Using Manipulatives to develop the rules for adding and ...



What are the rules for adding and subtracting positive and negative numbers? (

There are two activities here. I suggest that instructors do the first activity (adding positive and negative numbers) on one day, and then on another day do the second activity (subtracting positive and negative numbers). In this way, students will be familiar with the demonstration process before they develop the more difficult rule (subtracting positive and negative numbers). The directions are the same for both activities.

Materials Needed

Activity sheets: “Adding positive and negative numbers” and “Subtracting positive and negative numbers”

Manipulatives: counters to represent +1 and ─1. You could use poker chips in two colours, or squares of card in two colours, or squares marked + and ─ . Each team of students will need about 15 positive counters and 15 negative counters.

Optional: a place mat for each team.

You can ask students to work on their own, or in pairs, or in teams of three. Give the following directions, and work through one or more examples until everyone understands the procedure.

1. Designate a workspace on the table to do the demonstrations in, so that the extra counters will not interfere with the demonstration. (Placemats are useful for defining the workspace.)

2. Start by putting 5 positive counters and 5 negative counters on your workspace. Notice that the total value is 0, since the positive and negatives cancel each other out. Before starting any question, make sure that you have several positive and negative counters on your workspace, and that their total value is 0.

3. Use the counters to show the adding or subtracting work on the activity sheets. For example:

+2 + +3 Start with an equal number of positive and negative counters on your workspace, to show a value of 0. Show +2 by putting 2 positive counters on the mat. Then add 3 positive counters. Your mat will now show a value of +5. Notice that the sign of each number shows you which counter to use, positive or negative, and the operation sign tells you whether to add or subtract them.

Write your answer on your sheet.

4. After you have done all the demonstrations, answer the questions at the bottom of the page.

Adding positive and negative numbers

Show each of these operations with the counters, and fill in the answers. Before you start any question, make sure that you have several positive and negative counters on your workspace, and that their total value is 0.

+2 + +3 = _______ +2 + +1= _______ ─3 + ─3 = _______

─3 + ─1= _______ ─2 + ─2 = _______ +4 + ─3 = _______

+2 + ─3 = _______ +2 + ─1= _______ +2 + ─4 = _______

─1 + +3 = _______ ─4 + +3 = _______ +2 + ─3 = _______

+5 + ─1 = _______ +3 + ─5 = _______ ─1 + +4 = _______

─4 + +3 = _______ ─2 + +4 = _______ +4 + ─6 = _______

1. Write the rule for adding two positive numbers without using the counters. What operation do you do, and what is the sign of the answer?

______________________________________________________

______________________________________________________

2. Write the rule for adding two negative numbers without using the counters. What operation do you do, and what is the sign of the answer?

______________________________________________________

______________________________________________________

3. Write the rule for adding a positive and a negative number without using the counters. What operation do you do, and what is the sign of the answer?

______________________________________________________

______________________________________________________

4. When you add positive and negative numbers, does it matter what order the numbers are in?

______________________________________________________

______________________________________________________

Subtracting positive and negative numbers

Show each of these operations with the counters, and fill in the answers. Before you start any question, make sure that you have several positive and negative counters on your workspace, and that their total value is 0. Notice that some of the questions ask you to add, and some ask you to subtract.

+3 ( +2 = _______ +3 + ─2 = _______

+2 ( ─1= _______ +2 + +1 = _______

─3 ( +3 = _______ ─3 + ─3 = _______

─3 ( ─1= _______ ─3 + +1= _______

─2 ( ─2 = _______ ─2 + +2 = _______

+4 ( ─3 = _______ +4 + +3 = _______

─5 ( ─1= _______ ─5 + +1= _______

─5 ( +2 = _______ ─5 + ─2 = _______

+4 ( ─3 = _______ +4 + +3 = _______

+4 ( ─4 = _______ +4 + +4 = _______

1. What do you notice about the two answers in each row?

__________________________________________________

2. Are the questions the same in each row? ____________________

3. In each row, what is the difference between the first question and the second question? ________________________________________

______________________________________________________

Hint: Is the first number the same or different? ____________

Is the operation sign the same or different? ___________

Is the second number the same or different? __________

4. Write the rule for subtracting positive and negative numbers:

______________________________________________________

______________________________________________________

______________________________________________________

( I was inspired to write this activity after reading “Seeing is Believing” by Charles Brover, Denise Deagan, and Solange Farina. The article is available online

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