 91 + 82 = 336, so she needs 425 - 336 = 89. Method 2: distances from 85 are -7, 0, +6, -3, a total of -4, so the fifth score must be 85 + 4 = 89. It makes sense: 89 is a possible score.
Two approaches, one answer (Grade 7 geometry)
A rectangular garden 12 m by 8 m has a path 1.5 m wide all around it. What is the area of the path?
Equation: Approach A: (15 × 11) - (12 × 8) = 165 - 96 = 69 m². Approach B: two long strips 15 × 1.5 = 22.5 m² each and two short strips 8 × 1.5 = 12 m² each, total 69 m². Both count each corner square once.
After Example 3, show Diagram 2. Ask: "What would a student who did not change course conclude?" (That the graph is broken, or that the rocket never comes down.) After Example 5, ask students to point to the part of Approach B that matches the 15 in Approach A. The long strips include the corners, which is why they are 15 m long, not 12 m.
Guided Practice15 minutes
Pairs solve one problem together, writing each stage of the cycle in a separate box on their paper. Stop the class after each stage to share.
Problem (Grade 7-8 geometry): A rectangle has a perimeter of 46 cm and an area of 120 cm². Find its length and width.
Make sense: the givens are the perimeter and the area; the constraint is that length and width are positive; the goal is both dimensions. Half the perimeter, 23 cm, is length + width.
Plan: conjecture that the sides are whole numbers. List pairs that add to 23 and test their products.
Monitor: if the list gets long, switch pathways and write an equation, w(23 - w) = 120.
Solution: 8 + 15 = 23 and 8 × 15 = 120, so the rectangle is 15 cm by 8 cm.
Check by a different method: w² - 23w + 120 = 0 factors as (w - 8)(w - 15) = 0, giving the same two numbers.
Ask: "Which pathway would you choose if the area were 125 cm²?" (No whole-number pair works, so the equation is the better path.)
Independent Practice15 minutes
Students work alone on a task that rewards persistence. Tell them that being stuck for a while is expected, and that the goal is to try a simpler case, not to finish fast.
Task (Grade 7-8): How many squares of all sizes are on a 5-by-5 grid? Count the 1-by-1 squares, then the 2-by-2 squares, and so on.
Simpler cases first: a 1-by-1 grid has 1 square, a 2-by-2 grid has 4 + 1 = 5, a 3-by-3 grid has 9 + 4 + 1 = 14.
Pattern: the counts are sums of square numbers.
For the 5-by-5 grid: 25 + 16 + 9 + 4 + 1 = 55 squares.
Students who finish write a sentence on how they would count the squares on a 10-by-10 grid without drawing it. Circulate and ask, "What did you try first? What did you change?"
Closure5-10 minutes
Exit ticket (3 questions):
Name one entry point you used today and the problem you used it on.
Describe one moment when you changed course. What told you the first plan was not working?
A student says the answer to a bus problem is 4.3 buses. Write the question you would ask the student.
Read a few answers aloud and connect them to the stages of Diagram 1.
Differentiation Strategies
For Struggling Students
Give a planning organizer with four boxes: "I know", "I need to find", "The answer must", "My first step"
Offer a smaller version of each problem first, such as 4 teams instead of 12 or a 3-by-3 grid instead of 5-by-5
Let students act out or draw the situation before writing any equation
For Advanced Students
Ask for a formula for the number of squares of all sizes on an n-by-n grid, and a check of the formula for n = 4
Ask students to write a problem in which the obvious first strategy fails, and to explain the better entry point
Challenge: find the dimensions of a rectangle with perimeter 46 cm and area 125 cm², and explain why guessing whole numbers cannot work
Assessment Guidance
What to Look For
Listen for planning language before calculation: "I need to find...", "The answer should be about...", "Let me try a smaller case." Strong students state a constraint and use it to reject an answer, change pathways when a plan stalls, and check with a second method instead of redoing the same steps. Watch for students who start computing with the first two numbers they see, who stop at a decimal number of people or buses, or who abandon a problem because the first graph window shows nothing.
