Trade Mathematics Course: Lessons 2026

Lesson 1: Fractions, decimals and the tape measure

The Government of Canada's Skills for Success model calls this skill numeracy: your ability to find, understand, use and report mathematical information in words, numbers, symbols and graphics. Trade work starts with measuring.

The official Red Seal exam guide says each question has four choices with one correct answer, the pass mark is 70%, and a calculator is provided if needed.

Reading an imperial tape

On most imperial tapes, each inch has 16 equal parts. The longest middle line is 1/2 in; shorter lines mark quarters, eighths and sixteenths. Find the last whole inch, then count the small marks past it: five marks past 2 in is 2 5/16 in.

Write fractions in lowest terms by dividing the top and bottom by the same number: 10/16 = 5/8.

Reading a metric tape

A metre has 100 centimetres and 1,000 millimetres. A tape that reads 1 m, 4 cm and 7 mm shows 1 + 0.04 + 0.007 = 1.047 m, or 1,047 mm. Before you add metric lengths, put them in the same unit: 1.25 m + 760 mm + 0.6 m = 1.25 + 0.76 + 0.6 = 2.61 m.

Comparing fractions and decimals

To change a fraction to a decimal, divide the top by the bottom: 7/8 = 7 ÷ 8 = 0.875. To change a decimal inch to sixteenths, multiply by 16: 0.3125 × 16 = 5, so 0.3125 in = 5/16 in. To compare sizes such as 5/8, 19/32 and 9/16, change them all to decimals (0.625, 0.594, 0.5625) and compare.

Adding and subtracting

Give the fractions the same bottom number, then add or subtract. 2 3/8 + 1 3/4 = 2 3/8 + 1 6/8 = 3 9/8 = 4 1/8 in. To subtract a fraction from a whole number, borrow 1: 96 − 35 5/8 = 95 8/8 − 35 5/8 = 60 3/8 in.

Multiplying and dividing

Change a mixed number to an improper fraction before you multiply: 1 5/8 = 13/8, so 6 × 1 5/8 = 78/8 = 9 3/4 in. To divide a length into equal spaces, use decimals, then change the leftover part to sixteenths: 72 ÷ 5 = 14.4 in; 0.4 × 16 = 6.4, which rounds to 6/16 = 3/8, so each space is about 14 3/8 in.

Decimal feet

For decimal feet, multiply only the decimal part by 12: 12.75 ft = 12 ft + (0.75 × 12) in = 12 ft 9 in, not 12 ft 7.5 in.

Saw kerf and finding the centre

Each saw cut removes a strip of material called the kerf. Add one kerf per cut.

To find the centre, halve the whole number and the fraction separately. Half of 47 3/8 is 23 1/2 (= 23 8/16) plus half of 6/16 (= 3/16), so the centre is at 23 11/16 in.

Themenquiz 1: Fractions, decimals and the tape measure

Lesson 2: Measurement, conversions and scale drawings

You will meet both metric and imperial units at work. The federal Weights and Measures Act says units of measurement in Canada are based on the International System of Units (SI, the metric system). Its Schedule II defines Canadian units such as the inch, yard, gallon and pound in metric terms, so these factors are exact.

Metric prefixes tell you the size: milli means one thousandth, centi one hundredth and kilo one thousand. So 1 m = 1,000 mm = 100 cm, and 1 km = 1,000 m.

How to convert

Multiply by the factor to go from the imperial unit to the metric unit. Divide by it to go back.

• 8 ft to metres: 8 × 0.3048 = 2.4384, about 2.44 m.
• 15 m to feet: 15 ÷ 0.3048 = 49.2 ft.
• 3/8 in to millimetres: 0.375 × 25.4 = 9.525, about 9.5 mm.
• 1,830 mm to inches: 1,830 ÷ 25.4 = 72.05 in, about 72 in = 6 ft 0 in.

Check the size: a length in feet is a bit more than 3 times the length in metres. If 15 m comes out as 4.6 ft, you multiplied instead of dividing.

Area and volume units

For areas, square the length factor. For volumes, cube it. 1 ft = 0.3048 m, so 1 ft² = 0.3048 × 0.3048 = 0.0929 m², and 1 m² = 10.76 ft². 1 yd = 3 ft, so 1 yd² = 9 ft² and 1 yd³ = 3 × 3 × 3 = 27 ft³. In metric, 1 yd³ = 0.9144³ = 0.7646 m³.

Scale drawings

A ratio scale such as 1:50 means 1 unit on the drawing is 50 of the same units in real life. Real length = drawing length × scale. Drawing length = real length ÷ scale.

• 1:50 plan, wall measures 84 mm: 84 × 50 = 4,200 mm = 4.2 m.
• 7.5 m beam on a 1:100 plan: 7,500 ÷ 100 = 75 mm.

Imperial drawings often use an architect scale such as 1/4 in = 1 ft. Every inch on the drawing is then 4 ft. A line 6 1/2 in long shows 6.5 × 4 = 26 ft.

