Lesson Guide
Explanation
The fence-post trap catches half of every class: from the 12th to the 16th is NOT 16 − 12 = 4 days — it's 5, because subtraction counts the gaps between days, not the days themselves. When events happen on BOTH end days, the rule is subtract THEN add 1. When you're counting nights or days-until, plain subtraction is right. Always test with a tiny case: 12th to 13th is two show days (12th and 13th) but one night's sleep. The tiny case tells you whether this question wants +1 or not.
Your Turn
A festival runs from the 3rd to the 9th, with events every day. How many event days? Check: 9 − 3 + 1 = 7.
Watch Out
'From … to …' with activity on both ends means +1. 'How many days until' means no +1. The words pick the formula.
Explanation
A theatre puzzle: shows run Monday the 3rd to Thursday the 20th, every evening, plus afternoon shows on Wednesdays and Saturdays. Split the count. Evenings: both end days have shows, so 20 − 3 + 1 = 18. Extra afternoons: LIST the Wednesdays and Saturdays — weekdays repeat every 7 days, so Wed 5, 12, 19 and Sat 8, 15: five extras. Total 18 + 5 = 23. Two lessons inside: split complicated counts into simple layers, and WRITE the list of special days rather than guessing how many fit.
Your Turn
A market runs Mon 1st to Sun 14th, open daily, with a night market every Friday. Fridays are the 5th and 12th. Total open sessions? Check: 14 daily + 2 night = 16.
Watch Out
Same-weekday dates differ by 7. If your listed Wednesdays aren't 7 apart, the list is wrong.
Explanation
Every-nth patterns: buses from 7:10 every 25 minutes run 7:10, 7:35, 8:00, 8:25, 8:50, 9:15… arrive at 9:05 and your wait is 10 minutes — but only if you WRITE the sequence; heads drop buses. When two patterns overlap (a gong every 6 minutes, a bell every 8), they coincide at the first number both reach: list multiples of the bigger (8, 16, 24…) and test each against the smaller — 24 works. Writing three extra terms costs ten seconds and rescues the whole minute.
Your Turn
Trains every 20 min from 6:15. You arrive 7:20. How long until the next train? Check: 6:15, 6:35, 6:55, 7:15, 7:35 → wait 15 min.
Watch Out
The pattern starts AT the first time — 7:10 counts as a bus. Patterns that start 'after 7:10' shift everything by one gap.
Explanation
The fence-post family wears many costumes. Pages: reading page 15 to page 40 covers 40 − 15 + 1 = 26 pages (both ends read). Ladder: climbing rung 3 to rung 11 touches 9 rungs but climbs 8 gaps. Cuts: 5 cuts across a log make 6 pieces — pieces are always cuts + 1. Every version is the same question: are you counting THINGS (add 1) or the GAPS between them (don't)? Say which one out loud before computing, and the costume stops fooling you.
Your Turn
A paper strip is cut 7 times. How many pieces? Check: 8 — pieces = cuts + 1.
Watch Out
The string-folding puzzle is this in reverse: 3 folds → 8 strands → 7 fold-marks… things and gaps again. Name which you need.
Explanation
The counting toolkit: decide THINGS OR GAPS before computing (+1 for both-end events, plain subtraction for nights and gaps); split layered counts into daily-base plus special-day extras; LIST special days using the every-7 rule instead of guessing; and verify your formula on a tiny case. These questions are free marks for careful students — the arithmetic is Year 2 level, the care is what's being tested. Quiz and exercises!
Your Turn
Capstone: a fair runs Fri 6th to Sun 15th, open daily, fireworks every Saturday (7th, 14th). Total open days + fireworks nights? Check: 10 days + 2 fireworks = 12 events.
Watch Out
When a count 'obviously' equals a subtraction, that's exactly when to run the tiny-case check. Obvious is where the trap lives.
📌 Key Points — this lesson in a nutshell
- Fence-post rule: events on BOTH end days → subtract then add 1 (12th to 16th = 5 days).
- Nights, gaps and 'days until' use plain subtraction — the words pick the formula.
- Layered counts: daily base + special-day extras, added at the end (18 + 5 = 23 in the theatre example).
- List special days with the every-7 rule (Wed 12 → 19 → 26); never guess how many fit.
- Every-nth sequences: write the terms out; two patterns coincide at the first shared multiple.
- Pieces = cuts + 1; rungs vs gaps; pages need +1 — always ask "things or gaps?"
Step 1: Introduction
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