Background Knowledge And Content Objectives
With the understanding that I will be teaching freshmen geometry during the 2026-2027 school year, I want to leverage the knowledge of student learning needs and guide students to make connections between their biology and geometry courses. This will include students engaging with content that supports their development of identifying points of confusion, engaging in critical thinking skills to develop questions, and then further their knowledge by processing information received through lessons, activities, and explorations within and outside of a school context. Instructional time during this curriculum unit will focus on two critical areas of geometry: (1) establish criteria for congruence of triangles based on rigid motion and (2) establish criteria for similarity of triangles based on dilations and proportional reasoning.
There are two of the eight Standards for Mathematical Practices (SMP) that are most relevant to this curriculum unit and specifically for engaging with geometry content. The first practice is SMP1 which states that students will need to make sense of problems and persevere in solving them. Within this practice, students will analyze given information or a problem and develop explanations for the meaning of the problem. Students will also work towards a level of questioning that keeps them focused on asking, “Does this make sense?” Grappling with this question will encourage students to understand various approaches to persevere in approaching solutions to problems. Rather than jumping to conclusions and seeking just answers, students will plan a solution pathway and monitor their progress to change their approach to a problem and solutions when necessary. Some questions that develop mathematical thinking with respect to SMP1 include, “How would you describe...What do you notice about…How else might you organize/represent/show…?” These questions help students access relevant knowledge and experiences to make appropriate use of them to work through problems. Another practice that is supportive of questioning strategies in the learning process is SMP 7 which is looking for and making sense of structure. Students will start with the application of foundational, general mathematical knowledge and rules to specific situations or problems. Students are encouraged to look for overall structure and patterns in mathematics. Another set of questions that develop mathematical thinking with respect to SMP 7 include, “What observations do you have about…What do you notice when…What patterns do you find in…How does this relate to…?” These questions require students to remain curious when approaching a problem and identify structures from previous experiences that help students better understand and approach new experiences and situations.
In conversation with Dr. Paul Turner during the Yale National Initiative seminar on Asking Questions in Biology: Discovery vs. Knowledge, he talked about science being an act of discovery through asking questions and hypothesizing that does not need to focus on correctness, and the ability to make sense of information with available data. He then shared a term with me that I had never heard: allometry. Allometry is the study of the physical traits and characteristics of organisms with respect to body mass and size, and addresses the change in relation to overall body size.[4] In geometry we discuss the concept of isometry and symmetry when working with congruent and similar geometric figures. Students also learn about scale factors of congruent and similar geometric figures that bring about understanding the difference between symmetry and asymmetry. This notion of allometry in biology has me considering how to support student understanding and further exploring asymmetry in geometry. Students would benefit from seeing a concept in geometry that also aligns with concepts in biology, leading to asking some critical thinking questions to further process content knowledge.
In the California Reveal Geometry course, students will learn about transformations and symmetry, including geometric isometry with congruent and similar geometric figures, particularly triangles and quadrilaterals. When applying rigid transformations, the size, shape, and angles of figures remain the same when applying translations, rotations, reflections, and compositions of transformations, known as glide reflections[5]. When dimensions are altered, this is a non-isometric dilation where the size of the figures is changed, creating asymmetry in geometric figures.
Transformations and Symmetry
Rigid motion is a transformation that is performed on a geometric figure or shape that preserves the distance between points on the figure and also preserves any angle measurements within the figure. Transformations involve processes that move or change a geometric figure, shape, or image on a coordinate plane by either altering the position, size, or orientation of the figure. The most common transformations that are taught in geometry courses include reflection, translation, or rotations. Each of these transformations create a type of symmetry within a coordinate plane.
Reflections
The content focus for reflections is the use of rigid motions to reflect figures on a coordinate plane using a line of symmetry. This line of symmetry provides the direction in which an original figure is reflected, or flipped, across the line while keeping the shape and size of the figure the same. To maintain the shape and size of the figure, first it is important to identify the structure and characteristics of the figure, which includes knowing the coordinates of each vertex within the original figure. Then, predict the results of a figure after a reflection and lastly apply rules of reflection on the original figure to arrive at the new figure.
