Developing Number Theory and Fraction Concepts

Many students can begin to feel challenged in math in middle school. Students who have been good at, and have even enjoyed, math suddenly look to their teachers, friends or parents for assistance. Why does this happen? If you look at the concepts that are significant in middle school grades (fractions, decimals and integers), you find that these concepts appear to break all the rules their teachers have told them up to this point.

Typically, students are taught that when you multiply two numbers, the product is always larger. When you divide two numbers, the quotient is always smaller. However, these rules apply to whole numbers, not fractions. When you multiply two fractions, the resulting product may be smaller! When you divide two fractions, the quotient may be larger!  Many students become frustrated, confused and give up on math. As teachers, we need to make sure our students understand concepts, not just memorize rules about them. Students need time to explore and discuss real life examples of the concepts we are teaching. Below are some high-level tasks that allow students to explore number theory and fraction concepts. As with all tasks, students should represent their work in numbers, pictures and/or words. They should have time to communicate their thoughts and findings with others.

Topic: Factors and Multiples

Task: Max is making table favors for a party. The candles come in boxes of 15 and the candleholders come in boxes of 9. Max does not want any leftover candles or holders. What is the fewest number of candles and candleholders he needs without any leftover? How many boxes of each should he buy? Task: At a day camp, there are 12 girls and 18 boys. The camp counselors would like to split the campers into teams. However, they must follow these rules: 1) All campers must be on a team; nobody can be left out, 2) all teams must have the same number of campers, and 3) each team can only have all boys or all girls; no boys and girls can be on the same team. What is the greatest number of camperseach team could have?

Topic: Understanding Fractions

Task: In Penny’s Pet Shop,  of the pets were dogs,  of the pets were cats,  of the pets were birds and the rest were gerbils. There were 48 pets in all. How many of each type of pet were there? Task: Ms. Kinny has  tank of gas in her Volkswagen Beetle. Miss Jamison has  tank of gas in her Ford Mustang. Dr. Beck has  tank of gas in her Honda Accord. Mrs. Hughey has  tank of gas in her Toyota Prius. Without finding common denominators, list the women in order from the person who has the least amount of gas in her car to the person who has the greatest amount of gas in her car.

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