MIDDLE SCHOOL GEOMETRY Michelle Corbin, NBCT Making Connections Conference Biloxi, MS June 11 th – 13 th, 2014

THE 8 MATHEMATICAL PRACTICES Directions:  Locate the Ziploc bag in the center of the table and empty it’s contents.  Work with your table mates to math the math practice with it’s big idea.

HOW CAN WE PROVIDE OPPORTUNITIES? Math Practice #1…Never give up!  Provide challenging problems and question effectively. Math Practice #2…Think through it!  Provide opportunities for students to work together.

HOW CAN WE PROVIDE OPPORTUNITIES? Math Practice #3…Argue and PROVE your work!  Provide opportunities for students to explain and justify their work. Math Practice #4…Manipulatives, equations, tables, etc.  Allow student to show multiple methods of problem solving.

HOW CAN WE PROVIDE OPPORTUNITIES? Math Practice #5…Know when and how to use tools!  Provide various “tools” regularly and encourage students to use them. Math Practice #6…Communicate all the details!  Question students about the details while they are explaining.

HOW CAN WE PROVIDE OPPORTUNITIES? Math Practice #7…Connections across math topics!  Point out ways to build mathematical understanding on prior knowledge. Math Practice #8…Identify patterns to create an algorithm for solving!  Provide student explorations BEFORE teaching algorithms.

WHAT are you teaching?!?! Otherwise known as the CONTENT STANDARDS in the Common Core

PARCC MODEL CONTENT FRAMEWORK Major Clusters  Major focus Supporting Clusters  Assessed through major content Additional Clusters  Instrumental for next grade level, may not have a heavy correlation with another standard to be considered “supporting.”

PARCC TEST BLUEPRINT (EOY/PBA) Calculators  Yes or No  Fluency? Math Practices  Process  How? Clarifications  Additional insight

ACTIVITY GOALS 1) Recognize ways to incorporate the standards for mathematical practice in lessons. 1) Identify CCSS-M content standards in an activity or lesson.

CALCULATING VOLUME  Take a moment and work with your table to calculate the volume of the figure to the right. Volume = l(w)(h) 5(6)(7) 210 in 3

UNDERSTANDING VOLUME Directions :  Locate the “Volume” envelope and the pack of cubes in the center of the table.  Work with your team to build a model that could represent the information given on the card.  Complete one card at a time.

VOLUME…ONE STEP FURTHER  Joaquin wants to find the volume of his cereal box, but he only has ½ inch cubes available. He measured the box and found that it was 7 ½ inches wide, 11" tall, and 2 ½ inches thick.  With your team, determine how many cubes it will take to completely fill the cereal box. 7.5(2) = 1511(2) = 222.5(2) = 5 15(22)(5) = 1,650 half-inch cubes

VOLUME STRATEGIES  How many cubes fit in a 1” cube?  Using your answer from the previous problem, what is the volume of the cereal box measured in cubic inches? 8 1,650 ÷ 8 = inches 3

VOLUME STRATEGIES  Look back at your strategy for determining the number of 1 inch cubes it will take to fill the box from our first problem, (slide 12).  Calculate the volume (measured in cubic inches) of Joaquin’s cereal box using this strategy. Did you get the same volume as in part (b) of this problem? Why or why not? 7.5(11)(2.5) = inches 3

REFLECTING ON OUR GOALS Which Mathematical Practice(s) were used in the activity? Provide a complete justification for each MP you identify. Which content standard(s) were addressed in these activities? 6.G.2

SAMPLE PARCC ITEMS EOY 6.G

SAMPLE PARCC ITEMS EOY 6.G

THREE-DIMENSIONAL FIGURES  A cross section of a three-dimensional solid (an object that has length, width, and height) is the figure obtained when the solid is intersected or “sliced” by a plane.

THREE-DIMENSIONAL FIGURES  Using clay, make a rectangular prism or cube. Each person in your team should make a slightly different prism.  Use your string to slice the rectangular prism by holding it tightly and pulling it through the solid. Place the newly created face made by the slice that came from the inside of the rectangular prism on your paper and trace the resulting shape. What is the name of your shape?

