Measurement and Geometry
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Curriculum Resource Guide - Mathematics Content: Measurement and Geometry
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| Used in conjunction with the Content Modules that describe the content (what is being taught) in general education they provide teachers with the necessary background knowledge to prepare students for the NCSC alternate assessment. |
| The Curriculum Resource Guides offer examples of how the content is taught in general education, ideas for real life use, examples of universal design for learning, and ways to promote college and career readiness. |
Questions about:
- Perimeter
- Area
- Volume
- Surface Area
- Classifying and Comparing Figures
- Converting Units of Measurement
- Pythagorean Theorem
1a. What is “perimeter” and how is it taught in general education settings?
1a.1 Essential knowledge in this content area
- The concept of perimeter refers to the distance around a polygon. (A polygon is a shape which is formed by line segments enclosing an area). The distance can be found by adding the lengths of all sides. Students first begin to explore this concept by laying a ruler around all of the sides of an object and then adding all of the lengths. Students can be encouraged to generalize perimeter into the following equation:
- For rectangles:
- P = 2l + 2w (where P = perimeter, l = length, and w = width)
- For triangles:
- Add all sides or P = a + b + c (where P = perimeter and a, b, c = the sides of the triangle)
For circles:
- It is called circumference (the distance around a circle). To find circumference students must know: radius (connects the center to any given point on the circle) or diameter (connects two points on the circle and passes through the center).
- The ratio of the circumference to the diameter (C/d) of any circle is the same for all circles. This ratio is called pi, or π. You can use this relationship to find a formula for circumference. Pi (π) is an irrational number that is often approximated by the rational number 3.14.
- The circumference (C) of a circle is π times the diameter (d), or 2π times the radius (r).
- C = πd OR C=2πr
1a.2 Common misunderstandings in this content area
- Students may have a hard time understanding the dual meaning of the word length. It is not only the distance measured of each line segment, but it also means the longer side of a rectangle (where the width means the shorter side of the rectangle). In addition, once area is introduced students may become confused between area and perimeter. A helpful analogy to use is that perimeter is the “fence” and area is the “lawn.”
1a.3 Prior knowledge/skills needed (can be taught concurrently)
- Identify a polygon
Polygon Nonpolygon
- Addition
- Draw or connect line segments that touch end to end to enclose an area
- Multiply by 2
1b. What is “area” and how is it taught in general education settings?
1b.1 Essential knowledge in this content area
- The concept of area focuses on determining the amount of space inside of a two-dimensional figure. This is typically done in elementary school with the process of tiling an entire region with a small square piece called a unit square. For example, if we took 1 cm by 1 cm square tiles and covered a rectangle with 6 tiles, we would say that the area of the rectangle is 6 square centimeters.
Students should have ample hands-on experiences with tiles to construct understanding that calculating area is the process of repeatedly tiling a two-dimensional shape with a unit square. Whether the units are centimeters, inches or other measures does not matter, as long as the unit is consistently used to measure an object. As students master understanding of the concept you can teach the following formula to find the area of a rectilinear (composed of right [90⁰] angles) figure:
- A = l x w (where A = area, l = length, and w = width)
Area of a circle is more complex. Students can estimate area of a circle by counting squares within, similar to rectangles. The area A of a circle is π times the square of the radius r.
- A = πr2
1b.2 Common misunderstandings in this content area
The most common misunderstanding is confusing the concepts of area and perimeter. The perimeter of a shape is the distance around, while the area is the amount of space inside. If teachers move too quickly to a formula to calculate these two concepts, then students become easily confused.
A composite shape is a flat shape composed of two different sized rectangles. Students must break the shape up into rectangles in order to determine the entire area.
1b.3 Prior knowledge/skills needed (can be taught concurrently)
In general education, the student typically will need to:
- Lay tiles on a surface with no gaps or overlaps
- Count tiles after covering an object
- Draw squares on a piece of grid (graph) paper OR count squares on a piece of paper
- Multiply numbers
- Understand concepts of length and width
1c. What is “volume” and how is it taught in general education settings?
