# Arithmetic

## An Age Proportion Problem: Multiple Methods

(A new question of the week) Some problems can be done either by algebra or by basic arithmetic methods and some creativity; and although algebra generally makes work easier by making it routine, sometimes special-purpose thinking (once you have thought it!) can be quicker. Here we have a problem where a creative method didn’t quite …

## Abraham Lincoln and the Rule of Three

For the last two weeks, we have examined new and old ways to think about proportions. This time, we’ll look at an old method called the Rule of Three (both “single” and “double”), and how you might have learned to solve these problems 200 years ago without algebra. Be prepared for a deep dive!

## Many Ways to Solve a Proportion

Last week we looked at a set of special rules for working with proportions, which have been largely replaced by the more general “tool” of algebra (the “Swiss army knife” of problem solving, which can do the job of many specialized tools), though the latter can still be useful. We still find that many students …

## Negative x Negative = Positive? Abstract Proofs

Last time we looked at explanations for the product of negative numbers in terms of various concrete models or examples. But it really requires a mathematical proof, as we’ll explain and demonstrate here, first with a couple different proofs, then with the bigger picture, giving the context of such proofs.

## Negative x Negative = Positive? Concrete Illustrations

One of the more common questions we’ve been asked is, How can the product of two negative numbers be positive? Between this post and the next, I’ll put together many of the answers we have given, starting here with examples from the “real world” (gradually getting more abstract), and next time we’ll look at proofs. …

## Negative vs Minus: Two Words, One Symbol?

Last time we looked into terminology related to negative numbers; one subtopic was too big to fit, so I’ve broken it out into a separate post. How are the concepts of “negative” and “minus” (subtraction) related? How much do we need to distinguish them? We’ll look at two questions, the first from a child focused …

## Talking About Negative Numbers

Last week we looked at what negative numbers mean; here we’ll consider a number of questions we’ve been asked about the terminology of signed numbers: what “negative” means, and other words for negative numbers. Up, down, and opposite This question from 1998 asks about translating words to signed numbers: Converting Words to Numbers Can you …

## How Real Are Negative Numbers?

This week we’ll look at some Ask Dr. Math questions like, “How can a number be less than zero?” and “Why do we need negative numbers?” We’ll see a number of examples of their use, and how negative numbers make life easier.

## Two Word Problems About Factors and Sums

(A new question of the week) A couple recent questions involved factoring numbers, in interesting ways. One involves the volume and perimeter of a block of cubes, and the other involves finding numbers with a given HCF (Highest Common Factor) and sum. Both illustrate thinking through a non-routine problem about factors.

## Multiplication, Division, and Powers of Ten

(A new question of the week) We’ve looked in the past at place values, but here we’ll see some tricks for doing multiplication and division with both decimals and large numbers by moving the decimal point around. The first question is primarily a matter of arithmetic, then the second extends it to the algebraic concept …