ILLINOIS TEACH & TALK

Consider two diving boards, one 40 feet high, the other 8 feet high.  Students in earlier grades learned to compare these heights in an additive sense – “This one is 32 feet higher than that one” – by solving additive Compare problems (2.OA.1) and using addition and subtraction to solve word problems involving length (2.MD.5).  Students in Grade 4 learn to compare these quantities multiplicatively as well: “This one is 5 times as high as that one” (4.OA.1, 4.OA.2, 4.MD.1, 4.MD.2).  In an additive comparison, the underlying question is what amount would be added to one quantity in order to result in the other.  In a multiplicative comparison, the underlying question is what factor would multiply one quantity in order to result in the other.

See Operations & Algebraic Thinking Table 3.  Multiplication & Division Situations.

Language can be difficult in Multiplication Compare problems.  The language used in the three examples in table 3 is fairly simple, e.g., “A red hat costs 3 times as much as the blue hat.”  Saying the comparing sentence in the opposite way is more difficult.  It could be said using division, e.g., “The cost of a red hat divided by 3 is the cost of the blue hat.”  It could also be said using a unit fraction, e.g., “A blue hat costs one-third as much as a red hat”; note however that multiplying a fraction is not an expectation of the Standards in Grade 4.  In any case, many languages do not use either of these options for saying the opposite comparison.  They use the terms three times more than and three times less than to describe the opposite multiplicative comparisons.  These did not use to be acceptable usages in English because they mix the multiplicative and additive comparisons and as ambiguous.  If the cost of a red hat is three times more than a blue hat that costs $5, does a red hat cost $15 (three times as much) or $20 (three times more than: a difference that is 3 times as much)?  However, the terms three times more than and three times less than are now appearing frequently in newspapers and other written materials.  It is recommended to discuss these complexities with Grade 4 students while confining problems that appear on tests or in multi-step problems to the well-defined multiplication language.