Where this sits. This is one of the five number families — the arithmetic that a little under half the test is made of, and that every algebra family is built out of. Every family page on this site has the same eight parts, in the same order, so once you know the shape of one you know the shape of all of them. Where a part is empty, it says so.
Start hereSigned NumbersIf positives and negatives are new, or the signs still trip you up, begin with this page. It starts from the beginning: the number line, the four cases of adding and subtracting, the three rules, and multiplying and dividing, with practice in every part. The quizzes, rooms and sheet below go further.Open Signed Numbers → DrillsTwenty-two drills for this family, one skill each, ten problems at a timeGo to the drills ↓The number line continues below zero — temperatures, debts, yards lost, floors below the lobby. Signed numbers have their own small set of rules for adding, subtracting, multiplying, and dividing.
The rules are short, and the test uses them constantly once algebra starts, because half the numbers in algebra are negative. Most of the mistakes people make in algebra are not algebra mistakes. They are sign mistakes, made here.
What the test asks in this family
- Where negative numbers sit, and which of two is biggerSigned Numbers · Quiz 20 · Negative Mazarine! · Signed Numbers Practice
- OppositesSigned Numbers · Quiz 20 · Signed Numbers Practice
- Absolute valueSigned Numbers · Quiz 20 · Signed Numbers Practice
- Adding signed numbersSigned Numbers · Quiz 21 · Negative Mazarine! · Signed Numbers Practice
- Subtracting signed numbers, and why it turns aroundSigned Numbers · Quiz 21 · Negative Mazarine! · Signed Numbers Practice
- Multiplying signed numbersSigned Numbers · Quiz 22 · Signed Numbers Practice
- Dividing signed numbersSigned Numbers · Quiz 22 · Signed Numbers Practice
All 7 have something on this site.
Six questions, one from each part of this family. Type an answer and press Check. You get the reasoning for every one, right or wrong.
- Which is colder, −8°F or −3°F? −8. Below zero the rule turns over: the bigger the digit, the smaller the number. −8 sits eight steps from zero on the cold side, −3 only three, so −8 is further down.
- What is |−17|? 17. Absolute value asks how far from zero, not which way. −17 is seventeen steps from zero, and a distance is never negative.
- −6 + 10 = 4. Start at −6 and walk ten steps to the right. Six of them get you to zero, and the four left over carry you past it.
- −9 − 4 = −13. Subtracting means turn around. You are already at −9, facing the cold side, and four more steps take you to −13. Taking away from a debt makes the debt bigger.
- 7 × (−3) = −21. Multiplying is doing the same journey several times: seven journeys of three steps to the left is twenty-one steps to the left. A positive times a negative is negative.
- −48 ÷ (−6) = 8. Division asks the multiplication backwards: what times −6 gives −48? It has to be a positive 8, because a negative times a negative is a positive. Two negatives in a division cancel each other out.
Missed one? The quiz for it is in Learn it just below, and the three rooms of Negative Mazarine! under Work it by hand prove the same four truths three different ways.
Want more than these six questions? Flyover I asks about all fourteen families in one sitting and ends in a grid that says where to start; Flyover II asks them again later, with different questions, so you can see what has moved.
- Quiz 20Below ZeroWhat a negative number is, and where it lives.
- Quiz 21Adding and Subtracting Signed NumbersThe two moves that cause most of the trouble.
- Quiz 22Multiplying and Dividing Signed NumbersTwo rules, and why they are true.
If signed numbers are new, start with Signed Numbers, in the Start here box at the top of this page, which introduces the topic from the beginning.
Twenty-two short drills, one skill each, in order. Each starts with two worked examples, then gives you ten problems (or five) and the reasoning for every answer. Each one prints, with the answer key on the last page. When you get all ten right, or all five right twice in a row, the drill gets a check mark on the Drills page.
- 2.01Which is larger? (including two negatives)
- 2.02Put signed numbers in order
- 2.03Opposites and absolute value
- 2.04Which is farther from zero?
- 2.05Name the case: add or subtract a positive or a negative
- 2.06The four cases with small numbers
- 2.07Add, same signs (Rule 1)
- 2.08Add, different signs, the positive farther from zero (Rule 2)
- 2.09Add, different signs, the negative farther from zero (Rule 2)
- 2.10Add, all kinds, including zero pairs
- 2.11Rewrite a subtraction as an addition (Rule 3)
- 2.12Subtract a positive, including a smaller number minus a larger one
- 2.13Subtract a negative
- 2.14Subtract, all kinds
- 2.15Add and subtract, mixed
- 2.16Three or more numbers in a row
- 2.17Multiply two signed numbers
- 2.18Divide two signed numbers
- 2.19Multiply three or more: count the negatives
- 2.20All four operations, mixed
- 2.21Order of operations with signed numbers
- 2.22Word problems: temperature, money, depth, yards · calculator allowed
- Signed Numbers · start here if signed numbers are new, or if the signs still trip you up: the number line and absolute value, how to tell whether the second number is positive or negative, the four cases of adding and subtracting, the three rules, every kind of subtraction, three or more numbers, and multiplying and dividing. A card with all the rules, timed practice rounds, and eighteen questions at the end.
- Negative Mazarine! · ups and downs, gains and losses. The same four truths about adding and subtracting negatives, proved three ways with your own hands. Each room talks, and each has a story, guided practice, and challenges. They go beside Quizzes 20 to 22.
- 1. The Math Path · walking up and down the number line.
- 2. The Ballast · weight added to a submersible, and weight dropped: down and up.
- 3. The Temperature Vat · hot cubes and cold cubes in a tank: warmer and colder.
- Signed Numbers Practice · 202 problems in the order of the Signed Numbers page: the number line and absolute value, naming the case, adding with the same and different signs, every kind of subtraction, three or more numbers, word problems, and multiplying and dividing. Answer key at the end, with each subtraction rewritten as an addition. Made to print.
- Khan Academy, math · free videos and practice. Search it for negative numbers, adding and subtracting integers. These are the clips we start from in class.
- Math Antics · short, plain videos on the basics.
- Light & Salt Learning · free GED classes online, with playlists by topic.
- Get Sum Math · GED math on video, one topic at a time.
- Kaplan GED Test Prep · Unit 2, Chapter 5, Lesson 1 — The Number Line and Signed Numbers, page 316 (page 835 of 2029 in the ebook). And the McGraw-Hill Mathematical Reasoning Workbook · look up signed numbers or integers in the contents.
On to fractions. A fraction is division wearing different clothes, and it turns up inside algebra as well — a slope is a fraction, a rate is a fraction — so the work you do there is never only about fractions.