# The Craft of Word Problems

## CONTENTS OF CURRICULUM UNIT 04.05.01

- Narrative
- Place Value
- Addition and Subtraction (within 10)
- Fact Families
- Addition and Subtraction (within 20)
- Addition and Subtraction with and without regrouping
- Strategies for solving problems
- Lesson 1
- Lesson 2
- Lesson 3
- Lesson 4
- Lesson 5
- Lesson 6
- In the Future
- Mathematics Standards
- Bibliography for Teachers and Students

### Unit Guide

## The Relationship between Addition and Subtraction in Problem Solving

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After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

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Problem solving and word problems seem to be a universal concern at all grade levels. Students do not have the tools needed to solve word problems. Students need to learn steps in problem solving. They need steps that can be used to solve both simple and complex problems.

It is important that students understand how to solve word problems. This unit is designed for my second grade class, but can be used and modified for grades two through three. I will focus on the relationship between addition and subtraction in word problems. I want students to have the tools to solve word problems. Problem solving is a “life skill.” It is a skill that students will use in their everyday activities and careers.

According to New Haven Public Schools curriculum standards, students must relate operations to real word experiences and problem solving activities. Students must develop strategies for solving word problems and be able to write their own story problems. In the second grade curriculum, students are required to solve a word problem each day. This problem is solved through a guided class discussion. They are also asked to solve a similar problem for homework. These problems tend to be single or two-digit addition or subtraction problems.

My students for this unit are second graders. My classroom is composed of 23 students, 14 male students and 9 female students. We are a very diverse classroom made up of 14 Hispanic students, 6 African American students, and 3 White students. 8 of the 23 students are LTSS, Language Transition Support Services. They are LTSS students because they were in bi-lingual classrooms for thirty months. After the thirty months, the students are exited and forced into mainstream classrooms, whether they are proficient or not.

My students vary in levels from very low to very high. Almost half the class is below level in reading. Unfortunately, their reading affects all subject areas. I have noticed that the students that are below level in reading, tend to display difficulty in math because they can not read the problems. My LTSS students seem to be my students who score low in math. On the other hand, the other 15 students in the class seem to really enjoy and excel in math. The students are comfortable and have been doing Saxon Mathematics now for two years.

Saxon Mathematics is the math program that has been adopted at our school. The program is designed so that the students practice math during a morning meeting for twenty minutes and a math lesson for sixty minutes. During the meeting, students practice answering questions about attendance/lunch count, the calendar (days of the week and months of the year), finish patterns (shapes, numbers, etc.), tell time, count money, graph weather/temperature, and complete the given daily count (begin at 49 and count by 2’s). The students always seem to be actively engaged in this meeting. I try to make it unpredictable by asking different questions, switching the order, etc. Later in the afternoon, we continue with math. The lessons are scripted for the teachers and tell the teachers what materials are needed to do the lesson. After the lesson, we complete one side of two sheets, fact practice and guided practice. The fact practice gives students the opportunity to master the given facts. The guided practice has about eight or nine different problems on it. As a class, we discuss each problem. All of the problems are related to previous lessons, so that the students are practicing what they learned. At the top of the guided practice sheet is a word problem. For homework each night, students complete the other side of the work sheets, fact practice and guided practice.

Saxon is a very thorough program. The students are constantly assessed. Every five lessons, students take a fact assessment and written assessment (similar to the guided practice sheets). Students also take an oral assessment every ten lessons. A math portfolio is kept on each student. In the portfolio, I analyze the students strengths and weaknesses. I use this information in my focus groups.

In the beginning of the year, we focused on addition facts and fact families. A fact family is composed of two addition facts and two subtraction facts, for example, 4+5=9, 5+4=9, 9-5=4, and 9-4=5. At the time, my class seemed to grasp the concept of fact families, relating addition and subtraction facts. Now, we are focusing on two-digit subtraction facts. Some students are doing well, some are having difficulty. Sadly, I noticed that they have forgotten about the relationship between addition and subtraction when solving problems. They struggle to solve 14-7 because they forgot to relate the “doubles fact” 7+7=14. Instead of relating this, they try to solve the problem by counting with their fingers. This also carries over to two-digit subtraction problem solving. Students complete a problem like 95-33=62 but they cannot check their work or relate the addition fact 62+33=95.

Somewhere along the line, students lost the important lesson that addition and subtraction go “hand in hand.” Therefore, I hope that this unit will reinforce the concept that addition and subtraction relate. The principal objective of this unit is to have creative, critical thinkers who are able to solve word problems. Students will learn strategies, tools to solve word problems and check their work. I want students to not only “know how” but also “why.”

For many of the ideas about teaching problem solving I have relied on my reading of
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Teaching Problems and the Problems of Teaching
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by Magdalene Lampert. I have also used
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Knowing and Teaching Elementary Mathematics
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by Liping Ma and
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Children’s Mathematics
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by T. Carpenter. I found it enlightening that Liping Ma compares and contrasts Chinese teachers to American teachers. All of these books opened my eyes to new ideas.

In
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Knowing and Teaching Elementary Mathematics
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, Liping Ma discusses what is called the “knowledge package.” She suggests that there is not one piece that the teacher is presenting, rather a group of ideas that are related. In this unit I have composed a “knowledge package.” These are the ideas that I feel relate to my topic and my students. Keep in mind that some ideas may not relate to you and your students, make adjustments as necessary.

There are many ideas that should be discussed in this unit. These ideas scaffold. If a student is having trouble in one area, it is because the student is lacking in another area. The topics that I suggest are as follows:

Place Value

Simple Addition and Subtraction (within 10)

Fact Families

Addition and Subtraction (within 20)

Addition and subtraction without regrouping

Addition and Subtraction with regrouping

Strategies for solving problems

- What facts are given?
- What do I need to know?
- How do I solve it?
- Is my answer right? CHECK IT.