Patterns with Triangular and Square Numbers- Free Lesson

It was the Greeks who originally discovered the patterns of triangular and square numbers by rearranging pebbles into the geometric shapes, and you can too- but we think blueberries may be more delicious than pebbles.

The students will create square numbers by adding together two triangular numbers by using small fruit as manipulatives.

At least 25 Blueberries per student
(Olives, Grapes, Cranberries, Cherries etc. can be substituted.)

It was the Greeks who originally discovered the patterns of triangular and square numbers by rearranging pebbles into the geometric shapes.

Begin with creating triangular numbers. For this activity the consecutive triangular numbers we will use will be 1-10. (1,3,6,10…) Triangular numbers grow as the bottom row increases by one, and so on by consecutive counting sequence.
To create the triangular numbers 1-10, start with one blueberry. One blueberry is the triangular number of One. Add two blueberries to the existing blueberry to create a pyramid shape. This is now the triangular number of Three. Continue to add another row to the triangle, to create the triangular number Six. Again, add another row of blueberries and create the triangular number Ten. After adding each row to the triangle allow student to count the number of blueberries in their triangle.

To create square numbers have an equal number of blueberries in rows going down and across. The consecutive square numbers to use for this activity are 1-16. (1,4,9,16…) Point out the pattern to students, that there are equal amounts of blueberries in each row of the square numbers.

The relationship between triangular and square numbers is that by adding together any two consecutive triangular numbers, you will get a square number. Using the blueberries as manipulatives, have students add together their triangular numbers of 1+3=4. The product is the square number of 4.

Repeat this step, but adding together two more numbers of the triangular sequence to manipulate: 3+6=9, 6+10=16.
For assessment, have students manipulate the next triangular number in the sequence following ten. (15) Do the same with the next square number, have students show the next number in sequence coming after 16. (25) They can derive the answer by adding together the triangular numbers, or by continuing the pattern of square numbers.

It was the Greeks who originally discovered the patterns of triangular and square numbers by rearranging pebbles into the geometric shapes, and you can too- but we think blueberries may be more delicious than pebbles.

Discuss the pattern students observed that create triangular and square numbers. Allow students to snack the blueberries, (or whichever fruit you have chosen) and talk about the benefits of this fruit to their health.