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Algebra 2 9-12 / Week 04 / Garden Day
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Week 04 · Rational Expressions and Equations

Garden Day

Garden Day: the family mixing table
// Plant food in the tank
⏱ about 20 min

Garden Day: The Family Mixing Table

Saturday the measuring pitcher comes home, and the kitchen table becomes a small Glass House.

"Each card has a tank," Comet says. "Liters, percent, and water added. You find the new percent."

The first card reads 20 liters at 10 percent, add 5 liters. Someone calls out "Eight!" at once.

"How did you know?" Wren asks. "Two liters of plant food in twenty-five," comes the answer.

Nova hovers above the cards with a soft hint. "Would you like to check it with the rule 200/(x + 20)?"

"200 over 25 is 8," Comet says. "The card and the rule agree."

"Next card is backwards," Wren says. "It gives the percent you want. You find the water. That is an equation."

  1. A rational expression is a polynomial over a polynomial. A zero denominator is never allowed.
  2. Simplify by factoring and canceling whole factors. Multiply by factoring first. Add with a common denominator.
  3. Divide to rewrite a(x)/b(x) as q(x) + r(x)/b(x).
  4. Solve a rational equation by clearing denominators, then check. Throw out extraneous solutions.
  5. Mixing and shared-work problems become rational equations: plant food over total, and rates that add.
◇ FAMILY MIXING CARDS
Write five tank cards: liters, percent and water added, like "20 L at 10%, add 5 L." Keep the numbers friendly.
Deal a card. Everyone finds the new percent two ways: plant food over total, and the rule 100 × food/(water + liters).
For a harder round, write the percent you want and ask how much water to add. Solve it as an equation.
Trade roles so everyone writes a card and everyone solves one. Check each answer with the pitcher if you can.
For a grown-up
This is a paper activity with water only. If the family mixes real plant food, a grown-up pours it and nobody tastes anything.
A wrong answer is a good moment: ask how many liters of plant food are in the tank, and start from there.
Every number on the cards is made up for the game, not a fact about any real plant food.
GARDEN DAY CHECK
  • Read the question.
  • Tap your answer.
A tank holds 25 liters of 8% plant-food solution. The crew adds 15 liters of plain water. What is the new percent?
The 20-liter, 10% tank gets 30 liters of water. What is the new percent?
A card asks for 8%: solve (200)/(x + 20) = 8. How many liters of water?
A card asks for 2% from the 20-liter, 10% tank. How many liters of water are added? Type the number.
WHY THIS EXERCISERunning the mixing rule backwards is a rational equation. Clear the denominator, solve, and check.
Draw two of the family cards with their tanks and write the new percent beside each one.
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