I can't take credit for the below image. I found it on twitter. It's located here and above under my Fav Problems. The variety of ideas and misconceptions with this problem make it a must-do warm-up in every classroom.
As you can see below, I treated it like a number talk. I had students share all their answers, without any reaction or judgment, and asked students to defend their answer.
As you can see in the top right, Juliana's idea was that there's 1 hexagon in the middle. But if there are 6 trapezoids on the outside, 2 of those trapezoids can be rearranged to make 1 hexagon. So 6 trapezoids would make 3 hexagons. Therefore she reasoned that 1/4 of the shape was yellow, because if the whole shape is 4 hexagons, only 1 of them is yellow.
Victor said that the yellow hexagon in the middle was 2 trapezoids, or 2 reds. So, if there's 6 reds on the outside, that's 8 total. Therefore, he said 2/8 are yellow.
I did want to acknowledge the wrong answer of 2/6, because that is the ratio of yellow to red, so there's something correct about that. It's just not the part/whole ratio that we are looking for when we say what fraction of the shape is yellow.
I also asked students how someone could say 1/7. They said they probably saw 1 yellow shape out of 7 total shapes. Students said you couldn't do this because they are not all the same sized shapes.
One student said 1/2 of the shape is yellow because a hexagon is half of a trapezoid... a good thought that wasn't extended though to the whole shape.
I had to take a picture of this one because all day this was the only person who thought of it this way. The student said you can make the whole shape triangles, and a hexagon is 6 y yellow triangles, and there would be 24 triangles total, so 6/24 is yellow.
Every period I mention the way I see it if no one has said it yet, which is noticing the shape is symmetrical, and if you cut it in half you have 1 yellow trapezoid out of 4 total trapezoids, for 1/4.
More images like this can be found on fractiontalks.com. We have also worked in some of the flag matches from that site which are pretty awesome and cross curricular.
This blog post was inspired to by a video @mathletepearce made and the conversations on Twitter that followed.
Fraction Constructs: Part-Part Relationships. This time, Doritos Roulette "Hot" or "Not"? #MTBoS #mathchat #edchat pic.twitter.com/vVX1eohceV— Kyle Pearce (@MathletePearce) December 22, 2016