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Reference sheet covering operation laws and sign rules with worked examples plus error-check exercises to find and fix mistakes.
This two-page reference sheet organizes the core laws of arithmetic and algebra that students use when simplifying expressions. Page one presents operation laws, sign rules, and worked examples in a compact layout. Page two is planned to provide eight error-check items where students locate a mistake and correct it; the four exercises below are a starter subset of that planned set. The focus is on recognizing valid transformations and catching common sign or order-of-operations errors, not on graphing or solving equations.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
The commutative law states that addition and multiplication can be reordered: a + b = b + a and a × b = b × a. Subtraction and division do not commute, so 5 − 3 is not 3 − 5, and 8 ÷ 2 is not 2 ÷ 8.
The associative law groups three or more terms without changing the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). This law applies only to addition and multiplication, not to subtraction or division.
The distributive law connects multiplication with addition or subtraction: a(b + c) = ab + ac and a(b − c) = ab − ac. It also works in reverse for factoring: ab + ac = a(b + c).
The identity and inverse laws give neutral and canceling elements: a + 0 = a, a × 1 = a, a + (−a) = 0, and for nonzero a, a × (1/a) = 1.
Adding two numbers with the same sign keeps that sign and adds the absolute values: (−4) + (−3) = −7 and 4 + 3 = 7. Adding numbers with opposite signs subtracts the smaller absolute value from the larger and keeps the sign of the larger absolute value: (−7) + 4 = −3.
Subtracting a number is the same as adding its opposite: a − b = a + (−b). So 5 − (−2) = 5 + 2 = 7, and −3 − 4 = −3 + (−4) = −7.
Multiplying or dividing two numbers with the same sign gives a positive result: (−6) × (−2) = 12 and (−8) ÷ (−4) = 2. Two numbers with opposite signs give a negative result: (−6) × 2 = −12 and 8 ÷ (−4) = −2.
When a negative sign precedes parentheses, it distributes to every term inside: −(a + b) = −a − b and −(a − b) = −a + b.
The standard order is parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. For example, 2 + 3 × 4 = 2 + 12 = 14, not 20.
A useful error-check method is substitution: replace each variable with a small integer such as 2 or 3 and evaluate both the original and the transformed expression. If the values differ, the transformation is invalid. If the values match, that is only partial evidence; test several values, including values that make signs and parentheses matter (avoiding 0 and 1 where possible), and note that substitution can only disprove equivalence, never prove it. Verify algebraically for certainty.
Another check is to test a special value such as zero or one, which often exposes sign errors quickly. For instance, substituting x = 0 into −(x − 3) gives 3, while −x − 3 gives −3, revealing that the correct expansion is −x + 3.
Simplify −2(3x − 4) + 5x and check the result by substituting x = 2.
−2(3x − 4) + 5x = −6x + 8 + 5x = −x + 8. Substituting x = 2: original gives −2(6 − 4) + 10 = −2(2) + 10 = −4 + 10 = 6; simplified gives −2 + 8 = 6. Both equal 6, so the simplification is consistent.
Find and fix the mistake: 4 − (−3) = 1.
4 − (−3) = 4 + 3 = 7.
Why: Subtracting a negative number is equivalent to adding its opposite, so the operation becomes 4 + 3, which equals 7. The original answer of 1 incorrectly treated the double negative as subtraction.
Check: Rewrite as 4 + 3 and confirm on a number line: starting at 4 and moving 3 units right gives 7.
Find and fix the mistake: −(2x − 7) = −2x − 7.
−(2x − 7) = −2x + 7.
Why: The negative sign distributes to both terms. The first term becomes −2x, and the second term becomes +7 because subtracting a negative inside the parentheses reverses to addition when the outer negative is applied.
Check: Substitute x = 0: original gives −(0 − 7) = 7; corrected gives −0 + 7 = 7. The incorrect version gives −7, so the correction is confirmed.
Find and fix the mistake: (−5) × (−2) = −10.
(−5) × (−2) = 10.
Why: Two negative factors produce a positive product. The absolute values multiply to 10, and the even number of negative signs makes the result positive.
Check: Count the negative signs: two negatives give a positive. Verify with 5 × 2 = 10 and attach the positive sign.
Find and fix the mistake: 6 + 2 × 3 = 24.
6 + 2 × 3 = 6 + 6 = 12.
Why: Multiplication must be performed before addition. The product 2 × 3 equals 6, and then 6 + 6 equals 12. The original answer incorrectly multiplied the sum 6 + 2 by 3.
Check: Apply the order of operations: multiplication first, then addition. Confirm that 2 × 3 = 6 and 6 + 6 = 12.
No. Subtraction is not commutative because changing the order changes the result. Rewrite subtraction as addition of the opposite before reordering.
Substitute a small integer such as 2 for each variable in both the original and simplified expressions. A mismatch proves the simplification is wrong. A match is only a partial check, since expressions can agree at some values without being equivalent; test several values (avoiding 0 and 1 where possible) to try to find a mismatch, but remember substitution can only disprove equivalence. Verify algebraically to confirm.
Multiplying by a negative reverses direction on the number line. Reversing twice returns to the original direction, so the product is positive.
Evaluate the parentheses first, then apply exponents to the result, then proceed with multiplication and division from left to right.