Open Middle Exercises

Tim Brzezinski, lianemoler, Mar 20, 2023

Ditch those worksheets! This book serves as a collection of digitally interactive [b][color=#1551b5]Open Middle exercises[/color][/b]. Be sure to check out Open Middle's site: [url]https://www.openmiddle.com/[/url]. This book also contains great resources from Steve Phelps and John Ulbright.

Table of Contents

  1. Arithmetic (K-6)
    1. Building Number Sentences
    2. Equality
    3. Add Some, Subtract Some
    4. Sums to 100
    5. Open Middle Sum Problem
    6. Close to 1000
    7. Open Middle: Comparing and Identifying Fractions on a Number Line
    8. Create A Pattern
    9. Operations with Time 1
    10. Order of Operations 4
    11. Multiplying Two-Digit Numbers
    12. Open Middle: Multiplying 2-Digit Numbers (V2)
    13. Close Products
    14. Fraction Size
    15. Multiplying Fraction by a 2-Digit Whole Number
    16. Multiplying Fractions 4
    17. Distributive Property (Open Middle)
    18. Order of Operations 2
    19. Decimal Product Close To 50 v3
    20. Open Middle Percent Modeling Problem
    21. Creating Equal Ratios: Open Middle Theme
    22. Building Number Sentences: Open Middle Theme (1)
    23. Building Number Sentences: Open Middle Theme (6)
  2. Algebra & Functions (5-12)
    1. Open Middle: Horizontal and Vertical Distances (V1)
    2. Open Middle: Horizontal and Vertical Distances (V2)
    3. Distributive Property: Open Middle Theme
    4. Writing Equivalent Expressions: Open Middle Theme
    5. Writing Equivalent Polynomial Expressions (V2)
    6. Writing Equivalent Polynomial Expressions
    7. Open Middle Multi-Step Equations Problem
    8. Make a Linear Function
    9. Open Middle: Slope and Points Exercise
    10. Open Middle Slope Exercise (V1)
    11. Open Middle Slope Exercise (V2)
    12. Open Middle: Linear Function to Cone
    13. Open Middle: Point-Slope Exercise (1)
    14. Open Middle: Point-Slope Exercise (2)
    15. Parallel Lines and Slope: Open Middle
    16. Open Middle: Parallel and Perpendicular Lines
    17. Open Middle: Equations of Parallel Lines
    18. Perpendicular Lines and Slope: Open Middle
    19. Perpendicular Lines and Slope
    20. Equations of Perpendicular Lines (Open Middle) - V1
    21. Equations of Perpendicular Lines (Open Middle) - V2
    22. Perpendicular Lines Through a Given Point
    23. How Close to Perpendicular Can You Get? Open Middle Challenge
    24. Open Middle: Solving Equations in Two Variables
    25. Open Middle Linear Functions Exercise
    26. Open Middle: Systems of Linear Equations (3)
    27. Open Middle: Systems of Linear Equations (1)
    28. Open Middle: Systems of Linear Equations (4)
    29. Open Middle: Systems of Linear Equations (2)
    30. Open Middle: Systems of Linear Equations (5)
    31. Open Middle: Systems of Linear Equations (6)
    32. FUNCTIONS
    33. Discriminant
    34. Function Notation
    35. Equivalent lines in slope-intercept and standard form
    36. Open Middle: Absolute Value Graphs (1)
    37. Open Middle: Absolute Value Graphs (2)
    38. Open Middle: Absolute Value Graphs (3)
    39. Open Middle: Absolute Value Graphs (4)
    40. Open Middle: Rational Functions Graph (1)
    41. Open Middle: Rational Functions Graph (2)
  3. Geometry (K -12)
    1. Constructing Rectangles: Open Middle Problem
    2. Unit Squares: Multipurpose Template
    3. Open Middle: Perimeter of a Rectangle
    4. Open Middle: Maximizing and Minimizing Area (1)
    5. Multipurpose Composite Figure Templates
    6. Open Middle: Area of a House
    7. Open Middle: Trapezoid Area Problem
    8. Open Middle Trapezoid Area Problem: Choose Your Own Fixed Perimeter
    9. Open Middle: Pythagorean Theorem (1)
    10. Open Middle: Pythagorean Theorem (2)
    11. Creating Rectangular Prisms with Given Surface Area
    12. Open Middle (Revamped): Creating Rectangular Prisms with a Given Surface Area
    13. Maximizing Surface Area: Open Middle Cube Problem (V1)
    14. Maximizing Surface Area: Open Middle Cube Problem (V2)
    15. Open Middle: Linear Function to Cone