Themenquiz 2: Measurement, conversions and scale drawings

Lesson 3: Perimeter, area and volume

Perimeter is the distance around a shape, area is the surface inside it (square units) and volume is the space a solid fills (cubic units). Put every measurement in the same unit before you multiply.

Use π ≈ 3.1416. The radius is half the diameter; mixing them up is a common mistake.

Perimeter

A room 4.2 m by 3.6 m has a perimeter of 2 × (4.2 + 3.6) = 15.6 m. For baseboard, subtract doorways: a 14 ft by 11 ft room has 50 ft of wall, minus a 3 ft door = 47 ft.

Area

For walls, find the whole area and subtract openings. A 16 ft by 8 ft wall with a 3 ft by 4 ft window: 128 − 12 = 116 ft². For all four walls of a room, multiply the perimeter by the height: a 5 m by 4 m room with 2.4 m walls has 18 × 2.4 = 43.2 m² of wall.

A triangular gable 24 ft wide and 6 ft high is 24 × 6 ÷ 2 = 72 ft². For an L-shaped floor, add rectangles together, or take the full rectangle and subtract the missing corner: 10 × 8 − 4 × 3 = 68 m².

Circles

A 6 in round duct has a radius of 3 in, so its opening is 3.1416 × 3 × 3 = 28.27 in². A welder going around a pipe with a 4.5 in outside diameter travels the circumference: 3.1416 × 4.5 = 14.14 in.

Volume

A slab 6 m by 4 m and 100 mm thick: change 100 mm to 0.1 m, then 6 × 4 × 0.1 = 2.4 m³.

A round column form 300 mm across and 2.4 m tall: radius 0.15 m, so 3.1416 × 0.15 × 0.15 × 2.4 = 0.17 m³.

Cubic yards and litres

In imperial units, work in feet, then divide cubic feet by 27 to get cubic yards. A slab 20 ft by 12 ft and 4 in thick: 4 in = 1/3 ft, so 20 × 12 × 1/3 = 80 ft³, and 80 ÷ 27 = 2.96 yd³. A trench 40 ft by 2 ft by 18 in (1.5 ft) holds 120 ft³ = 4.44 yd³.

For liquids, 1 m³ = 1,000 L. A tank 1.2 m × 0.8 m × 0.5 m holds 0.48 m³ = 480 L. A 50 m water line with a 50 mm inside diameter holds 3.1416 × 0.025 × 0.025 × 50 = 0.0982 m³ = 98.2 L.

Themenquiz 3: Perimeter, area and volume

Lesson 4: Ratios, proportions and percentages

Ratios

A ratio compares amounts. A 1:2:3 concrete mix means 1 part cement, 2 parts sand and 3 parts gravel. Each part is multiplied by the same number: with 2 buckets of cement, you need 4 of sand and 6 of gravel.

When you know the total, add the parts first. A 1:3 mortar has 4 parts. For 0.24 m³ of dry mix, one part is 0.24 ÷ 4 = 0.06 m³, so cement is 0.06 m³ and sand is 0.18 m³.

Simplify a ratio by dividing both numbers by the same number, and keep the order the question uses: 4 journeypersons to 6 apprentices is 2:3.

Proportion

Direct proportion: twice the work takes twice the time. Find the rate first. 1,200 blocks in 3 days is 400 a day, so 2,000 blocks take 5 days. Inverse proportion: more workers take less time. 4 workers × 9 days = 36 worker-days; with 6 workers, 36 ÷ 6 = 6 days.

Percentages

• Percent of an amount: 10% of 420 = 0.10 × 420 = 42.
• Add a percentage: multiply by (1 + rate). 420 ft² plus 10% waste = 420 × 1.10 = 462 ft².
• One amount as a percentage of another: part ÷ whole × 100. 360 m of 480 m = 75%.
• Percentage change: change ÷ starting value × 100. A price falling from $0.80 to $0.68 is 0.12 ÷ 0.80 = 15% lower.

Markup, discount and GST

A markup is added to cost: $1,250 with a 15% markup is 1,250 × 1.15 = $1,437.50. A discount comes off the price: 20% off $640 is 640 × 0.80 = $512. Apply each step in the order given.

The Canada Revenue Agency lists the GST rate in Alberta as 5%, with no provincial sales tax. Other provinces charge HST or PST at different rates, so check the CRA rate for the place of supply. In Alberta, $2,380 plus GST is 2,380 × 1.05 = $2,499.00, and the discounted $512 tool above costs 512 × 1.05 = $537.60.

Slope and roof pitch

Slope as a percentage = rise ÷ run × 100, with both in the same unit. A drop of 30 mm over 1.5 m (1,500 mm) is 2%. To find the drop, multiply: a 12 m run at 2% drops 0.02 × 12 = 0.24 m = 240 mm. Your drawings or the code that applies to your work set the slope you need.

Roof pitch is written as rise in 12 of run. A 6 in 12 roof rises 6 in for every 12 in of run, so the rise is half the run: over 8 ft of run it rises 4 ft.

