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Math properties reference

Math properties reference sheet

A printable guide to arithmetic and equality properties, with definitions, equations, equivalent names and the nonzero conditions needed for reciprocals and division.

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Compare each definition with its equation. Explain which part changes and why the value or equality stays the same.

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US Letter

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Reference handout
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Definitions, examples and necessary conditions

The reference covers arithmetic and equality properties. It explains the difference between changing order and changing grouping, identifies equivalent names for addition with zero, and states the nonzero conditions for reciprocals and division.

There is no separate answer page or random question set for this fixed handout. Change its paper size, printed language or teacher note when needed.

Learning objective

What this activity builds

Connect each property name to a precise mathematical change, example and necessary condition.

Worked example

Are additive identity and the addition property of zero different rules?

They describe the same fact: a + 0 = a.

Zero leaves a number unchanged under addition. Different textbooks may use either name. Multiplication has a different identity, 1; multiplying by zero gives zero instead.

Teaching notes

Make the practice count

  • Keep the handout beside the identification practice at first.
  • Compare additive identity, multiplicative identity and multiplication by zero together.
  • Use actual numbers to introduce a symbolic example.
  • Point out the nonzero condition before using a reciprocal or equality-division rule.

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This fixed two-page reference covers all 19 named selections, states nonzero restrictions, and shows equivalent zero-addition names together.

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2 reference pages
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Math SheetNameScoreTeacherDateMathematical Properties ReferenceUse these definitions and examples to compare nineteen mathematical properties.Associative property of additionChanging only the grouping of three addends does not change their sum.(a+b)+c=a+(b+c)Associative property of multiplicationChanging only the grouping of three factors does not change their product.(a×b)×c=a×(b×c)Commutative property of additionChanging the order of two addends does not change their sum.a+b=b+aCommutative property of multiplicationChanging the order of two factors does not change their product.a×b=b×aDistributive propertyMultiplying a sum by a factor equals adding the products of that factor and each addend.(a+b)×c=a×c+b×cAdditive identity propertyAdding zero leaves a number unchanged.a+0=aEquivalent name: Addition property of zeroMultiplicative identity propertyMultiplying by one leaves a number unchanged.a×1=aAdditive inverse propertyA number plus its opposite equals zero.a+(-a)=0Multiplicative inverse propertyA nonzero number times its reciprocal equals one.a0a×(1a)=1Addition property of zeroAdding zero leaves a number unchanged.0+a=aEquivalent name: Additive identity propertymath-sheet.comPage 1 / 2
Math SheetNameScoreTeacherDateMathematical Properties ReferenceUse these definitions and examples to compare nineteen mathematical properties.Multiplication property of zeroAny number multiplied by zero equals zero.a×0=0Property of equalityAn equation is true when both sides have the same value.3+4=5+2Addition property of equalityAdding the same quantity to equal expressions preserves equality.a=ba+c=b+cSubtraction property of equalitySubtracting the same quantity from equal expressions preserves equality.a=ba-c=b-cMultiplication property of equalityMultiplying equal expressions by the same quantity preserves equality.a=ba×c=b×cDivision property of equalityDividing equal expressions by the same nonzero quantity preserves equality.a=b,c0ac=bcReflexive property of equalityEvery expression is equal to itself.a=aSymmetric property of equalityIf one expression equals another, the equality may be reversed.a=bb=aTransitive property of equalityExpressions equal to the same expression are equal to each other.a=b,b=ca=cmath-sheet.comPage 2 / 2

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