Home Mathematics Modeling mathematical ideas: developing strategic competence in elementary and middle school

# LESSON VIGNETTE: WHAT WOULD YOU CHOOSE? ANALYZING CHANGE IN NUMBER PATTERNS

Functions using an input-output machine like the magic pot can be a great precursor to functional thinking for later grades. Functions can explore the idea of change in a real-life context. Students experience many situations in their everyday lives that represent constant or varying change. Plant growth and temperature change represent varying change, while the cost of an international call can show constant change.

Students analyze the structure of numerical and geometric patterns (how they change or grow) and express the relationship using words, tables, graphs, or a mathematical sentence. The goal is for students to investigate and describe the concept of variable. They practice using a variable expression to represent a given verbal quantitative expression involving one operation, and write an open sentence to represent a given mathematical relationship, using a variable.

5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

In the following research lesson called Your Dream Job, lesson objectives focused on students

a. describing, extending, and making generalizations about geometric and numeric patterns and;

b. identifying and describing situations with constant or varying rates of change and comparing them.

Before students began, the teacher asked the students to write their name on an individual Post-It note and “vote” on the option he/she thought would get to \$1000 first. The students were told that they could change their answer at any time during the lesson. Students turned to their classmates and turned and talked about the various reasons for choosing either Option 1 or 2. As the students worked on their choices, the teacher asked students to stop and think about their choice after they reached Day 3 and asked if they want to change their original “votes” and why/why not.

Text Box 5.2 A Math Happening 5c: Your Dream Job

You just got a dream job and you have two options for your pay.

Option 1: Your salary will be doubled each day: you will earn \$1 the first day, \$2 the second day, \$4 the third day, \$8 the fourth day, and so on.

Option 2: Your salary will increase \$3 each day: you will make \$3 the first day, \$6 the second day, \$9 the third day, \$12 the fourth day, and so on.

Which of the two options will get your salary to \$1000 the fastest?

If it is possible, the students can also be asked to represent their results using a graph after Day 3 to help them see any emerging trends spatially. One can anticipate a majority of students will choose Option 2 because on Day 3, the total earnings for Option 2 (\$9) is more than the total earning for Option 1 (\$4). Having students share their choices independently (or in groups) helps to build a collaborative problemsolving environment in the classroom.

89

Modeling Math Ideas with Patterns and Algebraic Reasoning

The teacher repeated this process once again after Day 6, and the students were asked the same question whether they intended to change their votes or not. The teacher used a graph to show the result adding in the new results to the graph after Day 3. Several students noticed that Option 1 (\$32) was growing faster than Option 2 (\$18), especially after reviewing the graph at Day 6. The iterations for the first four days for each option are plotted on a graph shown. This can generate a lot of discussion in the classroom and can lead to an initial exposure to linear and nonlinear functions or graphs of those functions; terminology such as exponentially increasing helps them understand the concept of steepness in functions or rate of change.

The lesson ended with the teacher reading the book One Grain of Rice and then discussing the concept of doubling with students. One Grain of Rice by Demi (1997) is about a young village girl who outsmarts a selfish king by asking him to double a portion of rice every day for 30 days in order to feed the hungry. Students were amazed at the surprising power of doubling to win more than one billion grains of rice from the king.

Another lesson study team chose to modify an existing related lesson called the MP3 purchase plan. In order to complete this activity, students needed some background knowledge of how to write linear equations, complete a function table, graph on the coordinate plane, and analyze data. The problem was as stated,

You have decided to use your allowance to buy an mp3 purchase plan. Your friend Alex is a member of i-sound and pays \$1 for each download. Another one of your friends, Taylor, belongs to Rhaps and pays \$13 a month for an unlimited number of downloads. A third friend, Chris, belongs to e-musical and pays a \$4 monthly membership fee and \$0.40 a month per download. Each friend is trying to convince you to join their membership plan. Under what circumstances would you choose each of these plans and why?

It is important that students grapple with problem solving and try various solution strategies to find the most efficient one. This process also allows students to explore multiple approaches to problem solving. Some students used number sentences, simple charts, or tables to organize their thinking. After the students worked on the problem individually, they talked to their tablemates and discussed which approach was a better choice. Initially, students will verbalize the mathematical relationship informally; this process is the precursor to translating the verbal description into an algebraic formula.

Once students become comfortable describing patterns and relationships between numbers, using symbols to represent the variables becomes easier. To provide learners access to this problem, teachers developed in advance some questions to ask— for example, “How could you track the money you get each day from choice A?” Organizing information on a table can help students analyze the mathematics in the problem and become efficient problem solvers. Creating a table with the number of days listed from the smallest to the largest number gives students an opportunity to recognize a recursive or iterative pattern. One can also use technology to illustrate the connection between tables and graphs, allowing for a discussion of how each representation shows the rate of change. Relating this problem to banks or allowances can make this mathematics activity realistic and engaging.