New Jersey Mathematics Curriculum Framework

## STANDARD 11 - PATTERNS, RELATIONSHIPS, AND FUNCTIONS

 All students will develop an understanding of patterns, relationships, and functions and will use them to represent and explain real-world phenomena.

## Standard 11 - Patterns, Relationships, and Functions - Grades K-2

### Overview

The development of pattern-based thinking, using patterns to analyze and solve problems, is an extremely powerful tool for doing mathematics, and leads in later grades to an appreciation of how functions are used to describe relationships. The key components of pattern-based thinking at the early grade levels, as identified in the K-12 Overview, are recognizing, constructing, and extending patterns, categorizing and classifying objects, and discovering rules.

"Looking for patterns trains the mind to search out and discover the similarities that bind seemingly unrelated information together in a whole. . . . A child who expects things to 'make sense' looks for the sense in things and from this sense develops understanding. A child who does not see patterns often does not expect things to make sense and sees all events as discrete, separate, and unrelated."

- Mary Baratta-Lorton (cited on p.112 of About Teaching Mathematics by Marilyn Burns)

Children in the primary grades develop an awareness of patterns in their environment. Those who are successful in mathematics expand this awareness into understanding and apply it to learning about the number system. Children who do not look for patterns as a means of understanding and learning mathematics often find mathematics to be quite difficult. Thus, it is critical in the early grades to establish an early predisposition to looking for patterns, creating patterns, and extending patterns.

Children should recognize, construct and extend patterns with pattern blocks, cubes, toothpicks, beans, buttons and other concrete objects. Children in kindergarten can recognize patterns in motion, color, designs, sound, rhythm, music, position, sizes, and quantities. They are very aware of sound and rhythm, and can clap out patterns that repeat, such as clapclapclappause, clapclapclappause, etc. They can sit in a circle and wear colored hats which make a pattern, such as redwhiteblue, redwhiteblue. One child can walk around the circle and tap successive children in an armshoulderhead pattern. The teacher may ask the class who the next person to be tapped on the head would be if the pattern were to be continued. In addition to repeating patterns, students should have experiences with expanding patterns. They can indicate such a pattern by using motion: skipjumpturn around, skipjumpjumpturn around, skipjumpjumpjumpturn around, and so on. Songs are excellent examples of repetition of melody or of words, as well as of rhythmic patterns. Children's literature abounds with stories which rely on rhythm, rhyming, repetition and sequencing. As students move on to first and second grade, they should start to create their own patterns and develop pictorial and symbolic representations of those patterns. The transition will be from working with patterns using physical objects to using pictures, letters, and geometric figures in two and three dimensions, and then to using symbols, such as words and numbers, to represent patterns.

Categorization and classification are also important skills for students in the primary grades. Kindergartners should have numerous opportunities to sort, classify, describe, and order collections of many different types of objects. For example, students might be asked to sort attribute shapes, buttons, or boxes into two groups and explain why they sorted them as they did. This area offers an excellent opportunity for students to verbalize their thought processes and to integrate learning in mathematics and science as they sort natural objects such as shells, rocks, or leaves.

Discovering a rule and inputoutput games are two other settings in which primary children can enhance their work and their skills with patterns. The children might be asked to solve the mystery of the crackers as the teacher slowly and deliberately gives every boy two crackers and every girl four crackers one day during snack time. The inequity is addressed, of course, as soon as the children solve the mystery by discovering the rule that the teacher was using. On a different day, first graders can be told that they may request between 3 and 5 crackers for snack. But then each child is actually given two crackers less than his or her request. Again, as soon as the children verbalize the relationship between the request (input) and the portion allotted (output), they receive the missing crackers.

Establishing the habit of looking for patterns is exceedingly important in the primary grades. By studying patterns, young children develop necessary tools to become better learners of mathematics as well as better problem solvers. In addition, patterns help students to appreciate the beauty of mathematics and to make connections within mathematics and among mathematics and other subject areas.

## Standard 11 - Patterns, Relationships, and Functions - Grades K-2

### Indicators and Activities

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in kindergarten and in grades 1 and 2.

Experiences will be such that all students in grades K-2:

1. Reproduce, extend, create, and describe patterns and sequences using a variety of materials.

• Students make a collage with examples of patterns in nature.

