Linear Expressions with Tables and Chairs

Linear Expressions with Tables and Chairs

Author(s):

Kellie Gentry Section 18286, Nicole Weaver 18286, Samuel E. Zordak


Grade Level:

5th Grade
Timeframe:

6 days, 1 hour periods
Lesson Description or Explanation

Using a context of chairs around square tables, students will be exposed to three different linear patterns in this lesson. The patterns vary slightly from situation to situation, and the third situation allows students to determine a solution in multiple ways, in the end leading to an intuitive understanding of perimeter.
Present the following situation to students:
At Pal-a-Table, a new restaurant in town, there are 24 square tables. One chair is placed on each side of a table. How many customers can be seated at this restaurant?
Show an arrangement of one table with four chairs. If your room contains large square tables at which students work in groups, use them as a demonstration. If not, you can draw a picture on the chalkboard, or you can use pattern blocks or other transparent manipulatives on the overhead projector.


When all students understand how chairs are placed, ask, "If there were 24 tables in a room, how many chairs would be needed?" Depending on students’ understanding of multiplication, they may immediately realize that the answer is 24 × 4 = 96. If not, work with the class to complete a table, as follows:

Tables
Chairs
1
4
2
8
3
12
4
16
5
20

From this table, students should realize that the number of chairs is equal to four times the number of tables. Alternatively, they might recognize that each time a table is added, four chairs are added. If there are some students who use each approach, this is a good opportunity to reinforce the connection between multiplication and repeated addition. That is,
2 × 4
=
4 + 4
3 × 4
=
4 + 4 + 4
4 × 4
=
4 + 4 + 4 + 4
5 × 4
=
4 + 4 + 4 + 4 + 4
and so on.
Ask students to explain their observations. "What is the pattern? How can you find the number of chairs for any number of tables?" [Multiply the number of tables by 4. If there are 24 tables, for instance, the number of chairs is 96. If there are n tables, the number of chairs is 4n.]
After the original problem has been solved ("How many people can be seated at this restaurant?"), explain to students that the restaurant needs some additional help. Pose the following problem:
Pal-a-Table has a problem. For large groups, they must push some of the tables together to make a longer table. As before, they place one chair on each side of the table. How many tables would be needed for a group of 18 people?"
Again, show students an example of the situation. Explain that for just one square table, again four chairs are needed, but when two tables are pushed together, six chairs are needed.


Ask, "How many chairs would be needed if three square tables were pushed together? What about four tables? …five? How would you determine the number of chairs needed for any number of tables?"
At this point, divide students into groups. Allow them to explore various arrangements of tables and chairs using pattern blocks or the Chairs Around a Table applet. This online activity allows students to explore the various situations described in this lesson.

Chairs Around a Table Activity
As students work, they should keep an organized record of their data. Before they begin to explore, you may wish to discuss how a table or chart could be used to keep their data organized. Allow students to investigate the relationship between tables and chairs, and circulate as they explore. Make sure that every group is working toward finding a general relationship between the number of chairs and the number of tables.
After the exploration, conduct a class discussion to reveal the relationship between chairs and tables. Ask, "If you know the number of tables, how can you determine the number of chairs?" Allow students to suggest relationships, and keep a record of their suggestions on the overhead projector. Then, allow others to agree or disagree with the suggestions. Continue until the class reaches a consensus on which relationships are correct. (If students believe that a suggested relationship is correct, they should be able to show that it is true for the data they collected or provide an explanation of how they know it is true.)
As students suggest answers using words, convert their answers to algebraic expressions using variables. For instance, if a student says, "You find the number of chairs by multiplying the number of tables by 2, and then adding 2," then you might write either of the following on the overhead projector:
chairs = 2 × tables + 2
or
c = (2 × t) + 2
Although students in grades 3‑5 are not expected to understand symbolic manipulation, these examples will lay the conceptual foundation for understanding the use of variables.
Students may suggest several correct rules for determining the number of chairs. One student may realize that there are two chairs on the sides of each table, as well as an additional chair on each end, leading to the rule c = (2 × t) + 2. This is shown in the first diagram below. Another student may realize that there are three chairs around each end table, and there are two chairs on the sides of each middle table, leading to the rule c = 3 × 2 + 2 × (t - 2). This is shown in the second diagram below. Although students may not be able to verify that these relationships are algebraically equivalent, they should realize that both yield the same solutions.



