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Hong KongPerimeter, area and volume
No skill in the rule says "calculate area". Skill 3 of Competency 2 asks for an appropriate measurement unit and form for problems involving length, area, volume or mass, and skill 4 asks you to solve real-world measurement problems. Formulas come from the supplied reference sheet.
Search the Mathematics competencies for the word formula and you will not find it. Search for area and you find it once, inside a skill about units.
What is actually asked
| The rule says | What that means in an item |
|---|---|
| Determine an appropriate measurement unit and form for problems involving length, area, volume, or mass | Which unit fits, and how to express the quantity |
| Solve real-world measurement problems including fundamental units, derived units, and unit conversions | A situation that requires a measurement calculation |
| Identify and classify simple two- and three-dimensional figures according to their mathematical properties | Recognize the shape you are dealing with |
So a measurement item is a word problem with a shape in it, and a reference sheet is supplied. That means the difficulty sits in setting the problem up, not in recalling how to compute a cylinder.
The three dimensions and their units
- Perimeter is a length, so its unit is a plain unit: centimeters, feet, meters
- Area is two dimensions multiplied, so its unit is squared
- Volume is three dimensions multiplied, so its unit is cubed
That is the fastest check available on this competency and it costs nothing. If a question asks for the amount of carpet needed and an option is given in feet rather than square feet, the option is wrong before you calculate anything.
A rectangular room measures 20 feet by 5 feet. A teacher wants to run a border strip around the walls and also carpet the floor. Which pair of quantities does she need?
- Perimeter 50 feet and area 100 square feet
- Perimeter 100 feet and area 50 square feet
- Perimeter 40 feet and area 100 square feet
- Perimeter 50 feet and area 20 square feet
Composite shapes
A real-world measurement problem often uses a shape that is two rectangles joined, or a rectangle with a semicircle on one end. The method is always the same: cut it into shapes you recognize, compute each, then add or subtract.
Subtraction is the part people forget. A path around a pool is the area of the big rectangle minus the area of the pool, and an option offering the big rectangle alone will be sitting there waiting.
Surface area against volume
The question tells you which one it wants and it rarely uses the words. Anything about wrapping, painting, covering or lining is surface area. Anything about filling, holding or capacity is volume.
Translate the verb before you look for a formula. That habit is worth more than knowing the formula, because the formula is supplied and the translation is not.
Scaling, and the trap in it
Double every side of a rectangle and the perimeter doubles, the area goes up four times, and the volume of the corresponding solid goes up eight times. Lengths scale by the factor, areas by its square, volumes by its cube.
That is the single most testable idea in this competency, because the intuitive answer is wrong and the correct answer is one line of reasoning away. Scale drawings and similar figures works through it.
How much of the subtest this is
Competency 2 is about seven of the 35 questions on our split, spread across four skills, and only part of that is measurement calculation. Call it a handful. The estimate is our arithmetic on the rule's skill counts rather than a published weighting.
Our advice is to spend the time on units and on the scaling rule rather than on formulas. You are given the formulas. You are not given the judgment.
Common questions
Do I need to memorize area and volume formulas for the FTCE?
A mathematics reference sheet is supplied for the Mathematics subtest, per the Test Structure document. The rule asks you to solve real-world measurement problems and to choose appropriate units, so setup and judgment matter more than recall.
How do units tell me if an answer is wrong?
Perimeter is a length and takes a plain unit. Area is two dimensions and takes a squared unit. Volume is three and takes a cubed one. An option offering feet where the question needs square feet is wrong before any arithmetic.
What happens to area if I double the sides?
It goes up four times, not two. Lengths scale by the factor, areas by its square and volumes by its cube. The intuitive answer is wrong, which makes this the most testable single idea in the measurement skills.
How do I know whether a question wants surface area or volume?
Read the verb. Wrapping, painting, covering and lining are surface area. Filling, holding and capacity are volume. Translating the verb before you look for a formula is worth more than the formula, since the formula is supplied.