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Ratios and Data 6-8 / Week 08 / Monday
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Week 08 · Multiplying and Dividing Signed Numbers

Monday

Repeated hops
// Repeated hops and the sign rules
⏱ about 20 min

Monday: Repeated Hops

Rocket drops a crumpled paper ball into the bowl on the Numbers Desk. "New Field Day game. Every miss is -2 points."

"I missed three times in practice," he says. "So that is -2, -2, -2. Which is... -2 three times."

Raven writes 3 × (-2) on the whiteboard. "What do you notice about the hops?"

"Three hops, each one 2 to the left," Rocket says. "I land on -6."

Nova hovers over the taped number line, her light blinking once at each landing spot. "Multiplying is hopping the same hop again and again," she says.

"So four misses is 4 × (-2) = -8," Rocket says. "I should miss less."

Raven grins. "Or we find out what (-2) × 3 means. This week, the hops repeat."

Rocket, Raven and Nova at the Numbers Desk. A bowl of crumpled paper balls sits on the table beside a taped number line. Nova hovers with her light on.

Multiplying is repeated hopping

You already know 3 × 2 means three groups of 2, or three hops of 2 to the right: 6.

So 3 × (-2) means three hops of -2. Each hop goes 2 to the left. Start at 0 and land on -6.

A positive number times a negative number is negative. The size is the same as 3 × 2, only the direction changes.

What about (-2) × 3? Multiplication can be done in either order, so (-2) × 3 = 3 × (-2) = -6.

A number line from -7 to 1 with hops landing on -2, -4 and -6

Think of a miss worth -2 points. Three misses give 3 × (-2) = -6 points. Five misses give 5 × (-2) = -10.

A thermometer that drops 3 degrees every hour for 4 hours changes by 4 × (-3) = -12 degrees.

Every repeated loss is a multiplication with a negative number.

Dividing undoes multiplying

If 3 × (-2) = -6, then -6 ÷ 3 = -2. Dividing asks: what hop, repeated 3 times, lands on -6?

And -6 ÷ (-2) = 3 asks: how many hops of -2 reach -6? Three hops.

Rocket lost 10 points in 5 misses. Each miss was worth -10 ÷ 5 = -2 points.

  1. Multiply the sizes as if both numbers were positive.
  2. Count the negative numbers. One negative number means the answer is negative.
  3. Picture the hops to check: a negative hop repeated lands left of 0.
REPEATED HOPS
  • Read the question.
  • Tap your answer.
3 × (-2) = ?
5 × (-4) = ?
(-6) × 2 = ?
7 × (-1) = ?
UNDO THE HOPS
  • Read the question.
  • Tap your answer.
-6 ÷ 3 = ?
-20 ÷ 4 = ?
-6 ÷ (-2) = ?
-15 ÷ (-5) = ?
FIELD DAY MISSES
  • Read the question.
  • Tap your answer.
A miss is -2 points. Rocket misses 4 times. How many points is that?
The thermometer drops 3 degrees each hour for 6 hours. What is the total change?
Raven lost 14 points over 7 misses. How many points was each miss?
Number sentenceTrue or false?
3 × (-2) = -6?
(-2) × 3 = 6?
-6 ÷ 3 = -2?
-6 ÷ (-2) = -3?
WHY THIS EXERCISEThe second and fourth rows are the two sign slips you will meet most often this week.
A miss is -2 points. Rocket misses 8 times in a round. How many points is that? Type the number with its sign.
WHY THIS EXERCISERepeated losses are the most common signed multiplication at Field Day.
Try it
Lay out counters: three groups of two negative counters. Count them all. That is 3 × (-2).
Now lay out -12 as twelve negative counters and share them into 4 equal groups. That is -12 ÷ 4.
Draw a number line from -12 to 0. Show 4 × (-3) as four equal hops with arrows.

Good hopping. Tomorrow you meet the puzzle of the week: what is a negative times a negative?