Integers with manipulatives.

Activities for teachers.

Models Activities A middle school unit

Models:

Two models for integers are especially common:
 
One is based on measurement (e.g., temperature). It works fairly well for addition and subtraction, but not so well for multiplication and division, and makes connections with the real world.

 Something to think about: How would you describe addition in a temperature model? Are the two numbers playing the same role?

 The other model is based on objects which represent positive and negative units. A science connection can be made to charges on electrons and protons.
 
 

Activities:

Begin with some small objects which come in two colours: poker chips, bingo chips, the units from Alge-tiles, whatever. If one colour stands for positive units and the other for negative units, how would you represent +5? –3? 0? Think of several different representations for each number. Draw the representations and label them.

 How would you represent adding? Describe the actions associated with each of these additions:

+5 + +7
–6 + –3
+5 + –3
+5 + –6
–6 + +7

Some subtractions can be done by taking away:

–6 – –3
+7 – +5

Other subtractions require adding zeros. Describe the actions associated with each of these subtractions:

+5 – +7
–3 – –6
+5 – –3
–6 – +7

Students will need some time working with these objects to become comfortable with them, and to begin to abstract the number rules from them. There is no need to push the number rules. If kids want to use objects from a long while, that's fine. At the same time, including bigger numbers encourages the kids who are ready to abstract. Objects can also be used to represent things in word problems directly, rather than translating the words into symbols and then into actions, they can translate the words directly into actions on objects.

Multiplication is tricky in any model. It is not surprising that in the history of mathematics the status of negative numbers and especially products of negative numbers was debated by professional mathematicians well into the 19th century.

Two places to begin are repeated addition and patterns in number facts.

Repeated addition: +5 + +5 + +5 = +3 ´ +5 so –5 + –5 + –5 = ?

 +3 + +3 + +3 + +3 + +3 = +5 ´ +3 so –3 + –3 + –3 + –3 + –3 = ?

 These can be modelled with objects before being presented symbolically as I have done. Describe a repeated addition as an action on objects.
 

Patterns:
 
+3 ´ +5 = +15
+2 ´ +5 = +10
+1 ´+5 = +5
0 ´ +5 = 0
–1 ´ +5 = –5
–2 ´ +5 = –10		 +3 ´ –5 = –15
+2 ´–5 = –10
+1 ´ –5 = –5
0 ´ –5 = 0
–1 ´ –5 = +5
–2 ´–5 = +10

How would you show these patterns using objects?

How would you model division with objects?

Repeated addition and patterns use the “reduced” form of each integer, involving only one kind of chip.  Multiplication can also be done in an area model, with any representation (similarly to what has been done with base 10 blocks, and what will be done with algebra tiles). In a multiplication table we write the product of two numbers at the intersection of a row and a column:
 

´
4
3
12

We can do the same thing with objects, and it looks like an area model for multiplication:
 

´ * * * *
*
*
*		 * * * *
* * * *
* * * *

If we know that * ´  * = *; o´ o = *; * ´ o = o; and o´* = o then we can do multiplication on a table with any representation.
 

´ * * * * * * * * * * o o o o o o
*
*
*
*
*
*
*		 * * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *		 o o o o o o
o o o o o o
o o o o o o
o o o o o o
o o o o o o
o o o o o o
o o o o o o
o
o
o
o		 o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o 		 * * * * * *
* * * * * *
* * * * * *
* * * * * *

What multiplication is the table for? Make a table of your own for a multiplication, using un-”reduced” integers. How would you divide using a multiplication table?

Why is a negative times a negative a positive?

Hint: What does zero times zero look like on a table? What would happen if a negative times a negative was not a positive?
 

A middle school unit on integers:

Outline

Day 1: Discussion of integers students know about. Introduction of chips to represent integers (reduced form only). Representing simple addition and subtraction. Seatwork: Writing a story about changes in temperature, using integers.

Day 2: Representing zero with chips. Other ways to represent integers. Representing addition and subtraction with adding zeros. Seatwork: Addition and subtraction exercises.

Day 3: Problem solving and larger numbers. Seatwork on problems. Homework on larger numbers if ready.

Day 4: Multiplication by repeated addition. Have kids develop ideas for multiplication. Seatwork: Exercises in multiplying all except negative times negative. Homework: reviewing addition and subtraction, with a few multiplication.

Day 5: Negative times negative. Take kids' suggestions. If patterns are suggested go with that. Else develop tables idea. Seatwork on multiplying. Homework on multiplying.

Day 6: Division. Work from kids’ suggestions. Use repeated subtraction or tables as appropriate. Seatwork on division. Homework a review.

Day 7: Quiz: a few examples of each operation, and one longer problem.
 
 


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