Measurement and Geometry

**1MG.1.0 Students use direct comparison and nonstandard units to describe the measurements of objects:
**1MG.1.1 Compare the length, weight, and volume of two or more objects by using direct comparison or a nonstandard unit.

1MG.1.2 Tell time to the nearest half hour and relate time to events (e.g., before/after, shorter/longer).

**1MG.2.0 Students identify common geometric figures, classify them by common attributes, and describe their relative position or their location in space:
**1MG.2.1 Identify, describe, and compare triangles, rectangles, squares, and circles, including the faces of three-dimensional objects.

1MG.2.2 Classify familiar plane and solid objects by common attributes, such as color, position, shape, size, roundness, or number of corners, and explain which attributes are being used for classification.

1MG.2.3 Give and follow directions about location.

1MG.2.4 Arrange and describe objects in space by proximity, position, and direction (e.g., near, far, below, above, up, down, behind, in front of, next to, left or right of).

**Comments:** none

**Emphasis:** 1.0 1.1 1.2 **1.3** 1.4 1.5 **2.0 2.1 2.2**

**2MG.1.0 Students understand that measurement is accomplished by identifying a unit of measure, iterating (repeating) that unit, and comparing it to the item to be measured:
**2MG.1.1 Measure the length of objects by iterating (repeating) a nonstandard or standard unit.

2MG.1.2 Use different units to measure the same object and predict whether the measure will be greater or smaller when a different unit is used.

2MG.1.3 Measure the length of an object to the nearest inch and/ or centimeter.

2MG.1.4 Tell time to the nearest quarter hour and know relationships of time (e.g., minutes in an hour, days in a month, weeks in a year).

2MG.1.5 Determine the duration of intervals of time in hours (e.g., 11:00 a.m. to 4:00 p.m.).

**2MG.2.0 Students identify and describe the attributes of common figures in the plane and of common objects in space:
**2MG.2.1 Describe and classify plane and solid geometric shapes (e.g., circle, triangle, square, rectangle, sphere, pyramid, cube, rectangular prism) according to the number and shape of faces, edges, and vertices.

2MG.2.2 Put shapes together and take them apart to form other shapes (e.g., two congruent right triangles can be arranged to form a rectangle).

**Comments:** Although Standard 2MG.1.3 listed below from the Measurement and Geometry strand is important, more emphasis should be given to the topics in Standard 2MG.2.0.

**2MG.1.3** Measure the length of an object to the nearest inch and/or centimeter.

**2MG.2.0** Students identify and describe the attributes of common figures in the plane and of common objects in space. Because understanding spatial relations will be more difficult for some students than for others (especially the concepts involving three-dimensional information), teachers should carefully assess how well students understand these shapes and figures and their relationships.

**Emphasis:** 1.0 1.1 **1.2 1.3** 1.4 2.0 **2.1 2.2 2.3** 2.4 2.5 2.6

**3MG.1.0 Students choose and use appropriate units and measurement tools to quantify the properties of objects:
**3MG.1.1 Choose the appropriate tools and units (metric and U.S.) and estimate and measure the length, liquid volume, and weight/mass of given objects.

3MG.1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them.

3MG.1.3 Find the perimeter of a polygon with integer sides.

3MG.1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes).

**3MG.2.0 Students describe and compare the attributes of plane and solid geometric figures and use their understanding to show relationships and solve problems:
**3MG.2.1 Identify, describe, and classify polygons (including pentagons, hexagons, and octagons).

3MG.2.2 Identify attributes of triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle).

3MG.2.3 Identify attributes of quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square).

3MG.2.4 Identify right angles in geometric figures or in appropriate objects and determine whether other angles are greater or less than a right angle.

3MG.2.5 Identify, describe, and classify common three-dimensional geometric objects (e.g., cube, rectangular solid, sphere, prism, pyramid, cone, cylinder).

3MG.2.6 Identify common solid objects that are the components needed to make a more complex solid object.

**Comments:** In the first Measurement and Geometry standard, Standards 3MG.1.2 and 3MG.1.3 should be emphasized:

**3MG.1.2** Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them.

**3MG.1.3** Find the perimeter of a polygon with integer sides. The idea that one cannot talk about area until a square of side 1 has been declared to have unit area and is then used to measure everything else is usually not firmly established in standard textbooks. Analogies should be constantly drawn between length and area. For example, a line segment having a length 3 means that, compared with the segment L that has been declared to be of length 1, it can be covered exactly by 3 non-overlapping copies of L. Likewise, a rectangle with sides of lengths 3 and 1 has an area equal to 3 because it can be exactly covered by three non-overlapping copies of the square declared to have length 1.

In the second Measurement and Geometry standard, Standards 3MG.2.1, 3MG.2.2, and 3MG.2.3 are the most important.