02
Classroom Activities
3 Activities
1
Three Reads and a Plan
20 minGroups of 3-4
Each group gets a problem card with the question covered by a sticky note. The group reads the situation three times, each time for a different purpose, and writes a plan before it uncovers the question and solves.
The Three Reads
Read 1: What is the situation about? Retell it in one sentence, with no numbers.
Read 2: What are the quantities? List the givens, their units and any constraints.
Read 3: Uncover the question. Name the goal and make a conjecture about the answer: its size, units and form.
Problem Cards (with answers for the teacher)
Card A (Grade 6-7 ratios): Lemonade is mixed 1 part concentrate to 4 parts water. How many 250 mL cups can be filled from a 2.5 L bottle of concentrate? (12.5 L of lemonade, so 50 cups.)
Card B (Grade 8 geometry): A 13 ft ladder leans against a wall with its foot 5 ft from the wall. How high up the wall does it reach? (12 ft.)
Card C (Grade 6-7 statistics): The mean of 24 test scores is 72. One score of 96 was entered as 69. What is the correct mean? (The sum rises by 27, from 1,728 to 1,755, so the mean is 73.125, about 73.1.)
Card D (Grade 8 functions): Gym 1 charges $25 to join plus $15 per month. Gym 2 charges $40 per month and no fee. After how many months is Gym 1 cheaper? (25 + 15m < 40m when m > 1, so from month 2 on; both cost $40 for one month.)
Procedure
Groups record each read on chart paper before solving
After solving, the group compares the answer with its conjecture from read 3 and writes one sentence on what changed
Rotate cards so every group does at least two
Discussion Questions
On which card did the second read change what you thought the problem was about?
Which card had a constraint that ruled out an answer? (Card D: one month is a tie, not a win.)
How close were your conjectures to the final answers?
2
Viewing Window Detectives
20 minPairs
Pairs graph three functions in the standard window (x and y from -10 to 10), notice that each graph hides its important features, and change course: they rewrite an expression or use a table to choose a better window. This is the "older students" example in the text of MP1.
The Three Functions (Algebra I)
f(x) = x² - 36x + 260. In the standard window only a short, steep piece shows near the right edge, ending at (10, 0). Completing the square gives (x - 18)² - 64, so the vertex is (18, -64) and the zeros are 10 and 26. A good window: x from 0 to 30, y from -80 to 280.
g(x) = -2x + 90. Only a blank screen appears. A table shows g(0) = 90 and g(45) = 0. A good window: x from 0 to 50, y from 0 to 100.
p(x) = 1,000(1.05)ˣ, the value of $1,000 growing 5% per year. The screen is blank. A table shows p(10) is about 1,629. A good window: x from 0 to 20, y from 0 to 3,000.
Procedure
Graph each function in the standard window first and write what you see
Write how you know the window is hiding something (a table value, the expression's structure, or the context)
Choose a new window, sketch the graph and label one key feature with its coordinates
Discussion Questions
For which function did a table help more than the expression?
What would a student who never changed the window say about g(x)?
How does the context of p(x) tell you which x-values make sense?
Modification for Distance Learning
Pairs use a free online graphing tool in a shared screen and paste a screenshot of the first and the final window into a shared slide.
3
Two Roads to One Answer
20 minPairs, then groups of 4
Partners solve the same problem by two different methods, then a second pair explains how each step of one method matches a step of the other. Students practice checking by a different method and understanding the approach of others.
The Problem (Grade 8 / Algebra I)
A theater has 20 rows. The first row has 12 seats and each row has 2 more seats than the row in front of it. How many seats are there in all?
Answer Key for the Teacher
Method 1 (list and add): 12, 14, 16, ..., 50. The last row has 12 + 19 × 2 = 50 seats; adding the 20 numbers gives 620.
Method 2 (pair the rows): row 1 + row 20 = 12 + 50 = 62, row 2 + row 19 = 62, and so on, giving 10 pairs, so 10 × 62 = 620.
Correspondence: the 62 in Method 2 is the sum of the first and last numbers of the list in Method 1, and the 10 is half of the 20 rows.
