Reflection: Lesson Planning Are they proportional? - Section 3: Exploration


Sometimes I find it difficult to respond to mistakes or explanations I did not anticipate. I plan my lesson differently now that I expect and encourage multiple methods. When I ask students to think for themselves I suddenly have all these ideas to work with and it helps to anticipate what might come up even if I can't predict every possible thing. I ask myself different questions about the content:

  • how do I think my students might understand this content?
  • what methods/models might they use?
  • what mistakes might they make?

Once I think about these I will spend some time thinking about how I might respond to the mistakes, so I can be a little more prepared. I don't want to respond with telling and fixing, but instead in a way that helps them discover and correct the mistake themselves in such a way that they are still uncovering meaning.

One mistake that my students made was simplifying incompletely. One student simplified 8:12 to 4:6 and left 10:15 unsimplified and so didn't recognize that they were equivalent. I expect this from students who are not fluent with their multiplication facts. In this case I would choose questions from questions like the ones below, not necessarily in this order.


  • Tell me about what you have done so far.
  • It looks like you simplified the ratio 8:12, how did you do that?
  • What do the numbers represent in the ratios? 
  • How do you know 8:12 and 4:6 are proportional?
  • Could you use that same strategy on the other ratios?
  • If the ratios are proportional what would you expect to happen when you simplify them?
  • How will simplifying help you decide which designs are proportional?


Having them explain what they did and why they did it and paraphrasing it back to them is often all they need in order to continue. Often I will include the whole table group in the conversation for their feedback and to get them used to discussing each other's ideas.

Another mistake my students made was trying to draw straight lines not through zero on the graph. This is a mistake I didn't anticipate, but some of the questions from above were still helpful. I could still ask them what they did and why they did it. And even though some of the questions above are specific to simplifying I could still use the same question style and ask:


  • How would straight lines help you decide which designs are proportional?
  • If the points represent proportional designs what would you expect to happen with the straight lines?
  • What do the numbers [8:12, etc.] represent? [for every ____ there are ____]
  • I could also ask how they know which direction the line will go without another point.


I know that the missing information is the fact that it must pass through zero. I also know that the "stair pattern" or simplifying would help them as well. After completing the sentence frame 'for every...' I might ask them how many of each tile they would have if the added 8 more black tiles and have them plot that point. With the additional point they would know what direction the line would go. I could have them look at other student strategies and switch gears and then come back and ask the above questions about the graph. Having not anticipated this mistake I might not have been able to come up with these questions on the spot, but I will have them ready for next time.

  Lesson Planning: Responding to mistakes
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Are they proportional?

Unit 6: Proportionality on a graph
Lesson 5 of 10

Objective: SWBAT identify proportional relationships using a graph & using equivalent ratios.

Big Idea: Proportional relationships will graph into a straight line through the point (0,0) and will have equivalent ratios.

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9 teachers like this lesson
Math, Number Sense and Operations, Graphing, equivalent ratios, real world, testing for proportionality, ratios, pattern
  49 minutes
using rulers
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