Demonstration 1 of 4
Start, change, end
A marker starts at -4. Where does a positive move leave it: left of zero, on zero, or right of it?
Addition reads as start + change = end. The arrow begins at the start and is as long as the change. Zero is just one more mark it may cross, reach, or never touch.
\[-4 + 11 = 7\]
-4 is the start, a position four steps left of zero. The signed change says how far to move and in which direction: plus means right, minus means left. The end is where the move lands.
Predict first. From -4, a change of +4 ends where?
Choose an example
Constructed example: the chapter's move from -4 by +11, with other changes from the same start.
Calculated values
- Start
- -4
- Change
- +11
- End
- 7
- Where it lands
- crosses zero and ends right of it
start + change = end: -4 + 11 = 7. The move is 11 steps to the right, and it crosses zero and ends right of it. Four steps right reach zero and 7 more reach 7: 4 + 7 = 11. Check by subtraction: 7 - (-4) = 11.
Use the idea
When a value moves, name the start, the change and the end before writing any signs. The equation then records what you already said.
Where the conclusion applies
Equally spaced marks and right as the positive direction. If a scale were drawn with uneven spacing, step counting would no longer match the arithmetic.
Check your understanding: A marker starts at -6 and moves +9. Where does it end, and does it cross zero?
Chapter 6 source: section "Start, change, and end". Demonstration C06-D01.
Demonstration 2 of 4
Subtract by adding the opposite
Why does -5 - 3 move left while -5 - (-3) moves right?
Subtraction keeps the first number and adds the opposite of the second. Removing a leftward move has the effect of a rightward move, so the arrow turns around.
\[-5-3=-5+(-3)=-8\]
\[-5-(-3)=-5+3=-2\]
The first minus sign in each line means subtract. A minus sign inside parentheses belongs to the number. The opposite of 3 is -3, and the opposite of -3 is 3.
Predict first. Starting at -5 and subtracting -3, will the answer be more or less than -5?
Choose an example
Constructed example: the chapter's two subtractions from -5, plus the same pair from 2.
Calculated values
- Expression
- -5 - 3
- Rewritten
- -5 + (-3)
- Answer
- -8
- Check
- -8 + 3 = -5
-5 - 3 = -5 + (-3) = -8. Here subtracting positive 3 undoes a rightward move, so travel left: three steps left. Check by adding back the number that was subtracted: -8 + 3 = -5.
Use the idea
When two minus signs sit side by side, say both roles aloud before moving: "minus" for the operation, "negative" for the number.
Where the conclusion applies
Signed numbers on one number line with right as positive. The rule rewrites subtraction only; adding a negative number stays an addition.
Check your understanding: What is 8 - (-3), and how do you check it?
Chapter 6 source: section "Build signed arithmetic from zero". Demonstration C06-D02.
Demonstration 3 of 4
Change or distance?
Between -4 and 7, when is the answer +11, when -11, and when just 11?
Reversing start and end reverses the arrow, so the signed change flips sign. The distance bar has no arrowhead, so its length does not care about order.
\[7 - (-4) = 7 + 4 = 11\]
Signed change is end - start and keeps a direction sign. Distance is how far apart two positions are, always nonnegative, with no direction.
Predict first. If the marker moves from 7 back to -4, what signed change will you see?
Choose an example
Constructed example: the chapter's positions -4 and 7, taken in both orders.
Calculated values
- Start
- -4
- End
- 7
- Question
- signed change
- Answer
- +11
Signed change is end - start: 7 - (-4) = 7 + 4 = 11. The sign reports direction: rightward. Check: -4 + 11 = 7.
Use the idea
Before subtracting, ask which question is being asked: a move with direction, or a separation without it. The word "difference" alone does not say.
Where the conclusion applies
Two labeled positions on one line with right as positive. If the order of start and end is not stated, a signed change cannot be given.
Check your understanding: A reading moves from -3 units to 5 units. What are the signed change and the distance?
Chapter 6 source: section "The move from negative four to seven". Demonstration C06-D03.
Demonstration 4 of 4
Crossing zero, checked by addition
How can you tell that 12 - 19 is -7 and not 7?
Twelve steps left reach zero, and the rest carry on past it. Adding back what was subtracted must return to the start, so a wrong sign shows up as a missed landing.
\[12 - 19 = -7\]
\[-7 + 19 = 12\]
Subtracting positive 19 means nineteen steps left from 12. The check arrow adds 19 back to a proposed answer; it must land on 12.
Predict first. If someone proposes 7 for 12 - 19, where will the check arrow land?
Choose an example
Constructed example: the chapter's worked example 12 - 19, with smaller subtractions from 12.
Calculated values
- Subtraction
- 12 - 19 = -7
- Proposed answer
- -7
- Addition check
- (-7) + 19 = 12
- Check passes?
- yes
12 - 19 = -7. Twelve of the 19 leftward steps reach zero; 7 remain, so the end is -7. Check the proposed answer by adding 19 back: (-7) + 19 = 12. The check lands on 12, so -7 is confirmed.
Use the idea
After any subtraction with signs, add the subtracted number back to your answer. If you do not land on the start, look at the sign first.
Where the conclusion applies
Whole numbers and an exact check. The check catches a wrong answer but does not say which step went wrong.
Check your understanding: Using the addition check, is 5 - 13 equal to 8 or to -8?
Chapter 6 source: section "Worked example: subtraction crossing zero". Demonstration C06-D04.