standard_description: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
In earlier activities, students wrote expressions to describe visual patterns by first creating tables and reasoning repeatedly about how the pattern changes at incrementally greater step numbers. Here they write an equation to describe a pattern without creating a table of values. They look more directly at the structure of the visual pattern and its connections to the parts of the equations, and then reason in the other direction: given an equation, they generate a visual pattern. Students also begin to frame the relationship between the quantities as a function with inputs and outputs leading to a definition of
quadratic function
.
Some students are likely to view the
\(n\)
th step of the given pattern as an
\(n\)
-by-
\(n\)
square with four small squares added at the corners, leading to the expression
\(n^2 + 4\)
for the total number of small squares. Others may view this as an
\(n+2\)
by
\(n+2\)
square with 4 rectangles removed, each consisting of
\(n\)
squares. The latter leads to the (equivalent) expression
\((n+2)^2 - 4n\)
. In the next activity, students will look closely at how equivalent expressions arise when analyzing geometric patterns, but if equivalent expressions come up in this activity, consider inviting students to share them.
This prompt gives students opportunities to see and make use of structure (MP7) when writing an equation from a visual pattern. The specific structure they might notice is an inner square with a side length equal to the step number along with a small square on each corner. Later, generating a pattern given an equation requires them to reason abstractly and concretely (MP2).
Launch
Give students a moment to observe the pattern from the activity and ask them what they notice and what they wonder. Then, ask students to sketch the next step in the pattern and share their sketch with a partner.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Provide students with access to a simple two-column graphic organizer in which one column is for what they know and one column is for what they wonder. Allow students to organize their thinking this way for each step of the visual pattern.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
[IMG]
If the pattern continues, what will we see in Step 5 and Step 18?
Sketch or describe the figure in each of these steps.
How many small squares are in each of these steps? Explain how you know.
Write an equation to represent the relationship between the step number
\(n\)
and the number of squares
\(y\)
. Be prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.)
Sketch the first 3 steps of a pattern that can be represented by the equation
\(y = n^2 - 1\)
.
Student Response
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Student Facing
Are you ready for more?
For the original step pattern in the statement, write an equation to represent the relationship between the step number
\(n\)
and the perimeter,
\(P\)
.
For the step pattern you created in part 3 of the activity, write an equation to represent the relationship between the step number
\(n\)
and the perimeter,
\(P\)
.
Are these linear functions?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may wonder how to draw a pattern given the equation
\(y=n^2-1\)
.  Show them the warm-up problems where they subtracted 1 to remove the small square in Figure B and added 1 when there was an extra small square in Figure C.  Prompt them to describe subtracting 1 as removing one small square from each step.  Some students may find it easier to start drawing step 2 or step 3.  They can work backwards to draw step 1 which would have 0 squares since
\(1^2-1=0\)
.  Emphasize that making a table can help them figure out exactly how many small squares are needed in each step.
Activity Synthesis
Make sure students see the connection between the equation
\(y=n^2 +4\)
and composition of the squares in the pattern: that regardless of what
\(n\)
is, the figure at Step
\(n\)
is composed of a square that is
\(n\)
by
\(n\)
, plus 4 small squares at each corner.
Next, help students relate the work so far to the idea of functions. Discuss with students:
“In each pattern we’ve seen, is the relationship between the step number and the number of squares a function? How do you know?” (Yes. The step number is the input, and the number of squares is the output. For every step number, there is a particular number of squares.)
“How can the relationship be expressed using function notation?” (If the function
\(f\)
gives the number of squares at step number
\(n\)
, we can define it as
\(f(n) = n^2+4\)
.)
Introduce
quadratic function
as a function that is defined by a quadratic expression. Like other functions, it can be represented with an equation, a table of values, a graph, and a description.
Conversing: MLR2 Collect and Display.
Prepare students for the discussion by inviting them to first explain to a partner how each part of their equation relates to the pattern. Record the words, phrases, and diagrams students use to explain the equations they wrote to represent the pattern. Display the collected language for all to see, and invite students to borrow from, or add more language to the display throughout the remainder of the lesson. This will help students read and use mathematical language during partner and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


aligns: no