6. Number Play

7 topics1 h

Chapter 6 of NCERT Mathematics for Class 7: Number Play. Four sections — Numbers that describe an arrangement, Parity, Grids and magic squares and Sequences and hidden digits. Seven videos, 1 h in all.

Worked answers to the 26 exercise questions in this chapter →

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Part 1

Numbers that describe an arrangement

9 minEncoding a line-up as a sequence of numbersSeven children in a line, each saying how many people ahead of them are taller. Those seven numbers throw almost everything away — and lose nothing.9 min · 12 partsहिंदी

Part 2

Parity

9 minOdd and even as "can this be arranged in pairs"Five boxes, plus signs between them, thirty after the equals. It cannot be done, and you can know that without trying a single card.9 min · 12 partsहिंदी10 minThe parity of a sum or product is fixed before you compute itA table with seven rows, three filled in and four blank. The blank ones are the subtraction rules, and the explanation refuses to just hand them over.10 min · 12 partsहिंदी

Part 3

Grids and magic squares

10 minUsing row and column sums to prove a grid cannot be filledA three by three grid holding one to nine, with six numbers ringed round the outside and nobody saying what they are.10 min · 12 partsहिंदी10 minConstructing magic squares, and where they came fromThere are 362,880 ways to fill a three by three grid with 1 to 9. Checking one a second would take you over four days.10 min · 12 partsहिंदी

Part 4

Sequences and hidden digits

10 minVirahāṅka–Fibonacci numbers, and the poetry-counting problem that produced them1, 2, 3, 5, 8, 13. The usual story is that somebody stared until they noticed the pattern. What actually produced them was Prakrit poetry.10 min · 12 partsहिंदी10 minCryptarithms: recovering digits from constraints aloneT + T + T = UT. You are told nothing about either letter, and yet exactly one pair of digits works.10 min · 12 partsहिंदी

What you will be able to do

  • State the rule that turns a height arrangement into a sequence of numbers, and apply it to a drawn line-up
  • Represent a given whole number as dots laid out in pairs, and read off from the picture whether it is even or odd
  • Complete the parity table for addition and for subtraction, and justify each row from the dot picture rather than from examples
  • Read a grid whose row totals and column totals are printed around its edge, and say which cells each circled number controls
  • State the defining condition of a magic square, including the diagonals, and name the magic sum
  • State the beat values of a short and a long syllable, and read a rhythm as a sum of ones and twos
  • Read a letter-for-digit addition and say what each column asserts

What this chapter assumes you already know

  • Comparing two lengths or heights directly, without measuring either
  • Counting a subset of a group ("how many of these are taller than that one")
  • Reading a position in a list as first, second, third … and knowing that position matters
  • Sorting a claim into always true / sometimes true / never true, and knowing that one counterexample settles "always"
  • Even and odd numbers from earlier classes, at least as a familiar sorting
  • Skip counting in twos, and reading a picture of dots as a number
  • Adding several numbers, and the fact that order of addition does not matter
  • Consecutive whole numbers: what "one more than" means

It builds directly on Comparing two expressions by reasoning, not by evaluating, Why whole units are not enough to measure with, The four angles at a crossing: vertically opposite and linear pairs, Brackets decide which operation happens first, Answering "could this possibly fit?" by estimating in stages and Predicting how many digits a product will have, before multiplying.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Saying 0 means you are the tallest."
  • "The biggest number belongs to the shortest child."
  • "Any list of seven numbers could be somebody's line-up."
  • "The numbers tell you how tall the children are."
  • "You would have to guess to get the line-up back."
  • "A child standing in the middle cannot say 0."
  • "Odd means it leaves remainder 1 when you divide by 2, and that is all there is to say."
  • "I could not find five cards making 30, so probably there is a clever set I missed."