Extending the Mathematical Learning Progressions for Expressions and Equations

Text Box 6.1 A Math Happening 6a: Saving for Mom's Present

Dan and Nick want to buy their mother a birthday present next month. Dan has been saving dimes and Nick has been saving nickels. Dan has saved 18 dimes and Nick has saved 22 nickels. The brothers agree to take a coin out of their wallet each day and put it in a piggy bank for their mother’s birthday present. One day, when they looked into each other’s wallets, they saw that Nick had more money than Dan. When this happened, how many days had they been saving for their mother’s gift?

lesson study vIGNETTE: SETTING A MATH learning agenda

The following lesson study vignettes explore important algebraic concepts with strategies and tasks for modeling the mathematical ideas and developing algebraic thinking in the early grades. A focus in this chapter is to consider how establish mathematics goals can focus the mathematical learning agenda. This lesson was part of a lesson study with a vertical team of third, fifth- and eighth-grade teachers. Originally, this lesson was written and published as an eighth-grade research lesson in Japan.

We were introduced to this problem as the Ichiro problem because it was stated as, “It has been one month since Ichiro’s Mother has entered the hospital. He has decided to pray with his younger brother at a local temple every morning so that she will get better soon. There are 18 ten-yen coins in Ichiro’s wallet and just 22 five-yen coins in the younger brother’s wallet. They have decided to take one coin from each wallet everyday and put them in the offertory box and continue to pray until either wallet becomes empty.

One day, when they looked in to each other’s wallets when they were done with their prayer, the younger brother’s amount was greater than in Ichiro’s. When this happened, how many days had it been since they started their prayers?” The original lesson goal was to have students express the relative size of quantities in inequalities and use inequalities to solve the problems as a procedure: (1) express the less than/ greater than relationship of quantities as an inequality. Make them understand the concept of the inequality and its solution; (2) understand the characteristics of the inequality and be able to use them to solve simple linear inequalities. Make them able to apply the use of inequalities to solve word problems.

As we planned for a vertical lesson study, the third-grade teachers set the goal of the problem to develop algebraic thinking as students look for a pattern as the value of the coins continue to decrease. Teachers anticipated students using repeated subtractions as a method of showing the pattern of change. In the sixth-grade lesson, the goal was to see how students organize their information and work through this multi-step problem and observe whether students would use a table or graph to illustrate the change. The eighth-grade teachers had been working with variables so they wanted to see if some groups could come up with an algebraic expression to model the problem (see Table 6.1).

As teachers planned for the research lesson, they were most interested in studying the ways students make sense of algebraic reasoning in the K-8 curriculum by analyzing patterns, and change, solving multistep problems, representing linear relationships, and understanding linear inequalities using multiple representations such tables, graphs, rules, and words. While the third-grade teacher’s lesson focused on analyzing a pattern of change, the preplanning session also provided middle-grade teachers an opportunity to share possible student responses. One area that the teachers discussed as a possible student difficulty was with identifying the variables.

Table 6.1 Mapping the learning progression for "Saving for Mom's Present"

Saving for Mom's Present Problem

Dan and Nick want to buy their mother a birthday present next month. Dan has been saving dimes and Nick has been saving nickels in their piggy bank. Dan has saved 18 dimes and Nick has saved 22 nickels. The brothers agree to take a coin out of their wallet each day and put it in a piggy bank for their mother's birthday present. One day, when they looked into each other's wallets, they saw that Nick had more money than Dan. When this happened, how many days had they been saving for their mother's gift?

Learning Progressions

Grades 3-4

Grades 5-6

Grades 7-8

Grades 3-4: Math Agenda The student will recognize and describe a variety of patterns formed using numbers, tables, and pictures, and extend the patterns, using the same or different forms.

The student will solve single-step and multistep problems involving the sum or difference.

Grades 5-6: Math Agenda The student will describe the relationship found in a number pattern and express the relationship using tables, graphs, number sentences, and symbols.

Students will solve multistep practical problems involving linear equations.

Grades 7-8: Math Agenda Students will use problem-solving methods of inequalities by comparing and applying the characteristics of solving a simple linear equation.

Students will represent linear

relationships with tables, graphs, rules, and words and make connections between any two representations (tables, graphs, words, and rules) of a given relationship.

Table 6.2 Related Task to the Ichiro Problem, also known as "Saving for a Present" task

Related Task— Buying MP3s

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?

Related Task—Carlos' Cell Phone

Carlos is thinking of changing cell phone plans so he is comparing several different plans.

Plan 1) Pay as you go plan $0.99 per minute Plan 2) $30 monthly fee plus $0.45 per minute Plan 3) $40 monthly fee plus $0.35 per minute Plan 4) $60 monthly fee plus $0.20 per minute Plan 5) $100 monthly fee for unlimited minutes

What will Carlos need to consider to make his decision? How can Carlos figure out which plan is best for him?

For example, how will students make sense of the difference between the number of coins versus the amount and value of the coins and the number of days. By having teachers from multiple grade levels plan the lesson, teachers unpacked the standards that aligned to the problem and discussed in detail the prior learning building blocks and the connections to related concepts, which helped them see the pattern in the development of ideas. Table 6.1 presents a mapping of the learning progression for a problem related to the original Ichiro problem that was used in the vertically articulated lesson study.

Inspired by this vertical lesson study, our instructional team decided that having a collection of related tasks with vertical connection was important for both teachers as a resource tool but more importantly to expose students to similar and related tasks for deeper learning. This inspiration led us to creating a collection of tasks that we call the “Family of Problems” (see Appendix MMI Toolkit 7). The rationale for the “Family of Problems” was to create a collection of related tasks that were bound by a similar big idea. The following related task all had a conceptual connection to the big idea of relationships and patterns, including (see Table 6.2):

1. Solving linear equations (or inequalities) both algebraically and graphically

2. Slope as a rate of change and y-intercept as an initial amount

3. Writing equations of lines in slope-intercept form