Use the distributive property to expand and evaluate 1/2(1/6 + (-7/6)).
Rewrite with the named strategy, then evaluate exactly.
CCSS 7.NS.A.1d & 7.NS.A.2c · operation strategies
Properties of Operations with Rational Numbers Worksheets
Use properties to make rational-number computation easier instead of merely naming a property from a symbolic template. Problems ask students to pair inverses, pair reciprocals, rewrite subtraction, or distribute before evaluating.
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Properties with Rational Numbers
Properties as strategies · Grade 7 standard · Seed 271701
1.
2.
Use the distributive property to expand and evaluate 3/2(1/4 + 7/4).
Rewrite with the named strategy, then evaluate exactly.
3.
Use commutative and associative properties to evaluate (2 + 4/3) + (-2) by pairing additive inverses.
Rewrite with the named strategy, then evaluate exactly.
4.
Use commutative and associative properties to evaluate (-8/5 × 1) × (-5/8) by pairing reciprocals.
Rewrite with the named strategy, then evaluate exactly.
5.
Use commutative and associative properties to evaluate (-2 + 4/5) + 2 by pairing additive inverses.
Rewrite with the named strategy, then evaluate exactly.
6.
Use commutative and associative properties to evaluate (-1 + 2/3) + 1 by pairing additive inverses.
Rewrite with the named strategy, then evaluate exactly.
7.
Use commutative and associative properties to evaluate (4/3 × 7/4) × 3/4 by pairing reciprocals.
Rewrite with the named strategy, then evaluate exactly.
8.
Use the distributive property to expand and evaluate 2(1/3 + (-4/3)).
Rewrite with the named strategy, then evaluate exactly.
9.
Use commutative and associative properties to evaluate (1/6 × (-1/2)) × 6 by pairing reciprocals.
Rewrite with the named strategy, then evaluate exactly.
10.
Rewrite subtraction as addition of the additive inverse, then solve: 1/3 - (-2/5).
Rewrite with the named strategy, then evaluate exactly.
Exact-construction rule: operations use normalized integer numerator/denominator pairs, number-line positions use exact ticks, repeating decimals come from remainder cycles, and money contexts use exact cents. No answer depends on binary floating-point rounding.
Using properties to make arithmetic easier
Properties are most useful as strategies, not vocabulary. The problems here ask students to pair inverses, pair reciprocals, rewrite subtraction, or distribute before evaluating, so the property has to make the work shorter to count.
Checking that the property helped
Solve one problem both ways, directly and with the property, and compare the effort. If the shortcut really is shorter, the property has done its job. Keep the answer key open so they can see both routes reach the same value.
Properties in use
- (−5/6) + 2 1/3 + 5/6: reorder to (−5/6 + 5/6) + 2 1/3 = 2 1/3.
- (−4/5) × 7 × (−5/4): pair the reciprocals to get 1, then 1 × 7 = 7.
- 12 × (−2/3 + 1/4): distribute to get −8 + 3 = −5, which matches 12 × (−5/12).
Misuses to catch early
- Applying the commutative property to subtraction or division, which does not commute.
- Distributing a multiplier to only the first term.
- Naming the property correctly but not using it to simplify the calculation.
Teaching note 1
Commutative and associative properties can bring additive inverses or reciprocals next to each other.
Teaching note 2
Example: 8 + (-3) + (-8) reorders to 8 + (-8) + (-3), which is 0 + (-3), or -3.
Teaching note 3
Subtraction can be rewritten as addition of the additive inverse before applying addition strategies.
Teaching note 4
The distributive property should preserve exact rational values on both sides of the rewrite.
Continue the 7.NS progression
Move to another Grade 7 rational-number skill
Frequently asked questions
Is this only property identification?
No. The prompts use properties as computation strategies and require an evaluated exact result.
Which properties appear?
The set uses commutative, associative, distributive, additive-inverse, and reciprocal reasoning.
Which standard does this practice?
CCSS 7.NS.A.1d and 7.NS.A.2: apply properties of operations as strategies to add, subtract, multiply, and divide rational numbers.
Do students have to name the property they used?
Not for the answer key, which checks the exact result. Naming the property is a useful scaffold when the goal is explaining why the strategy is valid.