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

Wednesday

Glass House Lab: the mixing jug
// Plant food in the tank
⏱ about 20 min

Wednesday: Glass House Lab, the Mixing Jug

Comet sets a clear jug and a stack of measuring cups on the potting bench. "Lab day. We mix for real."

A grown-up pours one cup of the crew's plant-food mix into the jug and screws the bottle shut.

"Call that full strength," Wren says. "Now add plain water one cup at a time. What do you notice?"

Comet adds a cup. "Two cups total. The mix is half as strong." Another cup. "Three cups. A third."

Nova projects the fractions above the jug as they go: 1, 1/2, 1/3, 1/4.

"Would you like a hint? Write the strength after x cups of water as one fraction."

"One over one plus x," Comet says. "Our tank rule again, with cups instead of liters."

"Then the lab and the chalkboard agree," Wren says, and writes the table in the log.

What you need

  • A clear jug, a set of measuring cups and a measuring pitcher.
  • One cup of plant-food mix from the Glass House shelf, poured by a grown-up, and plain water.
  • Your Glass House Log, a pencil and a towel for spills.
Safety first
A grown-up handles the plant-food bottle and pours the first cup. Plant food is for plants: nothing goes in the mouth.
Wash your hands after the lab, and wipe up any spill right away so no one slips.
Work at a bench, not on a ladder. Nothing heavy is lifted alone; a full jug is a two-person carry.

Run the lab

  1. Have the grown-up pour 1 cup of plant-food mix into the jug. Call this strength 1, or full strength.
  2. Add 1 cup of plain water. Stir gently. The jug now holds 2 cups, and the mix is 1/2 as strong.
  3. Add another cup and record the total cups and the strength as a fraction of the start.
  4. Keep going to 5 cups of water added. Fill in the table as you go.
  5. Write the rule for the strength after x cups of water. Check it against two rows.

The crew's lab table

These are the crew's own readings from Nova's log, made up for the Glass House. Yours should match, cup for cup.

Water added x (cups)Total (cups)Strength (fraction of start)
011
120.5
231/3
340.25
450.2
561/6

The strength after x cups of water is 1/(1 + x). After 4 cups it is 0.2, and after 5 cups it is 1/6.

The tank on Monday followed the same shape: (200)/(x + 20) percent, because 20 liters were there to start with.

Both rules are rational expressions. The bottom grows as you add water, so the strength falls but never reaches zero.

READ THE JUG
  • Read the question.
  • Tap your answer.
A tank holds 18 liters of 10% plant-food solution. The crew adds 6 liters of plain water. What is the new percent?
A tank holds 15 liters of 6% plant-food solution. The crew adds 5 liters of plain water. What is the new percent?
A tank holds 30 liters of 12% plant-food solution. The crew adds 10 liters of plain water. What is the new percent?
After 4 cups of plain water are added to 1 cup of mix, what fraction of the starting strength is the jug? Type the fraction.
WHY THIS EXERCISEThe rule 1/(1 + x) is a rational expression you can test with a jug. The lab and the algebra agree.
After 2 cups of water, the strength is what fraction of the start? Type the fraction.
After 5 cups of water, the strength is what fraction of the start? Type the fraction.
How many cups of water make the strength 1/4 of the start? Type the number.
StatementTrue or false?
After 3 cups of water, the jug holds 4 cups and the mix is 1/4 as strong.?
1 ÷ (1 + 5) = 1 ÷ 6?
Adding more water makes the strength rise.?
The strength rule 1/(1 + x) can reach exactly 0 for some number of cups.?
200 ÷ (5 + 20) = 8?
WHY THIS EXERCISEThe lab shows what the rule means: a fixed amount spread through a growing total.
Draw the jug after 0, 1, 2 and 3 cups of water. Write the strength fraction under each one.

Great lab work. Tomorrow you solve rational equations and meet a solution that is not really a solution.

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