Number patterns |

**Author**: Judy Scotchmoor

**Overview**: In this lesson, students are challenged to discover the relationship among six numbers. The objective of this activity is to engage students in a problem-solving situation in which they practice aspects of the process of science: observation, conversation, questioning, developing expectations/predictions, formulating explanations, testing their ideas; modifying their initial ideas, and sharing their results with others. Students are then asked to reflect on whether they were doing science. The activity can serve as an effective introduction to or reminder about the process of science, as well as provide an opportunity for students to reflect on the basic characteristics that help delimit the scientific enterprise.

**Lesson concepts**:

- The process of science involves observation, exploration, discovery, testing, communication, and application.
- Scientists try to come up with many different natural explanations (i.e., multiple hypotheses) for the patterns they observe.
- Scientists test their ideas using multiple lines of evidence.
- Test results sometimes cause scientists to revise their hypotheses.
- Scientists are creative and curious.
- Scientists work together and share their ideas.

**Grade span**: 6-12

**Materials**:

- Scratch 8.5 x 11" paper cut into sixths for displaying predictions
- A surface upon which to write the numbers — e.g., chalk board

**Time**: 15-20 minutes

**Grouping**: Small groups of 2-4 students and whole group discussion

**Teaching tips**: Only one example of a number pattern is given below (a, b, c, c-a, c-b, c-c). You may wish to begin with a much simpler challenge for younger students and then build up to this one, or you may wish to raise the level of difficulty! Great rainy day activity!

**Procedures**:

- Place 6 lines on the chalkboard and explain to students that you are going to fill in the first three blanks and their job is to fill in the last three, one at a time. There is a relationship among all six numbers. Their job is to figure out what that relationship is.
- Fill in the first three numbers as follows:
2 4 6 - Ask students to predict what the next number is. They should talk with members of their team and decide what it should be, then write the number on the scratch paper and hold it up for you to see. Most students will suggest an 8. Once all groups have a paper raised, reveal the next number as follows:
2 4 6 4 - After the groans have died down, ask the students: Based on what you see now, what do you think the 5th number will be? Proceed as above and when all groups have a paper raised, reveal the 5th number as follows:
2 4 6 4 2 - Do not worry if there is some frustration at this point. And maybe some students will have guessed right! Just continue to be positive, and ask the students: Based on what you see now, what do you think the last number will be? Proceed as above and when all groups have a paper raised, reveal the 6th number as follows:
2 4 6 4 2 0 - At this point, reassure the students that they will eventually figure this out and you will help them by giving them another set of three numbers. The same relationship will hold true. So just as before, you will give them the first three numbers and they are to figure out the 4th, then the 5th, then the 6th. You can go with any three numbers, but the following works well:
3 5 7 Followed by: 3 5 7 4 Followed by: 3 5 7 4 2 Followed by: 3 5 7 4 2 0 - For the 3rd round, you can go with any three numbers, but something like the following works well:
5 8 11 Followed by: 5 8 11 6 Followed by: 5 8 11 6 3 Followed by: 5 8 11 6 3 0 - Continue with any three numbers. As the rounds proceed, eventually a group or two will think they have the relationship—but don't let them tell the whole class. At that point, ask one of those groups how they could test their idea. This encourages students to think about how ideas are tested. Students may need help here, but you can prompt them: Thus far, I have been giving the first three numbers, what would happen if you give the first three numbers? How could that act as a test? Let them know that they can give you any three whole numbers, but not to make it too hard on you! Ask the group to make a prediction at this point: what do they expect to happen based on their idea? Proceed exactly as above, using their three numbers and let the entire class participate. If they were correct or incorrect, find out if any other group thinks they know the relationship, and let them test their idea with three numbers. Eventually as more groups "get it," ask a group to explain the relationship. Then ask another group to suggest three numbers that would provide a good test for that idea. And proceed as above.
- Eventually the relationship will be revealed and you can express it as follows:
a b c c-a c-b c-c - Have students reflect on what they were doing that scientists do. This could be prompted by the questions: "Were you doing science? What were you doing that was like what scientists do?" Discussion should reflect the concepts listed above.
If this activity is used as an introduction to the nature and process of science, then it would be helpful to use students' comments to initiate a list of what scientists do as they engage in scientific investigations. This list can then be referenced as they read about scientists and their work or as the students participate in future investigations. Their list can also be compared to those represented in the Science Flowchart.

An Understanding Science lesson

© 2010 The University of California Museum of Paleontology, Berkeley, and The Regents of the University of California