Procedure
Partner A lists and adds; Partner B pairs the rows. Neither looks at the other's work until both finish
Partners compare answers. If they differ, each checks the other's method, not their own
Join another pair: each pair explains the other pair's method in its own words
Challenge Variation
Groups write a formula for the number of seats in n rows and test it with n = 20 and n = 1.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Problem-Solving Cycle Behind MP1
MP1 is not a straight line from question to answer. Students make sense of the problem, plan, carry out the plan and keep monitoring. The red loop is "change course if necessary"; the green loop is checking with a different method and asking "Does this make sense?"
Diagram 2: Changing the Viewing Window for h = -5t² + 30t + 1
The same toy-rocket model, h = -5t² + 30t + 1 (h in meters, t in seconds), drawn to scale in two windows. In the standard window the curve leaves the top of the screen, so a student might give up. Rewriting the model as h = -5(t - 3)² + 46 shows the highest point, so the window t from 0 to 7 and h from 0 to 50 shows the whole flight.
04
Homework Assignment
~30 min
MP1 Homework: Make Sense, Plan, Persevere, Check
Directions: For every problem, write the givens, the constraints and the goal before you calculate. Show any simpler cases you try. End each problem with one sentence that answers "Does this make sense?"
Part 1: Make Sense and Plan (Problems 1-3)
A school is taking 214 students and 16 adults on a trip. Each bus holds 48 people, and each bus must carry at least 2 adults. List the givens, the constraints and the goal. How many buses are needed? Check that the adult constraint is met.
Before you calculate, conjecture whether a polygon with 15 sides has more or less than 100 diagonals. Then count the diagonals of polygons with 4, 5 and 6 sides, find the pattern, and find the number of diagonals of the 15-sided polygon.
A water tank is draining at a steady rate. The table shows minutes t and liters V: (0, 500), (2, 450), (4, 400), (6, 350). Describe the situation in words, write an equation for V, and use the equation, the table or a graph to find when the tank is empty. Explain how the 25 in your equation shows up in the table.
Part 2: Monitor, Check and Compare (Problems 4-6)
Graph y = x² - 24x + 140 in the standard window (x and y from -10 to 10) and describe what you see. Then rewrite the expression by completing the square, give the vertex and the zeros, and choose a window that shows all three points.
Jordan spends 3/8 of his money on a video game and then 1/3 of what is left on a book. He has $30 left. How much money did he start with? Solve with an equation, then check by working backward from $30.
A 10-by-10 square of white tiles has a border of gray tiles one tile wide. Ana counts the border as 4 × 10 + 4. Ben counts it as 12 × 12 - 10 × 10. Find both results, explain what each part of each expression counts in the picture, and write both methods for an n-by-n square. Show that the two expressions are equal.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Making Sense
Givens, constraints and goal are listed correctly before calculating
Some are listed or one is wrong
Not listed
Strategy and Persistence
A clear plan, simpler cases where useful, and a change of course when needed
A plan but no simpler cases or no response to a stall
No plan
Accuracy
All answers correct with units
One or two errors
Many errors
Checking
Each answer checked by a different method and judged against the context
Some answers checked, or checked by repeating the same steps
No checks
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Each question asks you to use one part of MP1: making sense of a problem, planning, trying simpler cases, changing course or checking. 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
A bike-share costs $2 to unlock plus $0.15 per minute. Lena has $8. The question is: what is the longest ride she can take? Which is the best first step?
Answer: B
Restating the goal names what to find (minutes) and the constraint (total cost at most $8), which points to 2 + 0.15m ≤ 8. Choice A starts computing with the first two numbers seen, and the product 1.2 has no meaning here. Choice C treats the $2 fee as if it were charged every minute.
Question 2 of 20 · Multiple Choice
A rectangular dog pen will use 60 feet of fencing on three sides; a barn wall forms the fourth side. The owner wants the largest possible area. Which statement is a constraint, not the goal?
