6.EE.5, 6.EE.6, 6.EE.7, 7.EE.4a, 8.EE.7, A-CED.1
Students are often confused by the appearance of 0 in an equation and frequently interpret it to mean that the solution of the equation is also zero. For example, if they arrive at 0x=7 (see below), they might conclude that x=0.
2x -3 = 2x + 4
|Students get 0x = 7. 7/0 is undefined, but students often write zero.|
4x + 8 = 2(2x+4)
|Students arrive at 0x = 0 and often think that the answer is just zero; however, the original equation, 4x + 8 = 4x +8 is an identiy, which means the solution set is the set of all real numbers.|
|3x + 3 = 2x + 3||x = 0, zero is the only solution that makes this true.|
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Thus they specify two possible types of equation: a conditional equation; and an identity and, for example, 2x + 1 = 6 would be a conditional equation, but 2(2x + 1) = 4x + 2 would be an identity.
● These equations emphasize the need for reasoning and checking the solution to see if their solution is correct.
- If students get 0x=7 (for example) and conclude that x=7/0=0, teachers could ask them whether they can they take 7 items and share them with 0 people as a way of explaining why division by zero is undefined.
- Asking the students to justify their steps forces them to do only steps that they can explain.
- Students should be expected to check the validity of their solutions for confidence improvement.
- Connecting the solutions to real-life situations such as sharing cookies can help with understanding.
- The checking process becomes important in contextual problems because the meaning that is associated with the check will confirm the correct interpretation of the solution.
- Students should not align work horizontally (e.g., 8x+4=28 8x=24 x=3). To show the progression of ideas leading to a solution more clearly, they should align their work vertically.
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- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?