Math Review Units 3 and 4

Unit 3: Linear Measurement My comfort Level Topic Module found in 3.1 Identify a referent for a given common SI or imperial unit of linear measurement.A 3.2 Estimate a linear measurement, using a referent. A 3.3 Measure inside diameters, outside diameters, lengths, widths of various given objects, and distances, using various measuring instruments. A 3.4 Estimate the dimensions of a given regular 3-D object or 2-D shape, using a referent; e.g., the height of the desk is about three rulers long, so the desk is approximately three feet high. B 3.5 Solve a linear measurement problem including perimeter, circumference, and length + width + height (used in shipping and air travel). B 3.6 Determine the operation that should be used to solve a linear measurement problem. B 3.7 Provide an example of a situation in which a fractional linear measurement would be divided by a fraction. C 3.8 Determine, using a variety of strategies, the midpoint of a linear measurement such as length, width, height, depth, diagonal and diameter of a 3-D object, and explain the strategies. C 3.9 Determine if a solution to a problem that involves linear measurement is reasonable. C

Linear Referents Characteristics of a good linear referent: (Linear means straight line) – Uniform – always the same – Portable – easy to move around – NOT a usual measuring device Which is the best for measuring the length of a swing set? branchHockey stickhula hoop

Measuring using other Tools If we don’t have a referent, we may need to use other tools to find the size of something, especially smaller items: – Callipers – Compass – Adjustable – Wrench – Scissors

Perimeter of objects Perimeter – Linear Sum of all sides – Triangle = a+b+c – Rectangle = 2(l) + 2(w) Circumference – The perimeter of a circle: C= πd a b c --length— width diameter

Using Circular referentsCircular referents We can find the linear measurement of a place by rolling a circular object down its length. We then multiply the number of times the object went around by the objects circumference. If a barrel has a circumference of 1.5 meters, and we rolled it down the yard 11 times, how long is our yard?

Linear Measurement Used in shipping 3D (three dimensional) objects Rectangular objects: length + width + height Cylindrical objects diameter + length Width Height Length Width = diameter

Linear Midpoints Midpoints are the geometric centre of an object – find the middle.find the middle. The usual rule for finding the midpoint of one side is to divide the side by ½ of we have the measurement. Another way to find a midpoint is to use a compass and draw two arcs from each end, where they join will be the midpoint

2D Midpoints To find the midpoint of a 2 dimensional object you need to find the midpoint of each side, then draw a line from that midpoint to the opposite side. Where those lines intersect inside the object will be the midpoint.

3D Midpoints Using the same principle, finding the midpoint of each side, we now intersect the three dimensions inside of the object and we will find its midpoint.

Unit 4: Measurement Systems My comfort Level TopicModule 1. 1Explain how the SI system was developed, and explain its relationship to base ten. A 2.1 Explain how the imperial system was developed. A 1.2 Identify the base units of measurement in the SI system, and determine the relationship among the related units of each type of measurement. A 2.2 Identify commonly used units in the imperial system, and determine the relationships among the related units. A 1.3 Identify contexts that involve the SI system. A 2.3 Identify contexts that involve the imperial system. A 1.4 Match the prefixes used for SI units of measurement with the powers of ten. B 1.5 Explain, using examples, how and why decimals are used in the SI system. B 2.4 Explain, using examples, how and why fractions are used in the imperial system.B 2.5 Compare the American and British imperial measurement systems; e.g., gallons, bushels,B 1.6 Provide an approximate measurement in SI units for a measurement given in imperial units; e.g., 1 inch is approximately 2.5 cm. AND2.6 Provide an approximate measure in imperial units for a measurement given in SI units; e.g., 1 liter is approximately 1 4 US gallon. B and C 1.7 Write a given linear measurement expressed in one SI unit in another SI unit. 2.7 Write a given linear measurement expressed in one imperial unit in another imperial unit. C 1.8 Convert a given measurement from SI to imperial units by using proportional reasoning (including formulas); e.g., Celsius to Fahrenheit, centimetres to inches. AND 2.8 Convert a given measure from imperial to SI units by using proportional reasoning (including formulas); e.g., Fahrenheit to Celsius, inches to centimetres. C

Things we measure Length – how long something is Mass – how much something weighs Area – how much space something takes up in two dimensions (flat) Capacity – how much an object will hold (in three dimensions) Volume –how much space something takes up in three dimensions (Square)

SI/MetricImperial Kilometre kmMile mi Metre mYard yd Decimetre dmFoot ft or ‘ Centimetre cminch in or “ Millimetre mm1/6 of an inch Linear Measurement

Mass SI/MetricImperial Metric tonne tTon T Kilograms kgPounds lb Gram gOunces oz Milligram mgOunces oz Capacity SI/MetricImperial Kilolitre kLBushel bsh Gallon gal Litre LQuart qt Centilitre cLPint pt Cup c Millilitre mLFluid ounce fl oz

