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    In a four‑digit number, the sum of the digits is 10. All the digits are different. What is the largest such four‑digit number?
    Problems
    Grades: 3rd to 5th, 9th to 12th, 6th to 8th
    Num & Ops Base Ten
    Mathematical Practices
    Generalize place value understanding for multi-digit whole numbers.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, 4.NBT.A.2

    Place a number in each of the following empty boxes so that the sum of the numbers in any 3 consecutive boxes is 2013. What is the number that should go in the box with the question mark?

         607                       724   ?                 
    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Num & Ops Base Ten
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    4.NBT.B.4, 3.NBT.A.2, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2

    There are 4! = 24 ways to rank four objects. However, a friend told me that if ties are allowed, the number increases to 75.

    I attempted to list all the possibilities by first listing the 24 orderings of four objects, then using brackets to group ties involving two players, then group ties involving three players, and finally the single case in which all four objects are tied. But something has gone wrong; my list includes just 69 possibilities, not 75.

    What happened? Did I miss something, or was my friend mistaken?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Attend to precision.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6

    The rectangle shown consists of eight squares. The length of each side of each square is 1 unit. The length of the shortest path from A to C using the lines shown is 6 units. 

    How many different six-unit paths are there from A to C?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Attend to precision.
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6
    The length and width of a rectangle are whole numbers of centimeters. Neither is divisible by 6. The area of the rectangle is 36 square centimeters. What is the perimeter of the rectangle, in centimeters?
    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Algebraic Thinking
    Measurement & Data
    Make sense of problems and persevere in solving them.
    Gain familiarity with factors and multiples.
    Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
    Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
    Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
    4.MD.A.3, 3.MD.C.7b, 3.MD.D.8, 4.OA.B.4, CCSS.Math.Practice.MP1

    A calendar year is typically referred to as a four‑digit number, as in 2008, or as a two‑digit number, as in ’08. Sometimes, the two‑digit number divides evenly into the four‑digit number, with no remainder.

    How many times did this happen during the twentieth century?

    Problems
    Grades: 6th to 8th, 3rd to 5th, 9th to 12th
    The Number System
    Num & Ops Base Ten
    Mathematical Practices
    Algebraic Thinking
    Compute fluently with multi-digit numbers and find common factors and multiples.
    Perform operations with multi-digit whole numbers and with decimals to hundredths.
    Attend to precision.
    Make sense of problems and persevere in solving them.
    Multiply and divide within 100.
    3.OA.C.7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6, 5.NBT.B.6, 6.NS.B.2

    What is the smallest positive number with exactly ten positive integer divisors?

    And what is the next one after that?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Algebraic Thinking
    Make sense of problems and persevere in solving them.
    Gain familiarity with factors and multiples.
    4.OA.B.4, CCSS.Math.Practice.MP1

     7772 means 777 × 777,
    7773 means 777 × 777 × 777,
    and so on.

    Suppose 7777 is completely multiplied out. What is the units digit of
    the resulting product?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Num & Ops Fractions
    Algebraic Thinking
    Expression/Equation
    Attend to precision.
    Make sense of problems and persevere in solving them.
    Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
    Multiply and divide within 100.
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    6.EE.A.1, 3.OA.C.7, 5.NF.B.5a, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
    Would you rather work seven days at $20 per day or be paid $2 the first day and have your salary double every day for a week?
    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Algebraic Thinking
    Functions
    Num & Ops Base Ten
    Look for and make use of structure.
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    Solve problems involving the four operations, and identify and explain patterns in arithmetic.
    Interpreting Functions
    Generalize place value understanding for multi-digit whole numbers.
    4.NBT.A.2, HSF-IF.A.3, 3.OA.D.9, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP7
    There are 5 houses on a street: house A, B, C, D and E. The distance between any two adjacent houses is 100 feet.  There are 2 children living in house A, 3 children living in house B, 4 children living in house C, 5 children living in house D and 6 children living in house E.   If the school bus can only make one stop on that street, in front of which house should the bus stop so that the sum of walking distance among all children will be the least?
    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Attend to precision.
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6
    Jada has 1 penny, 2 nickels, and 1 dime. How many different sums of money can she make, if she uses at least one coin?
    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Attend to precision.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
    If the sum of three numbers equals zero, and the sum of their cubes equals 90, what is their product?
    Problems
    Grades: 6th to 8th, 9th to 12th, 3rd to 5th
    Expression/Equation
    Mathematical Practices
    The Number System
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Attend to precision.
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
    Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
    7.NS.A.3, 7.NS.A.1d, 7.EE.B.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, 6.EE.A.1
    What is the smallest integer n > 1 for which 3n > n9?
    Problems
    Grades: 6th to 8th, 3rd to 5th, 9th to 12th
    Expression/Equation
    Num & Ops Base Ten
    Mathematical Practices
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Generalize place value understanding for multi-digit whole numbers.
    Attend to precision.
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, 4.NBT.A.2, 6.EE.A.1

