Number Play
This chapter invites you to tinker with numbers and, while tinkering, to uncover why the patterns you notice hold. We open with a girl stacking up runs of consecutive numbers and bump into the idea of parity, what makes a number even or odd, and how that even-or-odd quality is carried through addition, subtraction and multiplication. We then ask exactly when a sum lands cleanly on a multiple of 4, of 6, or of any chosen number, and we learn to settle the question with algebra and with drawings instead of testing one case after another. We tighten our divisibility shortcuts for 3, 9 and 11, and, importantly, see why they work, even adding a clever test for 7. We meet digital roots, take apart a handful of cryptarithms (puzzles where letters stand in for digits), and close with a traditional Indian board game. The thread running through it all is the habit of asking “What if…?” and “Will this hold every time?”
Learning Outcomes
By the end of this chapter, you will be able to:
- write numbers as sums of consecutive numbers and reason about which numbers admit such a sum;
- use the laws of parity to predict whether an arithmetic or algebraic expression is even or odd, without evaluating it;
- decide whether a claim about factors and multiples is always, sometimes, or never true, and back it up with algebra and a picture;
- apply and explain the divisibility shortcuts for 3, 9, 11 (and a bonus test for 7), and combine factor tests for numbers like 6 and 24;
- compute the digital root of a number and link it to remainders on division by 9 and 3;
- crack cryptarithms using number sense, parity and place value.