Introducing Hardy-Weinberg equilibrium into the high school or college classroom can be difficult because many students struggle with the mathematical formalism of the Hardy-Weinberg equations. Despite the potential difficulties, incorporating Hardy-Weinberg into the curriculum can provide students with the opportunity to investigate a scientific theory using data and integrate across the disciplines of biology and mathematics. We present a geometric way to interpret and visualize Hardy-Weinberg equilibrium, allowing students to focus on the core ideas without algebraic baggage. We also introduce interactive applets that draw on the distributive property of mathematics to allow students to experiment in real time. With the applets, students can observe the effects of changing allele frequencies on genotype frequencies in a population at Hardy-Weinberg equilibrium. Anecdotally, we found use of the geometric interpretation led to deeper student understanding of the concepts and improved the students' ability to solve Hardy-Weinberg-related problems. Students can use the ideas and tools provided here to draw connections between the biology and mathematics, as well as between algebra and geometry.

## Introduction

Incorporating Hardy-Weinberg equilibrium (HWE) into a classroom allows teachers to integrate mathematics and biology while empowering students to use data to investigate population genetics and the relationship between gene frequencies and evolution. Data-driven hypothesis testing and integration across subject areas are key skills emphasized in the Next Generation Science Standards (NGSS, 2013) and the Mathematics Common Core State Standards (NGA et al., 2010). Bridging the biological and mathematical concepts of HWE is notoriously difficult, primarily because many students fear and struggle with math.

Here we describe a geometric approach for evaluating HWE that clearly and intuitively links biology and mathematics. We will not discuss in detail the assumptions and requirements for HWE, which was done in depth in a recent American Biology Teacher article (Smith & Baldwin, 2015). We will illustrate the geometric visualization of HWE and then apply this method to example problems, including interactive applets. In our experience this approach leads to deeper student understanding and greater procedural skill, compared to simulations and equations alone.

## The Mathematics of Hardy-Weinberg: Finding Equilibrium

Allele frequency change over time is a major contributor to the evolution of organisms. When a population is in HWE, the frequencies of alleles and genotypes remain the same across successive generations, and thus, the population is in genetic stasis (i.e., not evolving). HWE is a null hypothesis describing expected gene frequencies in the absence of the main forces of evolution: mutation, gene flow, nonrandom mating and survival (i.e., sexual and natural selection), and genetic drift. In the natural world, these evolutionary forces are always present and populations cannot be at HWE. However, investigating how and when populations deviate from HWE allows us to understand the mechanisms that lead to evolution. The Hardy-Weinberg equations provide a mathematical formalism for expressing this type of equilibrium. We use these formulas to identify genotype and allele frequencies where HWE can occur.

## The Distributive Property of Mathematics

The core of the Hardy-Weinberg equations lies in a clever application of the distributive property in mathematics, frequently written *a*(*b* + *c*) = *ab* + *ac*. The same principle applies to more complicated settings, such as (*a* + *b*)(*c* + *d*) = *a*(*c* + *d*) + *b*(*c* + *d*) = *ac* + *ad* + *bc* + *bd*, sometimes given the mnemonic FOIL (first, outer, inner, last). We can visualize the distributive property geometrically as well (Figure 1). Take a rectangle with one side measuring *a* + *b* and another *c* + *d*. The area of the rectangle is given by (*a* + *b*)(*c* + *d*) (Figure 1a). By considering each of the four smaller rectangles (Figure 1b), we see that the area can also be written as the sum (*a* + *b*)(*c* + *d*) = *ac* + *ad* + *bc* + *bd*, just as was shown by the algebraic interpretation. This geometric interpretation is a method frequently used to justify the distributive property in mathematics courses.

## Hardy-Weinberg Equilibrium: A Special Case of the Distributive Property

Consider the Mendelian inheritance of a single gene with two alleles represented by *A* and *a*, where *p* is the frequency of *A* and *q* is the frequency of *a* in a population. As we have only two alleles, it follows that *p* + *q* = 1, which is the first of the Hardy-Weinberg equations. Squaring the first equation, (*p* + *q*)(*p* + *q*) = 1^{2}, and then applying the distributive property, leads to the second Hardy-Weinberg equation: 1 = *p*^{2} + 2*pq* + *q*^{2}. These equations can be used to: (1) identify expected gene and genotype frequencies of a population at HWE (the actual genotype frequencies can be compared with expected frequencies generated from the Hardy-Weinberg equations using a chi-squared test), or (2) make predictions about genotype or allele frequencies in a population that meets the assumptions of HWE. However, it can be difficult for students to relate the biology of HWE to the application of these equations.

We can build the geometric visualization of the distributive property specific to HWE while explaining it in the context of the biology, and accomplish the same goals as the equations without writing out the equations. One assumption of Hardy-Weinberg is random mating (Smith & Baldwin, 2015), meaning it is equally likely for any pair of individuals to mate. Since the parents are chosen at random from the population, the probability of each parent contributing *A* or *a* is equal to the frequency of each allele in the population: the probability of contributing *A* is *p* and *a* is *q*.

