Talking Like a Mathematician
Kim Dinh has students explain a math process in order to better understand linear equations and graphing.
Teacher: Kim Dinh
School: Health Sciences High and Middle College, San Diego, CA
Discipline: Mathematics (Algebra 1)
Lesson Topic: Word problems involving linear inequalities
Lesson Month: February
Number of Students: 35
Featured Lesson’s Student Goals:
- Content objectives – Use understanding of inequalities and equations to solve word problems involving linear inequalities
- Literacy/language objectives – Understand the term “systems of linear inequalities”; relate systems of linear equations and linear inequalities; use the BUCK strategy to solve word problems; explain thinking to a partner (think-pair-share during the word problems and whole-class discussion) and via writing (in the warm-up) using sentence starters
- Student engagement/interaction objectives – Discuss strategies and thinking while collaborating with peers on how to solve word problems
Common Core State Standards for Mathematics
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Model with mathematics.
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
This 10-day unit focused on systems of equations and inequalities. It came toward the start of the second half of the academic year following a unit on linear functions. The featured lesson was an introduction to systems of inequalities. Following this unit, students learned about quadratics.
Before the Video
Before this lesson, Mr. Dinh’s students learned how to graph and understand the features of linear functions. Throughout the school year, and in nearly every unit, they had been practicing the BUCK strategy so that they had a tool they could use to begin and finish word problems. They then proceeded to solve systems of linear equations by graphing, elimination, and substitution. Right before this lesson, students had learned how to graph inequalities, which was the warm-up addressed in the featured lesson.
During the Video
The focus of this lesson was to introduce students to how systems of linear inequalities and systems of equations relate to one another. Mr. Dinh gave students a set of five warm-up problems similar to ones they had solved the previous day. The goal was for students to refresh their understanding of linear inequalities. For the fifth problem, students were asked to verbalize their process for solving the problem.
Next, Mr. Dinh used guided instruction as he began the core lesson of looking at word problems focusing on inequalities. Mr. Dinh wrote word problems that fit a real-world situation—his upcoming wedding—to engage students. The first problems focused on single linear inequalities. Mr. Dinh encouraged students to use the BUCK strategy to identify the information they needed to solve the problems and to ignore any unnecessary information). After students completed the problems, they reported out to the class. Mr. Dinh put a graph on the board and had students plot points that fit the inequality and look for a trend.
Students worked through the problems. The complexity increased when two problems were combined to become a system of linear inequalities. Students began initial work on the pair of problems, which continued into the next day.
After the Video
After this lesson, Mr. Dinh had students finish the pairs of problems and plot the graphs. He also went over any questions students had about this introduction. Afterward, he modeled examples of how to graph systems of linear inequalities and had students practice. Within a couple of days, students could do more word problems on the topic, with less guidance. The ultimate goal was for students to be able to relate the situations of the word problems as linear inequalities, graph them, and solve the problems individually.
In order to prepare for this lesson, Mr. Dinh thought about an engaging, real-life example of systems of inequalities that could be graphed simply and quickly while not confusing students too much. He graphed and solved these linear inequalities in advance to make sure everything worked out well. He also came up with sentence starters for the warm-up question involving the description of how to graph a linear inequality. Finally, he planned questions to ask students throughout the lesson to engage and assess their thinking.
Students needed to know how to graph linear functions and linear inequalities, solve equations, understand slope, understand the y-intercept, have some understanding of standard form, and understand that the intersection of two functions is a solution.
Students had been practicing think-pair-share, group discussions, and the BUCK strategy throughout the school year; these skills were necessary for the flow of the lesson.
Mr. Dinh answered students’ questions and clarified their misunderstandings individually or as a class, as needed. As he polled students about their work or walked around and saw what they did or did not understand, he determined how to meet their individual needs. Mr. Dinh also had students explain content to each other individually or to the class. He believed that having students identify possible solutions for the word problems allowed them to solve them in a variety of ways, thus giving them different ways to approach a problem depending on their learning style.
As they worked on the word problems, students engaged in think-pair-share with their partners and also discussed the problems together as a whole class. This gave them an opportunity to be heard by their classmates and to listen to others in a productive fashion.
Resources and Tools
- Linear Inequalities Warm-Up handout
- Linear Inequalities Word Problems handout
- Sentence starters for Warm-Up #5
- Document camera
During the warm-up, Mr. Dinh assessed students as they explained the solutions to the problems on the whiteboard. Then, as students worked on the word problems with their partners, Mr. Dinh walked around, answering questions and clarifying misunderstandings. Through observing and assessing students' understanding, he could decide whether to extend the lesson or allow students to move on.
When students worked in groups and reviewed answers with the whole class, they had the opportunity to check whether their own answers were correct and thus self-assess their comprehension of the concepts.
Mr. Dinh also conducted several informal polls with students (e.g., five-finger poll, thumbs up/thumbs down) after the warm-up and each set of word problems about their understanding of the concepts. This gave students an opportunity to gauge how well they were doing and to pose any questions they still had about the concepts.
The summative assessment for the featured lesson consisted of the practice on word problems and systems of linear inequalities that students worked on individually. The summative assessment for all of the units was a group competency in which students collaborated with one another on challenging problems (involving a number of word problems) as well as an individual competency focused on the skills of the unit.
Impact of Assessment
Mr. Dinh shaped the next day's lesson according to his assessment of students' understanding of linear inequalities from their work on the word problems. Because he could see that some students were still struggling with the concepts, he continued to use guided instruction to help them work through similar problems during the next lesson. He continued to assess students as they worked on the problems to determine when they could work more independently and also shift to writing their own problems.