Answer: D
A constraint is a limit the answer must respect: only 60 feet of fencing, on three sides. Choices A and B describe the goal. Choice C is a relationship between quantities, not a limit.
Question 3 of 20 · Multiple Choice
Two hoses fill a pool. The first alone takes 6 hours; the second alone takes 3 hours. Before solving, which conjecture about the time with both hoses makes sense?
Answer: C
Adding a second hose can only speed things up, so the time must be less than the faster hose's 3 hours (it is 2 hours: 1/6 + 1/3 = 1/2 pool per hour). Choice A averages the times, and choice D assumes the answer lies between the givens, a common error when students do not picture the situation. Choice B adds times instead of rates.
Question 4 of 20 · Multiple Choice
To find the sum of the first 50 odd numbers, Priya computes 1, 1 + 3, 1 + 3 + 5 and 1 + 3 + 5 + 7. What strategy is she using, and what does it suggest?
Answer: B
Priya is solving simpler forms of the problem. The sums 1, 4, 9 and 16 are square numbers, so the first 50 odd numbers add to 50² = 2,500. Choice A multiplies the last odd number by the count without halving or pairing. Choice C confuses the odd numbers, which grow by 2, with their sums, which do not.
Question 5 of 20 · Multiple Choice
Kai graphs y = 2x² - 48x + 290 in the window x from -10 to 10, y from -10 to 10, and sees almost nothing. What is the best change of course?
Answer: C
Completing the square gives 2(x - 12)² + 2, so the vertex is (12, 2) and the new window shows the vertex and both sides of the curve. Choice D halves 48 without first factoring out the leading 2 and keeps 290 as the constant, so its expansion does not equal the original. Choice A gives up, the opposite of persevering.
Question 6 of 20 · Multiple Choice
A table shows x = 0, 1, 2, 3 with y = 5, 8, 11, 14. Which description and equation both match the table?
Answer: A
At x = 0, y = 5, and y grows by 3 each time x grows by 1, so y = 3x + 5. Choice B swaps the starting value and the rate. Choice C would give 15 at x = 1, not 8. Choice D starts at the value for x = 1.
Question 7 of 20 · Multiple Choice
Sam solves 4(x - 3) = 2x + 10 and gets x = 11. Which check uses a different method from solving the equation again?
Answer: A
Substituting tests the answer in the original equation, which is a different method from solving. Both sides equal 32, so x = 11 is right. Choice B repeats the same method, so it can repeat the same mistake. Choice C uses a false rule: answers need not be whole numbers.
Question 8 of 20 · Multiple Choice
292 students and adults are going on a trip, and each bus holds 40 people. A student divides and gets 7.3. What is the most sensible answer to "How many buses are needed?"
Answer: C
Seven buses carry 7 × 40 = 280 people, which leaves 12 people behind, so 8 buses are needed. Choice A rounds without asking whether the answer fits the situation. Choice B gives a number of buses that cannot exist.
Question 9 of 20 · Multiple Choice
An L-shaped floor is a 10 m by 8 m rectangle with a 4 m by 3 m rectangle cut out of one corner. Ana splits the floor into a 10 m by 5 m piece and a 6 m by 3 m piece. Ben subtracts the cut-out from the big rectangle. Which statement is true?
Answer: C
Ana: 50 + 18 = 68 m². Ben: 80 - 12 = 68 m². The top 3 m strip is 10 × 3 = 30 m² in Ben's picture; taking away the 4 × 3 cut-out leaves Ana's 6 × 3 = 18 m² piece, so the methods correspond. Choice A misreads Ben's method. Choice D adds 50 and 4 × 3 = 12, using the cut-out instead of Ana's 6 by 3 piece.
Question 10 of 20 · Multiple Choice
"A 12-ounce bottle of shampoo costs $4.20 and a 20-ounce bottle costs $6.60. The store is open from 9 a.m. to 9 p.m. Which size costs less per ounce?" Which given is NOT needed?
Answer: B
Analyzing the givens shows the store hours have nothing to do with the goal. The other three are needed to find the unit prices: $0.35 per ounce for the small bottle and $0.33 per ounce for the large one.
