6. Signed Change, Distance, and Tolerance

Session 6: Signed Change, Distance, and Tolerance

Learning targets

• Add and subtract signed values using meaning.
• Find change between two positions.
• Distinguish net change from total distance.
• Use absolute difference to describe error and tolerance.

Why this matters

This session turns signed-number meaning into measurement of change. It focuses on situations in which direction, distance, and acceptable error must not be confused.

Core lesson

Adding a signed value combines a movement with a starting position. Like-signed changes move in the same direction; unlike-signed changes compete, so subtract magnitudes and keep the direction of the larger magnitude. Subtraction can be rewritten as addition of the opposite:
\[ a-b=a+\(-b\). \]

Change from an initial value to a final value is \(\text{final}-\text{initial}\). From \(-6^\circ\) to \(3^\circ\), the change is \(3-(-6)=9^\circ\). Distance between two values is the absolute difference and is never negative. Tolerance compares a measured value with a target: \(|\text{measured}-\text{target}|\). Net change can be small even when total distance is large.

Foundations in depth

Position, change, and distance must be kept separate. Position tells where a value is relative to a reference. Change is directional and is calculated as final minus initial. Distance is nonnegative and is found with an absolute difference. A temperature moving from −8°C to 5°C has a change of +13°C and travels a distance of 13 degrees; a move from 5°C to −8°C has a change of −13°C but the same distance.

Tolerance problems compare an actual measurement with a target. The expression |actual − target| measures the error regardless of which value is larger. The measurement is acceptable when this error is less than or equal to the allowed tolerance. On the GED®, underline whether the question asks for final position, net change, total distance, error, or an acceptable interval before calculating.

A reliable method

1. Label initial value, final value, and direction.
2. Write final minus initial for change.
3. Use absolute value if only distance or error is requested.
4. Check the sign and unit against the wording.

Ten GED®-style worked examples

Worked example 1

GED®-style problem. The temperature changes from \(-6^\circ\) to \(3^\circ\). Find the change.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Label initial, final, target, and tolerance values.

Step 3 — Solve carefully. \(3-(-6)=9\), so the temperature rises \(9^\circ\).

Step 4 — Verify and state the answer. Use final minus initial for change and absolute difference for distance or error.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 2

GED®-style problem. Find the distance between \(-3\) and \(5\).

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Label initial, final, target, and tolerance values.

Step 3 — Solve carefully. \(|5-(-3)|=8\), so the distance is 8 units.

Step 4 — Verify and state the answer. Use final minus initial for change and absolute difference for distance or error.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 3

GED®-style problem. A part should be \(12.00\) mm and measures \(11.94\) mm. Find the absolute error.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Label initial, final, target, and tolerance values.

Step 3 — Solve carefully. \(|11.94-12.00|=0.06\) mm.

Step 4 — Verify and state the answer. Use final minus initial for change and absolute difference for distance or error.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 4

GED®-style problem. The temperature rises from −6°C to 11°C. Find the change.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Use final minus initial and preserve both signs.

Step 3 — Solve carefully. Change = 11−(−6)=11+6=+17°C. The positive sign means an increase.

Step 4 — Verify and state the answer. Starting at −6 and moving up 17 reaches 11.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 5

GED®-style problem. An elevator moves from floor 8 to floor −2. Find the signed change and distance.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Calculate final minus initial for change, then absolute value for distance.

Step 3 — Solve carefully. Change = −2−8=−10 floors. Distance = |−10|=10 floors.

Step 4 — Verify and state the answer. Moving down ten floors from 8 lands at −2.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 6

GED®-style problem. A machine target is 25.00 mm and a part measures 24.93 mm. Find the absolute error.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Subtract actual and target, then take absolute value.

Step 3 — Solve carefully. Error = |24.93−25.00|=|−0.07|=0.07 mm.

Step 4 — Verify and state the answer. Error is a distance from target, so the reported value is nonnegative.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 7

GED®-style problem. Parts are acceptable within 0.05 mm of 12.00 mm. Is 12.04 mm acceptable?

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Compare absolute error with the allowed tolerance.

Step 3 — Solve carefully. |12.04−12.00|=0.04 mm. Since 0.04≤0.05, the part is acceptable.

Step 4 — Verify and state the answer. The acceptable interval is 11.95 to 12.05 mm, and 12.04 lies inside.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 8

GED®-style problem. Find all measurements acceptable within 0.3 kg of a 7.5 kg target.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Translate tolerance into a lower and upper bound.

Step 3 — Solve carefully. Lower bound: 7.5−0.3=7.2 kg. Upper bound: 7.5+0.3=7.8 kg. Acceptable measurements satisfy 7.2≤m≤7.8.

Step 4 — Verify and state the answer. Each endpoint is exactly 0.3 kg from the target.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 9

GED®-style problem. A hiker travels from elevation 120 m to 85 m, then to 140 m. Find net change and total vertical distance.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Use final minus initial for net change and add absolute leg changes for distance.

Step 3 — Solve carefully. Net change = 140−120=+20 m. Distance = |85−120|+|140−85|=35+55=90 m.

Step 4 — Verify and state the answer. The hiker can travel 90 vertical meters while ending only 20 meters above the start.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 10

GED®-style problem. A stock changes −$3, +$5, −$4, and +$7. Find the net change.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Add the signed changes, grouping positive and negative totals.

Step 3 — Solve carefully. Positive total = 5+7=12. Negative total = −3−4=−7. Net change = 12+(−7)=+$5.

Step 4 — Verify and state the answer. Applying the changes from any starting price leaves a final price $5 higher.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Adult-life and GED® applications

This session's reasoning applies to:

• temperature and elevation changes
• manufacturing tolerance
• profits, losses, and net movement

When the setting changes, keep the mathematical relationship and unit logic visible. Do not choose an operation from a keyword alone. First decide what the quantities mean and how they relate.

Common traps and repairs

• Reporting a negative distance: Use absolute difference for distance.
• Subtracting in an arbitrary order: Change is final minus initial.
• Adding distances when net change is asked: Net change keeps direction; distance does not.

Retrieval and mixed practice

Retrieve opposites and absolute value from Session 5, then explain why subtracting a negative increases a value.

After recalling an answer, compare it with the worked reasoning and correct the explanation, not only the final number. Then mix this session's skill with at least one earlier topic so method selection becomes part of the practice.

Session check

Exit reflection

Complete these two sentences: The main decision in this session is... and I can check my answer by... Keep both statements specific enough to use on a new problem.