visit this web-site In Susan Weinberg’s *Teaching Children Mathematics *article, *Links to Literature:* *Going Beyond Ten* *Black Dots*, she describes second graders’ responses to finding the sum of the numbers 1 through 10 so that they can make their own *Ten Black Dots* book. I use this same context to challenge older students (anywhere from Algebra 1 to Pre-Calculus) to find the sum of the first *n* whole numbers.

### Warm Up

Open the lesson introducing the book and have students “warm up” by doing the same activity as the second graders – how many dots would you need to make your own *Ten Black Dots *book? We assume that each page has that number of dots: page 1 has 1 dot, page 2 has 2 dots, page 3 has 3 dots, and so on.

### Emphasize the need to justify with a representation

Then challenge your students to represent their ideas – how can we *show* this?

The problem I posed was:

How many dots must the second graders use to make their own

can you buy viagra in pattaya Ten Black Dotsbook? What if you wanted to make the bookOne Hundred Black Dots, how many dots would you need?Justify your answer using words

ANDa picture or diagram.

Now this was tough for my students at first, but I gave them the two-colored counters (sometimes referred to as integer chips) to give them something tangible to play around with.

### Extending the Pattern

Finally challenge your students to figure out how many dots would be needed if they had *n* black dots. Does their representation help them to make sense of this generalization (forumla)? How can they see the formula in the representation?

### Connect to the Standards for Math Practice 7 & 8

In this activity students are both "looking for and making use of structure" (MP 7) and "looking for and expressing regularity in repeated reasoning" (MP 8). To achieve MP 7, students are looking at the structure of the problem - what are they adding and how does the structure of the numbers help them find a pattern to solve the problem? To achieve MP 8, students are using what they did in one situation and applying that logic/reasoning to a new situation (from 10 dots to 100 dots to *n* d0ts).