The reflection rules (see Figure 1) determine the direction of the reflection so that a new figure is flipped over the line of reflection creating a mirror figure of the original figure. Reflection over the x-axis (horizontal line of reflection) causes the x-coordinate of the original figure to remain the same and the y-coordinate to take the opposite (multiply the y-coordinate by -1). Reflection over the y-axis creates a new coordinate point where the x-coordinate takes the opposite and the y-coordinate remains the same as the original coordinate. These are the most common lines of reflection, including the use of y = x where we interchange the original x- and y-coordinates in the new coordinate point, and y = -x where we interchange the original x- and y-coordinates and take the opposite. Corresponding points on the original figure and the new figure are the same distance from the line of reflection
Translations
The translation of an original figure requires an understanding of adding units to or subtracting units from an original coordinate. Each point of the original figure is moved the same distance in the same direction using a translation vector. The translation vector (see Figure 1) describes the translation from the original figure to the new figure using the notation <a, b> → (x + a, y + b). A positive value in a translation vector indicates movement right for a x-coordinate or up for a y-coordinate, and a negative value in a translation vector indicates movement left for an x-coordinate or down for a y-coordinate. An example of applying the translation vector would be for an original figure to have the coordinates (5, 1) and the translation vector is < 5, 2 >, then the translated point on the new figure would be (5+5, 1+2) → (10, 3). This would indicate the original coordinate of (5, 1) would be translated right 5 units and up 2 units to the new position of (10, 3). The coordinates of an original and new figure can also be used to determine and confirm a translation vector within a coordinate plane. What is important to remember is translations to the left or right impact the x-coordinate and translations up or down impact the y-coordinate to slide across the coordinate plane and create the new image.
Rotations
Teaching about rotation requires students to understand how to use rigid motion to rotate a figure around a fixed point called the center of rotation. The center of rotation is often the origin, yet can be other points within a coordinate plane. Rotation is accomplished by applying the rotation rules (see Figure 1), which starts at a center of rotation and continuing with the transformation through the point or rotational symmetry in a specific direction (either clockwise or counterclockwise). When the point of rotation is at the origin with coordinates (0, 0), then the vertices of the figure are rotated counterclockwise about the origin. The most common angles of rotation include 90°, 180°, 270°, and 360°, and rotations often move counterclockwise. Corresponding points on the original image and the new image are the same distance from the center of rotation.
Figure 1. Transformation Rules: Reflection, Translation, Rotation
Compositions of Transformations
When teaching about transformations, the first step is to ensure the coordinates are known for each vertex of the original figure, and then recognize which transformation is being applied to the figure to arrive at the new image. It is helpful for students to hear the terms flip, slide, and turn as a way to understand the direction in which a figure should be positioned or oriented to go from an original figure to a new figure. Flip refers to a reflection as the letters “fl” appear in the term and makes a connection that when reflecting a figure, it is flipped over a line of reflection. Slide references the translation of a figure as the letters “sl” appear in the term and is conducted by sliding the original figure from one location on the coordinate plan to another. Rotation contains the letter “t” twice in the term and reminds students to turn a figure about a point of rotation to move from an original location to a new location on the coordinate plane. One of the reasons these shortened terms are provided is when we begin applying compositions of transformations, students can visualize the position or orientation of a figure from its original position to a new position in a particular order of transformations. An example of a composition of transformations would be to start with a triangle that is placed in quadrant I of a coordinate plane and it needs to first be reflected over the x-axis and then translated to specific units within the coordinate plane. A shorthand way for students to visualize what will happen to this original figure is to flip, then slide the figure across the coordinate plane.
Similarity
Once students have a grasp of transformations and compositions of transformations where the dimensions of the geometric figures remain the same, then they will explore when the geometric figures change into similar figures with non-rigid motion. Non-rigid motion is a transformation that changes the dimension of a given figure. Content objectives aligned with similar figures include learning about the application of scale factor through dilations of similar figures, including triangles using Angle-Angle (AA), Side-Side-Side (SSS), and Side-Angle-Side (SAS) similarity.
Dilations
When there is an enlargement or reduction in size of a figure proportionally by the same scale factor, the use of dilation has occurred and similar figures are created. The scale factor (k) is a ratio of the length of a new figure to the corresponding length to the original figure. The equation that aligns with scale factor as a ratio of new image to original image (k = new / original). A dilation that creates a larger new image in comparison to the original image where the scale factor is greater than 1 (k > 1) is dilation through enlargement. Dilation through reduction is when the original image is smaller than the new image and the scale factor is between zero and 1 (0 < k < 1). The scale factor is used to calculate the dimensions of the new dilated image. When creating a dilation in a coordinate plane, the coordinates of the vertices of the figure are multiplied by the scale factor and the center of dilation is often at the origin (0, 0). The new image will either be an enlargement of the original image if the scale factor was greater than one and a reduction of the original image if the scale factor is between zero and one.
Similar Figures
Similarity transformation is composed of a dilation on an original figure, or a dilation and then a rigid transformation, to form a new figure. The original and new figures are then classified as being similar figures - two figures that have congruent angle measurements and proportional side lengths. The scale factor between the two figures is used as a similarity ratio of the length of the new figure to the corresponding length on the original figure. This provides the means to use a proportion, which is an equation setting the scale factor equal to the ratio of corresponding sides of similar figures. The use of similarity statements identifies the corresponding proportional sides and congruent angles of similar figures.
