Fill In the Fractions, by Michael Naylor

These activities develop fraction sense

Fill up those odd moments of the day with some of these fill-in-the-blank fraction puzzles. Each of these activities is full of fraction, number and operation sense and can help you use your time to the fullest.

Fill in the Fraction Hexagon (Grades K-2)
A hexagon pattern block piece makes a great base on which to build fractions. Prepare for this activity by tracing around 12 hexagons on a piece of paper, then printing a copy for each student. These outlines will be used to record solutions.

Tell your students the hexagon is "one" and, as a class, discuss what fraction is represented by the triangle, rhombus and trapezoid pieces. (Note: Don't use the square or the narrow rhombus.) Have the students cover hexagons with six triangles to show sixths, three rhombi to show thirds and two trapezoids to show halves.

The challenge: How many different ways can your students fill up a hexagon with these fractions? They should record their solutions on the paper by drawing lines and coloring. You may want to ask them to color the shapes according to the colors found on pattern blocks; triangles are usually green, rhombi blue, trapezoids red and hexagons yellow. If appropriate for your students, have them also record their solutions using fraction words such as "one half and three sixths" (below left) or "two thirds and two sixths" (below right).

Fill in the Double Hexagon (Grades 2-5)
Using pattern block pieces (minus the squares and the narrow rhombi), how many ways can students create hexagons that are twice the length of the small hexagons in the previous activity?

Call the big hexagon "one." Challenge your students to decide what fraction each of the smaller pieces represents. In the illustration below, twelve twelfths is on the left; six eighths and one fourth is at right.

Students should come up with six different arrangements to fill the big hexagon. The possibilities are many, so students can look for beautiful and interesting arrangements. Students should record their arrangements both in pictures and words. Afterwards, have them add their fractions together to verify that their sum is one.

Fill in the Digits (Grades 4-6)
This series of puzzles builds fraction sense and good computation skills. After students have worked on these puzzles, be sure to discuss, as a class, different students' computation strategies as well as their strategies for selecting numbers. The examples shown here use addition and subtraction, but you can invent variations for multiplication and division as well.

Using four different digits (from 1-9), arrange them in the boxes below to make:

• a sum as close to one-half as possible

• a sum as close to three as possible

• the greatest sum possible

• the least sum possible

Using four different digits (from 1-9), arrange them in the boxes below to make:

• a difference as close to one-half as possible

• a difference as close to one as possible

• the smallest difference greater than zero

• the greatest difference possible

Fill in the Fraction Triangle (Grades 7-8)
Building this fraction triangle offers practice in adding and subtracting fractions. It's also a rich source of patterns and has important historical connections. On the board, draw the left and right edges of the triangle as shown below. The number 1 is on top, then down each side we see 1/2, 1/3, 1/4, 1/5 and 1/6. Draw empty boxes in the middle; these will be used to fill the triangle.

Challenge your students to fill in the boxes so every fraction in the triangle is the sum of the two fractions below it. Demonstrate the first box for your students: below 1/2 is 1/3 and an empty box, so 1/3 plus the fraction in the box must equal 1/2. The first empty box must contain 1/6, since 1/3 + 1/6 = 1/2. Your students will notice patterns as they fill in the boxes. If they continued the triangle, would the patterns continue?

The answers for the triangle shown above are: first row, 1/6; second row 1/12 and 1/12; third row 1/20, 1/30 and 1/20; fourth row 1/30, 1/60, 1/60 and 1/30. If you extend the triangle one more row to a row that begins and ends with 1/7, the answers are 1/42, 1/105, 1/140, 1/105 and 1/42.

This triangle was invented by one of the mathematicians who invented calculus in the 1700s, a German mathematician named Gottfried Leibniz (1646-1716). It's called "Leibniz's Harmonic Triangle." You and your students may be able to find some interesting connections between Leibniz's triangle and its better-known relative, Pascal's triangle.

I hope you've enjoyed exploring fractions with me.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.