1. The One Penny Whiteboard
Ongoing, “in the moment” assessments may be the most powerful tool teachers have for improving student performance. For students to get better at anything, they need lots of quick rigorous practice, spaced over time, with immediate feedback. The One Penny Whiteboards can do just that.
©Bill Atwood 2014

2. To add the One Penny White Board to your teaching repertoire, just purchase some sheet protectors and white board markers (see the following slides). Next, find something that will erase the whiteboards (tissues, napkins, socks, or felt). Finally, fill each sheet protector (or have students do it) with 1 or 2 sheets of card stock paper to give it more weight and stability.
©Bill Atwood 2014

3. ©Bill Atwood 2014

4. ©Bill Atwood 2014

5. On Amazon, markers can be found as low as $0. 63 each
On Amazon, markers can be found as low as $0.63 each. (That’s not even a bulk discount. Consider “low odor” for students who are sensitive to smells.)
©Bill Atwood 2014

6. I like the heavy-weight model.
©Bill Atwood 2014

7. On Amazon, Avery protectors can be found as low as $0.09 each.
©Bill Atwood 2014

8. One Penny Whiteboards and
The Templates
The One Penny Whiteboards have advantages over traditional whiteboards because they are light, portable, and able to contain a template. (A template is any paper you slide into the sheet protector). Students find templates helpful because they can work on top of the image (number line, graph paper, hundreds chart…) without having to draw it first. For more templates go to
©Bill Atwood 2014

9. Using the One Penny Whiteboards
There are many ways to use these whiteboards. One way is to pose a question, and then let the students work on them for a bit. Then say, “Check your neighbor’s answer, fix if necessary, then hold them up.” This gets more students involved and allows for more eyes and feedback on the work.
©Bill Atwood 2014

10. Using the One Penny Whiteboards
Group Game
One way to use the whiteboards is to pose a challenge and make the session into a kind of game with a scoring system. For example, make each question worth 5 possible points.
Everyone gets it right: 5 points
Most everyone (4 fifths): 4 points
More than half (3 fifths): 3 points
Slightly less than half (2 fifths): 2 points
A small number of students (1 fifth): 1 point
Challenge your class to get to 50 points. Remember students should check their neighbor’s work before holding up the whiteboard. This way it is cooperative and competitive.
©Bill Atwood 2014

11. Using the One Penny Whiteboards
Without Partners
Another way to use the whiteboards is for students to work on their own. Then, when students hold up the boards, use a class list to keep track who is struggling. After you can follow up later with individualized instruction.
©Bill Atwood 2014

12. Keep the Pace Brisk and Celebrate Mistakes
However you decide to use the One Penny Whiteboards, keep it moving! You don’t have to wait for everyone to complete a perfect answer. Have students work with the problem a bit, check it, and even if a couple kids are still working, give another question. They will work more quickly with a second chance. Anytime there is an issue, clarify and then pose another similar problem.
Celebrate mistakes. Without them, there is no learning. Hold up mistakes and say, “Now, here is an excellent mistake–one we can all learn from. What mistake is this? Why is this tricky? How do we fix it?”
©Bill Atwood 2014

13. The Questions Are Everything!
The questions you ask are critical. Without rigorous questions, there will be no rigorous practice or thinking. On the other hand, if the questions are too hard, students will be frustrated. They key is to jump back and forth from less rigor to more rigor. Also, use the models written by students who have the correct answer to show others. Once one person gets it, they all can get it.
©Bill Atwood 2014

14. Teachers: Print the following slides (as needed) and then have students insert them into their whiteboards. They will need them for slides 42 and above.
©Bill Atwood 2014

15. ©Bill Atwood 2014

16. ©Bill Atwood 2014

17. ©Bill Atwood 2014

18. ©Bill Atwood 2014

19. Questions
When posing questions for the One Penny Whiteboard, keep several things in mind:
Mix low and high level questions
Mix the strands (it may be possible to ask about fractions, geometry, and measurement on the same template)
Mix in math and academic vocabulary (Calculate the area… use an expression… determine the approximate difference)
Mix verbal and written questions (project the written questions onto a screen to build reading skills)
Consider how much ink the answer will require and how much time it will take a student to answer (You don’t want to waste valuable ink and you want to keep things moving.)
To increase rigor you can: work backwards, use variables, ask “what if”, make multi-step problems, analyze a mistake, ask for another method, or ask students to briefly show why it works
©Bill Atwood 2014

