Cumulative tales and rhymes such as The Napping House illustrate growing patterns, typically an increase or decrease by one on each page. As the growing pattern is revealed through the story, children get excited because they can figure out “what comes next.” Suddenly a nice pattern book for kindergarten or preschool becomes an adventure. As we point out in our book Big Ideas of Early Mathematics, children are naturally “tuned in” to patterns, which can deepen their understanding of them.

Like all patterns, growing patterns feature repetition and regularity. In this case, it’s not a sequence of elements that repeats such as in the AB pattern of a checkerboard, but a quantitative change that repeats. Our counting system is a basic growing pattern because each number is one more that the number that precedes it. The plus-one rule is what brings predictability to our count sequence.

Because children will be so engaged by these stories and songs, you can guide them to notice the growing pattern, using geometric models such as stacking blocks or pictographs to show how each repetition has a constant change in quantity. For example, if you stack block towers in order of size, children will see the resulting stair-step structure of a plus-one growing pattern. Acting outing out the stories and songs is another way to make the pattern structure come to life!

Here are a few of our favorite mathy books and songs that illustrate all the math going on in growing patterns.

1. We Had a Picnic This Sunday Past by Jacqueline Woodson

   This tale vibrantly captures an annual family picnic. As guests arrive with food—and Teeka and Grandma dread the arrival of Cousin Martha’s dried out apple pie—a growing story emerges! There are two sets: people at the party and food on the table.

   With preschool children, a reader can pause and comment on how there are more people and more food! They may also be interested in sorting the different types of food into binary sorts by attributes such as “food I like” vs. “food I don’t like” or “dessert” vs. “not dessert.”

   Kindergarten and first grade children may wish to keep track of the quantity of each set and discover that the change does not always follow a growing pattern – especially when Cousin Trevor comes empty handed, and Teeka complains, “you can’t eat air!” The delicious surprises in this book are a result of disrupting the pattern and are what make it such a rewarding read. After enjoying this fun read-aloud many times, children can write their own growing stories about their families’ traditional gatherings.

2. The Waterhole by Graeme Base

   In this gorgeously illustrated tale of animals convening to drink from a watering hole, author and illustrator Graeme Base takes a counting book and does something magical with it. As each animal comes to drink from the watering hole one by one, the water in it diminishes each time. It is a demonstration of a growing pattern as well as an example of cause and effect, a decreasing or shrinking pattern paired with the growing pattern, and an exploration of volume all in one unique story. What a great way to spark a conversation with children.

3. Two of Everything by Lilly Hong

   Not all growing patterns use the same plus-one rule for growing. For children ages 6-8, introducing more advanced growing patterns through books can help provide a context for understanding multiplicative change. Two of Everything is a great Chinese folktale that introduces a doubling pattern with wit and wisdom. You can pair this book with activities that involve input and output such as function machines or in/out boxes. Whatever goes in comes out double!

4. One Grain of Rice by Demi

   Another doubling pattern can be found in this Indian folktale, in which a clever girl, Rani, outsmarts a greedy ruler who is hoarding the people’s rice harvest for himself. Despite her hunger, Rani returns some spilled rice to the raja instead of keeping it and is rewarded for her honesty. What payment does she ask for? One grain of rice, but with the condition that the grains of rice will double each day for 30 days. Thinking this a meager amount, the raja agrees and is surprised to see that the reward doubles to more than a billion grains of rice in 30 days! The beautiful illustrations capture the growing pattern in a powerful way as the volume of rice given to Rani takes up more and more of the page, eventually taking a fold-out, 4-page spread to show it all.

   A chart at the back of the book shows the number of grains of rice given to Rani on each of the 30 days. First and second graders learning about the units of our base-10 system can practice reading these large numbers. Make a photocopy of the chart, cut it apart, and have children put the 30 days back in order according to the doubling pattern.

5. Songs for Growing or Decreasing Patterns

   The classic song Over in the Meadow uses fun animal sounds and a catchy tune to make its growing pattern. It creates a great opportunity to allow children to illustrate what they hear happening in the song and the change that happens with each new verse.

   Ten in the Bed is another song that has a decreasing pattern that children love to sing and act out. Imagine an activity that gets kids moving while visually demonstrating how there is one less in the bed while also creating an increasing set of people on the floor along the way. It’s a familiar and fun song that can spark great dialogue around the concepts of constant change.