    16. Open Middle: Maximizing Rectangular Prism Surface Area
    17. Area of Rectangles
    18. Total Area
    19. Open Middle: Surface Area and Percent
    20. Open Middle: Surface Area and Percent (no feedback version)
    21. Open Middle: Same-Side Interior or Exterior Angles
    22. Open Middle Challenge: Parallel Lines and Related Angles
    23. Similar Shapes
    24. Open Middle Perpendicular Bisector Problem
    25. Equidistant Points 2
    26. Open Middle Problem: Complementary and Supplementary Angles
    27. Open Middle: Supplementary Angles
    28. Open Middle: Triangle Angles
    29. Open Middle: Interior and Exterior Angles of Triangles
    30. Open Middle Triangle Angles Problem
    31. Triangle Inequality: Open Middle Problem (V1)
    32. Triangle Inequality: Open Middle Problem (V2)
    33. Open Middle Triangle Angle Sum Problem
    34. Open Middle Triangle Angle Sum Problem (V2)
    35. Classifying Triangles by Angles
    36. Open Middle: Creating Triangles
    37. Pythagorean Theorem: Building Right Triangles
    38. Creating Distances in the Coordinate Plane: Formative Assessment with Feedback
    39. Open Middle: Triangle Area Problem
    40. Create a Cylinder with Given Lateral Area
    41. Open Middle: Diagonal of a Rectangular Prism
    42. Creating Distances in 3 Space: Quick Formative Assessment
    43. Creating Distances in 3 Space
    44. Equations of Circles (V1): Open Middle Problem
    45. Equations of Circles (V2): Open Middle Problem
    46. Equations of Circles (V3): Open Middle Problem
    47. Equations of Circles (V4): Open Middle Problem
    48. Open Middle: Circle Tangent to Line (1)
    49. Position the Line Tangent to the Circle
    50. Open Middle: Triangle Area Problem (V2)
  4. Coordinate Geometry (7-12)
    1. Open Middle Midpoint Exercise (V1)
    2. Open Middle Midpoint Exercise (V2)
    3. Plotting Points via Distance in the Coordinate Plane
    4. Open Middle: Distance in the Coordinate Plane
    5. Open Middle: Distance in the Coordinate Plane (2)
    6. Open Middle: Distance in the Coordinate Plane (3)
    7. Open Middle: Distance and Midpoint Exercise
    8. Open Middle: Distance, Midpoint, Slope (1)
    9. Open Middle: Distance, Midpoint, Slope (2)
    10. Open Middle: Distance, Midpoint, Slope (3)
    11. Transformations: Possible PreImages?
    12. Open Middle: Building Parallelograms
    13. Open Middle: Building Rectangles
    14. Open Middle: Building Rhombuses
    15. Open Middle: Building Squares
    16. Open Middle: Building Trapezoids
    17. Open Middle: Building Isosceles Trapezoids
    18. Open Middle: Building Kites
    19. Building Parallelograms in the Coordinate Plane (Version 1)
    20. Building Parallelograms in the Coordinate Plane (Version 2)
    21. Creating 3 Lines to Form a Right Triangle
    22. Create a Rectangle with Given Area Whose Vertices Have No Repeated Coordinates
    23. Creating Squares
    24. Creating Right Triangles!
  5. Statistics (6-12)
    1. Open Middle: Two Way Table Setups
    2. Open Middle: Two Way Table (2)
    3. Open Middle: Two Way Table (3)
    4. Open Middle Histogram Exercise (1)
    5. Open Middle Histogram Exercise (2)
    6. Open Middle: Basic Statistics Exercise (2)
    7. Open Middle Box Plots
    8. Open Middle: Basic Statistics Exercise (1)
  6. Trigonometry (HS)
    1. Open Middle: Bearings Exercise
    2. Basic Trig Graphing Open Middle Problem
    3. Open Middle: Graphing Trig Functions Exercise
    4. Open Middle: Graphing Trig Functions Exercise (2)
    5. Open Middle: Trigonometric Equation (1)
    6. Open Middle: Trigonometric Equation (2)
    7. Open Middle: Creating Trig EQ's Exercise Set
    8. Open Middle: Ambiguous Case Setups
    9. Open Middle Problem: Law of Cosines (V2)
    10. Law of Cosines: Open Middle Problem (V1)
  7. Calculus (HS)
    1. Open Middle Calculus Problem
    2. Tangent to Cubic: Open Middle Problem
    3. Open Middle: Definite Integral Problem
    4. Open Middle: Definite Integral Problem (2)
    5. Definite Integral 3
  8. Computer Science (HS)
    1. JavaScript – For Loop 1

