Algebraic Thinking 5 th Grade Guided Instruction Finding Rules and Writing Equations For Patterns

Mr. Kelly is starting an after school cooking class. He wants students to learn how much fun it is to cook. The first week four students signed up for his class. He found a square table that fit four students nicely. Table 1 Arrangement

Table 1 Arrangement All the students loved Mr. Kelly’s cooking class so much that the word spread around the 5 th grade. The second week two more students signed up for his class. Mr. Kelly added another table to make room for two more students. Table 2 Arrangement

Table 1 Arrangement Wow, Mr. Kelly’s class was so much fun that more students signed up the next week. He added one more table and some more chairs. How many students joined the class in week three? How many students are in Mr. Kelly’s cooking class the third week? What do you notice about the number of tables and the number of chairs from week to week? Table 2 Arrangement Table 3 Arrangement

If this pattern continues, what will the Table Arrangement look like for week 4?Table 4 Arrangement How many chairs will fit around arrangement four? Table 4 Arrangement What do you notice about the ends of the table arrangements each week? Is there a connection between the Table Arrangement, the number of tables, and the week? Discuss this idea with your partner or group.

You got it! The weeks, the Table Arrangement Number, and the number of tables in the arrangement are all have the same number.

Wow, here is a challenge question for you. If the pattern continues……… How many chairs will Mr. Kelly need for his table the 9 th week? When I start to draw my picture like before, I run out of room. Your paper might be big enough but I need to come up with a new strategy.

I can make a table to organize my information. Now let’s fill in what we know about the pattern. Now let’s continue to fill in the chart using the pattern. What is happening in the pattern? Number of Tables Number of Chairs people can fit around table 9!

We can also organize my information on a coordinate grid. The first thing we have to do is label our scale. We will number the tables on the X axis. We will number the chairs on the Y axis Number of Tables Number of Chairs

Now, we will take each pair of numbers in our chart and use them as our ordered pairs to plot on our coordinate grid. Table Number Number of Chairs Let’s start with (1, 4). (2, 6)Now you finish the rest! Remember, we go across first and then up Number of Tables Number of Chairs 0

Number of Tables What happens to the pattern when we go down the column showing the number of tables? What happens to the pattern when we go down the column showing the number of chairs? Is there a pattern when we move from the Number of Tables column to the Number of Chairs column? How can we figure out the number of chairs needed if we know the number of tables? OR How can we figure out the number of tables if we know the number of chairs?

To make this connection, lets go back to our picture.Table 4 Arrangement Think back to all your Tables Arrangements. What is the same no matter how many tables there are? You got it!!!! There is one chair on each end. That is a total of two chairs. So guess what? We are going to take them away!

Table 4 Arrangement Now, what do you notice about the Table Arrangement, the number of tables, and the number of chairs that are left? To help you, let’s look at some other arrangements that have the two end chairs taken away. Table 1 Arrangement Table 2 Arrangement Ok, look carefully. What do you notice?

Table 4 Arrangement Table 1 Arrangement Table 2 Arrangement That’s right! The number of tables is the same as the Table Arrangement Number, which is also the same as the number of chairs on one side. But wait, there is a second set of chairs that match the Table Arrangement Number.

Table 4 Arrangement Table 1 Arrangement Table 2 Arrangement So you could say that there are two groups of chairs that have the same amount of chairs in each group as the Table Arrangement Number. One group of 4 Second group of 4 One group of 1 One group of 2 Second group of 1 Second group of 2

Table 4 Arrangement Table 1 Arrangement Table 2 Arrangement You could say that the number of chairs is 2 times the number of tables. Number of Chairs =2 x 2. Number of Chairs = 2 x 1 Number of chairs = 2 x 4. Number of chairs = 8. Number of Chairs = 2 Number of Chairs = 4 Number of Chairs = 2 x Number of tables

Can you find a time that this pattern does not work? Work with your partner or group to try to come up with an example that does not follow this pattern.

Did you find any? I didn’t! I tried and tried but I never did.

Ok, lets bring it all together. What do we know about the number of chairs in each Table Arrangement? 2.The Table Arrangement Number tells us the number of chairs we have on each side. There are two sides. So, we have two groups of chairs with the same number as the Table Arrangement Number. Table 3 Arrangement 1. Each arrangement has a chair at each end. There are two ends so there are two end-chairs for every arrangement.

So, if this patterns continues, we can use the information we discovered about the pattern to figure out the number of chairs we will have for any Table Arrangement Number. Ok, lets try one. How many chairs will I have at the 15 th Table Arrangement? Great, I have 15 chairs on each side and 2 end chairs = 32 Wow, that was easy. I will have 32 chairs at the 15 th table. 15 chairs 1 end chair = 32 OR (2 x 15) + 2 = 32

Hmm, I wonder how many chairs would fit around the 48 th Table Arrangement? Or the 100 th table? Gosh you would have to be real smart to figure out how many chairs would fit around the 500 th Table Arrangement. Can you find the answers to these problems? Use what we discovered about the pattern.

48 th table: 48 side chairs + 48 side chairs + 2 end chairs + = 98 total chairs 100 th table: 100 side chairs side chairs + 2 end chairs = 202 total chairs Wow, Mr. Kelly would have some big cooking classes if he had this many chairs. 500 th table: 500 side chairs side chairs + 2 end chairs = 1002 total chairs 48 chairs 1 end chair (2 x 48) + 2= 98 chairs (2 x 100) + 2 = 202chairs (2 x 500) + 2 = 1002 chairs

Hmm, what if I didn’t know the number of tables Mr. Kelly had but I knew the total number of students for the week? Lets’ think about this. Talk to your partner or group. Can you come up with a way? How could I find the number of tables he would need for his class of 14 students?

Ok, lets start with what we know. We know there will be a Table Arrangement. We also know that there are two end chairs. Since we know that the two chairs are on the ends, lets subtract them from the total number of chairs. 14 chairs – 2 chairs = 12 chairs We now have twelve chairs left. They must fit on the sides of the Table Arrangement in two equal groups. 2 x _____ = 12 or 12 ÷ 2 = 6 6 Oops, we need more room.

Since we know that the number of chairs on one side is = to the number of tables, we know there has to be six tables.

Hey, can you figure out these problems? How many tables will Mr. Kelly need for 52 students? How many tables will Mr. Kelly need for 102 students?

52 – 2 (two end chairs) = 50 chairs ½ of fifty is 25. So Mr. Kelly will need 25 tables for 52 students. 25 chairs (2 x 25) + 2 = 52

102 – 2 (the two end chairs) = 100 ½ of hundred is 50. So Mr. Kelly will need 50 tables for 102 students. 50 chairs (2 x 50) + 2 = 102

Mathematicians, like to write rules that describe the relationships in patterns. Since the number of tables in the arrangement is not always the same, we can use a variable to stand for the number of tables. Let’s say we use the variable n to stand for the number of tables in the arrangement. n = the number of tables in an arrangement We know that each side of the Table Arrangement has the same number of chairs as tables, therefore there are n chairs on one side.

Total number of chairs = (2 x n) + 2 This equation would work for any Table Arrangement Number. We looked at the pattern, found what happened in the pattern and then made an equation that will work for any Table Arrangement Number. Since the arrangement has two sides, that gives use 2 x n. Don’t forget to add on the 2 end chairs.

Try this equation for any other Table Arrangement Number. See if it works for your number. See if you can find any number that it doesn’t work for?

Try this process on another pattern. Can you find the rule? Can you use the information, find a pattern, and write an equation for the “nth” number?