We created a list of favorite cereals, and looked at different sized boxes of cereals. This is something very simple, but I need my students to "buy into" the lesson. My students get very enthusiastic when something is easy for them, and I need to make them equate being successful (not things being easy necessarily) with math as well. This really only takes a minute, so there's no loss in instructional time here. More importantly though, is viewing the various sizes of cereal boxes, so that the students have a frame of reference.
You tell us something very critical here - that the students will reason this problem from the front end - If I need to halve the volume, I need to halve the dimensions (did I get that correct?). Perhaps an example will help us understand this critical information.
And what do you do in response to this? Do you let them figure it out? Or do you nudge them in a different direction?
I open guided practice by showing several cereal boxes, and asking students to estimate the dimensions of a randomly selected box. I then choose a few students to measure the dimensions of this cereal box to find out the dimensions; this provides students with a frame of reference.
explain what you say, and how you frame the task with students.
Students may believe that in order to make the boxes “half the size” or “three times the size” they need to adjust each dimension (length, width, height) by half or three times. They need to investigate how the total volume is affected by changing the dimensions and determine “half” and “three time” by calculating total volume.
To accompany each assessment quiz (the next lesson), I often provide students with the opportunity to express their knowledge/skills through a performance based assessment. Students who might struggle demonstrating understanding through fill-in-the-blank type/multiple choice tests, can often draw or use other tools to demonstrate understanding. It's my responsibility to provide accessible opportunities to demonstrate knowledge/skills. In this project, the students design their own cereal boxes.
I had one student incorrectly label the dimensions of his "regular" sized box, and so it affected the rest of his boxes. In explaining his reasoning to me, he noticed this, and was able to self-correct his errors. He asked for help drawing the dimensions of the boxes, and some of my students are shaky in doing in. I obliged for one, and encouraged him to trace another that I outlined for him.
One thing I took for granted is the "value" sized. Several of my students didn't understand why, if the value costs more (inferred), why it was in fact a value. I explained that if you divided out the cost by the volume then the "Value" size was actually a better deal in the long run. This is actually is teaching a real-life skill about saving money, and getting the most "bang for your buck".
I graded students on successfully being able to construct the mini cereal box and the value cereal box with the correct dimensions respectfully. Students were able to choose their favorite cereal to use, and therefore were given a personal stake in the learning context.
To close out, students report to their partners which size box they would rather buy and why that's their choice. Some students want the largest box because they love the cereal. Some students choose the mini box, to try the "new" cereal to see if they like it before "wasting" money on "yucky" cereal.