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Ratios and Data 6-8 / Week 08 / Thursday
4/6
Week 08 · Multiplying and Dividing Signed Numbers

Thursday

The sign-up sheet
// Repeated hops and the sign rules
⏱ about 20 min

Thursday: The Sign-Up Sheet

Raven spreads the Field Day sign-up sheet across the Numbers Desk. "The kids on our street picked their favorite games."

"Three eighths chose ring toss," she reads. "One third chose paper ball toss."

"Which is more?" Rocket asks. "Eighths and thirds do not line up."

"What do you notice if we write them as decimals?" Raven asks.

Rocket divides. "3 ÷ 8 is 0.375. It comes out even. But 1 ÷ 3 is 0.333 and the 3s never stop."

Nova hovers over the long division, her light pulsing with each repeated 3. "Some decimals end," she says. "Some repeat forever."

"0.375 is bigger than 0.333," Rocket says. "Ring toss wins. Also, my eyes hurt from all the threes."

Fractions become decimals

Any fraction is a division. 3/8 means 3 ÷ 8. Write 3 as 3.000 and divide: 0.375. The division comes out even.

Now 1/3 means 1 ÷ 3. Divide: 0.3, remainder 1. Again: 0.33, remainder 1. The remainder keeps coming back.

When a remainder repeats, the digits repeat forever. We write 1/3 = 0.333... and say the 3 repeats.

Every fraction gives a decimal that either ends or repeats. There is no third kind.

The crew's sign-up sheet

These fractions are from the crew's own sign-up sheet for the kids on their street, not from any real survey.

GameFraction of kidsDivisionDecimalEnds or repeats
Ring toss3/83 ÷ 80.375ends
Paper ball toss1/31 ÷ 30.333...repeats
Relay7/207 ÷ 200.35ends
Seedling table5/65 ÷ 6??

Ring toss: 3/8 = 0.375. Relay: 7/20 = 0.35. Both end.

Paper ball toss: 1/3 = 0.333.... The remainder keeps coming back, so the 3 repeats.

WRITE THE DECIMAL
  • Read the question.
  • Tap your answer.
Write 3/8 as a decimal.
Write 5/6 as a decimal.
Write 7/20 as a decimal.
1 ÷ 8 = ?
ENDS OR REPEATS?
  • Read the question.
  • Tap your answer.
As a decimal, does 1/3 end or repeat?
As a decimal, does 7/20 end or repeat?
As a decimal, does 5/6 end or repeat?
As a decimal, does 2/11 end or repeat?

Scores with every operation

Thursday's scoreboard mixed everything. Rocket started a round with 10 points, lost 3 points twice, then gained 4.

Write it as one number sentence: 10 + 2 × (-3) + 4. Multiply first: 10 + (-6) + 4 = 8.

Nova's tally dropped by a total of 12 over 4 rounds. The change per round was -12 ÷ 4 = -3.

MIXED OPERATIONS
  • Read the question.
  • Tap your answer.
Rocket starts at 10, loses 3 points twice, then gains 4. What is his score?
A tally drops a total of 12 over 4 equal rounds. What is the change per round?
Raven scores 3 hits, 2 misses, then a 5 point penalty. Hits are 3, misses are -2. Total?
A thermometer fell 9 degrees over 4 hours at a steady rate. What was the change each hour?
Nova's tally changed by -15 over 5 rounds at a steady rate. What was the change per round? Type the number with its sign.
WHY THIS EXERCISEFinding the change per round is dividing a signed number, the skill the scoreboard needs most.
StatementTrue or false?
3 ÷ 8 = 0.375?
The decimal for 1/3 ends after three digits.?
10 + 2 × (-3) + 4 = 8?
-12 ÷ 4 = 3?
Every fraction gives a decimal that ends or repeats.?
WHY THIS EXERCISEReading a sheet of fractions as decimals lets you compare them at a glance.
Draw the long division for 1 ÷ 3 to four decimal places. Circle each remainder of 1.

Sharp work with the sign-up sheet. Tomorrow you find signed products in everyday life, and review the week.

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