Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Lessons

The first lesson will last five days, during which the students work for 20 minutes each day, and it is organized in the following way.

• The students copy two-word problems into their notebooks and will start solving them individually. These problems will be from the first set involving only whole numbers, where I used the simplified version for equations 1 and 2, with c = 0, leading to equations 1b and 2b:

ax + b = d (Eq. 1b)

• The teacher asks for a volunteer to read aloud the first problem.
• The teacher asks for a volunteer that gives a hint on how to approach the word problem.
• The teacher solves the problem, step by step, with the help of the students.
  • The teacher will give the students more independence as they progress in solving the problems during the five days of the lesson.
• The teacher asks for a volunteer to reread the problem and explain the solution.
• On the last day of the lesson, the teacher will reveal how all the problems can be solved with the general equation 2.

The second lesson will last another five days, where the students work for 20 minutes each day, and it is organized in the following way.

• The students copy two-word problems, from the second set involving rational numbers, into their notebooks and will start solving them individually. These problems will be from the second set involving rational numbers, where I used the simplified version for equations 3 and 4, with v = 0, leading to equations 3b and 4b:

• The teacher asks for a volunteer to read aloud the first problem.
• The teacher asks for a volunteer that gives a hint on how to approach the word problem.
• The teacher solves the problem, step by step, with the help of the students, reaching the general equation that involves the division of fractions, which will be considered the final answer for now.
  • The teacher will give the students more independence as they progress in solving the problems during the five days of the lesson.
• At this point, the teacher asks for a volunteer to reread the problem and explain the solution.
• On the last day of the lesson, the teacher will reveal how all the problems can be solved with a general equation, similar to what was done for the first set of problems.

The third lesson will last another five days, where the students work for 20 minutes daily, and it is organized in the following way.

• The students copy two-word problems of the second set involving only rational numbers and the solution at which they arrived the previous week, which is written in terms of the division of fractions.
• The teacher will review and illustrate using the rectangular model and the number line to finalize solving the problem.
• The teacher will also show how to use the symbolic model, invert and multiply, compare the three ways to solve the problem and ask for students’ feedback on their preferred approach.
  • The teacher will give the students more independence as they progress in solving the problems during the five days of the lesson.
• At this point, the teacher asks for a volunteer to reread the problem and explain the solution.

The fourth lesson will last another five days, where the students work for 20 minutes daily, and it is organized in the following way.

• The students copy two-word problems of the third set involving only rational numbers, which can be represented by Eq. 3 and whose solution is given by Equation 4.

• The teacher will review and illustrate using the rectangular model, the number line, and the symbolic model to finalize solving the problem.
• The teacher will compare the three ways to solve the problem and ask for students’ feedback on their preferred approach.
  • The teacher will give the students more independence as they progress in solving the problems during the five days of the lesson.
• At this point, the teacher asks for a volunteer to reread the problem and explain the solution.