Accentuate the Negative
By Michael Naylor

For more of Michael's "Multiplication Models" click here.

Negative numbers extend our number line and greatly simplify our calculations, but sometimes students struggle with the concepts. These fun activities will help kids to make sense of the "other half" of our number system.

Count down
(Grades K-2)
Test your calculators to see if they have a "constant" function: Enter 10 – 1 and then press = repeatedly. On many calculators, the display will decrease by one with each press of the equals button. This is useful for looking at many number patterns and is great for introducing negative numbers as a count "below zero."

Ask your students to enter 10 – 1 on their calculators and count down as a class. Continue counting aloud, "Zero, negative 1, negative 2, negative 3…" When your students are surprised that you've continued, explain that negatives are numbers that come before zero.

Continue the count to –10, then have students enter "+1" on their calculators. Now count upwards back to 0 and continue up to 10.

Blast to the sea
(Grades K-2)
Draw a line to represent the water and label it 0. Write a vertical number (a y-axis) to show values from –5 (below the water) to +5 above the water.

Use a cut-out rocket to raise and lower and ask students to call out the altitude in unison as you go along. Bring the rocket down into the water to become a submarine with negative altitude.

Number walk
(Grades 3-5)
Write the numbers from –10 to +10, including 0, on separate sheets of paper and set them up in a row in front of the class. Call students up one at a time to model addition and subtraction. Have the first student stand at 0 and face in the positive direction. Ask the first student to show how to demonstrate 4 + 3, and then 7 – 5 by taking steps, but not turning around. Your class will quickly understand that addition is stepping forward and subtraction is stepping backwards. Now ask your students to model number sentences like 3 – 5, 4 – 10, –2 + 7, and –3 – 5. Call on students to write number addition and subtraction problems on the board, then ask other students to model it.

Number walk 2: adding and subtracting negatives
The Number walk model also works with sentences like 3 + (–5) and 5 – (–4). Tell your students that when adding or subtracting a negative, they should turn around. So to show 3 + (–5), the student stands at 3 facing in the positive direction. Addition means step forward, but since we're adding a negative amount, we turn around and then add five steps in the negative direction.

Counter models
Use two different color counters to model numbers where one color chip (perhaps blue) is worth +1 and the other color (perhaps red) is worth –1.

Demonstrate to your students how pairs of chips, one blue and one red, are worth a total of zero, and demonstrate a few different ways to make zero (three blue and three red, for example). Now you're ready to begin some interesting modeling.

5 + (–3) becomes five blue chips and three red chips. What is its value? Remove pairs of blue and red chips until only two blue chips remain. Your students will have fun modeling several of these.

Show 6 – 4 = 2, emphasizing the removal of chips is subtraction. Now put up five red chips and show –5 – (–4) = –1: you simply remove four of the red chips to subtract the –4. Easy!

Now show six blue chips and ask, "How can we show 6 – (–3)? There are no red chips to take away?" The solution, bring in three pairs of chips (three blue and three red), which is the same as zero and doesn't change the value of the 6. Now remove the three red chips to subtract –3. The result is 9.

IOU games
(Grades 4-8)
Supply small groups of students with decks of cards that have the face cards removed. Each group member gets five cards. Black cards are "income" and red cards are "IOUs." Each student should figure out his or her total. Some will be positive and some will be negative. You can then ask your students to write a number sentence to describe their situation.

Now have them write their total and remove one of their cards, then complete a number sentence to show what happened. For example, if you have a total of +4 dollars, and you subtract a red 7, your hand is now worth $11. The sentence is 4 – (–7) = 11. Challenge your students to come up with three different situations of adding and subtracting IOUs and write their own number sentences to describe each one. Have fun!

For more of Michael's "Multiplication Models" click here.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.