1c.1 Essential knowledge in this content area
- Volume is the amount of space a three dimensional (3-D) figure takes up. Examples of 3-D figures that take up space include rectangular prisms, spheres, cubes, cones, cylinders, and pyramids. Students should understand figures that have volume are 3-D rather than flat, two-dimensional (2-D) objects. Rectangular prisms and cubes have three dimensions: length, width, and height. A flat surface of a 3-D figure is a face. An edge is where two faces meet. A vertex is where the figure comes to a point. The base is the shape used to classify the figure. Students should be provided with numerous hands-on opportunities to explore items that take up space and label the attributes of the figures.
- The units used to measure volume are called cubic units. Previously, students used tiles to measure area, and now they will be using cubes to measure volume. Volume is an extension of area. To determine area, students tiled a flat surface with unit squares. To determine the volume of a rectangular prism or cube, students begin by filling the object with unit cubes. The Common Core State Standards for Math refer to rectangular prisms and cubes as rectilinear shapes, since the shapes are composed of all right angles, and every face can be composed of 1 or more rectangles. Again, whether the units of the cubes are centimeters, inches, or other measures does not matter, as long as the unit is consistently used to measure an object. Students can practice measuring volume by using cubes to make a replica of a given shape and then count the cubes to measure the volume. They can also begin to identify the length, width, and height of the object and its numerical value. As students master understanding of the concept you can teach the following formulas:* *Rectangular Prisms
- The volume of a prism is the area of the base (B) [length times width] times the height (h).
- V = l x w x h OR V = B x h
- Cylinders
- The volume of a cylinder is the area of the base (B) [πr2] times the height (h).
- V = B x h OR V = πr2 x h
- Spheres
- The volume of a sphere is 4/3π times the cube of the radius (r).
- V = 4/3π x r3
1c.2 Common misunderstandings in this content area
Students may think that if items have the same volume, they must be the same shapes (i.e., “This square has 8 cubic units of volume, therefore all objects with 8 units of cubic volume are squares.”)
Example of two different shapes with the same volume.
1c.3 Prior knowledge/skills needed (can be taught concurrently)
In general education, the student typically will need to:
- Identify dimensions of length, width, and height
- Calculating area using a formula
- Count cubes
1d. What is “surface area” and how is it taught in general education settings?
1d.1 Essential knowledge in this content area
- Surface area is the sum of all the faces of an object. Students can first learn to add the areas of the faces to find the surface area. Nets will help with this. See: http://illuminations.nctm.org/LessonDetail.aspx?id=L570 for further information and examples. Students should be encouraged to engage in activities using nets to deepen their understanding of surface area (e.g., matching net to corresponding 3-D object).
pyramid and net rectangular prism and net Then the following formulas can be taught to find the surface area of rectangular prisms, cubes, and square pyramids:
- Rectangular prisms: SA = 2lw + 2lh + 2wh (where SA = surface area, l = length, w = width, and h = height)
- Cubes: SA = 6s² (where SA = surface area, and s = length of sides)
- Square pyramid SA = B + ½ Pl (where SA = surface area, B = base area, P = perimeter of base, l = slant height
1d.2 Common misunderstandings in this content area
Students may believe that they have identified surface area by merely counting all the faces of the object.
1d.3 Prior knowledge/skills needed (can be taught concurrently)
In general education, the student typically will need to:
- Understand the concept of the “face” of objects
- Count the number of “faces” on an object
- Solve equations using order of operations
- Count cubes
- Identify cubes and rectangular prisms
- Understand how to square a number (e.g. 3²)
1e. What is “Classifying and Comparing Figures” and how is it taught in general education settings?
1e.1 Essential knowledge in this content area
- Students should learn to identify 2-D and 3-D figures including: rectangles, squares, triangles, trapezoids, quadrilaterals, pentagons, hexagons, triangles, prisms, cones, cylinders, pyramids, cubes, and spheres. Given pattern blocks, students should be able to compose identified figures. They need to know the salient (defining) and non-salient (non-defining) features of figures. For instance, the defining features of rectangles are: they have four sides, they are closed figures (all lines touch end to end), opposite ends are parallel, and all four angles enclosed are right angles. Non-defining features of a rectangle would be its size, color, or orientation. Knowing the defining features of figures will enable students to compare figures based on attributes.
