Using a Scale
Scale: A scale can tell you how much bigger or smaller an enlargement or reduction should be, or
it can help you find the true size of objects from scale diagrams. A scale is often a ratio which
compares the size of something on a real object to the size from the diagram.
We can use a construction drawing with a scale to determine the true size of objects by
following these steps:
a. Measure the line used for the scale
b. Create a ratio (your measurement : true size)
*If you have a fraction, multiply the bottom of it by both sides of your ratio to make
the rest easier*
Math Essentials 12
Reference Material and Formula Package
c. Measure things on paper, and multiply them by the ratio to get the true size.
Rate and Slope
Rate is when a value needs to be expressed with two different units. Rate of speed is km per hr
(km/hr) , heart rate is beats per minute (bpm), and rate of pay is dollars per hour.
Ex: What is the rate of speed when someone drives 42 km in 30 mins?
*30 mins = 0.5 hr
42km ÷ 0.5hr = 84km/hr
Slope is something that you get from looking at a line on a graph, and a slope can also represent
a rate.
Ex:
Pay
Gross Pay: Pay before deductions.
Net Pay: Pay
Measurement Definitions
Accuracy means whether something is correct. In order for a measurement to be correct, it must
be measured with a standard tool and unit of measurement.
Precision means how exact a measurement is. More decimals means more precise, and on a
ruler the smallest unit of measurement tells us how precise measurements made with that tool will
be.
Metric units include meters, litres, and grams, and are used by most countries in the world. All
measurements with kilo, centi, mili, etc. are metric. *Decimals for partial cm!
Imperial units include feet, gallons, and pounds, and are only used in the trades and by the USA
and a couple other countries. Inches, yards, and miles are all imperial.
Fractions of an inch
The representation of all fractions within an inch will help you to accurately record measurements
in inches. The values that are not circled can be reduced to a circled value.
Percents, Ratios, Fractions, and Decimals
Fraction:
We can take a picture and write shaded to get a fraction
Total
Fraction = 3
10
Ratio:
We can write shaded : not shaded to get a ratio
Ratio = 3 : 7
Decimal:
We can divide the top and bottom of the fraction to get a decimal
3 ÷ 10 = 0.3
Ex:
Percent:
We can multiply the decimal by 100 to get a percent
0.3 X 100 = 30%
Fraction to Ratio:
Top : (bottom - top)
Ratio to Fraction:
First_____
First + Second
Pythagorean Theorem Example
Place Value
Area refers to how many square units will fit on a given 2D surface (cm2, in2, ft2).
Volume refers to how many cubed units will fit inside a given 3D space. (m3, in3, ft3).
Area Formulas
Shape
Area
Area = Length X Width
A=LXW
Rounding
Ex: Round 7.456 to the nearest tenth:
Step.1: Under line “tenth” place (7.456)
Step.2: Check next number :
(0-4 means the rest of the number gets removed with no changes.)
(5-9 means the underlined number is increased by 1.)
*The next number is a 5, so our 4 will increase.
Step.3: Answer = 7.5
Area = Base X Height ÷ 2
A=BXH
2
Pythagorean Theorem
Area = � X r X r
A=�Xr2
Volume Formulas
Area, Length, and Volume
Length refers to a distance measured in standard units (mm, cm, m, km, in, ft, mi).
Shape
Volume
Volume = Length X Width X Height
V=LXWXH
bigger than a
unit
bigger than a
unit
times bigger
than a unit
Litre
Gram
smaller than
a unit
smaller
than a unit
smaller
than a unit
Multiply when moving to a smaller unit (or move decimal to the
Divide when moving to a bigger unit (or move
Common Unit Conversions
Metric
Volume = � X r X r X h
V=�Xr2 Xh
Imperial
Metric and Imperial
10 mm = 1 cm
12 in = 1 ft
2.54 cm = 1 in
100 cm = 1 m
36 in = 1 yd
1 m = 3.28 ft
1000 m = 1 km
3 ft = 1 yd
1 mi = 1.61 km
1760 yd = 1 mi
Unit Conversion Examples
1. Metric Conversion: Move the Decimal
2. Metric Conversion: Unit Method
450 mm = ____cm
1.2km = ____m
-from mm to cm is one space left on chart
-Draw a decimal at end of 450.
-1.2km X 1000m =1.2 X 1000 = 1200 m
-Moving it one space left gives 45.0
1km
Answer: 450mm = 45.0cm
3. Metric and Imperial: Unit Method
14 cm = ______in
-14cm X 1 in_
= 14 ÷ 2.54 = 5.5 in
2.54cm
4. Imperial Conversion: Simplifying
75 in = _____ft _____in
-75 ÷ 12 = 6.25 (6 full feet)
-6 X 12 = 72 (the 6 ft used 72 inches)
-75 - 72 = 3 (3 inches left)
So 75 in = 6 ft 3 in
Metric Unit Conversion Chart
Kilo
Hecto
Deca
Unit
Deci
Centi
Milli
1000 times
100 times
10
Meter
10 times
100 times
1000 times