Patterns in Mathematics · Lesson 3 of 7
Building Squares from Sums
“Explain several ways that simple additions build triangular numbers and perfect square arrays.”
• Use odd-number layers to explain square numbers. • Find large sums of successive odd numbers without adding every term separately. • Explain why counting up and back down also produces a square. • Connect sums of 1s and counting numbers to familiar sequences. • Combine consecutive triangular arrangements into a square.
Adding odd numbers reveals a new sequence
Start with 1 and add the next odd number each time. The totals are 1, 1 + 3 = 4, 1 + 3 + 5 = 9, and 1 + 3 + 5 + 7 = 16. These totals are square numbers. Calculating several cases helps us notice the pattern, but a picture can explain why it keeps working.
Draw one dot. To turn it into a 2-by-2 square, add an L-shaped group of 3 dots around two adjoining sides. To turn that square into a 3-by-3 square, add an L-shaped group of 5 dots. The old square stays inside, and the new dots extend both its width and its height.
When a square grows from side length 5 to side length 6, the new bottom row has 6 dots and the new right column has 6 dots. Their corner is the same dot, so the total added is 6 + 6 − 1 = 11. More generally, a layer that completes a square of side length n contains n + n − 1 dots, the nth odd number. Here n simply means the number of dots along the new square’s side.
The construction works for any chosen size: one dot begins the square, and each next odd-sized layer completes the next larger square. This is an explanation that extends beyond the small cases we drew. The number of odd terms added equals the number of dots along the completed square’s side.
Problem
Find 1 + 3 + 5 + 7 + 9 + 11, and explain the result.
- 1.There are six successive odd terms starting at 1.
- 2.The six layers fill a square with 6 rows and 6 columns.
- 3.The total is 6 × 6 = 36. The arrangement explains the total, rather than only checking the arithmetic.
Problem
Find the sum of the first 10 odd numbers and the sum of the first 100 odd numbers.
- 1.Ten successive odd layers fill a 10-by-10 square, so the first total is 10 × 10 = 100.
- 2.One hundred such layers fill a 100-by-100 square, so the second total is 100 × 100 = 10,000.
- 3.We do not need to draw all 10,000 dots. The same construction determines the count for either size.
Use the number of terms as the square’s side length. In 1 + 3 + 5 + 7 + 9, the last term is 9, but there are five terms, so the sum is 5 × 5 = 25. Also, the rule requires consecutive odd numbers beginning at 1; it does not apply unchanged to any collection of odd numbers.
Count up and back down
There is another route to square numbers: 1; 1 + 2 + 1; 1 + 2 + 3 + 2 + 1; and 1 + 2 + 3 + 4 + 3 + 2 + 1. The totals are again 1, 4, 9, and 16. This time the addends increase by 1 to a peak, then decrease by 1.
Take a square dot grid and turn the picture so it stands on a corner. Count along the horizontal diagonal rows. They grow from one dot up to the side length, then shrink to one dot. Turning the square has not changed the number of dots, so the up-and-down sum must equal the square’s original row-by-column count.
Problem
Find 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1.
- 1.The numbers rise to 100, then fall to 1. The peak 100 is used once.
- 2.Imagine the diagonal rows of a turned 100-by-100 dot square.
- 3.The total is 100 × 100 = 10,000, even though drawing the whole square would be inconvenient.
Other running totals
Adding terms one after another creates a new sequence of running totals. The first total uses one term, the second uses two, and the third uses three. The sequence of original terms and the sequence of running totals describe different quantities, so label them carefully.
| Original terms | Running totals | Resulting sequence |
|---|---|---|
| 1, 1, 1, 1, … | 1; 1 + 1; 1 + 1 + 1; … | 1, 2, 3, 4, …: counting numbers |
| 1, 2, 3, 4, … | 1; 1 + 2; 1 + 2 + 3; … | 1, 3, 6, 10, …: triangular numbers |
| 1, 3, 5, 7, … | 1; 1 + 3; 1 + 3 + 5; … | 1, 4, 9, 16, …: square numbers |
What if the all-1s sequence is added up and down? A peak position of 1 uses one 1; a peak position of 2 uses 1 + 1 + 1; a peak position of 3 uses five 1s. There are n positions going up and n − 1 coming down, making 2n − 1 ones in total. The results are 1, 3, 5, 7, …: odd numbers. The entries themselves remain 1 throughout; it is the number of entries used that changes.
Running totals of counting numbers are triangular because each new term becomes the next longer row of dots. A right-angled triangular arrangement gives the same count as the more balanced triangular picture: rows of 1, 2, 3, and 4 still contain 10 dots. The total depends on the row lengths, not on how far each row is shifted sideways.
Two triangular arrangements make a square
Now add pairs of neighbouring triangular numbers: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, and 10 + 15 = 25. These are square numbers starting at 4. To see why, split a square along a diagonal into two dot groups. Give the diagonal dots to one group so that no dot belongs to both.
In a 4-by-4 square, one group has rows of 1, 2, 3, and 4 dots, while the other has rows of 1, 2, and 3 dots. Their totals are 10 and 6. The larger triangle includes the diagonal; the smaller fills the remaining spaces. The same division works in a square of any size.
Problem
Find 15 + 21 using a triangular arrangement.
- 1.15 is the triangular count through row 5; 21 is the count through row 6.
- 2.Arrange the 21-dot triangle as one half of a 6-by-6 square, including its diagonal. The 15-dot triangle fills the other half without the diagonal.
- 3.Together they fill 6 × 6 = 36 positions, so 15 + 21 = 36.
Use counters or dots to make three explanations: odd layers filling a square, diagonal rows of a turned square, and two consecutive triangles filling a square. For each, say which objects are counted in each addend and why nothing is left out or counted twice.
Quiz
What is the sum of the first 8 odd numbers?
How many dots are added when a 5-by-5 square grows into a 6-by-6 square?
What is 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1?
What sequence comes from the running totals of 1, 2, 3, 4, …?
Adding the all-1s sequence up and down gives 1, 3, 5, … because:
Which pair of triangular numbers fills a 5-by-5 square?
Practice Problems
- Draw odd-number layers for a 5-by-5 square. Label the five layer counts and find their sum.
- Find the sum of the first 12 odd numbers. State the final odd term as well as the total.
- Without adding every term, find 1 + 2 + … + 19 + 20 + 19 + … + 2 + 1. Explain a picture that supports your answer.
- Write the first six running totals of the all-1s sequence and the first six up-and-down totals. Explain why the resulting sequences differ.
- Show 1 + 2 + 3 + 4 + 5 = 15 using both a balanced triangle and a right-angled triangle of dots.
- Draw two consecutive triangular arrangements with totals 21 and 28. Fit them into a square and find the total.
- A student writes 1 + 3 + 5 + 7 = 7². Explain the error and correct it.
- Compare the three ways of making the square number 36: odd layers, counting up and down, and two consecutive triangular arrangements.
Key Takeaways
• Successive odd numbers beginning at 1 form the layers of a growing square. • The sum of the first n odd numbers is n × n. • Counting from 1 up to n and down to 1, with one peak, also totals n × n. • Running totals of 1s give counting numbers; up-and-down totals of 1s give odd numbers. • Running totals of counting numbers give triangular numbers, and consecutive triangular numbers combine into squares.