We can visualize this using a square, where the length of each side is the probability of a parent chosen at random contributing an allele, which is 1 (Figure 2). Each randomly selected parent will contribute either an *A*(*p*) or *a*(*q*), and so each side of the square will be made up of two segments of lengths *p* and *q* which will add up to 1. We can now subdivide the square by drawing lines that represent the division by each of the parents' probabilities (Figure 2). There are now four smaller rectangles, where the area of each rectangle represents the frequency of each genotype in the population, such that the frequency of *AA* equals the area *pp*, *Aa* equals the area *pq* + *pq*, and *aa* equals the area *qq*. There are two *pq* rectangles, and therefore we must add the areas of these rectangles together to obtain the frequency of the *Aa* genotypes (equivalent to the 2*pq* term in the second Hardy-Weingberg equation). By framing the Hardy-Weinberg equations in this way, we can encourage students to make geometric sense of the equations without memorizing, while keeping all rigor inherent in the algebraic representations.

While these images may look similar to Punnett squares, there is a key difference: the length of each segment, *p* and *q*, can change with the frequencies of each allele, whereas for Punnett squares, segment length is constant. Each side of a Punnett square represents a single parent, and the probability of inheriting either allele is 50 percent, therefore, Punnett squares are split into four equal parts.

## Applications to the Classroom

When introducing Hardy-Weinberg problems in class, we recommend introducing the geometric method first or alongside the algebraic approach. In our classes, we began by presenting a problem and drawing a version of Figure 2 on the board. We described the biological significance of each variable and emphasized that the length of each segment on the side of the square, *p* and *q*, represents the gene frequencies from the population, and the areas of the inner rectangles represent genotype frequencies. We then substitute the values from the example problem into the square, adjusting the lengths of the line segments and areas of the internal rectangles to match. We also solved the same problem using the more common algebraic approach. In subsequent example problems, we displayed each method side by side, pointing out how they accomplish the same goal.

To assist in the instruction of the geometric approach, we created interactive applets (links below) that offer an online manipulative (accessible by phones, tablets, and laptops) to help students understand how shifting values of *p* and *q* can change genotype frequencies in a population. When teaching, these applets could be projected onto a screen in place of drawing on the board. Students can also use these applets to explore HWE or solve additional Hardy-Weinberg problems independently.

We found that students who were stronger in math often preferred using an algebraic approach to solve HWE problems, whereas students who expressed a dislike or fear of math preferred the geometric approach. (We had to break it to them that it was still math and they were good at it.) There was mathematical value for students who saw both methods and, although they may have had a preference, understood that these methods represented the same process, included the same numerical values, and achieved the same goal. With emphasis in the Common Core State Standards on student understanding and justification of algorithms, this kind of flexible understanding of geometric and algebraic thinking is invaluable.

## Sample Problem with Interactive Applets

### Problem

In a population of plants, the allele for yellow fruits is dominant over the allele for green fruits. You know the frequency of the dominant allele is 0.4. If this population of plants was in Hardy-Weinberg equilibrium, what proportion of the population would be yellow? What proportion of the population would be both yellow and heterozygous for fruit color?

### Solution

To solve geometrically, we first draw the square where we know that the length of each side sums to 1 (Figure 3). We know the frequency of the dominant allele (*p* = 0.4). Therefore, the segment of the square *q* must be a length of 0.6 because 1 – 0.4 = 0.6. We can then designate the segment lengths for *p* and *q*, and multiply the segments to obtain the area of the rectangles for each genotype (Figure 3). Genotypes with two dominant alleles (homozygous dominant) or one dominant and one recessive allele (heterozygous) will all be yellow. Thus, we can sum the areas of the rectangles representing *pp* and *pq*: 0.16 + 0.24 + 0.24 = 0.64. Therefore, 0.64 of the population is yellow. To determine the proportion that is both yellow and heterozygous, we sum the areas of the rectangles representing *pq*: 0.24 + 0.24 = 0.48.

We have created two interactive applets (Figure 4), one providing the frequencies of individuals expressing either the dominant or recessive trait (http://ggbm.at/pAZXe5rg) and one providing only the frequencies of each genotype in the population (http://ggbm.at/a7sffXGy), both created using Geogebra (geogebra.org). In these applets, the student can use a slider to adjust values for *p* and it calculates *q*. At the same time, the areas of rectangles representing the genotypes will respond to the changes in *p* and *q*.

### Extension

Use of geometric tools extends well to more complicated examples involving more than two alleles, where the necessary algebra for such problems could be prohibitive for some students. For example, consider a population with a dominant gene, *A*, a recessive gene, *a*, and a mutation of the recessive gene, *a*′. It would be reasonable to consider this situation using the same geometric framework (Figure 5). An interactive applet for this example can be seen at http://ggbm.at/zxC827Cc. The applet allows independent adjustment of the frequencies of the *A*(*p*) and *a*(*q*) alleles, and the third allele, *a*′(*r*), is assigned the remaining value (calculated as 1 – *p – q*).

## Conclusion

This framework provides a way for students to make the abstract concept of HWE more concrete and accessible. It can also be used to justify the formalism of the algebraic interpretation of the HWE and combine it with a geometric interpretation. Virtual manipulatives like those shown here can be easily accessed by phones, tablets, or computers, and expose students to an application of mathematics by exploring the relationships in their biological context.

## Acknowledgments

Thanks to M.G. Javier, K. Boerup, and A. Baffert for feedback on an early draft of this article.