# Hungry Countries Math Exploration

The last time I looped between grade 6 and 7, I noticed a gap existing in students understanding of equivalency and their ability to make connections between ratios, fractions, and decimals. This was after teaching the same group of students for two years, and placing a large emphasis on this strand. Through conversations with colleagues, we recognized that this is an area many students find challenging, in all grade levels. I began to think about other ways of teaching these concepts that would build a deeper, longer-lasting understanding.

While working in our Math lesson study with Galileo, a problem began to take shape where students would use fictitious countries (named Country A through I) which would each be assigned a number representing “People” and a number representing “Food Units”. Each country was assigned a people to food ratio where we purposefully built in combinations that would bring out student misconceptions (see the chart below). More about that in a minute.

In table groups, students were given a cup that contained the number of food and people present in their assigned country. This was represented by two different colors of tiles. Students had to decide, based on their population if their country was well fed or not. Groups then began to fill in a chart on the Smartboard and were asked, with their groups members, to rank the countries in order of best fed to worst fed. This generated a great deal of discussion within groups and as each group put their ranking up on the board, prepared to defend their choices. This is when the misconceptions began to appear. Students wrestled between countries C and G, finding it difficult to prove which country was better fed with their newly forming understanding of ratios.

Once a ranking was established that the entire class could agree with, students were asked to, using manipulatives, create a ‘twin country’ to each of the countries above that would have the same ratio of food to people, but a different population and to model this. Through this task, it was very apparent which students had a deep understanding of equivalency and who was able to represent their thinking concretely and pictorially. Students also began to generate other ways, besides using manipulatives of demonstrating and explaining equivalent ratios and fractions such as symbolic representation.

This problem led students into additional problems working more directly with fractions.

 Country People Food A 1 4 B 4 4 C 7 31 D 2 5 E 3 12 F 2 30 G 6 32 H 5 1 I 15 60

[vimeo www.vimeo.com/35023465 w=600&h=338]