Creating Largest and Smallest Quotients
Creating Largest and Smallest Quotients is a Grade 5 math skill from Illustrative Mathematics Chapter 3 (Multiplying and Dividing Fractions) that challenges students to strategically arrange a set of numbers in the expression W ÷ (1/d) to produce the largest or smallest possible quotient. To maximize, use the largest number as the whole number dividend and the next largest as the denominator. To minimize, use the smallest numbers in those positions.
Key Concepts
Property To create the largest possible quotient from a set of numbers for the expression $W \div \frac{1}{d}$: Use the largest available number for the whole number dividend ($W$). Use the next largest available number for the denominator of the unit fraction divisor ($d$). To create the smallest possible quotient: Use the smallest available number for the whole number dividend ($W$). Use the next smallest available number for the denominator of the unit fraction divisor ($d$).
Examples Given the numbers 3, 4, and 5 to fill in the blanks for $W \div \frac{1}{d}$: To make the largest quotient: Use the largest number (5) for $W$ and the next largest (4) for $d$. $$5 \div \frac{1}{4} = 20$$ To make the smallest quotient: Use the smallest number (3) for $W$ and the next smallest (4) for $d$. $$3 \div \frac{1}{4} = 12$$.
Explanation This skill challenges you to use your understanding of division to create expressions with the greatest or least possible value. To get the largest result when dividing a whole number by a unit fraction, you need to start with the largest whole number and divide it into the smallest possible fractional pieces. Smaller fractional pieces have larger denominators. To get the smallest result, you should start with the smallest whole number and divide it into the largest possible fractional pieces (which have smaller denominators).
Common Questions
How do you create the largest quotient when dividing a whole number by a unit fraction?
Use the largest available number as the whole number (W) and the next largest as the denominator (d) in the expression W ÷ (1/d). Since W ÷ (1/d) = W × d, larger values for both W and d produce a larger quotient.
How do you create the smallest quotient when dividing by a unit fraction?
Use the smallest available number as the whole number and the next smallest as the denominator. This minimizes both W and d in the expression W × d, producing the smallest possible product.
What chapter covers creating largest and smallest quotients in Illustrative Mathematics Grade 5?
Creating largest and smallest quotients is covered in Chapter 3 of Illustrative Mathematics Grade 5, titled Multiplying and Dividing Fractions.
What is an example of maximizing a quotient with unit fractions?
Given digits 3, 4, and 5, to maximize W ÷ (1/d): use W=5 and d=4 to get 5 ÷ (1/4) = 5 × 4 = 20. Using W=4 and d=5 gives 4 ÷ (1/5) = 20 as well, but 5 ÷ (1/4) typically beats other arrangements.
Why does a larger denominator in the unit fraction produce a larger quotient?
Dividing by a smaller fractional piece means more pieces fit into the whole. A larger denominator means a smaller piece size, so more of them fit into the whole number, resulting in a larger quotient.