Geometry: Grade 8
Prompt: The figures shown have all undergone a different type of transformation. How are these transformations the same and how are they different?
By including a dilation, this hook allows students to identify the congruent properties of translations, reflections and rotations. Students may begin by stating "the shape stays the same" in the first three translations but we can encourage them to be more specific than this by supplying measuring tools and asking "How do we know the shape remains the same?", "What parts remain the same?" or "Can you prove the shape remains the same? How?". Diving deeper into congruence gets to the heart of the standard and gives students an opportunity to be precise and think critically about what is needed mathematically to prove congruence.
Students will most likely see differences in the movements of the transformations but may have difficulty articulating these differences. One way to help students hear articulation is to listen to different groups' discussions as you monitor the room and then sequence your class discussion in a way that begins with the least clear descriptions and moves its way up to the most articulate. Students will hear on their own the increased clarity in descriptions and have exemplars to think of in the future when describing basic rigid motions.
For those who remember descriptions from earlier grades such as slide, flip and turn and are in need of a challenge, we can ask them "How?" we slide, flip or turn on a coordinate plane. Can they articulate lines of reflection, centers and vectors? In short, challenge them to be mathematicians.
Video: Transformations and Code Breaking (3:52)
Hook A 8.G.A.2
Prompt: Which pairs of faces are exactly the same only transformed?
Pairs D and E are not congruent. In D, the smile has been translated while the eyes have been reflected. In E, the eyes have been translated while the smile has been reflected. In all other pairs, the second face can be created by performing a transformation or combination of transformations on the first. While students work to find matches, we can supply them with word banks or sentence frames to strengthen their vocabulary usage. Ultimately, this hook provides students the challenge of applying the geometric definition of congruence before actually learning the term "congruence" or its formal definition.
Video: Reflections in magic (1:46)
Hook B 8.A.G.2
Prompt: Which cap fits the boy?
Kids will more than likely jump all over the blue cap but the proof is where we should concentrate their discussion. One way to generate this conversation would be to discuss how we buy clothes. Usually, we try them on. So challenge kids who "know the answer" to help the boy "try on" his cap by using transformations. Ignoring the visor, the semi-circles of the boy's head and the blue cap are congruent, which can be proven with more than one combination of basic rigid motions.
Video: Crop Circles (2:35)
Prompt: The original image has gone through three different transformations. Identify each transformation.
Students can apply prior knowledge to graph each image, which makes for a terrific proof. However, look for students who notice patterns in the coordinates and ask them to share out their ideas. More than likely, they will have difficulty expressing the patterns concisely but their observations can generate a class discussion to identify and then further prove the effect transformations have on the coordinates of figures.
Video: Geometry in Islamic Artwork ( 5:06)
Prompt: The triangles in Group 1 are similar. Which 2 figures in Group 2 do you think are similar? Explain.
The concept of similarity may be new for some students but this prompt should generate ideas and give students the opportunity to use tools (ruler, protractor) to discuss those ideas as mathematicians. Students may intuitively choose figures C and F correctly. Pushing for proof, once again, is the key to building and deepening students' understanding of the concepts at hand. "What proof can you supply for your answer?", "What patterns do you see in your measurements?", and "How do you know C and F are the only similar pairs?" are all questions that force students to go beyond their visual intuitions and to think mathematically about the nature of similarity.
Video: Time Dilation (1:22)
Hook A 8.G.A.5
Prompt: Which labeled angles appear to be congruent? Do you notice any patterns in the placement of the congruent angles?
The labels purposefully eliminate vertical angles from consideration to simplify the conversation. More angles and/or lines can be added to heighten the challenge and consider other facets of this standard, such as the angle sum and exterior angles of triangles. Obviously, there's ample opportunity to review simpler angle properties and categories here. A good beginning point might be to make observations without measurements by identifying acute, obtuse and right angles. Encouraging students to verify these observations using a protractor deepens the activity by allowing for practice of using tools, practice of precision and the verification of the angle properties created by parallel lines cut by a transversal. If the conversation stalls, ask students to circle angle pairs they believe are congruent and look for a pattern in the placement of congruent angles. This may lead to them identifying corresponding or alternate exterior angles. Directing the class discussion toward the idea of congruent angles created by parallel lines being a translation could help enrich students' understanding as well. This idea connects to prior knowledge and creates a convincing visual on a digital device when one parallel line is slid atop the other to align congruent angle pairs. In short, there's a lot to do and discuss in this hook and the more student-generated the discussion the more powerful the lesson.
