Grade6 ratios, percentages, expressions and equations
Ratios preserve a named relationship
A ratio compares quantities in the stated order. Four cats to seven dogs is not the same relationship as seven cats to four dogs. Distinguish part-to-part from part-to-whole: four to seven has eleven total equal parts when the categories exhaust the collection. Equivalent ratios multiply or divide both quantities by the same factor.
A unit rate describes an amount per one unit. If three pounds of grapes cost 4.50 dollars, the supplied unit price is 1.50 dollars per pound. Keep the units when dividing so that price per pound is not confused with pounds per dollar. Compare offers using equivalent quantities rather than the smaller total price alone.
Percentages describe a base
Percent means per hundred. Fifteen percent of sixty is 0.15×60=9. A percentage statement needs a reference quantity: increasing an original amount and then decreasing the new amount by the same percentage need not return to the starting amount. At this foundation stage, identify the base before choosing multiplication or division.
A table can represent a proportional situation when equivalent ratios are maintained. Do not assume every increasing table is proportional. A fixed starting charge plus a per-unit charge creates a different relationship from a price that begins at zero and grows at a constant rate.
Expressions represent quantities; equations assert equality
An expression such as 3 x+2 has a value when x is specified. An equation states that two expressions have the same value. Substitute carefully: for x=4,3 x+2=14. Do not treat juxtaposition as digit concatenation or combine unlike terms as if they described the same units.
To solve x+7=12, subtract seven from both sides and obtain x=5. Checking 5+7=12 tests the proposed solution. For a simple inequality, the solution can include several numbers rather than one; interpret the boundary and comparison symbol in context.
In a word problem, define the unknown, describe the relationship and only then select an operation. A numerically correct calculation can still answer the wrong question. This practice develops common grade 6 reasoning without claiming that another state's benchmark list or official task format is identical.