One of the most useful things a family can hear is that a Grade 11 student finding Pre-Calculus heavy going is frequently well able to handle Pre-Calculus itself. Their attention is going into fraction arithmetic, and there is very little left over for the new idea being taught.
It is a small cause with a large effect, and it is quick to put right.
Where the assumption shows up
Senior mathematics assumes fractions are automatic. That assumption is present in more places than it first appears:
- Rational expressions. The entire unit is fraction arithmetic with polynomials in place of numbers. Common denominators, cancelling and restrictions all behave exactly as they did in Grade 7.
- Slope and linear relations. Rise over run is a fraction, and comparing, simplifying and using it fluently is assumed rather than taught.
- Solving equations. Multiplying through by a denominator is a routine early step, and a student who hesitates there hesitates on every question of that type.
- Trigonometric ratios. Sine, cosine and tangent are ratios, and exact values in later courses are written as fractions with radicals in them.
- Rational exponents. The exponent itself is a fraction, and reading it confidently is what makes the notation make sense.
A student who pauses at each of those points loses time on an exam and confidence during a lesson, even though every individual step is one they can do.
Why it persists quietly
Fraction work is usually assessed thoroughly in Grades 6 to 8 and then assumed thereafter. A student who was reliable rather than fluent will pass those assessments, and the difference between reliable and fluent only becomes visible years later, when speed starts to matter.
Calculators also mask it. A calculator handles the arithmetic beautifully and teaches nothing about structure, which is exactly what rational expressions require.
The good news about the fix
Rebuilding fraction fluency in a senior student is one of the fastest pieces of work in tutoring. The concepts are already familiar; what is needed is focused practice until the steps happen without deliberation. In our experience it takes a few sessions rather than a term.
The effect afterwards is noticeable across the whole course rather than in one unit, because the attention that was going into arithmetic becomes available for the mathematics being taught.
What to practise, in order
Common denominators first, because everything else depends on them. Then multiplication and division, which are quicker than addition once the structure is clear. Then negative fractions, which are where most of the hesitation lives. Then fractions inside a larger expression, which is the form senior mathematics actually presents them in.
Short and frequent beats long and occasional here. Ten minutes several times a week for a fortnight is usually enough to make the difference visible.
How to tell whether it applies
A quick check at home: ask the student to add three-quarters and two-fifths, then to simplify a fraction with a negative sign in the numerator, then to divide one fraction by another. Watch for hesitation rather than errors. Hesitation is the signal.
If it is there, it is genuinely good news, because it is the most fixable thing on the list of possible causes.
How we approach it
Our first sessions with a senior student are diagnostic for exactly this reason. We confirm what is secure, rebuild what needs it, and then move at the pace of the class. Parents are told plainly what we found and how long we expect the repair to take.
Families who come to us looking for a math tutor in Surrey often expect us to start with the current chapter. Starting two years earlier for a fortnight is frequently the faster route to the mark they want.
We teach Grades 1 to 12 at 8806 141A Street, Surrey, BC V3V 7W5 and online over Zoom. Every new student starts with a $50 trial class. Call 604-725-6845.