Video: How to Make a Periscope (3:14)
Hook A 8.G.B.6
Prompt: Explain how the diagrams above prove Pythagorean's Theorem.
This is a great exercise to deepen students' understanding of the Pythagorean Theorem because it connects their algebraic understanding of the theorem to its geometric proof. The two square frames pictured are congruent. Each square frame contains four triangles with side lengths of a, b and c. In the first square, side length c is squared. In the second frame, side lengths a and b are squared. Because the 4 triangles in each square are congruent and therefore must have an equal area, the picture proves that the remaining areas, a squared + b squared in the first frame and c squared in the second frame, must be equal. It will take time and some leading questions for students to grasp this. Ask them what they notice about the two large square frames. Ask them what they notice about the triangles, their measures, and the number of triangles in each frame. Ask them to label the triangles' side lengths a,b and c. Ask them what they know about the the areas of the squares created. Have them label each square a squared, b squared and c squared. In my experience, once a few students make the connection they help the others and the "aha!" moment spreads like wildfire. Some curricula have students cut out the individual squares and triangles like a puzzle and then work to fit the shapes into the congruent, blank square frames. The action of placing the pieces themselves may help students understand the underlying proof more independently.
Hook A 8.G.B.7
Prompt: Find a relationship between the side lengths of the triangle by creating an equation with the terms and symbols given to achieve an answer of 5.(Note: Symbols may be used more than once)
Full disclosure, this hook was inspired by this amazing video. The video not only exemplifies the use of this hook into Pythagorean theorem (which is simplified and altered slightly from the much larger exercise used in the video) but also gives evidence to the power of inquiry before instruction. While the Pythagorean theorem is a simple algorithm, this hook turns the equation into a challenge for students to discover themselves. More than likely, few students will write the correct algorithm, just as in the video. However, the opportunity for students to "productively struggle" not only engages them in a pure math activity but also helps the algorithm stick because students become intrigued by an answer that eluded them.
Hook A 8.G.B.8
Prompt: About how far is the football pass? About how far is Point A from Point B?
A common error associated with finding diagonal distances on a coordinate plane is that students count units either horizontally or vertically and then assign this distance to the diagonal distance between the points The visual of the football play places the question in the real world, which may help some see the incorrect logic of answering using either the vertical or horizontal distance. However, an in-class demonstration (or on playground) may be helpful for two reasons: 1.) To clarify the concept of a football play to those students unfamiliar with football. 2.) To illustrate that a diagonal distance can not be equal to either a horizontal or vertical distance. This second concept can be shown simply with a measuring tape, rope or string by measuring a horizontal distance from a point and then displaying how that distance is not accurate once we move to a diagonal location from the original point.
Hook A 8.G.C.9
Prompt: What is the same in each volume formula? What is different and why?
The match boxes and tuna cans are given to get students thinking about the formula of volume as the multiplication of the volume of the base times the height, though it is doubtful they will use this language on their own. Some will recognize both formulas have h, height, which leaves them to compare "length times width" to "pi times radius squared". No doubt, leading questions will be helpful for students struggling. Possible queries are "Describe your process of calculating the volumes for the stacks of match boxes and tuna cans", "What is the difference between the tuna can and the match box?", "How would these differences appear in the formula?, "What do the variables in each formula represent?", "Does any part of the formula look familiar to you?". All of these questions encourage students to move their thinking beyond the volume formulas and to consider the meaning and logic of each variable component. Ultimately, the hope is that students recognize length times width as the area of the rectangular prism's base and pi radius squared as the area of the base of the cylinder.