Patterns and Functions: What Comes Next?
Content Guide - Andee Rubin
Supplies Needed for Workshop #1:
pencils, paper, scissors, rulers, calculators, tape, a variety of colored markers
About the Workshop
What is the theme of the workshop?
In a sense, doing mathematics requires looking for patterns everywhere and expecting to find them - believing that the world is often predictable. This workshop explores how the idea of predictability forms the basis of mathematics, and introduces some mathematical activities in which patterns are central.
Whom do we see? What happens in the videoclips?
We'll see students from pre-kindgergarten through middle school working with patterns, and we'll consider what kinds of pattern-related activities support their development - activities ranging from simple-patterned songs and stories to generalized functions represented by mathematical expressions and graphs.
What issues does this workshop address?
One issue we'll consider is how to make activities accessible to students at different points in their mathematical development. We'll do several activities together that we hope you will find challenging, and we'll think about how to modify these activities for younger and older students.
What teaching strategy does this workshop offer?
We'll discuss the role of group work in math class – many of the students we'll watch will be working in groups, and the teacher will be playing the role of active observer and questioner.
To which NCTM Standards does this workshop relate?
tandard emphasizes a variety of ways students can interact with patterns: "recognize, describe, extend, and create a wide variety of patterns." In grades 5-8, the corresponding standard is Standard 8: Patterns and Functions, which recommends that students, in addition, be able to use tables, graphs, and rules to represent functional relationships. In this workshop, we see examples of all of the above ways of working with patterns, and we see how students progress from pictorial representations to more algebraic forms of functions.
In addition, this workshop is related to the general standards: Mathematics as Problem Solving and Mathematics as Communication. The students are solving problems and communicating about them verbally, in pictures, in graphs, and even in song.
Suggested Classroom Activities
Guess My Rule
You may structure this activity as presented in the Workshop: have one student (or a group of students) make up a rule and give a three-number set that follows the rule, and have other students in class try to guess the rule by suggesting other three-number sets that they think follow the rule. Each guess must be accompanied by a reason. Other ways to play the game include:
- Pair students (or groups) up and have them alternate making up and guessing a rule.
- For younger students, the rule can be simpler -- just taking one number and giving back a different one. For example, what is the rule if I give you two and you give me five; then, I give you three and you give me seven, etc. You can act this out, with a student being the function machine that changes any number to another based on a rule.
- For older students, rules can be quite complex, involving more than one operation and including operations such as squaring, dividing, etc.
This activity provides a strong basis for the concept of functions, as it captures one of the most important characteristics of a function: that it is defined by a rule that will always give one — but only one — answer for every number you put into it.
Patterns in Poems and Songs
Challenge students to come up with songs that have the same kind of pattern as a selected song. Examples from the program include: rounds (e.g., "Row, Row, Row Your Boat"), repeat with variation (e.g., "Hokey Pokey"), and growing (e.g, "There Was an Old Lady Who Swallowed a Fly"). You might also try structures such as verse and chorus (e.g., "Puff the Magic Dragon"), or shrinking (like "B-I-N-G-O"). Look for similar patterns in children's stories.
Patterns in poems usually have to do with the rhyme scheme. Choose a poem and have students find others with the same rhyme scheme. Some verses of songs count as poems, too. See if all of the verses of a song follow the same rhyming pattern.
These activities are accessible to younger students and introduce them to the idea of predictability in an entertaining way.
Students make up a code that matches numbers to letters or letters to other letters. They write messages in code to other students, who must decode them in order to read them. The decoder might be told what the code is, or for a more advanced activity, might have to figure it out. For example, a code might be: a=c, b=d, c=e, . . . ,y=a, z=b. Some students may be able to figure out the code based on common letter patterns; others might need some clues.
These codes are actually functions - the rule specifies how a letter is transformed to another. The rule works in the same way for all letters, just as functions work in the same way for all numbers.
Figurate numbers are those that can be described geometrically. Square numbers are 1, 4, 9, 16, etc. Triangular numbers are 1, 3, 6, 10, etc. "L" numbers are 1, 3, 5, 7, etc. Finding a pattern for square numbers, for example, could be noticing that the differences between them are all the odd numbers (3, 5, 7, 9, etc.)
These kinds of patterns are especially interesting because they can be looked at both numerically and geometrically; different students are likely to approach the problem in different ways. There are many other problems of this general type, such as the Tiles problem from the Workshop.
The Guess My Ruleactivity puts students in both the "expert" role and the "guesser" role. This is an important shift away from students always trying to answer the teacher's questions. Try this strategy for some math activity - either the Guess My Ruleactivity or some other —and watch what students can learn in each role.
- Bonnie Edwards connects patterns with addition and multiplication when she asks a student how many balloons it would take if the pattern repeated twice. How might you use patterns to work on other topics such as multiples, factors, even and odd numbers, measurement, fractions, etc.?
- Many of the students in Lilia Olivas's class had trouble with the valentine exchange problem. How might you have made the problem simpler and/or given students some help with it? What have you done when a problem that you have posed seems too difficult for most of your students?
- Both Bonnie and Lilia have their students work in groups. Is this an appropriate arrangement for these particular problems? In general, what kinds of problems work well in groups? How do you bring closure to a problem that groups have been working on separately?
Pre-Workshop Assignment for Workshop #2
Please administer the "really quick" survey to your class. Copy enough for your class (e.g., seven if you have a class of 28) and cut them into individual forms.
Explain to your class that they are participating in a survey of students from around the country as a part of a workshop you will be attending. If you wish, you can explain that no one will know what they answer (it's anonymous), and that the purpose of the survey is to help the teachers in the workshop learn about different ways to display data.
Pass out the forms, asking students to fill them in. Collect the forms. This should take less than five minutes.
Important: If students should ask whether they can mark more than one box per question ("What if we like both dogs and cats?") or none ("I only like soccer. What if I hate both basketball and football?"), DO NOT answer the question directly. Instead, PLEASE say simply, "Follow the instructions on the survey as best as you can." If necessary, explain that your directions indicated that you can only tell them to follow the instructions on the form. In a pinch, show them these instructions!
Afterwards, you may, if you wish, share the data with the class and have them try to figure out a good way to display it.
If you have students who are non-readers or ESL students, you may adapt this questionnaire any way you think will give us the same data. The easiest way is to read it aloud or to make a new questionnaire with pictures.
If you are an administrator or do not have direct access to a classroom, we encourage you to submit your own data, or poll some students on your own.
Mathematics: What's the Big Idea?