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Geometry and measurement

Mathematics Generalist EC-6 Standard III. Geometry and measurement. Length. Length is defined as the measurement of a distance between two points. For example, the length of the line below is 9cm.

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Geometry and measurement

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  1. Mathematics Generalist EC-6 Standard III Geometry and measurement

  2. Length Length is defined as the measurement of a distance between two points. For example, the length of the line below is 9cm. It is the linear extent in space from one end to the other or the longest dimension of a two-dimensional object. In the rectangle below, the length is 5cm. 0cm 9cm 2cm 5cm

  3. Perimeter The perimeter is the total distance around the outside of a 2D shape. You can calculate it by adding together all the lengths of a shape. In summary, the perimeter of a polygon is the sum of the lengths of all its sides. 12cm The perimeter= 12cm+5cm+12cm+5cm =34cm 5cm 5cm 12cm

  4. Perimeter Formulas Because a rectangle is a shape that has 2 pairs of equal parallel sides, you can use a formula to calculate its perimeter: P=2l+2w This means 2 times the length plus 2 times the width. For the example used in the previous slide the formula would be solved as follows: P=2(12)+2(5)=34 This formula works for any shape with 2 equal pairs of parallel sides; such as a rhombus or a parallelogram.

  5. Area The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square centimeters, square inches, or square kilometers . Therefore, area is defined as the number of square units needed to cover a surface. 6cm 2cm The area= 6cm x 2cm = 12 square cm. We write it as 12cm². If you count the squares, they equal 12.

  6. Area Formulas To calculate the area of a rectangle, you can multiply the length by the width; which is expressed by the following formula: A=lw To calculate the area of a square the formula is A=s²; which means side times side. The area of a triangle is calculated as follows: A=ab/2 d=11cm c=11cm a=10cm b=10cm Triangle Area A=ab/2 =10(10)/2 =50cm²

  7. Volume Volume is the amount of 3-dimensional space occupied by an object. Example: volume of a rectangle prism Volume V= lwh =Length x width x height =(12 x 6)(1) 72cm³ w=6cm h=1cm l=12cm

  8. Volume Formulas

  9. Making a Connection Length is a measure of one dimension, whereas area is a measure of two dimensions and volume is a measure of three dimensions. One dimension= length Two dimensions= length and width Three dimensions= length, width, and height

  10. Units of Measurement • A unit of measurement is a universally accepted definite amount of a physical quantity that is used as a standard for measurement of the same physical quantity of any amount. • Some examples of physical quantities include: • Temperature • Money • Percent • Speed • Acceleration

  11. Temperature • The metric system uses the Celsius scale to measure temperature. However, temperatures are still measure on the Fahrenheit scale in the United States. • Water freezes at 0°Celsius and boils at 100° Celsius which is a difference of 100°. Water freezes at 32° Fahrenheit and boils at 212 ° Fahrenheit which is a difference of 180 °. Therefore, each degree on the Celsius scale is equal to 180/100 or 9/5 degrees on the Fahrenheit scale.

  12. Temperature cont. • How to convert Celsius temperatures to Fahrenheit: • Multiply the Celsius temperature by 9/5 • Add 32 ° to adjust for the offset in the Farenheit scale • Example: Convert 37 °C to Fahrenheit • 37 x 9/5 = 333/5 = 66.6 • 66.6 + 32 = 98.6 °F

  13. Temperature cont. • How to convert Fahrenheit temperatures to Celsius: • Subtract 32° to adjust for the offset in the Fahrenheit scale. • Multiply the result by 5/9 • Example: Convert 98.6 °F to Celsius • 98.6 - 32 = 66.6 • 66.6 x 5/9 = 333/9 = 37°C

  14. = = Money • Converting between Dimes, Nickels, and Pennies • Converting between coins involves finding a coin or group of coins that have the same value in cents. • A dime is worth 10 cents and is equal to 2 nickels or 10 pennies. • A nickel is worth 5 cents and is equal to 5 pennies. Two nickels have the same value as 1 dime. • A penny is worth 1 cent. Five pennies have the same values as 1 nickel. Ten pennies have the same value as 1 dime. = = = =

  15. Percent • To find the percentage of a number: • For example: Find the 68% of 87 • Multiply the number by the percent • 87 x 68 = 5916 • Divide the answer by 100 • 5916 / 100 = 59.16 • 59.16 is the 68% of 87

  16. Percent cont. • To determine percentage: • For example: 68 is what percent of 87? • Divide the first number by the second • 68 / 87 = 0.7816 • Multiply the answer by 100 • 0.7816 x 100 = 78.16 • Follow the answer with the % sign • 78.16% • 68 is the 78.16% of 87

  17. Fraction to Percent • To convert a fraction to a percent: • For example: Convert 4/5 to a percent • Divide the numerator of the fraction by the denominator • 4 / 5 = 0.80 • Multiply the answer by 100 • 0.80 x 100 = 80 • Follow the answer with the % sign • 80% • 4/5 = 80%

  18. Percent to Fraction • To convert a percent to a fraction: • For example: Convert 83% to a fraction • Remove the percent sign • 83 • Make a fraction with percent as the numerator and 100 as the denominator • 83/100 • Reduce the fraction if possible • 83% = 83/100

  19. Decimals and Percents • To convert a decimal to a percent: • For example: convert 0.83 to a percent • Multiply the decimal by 100 • 0.83 x 100 = 83 • Add a percent sign after the answer • 83% • 0.83 = 83% • To convert a percent to a decimal: • For example: convert 0 83% to a decimal • Remove the percent sign • 83 • Divide the percent by 100 • 83 / 100 = 0.83 • 83% = 0.83

  20. Symmetry • A pattern is symmetric if there is at least one symmetry ; rotation, translation, or reflection, that leaves the pattern unchanged. • Plane symmetry involves moving all points around the plane so that their positions relative to each other remain the same, although their absolute positions may change. Symmetries preserve distances, angles, sizes, and shapes.

  21. Rotation • To rotate an object means to turn it around. Every rotation has a center and an angle.

  22. Translation • To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance.

  23. Reflection • To reflect an object means to produce its mirror image. Every reflection has a mirror line. A reflection of an "R" is a backwards "R".

  24. Works Cited • Geometry information from: • http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/perimeter_and_area/index.html • http://www.mathleague.com/help/geometry/area.htm • http://staff.argyll.epsb.ca/jreed/math9/strand3/formulae.htm • Measurement information from: • http://www.aaastudy.com/mea.htm • http://en.wikipedia.org/wiki/Units_of_measurement • Symmetry information from: • http://mathforum.org/sum95/suzanne/symsusan.html

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