• Students create visual patterns with objects, colors, or shapes using materials such as buttons, macaroni, pattern blocks, links, cubes, attrilinks or attribute blocks, toothpicks, beans, or teddy bear counters. They challenge other students to describe or extend their patterns.

• Students sort objects such as leaves, buttons, animal pictures, and blocks, using categories corresponding to characteristics like number of holes, number of sides, shapes, or thickness.

• One child walks around the outside of a circle and taps successive children in a headshouldershoulderhead pattern. The teacher asks who the next person to be tapped on the head would be if the pattern were to be continued. The children sing and act out the song, Head, shoulders, knees and toes.

• Students describe patterns made from circles, triangles, and squares, and select the next shape in the pattern.

• Students make patterns with letters and extend the sequence.

• As an assessment task, students use letters to translate patterns they have created with objects - for example, RRBRRB for a Unifix pattern of redredblueredredblue, or ABBCABBC for a shape pattern of square - circle - circle - triangle - square - circle -circle - triangle.

• Students connect the dots to make a picture by following a number sequence, such as 2, 4, 6, 8, ... .

• Students create one more and one less patterns.

• Students create patterns with the calculator. They enter any number such as 10, and then add 1 for 10+1= = = ... . The calculator will automatically repeat the function and display 11, 12, 13, 14, etc. Some calculators may need to have the pattern entered twice: 10+1=+1= = = ... . Other calculators will need 1++10= = = ... . Students may repeatedly add or subtract any number.

• Students name things that come in pairs (or 4s or 5s): eyes, ears, hands, arms, legs, mittens, shoes, bicycle wheels, etc. They work in pairs to find how many people there are if there are 20 eyes.

• Students count by 2, 5, or 10 using counters or creating color patterns with Unifix or Linker cubes; they repeat this using skip counting on a number line.

• Students use skip counting or calculators to find multiples of numbers and then color them on the hundreds chart. Linking cubes or Unifix cubes can be used to build towers or trains with every other cube or every third cube a certain color to illustrate, recognize, and practice skip counting patterns.

• Students write their first name repeatedly on a 10x10 grid, and then color the first letter of their name to create a pattern. They discuss the patterns formed.

• Students identify the same pattern in a variety of contexts. For example, blackwhite-blackwhite is like sitstandsitstand and ABAB and updownupdown and straightcurvestraightcurve.

• Students identify patterns on a calendar using pictures or numerals. For example, in November, even dates might be marked with a snowflake, and odd dates with a picture of a turkey. Or, they might mark each date with the day of the week.

• Students create a pattern using various rubber stamp blocks or picture designs.

• Students use or create patterns with geometric figures (circles, triangles, squares, pentagons, hexagons, etc.) and record how many of each shape exist after each repeating cluster.

• Students create a mosaic design (tessellation) made of different shapes using objects such as pattern blocks. They color congruent shapes of a mosaic design with the same color.

2. Use tables, rules, variables, open sentences, and graphs to describe patterns and other relationships.

• Students complete a table given several starting numbers and a verbal rule.

• Kindergartners look at Anno's Counting House by Mitsumasa Anno to see if they can figure out the pattern that is used in moving from one set of pages to the next. The people in this book move, one by one, from one house to another.

• Students describe the pattern illustrated by the numbers in a table by using words (e.g., one more than), and then the teacher helps them to represent it with symbols in an open sentence ( = + 1).

• Students use colored squares to make a graph showing the multiples of 3 and relate this to a table and an expression involving a variable, such as 3 x .

3. Use concrete and pictorial models to explore the basic concept of a function.

• Students study the pictures in Anno's Math Games II by Mitsumasa Anno. As they do, they try to figure out what happens to the objects as the elves put them into the magic machine. Sometimes the number of objects doubles, sometimes the objects grow eyes, and sometimes the objects turn into circles.

• Students put numbers into Max the Magic Math Machine and read what comes out. (The teacher acts as Max.) Then they describe what Max is doing to each number. The teacher pays careful attention to the students' responses to assess their levels of understanding.

• Students investigate a hole-making machine that puts 4 holes into buttons. They make a table that shows the number of buttons put into the machine and the total number of holes that must be made in them. Then they write a sentence that describes how the total number of holes changes as new buttons are added.

• Students play Guess my Rule. The teacher gives them a starting number and the result after using the rule. She continues giving examples until students discover the rule.

