The first step to the final goal of student generated problems is teacher instruction. The teacher must model the process and explain the concepts explicitly.
This portion of the lesson is teacher directed and students are guided to discover what methods and strategies will aid them in solving word problems. Discussion is a focal point in this portion because the ultimate goal is for the students to grasp the knowledge of how to solve word problems. By discussing the possibilities, teacher and students can share their ideas and concerns for each set of problems. This class brainstorming is meant to help the students, not hinder their development. By making this a non-threatening environment where all ideas are accepted and evaluated, no answer or reason given by a student is stupid. Students need to take risks within the classroom and it is the teacher’s responsibility as a facilitator for learning, to make every student feel comfortable enough to share his/her own ideas. It also should be noted that to lead a discussion, the teacher must be comfortable with the topic and evaluate the problems personally in order to be prepared to share his/her own reasoning. Word problem modeling must not only occur in a pencil-paper type environment, but also in the area of discussion. The teacher should set the stage for the students, establishing a comfort zone in which ideas can be expressed and opinions shared. Once this has been modeled, the teacher can integrate the students into the conversation, giving assistance when necessary.
It is necessary for students to not only know how to perform basic operations such as addition, subtraction, multiplication and division, but also to know that the operations are connected to each other. They must realize that addition and subtraction are inverse operations, meaning if you know that 1 + 1 = 2, then 2 - 1 = 1. The same principle can also be shown with higher numbers to show that if you know that 734 + 89 = 823, then 823 - 89 = 734. To be more exact, a given number (from any number) is the inverse to adding the given number: (a + b) - b = a. Multiplication and division are inverse operations as well, so that if: 3 x 5 is equal to 15 then 15 / 3 must then equal 5.
The following problems are a model for the teacher to follow and should walk the students through them. They are arranged in a Step Chart that includes the word problem (along with its inverse problem), the givens in the problem, steps to solve, and the rationale behind the methods used to solve it. The problems are arranged in this manner so the students become used to the format that they will later use when devising their own problems. The format also allows the students to lay out the information methodically to aid in their journey to the solution. Also note that the activity of analyzing word problems can be carried further by incorporating more complex problems based on the needs and skills of the students and should not be limited by the problems given within this unit.
Before each set of problems, a rationale for each is provided. This will outline the reasoning behind the ordering of the problems and the format they are in. In some rationales the term “basic” is utilized. This term indicates facts that a third grade student should, in most instances, be able to solve due to prior instruction and lessons. The goal of these word problems are not to stump the students on the equations, but to teach them how to read the problem, find out what the problem is asking for, solve it, and then be able to generate their own. To use extremely complex equations would defeat the purpose of the lesson and only frustrate the student.
It is also important to note that following work is the teacher’s preparatory work. Since the teacher must be familiar with how to solve the problem and also must be comfortable with explaining the routes to the solution, this framework should be used as a teacher’s guide before teaching the lesson. The teacher’s rationales and connections made trying to solve the problem may not always be identical to those of the students. The explanations given here are for the teacher model. The students are not asked to, nor are expected to, give the explanations below verbatim. What the teacher’s expectation should be is to have a class that is actively listening and adding their own ideas when they are inspired to. As long as the students are participating, discussing, and being led in the right direction, then the given rationale can take any form and go any route.
Problems A.1. through A.11.
Intended for students on-grade level
The problems listed for students on grade level increase in difficulty as they progress. It should be noted that the teacher should focus on each type of problem and may use the given problems. It is a strong possibility that a student may require more practice, so the teacher should then create his/her own parallel set of problems for each type before in fact moving on to the next set. This will allow students to practice each type until they are able to master the skills.
A.1. & A.2. skills: Addition and subtraction
The rationale behind this first set of problems is that they are formulated from basic addition and subtraction facts. The equations require regrouping of two-digit numbers, a skill in which third grade students learn the first few months of the year. Students appear to be more comfortable with addition, so A.1. is an addition problem. It is important for students to feel a level of success when then are learning new concepts and performing tasks. That confidence will then aid them in more complex problems, which they can then tackle without a high level of frustration. A.2. requires the inverse operation of A.1. to solve and when students become aware of the connection, the method to solve the second problem will become clear to them. The problems in Appendix A are one-step problems that require only one equation to arrive at the solution and belong to the same “addition-subtraction family.”
A.3. & A.4. skills: Multiplication and division
The rationale behind A.3. and A.4. stems from the fact that third grade students are introduced to multiplication and division during the second half of the year. They practice one-digit multiplication, and are able to divide two-digit numbers by one-digit numbers evenly (without a remainder). A.3. and A.4. have the students practicing those skills. They must also get used to all types of vocabulary within the word problem and realize that when certain words appear, such as: times, groups of, equal parts, then multiplication and division are most likely to follow. A.3. and A.4. in Appendix B are still one-step problems that just deal with different operations than those that preceded.
A.5. & A.6. skills: Multiplication, division, addition, subtraction, and multiple steps
The next set of problems are the more complex that those before and require multiple steps. A tendency of some students when they encounter a word problem is to just add all of the numbers that appear. This is a habit that we as teachers try desperately to break because it is when they come upon problems like these that the students become lost. It also indicates a need for careful analysis in a while class discussion. A.5. starts with a basic multiplication equation and then an equation that requires basic regrouping. A.6. then requires basic subtraction with regrouping and then a basic division equation. A.5. and A.6. in Appendix C are the first two-step problems that arise in the set.
A.7., A.8., & A.9. skills: Subtraction, addition, multiplication, division, and multiple steps
The word problems are still becoming more complex. The information within the problems is changing and is now offering the students more givens that they must identify and filter according to what they need to use to solve it, and the manner that they will use them in. A.8. and A.9. in Appendix D are the first three-step problems to appear, but have been gradually introduced within the course of problems.
A.10. & A.11. skills: Addition, subtraction, multiplication, division, and multiple steps
These last problems involve not only multiple-steps, but also multiple questions. Students must be able to identify not only the givens in the problems, but all of the questions that the problem is asking. If students have a method in which they organize the information that is provided in the problem, then they will be able to clearly see what the question is, and if there are multiple questions asked. Some confident students like to show their mathematical skill by finishing the problem quickly before their peers. On occasion this results in missed portions and skipped questions. A.10. and A.11. give the students the opportunity to witness problems that are multi-step and have multiple questions. The problems in Appendix E will also give them an example on what to strive for in the creation of their own generated problems.
Varying the Level
Since classrooms are not homogeneously grouped, problems must be altered to meet the needs of the individuals who do not reach the expectations to be considered on-grade level. The following examples are word problems intended for students who are below-grade level. The word problems have been altered by reducing the number of problems, smaller digits, and less steps. These students may also need the assistance of manipulatives to aid them in finding the ultimate solutions. Some manipulatives they can use to help could be base ten blocks and/or counters. Problems A.1. through A.11. are models of how to record all of the steps to solve a word problem and the same process can be applied to problems B.1. through B.6.
Problems B.1. through B.6. in Appendix F
Intended for students below-grade level
Just as problems are needed for students who are below-grade level, students who are above-grade level need to have their needs met as well. The following problems are designed to challenge them and involve higher numbers that require regrouping as well as problems that have multiple steps. Depending on the mathematical skills of the students, the numerical values in the problems given may be increased to suit the needs of the above-grade level students. The given problems are meant to be adapted for each individual classroom, so if the need presents itself, more complex and challenging numbers may be substituted.
Problems C.1. - C.10. in Appendix G
Intended for students above-grade level