Themenquiz 4: Ratios, proportions and percentages

Lesson 5: Angles, right triangles and trigonometry

Angle facts

• A full turn is 360°. A straight line is 180°. A square corner (right angle) is 90°.
• The three angles in any triangle add up to 180°. In a right triangle with a 35° angle, the third angle is 180 − 90 − 35 = 55°.
• Where two pieces meet at a corner, each piece is cut (mitred) at half the corner angle: a 90° corner needs two 45° cuts.
• For a frame with equal sides, each mitre is 180° ÷ number of sides. An eight-sided frame needs 180 ÷ 8 = 22.5° cuts.

The Pythagorean theorem

In a right triangle, the longest side (the hypotenuse) is opposite the 90° angle. The rule is c² = a² + b². Legs of 9 ft and 12 ft give c² = 81 + 144 = 225, so c = 15 ft. A rafter with a 12 ft run and 5 ft rise is √(144 + 25) = 13 ft long.

To find a shorter side, subtract: a = √(c² − b²). A 6 m ladder with its foot 1.5 m out reaches √(36 − 2.25) = 5.81 m up the wall.

Squaring a layout

The 3-4-5 rule comes from Pythagoras: 3² + 4² = 5². Any multiple works: 6-8-10, 9-12-15, or 1.2 m, 1.6 m and 2.0 m. Measure along both sides from the corner; if the diagonal between the marks is right, the corner is 90°. For a full rectangle, measure both diagonals. They are equal only when it is square. A 6 m by 4.5 m form should have diagonals of √(36 + 20.25) = 7.5 m.

Sine, cosine and tangent

From the angle you use, the opposite side faces it, the adjacent side touches it, and the hypotenuse is the longest side. Remember SOH CAH TOA:

• sin = opposite ÷ hypotenuse
• cos = adjacent ÷ hypotenuse
• tan = opposite ÷ adjacent

Set your calculator to degrees. To find a side, multiply: a 10 m conveyor at 20° rises 10 × sin 20° = 3.42 m. Standing 15 m from a tower with a 40° angle to the top, its height is 15 × tan 40° = 12.59 m. To find an angle, use the inverse key: a ramp rising 4 ft over 12 ft of run is at tan⁻¹(4 ÷ 12) = 18.4°, and a 6 in 12 roof is at tan⁻¹(0.5) = 26.6°.

Pipe and conduit offsets

An offset moves a run sideways with two equal bends or fittings. The distance between bends (the travel) = offset ÷ sin(bend angle), so 1 ÷ sin is a handy multiplier.

Plumbing: a 300 mm offset with 45° fittings has a travel of 300 × 1.414 = 424 mm. Electrical: a 10 in offset with two 30° bends needs the bends 10 × 2 = 20 in apart.

Themenquiz 5: Angles, right triangles and trigonometry

Lesson 6: Estimating materials and costs

Skills for Success lists measuring, calculating and estimating among numeracy operations, and checking that an answer is the expected size. The Red Seal exam guide also says to estimate a calculation answer first.

Items spaced along a length

Divide the length by the spacing to get the number of spaces. You need one more item than spaces when there is an item at both ends.

• Studs at 16 in on centre along a 12 ft (144 in) wall: 144 ÷ 16 = 9 spaces, so 10 studs (before extras for corners and openings).
• Rebar no more than 300 mm apart across 4 m: 4,000 ÷ 300 = 13.3, so round up to 14 spaces and use 15 bars.

Rounding up

You cannot buy part of a sheet, a length or a box, so round up. 47 m of pipe in 6 m lengths is 7.83, so buy 8. Concrete sold in 0.25 m³ steps: 2.52 m³ becomes 2.75 m³, because 2.50 m³ would be short.

Coverage and waste

Find the area, add waste, then divide by what one unit covers. A 3.6 m by 2.4 m floor is 8.64 m²; with 10% waste it is 9.504 m²; boxes covering 1.44 m² each give 6.6, so buy 7 boxes. For paint, multiply by the number of coats: 43.2 m² × 2 coats ÷ 10 m² per litre = 8.64 L, which is 2.16 four-litre cans, so buy 3.

Roofers count in squares: one square covers 100 ft². A 1,650 ft² roof with 10% waste is 1,815 ft² = 18.15 squares. If the shingles need 3 bundles per square, that is 54.45, so buy 55 bundles.

Unit prices and labour

To compare packages, find the price per unit: a 50 m roll at $62 is $1.24 a metre; a 75 m roll at $89 is about $1.19 a metre, so the larger roll is cheaper by about 5 cents a metre. Labour = number of workers × hours × rate: 3 workers × 7.5 h × $34.50 = $776.25.

Building a job price

1. Total the materials (quantity × unit price for each line).
2. Add labour (hours × rate).
3. Add any markup the company uses.
4. Add the sales tax that applies to the whole subtotal.

Example in Alberta: materials $1,840 plus 26 h × $68 = $1,768 labour gives a subtotal of $3,608. With 5% GST: 3,608 × 1.05 = $3,788.40.

Quick estimates

Round to easy numbers: 19 sheets at $18.95 is about 20 × $19 = $380. The exact $360.05 is close, so $3,600 would be a clear mistake.

Themenquiz 6: Estimating materials and costs