20. Each of these can be solved on the Ten Cent Whiteboard.
Examples
What follows are some sample questions involving operations with exponents, multiplying binomials, and applications with perimeter, area, surface area, volume.
Each of these can be solved on the Ten Cent Whiteboard.
To mix things up, you can have students “chant” out answers in choral fashion for some rapid fire questions. You can also have students hold up fingers to show which answer is correct.
Remember, to ask verbal follow-ups to individual students: Why does that rule work? How do you know you are right? Is there another way? Why is this wrong?
©Bill Atwood 2014

21. ©Bill Atwood 2014

22. Before we begin, let’s review the rule of multiplying with same bases and the reason it works…
Students might copy the middle line of this rule on their whiteboards just as a way of warming up.
©Bill Atwood 2014

23. (23)(22)(24) = (29) 2(3+2+4)=29 (2)(2)(2)(2)(2)(2)(2)(2)(2) =29
Solve this on your whiteboard then check with your partner. On my signal, hold up your whiteboards. Then I will show the answer and reason why… (Give your answer in exponential form!)
(23)(22)(24) =
(29)
2(3+2+4)=29
(2)(2)(2)(2)(2)(2)(2)(2)(2) =29
©Bill Atwood 2014

24. (42)(43)(4) = (46) (3)(35)(32) = (38) (2)(35)(32) = (2)(37)
Simplify each. Leave answers in exponential form!
(42)(43)(4) =
(46)
Don’t forget 4 = 41
This is really (4*4)*(4*4*4)*(4)
(3)(35)(32) =
(38)
(2)(35)(32) =
(2)(37)
Bases must be same!
(x)(x3)(x0) =
(x4)
X0 = 1
(x2)(x-3)(x3)(y) =
(x2y)
Remember rules for adding integers!
©Bill Atwood 2014

25. A student made a mistake below. Correct it.
Raise you hand to respond: Why do you think this student make this mistake?
©Bill Atwood 2014

26. (x9) (x3)(x2)(y4) = A student made a mistake below. Correct it.
Raise you hand to respond: Why do you think this student make this mistake?
©Bill Atwood 2014

27. (x2)(x3) = x5 (x2)(x3)(x)(y2) = x6y2 (x5)(x4)(x-2) = x7
Solve these on your whiteboard
(x2)(x3) = x5
(x2)(x3)(x)(y2) =
x6y2
(x5)(x4)(x-2) = x7
(x3)(x)(y4)(y3 )=
x4y7
©Bill Atwood 2014

28. (x2)(x-2) = X0 = 1 (3x2)(2x3) = 6x5 (2x)(2x)(2x) = 8x3
Solve these on your whiteboard
(x2)(x-2) =
X0 = 1
(3x2)(2x3) =
6x5
Don’t forget to multiply coefficients
(2x)(2x)(2x) =
8x3
(2x3)(3x)(4y4)(2y3 )=
6x48y7
48x4y7
©Bill Atwood 2014

29. (x2)(x5)(x) = x8 (x)(x3)(x0) = x4 (x)(x0)(x2) = x3 (x-1)(x-3)(x3) =
Hold up the correct number of fingers to show the exponent.
(x2)(x5)(x) = x8
(x)(x3)(x0) = x4
(x)(x0)(x2) = x3
(x-1)(x-3)(x3) =
x-1
©Bill Atwood 2014

30. (x-2)(x5) = x3 (x2)(x4)(x-2) = x4 (x-2)(x-1)(x-2) = X-5
Hold up the correct number of fingers to show the exponent.
(x-2)(x5) = x3
(x2)(x4)(x-2) = x4
(x-2)(x-1)(x-2) =
X-5
(x-1)(x-3)(x3) =
x-1
©Bill Atwood 2014

