When students start learning about proving theorems, they already have some knowledge of what a theorem is. For example, they’ve come across the Pythagorean Theorem in earlier grades. Much of their work in geometry will consist of proving theorems.
And even for the best of math students, this can be a bit of a challenge! This is because proving theorems essentially relies on logical arguments and deductive reasoning, in addition to a solid knowledge of previous math axioms.
So if you’re a math teacher looking for guidance on how to teach this challenging topic, look no further! In this article, we’ve compiled awesome guidelines on teaching how to prove theorems in geometry! Read on to learn more.
What Is Proving Theorems in Geometry?
When we talk about a postulate in geometry, we’re referring to a statement that is assumed to be true without proof. An example of a postulate is this statement: “a line contains at least two points”.
Postulates are used to explain undefined terms, and also, to assist us in proving other statements. Thus, we use postulates and previously proven theorems to prove theorems. Unlike a postulate, a theorem is a statement that we can demonstrate and prove to be true.
But what does it mean to “prove” something? By now, students have already been asked in different grades to “justify” or “explain their answer”. To explain what we mean by proving theorems, you can start by reminding them of expressions like these.
Point out that by explaining how we reached our answers, we’re showing our reasoning to others so that we convince them of the validity of each step and get them to agree with us, ideally.
Proving a theorem is just a formal way of justifying your reasoning and answer. A proof is a set of logical arguments that we use when we’re trying to determine the truth of a given theorem. In a proof, our aim is to use known facts so as to demonstrate that the new statement is also true.
How to Teach Proving Theorems
Ways of Proving a Proof
Explain to students that there are two types of proof: direct proof, where we’re assuming that the statement to be proved is true, and indirect proof, where we’re assuming that the statement to be proved is untrue.
Explain to students that there are three ways of proving a proof:
- flow-chart proof
- paragraph proof
- two-column proof
Point out that all three ways rely on statements, or the claims that we’re making while proving a theorem, and reasons or the justifications we provide for the statements.
Flow-Chart Proof
Explain that flowchart proofs are proofs in which we use boxes and arrows to organize the statements. We write every statement inside a box and provide a reason under this box. In this proof, we can make subsequent statements from each statement.
In this type of proof, we show how the sequence of statements takes place with the help of arrows, i.e. boxes are connected with arrows. The flow-chart proof is praised for its visualization component, as it enables us to easily visualize how the conclusion results from the progression of statements.
Paragraph Proof
You can define the paragraph proof as a type of proof where, as the name suggests, we use a paragraph to prove a theorem. This is typically a detailed and lengthy paragraph where we explain our reasoning in detail.
Two-Column Proof
Two-column proofs are proofs that contain two columns – in the first column, we place the statements, whereas in the second column we place the reasons, i.e. the justifications of the statements.
Since two-column proofs are highly structured, they’re often very useful for analyzing every step of the process of proving a theorem. Two-column proofs are a good starting point for students in geometry and are most frequently used in geometry classes.
Point out to students that you will be using two-column proofs in this lesson. Use this video as an illustration of the two-column proof. Add that there are a few things that we need to pay attention to when we’re using two-column proofs, such as:
- We assign a number to every single step
- We always begin with “the given” (i.e. the information we have)
- In cases where we have statements with the same reason, we can also decide to place them in one step
- We must provide a reason for each and every statement
- We need to make sure that the order of statements contained in the proof is logical, even though it may not be fixed
- We use postulates, definitions, properties, and theorems that have already been proven as reasons
- The only time we can use “given” as a reason is if the problem gave us the info from the column with the statement
- We can replace certain words inside the proofs (such as angle or congruent) with symbols or abbreviations.
Examples
You can use videos in your lesson to provide examples of proving a theorem and thus make the lesson more engaging. For instance, use this video to demonstrate to students how to prove that lines are parallel.
The video contains a fun illustration of two snails (that are creating lines while moving in the woods). They’re concerned that their paths might cross. However, we can easily see from the video that their paths are parallel lines, and we know that parallel lines never cross. The only thing left to do is prove it!
In addition, you can use this video in your class to show how we can prove that opposite angles of a parallelogram are congruent, that is, the opposite angles of a parallelogram have the same angle in degrees.
Additional Resources for Proofs
You may also want to check out our article on reasoning and proof in geometry for further guidance on how to approach this subject with your students. It contains a bundle of free resources and exercises on the difference of inductive and deductive reasoning, which we apply in proofs.
Activities to Practice Proving Theorems in Geometry
Uno Cards Proof
To practice proving theorems, you can gamify your classroom! This is always a fun way for children to learn and it’s super useful for creating an engaging classroom atmosphere!
Besides, most students already know the game Uno.
Check out our article on how to use Uno Cards to prove theorems. It contains a bundle of free resources, inducing a free-of-charge full deck of digital Uno cards. Alternatively, you can also create math problems on the cards on your own.
Divide students into groups of 3 or 4. Print out the cards and hand them out to each group. The postulates are the rules of the game and the first card is usually the problem that students should solve. This card is known as “the given”.
Each student writes down the “given” information and forms a theorem that they try to prove. They justify their reasoning in a 2-column format, placing true statements in the first column and reasons why the statement is true in the second column.
As each student plays, they’re supposed to demonstrate their logic of how they’re proving their theorem step by step to the others in the group. The other members of the group can disagree with a given step or ask questions, which should open a discussion. Students may also come to the conclusion that there are a number of proofs that can be used to prove a theorem.
Angle Proofs Game
You can use this online game by IXL for practicing proofs that involve angles. Students are given a set of questions and have some missing information that they need to fill in. Typically, this is a proof that they need to complete.
For instance, they need to write the missing reason that justifies a given statement. A drop-down menu is provided to facilitate this process where students select the appropriate reason, such as a property, definition, or postulate.
Students can play this game individually or in pairs. The only thing you’ll need is a sufficient number of devices, as well as a stable internet connection.
Before You Leave…
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Unit 2 – Reasoning and Proof