# Same and Different

I’ve been using “Same and Different” as an inquiry strategy with my students for several years. Read on for more about this thinking routine and some of my favorite prompts.

##### What is “Same and Different” ?

“Same and Different” is an inquiry strategy sometimes known as “Compare and Contrast” or various other names (see Resources links below).  This powerful strategy asks students to compare and analyze features of two mathematical situations. They may require different solution strategies, be similar except for one feature, or have mathematically meaningful nuances to notice.

The routine is launched by presenting two or more math situations, then have students examine and note how they are the same and how they are different.  This is a great opportunity to use technology to illuminate some of the numerical and graphical differences, although technology is not required.

The examples that follow are grouped (loosely) by theme, and many can be used in several math levels. Keep in mind that not all students will find all the similarities and differences, and you can highlight the details that are pertinent to your curriculum.

##### Representation Examples

These examples begin with a visual representation to tap into students’ interpretation of the graph or diagram.

Example 1: (PreAlgebra or Algebra 1) The following two images show these subtraction problems: 5 – 2 (top) and 2 – 5 (bottom). I like beginning with the number line representation because it spurs more discussion than simply presenting the numerical expressions.

Example 2: (Algebra 1) These images get students talking about slope and y-intercept of linear equations.

Example 3: (Algebra 1 or 2) These graphs of quadratic functions show various transformations of the parent function $y=x^2$ (graphs 1 & 3). Discussion can highlight the types of transformations, how they impact the equation, and what the graphs have in common.

Example 4: (Geometry) Continuing with the idea of transformations, students can interpret these geometric transformations on the coordinate plane. Discussion can analyze what is the same and different about the pre-image and image, contrast the two transformations, and generalize a coordinate rule for these motions.

##### How to Solve? Examples

The next set of examples presents problems that can be solved with a variety of methods. Students can debate pros and cons of the available techniques.

Examples 5, 6, 7: (Algebra 1) Solving linear equations: Which inverse operations are appropriate? Is it beneficial to distribute first or not?

Example 8: (Algebra 1 or 2) Solving systems of equations. Which method would you choose: graphing, substitution, or elimination?

Example 9: (Algebra 1 or 2) Solving quadratic equations. Would you factor, use inverse operations, or use the quadratic formula? Or something else?

Example 10A: (Algebra 2 or Precalculus) Solving equations involving exponents. Is the next step to take a root or a log of both sides, or is there another technique available?

##### Features of Functions Examples

In these examples, think about how graphs or numerical tables of values can be used to determine the similarities and differences between the functions. Don’t forget to zoom in or out on the graph to see key features and end behavior, and change the table increment if warranted.

Example 10B: (Algebra 1 or 2) Linear vs. Exponential functions. For more about this scenario, check the first part of my post Table Techniques, with student activity sheet here.

Examples 11, 12, 13: (Algebra 2 or Precalculus) Students analyze the features of power functions, exponential functions, and logarithmic functions. For examples 12 & 13, what happens with other bases?

Example 14: (Algebra 1 or 2) Forms of Quadratic Functions. These are all actually the same—but what information about the parabola graphs is most clearly visible in each of these equations?

There’s another quadratic function Bonus prompt in the Google Drive Folder of all the images.

Example 15: (Algebra 2 or PreCalculus) Rational Functions often can be simplified. Is the simplified version exactly the same as the original?

My favorite way to examine the above situation is to use the TI-84+ ZoomDecimal window; since it equalizes the pixels, a hole is visible in the graph of the rational function. Then, use the “tracer ball” graphing style to graph the simplified linear function–it goes ‘through’ the hole, demonstrating that the domains of the two graphs are different.

Another nice question is how does the rational function’s graph compare to the numerator quadratic alone?

##### Simplification and Notation Examples

Examples 16 & 17: (Algebra 1 or 2) Simplifying fractions and radicals. What is possible, what is necessary in each situation? Use a calculator or grapher to check.

Examples 18 & 19: (Algebra 1 or 2) Dealing with an exponent of –1. Does it “go to the denominator” or does it “flip”? When does an exponent “distribute”?

Example 20: (Algebra 2 or Precalculus) I’ve found that –1 is one of the most challenging notations for students, because it means different things in different situations. How do you read these notations aloud? What does each mean? What other ways can each be written?

