If you try this activity with your student(s), we’d love to see what you do. Share your journey via the #Inspired2Learn hashtag on your preferred social platform.
Created by: Aleta Margolis
Discipline: Math (specifically geometry)
Age level: 5th grade and above
Time: 50 minutes
Materials: lots of sticky notes, chart paper, markers, several circular objects and tools for measuring them these should include string or yarn, rulers, measuring tape, etc.
Often when we learn about “absolutely true things” like the fact that 3.14 is pi we never question WHY. We are simply told this is so and we proceed from there. And when we are calculating the circumference of a circle we plug that number into an equation with a circle’s radius or diameter and voila the calculation works. But why 3.14? Where does that come from? This activity helps students to answer that question and it can be used as a lesson for that purpose OR it can be used as a tool for encouraging students to question assumptions and figure out why things work the way they do. Understanding where the number pi comes from may not be the key to understanding geometry, but it teaches students that math actually describes the world around us and that understanding can build motivation to learn.
What to do:
Prepare the room with a wide variety of circular objects and measuring materials. Put students in pairs or small groups. Start the lesson with this prompt either spoken or written on the board: “Using the materials in the room, find the diameter and the circumference of at least 5 objects.”
Give groups about ten minutes to measure the circumference and diameter of the objects. Be sure not to tell them how to measure the objects. You want them to discover that they can use the string to wrap around the object and then measure its length on the ruler. You may guide them in this direction by asking questions like, “Is there anything in your set of tools that is flexible?” Groups may even come up with another method based on the tools you give them.
As groups are measuring, circulate and ask questions like:
- How did you measure the diameter?
- How did you measure the circumference?
- What challenges did you face in measuring the circumference?
Now share this prompt: “Using your measurements, find the relationship between circumference and diameter and write the relationship on a post-it.”
Do not be specific about what you are looking for. Learners may represent the relationships in many different ways – an equation, a ratio, a number, etc. As they are doing this, challenge groups to come up with as many representations for this relationship as possible. When the energy in the room begins to slow down, ask one person from each group to bring up the post-its to the board or to a blank wall – all post-its should be placed there.
Invite the class to huddle around the posted post-its. Invite them to share their thoughts with a partner who was not in their original group reflecting on what they learned. During the discussion, you can use the questions below to start the conversation, but you want to build on the students’ responses and let them really guide the discussion. Eventually you want them to understand that the ratio of circumference to diameter for every circle is pi. This will also lead them to the conclusion that the equation C=(pi)(D) also holds true for every circle.
Finally, bring the class back together as a whole to debrief. Discuss the results to see what the participants have discovered.
- What relationship did you discover?
- What did you notice in the work posted on the wall? How did different groups interpret the relationship?
- How did your group choose to represent that relationship? Why did your group choose to represent it that way?
- Is there another way you could have represented it?
- Was the relationship the same for all the different objects?
Inspired Teaching Connection:
Learners are focused on understanding a concept central to mathematical thinking which gives this activity Purpose. The process of discovery requires both Persistence and Action. The range of descriptions students create to explain the relationship between circumference and diameter will offer Wide Ranging Evidence of Student Learning, as will the constructive discussions you hear them having throughout the process. Sharing their ideas and putting them in the position of discoverers rather than passive recipients of knowledge positions the Students as Experts. This taps into their Intellects, and the whole experience involves Inquiry.
See our instructional model here.
Standards Addressed by this Activity
Common Core Standards for Mathematical Practice
Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP1 Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP2 Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP3 Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Model with mathematics.
CCSS.MATH.PRACTICE.MP4 Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP5 Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Attend to precision.
CCSS.MATH.PRACTICE.MP6 Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
Look for and make use of structure.
CCSS.MATH.PRACTICE.MP7 Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Look for and express regularity in repeated reasoning.
CCSS.MATH.PRACTICE.MP8 Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Common Core College and Career Readiness Anchor Standards for Speaking and Listening
Comprehension and Collaboration:
CCSS.ELA-LITERACY.CCRA.SL.1 Prepare for and participate effectively in a range of conversations and collaborations with diverse partners, building on others’ ideas and expressing their own clearly and persuasively.
CCSS.ELA-LITERACY.CCRA.SL.2 Integrate and evaluate information presented in diverse media and formats, including visually, quantitatively, and orally.
Presentation of Knowledge and Ideas:
CCSS.ELA-LITERACY.CCRA.SL.4 Present information, findings, and supporting evidence such that listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience.
Collaborative for Academic, Social, and Emotional Learning Competencies
Social awareness: The abilities to understand the perspectives of and empathize with others, including those from diverse backgrounds, cultures, and contexts. This includes the capacities to feel compassion for others, understand broader historical and social norms for behavior in different settings, and recognize family, school, and community resources and supports.
Responsible decision-making: The abilities to make caring and constructive choices about personal behavior and social interactions across diverse situations. This includes the capacities to consider ethical standards and safety concerns, and to evaluate the benefits and consequences of various actions for personal, social, and collective well-being.
Relationship skills: The abilities to establish and maintain healthy and supportive relationships and to effectively navigate settings with diverse individuals and groups. This includes the capacities to communicate clearly, listen actively, cooperate, work collaboratively to problem solve and negotiate conflict constructively, navigate settings with differing social and cultural demands and opportunities, provide leadership, and seek or offer help when needed.