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Chapter 3 · Number Play

Clock times and calendar dates: patterns the format lets you have

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11 min.

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

What they should be able to do

  • Explain why a "pretty" time is a fact about the display rather than about time
  • List every time of a stated pattern on a 12-hour clock, and argue the list is complete
  • Say how writing a leading zero on single-digit hours changes those lists
  • Find the number of minutes from one palindromic time to the next
  • Recognise a date whose digits repeat in blocks, and one that reads the same both ways
  • Explain why palindromic dates are scarce, in terms of what each slot may hold
  • State what the chapter asks about reusing a calendar, without overstating what it answers
  • Use the fact that a common year is one day more than fifty-two weeks
  • Given four digits, build the largest and smallest four-digit numbers and compute their sum and their difference
  • Predict how changing one digit moves that sum and that difference

Where it usually goes wrong

  • "12:21 is special." The time is not; the display is. At the same instant a 24-hour clock shows something quite ordinary. Say this early or the whole section reads as numerology.
  • "There must be lots of pretty times, we just have to find them." There are very few and they can all be written down. Completeness is the skill here, not discovery.
  • "6:66 is a time." The second slot pair cannot exceed 59. Students extending 1:11, 2:22, 3:33 will run straight off the end of the minute range, and that is the right moment to make the ranges explicit.
  • "Every clock writes the hour the same way." Some displays print 01:10 and some print 1:10. The lists of palindromic and repeating times are different in the two cases, and the book's own sample set is consistent with the leading-zero style.
  • "A palindromic date is just a date with repeated digits." 20/12/2012 repeats a block; 11/02/2011 is a palindrome. The book prints one of each precisely because they are different conditions, and students merge them.
  • "The chapter says a calendar repeats after so many years." It does not. It asks. Any figure the explanation gives must be presented as reasoning the class does, with the weekday-shift argument shown.
  • "Rearranging digits changes the sum and the difference unpredictably." Both are fixed by the four digits, and the structure above says exactly how. Part (a) to part (d) become one decision instead of four searches.

Questions to check understanding

  • List all times of a stated pattern on a 12-hour clock
  • "How many minutes until the next palindromic time?"
  • Identify whether a given date repeats a block or reads the same both ways
  • Produce a past date of a stated shape
  • Given four digits, write the largest and smallest four-digit numbers and find their sum and difference
  • Choose four digits so that the sum or the difference beats a stated value
  • Reasoning items about whether a calendar can be reused, marked on the argument rather than on the number
  • Items on the sum and difference of the extreme palindromes of a given length

Examples worth working on the board

  • The three sample times (§3.7, p.64). 4:44, 10:10 and 12:21. The book presents these as three types and asks the class to find every time of each type. They are three genuinely different patterns.
    • the hour digit repeated in both minute slots
    • the hour block copied into the minute block
    • the whole display reading the same in both directions
  • What each slot allows. Hours run 1 to 12. The first minute slot runs 0 to 5; the second runs 0 to 9.
  • Manish's birth date (§3.7, p.64). 20/12/2012, in which the digits 2, 0, 1 and 2 come round again in the same order. The date itself is drawn inside a decorative illustration and does not extract. The book asks for other dates of the same shape.
  • Meghana's birth date (§3.7, p.64). 11/02/2011, whose eight digits read the same from either end. Also inside the illustration, also read from the printed page. The book asks for every past date of this shape.
  • Jeevan's calendar question (§3.7, p.64). Why a new calendar every year, and could an old one be reused? Then: will any year's calendar ever come round again with every date on the same weekday? The book asks and does not answer.
  • The palindromic-time question (§3.7, p.65, question 3). Starting from 10:01, how many minutes to the next palindromic time, and to the one after that.
  • Pratibha's digits (§3.7, p.64, question 1). From 4, 7, 3 and 2 she builds 2347 and 7432. The book prints their difference, 5085, and their sum, 9779. The four parts then ask for four digits making the difference bigger, the difference smaller, the sum bigger and the sum smaller.
  • The structure behind Pratibha's question. Sort the four digits from biggest to smallest. Subtracting the smaller arrangement from the bigger one leaves 999 lots of (biggest − smallest digit) plus 90 lots of (second − third). Adding them gives 1001 lots of (biggest + smallest) plus 110 lots of (second + third). Check both against the printed numbers before using them. This is not in the book — it is the structure the four parts are probing, and it turns guess-and-check into control.
  • The palindrome extremes (§3.7, p.65, question 2). The sum and the difference of the largest and the smallest five-digit palindrome. Inputs only: five digit slots, the first and last equal, the first not 0.

Figures to have open

  • Three clock faces or digital displays showing 4:44, 10:10 and 12:21. Standard schematic; the book prints the times as text, not as clocks.
  • The two birth dates, 20/12/2012 and 11/02/2011, set in a fixed-width display so the repeating block and the mirror line up. The textbook's own illustration is decorative and hard to read; redraw.
  • A slot diagram of a date — two boxes, two boxes, four boxes — with the permitted range under each. Standard schematic, needed for section 10.
  • A minute-axis timeline from 10:00 to 12:30 for section 7.
  • A column layout for Pratibha's addition and subtraction. Standard schematic.
  • No table, dataset or photograph from the textbook is required.

Where this sits in the book

The book

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