This activity challenges students to draw on their work with patterns and their explorations in Consecutive Sums to identify general rules for two important patterns.

Question 1 involves consecutive sums starting with 1. Students investigate the relationship between the height of a stack of squares and the number of squares in the stack. The stack of squares is arranged in a triangular pattern, with each row of squares one unit longer than the one above. A 1-high stack contains 1 square, a 2-high stack contains 1 + 2 = 3 squares, and a 3-high stack contains 1 + 2 + 3 = 6 squares. The numbers 1, 3, 6, and so on are called the triangular numbers.In general, an n-high stack will contain 1 + 2 + 3 + ... + n squares.

Question 2 involves an analogous idea for consecutive products starting with 1. It poses a combinatorial question: How many ways are there to arrange n scoops of ice cream on a cone? There is 1 way to arrange 1 scoop and 2 ways to arrange 2 scoops. However, there are 6 ways to arrange 3 scoops. To make the problem easier to think about, imagine that each scoop is a different flavor. For 4 scoops, once the first flavor is chosen, we know there are 6 ways to arrange the rest, and with 4 ways to choose the first flavor, there are 4(6) = 4(3)(2) = 24 arrangements altogether. In general, there are n(n – 1)(n – 2)...(2)(1) = n! (read “n factorial”) ways to arrange n flavors.

Each question presents the first few rows of an In-Out table. Students are asked to predict the values in subsequent rows and then to generalize the patterns they used to make their predictions.

The activity is particularly appropriate for small-group exploration. The activity Diagonally Speaking follows a similar numeric approach and then challenges students to identify why the rule they discover must always hold.

20 minutes

Groups

Tell students that they will now explore two very important number patterns—patterns that they will see repeatedly, and in surprising places, in their future mathematics work.

In Question 1, students might see a vertical recursive pattern, in which a value in the second column is found from the previous value, and a “zigzag” addition pattern.

Encourage groups to write a general rule for the patterns they find, but allow that they do not necessarily have to be written with symbols alone. Encourage use of words and sentences as well. Some students may recognize that the number of squares in an n-high stack is equal to the sum of the numbers 1 to n. If so, you might remind them of summation notation, which was introduced during Consecutive Sums.

None of these patterns is optimal when searching for, say, row 40, or for row n. In these cases, a rule that relates the In value to the Out value is best.

For Question 2, students might notice an analogous zigzag pattern, in this case a multiplicative one. If groups focus on the recursive pattern and you decide to challenge them to identify a functional pattern, you might turn their attention to Question 2c. In their solutions, even if they begin with their answer for ten scoops, they will probably say something like, “Multiply this by 11, then by 12, then by 13, and so on, all the way up to 100.”

If students recognize some connections, you might remind them of factorial notation, mentioned briefly in 1-2-3-4 Puzzle. The notation for products is analogous to summation notation, using the uppercase Greek letter pi (∏) in place of sigma for sums. For example,

means 3⋅4⋅5⋅6⋅73⋅4⋅5⋅6⋅7 size 12{3 cdot 4 cdot 5 cdot 6 cdot 7} {}.

There is no need for a formal debriefing of this activity. You might invite posting of solutions or some class discussion if students wish to examine other groups’ work.

Depending on students’ interest, you might identify the common name for the set of numbers in the Out column of Question 1: the triangular numbers.

Also of note, the patterns in these In-Out tables would be much harder to find if the entries were not arranged sequentially. You might mention that while this is a good principle for analyzing information, the entries of an In-Out table do not, in general, have to be arranged in any particular order.

Is 1.5 an appropriate input for either of these tables?

What do you call the set of possible inputs for an In-Out table?

What other examples have you seen in which only certain inputs were allowed?

1.1.S.23 From One to N (extension) asks students to find a simple expression in terms of n that allows one to find a sum without repeated addition. If students find such an expression, they look for a proof that their answer is correct.