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Seeing Dots


Alignments to Content Standards: A-SSE.A.1 A-SSE.A.2

Task

Consider the algebraic expressions below:

$$ (n + 2)^2 - 4 \qquad \text{and} \qquad n^2 + 4n. $$

  1. Use the figures below to illustrate why the expressions are equivalent:

    Dots_975403f4ca152a8c9b57dc205eb44b92
  2. Find some ways to algebraically verify the same result.

IM Commentary

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students are asked to notice a pattern and connect the pattern to the algebraic representation. A potential source of scaffolding is to help students make this connection by labelling the "$n$-th" figure in the sequence. On the other hand, a potential source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Some students might show the equivalence algebraically from the start, either by expanding or by factoring. The algebraic approach should be rewarded, not discouraged. A student who expands could be asked if there is another algebraic method; a student who factors could be asked if there is a way of relating this form to the figure. Indeed, as an alternate solution to the one given in the solution, we observed that the factored form, $(n+4)n$, can be related to the figures as follows: If you take the top and bottom borders, turn them vertically, and place them next to the rest of the figure, you get an $(n+4) \times n$ rectangle of dots.

There is also an opportunity here to discuss the process of justifying an algebraic identity. For one, the algebraic solution in part (b) applies to all real numbers $n$, whereas the proof by pictures only directly applies to the case that $n$ is a positive integer (though students could be encouraged to replace "numbers of dots" with "areas of regions" to give a version of the geomtric proof that works for all positive real numbers.)

Solution

  1. A geometric approach to the problem proceeds by identifying, somewhere in the $n$-th figure, the value $n$, and seeing two ways of looking at the dots, giving both $(n+2)^2 - 4$ and $n^2 + 4n$. One such approach (among many) is below.

    Let $n$ be the number of the figure, with $n=1$ at the left. We count the dots in each figure in terms of $n$ in two different ways. One represents $n^2 + 4n$ and the other represents $(n+2)^2 - 4$.

    Visualizing $n^2 + 4n$:

    • $n^2$ is the inside full square.
    • $4n$ is the four outside borders with $n$ in each.

    Visualizing $(n+2)^2 - 4$:

    • Imagine the larger square with the four additional dots filled in at the corners. Then $(n+2)^2$ is the number of dots in the larger square.
    • 4 is the number of dots added.

  2. Perhaps most directly, we have $$ (n+2)^2-4=(n^2+4n+4)-4=n^2+4n. $$ Alternatiely, reversing the steps in this series of equalities is precisely the process of completing the square for the expression $n^2+4n$. Similarly, the left-hand side could be viewed as a difference of two squares, in which case we can reason: $$ (n+2)^2-4=((n+2)+2)((n+2)-2)=(n+4)(n)=n^2+4n. $$

sdaayoung says:

about 4 years

I wonder if it would be useful to number the cases. I've had some teachers wonder just what on earth was going on and it might be helpful for them to see that this is supposed to be a progression.

Donna

Cam says:

about 4 years

Hi Donna,

Thanks for the feedback. We found that including explicit numbering "gave away the ballgame" to an extent, and the struggle to make that connection is worthwhile (see, e.g., Levi's comments). As a compromise, I've included some language in the commentary referencing your suggestion as potential scaffolding.

Thanks again,

Cam

Levi says:

about 4 years

I've used this task with educators to illustrate what I believe to be one of the best example of "Seeing Structure in Expressions." Instructionally, I use this task first without the expressions, following the next step, 10th step, nth step progression usually associated with this type of task. This way, I'm almost certain to get these two expressions, but I also get some very interesting variations. My favorite so far was 2(n(n+2)) - n^2 . It has been incredibly powerful for educators to have to think about where this expression "appears" in the dots. If they don't believe me that it is there, I usually have them start by algebraically verifying it. Then the interest is peaked. I'll leave the verification for you to visualize on your own.

Also, if you like this task you might really like VisualPatterns.org.

Levi says:

about 4 years

I'll also say this type of expression is just too perfect not to connect to function notation such as d(n) or otherwise (see F.BF.A).

Cam says:

about 4 years

Hi Levi,

Very interesting -- thanks! I'd argue that while function notation could be used here, it's also important to have tasks that let students (or teachers) view expressions as things in and of themselves, that can be used to describe quantities.

Gary Einhorn says:

over 4 years

From your solution to (b), we see what is meant by "deductions" but the word alone can denote a wide range of concepts. If the point of (b) is to have students prove algebraically that the two expressions are equivalent, would it make more sense to just say that?

Cam says:

over 4 years

Good idea. Thanks.

NEB says:

over 4 years

The task is great. Love the commentary. Please help me with the tagging. I just am not connecting to 'factoring to reveal the zeroes of the function it defines', which is A-SSE.B.3.a. Where did we reveal the zeroes?

Cam says:

over 4 years

Great comment, thank you. I concur completely that that was an errant tag. The most significant aspects of this task are the process of connecting the algebraic expression and its subparts to the geometric representation, and the manipulation of that algebraic expression to see the equivalency.

Kathleen Smith says:

over 4 years

I found the use of the term "following" confusing. I would replace it in the first instance with "Consider the algebraic expressions BELOW" and in the second instance with "...why the expressions ABOVE..." I had to clarify this when using the task recently. It just so happens we were discussing this standard along with F-IF.3 and this worked great! Thanks

Cam says:

over 4 years

Good point, thanks. In fact, I think the "above" might be unnecessary. See how it reads now.

Kathleen Smith says:

over 4 years

It reads very well now. I have used this activity twice recently with teachers in PD and it has created some very interesting conversation. All the teachers have said they plan to use the task. Good Job!

p.s. These types of problems also work very well when trying to teach proof by induction.
You can show by induction that the explicit and recursive formulas actually create the same sequence.

Ben McAuslan says:

over 4 years

Oh, my

Kristin says:

over 4 years

Hmmm... that's an interesting response. Perhaps you could elaborate?