**3MG.2.1** Identify, describe, and classify polygons (including pentagons, hexagons, and octagons).

**3MG.2.2** Identify attributes of triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle).

**3MG.2.3** Identify attributes of quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square). All of these standards can be difficult to master if they are presented too generally. A principal difficulty with geometry at all levels is the need of precise definitions of geometric concepts. Even in grade three, we need a workable definition of a polygon, which textbooks usually do not supply. One may define a polygon as a finite number of line segments, joined end-to-end, so that together they form the complete boundary of a single planar region. It is strongly recommended that the skills for this grade level be limited to such topics as finding the areas of rectangles with integer sides, right triangles with integer sides, and figures that can be partitioned into such rectangles and right triangles. A few examples in which the sides are not whole numbers should also be provided. Estimation should be used for these examples. Implicit in Standards 3MG.2.4 and 3MG.2.5 is the introduction of the concept of an angle. But this topic should not be emphasized at this time.

**Emphasis:** 1.0 1.1 1.2 1.3 1.4 **2.0 2.1 2.2 2.3**

**4MG.1.0 Students understand perimeter and area:**

4MG.1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2).

4MG.1.2 Recognize that rectangles that have the same area can have different perimeters.

4MG.1.3 Understand that rectangles that have the same perimeter can have different areas.

4MG.1.4 Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes.

**4MG.2.0 Students use two-dimensional coordinate grids to represent points and graph lines and simple figures:
**4MG.2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation

4MG.2.2 Understand that the length of a horizontal line segment equals the difference of the

4MG.2.3 Understand that the length of a vertical line segment equals the difference of the

**4MG.3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems:
**4MG.3.1 Identify lines that are parallel and perpendicular.

4MG.3.2 Identify the radius and diameter of a circle.

4MG.3.3 Identify congruent figures.

4MG.3.4 Identify figures that have bilateral and rotational symmetry.

4MG.3.5 Know the definitions of a right angle, an acute angle, and an obtuse angle. Understand that 90°, 180°, 270°, and 360° are associated, respectively, with 1/4, 1/2, 3/4, and full turns.

4MG.3.6 Visualize, describe, and make models of geometric solids (e.g., prisms, pyramids) in terms of the number and shape of faces, edges, and vertices; interpret two-dimensional representations of three-dimensional objects; and draw patterns (of faces) for a solid that, when cut and folded, will make a model of the solid.

4MG.3.7 Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their attributes.

4MG.3.8 Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid).

**Comments:** The Measurement and Geometry strand for the fourth grade contains a few key standards that students will need to understand completely.

The first standard (**4MG.1.0**) relates to perimeter and area. The students need to understand that the area of a rectangle is obtained by multiplying length by width and that the perimeter is given by a linear measurement. The intent of most of this standard is that students know the reasons behind the formulas for the perimeter and area of a rectangle and that they can see how these formulas work when the perimeter and area vary as the rectangles vary. A more basic standard is the second one:

**4MG.2.0** Students use two-dimensional coordinate grids to represent points and graph lines and simple figures. Although the material in this standard is basic and is not presented in depth, this concept must be presented carefully. Again, students who are confused at this point will very likely have serious difficulties in the later grades-not just in mathematics, but in the sciences and other areas as well. Therefore, careful assessment is necessary. We call special attention to the need of students to understand the graphs of the equations x = c and y = c for a constant c. These are what are commonly called vertical and horizontal lines, respectively. What has to be done is to get hold of some points on these graphs strictly according to the definition of the graph of an equation as the set of all points (x, y,) whose coordinates satisfy the given equation. Unless this is painstakingly done, these graphs will continue to be nothing but magic through the rest of students' schooling. In connection with Standard 4MG.3.0, teachers should introduce the symbol ^ for perpendicularity. Incidentally, this is the time to introduce the abbreviated notation ab in place of the cumbersome a × b.

**Emphasis:** 1.0 **1.1 1.2 1.3** 1.4 2.0 **2.1 2.2** 2.3

**5MG.1.0 Students understand and compute the volumes and areas of simple objects:
**5MG.1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).

5MG.1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.

5MG.1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], cubic yard [yd3]) to compute the volume of rectangular solids.

5MG.1.4 Differentiate between, and use appropriate units of measures for, two-and three-dimensional objects (i.e., find the perimeter, area, volume).

5MG.2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.

5MG.2.3 Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.

**Comments:** Finally, in Measurement and Geometry these three standards should be emphasized:

**5MG.1.1** Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).

**5MG.2.1** Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).

**5MG.2.2** Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems. Students need to commit to memory the formulas for the area of a triangle, a parallelogram, a rectangle, and the volume of a rectangular solid. The fact that the angle sum of a triangle is 180° is one of the basic facts of plane geometry, but for students in grade five, it is more important to convince them of this fact through direct measurements than to give a proof.

