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Lesson 10 of 10

The Other Side of Zero · Lesson 10 of 10

Chapter Summary and Practice

“Connect integer meanings, models, rules, applications, and puzzles in a complete chapter review.”

Learning Objectives

• Connect signed positions, signed changes, and reference values. • Compare integers and recognise additive inverses. • Explain addition and subtraction with number lines and tokens. • Choose an efficient method for larger integer calculations. • Interpret integers in money, height, temperature, and time. • Use rules and patterns to justify mixed-problem solutions.

One chapter, several connected models

The chapter began by asking whether numbers could exist below zero. Floors below ground made those numbers meaningful. Lift movements gave us addition and subtraction, the number line extended them to any integer, and tokens made cancellation visible. These models are different ways to explain the same arithmetic.

A signed number may describe a position or a change. In a lift equation, −3 could be a floor three levels below the entrance or a downward movement of three levels. In a bank record, it could be a negative balance or a debit. Read the context before deciding what the number represents.

Chapter ideaMeaning or methodConnection to another idea
Signed numbers and zeroDescribe opposite sides of a chosen reference; zero is neither sign.Ground level, sea level, and 0°C are different reference meanings.
Integers…, −2, −1, 0, 1, 2, … continue both ways.Whole signed levels become positions on a full number line.
ComparisonHigher on a vertical scale or farther right on a horizontal line means greater.−4 < −3 because −4 lies lower or farther left.
AdditionApply a movement, combine changes, or join token collections.Start + movement = target.
Additive inversesEqual opposite values sum to zero.Zero pairs and return journeys show cancellation.
SubtractionFind a required change or remove the specified tokens.Target − start = movement; subtracting equals adding the inverse.
Large integersUse meaningful jumps on an unmarked number line.The same relations work without drawing every unit tick.
ApplicationsInterpret credits, debits, heights, and temperatures relative to their references.A negative value is meaningful in its situation.
Calendar yearsTraditional BCE/CE labels have no year 0.Count elapsed years across the boundary carefully.
Puzzles and patternsCheck border conditions, selections, possible sums, and repeated groups.Arithmetic helps explain why a pattern works or fails.
History and playBrahmagupta’s rules consolidate integer arithmetic; signed dice require choices.Models explain the rules, and rules support strategic calculation.

Read the signs before calculating

A sign in front of a number belongs to that number. An operation sign between numbers tells us how to combine them. Parentheses help keep a negative number together: 7 − (−3) means subtract the whole number −3. It is different from 7 + (−3), and both can be explained by a model.

Example — Reading three similar expressions

Problem
Compare 7 + (−3), 7 − (+3), and 7 − (−3).

  1. 1.For 7 + (−3), start at 7 and move three units left: the result is 4.
  2. 2.For 7 − (+3), replace subtraction of +3 with addition of −3: the result is also 4.
  3. 3.For 7 − (−3), add the inverse +3: 7 + 3 = 10.
  4. 4.The first two describe the same result, but changing the sign of the subtracted number changes the third.
Two journey relationshipsLaTeX
Identify which quantity is unknown before choosing the operation.
The inverse relationshipLaTeX
The second number is replaced by its additive inverse. This works for positive, negative, and zero values.
Example — One journey, two equations

Problem
A lift starts at −4 and arrives at +3. Write both the addition and subtraction statements.

  1. 1.From −4 to zero needs +4; from zero to +3 needs +3.
  2. 2.The combined movement is +7.
  3. 3.Addition: −4 + (+7) = +3. Subtraction: +3 − (−4) = +7.
  4. 4.Seven is the movement size; +3 is the final floor. They answer different questions.
−6−5−4−3−2−10123456+7 movement
A complete journey across zero— The same trip supports −4 + 7 = 3 and 3 − (−4) = 7.

Choose a model that explains the question

A number line is useful for direction, comparison, and a change between positions. Tokens are useful when you want to see cancellation or the effect of removing negative values. Once the models are understood, the inverse rule can make calculations quicker. An answer is stronger when you can also explain it in another way.

Example — Checking by two representations

Problem
Find −6 − (−9) using an inverse and tokens.

  1. 1.Inverse method: subtracting −9 means adding +9, so −6 + 9 = +3.
  2. 2.Token method: begin with six negatives. Add three zero pairs to make nine negatives and three positives, still worth −6.
  3. 3.Remove the nine negatives; three positives remain.
  4. 4.Both methods give +3, showing how the rule follows from the model.
Starting value −6−−−−−−After three zero pairs+++−−−−−−−−−After removing −9+++Each + token is +1; each − token is −1.
Two equivalent subtraction pictures— Six negatives and the mixed middle collection have the same value. After the required negatives are removed, the result is +3.
Example — A large calculation without every tick

Problem
Find −125 + (−30), then explain why −99 − (−200) has a positive answer.

  1. 1.For −125 + (−30), both changes are negative: 125 + 30 = 155 units downward, so the result is −155.
  2. 2.For −99 − (−200), add the inverse: −99 + 200.
  3. 3.The 200 positive units cancel the 99 negative units, leaving 101 positives. The result is +101.
  4. 4.A negative first number does not guarantee a negative answer. The whole expression matters.

References give signs their meaning

In a geographical cross-section, zero means sea level. In a Celsius temperature record, zero is the freezing reference. In an account, zero means no net balance in our simplified model. The same operations apply, but the words and units tell us what the result measures.

Example — A changing temperature and a changing balance

Problem
A temperature rises from −4°C to 14°C. Separately, a −₹20 balance receives ₹200. Find the change or result in each case.