Attributes of figures students will need to know include:
- Recognize different types of angles:
right acute obtuse
- Understand parallelism and perpendicularity
Two examples of parallel lines Two examples of perpendicular lines Students need to understand congruency. Figures that are congruent have the same shape and size.
Example of congruent shapes Example of non-congruent shapes
- Once students understand defining attributes, given the following shapes students should be able to classify them by number of angles, number of sides, number of angles which are greater than ninety degrees, tell whether the shapes are congruent, identify which shape is a quadrilateral, etc.
1e.2 Common misunderstandings in this content area
Students may learn the label of a figure such as “square” and when asked if it is a quadrilateral say “no.” However, since the definition of a quadrilateral figure is that it has four sides then a square is a quadrilateral. Therefore students will need practice sorting shapes into categories. Also, students may have difficulty determining whether shapes are congruent when they are rotated (see example on previous page) and may need repeated practice with determining congruency of shapes with various rotations. 1e.3 Prior knowledge/skills needed (can be taught concurrently)
- Label 2-D and 3-D figures based on attributes (e.g., triangle, pyramid)
- Count (number of angles, sides, etc.)
- Understand same and different
1f. What is “Converting Units of Measurement” and how is it taught in general education settings?
1f.1 Essential knowledge in this content area
In order to be able to convert measurements, students will need to understand that objects can be measured to find length, weight, and capacity (of liquids). Students should know the units of measurement and the relationship between units of measurement in the same system. For example:
| Length |
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| U.S. Unit | Equivalent | Metric Unit | Equivalent |
|---|
| 1 foot (ft) | 12 inches (in) | 1 centimeter (cm) | 10 millimeters (mm) |
| 1 yard (yd) | 3 feet (ft) | 1 meter (m) | 100 centimeters (cm) |
| 1 mile (mi) | 1,760 yards (yd) | 1 kilometer (km) | 1,000 meters (m) |
1f.2 Common misunderstandings in this content area
- Students may find converting standard measures particularly difficult because: (a) the number to multiply or divide to convert will vary based on the given unit (e.g., feet to inches, multiply by 12; yards to feet, multiply by 3); (b) if the conversion does not equal an exact whole number unit of measurement then students may need to combine twounits to provide the answer (e.g., 5 feet, 11 inches) [This is only true with standard measure, with metric measurements decimals may be used.]; and (c) If conversions are more than one unit removed students will be required to perform multiple steps to find the correct amount (e.g., if converting weeks to minutes, students must convert weeks to days, then days to hours, finally hours to minutes).
1f.3 Prior knowledge/skills needed (can be taught concurrently)
- Knowledge of both metric and standard units of measurement
- Knowledge of measurement units for length, volume, mass, time
- Knowledge of equivalents (e.g., 1 foot = 12 inches)
- Students can be provided with a conversion chart if needed
- Multiplication and division
- Number identification
- Use of measurement tools (i.e., line up a ruler’s edge to the edge of the object being measured)
- Knowledge of fractions and decimals
1g. What is “Pythagorean Theorem” and how is it taught in general education settings?
1g.1 Essential knowledge in this content area
Right triangles have 1 right angle and 2 acute angles. The side opposite the right angle is called the hypotenuse and the other two sides are called legs. In any right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse – this is called the Pythagorean Theorem.
Below is a visual model of the concept of the Pythagorean theorem where the length of a = 3, b = 4, c = 5. a b c hypotenuse legs a2 + b2 = c2 a2 + b2 = c2 32 + 42 = 52 9 + 16 = 25
1g.2 Common misunderstandings in this content area
Students may forget to use squares to determine length of sides (e.g., compute a + b to find c instead of a2 + b2 to find c2). In addition, students may overgeneralize if not given enough practice examples. For example, if given an example such as 32 + 42 = 52, student may think that 62 + 72 = 82.
1g.3 Prior knowledge/skills needed (can be taught concurrently)
- Identify right triangles
- Number identification
- Multiplication and Division
- Knowledge of exponents, specifically squaring (x2) and finding the square root]