1.

Arithmetic (K-6)

  • 1. Building Number Sentences
  • 2. Equality
  • 3. Add Some, Subtract Some
  • 4. Sums to 100
  • 5. Open Middle Sum Problem
  • 6. Close to 1000
  • 7. Open Middle: Comparing and Identifying Fractions on a Number Line
  • 8. Create A Pattern
  • 9. Operations with Time 1
  • 10. Order of Operations 4
  • 11. Multiplying Two-Digit Numbers
  • 12. Open Middle: Multiplying 2-Digit Numbers (V2)
  • 13. Close Products
  • 14. Fraction Size
  • 15. Multiplying Fraction by a 2-Digit Whole Number
  • 16. Multiplying Fractions 4
  • 17. Distributive Property (Open Middle)
  • 18. Order of Operations 2
  • 19. Decimal Product Close To 50 v3
  • 20. Open Middle Percent Modeling Problem
  • 21. Creating Equal Ratios: Open Middle Theme
  • 22. Building Number Sentences: Open Middle Theme (1)
  • 23. Building Number Sentences: Open Middle Theme (6)

1.1.

Building Number Sentences

Source: [url=https://www.openmiddle.com/]Open Middle[/url]
Directions: Using the digits 1 to 9 at most one time each, drag a digit into each box to create a true number sentence.
Create ANOTHER number sentence that is DIFFERENT from the one you created above.
Create ANOTHER number sentence that is DIFFERENT from the two number sentences you created above.
Fill digits inside the boxes to make a true statement and so that the NUMBER AT THE BOTTOM is as LARGE AS POSSIBLE!
Fill digits inside the boxes to make a true statement and so that the NUMBER AT THE BOTTOM is as SMALL AS POSSIBLE!
– Tim Brzezinski, Steve Phelps

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

1.9.

–

1.10.

–

1.11.

–

1.12.

–

1.13.

–

1.14.

–

1.15.

–

1.16.

–

1.17.

–

1.18.

–

1.19.

–

1.20.

–

1.21.

–

1.22.

–

1.23.

–

2.

Algebra & Functions (5-12)

