Patterns in Mathematics · Lesson 2 of 7
Seeing Number Sequences
“Use dots, grids, groups, and cubes to make the growth of number sequences visible.”
• Represent simple number sequences using groups of dots. • Explain the names triangular number, square number, and cube number. • Arrange 36 objects as both a triangle and a square. • Extend the chapter’s hexagonal-number pictures. • Visualise powers of 2 and 3 through repeated copying.
A picture shows how a number is built
A list tells us how many objects there are. A picture can show how those objects are organised and what changes when the number grows. For example, the number 9 can be a single row of nine dots, three rows of three, or an uneven group. Each arrangement has the same count but may reveal a different relationship.
When you draw a sequence, preserve the feature that explains its rule. If every next group adds a pair, make the pair easy to see. If the objects form a square, keep the rows and columns equal. A useful mathematical picture makes the counting reliable and the growth understandable.
For the all-1s sequence, draw one dot in every picture. For counting numbers, add one dot each time. Odd-number pictures can be made from pairs plus one extra dot, and even-number pictures entirely from pairs. Redraw these sequences and build the next picture: 6 dots for counting, 11 for odd numbers, and 12 for even numbers.
Problem
Show why 9 is odd and 10 is even using dots.
- 1.Arrange 9 dots into four pairs and one separate dot: 2 + 2 + 2 + 2 + 1 = 9.
- 2.Add one dot beside the separate dot. There are now five complete pairs: 2 + 2 + 2 + 2 + 2 = 10.
- 3.The unpaired dot distinguishes the odd arrangement from the even one.
Triangles, squares, and cubes
Some sequences take their names from the shapes that their objects can make. Build a triangle by adding rows of length 1, 2, 3, and so on. Build a square by using equal numbers of rows and columns. Build a cube by stacking equal square layers. These arrangements explain the names rather than merely illustrating them.
A number that can count dots arranged in successive rows of 1, 2, 3, and so on, forming a triangular arrangement.
A number obtained by multiplying a whole number by itself. In the displayed sequence, an n-by-n array has n rows with n dots in each row.
A number obtained by multiplying a whole number by itself three times. An n-by-n-by-n cube has n square layers, each containing n × n unit cubes.
The first triangular pictures contain 1; 1 + 2 = 3; 1 + 2 + 3 = 6; and 1 + 2 + 3 + 4 = 10 dots. The first square pictures contain 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, and 4 × 4 = 16 dots. A triangle grows by one new row; a square grows by extending both its height and width.
For cubes, imagine small blocks rather than flat dots. A 2-by-2 square layer contains 4 blocks; two such layers contain 8. A 3-by-3 layer contains 9 blocks; three layers contain 27. Count every block in the solid, including blocks hidden behind or inside it. Counting only the visible faces will not give the number of unit cubes.
Problem
A triangular arrangement has rows containing 1, 2, 3, 4, and 5 dots. How does the next picture grow?
- 1.Count the existing arrangement: 1 + 2 + 3 + 4 + 5 = 15.
- 2.Add a sixth row of 6 dots, keeping the triangular shape.
- 3.The new total is 15 + 6 = 21. The increasing row length explains the changing additions in the sequence.
Problem
How many unit cubes make a cube with 4 unit cubes along each edge?
- 1.One square layer contains 4 rows of 4 cubes: 4 × 4 = 16.
- 2.The solid has 4 such layers, so it contains 16 × 4 = 64 unit cubes.
- 3.This is the fourth displayed cube number: 1, 8, 27, 64.
One number can have several roles
A number does not belong to only one kind of arrangement. The same collection can be reorganised without adding or removing any objects. This is especially useful when we want to connect two sequences: one count may fit both descriptions.
Problem
Explain why 36 is both a triangular number and a square number.
- 1.A square with 6 rows of 6 dots has 6 × 6 = 36 dots.
- 2.A triangle with 8 rows has 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 dots. Pair the end rows: 1 + 8, 2 + 7, 3 + 6, and 4 + 5 each give 9.
- 3.The triangular total is 4 × 9 = 36. Both drawings therefore use exactly the same number of dots.
“Triangular” names the arrangement, not a test that the number itself has three digits or three factors. Also, a flat 3-by-3 square contains 9 dots, while a solid 3-by-3-by-3 cube contains 27 unit cubes.
Hexagonal growth around a centre
Begin with one centre dot. Surround it with a ring of 6 dots, then another ring of 12, then another of 18. The growing outline has six sides, so the chapter calls these hexagonal numbers. In this lesson, that name means exactly the displayed sequence 1, 7, 19, 37, and so on.
Count the additions separately from the running totals. The added rings are 6, 12, 18, 24, …, while the totals are 1, 7, 19, 37, 61, …. Each new ring is longer by 6 dots. To draw the next picture after 37, keep all existing dots and add the 24-dot outer ring; 37 + 24 = 61.
Pictures for doubling and tripling
Repeated copying makes powers visible. For powers of 2, take two copies of the preceding group each time. For powers of 3, take three copies. You can use dots, counters, or sketches of connected groups; what matters is that the count really doubles or triples.
Another picture for powers of 2 counts the corner points of familiar shapes: a point has 1, a segment has 2 endpoints, a square has 4 corners, and a cube has 8 corners. Two separate copies of the cube give 16 corner points. This is counting corners, whereas cube numbers count unit blocks in a solid; the two tasks count different things.
Extend the dot drawings for all 1s, counting, odd, even, triangular, and square numbers by one stage. Sketch a 6-by-6-by-6 cube and find its unit-block count. Then draw your own repeated-copying picture for powers of 3. In each case, describe what was added or copied.
Quiz
Which arrangement explains the triangular number 10?
How many dots are in a 5-by-5 square?
A cube has 3 layers, each with 3 rows of 3 blocks. How many blocks are there?
Which statement about 36 is correct?
What is added to the 37-dot hexagonal picture to reach the next picture?
Which action changes a 9-dot group into the next powers-of-3 group?
Practice Problems
- Draw the first five all-1s, counting, odd, and even groups. Extend each one stage and describe what changes.
- Draw triangular arrangements for 15 and 21. Show the new row that changes one into the other.
- Draw square arrays for 25 and 36. Explain why adding a single six-dot row to the 25-dot picture is insufficient.
- Show 36 as both an 8-row triangle and a 6-by-6 square. Check the counts without counting every dot separately.
- Explain how to count the unit cubes in a 5-by-5-by-5 cube. What about a 6-by-6-by-6 cube?
- Draw the hexagonal numbers 1, 7, 19, 37, and 61. Label the number of dots in each added ring.
- Create pictures for 1, 2, 4, 8, 16 and for 1, 3, 9, 27. Make the doubling or tripling visible.
- Choose another number, such as 12, and draw two different arrangements. Explain what each arrangement makes easy to notice.
Key Takeaways
• Pictures can show both a number’s count and the reason for its growth. • Triangular numbers count growing rows; squares count equal rows and columns; cubes count equal square layers. • The same number, such as 36, can belong to more than one sequence. • The hexagonal pictures grow by rings of 6, 12, 18, 24, … dots. • Doubling and tripling can be represented by making repeated copies of a group.