Table of Contents

Unit 5: Seeing Structure and Generalizing (MP7 and MP8)

5.1 Structure, Repeated Reasoning, and Generalization

Structure and Repeated Reasoning

You will now have the opportunity to work on a problem and examine student expectations at specific grade spans in the area of structure and repeated reasoning.

Time to Try

Work on the problem in the Consecutive Sums activity. As you work through this problem, consider how you use aspects of the Seeing Structure and Generalizing mathematical practices.

Consecutive Sums

Some numbers can be written as a sum of consecutive positive integers:

 6 = 1 + 2 + 3
15 = 4 + 5 + 6
                 = 1 + 2 + 3 + 4 + 5

Which numbers have this property? Explain.

Time to Read

For examples of what might be expected of students at each grade span when working on this problem, select the links below:

K–2 Example

3–5 Example

6–8 Example

9–12 Example
 

The structure of our number systems refers to the way they work. This includes the place value system in which we denote numbers and how different places are related to each other.

Equally important are the properties that our numbers satisfy. The Whole Numbers (0, 1, 2, 3…) satisfy Closure, the Commutative Property, and the Associative Property for addition and multiplication; 0 is the Additive Identity; and 1 is the Multiplicative Identity. The Distributive Property connects multiplication and addition. When we extend to integers we also have the Additive Inverse Property. When we extend our numbers to the Rational Numbers, we also have the Multiplicative Inverse Property.

See Properties in the glossary for additional information.

Time to Reflect

In your Metacognitive Journal, respond to the following questions:

 

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