  • 1. Open Middle: Horizontal and Vertical Distances (V1)
  • 2. Open Middle: Horizontal and Vertical Distances (V2)
  • 3. Distributive Property: Open Middle Theme
  • 4. Writing Equivalent Expressions: Open Middle Theme
  • 5. Writing Equivalent Polynomial Expressions (V2)
  • 6. Writing Equivalent Polynomial Expressions
  • 7. Open Middle Multi-Step Equations Problem
  • 8. Make a Linear Function
  • 9. Open Middle: Slope and Points Exercise
  • 10. Open Middle Slope Exercise (V1)
  • 11. Open Middle Slope Exercise (V2)
  • 12. Open Middle: Linear Function to Cone
  • 13. Open Middle: Point-Slope Exercise (1)
  • 14. Open Middle: Point-Slope Exercise (2)
  • 15. Parallel Lines and Slope: Open Middle
  • 16. Open Middle: Parallel and Perpendicular Lines
  • 17. Open Middle: Equations of Parallel Lines
  • 18. Perpendicular Lines and Slope: Open Middle
  • 19. Perpendicular Lines and Slope
  • 20. Equations of Perpendicular Lines (Open Middle) - V1
  • 21. Equations of Perpendicular Lines (Open Middle) - V2
  • 22. Perpendicular Lines Through a Given Point
  • 23. How Close to Perpendicular Can You Get? Open Middle Challenge
  • 24. Open Middle: Solving Equations in Two Variables
  • 25. Open Middle Linear Functions Exercise
  • 26. Open Middle: Systems of Linear Equations (3)
  • 27. Open Middle: Systems of Linear Equations (1)
  • 28. Open Middle: Systems of Linear Equations (4)
  • 29. Open Middle: Systems of Linear Equations (2)
  • 30. Open Middle: Systems of Linear Equations (5)
  • 31. Open Middle: Systems of Linear Equations (6)
  • 32. FUNCTIONS
  • 33. Discriminant
  • 34. Function Notation
  • 35. Equivalent lines in slope-intercept and standard form
  • 36. Open Middle: Absolute Value Graphs (1)
  • 37. Open Middle: Absolute Value Graphs (2)
  • 38. Open Middle: Absolute Value Graphs (3)
  • 39. Open Middle: Absolute Value Graphs (4)
  • 40. Open Middle: Rational Functions Graph (1)
  • 41. Open Middle: Rational Functions Graph (2)

2.1.

Open Middle: Horizontal and Vertical Distances (V1)

Creation of this Open Middle exercise was inspired by a problem found in the Grade 6 iReady math curriculum published by Curriculum Associates.
Using digits -9 to 9 no more than one time each, fill in the boxes below to make all statements true. Note: Only positive digits can be placed in the last 3 boxes. Points (soon to appear) are draggable.
Can you create another valid setup that is different from the one above? Try to do so!
What is the greatest possible value you can get for the DIFFERENCE BETWEEN the distances AB and CD (bottom right most box)? Try to make this value as large as you possibly can!
– Tim Brzezinski

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

2.7.

–

2.8.

–

2.9.

–

2.10.

–

2.11.

–

2.12.

–

2.13.

–

2.14.

–

2.15.

–

2.16.

–

2.17.

–

2.18.

–

2.19.

–

2.20.

–

2.21.

–

2.22.

–

2.23.

–

2.24.

–

2.25.

–

2.26.

–

2.27.

–

2.28.

–

2.29.

–

2.30.

–

2.31.

–

2.32.

–

2.33.

–

2.34.

–

2.35.

–

2.36.

–

2.37.

–

2.38.

–

2.39.

–

2.40.

–

2.41.

–

3.

Geometry (K -12)

  • 1. Constructing Rectangles: Open Middle Problem
  • 2. Unit Squares: Multipurpose Template
  • 3. Open Middle: Perimeter of a Rectangle
  • 4. Open Middle: Maximizing and Minimizing Area (1)
  • 5. Multipurpose Composite Figure Templates
  • 6. Open Middle: Area of a House
  • 7. Open Middle: Trapezoid Area Problem
  • 8. Open Middle Trapezoid Area Problem: Choose Your Own Fixed Perimeter
  • 9. Open Middle: Pythagorean Theorem (1)
  • 10. Open Middle: Pythagorean Theorem (2)
  • 11. Creating Rectangular Prisms with Given Surface Area
  • 12. Open Middle (Revamped): Creating Rectangular Prisms with a Given Surface Area
  • 13. Maximizing Surface Area: Open Middle Cube Problem (V1)
  • 14. Maximizing Surface Area: Open Middle Cube Problem (V2)
  • 15. Open Middle: Linear Function to Cone
  • 16. Open Middle: Maximizing Rectangular Prism Surface Area
  • 17. Area of Rectangles
  • 18. Total Area
  • 19. Open Middle: Surface Area and Percent
  • 20. Open Middle: Surface Area and Percent (no feedback version)
  • 21. Open Middle: Same-Side Interior or Exterior Angles
  • 22. Open Middle Challenge: Parallel Lines and Related Angles
  • 23. Similar Shapes
  • 24. Open Middle Perpendicular Bisector Problem
  • 25. Equidistant Points 2
  • 26. Open Middle Problem: Complementary and Supplementary Angles
  • 27. Open Middle: Supplementary Angles
  • 28. Open Middle: Triangle Angles
  • 29. Open Middle: Interior and Exterior Angles of Triangles
  • 30. Open Middle Triangle Angles Problem
  • 31. Triangle Inequality: Open Middle Problem (V1)
  • 32. Triangle Inequality: Open Middle Problem (V2)
  • 33. Open Middle Triangle Angle Sum Problem
  • 34. Open Middle Triangle Angle Sum Problem (V2)
  • 35. Classifying Triangles by Angles
  • 36. Open Middle: Creating Triangles
  • 37. Pythagorean Theorem: Building Right Triangles
  • 38. Creating Distances in the Coordinate Plane: Formative Assessment with Feedback
  • 39. Open Middle: Triangle Area Problem
  • 40. Create a Cylinder with Given Lateral Area
  • 41. Open Middle: Diagonal of a Rectangular Prism
  • 42. Creating Distances in 3 Space: Quick Formative Assessment
  • 43. Creating Distances in 3 Space
  • 44. Equations of Circles (V1): Open Middle Problem
  • 45. Equations of Circles (V2): Open Middle Problem
  • 46. Equations of Circles (V3): Open Middle Problem
  • 47. Equations of Circles (V4): Open Middle Problem
  • 48. Open Middle: Circle Tangent to Line (1)
  • 49. Position the Line Tangent to the Circle
  • 50. Open Middle: Triangle Area Problem (V2)