Once students have discovered acceptable relationships, they should use them to determine the number of chairs when the number of tables is known, and vice versa. To ensure student understanding, ask the following two questions:
How many chairs would be needed for 24 tables?
[Two chairs are needed on the sides of each table, so that is 2 × 24 = 48 chairs. An additional two chairs are needed, one on each end, for a total of 48 + 2 = 50 chairs.]
How many tables would Pal-a-Table have to push together for a group of 18 people?
[When tables are pushed together, one person can sit on each end. That leaves 18 ‑ 2 = 16 people to be seated on the sides of the tables. Two chairs can be placed on either side of each table, and 16 ÷ 2 = 8. Therefore, 8 tables are needed for a group of 18 people.]
Finally, allow students to consider arrangements in which square tables are connected to form the outside border of a rectangle. Pose the following problem to students:
Customers at Pal-a-Table like that tables can be combined for larger groups, but they don’t like that tables are only arranged end-to-end to form a long chain. One patron suggests that tables should instead be arranged in a rectangular pattern with chairs placed around the outside.
Demonstrate some examples on the overhead projector or chalkboard. An arrangement showing 14 tables and 18 chairs is shown below.


Ask, "How many chairs would you need when tables are arranged in rectangular patterns like this?" Again, allow students time to explore various arrangements in their groups using pattern blocks or the Patch Tool.

As with the previous problem, students may discover various relationships that allow the number of chairs to be determined. If students are familiar with perimeter, then they may realize that the number of chairs is dependent on two variables, namely length and width. The arrangement above can be thought of as a rectangle with length 5 and width 4. The length contributes five chairs on two sides, and the width contributes four chairs on two sides, so the total number of chairs is 2(5) + 2(4) = 18. (Coincidentally, the perimeter of a 5 × 4 rectangle is 18 units.) In general, students should realize that the number of chairs for such an arrangement is equal to twice the sum of the length and width.
Alternatively, students may approach this situation like the previous one and realize that each of the four corner tables has two chairs next to it, but all other tables have only one chair. That gives a total of 4(2) + 10(1) = 18 chairs.
Similarly, students could realize that all tables have one chair next to them, but the four corner tables have an "extra" chair, giving a total of 14(1) + 4(1) = 18 chairs.
If the number of chairs in the length and width of the arrangement are represented by m and n, these three approaches lead to the following symbolic rules, respectively:
c = 2m + 2nc = 8 + 2(m - 2) + 2(n - 2)c = t + 4
Note that the last of these symbolic rules indicates that the number of tables (t) is four less than the number of chairs. But the number of tables is given by t = 2m + 2n ‑ 4, so the last expression could be represented by:
c = (2m + 2n ‑ 4) + 4
A whole-class discussion or individual group presentations are important after this exploration. Because there are various ways that students could represent the relationship between chairs and tables, it is important that all students are exposed to other students’ strategies.


Indiana Curricular Standards


5.2.1 Solve problems involving multiplication and division of any whole numbers.
5.3.1 Use a variable to represent an unknown number.
5.3.2 Write simple algebraic expressions in one or two variables and evaluate them by substitution.
5.7.1 Analyze problems by identifying relationships, telling relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.
5.7.3 Apply strategies and results from simpler problems to solve more complex problems.
5.7.9 Note the method of finding the solution and show a conceptual understanding of the method by solving similar problems.


ISTE Standards

· Creativity and innovation: Students demonstrate creative thinking, construct knowledge, and develop innovative products and processes using technology.
o Apply existing knowledge to generate new ideas, products, or processes.
o Create original works as a means of personal or group expression.
o Use models and simulation to explore complex systems and issues.
· Digital Citizenship: Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior.
o Advocate and practice safe, legal, and responsible use of information and technology.
o Exhibit a positive attitude toward using technology that supports collaboration, learning, and productivity.
o Demonstrate personal responsibility for life long learning.
o Exhibit leadership for digital citizenship.
· Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems, and operations.
o Understand and use technology systems.
o Select and use applications effectively and productively.
o Troubleshoot systems and applications.
o Transfer current knowledge to learning of new technologies.
Assessments
Formative/Summative

Require each group of students to present their findings to the class. Each group could create a poster to explain how they determined the relationship between the number of chairs and the number of tables.
Have students write a letter to the restaurant explaining what they learned.


Prior Knowledge

Curricular Knowledge or Skills: Students already have extensive knowledge of basic mathematical operations such as addition, subtraction, multiplication, and division and know the order of operations. Students have also been introduced to the concept of variables in which one substitutes in a number for a variable to come up with a solution. Students have practiced solving for a variable as well.
Technology Knowledge: Students have several years experience using a computer and the internet. All of them know how to move and click a mouse. Students have also had experience using excel and recording and analyzing data using the Excel software. They are also have solid skills using PowerPoint to create presentations.

