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Thinking Algebraically with Hundred Boards and Arrow Paths

Fill in the missing numbers on these hundred board shapes.. . 15. . . . . . . . 47. . . Write a rule for finding the missing numbers on these shapes.. 27. . . 89. . . 52. Write a rule for finding the missing numbers on these shapes.. . 34. . 88. . 67. Write a rule for finding the missing numbers on these shapes..

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Thinking Algebraically with Hundred Boards and Arrow Paths

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    1. Thinking Algebraically with Hundred Boards and Arrow Paths Wade H. Sherard 2008 NCTM Annual Meeting Salt Lake City 11 April 2008

    3. Write a rule for finding the missing numbers on these shapes.

    4. Write a rule for finding the missing numbers on these shapes.

    5. Write a rule for finding the missing numbers on these shapes.

    6. Write a rule for finding the missing numbers on these shapes.

    7. Fill in the missing numbers on these hundred board shapes.

    8. If you know the center number ? of this shape, explain how to find the other numbers.

    9. These shapes will not fit on a hundred board. Explain why not.

    10. Fill in the missing numbers on these hundred board shapes.

    11. Write the operation you can use to find each missing number in the shape if you know the number ? in the center square.

    12. Find the missing numbers on these hundred board shapes.

    13. Find all numbers from 1 to 100 that can replace the ? so that the shape will not fit on a hundred board.

    14. This shape fits on a row of a hundred board. Fill in the missing numbers so that their sum is 54.

    15. This shape fits on a column of a hundred board. Fill in the missing numbers so that their sum is 45.

    16. Challenge: fill in the missing numbers of this hundred board shape so that their sum is 50.

    17. Each shape has an n in one square to represent a missing number. Write expressions for the other missing numbers in each square.

    18. Let n represent the number in the center square. Write an expression for each missing number in the hundred board shape.

    19. Now add the expressions from the nine squares to get a formula for the sum of the numbers in the nine squares. Explain how this formula helps you quickly to find the sum of all nine numbers in any such square on a hundred board.

    20. Arrows can show how to move from one number to another number on a hundred board. For example, ? ? ? ? ? means start at 45; move right to 46; move down to 56; move down to 66; move up to 56; and move left to 55.

    21. Find the number at the end of each arrow path. 82 ? ? ? ? 39 ? ? ? ? ? ? ? ? 76 ? ? ? ? ? ? ? ?

    22. Explain how each arrow changes a number. a. ? b. ? c. ? d. ?

    23. Create an arrow path that connects the numbers in this set. 17 27 37 36 35 25 24 34 44 b. starts at 53 and ends at 29.

    24. Find the number at the end of this arrow path. ? ? ? ? ? ? ? ? ? ? ? ? Now, create a shorter arrow path that joins these two numbers.

    25. Explain how each pair of arrows changes a number. ? ? b. ? ? ? ? d. ? ? ? ? f. ? ? g. ? ? h. ? ?

    26. Arrow Paths: Shapes and Alphabet Letters Create an arrow path with more than 10 arrows that starts and ends at 53 so that on a hundred board the path is a rectangle. Create an arrow path that starts at 12 and ends at 54 so that on a hundred board the path is the letter T.

    27. Other kinds of arrows can also show how to move from one number to another number on a hundred board. For example, 49 ? ? ? ? means start at 49; move diagonally up to 38; up to 28; diagonally down to 37;and left to 36.

    28. Find the number at the end of each arrow path. 65 ? ? ? ? ? 32 ? ? ? ? ? ? ? ? ? ? c. 79 ? ? ? ? ? ? ? ? ? ?

    29. Explain how each of these arrows changes a number. ? b. ? ? d. ?

    30. Create an arrow path that connects the numbers in this set. 67 56 46 37 27 38 49 59 58 starts at 42, ends at 13, has at least 8 arrows, and uses only these kinds of arrows: ? ? ? ?

    31. Explain how each pair of arrows changes a number. ? ? b. ? ? ? ? d. ? ? ? ? f. ? ? g. ? ? h. ? ?

    32. Arrow Paths: Shapes and Alphabet Letters Create an arrow path with more than 8 arrows that starts and ends at 53 so that on a hundred board the path is a triangle. Create an arrow path that starts at 22 and ends at 88 so that on a hundred board the path is the letter Z.

    33. Create the shortest arrow path by crossing out certain arrows. Explain how you decided which arrows to use. ? ? ? ? ? ? ? ? ? ? ? ? ? 56

    34. ALGEBRAIC THINKING CAN INVOLVE: recognizing, describing, creating, analyzing, or extending patterns; using mathematical relationships to describe change between variables or to express proportional reasoning; representing patterns, relationships, and functions; making generalizations from arithmetic; understanding and using equivalence;

    35. ALGEBRAIC THINKING CAN INVOLVE: using properties from the structure of algebra (e.g., the order of operations); focusing on number operations and relationships between numbers; inverting (or reversing) mathematical operations; modeling real-world relationships by using the language and tools of mathematics.

    36. Importance of Algebraic Thinking Problems on high-stakes tests Preparation for algebra one or its equivalent Algebra as a gate-keeper for courses in mathematics or other disciplines

    37. References Joyner, Jeane M. and Wade H. Sherard III. Thinking Algebraically with Numbers and Shapes, Level C. Dale Seymour Publications, 2003. Sherard, Wade H. III. Thinking Algebraically with Numbers and Shapes, Level D. Dale Seymour Publications, 2003. Sherard, Wade H. III. Thinking Algebraically with Numbers and Shapes, Level E. Dale Seymour Publications, 2003. Sherard, Wade H. III. Thinking Algebraically with Numbers and Shapes, Level F. Dale Seymour Publications, 2003.

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