Question 11 of 20 · Multiple Choice
Which simpler problem would help most with "What is the last digit of 7¹⁰⁰?"
Answer: A
The last digits of 7¹ to 7⁵ are 7, 9, 3, 1, 7, a cycle of length 4. Since 100 is a multiple of 4, the last digit of 7¹⁰⁰ is 1. Choice B treats the exponent as a factor. Choice C swaps the base and the exponent, a different number entirely.
Question 12 of 20 · Multiple Choice
Lee needs the width of a rectangle with area 84 cm² whose length is 5 cm more than its width. After ten minutes of guessing the widths 1 through 6, he has no answer. What is the best change of course?
Answer: D
Monitoring shows the plan is stuck, so Lee should change course: an equation, or a wider range of guesses. Choice A repeats a plan that has already failed. Choice C draws a conclusion from too few cases. Choice B changes the problem rather than the plan.
Question 13 of 20 · Multiple Choice
A graph starts at (0, 200), falls in a straight line and reaches (8, 0). Which situation matches it?
Answer: B
The graph drops 200 units over 8 units of time, a rate of 25 per unit, from a starting value of 200. Choice A uses the number of weeks as the rate. Choice C starts at 0 and increases. Choice D would drain 1,600 liters, not 200.
Question 14 of 20 · Multiple Choice
"The sum of three consecutive even numbers is 138. Find the numbers." Which plan is the best start?
Answer: D
Consecutive even numbers differ by 2, so n + (n + 2) + (n + 4) = 138 models the problem. Choice B uses consecutive whole numbers, which ignores the constraint "even". Choice C gives up after one case: 40 + 42 + 44 = 126, and the miss is a clue to try larger numbers, not to stop.
Question 15 of 20 · Short Answer
A painter will paint 4 walls, each 3 m tall and 4.5 m wide. One wall has a door 0.9 m wide and 2 m tall that will not be painted. One liter of paint covers 10 m², and paint comes in 4-liter cans. List the givens, the constraint on the cans and the goal, then find how many cans to buy for one coat.
Givens: 4 walls of 3 × 4.5 = 13.5 m², a door of 0.9 × 2 = 1.8 m², 10 m² per liter. Constraint: paint comes only in whole 4-liter cans. Goal: the number of cans. Area: 4 × 13.5 - 1.8 = 52.2 m², so 5.22 liters are needed. One can (4 L) is not enough, so buy 2 cans (8 L). This makes sense: 8 L covers 80 m², more than 52.2 m².
Question 16 of 20 · Short Answer
A row of 1 square uses 4 toothpicks, a row of 2 squares uses 7, and a row of 3 squares uses 10. How many toothpicks does a row of 25 squares use? Explain how the simpler cases helped.
Each new square adds 3 toothpicks to the first one, so a row of n squares uses 3n + 1 toothpicks. For 25 squares: 3(25) + 1 = 76 toothpicks. The cases 1, 2 and 3 showed the pattern of adding 3, which would be hard to see by drawing 25 squares.
Question 17 of 20 · Short Answer
Solve x/4 + x/6 = 15. Then check your answer by a different method.
Multiply both sides by 12: 3x + 2x = 180, so 5x = 180 and x = 36. Check by substituting: 36/4 + 36/6 = 9 + 6 = 15. Another check: x/4 + x/6 = 5x/12, and 5/12 of 36 is 15.
Question 18 of 20 · Short Answer
Two methods find a 15% tip on a $48 bill. Method A: 0.15 × 48. Method B: 10% of 48 is 4.80, half of that is 2.40, and 4.80 + 2.40 is the tip. Find the tip, and explain how the steps of Method B match Method A.
Both give $7.20. Method B splits 0.15 into 0.10 + 0.05. The 4.80 is 0.10 × 48, the 2.40 is 0.05 × 48 (half of 10% is 5%), and adding them is the same as multiplying by 0.15, by the distributive property.
Question 19 of 20 · Short Answer
A student graphs y = 0.5x + 40 in the standard window (x and y from -10 to 10) and sees a blank screen. Explain why, and give a window that shows both intercepts.