Volume SI/MetricImperial Metre cubed m 3 cubic yard yd 3 Centimetre cubed cm 3 Cubic foot ft 3 Millimetre cubed mm 3 Cubic inch in 3 Area SI/MetricImperial Kilometre km 2 acres Metre m 2 Yard squared yd 2 Decimetre dm 2 Foot squared ft 2 Centimetre cm 2 Inches squared in 2 Millimetre mm 2

SI System / Metric The Metric System or System International Based on natural lengths of the earth, the weight of water, and the decimal system Familiar names: Centimetres, liters, metric tonnes, cm 2, kilometres, hectares, m 3, kilograms Based on the Powers of 10Powers of 10

SI/ Metric System Invented in France 200 years ago Introduced to Canada in the 1970s

Prefixs in Metric Prefix Matching Game SI Conversion between units Use the Prefix system kilohectodecaBase unit Gram Litre Metre decicentimilli ÷ 10 x 10

Imperial System Used in USA and used to be used in Britain Developed from measurements decided upon by kings and queens – they determined the length of a yard Familiar names: Foot, yard, mile, bushel, acre, gallon, cup, ounces, pounds, No consistent way to convert, just have to memorize the relationships between them Ex. 1 foot – 12 inches

Imperial Conversion Factors Linear Units 1 inch is divided into 16 parts or 1/16ths 1 foot = 12 inches 1 yard = 3 feet or 36 inches 1 mile = 5280 feet or 1760 yards Weight 1 pound = 16 ounces 1 ton = 2000 pounds Capacity 1 cup = 8 ounces 1 pint = 2 cups 1 quart = 4 cups 1 gallon = 4 quarts Area 1 ft 2 = 144 inches 2 1 yard 2 = 9ft 2 1 mile 2 = ft 2 1 mile = 640 acres

Using conversion factors Use them to go from large units to small 12 miles is how many feet? 12 x 5280 = Use them to go from small units to large: 785 inches is how many feet? 785 / 12 inches =

Converting between SI and Imperial We can use estimates to get a close but not accurate measurement: 1 inch =about 2.5 cm 1 foot = about 30 cm 1 gallon = about 4L 1 kg = about 2 pounds 1m is about 1 yard 1 mile is about 2 km

But we also have to be accurate 1 inch = 2.54 cm 1 gallon = 4.56L 1 mile = 1.6 km 1 yard = cm 1 kg = 2.2 pounds 1 pint = 0.57 L

Converting between temperatures There is a simple formula for converting from Celsius to Fahrenheit and vice versa: T F = 9/5 x T C + 32 T C = 5/9 x (T F -32)

Unit 5: 2-D and 3-D Measurements My comfort Level TopicModule 4.1 Identify and compare referents for area measurements in SI and imperial units. A 4.2 Estimate an area measurement, using a referent. A 4.3 Identify a situation where a given SI or imperial area unit would be used. A 4.4 Estimate the area of a given regular, composite or irregular 2-D shape, using an SI square grid and an imperial square grid. B 4.5 Solve a contextual problem that involves the area of a regular, a composite or an irregular 2-D shape. B 4.6 Write a given area measurement expressed in one SI unit squared in another SI unit squared. A 4.7 Write a given area measurement expressed in one imperial unit squared in another imperial unit squared. A 4.8 Solve a problem, using formulas for determining the areas of regular, composite and irregular 2-D shapes, including circles. B 4.9 Solve a problem that involves determining the surface area of 3-D objects, including right cylinders and cones. C 4.10 Explain, using examples, the effect of changing the measurement of one or more dimensions on area and perimeter of rectangles. C 4.11 Determine if a solution to a problem that involves an area measurement is reasonable. C

Area Referents Must have length and width Should meet the criteria for other referents Grid Paper is also an area referent

Over-Under estimate Circle For use with objects that are circular or don’t line up with grids Over estimate: draw a square to include the entire circle Under estimate: draw a square inside the circle Find the area of each Add them together and find their average

Over-Under estimate other shapes For use with objects that are circular or don’t line up with grids Over estimate: count all the squares that are touched Under estimate: count only the whole squares Add the two together and find their average

Composite Shapes Shapes that consist of more than one whole shape we have looked at.

Finding area Rectangle = length x width Triangle = ½ base x height Circle = πr 2 Trapezoid= ½ (a+b) x height

Converting in the Metric System Use the Prefix system kilohectodecaBase unit Metre 2 decicentimilli For each step ÷ 100 For each step x 100

Converting in the Imperial system Memorize the units: 1 ft 2 = 144 inches 2 1 yard 2 = 9 ft 2 1 mile 2 = ft 2

3-D shapes Rectangular Prism Triangular Prism Sphere Cone Cylinder Square Pyramid Triangular Pyramid

To find the surface area Use a net