Investigating Equivalencies

How many of you have ever taught equivalent fractions to students? If that's you, try to complete this phrase:

"What you do to the top, you must do to...."

This teaching phrase is commonly used as a "rule" to help students remember how to simplify a fraction (or what we used to call "reducing"). If we allow students to investigate and explore fractions in a concrete manner, they can create their own rules as early as third grade.

Now, in 3rd grade, equivalencies are already being determined on a number line (hopefully through exploration and utilization of fraction tiles)! *See 3.3G

In 4th grade, however is where we find ourselves reading TEK 4.3C "Determine if two given fractions are equivalent using a variety of methods." And almost immediately we think 'Let's make an anchor chart that tells the students to multiply or divide the top and bottom number by the same numeral to get a larger or smaller fraction'.

A few problems occur when we teach this way:

1. We teach a rule that kids feel the "need" to memorize. As a result, they feel defeatedd when they cannot memorize or recall that rule. 

2. We impose on students (yet another) rule in math, causing them to believe math is made up of a lot of rules that are impossible to remember. 

3. We become the ones imparting knowledge rather than giving students an opportunity to discover the relationships (not rules) that math creates!

Cathy Seely, author of Smarter than we Think, writes "If experience influences a person's intelligence, then the mathematical tasks and problems we present to students, and how we present them take on critical importance." I love this picture she paints of expanding a students capacity by allowing them to struggle. This concept addresses all of the aforementioned problems. 

So let's pose this problem to 4th grade students. 

Build/construct/compose an array of objects to reflect 3 out of 12 (3/12ths). Find another way to divide this array in order to find a fraction that is equivalent to three-twelfths. 

You may find this "struggle task" vague and complex for students. But allow them to struggle. The concrete model should assist them in finding equivalencies. Some students who struggle a little longer and harder may need a little assistance.

Once students have found that 1/4 (one-fourth) is a fraction that is equivalent to 3/12ths, have them explain why (justify their solution). It may help to have them record the two fractions side by side to help them prove their answer. Some students may extend their exploration to find that two out of eight also equals 3/12. With all of these fractions sitting side by side, they should be able to compose their own rule (relationship) between the three fractions. 

If students can visually see that one green chip out of 4 chips doubled is the same as two green chips out of 8 total chips due to a multiplicative pattern, they will determine that equivalencies are built upon multiplication. They will have created the information necessary to create an anchor chart (together as a class).

(credit: lawlerjoyinthejourney.blogspot.com)

This method addresses what Cathy Seely calls the "upside down" strategy and problems #1 & 2 from above. The students take ownership of their learning therefore releasing them from the pressure of having to memorize rules. Likewise, the concrete investigation seals their understanding and gives them the power to build on their discoveries. As uncomfortable as this may make the teacher by changing their role from teacher to facilitator, it's also powerful because students no longer see us as pouring knowledge into them, but rather bringing out what they know. This, in turn, significantly increases student efficacy.

Happy investigating!!