1. Vacaville USD February 19, 2015
SIXTH GRADE CCSS-Math
Vacaville USD
February 19, 2015

2. AGENDA Problem Solving – Where are the Cookies?
Estimating and Measurement
Geometry
Area of parallelograms, triangles and trapezoids
Volume of rectangular prisms
Others???

3. Mrs. James left a tray of cookies on the counter early one morning
Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

4. Mrs. James left a tray of cookies on the counter early one morning
Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

5. Mrs. James left a tray of cookies on the counter early one morning
Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

6. Mrs. James left a tray of cookies on the counter early one morning
Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

7. Mrs. James left a tray of cookies on the counter early one morning
Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

8. Mrs. James left a tray of cookies on the counter early one morning
Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

9. Mrs. James left a tray of cookies on the counter early one morning
Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

10. Analyze Student Work For each piece of work:
Describe the problem solving approach the student used. For example, you might:
Describe the way the student has organized the solution.
Describe what the student did to calculate the number of cookies that started on the tray.
Explain what the student needs to do to complete or correct his or her solution.

11. Analyze Student Work Suggestions for feedback Common issues
Suggested questions and prompts

21. Kentucky Department of Education
Mathematics Formative Assessment Lessons
Concept-Focused Formative Assessment Lessons
Problem Solving Formative Assessment Lessons
Designed and revised by Kentucky DOE Mathematics Specialists
Field-­‐tested by Kentucky Mathematics Leadership Network Teachers

22. Where are the Cookies? Grades 4 – 6
Problem Solving Formative Assessment Lesson
Lesson Format
Pre-Lesson (about 15 minutes)
Lesson (about 1 hour)
Follow-Up (about 10 minutes)
Where are the Cookies?

23. Estimation

24. Estimation
How many cheeseballs are in the vase?
183

25. Estimation
How many cheeseballs are in the original container?
917

26. 441 Estimation How many peanut m&m’s are in the vase?
Are there more m&m’s than cheeseballs or less?
How do you know?
441

27. Ratios and Measurement
RP.3d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

28. Measurement Is a foot larger or smaller than a yard?
So suppose I tell you I have 9 feet but I need to know how many yards that is.
Will I have more than 9 yards or less than 9 yards?
How do you know?

29. Measurement
So what do we know about feet and yards?
3 feet
1 yard

30. Measurement
3 feet
1 yard
9 feet = ____ yards

31. Measurement
3 feet
1 yard
9 yards = ____ feet

32. Measurement
1 yard
3 feet
9 yards = ____ feet

33. Measurement So what do we know about meters and centimeters?

34. Measurement
50 m = ____ cm
70 cm = ____ m
100 centimeters
1 meter

35. Measurement
1 m
100 cm
50 m = ____ cm
70 cm = ____ m

37. How many green marshmallows will fit on the skewer?

38. How many green marshmallows will fit on the skewer?

39. How many green marshmallows are inside the glass?

40. How many green mallows are needed to complete the 4-leaf clover?

41. How many green mallows are needed to complete the 4-leaf clover?

42. What's the capacity of the tall vase?

43. What's the capacity of the wide vase?

44. Order the glasses from least to greatest in capacity.

45. How many Red Vines are in my hand?

46. How many Red Vines are in the container?

47. A Brief Discussion of Division

48. Dividing Decimals Why do we “move” the decimal before we divide?
Why are we “allowed” to move the decimal (mathematically speaking)?

49. Dividing Fractions
Why do we “invert and multiply” instead of dividing?
Why are we “allowed” to multiply by the reciprocal (mathematically speaking)?

50. Area

51. G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

52. Unpacking the Standard
Students continue to understand that area is the number of squares needed to cover a plane figure. Students should know the formulas for rectangles and triangles. “Knowing the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (area) and the figure. This understanding should be for all students.

53. G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Why a rectangle?

54. Parallelograms
Can you find a way to turn your parallelograms into rectangles?
Scissors, tape, grid paper
How can you use what you just discovered to develop a formula for finding the area of a parallelogram (without having to cut it up)?

55. Parallelograms

56. Parallelograms 1 NCTM Illuminations
Rectangles and Parallelograms
Quadrilateral Area Record Sheet

57. Rectangle A
Have students cut from the lower left corner to a point on the top edge that is three units in from the upper left vertex;
This cut will form a 45-degree angle, which divides each of the squares through which it passes exactly in half.

58. Rectangles B and C
Have students remove a triangle by cutting from the lower left corner diagonally to any point along the top edge.
Encourage students to choose a point along the top where the edge and one of the grid lines meet.
Students should then place the removed triangle at the other end of the rectangle.
Encourage students to make a different cut than other members of their group.

59. Rectangles D and E
Have students remove the right triangle on either the right or left side, and move it to the other side.
Students should realize that this modification changes the parallelogram to a rectangle with the same area. 