  1. 1.The temperature question asks for a change: 14 − (−4) = +18°C.
  2. 2.The account question gives the change and asks for the target: −20 + 200 = ₹180.
  3. 3.Use target minus start for the first and start plus movement for the second.
  4. 4.The number 200 is a credit, while 180 is a balance. These quantities must not be confused.
SituationMeaning of zeroPositive / negative meaningUseful model
BuildingGround or entry levelAbove / below ground; upward / downward movementVertical line or lift
Account recordZero net balanceCredit / debit, or positive / negative balanceRunning-total table or tokens
Geographical heightSea levelAbove / below sea levelVertical cross-section
Celsius temperatureFreezing referenceAbove / below 0°CThermometer or number line
BCE/CE calendarNo year 0 in this naming systemUse era labels and elapsed stepsTimeline with a direct 1 BCE → 1 CE step

Calendar questions need their own convention. From 2 BCE to 2 CE there are three yearly steps: 2 BCE → 1 BCE → 1 CE → 2 CE. Do not force the calendar labels onto an integer line by inserting a year called 0. Keep the stated era with the year number.

Explain a pattern rather than only extending it

A successful puzzle solution satisfies all its conditions. Four equal border sums are stronger evidence than one matching row. A fixed selection total needs an explanation using each row and column. An impossible dice sum must be ruled out by checking the permitted faces, not merely by saying you did not roll it.

Example — A repeated group with a remainder

Problem
Find the value of 102 tokens following +,+,+,−,− repeatedly.

  1. 1.Twenty complete groups use 100 tokens, with value +20.
  2. 2.The next two tokens are both positive, so they add +2.
  3. 3.The full value is +22. Grouping helps, but the incomplete final group must also be counted.
Example — A strategic operation choice

Problem
A player at +43 rolls +5 and −2. Which operation reaches +50?

  1. 1.Adding gives +5 + (−2) = +3, so the target would be +46.
  2. 2.Subtracting in the order +5 − (−2) gives +7, reaching +43 + 7 = +50.
  3. 3.The reverse subtraction gives −2 − 5 = −7, reaching +36.
  4. 4.Choose the operation only after relating its result to the present position.

The history in this chapter explains how people extended familiar arithmetic to include these numbers. Brahmagupta’s rules are useful because they treat positive values, negative values, and zero consistently. Our diagrams provide reasons for those rules, and the puzzles let us apply them beyond a rehearsed example.

Three errors to watch for

Do not compare negative integers by ignoring their signs. Do not assume every addition increases a value or every subtraction decreases it. Do not omit zero when listing possible outcomes: +5 + (−5) and −5 − (−5) both give zero.

Quiz

Quick check

Which list is increasing?

Quick check

What is the additive inverse of −543?

Quick check

From −4 to +3, what movement is needed?

Quick check

Which equals 7 − (−3)?

Quick check

What remains in token subtraction −6 − (−9)?

Quick check

What is the change from +1200 m to −1200 m?

Quick check

How many years elapse from 2 BCE to 2 CE?

Quick check

What is the value of 102 patterned tokens +,+,+,−,−?

Practice Problems

Practice Problems
  1. Describe −3 as a position and as a movement in a building.
  2. Arrange −12, +4, −1, 0, +9, and −7 in increasing order. Explain the two smallest numbers.
  3. Write the additive inverses of +8, −8, +543, −543, and 0.
  4. A lift starts at +6 and moves −11. Find the target and then check by subtraction.
  5. Find the required movement from −8 to +5 and write the matching addition equation.
  6. Evaluate −12 + (−7), +12 + (−7), −12 + (+7), and +12 + (+7).
  7. Evaluate +3 − (+8), −3 − (+8), −3 − (−8), and +3 − (−8). Explain one with tokens.
  8. Use an unmarked line for −100 − (+250) and explain the movement.
  9. A balance starts at ₹40, receives ₹25, pays ₹90, and then receives ₹35. Find each balance.
  10. Using the cross-section heights A=+1500, B=−500, C=+300, D=−1200, E=+1200, F=−200, G=+100 metres, order the points and find the change from D to A.
  11. A temperature moves from −6°C to +4°C and then to −1°C. Find each signed change and the overall change.
  12. Find the year 320 years after 680 BCE and the elapsed years from 3 BCE to 3 CE.
  13. Complete [[−10, blank, blank], [blank, empty, −5], [9, blank, blank]] with border sum +4 and check all four borders.
  14. Explain the fixed total of the first 4 × 4 selection grid using its top-row total 16 and row changes 0, −5, −2, −10.
  15. For dice with faces −1, 2, −3, 4, −5, 6, list the impossible totals from −10 to +12.
  16. Continue −40, −34, −28, −22 for three terms; continue 3, 4, 2, 5, 1, 6, 0, 7 for three terms. State the rules.
  17. Find the value of 103 tokens repeating +,+,+,−,−.
  18. At −43, roll +5 and −2 in the signed game. Choose an operation that reaches −50.
  19. Explain one addition rule and one subtraction rule using a diagram or tokens. Include an example with zero.
  20. Give an opposite-sign addition and a negative-minus-negative calculation that both give zero. Explain why zero must be included among possible outcomes.

Position: three floors below ground. Movement: three floors downward from the current position.

Key Takeaways

Key Takeaways

• Signed numbers describe positions and changes relative to a reference. • Integers include negative whole numbers, zero, and positive whole numbers. • Comparison follows position; additive inverses cancel. • Addition applies or combines changes, and subtraction finds a change or removes a value. • Adding the inverse explains every integer subtraction. • Applications, puzzles, historical rules, and games all use the same consistent integer arithmetic.

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How Integer Rules Developed

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