Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Learner Express: Modules for Teaching and Learning
Students work to answer the question: "How many cards would be needed if all 24 students in the class were to exchange cards with each other?" Solving the problem helps them develop algebraic thinking skills. Run Time: 00:04:20
Cupid raises his bow and arrow in Lilia Olivas' fourth grade classroom at The Carrillo Intermediate Magnet School in Tucson, Arizona. Valentine's Day has just passed, and students are presented with the problem: "How many cards would be needed if all twenty-four students in the class exchanged cards with each other?" Before beginning the whole class exchange, the teacher has students model what happens when two students exchange cards and when four students exchange cards. Then, students choose to investigate the problem alone, with a partner, or in a group. They also can choose any materials to represent their exchanges. Students are to try and determine what functional relationship, if any, exist between the number of people and the cards exchanged. Students are asked to share their problem-solving strategies with classmates.
(Practice Standard)—The Common Core Practice Standard #8—Look for and express regularity in repeated reasoning—is clearly evident in this lesson. One young girl working alone grasps the concept involved in this growing pattern by using repeated reasoning to determine the number of exchanges that would be required to satisfy the conditions of the problem. She is tentative in declaring her answer because none of her peers used repeated reasoning in their attempted solutions. These students are well on their way to becoming algebraic thinkers because they are able to act out simple cases of the problem (2, 3, and 4 exchanges). They try to model with blocks, pictures, and charts more complicated cases of the problem. What remains is for them to generalize the problem for any number of students.
(Content Standard)—The Common Core Practice Standard evident in this clip is Operations and Algebraic Thinking. Students are encouraged to generate a number that follows a given rule. Using repeated reasoning, they are expected to identify features of patterns that are not explicit in a rule. Students should be able to explain why numbers will continue to grow in the fashion identified by a repeating pattern.
The classic "Hello" problem is analogous to this Valentine problem: If each person in the room were to say "hello" to each other, how many "hellos" would be spoken, presuming that you do not talk to yourself? A modification is the "Handshake" problem: If each person in the room were to shake hands with each other, how many "handshakes" would be completed, presuming you do not shake hands with yourself? How are these problems, and the Valentine's exchange alike? How are they different? What problem-solving strategies can be exploited when introducing these problems to fourth graders?
8. Look for and express regularity in repeated reasoning
4.OA Operations and Algebraic Thinking