THREE-DIMENSIONAL FIGURES Which of the following shapes can you make by slicing your prism in different ways? Before making a new slice, reshape your prism. Triangle Square Rectangle Trapezoid Parallelogram Pentagon Hexagon Octagon Circle

THREE-DIMENSIONAL FIGURES Now make a rectangular pyramid. Again, each person in your team should make a slightly different pyramid. Using your string, what other shapes can you make by slicing your pyramid in different ways? What shapes are impossible to make? How do the types of shapes that you can make with the rectangular pyramid compare to the types of shapes that you made with the rectangular prism?

Three- Dimensional Figures Use the pyramid shown at left to answer the questions below.  If the pyramid were sliced vertically, what would the resulting cross sections look like? Sketch them.  If the pyramid were sliced horizontally, what would the resulting cross section look like? Sketch it.

REFLECTING ON OUR GOALS Which Mathematical Practice(s) were used in the activity? Provide a complete justification for each MP you identify. Which content standard(s) were addressed in these activities? 7.G.3

SAMPLE PARCC ITEMS EOY 7.G.2

PROVING PYTHAGORAS The picture to the left represents the grass area in my backyard. The width of the grass area is 10 yards and the length of the grass area is 10 yards. How much total grass do I mow? 100 square yards

PROVING PYTHAGORAS I want to have a pool put in the backyard as shown. If the total area of the pool is 20 square yards, how much grass will I be cutting after the pool is installed? 100 – 20 = 80 square yards

PROVING PYTHAGORAS What if I move the pool to this location? How much grass will I have to mow? The amount will stay the same since the area of the total grass and the area of the pool are staying the same. 100 – 20 = 80 square yards

PROVING PYTHAGORAS What if I wanted to split the pool into two equal sections? No matter how we move or cut a figure, as long as we have the same size, it will always have the same area.

PROVING PYTHAGORAS What is the area of the inner green square? What is the area of the outer square? Can we rearrange the triangles in some way to PROVE a 2 + b 2 = c 2 ? c 2 b a c b a c b a c b a c (a + b) 2

PROVING PYTHAGORAS Work with your team to figure out a way to rearrange and/or cut the triangles in some way to PROVE a 2 + b 2 = c 2 using the manipulatives on the tables. b a c b a c b a c b a c

PROVING PYTHAGORAS What is the area of the green space now? b a c b a c b a c b a c c 2

PROVING PYTHAGORAS What is the area of the green space now? b a c b a c b a c b a c c2c2

PROVING PYTHAGORAS What is the area of the green space now? Can you rename it in any way? b a c b a c b a c b a c c2c2 a 2 + b 2 b2b2 a2a2

PROVING PYTHAGORAS What area does this shape represent in relationship to a, b, and c? a 2 + b 2 b b a a

PROVING PYTHAGORAS  Use your original triangles as a tracing tool on your model of a 2 + b 2 as shown at the left.  Cut along the traced lines.  Now can you rearrange the shapes to model the area as c 2 ? b a c b a c

PROVING PYTHAGORAS Cut, move, and replace. b a c b a c

PROVING PYTHAGORAS Cut, move, and replace again. b a c b a c

REFLECTING ON OUR GOALS Which Mathematical Practice(s) were used in the activity? Provide a complete justification for each MP you identify. Which content standard(s) were addressed in these activities? 8.G.6

EVIDENCE TABLES FOR 8.G.6  Not specifically identified  It is found encompassed in the “C” claims  8.C.5.3  Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content Scope: Knowledge and skills articulated in 8.G.B  Mathematical Practices: 2, 3, 5  Calculator will be allowed

CREDITS AND RESOURCES College Preparatory Mathematics  PARCC 

CONTACT INFORMATION Michelle Corbin, NBCT Mathematics Specialist Desoto County Schools (662) office (901) cell