Fractions of Area?? In 3rd grade?

I'm sure some of these new math TEKS have taken many of you by storm! If you're not ready to throw in the towel, then you're a real trooper! As Math Educators, we are really on the "front line" out here because we are having to re-structure our thinking, all while teaching our students. It's almost the "blind leading the blind"...ALMOST! Hopefully, many of you are staying a few feet ahead of the students.

As an Instructional Specialist, I'm not suffering any less. I'm challenged to stay a few feet ahead of the teachers! So let's explore a TEK that shows up in the Geometry and Measurement Unit!

3.6E "Decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole and recognize that equal shares of identical wholes need not have the same shape."

In English please??? Right?

Layman's terms??: "Upon creating 2 flat shapes (using area models) that have the same product or area, separate/divide the shapes in different ways but so that both are separated equally. Do this so students understand that no matter how you separate a shape, its partner (same size, not same shape) shape will have the same amount of parts."

Let's explore this concept.

If I had 4 quarters and gave you two of them, I would have given you $0.50, right?
Well, if you had 10 dimes and you gave me five of those dimes, you would have given me 50 cents.
We both had the same value of money ($1.00); though we displayed our values in different ways (4 quarters, versus 10 dimes) we still both ended up with the same value ($0.50).

So how do we get our students to understand this concept?

Teacher: (Place students in pairs) Have students work with colored tiles (concrete) or colors and grids (representational) to display an array that creates 12. Encourage each partner in a pair to create an array different than their partner.

Students may make 1x12, 2x6, 3x4 and/or the commutative property of these arrays.

Teacher: Have students discuss whether they think dividing their perspective shapes in half will create the same amount of tiles. Simple YES or NO prediction. **Estimation opportunities are critical for students. It's a great way to get them to question their own thinking and justify their reasoning.

Students: Challenge students to divide their shapes in half (perhaps using a popsicle stick) and ask them to find out how many tiles are represented in this "half".

Teacher: Lead a discussion as to "why" each figure, though divided differently, might yield the same result in number of tiles? Have each student (in the pair) write a statement about their findings. Offer ELL students & struggling learners language support via sentence stems.

So when we look at the original version of the TEK, it may sound foreign to us and from that stand point a bit intimidating that we are essentially teaching our students how to find a fraction of a whole! And indeed we are! But here's the relief: we aren't teaching them an abstract concept without first laying the concrete foundation! One day the algorithm of multiplying a fraction by a whole number will make sense because they received a foundation in 3rd grade. Students are capable of understand myriads of things when given the opportunity to manipulate items. Are you giving your students that chance?

Here's a released STAAR question: 



Now, without this sample lesson I just shared, which answer do you think they would have chosen? I'm thinking A, due to spatial reasoning.
Students may be inclined (without the proper instruction and investigation) to assume that since the shaded portion in Figure M looks thicker than that in Figure N, that A is the answer.

But the true answer is C. With proper instruction (an investigative format coupled with a concrete standpoint), students can develop the cognitive reasoning that when both shapes have the same area and both have one-fourth selected (or separated) then the portion shaded in both figures represents an equal area. Third graders can "learn"; it is imperative that we, as teachers, remain lifelong learners so we can pull out the abilities of our students. Then and only then, will we witness the rewarding success of (not just our own practices, but) our students achieving what seems impossible!