Grade 5 Unit 5 Patterns and Relationships

In preparation for Quarter 2, Grade 5 teachers are beginning to plan instruction for Unit 5, Patterns and Relationships. Some of the key understandings for the units include students using lists, tables, and diagrams to find relationships between sets of data, writing rules that describe the relationships as they relate to input and output, extending patterns to solve problems, and describing the processes used to find the solutions.

The performance indication used to evaluate student mastery is as follows:

 

Create and extend a table of related numbers (e.g., sides on one pentagon, wheels on one bus, etc.). Write a rule with one unknown to represent the related number pairs in the table. Use the rule to find the output value when given the input value and find the input value when given the output value [e.g., (65 buses, ? wheels), (? pentagons, 100 sides), etc.]. Write a verbal description of the rule and how the rule works to predict other related number pairs.

 

Sample Performance Indicator:

• Samuel was creating a pentagon design using toothpicks as shown in the figure below.

He added more toothpicks to the figure as shown below.

He continues this sequence by adding toothpicks to the figure so that each pentagon added shares one side with an existing pentagon to form a line of pentagons. Create a table to show the relationship between the number of pentagons and the number of toothpicks up to 5 pentagons. Record a written description of the pattern in the output column (the number of toothpicks used) and why this pattern continues. Use the pattern to predict the number of toothpicks Samuel will use to create a figure with 8 pentagons.

 

Source: 2013 TCMPC from http://teksresourcesystem.net/

 

The standards for this unit include: 5.5A , 5.6 , 5.14A , 5.14C , 5.15A , 5.15B , 5.16B
and ELPS ELPS.c.1C , ELPS.c.4I , ELPS.c.5F , ELPS.c.5G .

 

The sample problem provided proves to be very difficult for students when it comes to finding the rule. The students can solve the problem pretty easily by creating the formations using the toothpicks and/or sketching their results and then recording the information in the table. They almost immediately recognize that the number of toothpicks increase by 4 each time a pentagon is added. They are able to extend the information in the table to find the number of toothpicks required to create 8 pentagons. But, they struggle to "see" the rule which defines the relationship between the number of pentagons and the number of toothpicks.

 

The rule for the data in this particular problem is not as clear cut as they were in Grades 3 and 4. The best way to prepare the students to "see" the pattern is to create experiences in which they create the models described in the problems, sketch the diagrams, and look for repeated groups within the diagrams.

 

Look at the work shown below:

 

 

 

Using different colored toothpicks helps to highlight how the toothpicks are added in groups of 4. Each time a pentagon is added, the students are actually only adding 4 new toothpicks since it shares one toothpick with the first pentagon. Only that first pentagon was made with 5 toothpicks which we can redefine as a group of 4 toothpicks plus 1 toothpick that will be shared with the second pentagon. Each successive pentagon after that shares one of the four added toothpicks, so there is only 1 extra toothpick from the first pentagon. The rule for the set of related data is: 4x + 1, or the number of toothpicks times 4 plus 1 equals the number of toothpicks.

 

In order to prepare students to be successful on this task independently, students should be given opportunities to solve similar problems using different shapes using the colored toothpicks. The students should also be expected to seek and explain the resulting patterns that they see. Participation in a lot of math talk is required to help the students "see" the emerging patterns.