31. (2x2)(x3) = 2x5 (3x2)(2x4) = 6x6 (3x-2)(3x4) = 9X2 (3x)(2x3)(y3) =
Think first. Then when I signal you or cold call you, call out the answer.
(2x2)(x3) =
2x5
(3x2)(2x4) =
6x6
(3x-2)(3x4) =
9X2
(3x)(2x3)(y3) =
6x4y3
©Bill Atwood 2014

32. This is the Power Rule. Can you see why it works?
©Bill Atwood 2014

33. (X)(X)(X)*(X)(X)(X) = X6
Don’t forget to apply exponent to coefficient!
(3x3)2 =
(32 )(X6) = 9X6
©Bill Atwood 2014

34. (2x3)3 = (23)x9 = 8x9 (3x3y2)2 = 9x6y4 2x3 = 2x3 (2x)3 = 8x3
©Bill Atwood 2014

35. (Xa)b = Xa*b (xa)(xb) = X a+b (23)(32) = 8*9= 72 (23)(22)(24) = 29
©Bill Atwood 2014

36. (x6) (x2)4 = A student made a mistake below. Correct it.
Raise you hand to respond: Why do you think this student make this mistake? How could you prove this is wrong?
©Bill Atwood 2014

37. Explain to your neighbor the difference between the power rule and the rule for multiplying exponents with the same bases. Be sure to use sound mathematical reasoning
(X2)3 and (x2)(x3)
©Bill Atwood 2014

38. (x3)4 = x12 28 = (24)2 36 = (32)(32)(3) x10 = (x5)2 (3x4y)2 9x8y2 =
Make up an exponent problem that goes with each solution. There are many possible answers.
28 =
(24)2
36 =
(32)(32)(3)
x10 =
(x5)2
9x8y2 =
(3x4y)2
©Bill Atwood 2014

39. Solve this: Can you explain why your answer makes sense?
©Bill Atwood 2014

40. Insert into white board
©Bill Atwood 2014

41. The following problems all say “show work” but after students demonstrate they can clearly show work (with empty formula) and if you are working on speed or trying to get through a bunch of problems quickly you can eliminate this requirement as needed.
©Bill Atwood 2014

42. For given rectangle below: Length = 2x5 width = 3x4
Write an expression for the area. Label figure and show work.
2x5
3x4
Area = l*w
Area = (2x5)(3x4)
Area = 6x9
©Bill Atwood 2014

43. Perimeter = 2(l + w) P = 2(2x5 + 3x4) P = 4x5 + 6x4
length = 2x5 width = 3x4
Write an expression for the perimeter. Show work.
2x5
3x4
Perimeter = 2(l + w)
P = 2(2x5 + 3x4)
P = 4x x4
©Bill Atwood 2014

44. Area = l*w Area = (3y3)(4y2) Area = 12y5 length = 3y3 width = 4y2
Write an expression for the area. Show work.
3y3
4y2
Area = l*w
Area = (3y3)(4y2)
Area = 12y5
©Bill Atwood 2014

45. Perimeter= 2(l + w) P = 2(3y3 +4y2) P = 6y3 +8y2
length = 3y3 width = 4y2
Write an expression for the perimeter. Show work.
3y3
4y2
Perimeter= 2(l + w)
P = 2(3y3 +4y2)
P = 6y3 +8y2
©Bill Atwood 2014

46. Area = 1/2 (bh) Area = 1/2(2y2)(6y2) Area = 1/2(12y4) Area = 6y4
base = 2y2 height = 6y2
Write an expression for the area of this triangle.
Label diagram and show work.
6y2
2y2
Area = 1/2 (bh)
Area = 1/2(2y2)(6y2)
Area = 1/2(12y4)
Area = 6y4
©Bill Atwood 2014

47. Area = 1/2 (bh) Area = 1/2(4y2)(12y2) Area = 24y4
base = 2y2 height = 6y2
Write an expression for the area a similar triangle where
The corresponding sides are double the length. Label diagram and show work.
12y2
4y2
Area = 1/2 (bh)
Area = 1/2(4y2)(12y2)
Area = 24y4
©Bill Atwood 2014
©Bill Atwood 2014