Example 21: (Algebra 2, Precalculus) Notation issues can get confusing with trigonometry, so I like this prompt to tease out the meanings of the 2 in each case. This is also helpful for calculus students who need a review! Discuss algebraic meaning and check the graphs for visual evidence.

##### Geometry Examples

Examples 22 & 23: (Geometry) What are the similarities and differences between polygons and regular polygons? What about convex and concave polygons?1

Example 24: (Geometry) What is the same and different about these 3 triangles? What can you determine about their angles?2

Example 25: (Geometry) Quadrilaterals of all types! If you are teaching special quadrilaterals, ask students to compare and contrast any two of them:

• Squares and Rectangles
• Parallelograms and Trapezoids
• Trapezoids and Isosceles Trapezoids
• Squares and Rhombuses (rhombi?)
• Rhombuses and Kites

Other geometric figures:

• Congruent triangles vs. Similar triangles
• Circles vs. Ellipses
• Cylinders vs. Prisms

Example 26: (Geometry) How does scaling a polygon by a factor of 2 change the measurements we can take, such as perimeter, area, interior angle measure, or number of vertices? Here is an animation from Mathigon Polypad3 showing what measurements are the same and what are different.

##### Bonus Examples: Absolute Value & Sequences/Series

Bonus Example 1: (Algebra 2) Absolute Value Equations and Inequalities. How can you represent each of these on a number line? How do you solve each of these? Can a graph in the coordinate plane help?

I prefer the “distance from zero on the number line” explanation for absolute value. So in the equation, we need any points that are exactly 2 units from zero (2 and –2). In the inequalities, we need values that are MORE than 2 units away (x $\leq$ –2 or x $\geq$ 2) or LESS than 2 units away from zero ( –2 $\leq$ x $\leq$ 2). In class, I wave my hands in the air along an imaginary number line to help students visualize the correct intervals.4

The coordinate plane can help students understand these situations: graph y = 2 and y = |x| at the same time. The intersection points are where the graphs are equal (top image below). Then look for x-values where y-values of the |x| graph is greater than 2 (middle image) or where y-values of the |x| graph is less than 2 (bottom image). [Read more about a coordinate plane graph visual in my post How Else Can We Show This? part 2 “Absolute Certainty”—with video!]

Bonus Example 2: (Algebra 2 or Precalculus) When studying sequences and series, students learn several formulas. The sums of geometric series formulas (finite and infinite) are actually the same formula for large values of n if r is between -1 and 1! Use this prompt and numerical examples to examine this situation. Why does this formula have a restriction on the size of common ratio r?

##### Wrapping Up

I’ve used Same and Different prompts as a lesson opener, exit ticket, stimulus for discussion, and during class review; they work well as formative assessment or consolidation anytime. Be sure to annotate and share student thinking and summarize important nuances with the whole class.

With a 3-part prompt, consider presenting two parts first, then add the third (in order to reduce the initial cognitive load of examining all three parts of the prompt).

I hope you’ve found some useful ideas here in this (longer than I expected) post. See below for more Same and Different resources.

##### Notes & Resources:

ALL OF THE IMAGES from this post are in this Google Drive Folder. Please use with your math classes and the Same and Different thinking routine.

Check out my follow-up post for Calculus topics: “Same & Different: Calculus Edition

Same and Different is also called “Same But Different”, “Same or Different?” and “Compare & Contrast”. Here are some great resources:

One more resource is the Math Routine Collaborative group of educators which meets periodically on Zoom to learn and discuss various math routines; moderated by Shelby Strong (@StrongerMath), Annie Fetter (@MFAnnie), and Annie Forest (@mrsforest). More information and register here: http://www.strongermath.com/mrc/.

1Tim Brzezinski (@TimBrzezinski) has a GeoGebra interactive applet on Convex vs. Concave polygons here: https://www.geogebra.org/m/knnPDMR3

2My GeoGebra activity Converse Pythagorean Theorem dives in to this topic more deeply. https://www.geogebra.org/m/zvdhmzv8

3The Mathigon Polypad is a wonderful playground for virtual manipulatives that includes both fun and serious math. Each of these measurements is accompanied by music and animation! Follow @MathigonOrg and David Poras (@davidporas) for more.

4There are many good GeoGebra visualizers for absolute value equations and inequalities graphed on a number line. One I found is here: https://www.geogebra.org/m/V4SwRtrb

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