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Grade 7 · Mathematics · 30 minutes

Unit 1 · Proportions and rational operations · Lesson 2 of 2

Rational Operations on a Trail

Interpret signed gains and losses using fractions and decimals.

Mr. S Teaches · Original starter mini-lesson · MS7-MATH-02

Learning objective

Compute signed sums, products and quotients and explain the sign of each result.

Before you begin

  • Grade 6 rational numbers
  • Fraction operations

Gather your materials

  • Paper and pencil
  • Ruler and graph paper

Language to know

additive inverse
A number that combines with another to make zero.
net change
The total change after gains and losses are combined.
signed product
A multiplication result whose sign depends on the factors.

Understand the concept

An uphill change can be positive and a downhill change negative. Adding combines changes; subtracting a change adds its opposite. Repeated equal negative changes give a negative product when the number of repetitions is positive. A negative divided by a positive remains negative; dividing two negatives is positive because the quotient must multiply by the divisor to recover the dividend.

Teach it step by step

  1. Draw a number line and combine an elevation change of +2.5 km with −3.25 km; discuss the net change rather than final elevation.
  2. Rewrite −1.5 − (−0.75) as −1.5 + 0.75 and model the movement.
  3. Compute three equal losses of −2/3 unit and connect repeated addition to multiplication.
  4. Check (−6) ÷ (−2) using the multiplication statement (−2) × 3 = −6.
  5. Have learners annotate the operation and sign before calculating each practice item.

Make the thinking visible

Worked example or model response

+2.5 + (−3.25) = −0.75 km net change. Three losses of −2/3 give 3(−2/3) = −2. For −1.5 − (−0.75), adding the opposite gives −0.75.

Guided practice

Calculate −3/4 + 1/2 and −2.4 ÷ 0.6. Explain the signs with a line or a multiplication check.

Independent practice

1. Compute 1.2 − 2.7. 2. Compute (−3/5)(−10). 3. Compute −7/8 − (−1/4).

Quick assessment

A learner says the quotient (−8) ÷ (−4) is −2. Use multiplication to test that claim.

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: −1/4; −4 because 0.6 × (−4) = −2.4.
  2. Independent: 1. −1.5. 2. 6. 3. −5/8.
  3. Quick check: (−4)(−2) = +8, not −8. The quotient is +2 because (−4)(2) = −8.

Look for: Look for an explanation of the choices, not only a final answer. Require units and a check against the context.

Make it work for your learner

More support

Use common denominators and number-line drawings before the symbolic calculation.

Extension challenge

Invent a two-step rational-number expression with a positive answer and a negative intermediate result.

Selected contributions · Official sources

Standards connections

Contributes to Common Core standard

7.NS.A.1.c

Common Core · Grade 7

Interpret subtraction as adding an opposite.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.

Contributes to Common Core standard

7.NS.A.2

Common Core · Grade 7

Calculate and check signed rational products and quotients.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.