6 Representing Conjectures Symbolically

6 Representing Conjectures Symbolically

· Students need to learn number sentences with multiple variables and equations in which variables appear more than once

· Then they have the necessary tools to represent conjectures using symbols

Replacing Words with Symbols

· The zero property for addition is a good place to start

· Students may require a little support to get started

· For an open number sentence to represent a conjecture, there is an implicit statement that the number sentence is true for every number

Open Number Sentences That Are Always True

· Challenge students to write an open number sentence that is true for all numbers

· This is a little more challenging than what has been done before, but is still doable

· Classes might generate one of the following open sentence conjectures:

o a – a = 0

o 0 x r = 0

o 1 x a = a

o a ¸ 1 = a

· Teacher needs to get students to see a connection between these newly generated conjectures and ones previously done with numbers around the room

· Students will realize almost immediately that the number sentences from previous work correspond to these newly generalized conjectures

· Important to make this link in their thinking

Taking Advantage of an Opportunity

· Sometimes situations arise that provide a natural forum for introducing symbols to represent conjectures

· Even in a 2nd grade class this can be done

o Class discussing whether or not zero plus a number equals that number and a number plus zero equals that number were different conjectures

o Can use symbols like * or ¨ instead of letters, even will work well here

o Following this discussion and introduction to writing open sentences, students were able to revisit previous conjectures and re-write them using open sentences

Advantages of Open Number Sentences for Representing Conjectures

· Open number sentences provide a precise statement of a given conjecture

· Students generally agree it is an easier way to write a conjecture than with words

· Open number sentences also make it possible to generate conjectures that are difficult to state in words

· Still, students may find some conjectures are not readily expressed using an open number sentence and words are easier

· In general, conjectures that represent fundamental properties of addition, subtraction, multiplication, and division can be represented with an open number sentence

· Open sentences provide younger students with an opportunity to use variables to express mathematical ideas

· Recognizing variables can be used to make these types of representations is an important mathematical skill

· For many older students variables are a mystery that is never really understood

· Lack of understanding of the concepts surrounding variables extremely limits a student’s mathematical perspective and causes problems as students develop more complicated ideas

· Students need multiple perspectives of what it means to use variables in ideas associated with arithmetic