3.1.

Constructing Rectangles: Open Middle Problem

Creation of this resource was inspired by this Open Middle problem from Bryan Anderson.
[size=150]Source: [url=https://www.openmiddle.com/]Open Middle[/url][/size][br]
Drag the squares around to form a rectangle. Be sure to use all the squares!
Can you create a rectangle different from the one you made above? Try to do so!
Can you create a rectangle different from the two you made above? Try to do so!
– Tim Brzezinski

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

3.8.

–

3.9.

–

3.10.

–

3.11.

–

3.12.

–

3.13.

–

3.14.

–

3.15.

–

3.16.

–

3.17.

–

3.18.

–

3.19.

–

3.20.

–

3.21.

–

3.22.

–

3.23.

–

3.24.

–

3.25.

–

3.26.

–

3.27.

–

3.28.

–

3.29.

–

3.30.

–

3.31.

–

3.32.

–

3.33.

–

3.34.

–

3.35.

–

3.36.

–

3.37.

–

3.38.

–

3.39.

–

3.40.

–

3.41.

–

3.42.

–

3.43.

–

3.44.

–

3.45.

–

3.46.

–

3.47.

–

3.48.

–

3.49.

–

3.50.

–

4.

Coordinate Geometry (7-12)

  • 1. Open Middle Midpoint Exercise (V1)
  • 2. Open Middle Midpoint Exercise (V2)
  • 3. Plotting Points via Distance in the Coordinate Plane
  • 4. Open Middle: Distance in the Coordinate Plane
  • 5. Open Middle: Distance in the Coordinate Plane (2)
  • 6. Open Middle: Distance in the Coordinate Plane (3)
  • 7. Open Middle: Distance and Midpoint Exercise
  • 8. Open Middle: Distance, Midpoint, Slope (1)
  • 9. Open Middle: Distance, Midpoint, Slope (2)
  • 10. Open Middle: Distance, Midpoint, Slope (3)
  • 11. Transformations: Possible PreImages?
  • 12. Open Middle: Building Parallelograms
  • 13. Open Middle: Building Rectangles
  • 14. Open Middle: Building Rhombuses
  • 15. Open Middle: Building Squares
  • 16. Open Middle: Building Trapezoids
  • 17. Open Middle: Building Isosceles Trapezoids
  • 18. Open Middle: Building Kites
  • 19. Building Parallelograms in the Coordinate Plane (Version 1)
  • 20. Building Parallelograms in the Coordinate Plane (Version 2)
  • 21. Creating 3 Lines to Form a Right Triangle
  • 22. Create a Rectangle with Given Area Whose Vertices Have No Repeated Coordinates
  • 23. Creating Squares
  • 24. Creating Right Triangles!