    Write the number 41 in a box. Now move in a counterclockwise direction, creating new boxes and each time adding 1 to the number inside. This spiral starts out as follows.

    Note that the numbers in bold — 41, 43, and 47 — are prime numbers (numbers whose only divisors are themselves and 1), and they occur along a diagonal. If you keep filling in the spiral, what is the first number that is not prime to appear along this diagonal?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Algebraic Thinking
    Make sense of problems and persevere in solving them.
    Gain familiarity with factors and multiples.
    4.OA.B.4, CCSS.Math.Practice.MP1

    The letters of EAT can be rearranged to become TEA by moving the third letter (T) to the first position and by moving the other letters one position to the right. This process could be described as 1→2, 2→3, 3→1. When this same process is applied again, then TEA becomes ATE.

    Similarly, the process 1→3, 2→2, 3→1, 4→4 will convert TONE to NOTE. When the process is applied again, NOTE returns to TONE. (Not very interesting, is it?)

    Your challenge begins with the five letters A, E, M, S, and T. Use them to form a common English word. Then, rearrange the letters to form a second common English word. Finally, apply the same process of rearrangement to form a third common English word.

    Can you do it?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2

    Label the ten points in the grid shown with the letters A-J so that

    AB < BC < CD < … < HI < IJ.

    Problems
    Grades: 9th to 12th, 3rd to 5th, 6th to 8th
    Geometry
    The Number System
    Mathematical Practices
    Expressing Geometric Properties with Equations
    Graph points on the coordinate plane to solve real-world and mathematical problems.
    Solve real-world and mathematical problems involving area, surface area, and volume.
    Apply and extend previous understandings of numbers to the system of rational numbers.
    Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
    Make sense of problems and persevere in solving them.
    Understand and apply the Pythagorean Theorem.
    8.G.B.8, CCSS.Math.Practice.MP1, 4.G.A.1, 6.NS.C.8, 6.G.A.3, 5.G.A.1, HSG-GPE.B.6, HSG-GPE.B.7
    Students in Mrs. Walker’s classroom had an estimation contest.  The student whose estimate is the closest to the number of marbles in a jar wins the contest. Vicki, who estimated 135 marbles, won the contest. Timothy, who estimated 150 marbles, got second place. Lyon, who estimated 152, got third place. And Quinn, who estimated 131, got fourth place. What is the exact number of marbles in the jar?
    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Attend to precision.
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6

    Three lines cut rectangle ABCD into six congruent (identical) squares. 

    The perimeter of rectangle ABCD is 30 cm. What is the area of the shaded region, in square centimeters?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Measurement & Data
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
    Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
    4.MD.A.3, 3.MD.C.6, 3.MD.C.5a, 3.MD.C.5b, 3.MD.C.7a, 3.MD.C.7b, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
    Suppose you found an old roll of 15¢ stamps. Can you use a combination of 33¢ stamps and 15¢ stamps to mail a package for exactly $1.77?
    Problems
    Grades: 6th to 8th, 3rd to 5th, 9th to 12th
    Ratio & Proportion
    Num & Ops Base Ten
    Mathematical Practices
    Algebraic Thinking
    Understand ratio concepts and use ratio reasoning to solve problems.
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    Look for and make use of structure.
    Attend to precision.
    Reason abstractly and quantitatively.
    Make sense of problems and persevere in solving them.
    Multiply and divide within 100.
    3.OA.C.7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP7, 4.NBT.B.5, 6.RP.A.3a
    How many different color patterns can be created by placing the circles onto the 4 × 4 grid such that each circle is placed on a square with the same number?
    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Mathematical Practices
    Attend to precision.
    Make sense of problems and persevere in solving them.
    CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
    1 - 20 of 144 results