Reflection: Staircase of Complexity Patterns in Addition - Section 2: Warmup

 

We had an opportunity here to dig a little deeper and not stop at a final answer. The real value in sharing multiple solutions or multiple representations is in comparing them to each other to uncover the structure and patterns in the math. This is what helps students understand foundational concepts and develop number sense. 

After several of my students shared their solutions to the first triangle I asked them to discuss "what was the same about all of them" or "what was true about all of them". Several responses included the idea that, in order to result in a sum of zero the two bottom numbers had to be "the same, but different signs". Thinking about the numbers in this generalized way helps my students think about the number relationships and develop number sense.

 I then asked another series of questions to get them to clarify by using more precise language:

  • "Why does it make sense that being 'the same with different signs' will have a sum that is equal to zero?"
  • "what would that look like on a number line?"
  • "Is there another way to represent that?"
  • "do we have to use numbers to show they are the same?"

By the end they were relating the 'sameness' and 'oppositeness' to distance on a number line, saying "the two numbers have to be the same distance on the number line in opposite directions" (absolute value). 

This precision made it easier to talk about the third number in the triangle. My students said the number had to be positive, so I followed up with questions like:

  • "How do you know?" or "what makes you say that?"
  • "Will any positive number work?" 
  • "Which positive numbers won't work?" and "why not?"
  • "how does the top number have to relate to the other two?"

After this conversation my students were able to tell me that this positive number had to be a greater distance on the number line from zero than the negative number.

Taking the time to discuss the puzzles and generalize will make the following explorations move faster. Students will notice the patterns more readily and be able to discuss their ideas more clearly. They may even begin asking themselves and each other questions to help them dig deeper.

  Staircase of Complexity: Digging deeper
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Patterns in Addition

Unit 4: Operations with Integers
Lesson 9 of 24

Objective: SWBAT manipulate an integer sum to make it positive, negative, or zero in order discover the pattern in the number relationships.

Big Idea: Students will create integer addition problems that will be negative, positive, or zero.

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Subject(s):
Math, Number Sense and Operations, Operations and Expressions, Integer Addition, puzzle, questioning, conceptual development
  54 minutes
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