The Language of Division: Rephrasing for Greater Understanding

by Mary Saltus, Diane Neison, and Chet Delani

Fluency with the operation of division is an important skill to bring to more complex mathematical challenges. Of the four operations, division can be the most tedious for students to learn and practice. However, when division is experienced in the context of dice games, students readily engage in division activities and actually want to practice--or as they would call it, "play the game."

Games involving division and dice are most productive when they focus on developing an understanding of the concepts of divisor, dividend, quotient, factor, and remainder, and this is the subject of Dice Activities for Division. Fluidity with division, as with any subject, comes only with practice based on understanding.

Using the Phrase "Has How Many" to Indicate Division

In our work with students involving the practice of division through dice activities, we learned that attaching the term "has how many" to the division sign creates a vivid image of the division process. "Has how many" makes sense to students in the context of their everyday language whereas "divided by" sounds abstract and technical, implying a procedure or process.

For example, instead of asking students "What is 20 divided by 10?" you could rephrase the question as "20 has how many sets of 10?" Many times, with this simple rephrasing, the meaning of the question is instantly clear, and students can successfully solve the problem without relying on memory, skip counting, or manipulatives.

Problems such as "27 divided by 3" can be rephrased as "27 has how many sets of three," but some students may still need to skip count by threes to 27 or take 27 tokens and divide them into sets of 3. Even so, including "has how many" in phrasing the task makes it clear to students what they need to do to solve the problem (i.e., grab 27 tokens and divide them into sets of 3).

The problem "3 divided by 5" is often confusing to students. If you ask,"How many fives are in 3?" their first response may be "None" or "They're aren't any." Using the phrase "has how many" and tokens will easily clarify the confusion, as illustrated in this sample dialog:

Show 3 tokens. Ask: "How many sets of 5 tokens do you see?"
Response: "None."
Ask: "What's left?"
Response: "Three."
Say: "So the solution is 0 sets of 5 tokens with a remainder of 3 tokens."

Using the Phrase "Of the" to Indicate Multiplication

Because division and multiplication are inverse operations, it makes more sense for students to use multiplication to arrive at a quotient than it does to subtract repeatedly, as they often do. Once again, careful rephrasing of the question usually brings clarity. For instance, the problem "100 divided by 25" can be rephrased as "How many of the 25s are in 100?" Attaching the phrase "of the" to the multiplication sign enables students to make sense of the question and quickly arrive at the answer, "4" (the quotient). It is worth noting that the phrase "of the" works well for multiplication problems, too. Rather than asking students to solve "5 times 3," try asking for "5 of the 3s" and see how quickly you get the answer.

In the same way that "has how many" clarifies division problems, the words "of the" invites students to mentally construct meaning from their experience rather than rely on memory to solve multiplication problems. When first-graders listen to the expression "2 of the 10s," they readily understand what is called for and will easily come up with "20" as the answer. In contrast, "2 times 10" implies an operation, and there is nothing in this language that suggests what the student must do to arrive at 20.

Rephrasing works for teaching multiplication with fractions, too. The phrase "one half of 8," for example, is much more accessible to students than the written form of the problem, 1/2 x 8. Multiplication allows more of a constructivist approach when the times sign is verbalized as "of" or "of the."

Sample Activities

The two sample activities from Dice Activities for Division included in this newsletter (see Downloadable Activities below) will provide you with opportunities to practice rephrasing division questions for greater understanding--and give students the chance to experience how much fun division can be when dice are part of the program!

In FACTOR TWO-DICE SWITCH, students who are not fluid with multiplication or division facts have little difficulty identifying factors. Students soon generalize that 1 is a factor of any number and that 2 is a factor of any even number. This activity can easily be used as an introductory activity to divisibility rules.

In FOUR-GRID TIC-TAC-TOE, the game element takes center stage. This Tic-Tac-Toe activity focuses on reinforcing division facts when the divisor is 6. As with any game involving the toss of dice, randomness often supersedes skill, but with FOUR-GRID TIC-TAC-TOE strategy can play a significant role in the outcome.

Mary Saltus, Diane Neison, and Chet Delani are co-authors of Didax's best-selling Dice Activities series. Mary and Chet are currently at work on the new book Dice Activities for Algebraic Thinking.

Division Downloadable Activity Pages

Dividends, divisors, quotients, and remainders will all be part of the fun when you grab a handful of dice and try out these activities from Dice Activities for Division.

	Download Factor Two-Dice Switch activity
	Download Four-Grid Tic-Tac-Toe activity

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