Geometry standards MATH 124

Kindergarten Identify and describe shapes. CCSS.Math.Content.K.G.A.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.CCSS.Math.Content.K.G.A.1 CCSS.Math.Content.K.G.A.2 Correctly name shapes regardless of their orientations or overall size.CCSS.Math.Content.K.G.A.2 CCSS.Math.Content.K.G.A.3 Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid").CCSS.Math.Content.K.G.A.3

Analyze, compare, create, and compose shapes. CCSS.Math.Content.K.G.B.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length).CCSS.Math.Content.K.G.B.4 CCSS.Math.Content.K.G.B.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.CCSS.Math.Content.K.G.B.5 CCSS.Math.Content.K.G.B.6 Compose simple shapes to form larger shapes. For example, "Can you join these two triangles with full sides touching to make a rectangle?"CCSS.Math.Content.K.G.B.6

Analysis Students are expected to be at Van Hiele level 0. ▫Students recognize and name shapes. ▫What shapes “look like” ▫They may not understand that orientation does not affect the shape. ▫Students at this level can begin to understand classifications of shapes.

First grade Reason with shapes and their attributes. CCSS.Math.Content.1.G.A.1 Distinguish between defining attributes (e.g., triangles are closed and three- sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.CCSS.Math.Content.1.G.A.1 CCSS.Math.Content.1.G.A.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. 1CCSS.Math.Content.1.G.A.2 CCSS.Math.Content.1.G.A.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.CCSS.Math.Content.1.G.A.3

Analysis A few of you made the observation about the relationship between geometry and fractions in the standards. This is probably just acknowledgment that when you are breaking up a square, circle, or rectangle into equal parts, you are also using geometric reasoning. There is a neat progression of this understanding between grades 1-3: halves and fourths in first grade, thirds in second grade, and other parts in third.

First grade still seems to be at Ven Hiele level 0, but moving toward level 1. The focus seems to be on manipulating shapes and developing visualization. However, understanding defining versus non-defining characteristics is level 1.

Second grade Reason with shapes and their attributes. CCSS.Math.Content.2.G.A.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. 1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.CCSS.Math.Content.2.G.A.1 CCSS.Math.Content.2.G.A.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.CCSS.Math.Content.2.G.A.2 CCSS.Math.Content.2.G.A.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.CCSS.Math.Content.2.G.A.3

Analysis The focus in second grade is still on using geometry to develop number and operation sense, so area us used to help understand multiplication, and students split regions into thirds and not just halves. Recognizing attributes is Van Hiele level 1, but no great geometric development happens in grade 2.

Third grade Reason with shapes and their attributes. CCSS.Math.Content.3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.CCSS.Math.Content.3.G.A.1 CCSS.Math.Content.3.G.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.CCSS.Math.Content.3.G.A.2

Analysis In third grade, it seems that students should be at Van Hiele level 1, progressing to level 2. At level 1, students begin to notice properties of shapes, as they are supposed to do in third grade with quadrilaterals. However, students at level 1 do not necessarily understand that different quadrilaterals are subclasses of one another, which is why third grade standards progress toward level 2.

Fourth grade Draw and identify lines and angles, and classify shapes by properties of their lines and angles. CCSS.Math.Content.4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two- dimensional figures.CCSS.Math.Content.4.G.A.1 CCSS.Math.Content.4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.CCSS.Math.Content.4.G.A.2 CCSS.Math.Content.4.G.A.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.CCSS.Math.Content.4.G.A.3

Analysis In fourth grade, many new terms are defined and used, but still with the intention of classifying shapes. No great geometric developments are made in grade 4, and students are still expected to be at level 1.

Fifth grade Classify two-dimensional figures into categories based on their properties. CCSS.Math.Content.5.G.B.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.CCSS.Math.Content.5.G.B.3 CCSS.Math.Content.5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties.CCSS.Math.Content.5.G.B.4

Analysis In fifth grade, all the work from K-4 grade comes together. Quadrilaterals and triangles are fully classified based on their properties. According to the standards, fifth grade students should be at Van Hiele level 2: they think about properties of geometric objects and are able to reason about them. Some of you noted that these standards are demanding.

It might be unreasonable to expect that students will go from level 0 to level 2 in five years. It is possible if they are exposed to a lot of hands-on activities where they can interact with objects of study, but unfortunately geometry is not the primary focus in elementary classrooms. It helps that some connections are made between geometry and operation standards. My recommendation: use a lot of hands-on activities with young children to help them progress through the Van Hiele levels.

Van Hiele level 3 High school geometry classes are taught at level 3, and if students have not progressed to level 2 in K-8 education, they will not be able to succeed in such a class.

Coordinate geometry? Graph points on the coordinate plane to solve real-world and mathematical problems. CCSS.Math.Content.5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y- coordinate).CCSS.Math.Content.5.G.A.1 CCSS.Math.Content.5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.CCSS.Math.Content.5.G.A.2

Activity (Level 2) If it is a square, then it is a rhombus. All squares are rectangles. Some parallelograms are rectangles. All parallelograms have congruent diagonals. If it has exactly two lines of symmetry, it must be a quadrilateral. All pyramids have square bases.

Area and geometry? Starting in 6 th grade, geometry standards start talking about area. Why not before? Because area does come up before, but in the measurement strand (starting in third grade). Geometry and measurement are related but different strands of mathematics. Both are taught in geometry courses. We will look at area in some detail in the next few weeks, and will revisit the measurement standards next week.

Congruence and similarity Because these are middle school topics, we are leaving them for the last week of classes as optional material.

Anything else?