📚 Topic Summary
Algebraic proof involves using algebraic manipulation to demonstrate that a statement is true for all possible values (within a defined set). Instead of just solving for a variable, you're showing a general truth. This often involves manipulating expressions to arrive at a conclusion that matches the statement you're trying to prove. Key words to look out for include 'prove', 'show', 'demonstrate', and 'always true'.
For example, you might be asked to prove that the sum of two consecutive integers is always odd. You'd represent the integers algebraically (e.g., $n$ and $n+1$), add them, and manipulate the expression to show it always results in an odd number.
🧮 Part A: Vocabulary
Match the term to its definition:
| Term |
Definition |
| 1. Identity |
A. A statement that is true for all values of the variable(s). |
| 2. Variable |
B. A symbol representing a quantity that can change. |
| 3. Expression |
C. A mathematical phrase containing numbers, variables, and operators. |
| 4. Coefficient |
D. A number multiplying a variable. |
| 5. Consecutive |
E. Following continuously. |
(Match the numbers 1-5 to the letters A-E)
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct words:
Algebraic proofs use __________ to show that a statement is __________ true. We often use __________ to represent unknown values. The goal is to __________ the expression to match the statement we are trying to prove. Key words like '__________' and '__________' indicate the need for proof.
(Word Bank: always, prove, variables, manipulation, show, demonstrate)
🤔 Part C: Critical Thinking
Explain in your own words why algebraic proof is important in mathematics. How does it differ from simply solving an equation?