**Emphasis:** 1.0 **1.1** 1.2 1.3 2.0 2.1 **2.2**

**6MG.1.0 Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems:**

6MG.1.1 Understand the concept of a constant such as p; know the formulas for the circumference and area of a circle.

6MG.1.2 Know common estimates of p (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.

6MG.1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base x height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.

**6MG.2.0 Students identify and describe the properties of two-dimensional figures:**

6MG.2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.

6MG.2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.

6MG.2.3 Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).

**Comments:** The following core standards are a part of the Measurement and Geometry strand:

**6MG.1.1** Understand the concept of a constant such as p; know the formulas for the circumference and area of a circle.

**6MG.2.2** Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.

One can define p in many different ways: p=area of the unit circle, p=circumference/diameter of any circle. The custom here is to define it as the ratio of the circumference to diameter. The latter is built on two concepts relatively new to students, ratio and length of a curve (circumference), whereas the former uses only the concept of area. Moreover, the area of the unit circle can be approximated directly by the use of (good) grid papers, and students have a good chance of getting p = 3.14 ± 0.05. This would not only create a strong impression on students but also deepen their understanding of both the number p and the concept of area.

Standard 6MG.1.3 is also important, and students should know that the volumes of three-dimensional figures can often be found by dividing and combining them into figures whose volumes are already known.

Emphasis: 1.0 1.1 1.2 **1.3** 2.0 2.1 2.2 2.3 2.4 3.0 3.1 3.2 **3.3 3.4** 3.5 **3.6**

**7MG.1.0 Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems:
**7MG.1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters).

7MG.1.2 Construct and read drawings and models made to scale.

7MG.1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.

**7MG.2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale:
**7MG.2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

7MG.2.2 Estimate and compute the area of more complex or irregular two-and three-dimensional figures by breaking the figures down into more basic geometric objects.

7MG.2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.

7MG.2.4 Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 144 square inches or [1 ft

**7MG.3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures:
**7MG.3.1 Identify and construct basic elements of geometric figures (e.g., altitudes, mid-points, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge.

7MG.3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.

7MG.3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.

7MG.3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.

7MG.3.5 Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms, and cones.

7MG.3.6 Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect).

**Comments:** The first major emphasis in the Measurement and Geometry strand is for the students to develop an increased sense of spatial relations. This topic is reflected in these two standards:

**7MG.3.4** Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.

**7MG.3.6** Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect). A critical part of understanding this material is that the students know the general definition of congruence-two figures are congruent if a succession of reflections, rotations, and translations will make one coincide with the other-and understand that properties of congruent figures, such as angles, edge lengths, areas, and volumes, are equal.

The next basic step is contained in the following standard:

**7MG.3.3** Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. The Pythagorean theorem is probably the first true theorem that the students will have seen. It should be emphasized that students are not expected to prove this result. But the better students should be able to understand the proof given by cutting, in two different ways, a square with the edges of length a + b (where a and b are the lengths of the legs of the right triangle). However, everyone is expected to understand what the theorem and its converse mean and how to use both. The applications can include understanding the formula that the square root of x2 + y2 is the length of the line segment from the origin to the point (x, y) in the plane and that the shortest distance from a point to a line not containing the point is the length of the line segment from the point perpendicular to the line. Although the following topics are not as basic as the preceding ones, they should also be covered carefully. Seventh grade students should memorize the formulas for the volumes of cylinders and prisms (Standard 6MG.2.1). Students at this point should understand the discussion that began in the sixth grade concerning the volume of "generalized cylinders." More precisely, they should think of a right circular cylinder as the solid traced by a circular disc as this disc moves up a line segment L perpendicular to the disc itself. The disc is replaced with a planar region of any shape, and the line segment L is no longer required to be perpendicular to the planar region. Then, as the planar region moves up along L, always parallel to itself, it traces out a solid called a generalized cylinder. The formula for the volume of such a solid is still (height of the generalized cylinder) x (area of the planar region). Height now refers to the vertical distance between the top and bottom of the generalized cylinder.The final topic to be emphasized in seventh grade Measurement and Geometry is as follows:

**7MG.1.3** Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer. This standard interacts well with the demands of the algebra standards, particularly in solving linear equations. Typically, the main difficulty in understanding problems of this kind is keeping the definitions and the physical significance of the various measures straight; therefore, care should be taken to emphasize the meanings of the terms in the various problems.

**Emphasis:** none given

Keywords: 1MG ; 2MG ; 3MG ; 4MG ; 5MG ; 6MG ; 7MG ; Geometry ; Measurement ; Measurement and Geometry Strand