An example of similar figures includes similar triangles, three-sided polygons that have congruent corresponding angles and proportional corresponding sides. Congruent statements are used to identify congruent corresponding angles and similarity statements are used to name similar triangles with corresponding proportional sides listed in a particular order. To further explain how to identify similar triangles, there are similarity postulates and theorems applied to prove specific parts of one figure are similar to another. Postulates are statements that are accepted as true without proof and theorems are conjectures that have been proven to be true[6]. Angle-Angle (AA) Similarity Postulate states that if two angles of one triangle are congruent to two angles of another, then the triangles are similar[7]. Side-Side-Side (SSS) Similarity Theorem states if the corresponding side lengths of two triangles are proportional, then the triangles are similar[8]. Side-Angle-Side (SAS) Similarity Theorem states if the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar[9]. Figure 2 provides a visual of the congruent angles and corresponding proportional sides that need to align when applying each postulate and theorem. The use of similar figures provides real life application of identifying similarity through proportionality.
Figure 2. Similarity Postulates and Theorems
Isometry and Allometry
The study of isometry in geometry engages the skill of congruency and proportionality while allometry causes a change in shape where there is disproportionate change. Allometry is seen when changes to the size and dimensions of the original figure creates a new, disproportionate figure. Positive allometry occurs when a body part grows faster than the body as a whole. An example of positive allometry is the growth of the claw of a fiddler crab where the claw size grows proportionally larger than the body of the crab[10]. Another example of positive allometry that students may better understand and have seen is the growth of deer antlers where the antlers surpass the size of the deer’s head during stages of growth (see Figure 3). Negative allometry occurs when the body part grows slower than the rest of the body. An example of negative allometry is the growth of the human head in comparison to the body where the head size is proportionality smaller than the growth of the body[11]. The growth development of a human head from early in life as an infant to a fully grown person is where this negative allometry is visible (see Figure 3). As the human body grows from an infant into a grown person, the head is disproportionately smaller than the rest of the body.
Figure 3. Examples of Positive and Negative Allometry
Transitioning into exploring allometry in real world situations, students will be presented with the idea that some individuals in nature show constant proportional change as the individual grows in size. One example mentioned by Dr. Turner was the shells of some creatures like land snails and the chambered nautilus. The land snail has a singular inner shell that is one continuous space that often has a visual spiral pattern, whereas the nautilus has partitioned internal chambers that become closed off as the nautilus grows into the larger chambers across its lifespan[12]. While visiting the Yale Peabody Museum of Natural History, I was also introduced to an ammonite, which has a similar shell pattern as the nautilus where the shell grows in a spiral. The direction of the internal chambers created by the spirals is how you can tell the ammonite and nautilus apart - the ammonite walls are convex and have a wavy pattern, whereas the nautilus walls are concave. With the ammonite, the newer shell grows completely over the older shell, creating a series of small chambers that increase in size as the shell continues to grow in a spiral pattern[13] - which is an indicator of positive allometry in which the new shell begins to grow disproportionately larger to the older shell. In contrast, Dr. Turner shared that the change in proportion is obvious in creatures like humans where the skull size does not change as much through development to adulthood, compared to the length of our limbs. This is an indicator of negative allometry when the human skull remains disproportionately smaller than the rest of the growing body. There are some consequences of mutations involved in the development process of the human body, such as dwarfism where limb length changes much less or sometimes equally to the change in skull size.
When reviewing variables used in describing scale factor in geometric dilations and variables used in describing allometry in biology, I noticed that the description of the variables has some similarities. With scale factor, we use k as the variable representing the mathematical relationship of a figure that is either enlarged (k > 1), reduced (0 <k <1), or the figure has remained exactly the same (k = 1)[14]. With allometry, we use b as the variable representing the mathematical scaling relationships of organism that has either positive allometry (b>1), negative allometry (b<1), or the organism has isometry (b = 1)[15].
Isometry will teach students about rigid transformations that preserve the shape and size of geometric figures, while allometry will have students critically think about and explore how proportions change as the size of an organism and overall shape changes. This exploration will guide students in understanding isometric and allometric growth across two content areas by considering how the growth of geometric figures can be accomplished isometrically by applying the same scale factor to all parts of the figure. When applying allometry to a geometric figure, such as a triangle, students would notice that when the dimensions change for only one or two sides of the triangle, then the scale factor is no longer applicable and the figure will grow at a different rate along all three sides, changing the shape of the original figure.
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