4.1.

Open Middle Midpoint Exercise (V1)

Creation of this resource was inspired by an Open Middle problem posted by Dane Ehlert.
Find locations for A and B so that M = (3,4) is the midpoint of the segment with endpoints A and B.
Find other locations for A and B so that M = (3,4) is the midpoint of the segment with endpoints A and B. Make sure your setup is different from the one you created above.
– Tim Brzezinski

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

4.8.

–

4.9.

–

4.10.

–

4.11.

–

4.12.

–

4.13.

–

4.14.

–

4.15.

–

4.16.

–

4.17.

–

4.18.

–

4.19.

–

4.20.

–

4.21.

–

4.22.

–

4.23.

–

4.24.

–

5.

Statistics (6-12)

  • 1. Open Middle: Two Way Table Setups
  • 2. Open Middle: Two Way Table (2)
  • 3. Open Middle: Two Way Table (3)
  • 4. Open Middle Histogram Exercise (1)
  • 5. Open Middle Histogram Exercise (2)
  • 6. Open Middle: Basic Statistics Exercise (2)
  • 7. Open Middle Box Plots
  • 8. Open Middle: Basic Statistics Exercise (1)

5.1.

Open Middle: Two Way Table Setups

Using digits 0-9, fill in the boxes to create a two-way table below. Design this table so that NO DIGIT REPEATS MORE THAN 3 TIMES.
Using digits 0-9, fill in the boxes to create another two-way table that is entirely different from your setup above. Design this table so that NO DIGIT REPEATS MORE THAN 3 TIMES.
Using digits 0-9, fill in the boxes to create another two-way table that is entirely different from your two tables above. Design this table so that NO DIGIT REPEATS MORE THAN 3 TIMES.
– Tim Brzezinski

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

6.

Trigonometry (HS)

  • 1. Open Middle: Bearings Exercise
  • 2. Basic Trig Graphing Open Middle Problem
  • 3. Open Middle: Graphing Trig Functions Exercise
  • 4. Open Middle: Graphing Trig Functions Exercise (2)
  • 5. Open Middle: Trigonometric Equation (1)
  • 6. Open Middle: Trigonometric Equation (2)
  • 7. Open Middle: Creating Trig EQ's Exercise Set
  • 8. Open Middle: Ambiguous Case Setups
  • 9. Open Middle Problem: Law of Cosines (V2)
  • 10. Law of Cosines: Open Middle Problem (V1)

6.1.

Open Middle: Bearings Exercise

Using digits 0-9 NO MORE THAN ONE TIME EACH, fill in the blanks to create a true statement.
Using digits 0-9 NO MORE THAN ONE TIME EACH, create another true statement that is different from the one you made above..
Using digits 0-9 NO MORE THAN ONE TIME EACH, create another true statement that is different from the two setups you made above.
Given the no repeated digits constraint, what is the LARGEST possible bearing (0° < bearing < 360°) you can create here?
Given the no repeated digits constraint, what is the LARGEST possible direction angle (0° < direction angle < 360°) you can create here?
– Tim Brzezinski

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

7.

Calculus (HS)

  • 1. Open Middle Calculus Problem
  • 2. Tangent to Cubic: Open Middle Problem
  • 3. Open Middle: Definite Integral Problem
  • 4. Open Middle: Definite Integral Problem (2)
  • 5. Definite Integral 3

7.1.

Open Middle Calculus Problem

Creation of this resource was inspired by this Open Middle problem created by Erin Stenger:
This app does not provide feedback re: correct solution(s), but rather the number of repeated coordinates. How many solutions can you find?
Explain or show your thought process below.
– Tim Brzezinski

7.2.

–

7.3.

–

7.4.

–

7.5.

–

8.

Computer Science (HS)

  • 1. JavaScript – For Loop 1

8.1.

JavaScript – For Loop 1

Directions: In JavaScript, using the digits 1 to 9 at most one time each, place a digit in each box to find then total would have the largest number printed to the screen. https://www.openmiddle.com/javascript-for-loop-1/
– Steve Phelps

Information