1. Reason with shapes and their attributes. 2. G. 2
Reason with shapes and their attributes. 2.G.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
What mathematical ideas are addressed in this standard?
Read Grade 2 Progressions:
Page 11 Paragraphs 2 and 3.
Measurement is one idea that is addressed in this standard. The process of identifying an attribute, finding a unit and iterating a unit is acutally the process of measuring.
Do you have to count to identify which shape is larger?” Could fill one and move over to cover another shape or – compare the fillings of one shape onto another shape.

2. Classification of Shapes Spatial Visualization
Essential Understandings of Geometry
Classification of Shapes
Features or properties of geometric shapes can be analyzed and described to define and refine classification schemes with growing precision.
Spatial Visualization
Spatial relationships and spatial structuring involves developing, attending to, and learning how to work with imagery, as well as to specify locations.
Geometry is the branch of mathematics that addresses spatial sense and geometric reasoning.
Transformations
Transformation involves working with geometric phenomena in ways that build on spatial intuition by explaining what does and does not change when moving and altering the objects and the space that they occupy.

3. Structuring Task: Student Work

4. Spatial Structuring: Student Work
Individually study the student work samples.
With a partner discuss,
What misunderstandings and understands can you identify in the students’ work?
What is your evidence for these misunderstandings and understandings?

5. PRR: Spatial Structuring
In your 3-5 Essential Understandings book… Read about Spatial Structuring on Pages 104 – 110. What did you learn about the importance of spatial structuring? What did you learn about Isaac? How does this information impact how your work with students?

6. Area Measurement Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Core Math
July 21, 2016
8:00 – 10:30 AM

7. Composing and Decomposing
Essential Understandings of Measurement
Developing Essential Understanding of Geometry and Measurement for Teaching Mathematics
(Grades PreK-2, 3-5, and 6-8). Reston, VA: National Council of Teachers of Mathematics
How Much
Measurement specifies “how much” by assigning a number that corresponds to a chosen unit to attributes such as length, area, volume, and angle.
Comparing
Some attributes can be compared directly, others are measured and compared indirectly, and some are computed from other measurements.
Motion
Motion is useful in coordinating attributes and measures of length, area, volume, and angle.
One way to analyze and describe geometric objects, relationships among them, or the space they occupy is to quantify—measure or count—one or more of their attributes.
Developing Essential Understanding of Geometry and Measurement (NCTM 2014)
Composing and Decomposing
Building or breaking apart objects facilitate measuring them; those decompositions and compositions clarify relationships between attributes and units of measure and are the basis to derive formulas.
V = B h

8. Learning Intention
We are learning to help children understand the process of area measurement.
We will be successful when we can
Explain measurement of area as an example of the general measurement process;
Explain why measuring the area of a rectangle can be reduced to multiplication;
Use the Moving and Combining Principles to find areas of simple geometric shapes

9. What is “Area”?
Answer this question (silently!) on your own Compare your answer with those of others at your table. Come to consensus on an answer to report out to the whole group.

10. CCSSM Area Measurement Standards
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement
a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
3.MD.6 Measure areas by counting unit squares.

11. CCSSM Area Measurement Standards
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement
a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
3.MD.6 Measure areas by counting unit squares.
How do standards 3.MD.5 and 3.MD.6 relate to the general measurement process?

12. What is the area of the Evil Sea?

13. What is the Area of a Rectangle?
Draw a 3 x 5 rectangle on grid paper (or make one on a geoboard).
Find the area of this rectangle, and be prepared to explain your reasoning.
Given what you read yesterday, would you do this on grid paper? Dot paper? Plain paper?

14. What is the Area of a Rectangle?
Draw a 1/2 x ½ rectangle on your white board.
What is the area of this rectangle?
Can you find the area of this rectangle by multiplication?
Justify using a visual representation.
Provide ¼ “ dot paper

15. What is the Area of a Rectangle?
Draw a 1/3 x 1/5 rectangle on a piece of paper.
Can you find the area of this rectangle by multiplication? Why or why not?
Justify using a visual representation.

16. CCSSM Area Measurement Standards
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.4b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
How does standard 5.NF.4b relate to the general measurement process?

17. What is the area?
Sam is planning a garden. He wants to build a fence around the garden and he needs to buy mulch to cover the garden to control weeds. It will have the shape shown below in the picture. You can see what he has measured so far. Be sure to include the units in your answer and to show your work!
Now we know how to find the area of rectangles we should be able to solve tasks like these

18. What is the Area of this Figure?
Find the area of the parallelogram in as many ways as you can. (At least two.)
Explain your reasoning to your group.
Have participants work on geoboards

19. What is the area of this Figure?
Standard units still are not

20. What is the area of this Figure?
Standard units still are not

21. What is the area of this Figure?
Standard units still are not

22. Properties of Area
“Moving principle”: the area of a shape is not changed under a rigid motion. (Congruent shapes have equal areas.)
“Combining principle”: the total area of two (or more) non-overlapping shapes is equal to the sum of their areas.
How have we seen moving and combining properties at work during our work on linear measurement?
These have been powerful principles in area measurement, Make the connection back to linear measurement (worm task)

23. Who Is Correct?
Joe and Beth are discussing these two triangles. Joe thinks ∆ABC has a larger area than ∆ABF. Beth thinks the triangles have the same area. Who do you think is correct and how do you know?

24. Area of a Right Triangle
Explain why the area of a right triangle must be equal to ½ the product of its two legs.
We could (should?) leave the area formula for triangles until after the “Who is Correct” student work analysis.

25. Area of an Arbitrary Triangle
Explain why the area of a general triangle must be equal to ½ times the product of one side (the “base”) and the perpendicular distance from that side to the opposite vertex (the “height”)

26. 6.G.1
Solve real-world and mathematical problems involving area, surface area, and volume.
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.

27. 6.G.1
Solve real-world and mathematical problems involving area, surface area, and volume. 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 is that standard written that way?
If a kid can invert and multiply to divide fractions, does the kid really understand how to divide fractions

28. “The geometric understanding that is developed in grades 3-5 lays the foundation for the geometric ideas that students will develop in grades 6-8.” - Putting Essential Understanding of Geometry and Measurement into Practice 3-5, p. 138
Turn to page 140 in your Essential Understandings Book.
Read the 1st and 2nd full paragraphs (begins “Students in grades 6-8…”).
How does the information you read support the quotation from above?

29. Learning Intention
We are learning to help children understand the process of area measurement.
We will be successful when we can
Explain measurement of area as an example of the general measurement process;
Explain why measuring the area of a rectangle can be reduced to multiplication;
Use the Moving and Combining Principles to find areas of simple geometric shapes

30. Analyzing Student Work
Go through your set of student work and identify trends of what you have noticed. Be prepared to share with your opposite table group.
Trio 1: Planning a Garden—area only (Grades 3-5)
Trio 2: Who is Correct? (Grades 6-7)
Structure:
½ table looking at “Who is correct?”
½ table looking at “Planning a garden”
Identify misconceptions and ways students are meeting the expectations.
Identify pieces of student work that show interesting thinking based on what we’ve learned so far.
How are kids making sense of these problems and these concepts?
MP7 & MP8

31. Core Mathematics Partnership Project
Disclaimer
Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.