48. Area = 1/2 (bh) Area = 1/2(3y2)(y2) Area = 1.5y4 or 3/2y4
base = 2y2 height = 6y2
Write an expression for the area a similar triangle where
The corresponding sides are half the length. Label diagram and show work.
3y2 y2
Area = 1/2 (bh)
Area = 1/2(3y2)(y2)
Area = 1.5y4 or 3/2y4
©Bill Atwood 2014
©Bill Atwood 2014
©Bill Atwood 2014

49. You might want to review some Pythagorean Theorem problems while you have the right triangle…
©Bill Atwood 2014

50. a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 5cm = c A B C
Write the Pythagorean theorem
Label the lengths: AB = 3cm BC = 4cm
c cm
Find length of AC
3 cm
a2 + b2 = c2
= c2 B C
4 cm
= c2
25 = c2
5cm = c
©Bill Atwood 2014
©Bill Atwood 2014

51. a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 100 = c2 10cm = c A B C
Write the Pythagorean theorem
Label the lengths: AB = 6cm BC = 8cm
c cm
Find length of AC
6 cm
a2 + b2 = c2
= c2 B C
8 cm
= c2
100 = c2
10cm = c
©Bill Atwood 2014
©Bill Atwood 2014
©Bill Atwood 2014

52. A
Write the Pythagorean theorem
Label the lengths: AB = 5cm AC = 13cm
13 cm
Find length of BC
5 cm
a2 + b2 = c2
52 + b2 = 132 B C
? cm
25 + b2 = 169
b2 =
b2 = 144
b = 12 cm
©Bill Atwood 2014
©Bill Atwood 2014
©Bill Atwood 2014
©Bill Atwood 2014

53. The following problems all say “show work” but after students demonstrate they can clearly show work and if you are working on speed or trying to get through a bunch of problems quickly you can eliminate this requirement as needed.
©Bill Atwood 2014

54. Perimeter= 5(s) P = 5(3x3) P = 15x3 Regular pentagon. Side = 3x3
Find perimeter. Show work
3x3
Perimeter= 5(s)
P = 5(3x3)
P = 15x3
©Bill Atwood 2014

55. Perimeter= 6(s) P = 6(4x4y) P = 24x4y Regular hexagon. Side = 4x4y
Find perimeter. Show work
4x4y
Perimeter= 6(s)
P = 6(4x4y)
P = 24x4y
©Bill Atwood 2014

56. Perimeter= 8(s) P = 8(2x5y3) P = 16x5y3 Regular octagon. Side = 2x5y3
Find perimeter. Show work
2x5y3
Perimeter= 8(s)
P = 8(2x5y3)
P = 16x5y3
©Bill Atwood 2014

57. Write an expression for the volume. Show work.
side length = 2x4
Write an expression for the volume. Show work.
2x4
Volume = s3
V = (2x4)3 or think 2x4 * 2x4 * 2x4
V = 8x12
©Bill Atwood 2014

58. Write an expression for the volume. Show work.
side length = 3x5
Write an expression for the volume. Show work.
3x5
Volume = s3
V = (3x5)3 or think 3x5 * 3x5 * 3x5
V = 27x15
©Bill Atwood 2014

59. ©Bill Atwood 2014

60. length = 2x2 width = 3x4 height = x5
Write an expression for the volume. Label prism then clearly show work. x5
3x4
Volume = l*w*h
V = (2x2)(3x4)(x5)
V = 6x11
2x2
©Bill Atwood 2014

61. length = 3y3 width = 2y3 height = 3y2
Write an expression for the volume. Label prism then clearly show work.
3y2
2y3
Volume = l*w*h
V = (3y3)(2y3)(3y2)
V = 18y8
3y3
©Bill Atwood 2014

62. length = 2xy3 width = 3xy2 height = 2y3
Write an expression for the volume. Label prism then clearly show work.
2y3
3xy2
Volume = l*w*h
V = (2xy3)(3xy2)(2y3)
V = 12x2y8
2xy3
©Bill Atwood 2014
©Bill Atwood 2014