Technology


Internet Resources: http://illuminations.nctm.org/ActivityDetail.aspx?id=144

Hardware: Desktop or Laptop Computers with a high speed internet connection

Software: Excel to record data (optional)

Procedure

Day One:
Mini-Lesson:
Present the following situation to students:
At Pal-a-Table, a new restaurant in town, there are 24 square tables. One chair is placed on each side of a table. How many customers can be seated at this restaurant?

Show an arrangement of one table with four chairs using pattern blocks and small base-10 cubes on a document camera. After the students understand how the tables and chairs are arranged, restate the question, “How many customers can be seated at this restaurant?”

Time for Exploration:

Allow students to work in groups of 4 or 5, and explain that all work must be shown. Give students time to work in groups to solve this problem.

Regroup the class:

“What is the pattern? How can you find the number of chairs for any number of tables?” Allow the groups more time to work together to solve the problem. The goal of this time to work is to come up with the algebraic sentence 4t = c, where t stands for tables, and c stands for chairs or some other

Math Congress:

Regroup the class back together again, and discuss the patterns and answers to the problem.

Day Two:

Mini-Lesson:

Present the problem:

Pal-a-Table has a problem. For large groups, they must push some of the tables together to make a longer table. As before, they place one chair on each side of the table. How many tables would be needed for a group of 18 people?

Again, the example of the situation will be shown using pattern blocks and base-10 blocks on a document camera. Explain to students that for just one square table, again four chairs are needed, but when two tables are pushed together, six chairs are needed. Ask the students, “How many chairs would be needed if three square tables were pushed together? What about four tables? …five? How would you determine the number of chairs needed for any number of tables?

Time for Exploration:

Divide the students into groups of 4 or 5 once again. Allow the students to explore various arrangements of tables and chairs using the Chairs Around a Table applet.

As students work they should keep an organized record in Excel of their data. Allow the students to investigate the relationship between tables and chairs, and circulate as they explore. Make sure that every group is working toward finding a general relationship between the number of chairs and the number of tables.

Math Congress:

After the exploration, conduct a class discussion to reveal the relationship between chairs and tables. Ask, “If you know the number of tables, how can you determine the number of chairs?” Allow students to suggest relationships and keep a record of their suggestions on a piece of paper displaying it to the class using a document camera. Continue until the class comes to a consensus on which relationships are correct. Evidence must be shown. Assist the students in converting the relationship into an algebraic expression using variables.

Day 3:

Mini-Lesson:

Begin by having the students solve on slates some of the acceptable relationships by inserting various numbers of chairs. Repeat with various numbers of tables.

Present the Problem:

Customers at Pal-a-Table like that tables can be combined for larger groups, but they don’t like that tables are only arranged end-to-end to form a long chain. One patron suggests that tables should instead be arranged in a rectangular pattern with chairs placed around the outside.

Again demonstrate some examples on the document camera using pattern blocks and small base-10 cubes.

Ask students, “How many chairs would you need when tables are arranged in rectangular patterns like this?”

Time for Exploration:

Allow students to work in groups of 4 or 5 with Chairs Around a Table as they did before. Also have students record their data in Excel.

Math Congress:

After the exploration, conduct a class discussion to reveal the relationship between chairs and tables. Ask, “If you know the number of tables, how can you determine the number of chairs?” Allow students to suggest relationships and keep a record of their suggestions on a piece of paper displaying it to the class using a document camera. Continue until the class comes to a consensus on which relationships are correct. Evidence must be shown. Assist the students in converting the relationship into an algebraic expression using variables.

Day 4:

Mini-Lesson:

Explain to students they will be presenting their suggestions of solving the problems to the class by creating a PowerPoint presentation. Explain that pictures need to be included to show details about the way they went about solving the problems.

Time for Exploration:

Give students time to prepare PowerPoint presentations.

Day 5:

Time for Exploration:

Give students more time to prepare PowerPoint presentations.

Day 6:

Present PowerPoint creations.





Differentiated Instruction

ESL
Students will be provided with written instructions in both English and their first language. ESL support from an ESL teacher will be provided to the students during one or two days of instruction.

Challenge/Extend

Allow students to work on other similar problems. For example, the tables could be hexagon, pentagon, triangular, or a circle with different diameters and widths for chairs.

Special Needs

Those with reading difficulties or visual problems will have access to WYNN. This program reads the text to the students, assists students in taking notes, provides a talking dictionary, and has an electronic highlighter. For those with motor problems, a trackball will be provided. The trackball has a stationary base with a mouse pointer that can be moved with a thumb, finger, palm, foot, or other body part.
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