• Students count the number of pennies (or nickels) in 1 dime, 2 dimes, 3 dimes and record their results in chart form. They study the patterns and discuss the rules observed.

• Students consider the cost of two or three candies if one candy costs one dime. They make a chart using the information.

• Students count the number of lifesavers in an assorted pack. They make a table showing the number of each color and the total number in one pack. Then, assuming all of the packs are the same, they make a table showing the total number of each color for 2 packs, 3 packs, 4 packs, and so on. They check their results with packs of lifesavers, which in general, have the same number of each color.

4. Observe and explain how a change in one physical quantity can produce a corresponding change in another.

• Students discuss how ice changes to water as it warms. They talk about how it snows in January or February but rains in April or May.

• Students plant seeds and watch them grow. They write about what they see and measure the height of their plants as time passes. They discuss how changes in time bring about changes in the height of the plants. They also talk about how other factors might affect the plants, such as light and water.

5. Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts.

• Students go on a pattern hunt around the classroom and the school, discussing the patterns they find.

• Students sing and act out songs like "Rattlin' Bog" (Bird on the leaf, and the leaf on the tree, and the tree in the hole, and the hole in the ground, . . .) and "Old MacDonald Had a Farm."

• In reading, students recognize patterns in rhythm, in rhyming, in syllables and in sequencing. Stories such as Ten Black Dots by Donald Crews, Five Little Monkeys Jumping on a Bed by Eileen Christelow, Jump, Frog, Jump by Robert Kalan, The Little Red Hen, and Dr. Seuss books offer such opportunities. Visual patterns can be shown using picture representations for children's books such as 1 Hunter by Pat Hutchins, Rooster's Off to See the World by Eric Carle, The Patchwork Quilt by Valerie Flournoy, and The Keeping Quilt by Patricia Polacco.

• Students identify every third letter of the alphabet; every fourth letter, etc. They use those sets of letters to see what words they can make.

• Students choose a day. Using a calendar, they identify the name of the next day, of theprevious day, and also the name of the day two days (or more) before and after. They select a date, and give the date of the next day and of the previous day, the name of the month, of the next month, and of the previous month. They give the name of the date two days before and after, and three days (or more) before and after.

• Students graph daily weather patterns, showing sunny, cloudy, rainy or snowy days. Then they discuss monthly or seasonal patterns.

• In social studies, students identify traffic patterns such as how many cars, trucks, or buses pass the front of the school during five minutes at different times of the day. They keep records for five days, organizing the information in chart form.

• In art, students observe patterns in pictures, mosaics, tessellations, and Escherlike drawings, as well as in wallpaper, fabric, and floor tile designs.

6. Form and verify generalizations based on observations of patterns and relationships.

• Students draw pictures of faces and make a table that shows the number of faces and the number of eyes. The teacher writes a sentence on the board that the class composes, describing the patterns that they find.

• Students observe that there are 12 eggs in a carton of eggs. These are called a dozen. They explain how to find the number of eggs in 2 cartons, 3 cartons, and so on.

• Students write a sentence or more telling about the patterns they have observed in a particular activity. They may use pictures to describe or generalize what they have observed. For example, after students have colored multiples of a certain number on the hundreds chart, they write about the geometric pattern they observe on the chart.

### References

Anno, Mitsumasa. Anno's Counting House. New York: Philomel Books, 1982.

Anno, Mitsumasa. Anno's Math Games II. New York: Philomel Books, 1982.

Burns, Marilyn. About Teaching Mathematics: A K-8 Resource. Sausalito, CA: Math Solutions Publications, 1992.

Carle, Eric. Rooster's Off to See the World. New York: Simon & Schuster Books for Young Readers, 1972.

Christelow, Eileen. Five Little Monkeys Jumping on a Bed. New York: Clarion Books, 1989.

Crews, Donald. Ten Black Dots. New York: Greenwillow Books, 1986.

Flournoy, Valerie. The Patchwork Quilt. New York: Dial Books, 1985.

Hutchins, Pat. 1 Hunter. New York: Greenwillow Books, 1982.

Kalan, Robert. Jump, Frog, Jump. New York: Greenwillow Books, 1981.

Polacco, Patricia. The Keeping Quilt. New York: Simon and Schuster, 1988.

Seuss, Dr. Most Dr. Seuss books exhibit appropriate patterns.

The Little Red Hen. Many versions are available.

### On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.