63. Volume = l*w*h V = (4xy3)(6xy2)(4y3) V = 96x2y8 4xy3
length = 2xy3 width = 3xy2 height = 2y3
Find the volume of a similar prism with side ratio of 1:2 (each dimension is double). Label prism then clearly show work.
4y3
6xy2
Volume = l*w*h
V = (4xy3)(6xy2)(4y3)
V = 96x2y8
4xy3
©Bill Atwood 2014
©Bill Atwood 2014
©Bill Atwood 2014

64. length = ___ width = ___ height = ___
Make up your own volume problem. Solve by writing an expression for the volume. Show work.
©Bill Atwood 2014

65. Write an expression for the surface area. Show work.
side length of cube = 2x4
Write an expression for the surface area. Show work.
2x4
SA = 6* s2
SA = 6* (2x4)2
SA = 6 *4x8
SA= 24 x8
©Bill Atwood 2014

66. Write an expression for the surface area. Show work.
side length of cube = 3y3
Write an expression for the surface area. Show work.
3y3
SA = 6* s2
SA = 6* (3y3)2
SA = 6 *9y6
SA= 54y6
©Bill Atwood 2014

67. ©Bill Atwood 2014

68. b1 = 6x2 b2= 4x2 height = 2x5 Area =( b1 + b2)/2 *h
Write an expression for the area of the trapezoid. Label drawing and show work.
4x2
2x5
Area =( b1 + b2)/2 *h
A= (6x2 + 4x2)/2 * 2x5
A = 5x2 * 2x5
6x2
A = 10x7
©Bill Atwood 2014

69. b1 = 4y3 b2= 2y3 height = y4 Area =( b1 + b2)/2 *h
Write an expression for the area of the trapezoid. Label drawing and show work..
2y3
Area =( b1 + b2)/2 *h
A= (4y3 + 2y3)/2 * y4
A = 3y3 * y4 y4
A = 3y7
4y3
©Bill Atwood 2014

70. ©Bill Atwood 2014

71. Diameter = 10y5 Area = π (r2) A= π (5y5)2 A = π (25y10) A = 25πy10
Write an expression for the area of the circle. Label drawing and show work.
10y5
Area = π (r2)
r = d/2
r= 10y5/2
r = 5y5
A= π (5y5)2
A = π (25y10)
A = 25πy10
©Bill Atwood 2014

72. Diameter of circle = 8y3 Area = π (r2) A= π (4y3)2 A = π (16y6)
Write an expression for the area of the circle. Label drawing and show work.
8y3
Area = π (r2)
A= π (4y3)2
A = π (16y6)
A = 16πy6
©Bill Atwood 2014

73. Diameter = 6x2 height = 10x4 V = (π)(r2)(h) V = (π)(3x2)2(10x4)
Write an expression for the volume of the circle. Label shape and show work.
Volume = (area of base)(h)
V = Bh
d =6x2
r = 3x2
V = (π)(r2)(h)
10x4
V = (π)(3x2)2(10x4)
V = (π)(9x4)(10x4)
V = (π)(90x8) or 90πx8
©Bill Atwood 2014

74. Patterns with negative exponents. Study this chart
Patterns with negative exponents. Study this chart. List two things you notice.
©Bill Atwood 2014

75. Patterns with negative exponents. Study this chart
Patterns with negative exponents. Study this chart. List two things you notice.
©Bill Atwood 2014

76. 3 31 = 3 (3)(3)(3) = (3)(3) (33) = (32) Solve these on your whiteboard
Remember anything divided by itself = 1 3/3 = 1
20/20= 1
x/x = 1
(33) =
(32)
31 = 3
©Bill Atwood 2014

77. x X1 = x (X)(X)(X) = (X)(X) (X3) = (X2)
Remember anything divided by itself = 1 3/3 = 1
20/20= 1
x/x = 1
(X3) =
(X2)
X1 = x
©Bill Atwood 2014

78. x3 X5-2 = x3 (X)(X)(X)(X)(X) = (X)(X) (X5) = (X2)
Remember anything divided by itself = 1 3/3 = 1
20/20= 1
x/x = 1
(X5) =
(X2)
X5-2 = x3
©Bill Atwood 2014

