Study Guides
Why Speed Is a Step Harder Than Percentage — Turning Two Different Kinds of Quantity Into One
"A bike going 300 metres a minute, and a bike going 12 kilometres an hour — which one is faster?" You may have watched your child's hand stop in front of a problem comparing two speeds given in different units. Unlike times tables or column calculation, where the stumble is not knowing the method itself, this is a stumble one step before the comparison even begins.
Percentage is another grade 5 topic, taught in the same year as speed. It compares two amounts of the same kind — a base amount and a compared amount. Speed, though, compares two amounts that are different kinds of thing to begin with: distance (km) and time (minutes). That's what makes speed harder than percentage.
Japan's Ministry of Education, in its commentary on the national curriculum guidelines for elementary mathematics (2017), states: "In grade 5's study of amount per unit quantity, including speed, this is the first time children compare quantities that don't have the basic properties of a quantity, so it is important to make sure they fully understand the meaning of comparing quantities that can be grasped as the ratio of two quantities of different kinds." (Grade 5, Change and Relationships)
I used to think of speed as just one more application of percentage. Looking into it, I found the commentary deliberately uses the word "first" — and learned that speed is a step harder an idea than percentage, not just another version of the same one.
The root cause: turning two different kinds of quantity into one
Cause 1: turning distance and time into "one quantity" is a new idea
Expressing speed takes two quantities: the length travelled and the time it took. The commentary also states: "If speed is understood as the length travelled per unit of time, it can be expressed as (speed) = (length) ÷ (time)." Travel 40 km in an hour, and that's 40 km/h; travel 300 m in a minute, and that's 300 m/min. Speed is a new quantity, neither length nor time, built as the length travelled "per hour" or "per minute." Memorise just the formula without swallowing this idea, and your child ends up guessing every time between distance ÷ time and time ÷ distance.
Cause 2: calculating 1 hour 20 minutes as 1.2 hours
The most common mistake is how time units get handled. Treating 1 hour 20 minutes as 1.2 hours stands out as an error. Time runs on base 60 — 60 minutes make an hour — while decimals run on base 10. Replace 20 minutes with "0.2," and the two counting systems get mixed together.
Cause 3: multiplying and dividing by 60 get swapped when converting units
When converting a speed per second into a speed per minute, or the other way around, dividing by 60 where it should be multiplying, or the reverse, keeps happening. If the answer's speed per second comes out bigger than the speed per minute, this is the mix-up.
How to teach it at home
Step 1: Settle on one formula — (speed) = (distance) ÷ (time)
Tell your child that the formulas for finding distance and finding time are nothing more than rearrangements of this one equation. Some schools use a diagram — a distance/time/speed triangle, for instance — but starting from what this one equation means, rather than memorising a diagram first, lets a child work out the rearranged formulas on their own.
Step 2: Convert time to either minutes or hours before you calculate
See "1 hour 20 minutes," and convert it to "80 minutes" first, before calculating. Building the habit of putting this one extra step before the equation heads off the Cause 2 mistake almost entirely.
Step 3: Confirm unit conversion with a picture of "how many metres in one minute"
If your child covers 5 m in one second, then in one minute (60 seconds' worth) that's 5 × 60, or 300 m. Rather than memorising whether to multiply or divide by 60, confirming with a picture how many times "one unit's worth" stacks up keeps it from flipping the wrong way.
What to avoid
Having your child memorise unit conversion by rote, the way times tables get memorised, is worth avoiding. Memorise it with the meaning of "times 60" left out, and there's nowhere to go back to once it's forgotten.
How to practise
- Speed calculations are built on division of decimals. If Division of Decimals feels shaky, settle that first
- The Speed unit runs through the three uses — finding speed, finding distance, finding time — then moves on to two-step problems, unit conversion, and comparing speeds, in that order
- Speed is one unit within a group called Amount per Unit Quantity. Practising it alongside other "how much per one" problems — crowdedness, population density — makes clear a structure that's hard to see from speed alone
Frequently asked questions
Q. Is it wrong to use a distance/time/speed triangle diagram?
A. There's no need to reject the diagram itself. But memorising it as the only tool tends to turn into plugging numbers in without checking which of distance and time is known and what's being asked for. It's more solid to set up the equation from (speed) = (distance) ÷ (time) first, and use the diagram afterward as a check.
Q. My child never seems to get the hang of calculating with base-60 time.
A. Keep a picture of a clock, or a real one, next to your child, and have them calculate while checking with their eyes what 20 minutes looks like. Once the fact that an hour is 60 minutes has sunk in physically, the rest is just a matter of practice.
Speed is the first unit in elementary math to handle "two different kinds of quantity." Your child is only stumbling on something that's genuinely hard to understand at this stage, so prioritise time spent confirming the meaning over sheer repetition.
This article is written by a parent who builds diglabo. I am not a teacher or an education specialist. The general guidance here is based on how the topic is taught in Japanese primary schools, checked against Japan's national curriculum guidelines.
diglabo's worksheets follow the Japanese curriculum. What that means if you are outside Japan