For every x from -10 to 10, y is between 35 and 45, above the top of the window. The y-intercept is (0, 40), and the x-intercept is where 0.5x + 40 = 0, at (-80, 0). A window such as x from -100 to 20 and y from -10 to 60 shows both. Choosing it is changing course after monitoring the first graph.
Question 20 of 20 · Short Answer
A recipe for 6 servings uses 2 1/4 cups of flour. Mia scales it to 20 servings and gets 0.675 cups. Does her answer make sense? Find the correct amount.
It does not make sense: 20 servings need more flour than 6, but 0.675 cups is less than 2 1/4. Mia multiplied by 6/20 instead of 20/6. Correct amount: 2.25 × 20/6 = 7.5 cups. Check: 7.5 cups for 20 servings is 0.375 cups per serving, the same as 2.25 ÷ 6.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does MP1 mean in math?
MP1 is the first Standard for Mathematical Practice: "Make sense of problems and persevere in solving them." It describes how students work, not what content they learn. They understand the problem first, plan a pathway, try simpler cases, change course when stuck and check the answer with a different method.
What are the eight Standards for Mathematical Practice?
They are MP1 Make sense of problems and persevere in solving them; MP2 Reason abstractly and quantitatively; MP3 Construct viable arguments and critique the reasoning of others; MP4 Model with mathematics; MP5 Use appropriate tools strategically; MP6 Attend to precision; MP7 Look for and make use of structure; and MP8 Look for and express regularity in repeated reasoning. They apply in every grade, alongside the content standards.
Is MP1 tested on state tests?
Not as a separate item. State tests are built on the content standards, but many multi-step and unfamiliar problems cannot be solved without making sense of them first. Check your state test's published blueprint to see how it describes these items.
What is an entry point to a problem?
An entry point is a first step that gets the work moving, such as finding a total, drawing a picture, making a table or solving a smaller case. In the trail-mix example, the entry point is the total amount of mix, 14 cups; once it is known, the ratio does the rest.
How do I teach perseverance without letting students stay stuck too long?
Plan the struggle. Give a task that has several entry points, set a short time to work alone, then ask what students tried. If a student is stuck, ask a question ("What would a smaller version look like?") rather than giving a step. Praise the change of course, not only the right answer.
What should a student do when stuck on a problem?
Reread the problem and restate the goal, list what is known, and try one of these: a simpler or special case, a picture or table, working backward, or an equation. If a graph shows nothing, change the window. Students can keep a short list of these moves in their notebook.
Does MP1 look different in middle school and high school?
Yes, in the tools, not in the habit. A grade 6 student might use a tape diagram or a table of ratios; an Algebra I student might complete the square or change a graphing window. The text of MP1 itself mentions that older students transform algebraic expressions or change the viewing window to get the information they need.
How can I assess MP1?
Assess the process as well as the answer. Ask students to write the givens, constraints and goal before solving, to show one simpler case, and to check by a second method. A short rubric with rows for making sense, strategy, accuracy and checking works well, like the one in the homework above.
Which content standards are good for practicing MP1?
Any rich, multi-step problem works. Good fits include ratio problems (6.RP.A.3), multi-step problems with rational numbers (7.EE.B.3), systems of linear equations (8.EE.C.8), rewriting expressions to reveal properties (HSA.SSE.B.3) and interpreting key features of graphs (HSF.IF.B.4).
How is MP1 different from MP4, Model with mathematics?
MP1 is about how students approach any problem: understand it, plan, persist and check. MP4 is about building a mathematical model of a real situation, with assumptions and approximations. A modeling task usually needs MP1, but a pure-math puzzle such as counting squares on a grid uses MP1 without any modeling.
07
Related Standards
5 standards
These content standards are good places to practice MP1: earlier standards to start with, standards at the same level, and later courses where the practice keeps paying off.
Before this lesson
6.RP.A.3Prerequisite
Use ratio and rate reasoning to solve real-world and mathematical problems