79. Think: Keep going… 24= 16 2-1= 1/2 = 1/21 23 = 8 2-2 = 1/4 = 1/22
Use patterns to explain to your neighbor (or write a quick explanation on your paper.
Why = 1/22
Think:
24= 16
23 = 8
22 = 4
21 = 2
20 = 1
Keep going…
2-1= 1/2 = 1/21
2-2 = 1/4 = 1/22
2-3 = 1/8 = 1/23
2-4 = 1/16 = 1/24
2-5 = 1/32 = 1/25
©Bill Atwood 2014
©Bill Atwood 2014

80. (X3) =
(X2)
X1 = x
(X3-2) =
X3-2 = x1 = x
©Bill Atwood 2014

81. ©Bill Atwood 2014

82. (4X5) =
(2X2)
2X5-2 = 2x3
©Bill Atwood 2014

83. (4X2) =
(8X3) 1_ 2x
©Bill Atwood 2014

84. (5X5) =
(10X8) 1_
2x3
©Bill Atwood 2014

85. (2X3)(4x4)(x2) =
(8X3) X6
©Bill Atwood 2014

86. (2X3)2 =
(5X3)
4x 6 =
5x3
4x 3 5
©Bill Atwood 2014

87. 6X3y3 =
3X2y
3x1y2 =
3xy2
©Bill Atwood 2014

88. Think: Keep going… 24= 16 2-1= 1/2 = 1/21 23 = 8 2-2 = 1/4 = 1/22
Again: Use patterns to explain to your neighbor (or write a quick explanation on your paper Why = 1/22
Think:
24= 16
23 = 8
22 = 4
21 = 2
20 = 1
Keep going…
2-1= 1/2 = 1/21
2-2 = 1/4 = 1/22
2-3 = 1/8 = 1/23
2-4 = 1/16 = 1/24
2-5 = 1/32 = 1/25
©Bill Atwood 2014
©Bill Atwood 2014

89. Use patterns to explain to your neighbor (or write a quick explanation on your paper.
Why does x3 = x3-2 x2
©Bill Atwood 2014
©Bill Atwood 2014
©Bill Atwood 2014

90. 2x5 Area= 6x9 w l * w = A w = (6x9) (2x5) l * w = A l l w = 3x4 w = A
length = 2x5 area= 6x9
Write an expression for the width. Show work.
2x5
Area= 6x9 w
l * w = A
w = (6x9)
(2x5)
l * w = A
l l
w = 3x4
w = A l
©Bill Atwood 2014

91. 3x2 Area= 18x5 w l * w = A w = (18x5) (3x2) l * w = A l l w = 6x3
length = 3x2 area= 18x5
Write an expression for the width. Show work.
3x2
Area= 18x5 w
l * w = A
w = (18x5)
(3x2)
l * w = A
l l
w = 6x3
w = A l
©Bill Atwood 2014

92. 12x3 Area= 6x4 w l * w = A w = (6x4) (12x3) l * w = A l l w = ½ x or x
length = 12x3 area= 6x4
Write an expression for the width. Show work.
12x3
Area= 6x4 w
l * w = A
w = (6x4)
(12x3)
l * w = A
l l
w = ½ x or x 2
w = A l
©Bill Atwood 2014

93. l = __ Area= ____ w Make up your own missing dimension problem
Solve. Show work.
l = __
Area= ____ w
©Bill Atwood 2014

94. length = 2x2 width = 3x4 Volume = 6x11
Write and expression for the height. Show work.
l = 2x2
w = 3x4
h = ?
Volume= 6x11
Volume = l*w*h
Volume = h
l*w
6x = h
2x2*3x4
X5 = h
©Bill Atwood 2014

95. length = 3x3 width = 2x2 Volume = 12x7
Write and expression for the height. Show work.
l = 3x3
w = 2x2
h = ?
Volume= 12x7
Volume = l*w*h
V = h
l*w
12x = h
2x3*3x2
2X2 = h
©Bill Atwood 2014

96. length = ___ width = ___ height = ____Volume = ___
Make up your own missing dimension problem. Solve.
©Bill Atwood 2014