Me: What’s a half?

Students: When you take something and cut it in half.

Me: So what’s a half then?

Students: When you take a piece of anything and cut it into 2 pieces.
Anticipating this response I quickly grabbed the scissors and cut a piece of paper into 2 obviously unequal parts.

Students:  That’s not a half because they’re not equal.

We formalized our definition of a half and this is what the students came up with:  “A half is when a shape or noun is cut, shaded, partitioned into 2 EQUAL parts.”

They told me I had to underline equal and I gladly obliged.  This is why I love what I do.

• Student 1 defined half using the term shape
• Student 2 said pizzas needed to be included in our definition
• Student 3 said a cake can be cut in half as well
• Last student said “well your bedroom should be included because my sister doesn’t keep her half clean and we get in trouble.”

And that is how “noun” got added into our 2nd grade definition.

Me:  So we all have an understanding of what half means, right?  Ok, take a look at the rectangles.  Circle the ones that are shaded in half.

MCC2.G.3 states that students must partition rectangles into two equal shares which was the concept I was hoping to drive home.  The only difference was I did the partitioning and they identified which were halves.  At first there were the obvious rectangles that every student identified but 2 students began to identify more halves outside the usual suspects.

Shading Halves (Student Sheet)

Student 1: This one is a half Mr. Fletch (pointing to column 2, 2nd one down)

Me: No way!  How do you know?
The student begins to explain his thinking but the other students are lost. 

Me: Can you explain it a different way?

Student 1: Can I borrow your scissors?

The student cuts out the rectangle and divides it into black and white sections.  He placed the black pieces on the white piece and explained that they completely cover each other so they had to be equal.

Out came the scissors and the class went to town.

Towards the end of the class the majority of students could prove that 9 of the 10 rectangles where shaded in half.  And then it happened…the same little guy from the beginning of the lesson gave me an ESPN Highlight of the Year moment.

Student: I know a rule for how to make a half for any rectangle.

Me: Really?!  What do you mean? (I was sincerely interested but thought he was going to show us how folding works to prove a half)

Student: if you find the middle of each side (pointing to a rectangle) and connect them you make a perfect diamond (I addressed it as a rhombus after).  If you color the diamond in black and leave everything else white you will have a half of the shape shaded.

Me: Prove it! (I handed him 3 different size rectangles)

He was right but we obviously need some intervention with measuring a midpoint…and I’ll take that!  So maybe it’s not quite proofs but it definitely has the underpinnings of proving a geometric conjecture and generalizing it.