60. Parallelograms 2

61. Triangles
Can you find a way to turn your triangle (or triangles) into a rectangle or a parallelogram?
Scissors, tape, grid paper
How can you use what you just discovered to develop a formula for finding the area of a triangle (without having to cut it up)?

62. Triangles

63. Triangles – Version 1

64. Triangles – Version 1

65. Triangles – Version 2

66. Triangles – Version 3

67. Triangles – Version 3

68. Triangles 1 NCTM Illuminations
Squares and Rectangles Activity Sheet

69. Triangles 1 Shapes A, B, and C
Using rulers, draw one diagonal in each of shapes
Cut each shape into two parts along the diagonal
Estimate the area of each triangle formed from each shape.

70. Shape D
Pick a point along the top of the rectangle
Using rulers, draw a line connecting the bottom corners of the rectangle to the selected point on the top
Cut each shape into 3parts along the lines
Estimate the area of the triangles formed
How does the area of the 2 small triangles relate to the area of the large triangle?

71. Triangles 2

72. Trapezoids
Can you find a way to turn your trapezoid (or trapezoids) into rectangles, parallelograms, or triangles?
Scissors, tape, grid paper
How can you use what you just discovered to develop a formula for finding the area of a trapezoid (without having to cut it up)?

73. Trapezoids – Version 1

74. Trapezoids – Version 3

75. Trapezoids – Version 3

76. Trapezoids – Version 2

77. Trapezoids 1 NCTM Illuminations
Centimeter Grid Paper
Trapezoids

78. Trapezoids 1
Ask students to suggest methods for finding the area of the trapezoid.
Prompt: "What other shapes could you use to help you? Are there any shapes for which you already know how to find the area?"
After students have had some time to discuss suggestions in their groups, ask them to share their ideas.

79. Trapezoids 2

80. Volume

81. G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

82. Volume Example:
V = base area x height
V = x
V = 60 cubic cm 15 4

83. V = 60 cubic cm Volume Example: V = length x width x height V = x x 53 4

84. Formative Assessment Lesson
This is a 5th Grade Formative Assessment Lesson on developing and understanding 2 formulas for finding the area of a rectangular prism:
V = base area x height
V = length x width x height

85. Formative Assessment Lesson
Mathematical goals
Recognizing volume as an attribute of three-dimensional space.
Measuring volume by finding the total number of same-size units of volume required to fill the space without gaps or overlaps.
Measuring necessary attributes of shapes, in particular the base area, in order to determine volumes to solve real world and mathematical problems.

86. Formative Assessment Lesson
Pre-Lesson:
Assessment Task: How Many Cubes?
As you work on the task, consider:
What are some common student errors or misunderstandings that you might expect to see?
What questions might you use to focus students’ attention on those issues?

87. Formative Assessment Lesson
Return task and pose questions; allow students about 10 min to revisit the task
Collaborative work (partner): card sort 1
Begin with a Task card from Card Set A.
Model this problem with the blocks first.
Then find a card from Card Set B that matches the model you built.

88. Formative Assessment Lesson
Card sort 2
Keep the pairs you have already sorted
Now look at Card Set C – Base Area
Match each of your sorted pairs with their correct base area figure

89. Formative Assessment Lesson
Card sort 3
Keep the sets you have already sorted
Now look at Card Set D – Formula Cards
Match each of your sorted sets pairs with the correct V = l x w x h formula; then fill in the missing number(s)
Whole class discussion and debrief
Improve individual solutions

90. V = 88.2 cubic cm Volume Example: V = base area x height V = x 17.64 5

91. V = 88.2 cubic cm Volume Example: V = length x width x height
V = x x 5
V = 88.2 cubic cm

92. Box of Clay
A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams of clay. A second box has twice the height, three times the width, and the same length as the first box. How many grams of clay can it hold?

93. Optimizing: Packing It In
FAL – The Shell Center
MATHEMATICAL GOALS
To help students to:
Reason precisely and defend their conclusions.
Use mathematics to model a scenario concerning volume.

94. Introduction The lesson unit is structured in the following way:
Before the lesson, students attempt the task Packing It In individually. You review their responses and formulate questions that will help them improve their work.
At the start of the lesson, students read your comments and consider ways to improve their work.

95. Introduction
In pairs or threes, students work together to develop a better solution, producing a poster to show their conclusions and their reasoning.
Then, in the same small groups, students look at some sample student work showing different approaches to the problem. They evaluate the strategies used and seek to improve the arguments given.

96. Introduction
In a whole-class discussion, students compare different solution methods.
Finally, students reflect individually on their learning.

98. Sample Work Analyze the sample student work
What common errors or misunderstandings do you see?
If these responses were typical of your class, what prompts or questions would you use to move the class forward?