After the rain

Work forward

Find one useful fact. Then ask what it lets you find next.

Amara adjusts a small greenhouse demonstration valve while Rowan prepares a planning board. Green coach CKSprout listens nearby.
The team tests its ideas before visitors arrive. The pictures set the scene; the problem cards give the exact amounts and rules.

Rain is clearing outside the community greenhouse. Amara and Rowan are preparing a demonstration about saving water for flowers. They have water models, a tour to organise and a watering plan to check. CKSprout is their coach.

Rowan “I can see lots of numbers. I can’t see the whole answer yet.”

CKSprout “What could you find from the facts you already have?”

Amara “One amount first. Then I’ll check which other part needs it.”

CKSTEM Thinking CompassStuck? Find your next move.

You don’t need the whole route yet. Start with one useful move, then notice what it tells you.

What do you know for certain? What are you trying to find?

The direction reflects your self-check. Use the lesson’s answer and reasoning to check further when you are ready.

One fact opens the next door

A model supply holds 42 litres of water. The team reserves 6 litres for later. All the remaining water is divided equally among 3 tanks.

Each tank then fills 4 identical pots equally, using all its water. There is enough space in the tanks and pots. No water is lost, and there are no other transfers.

How many litres does each pot receive? Explain a chain of useful facts that reaches that answer.

Known supply · unknown intermediate amounts

42 LWhole supply
Reserve 6 L
? LAfter reserve
Share equally into 3
? LIn one tank
Share equally into 4
? LFor one pot
Each arrow names a stated change. The two sharing steps refer first to tanks, then to pots from one tank.
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A question to help me begin

Which amount can you find before you know anything about one tank or one pot?

Show the answer and reasoning: one pot

Answer: Each pot receives 3 litres.

The whole answer is not needed to make a start. After reserving 6 litres, \(42-6=36\) litres remain. That new fact tells us how much can be shared among the three tanks.

Work forward · each result unlocks the next

42 LWhole supply
− 6 L
36 LAll three tanks
÷ 3 tanks
12 LOne tank
÷ 4 pots
3 LOne pot
The same unit, litres, describes a different whole or share at each step. Twelve pots receive three litres each.
\[\begin{aligned}42-6&=36,\\36\div3&=12,\\12\div4&=3.\end{aligned}\]

Amara “The 36 counts litres for all the tanks. The 12 counts litres in one tank.”

CKSprout “What does finding the amount in one tank let you work out next?”

Rowan “Its four equal pots. Twelve litres shared among four gives three litres in each.”

Check: the three tanks fill twelve pots altogether. \(12\times3=36\) litres go into those pots; adding the 6 reserved gives the original 42. Each step kept track of what its number counted.

Work forward

Use a fact you know in a relationship whose other needed information is also known. Find a justified new fact. Then ask what that fact unlocks.

Use a known fact. Which relationship can it help with?

Keep its meaning. Is it a total, a share, a count, a rate or a time?

Choose a useful next step. Does the deduction help reach the actual question?

A chain may split into branches or wait until two results can be brought together. You do not have to calculate in the order the clues are written.

Making a small, justified step is a useful way to begin when you feel stuck. But a calculation that has no connection to the question may not help. If a needed fact is missing, keep it unknown instead of inventing it.

Start with what you know helps choose a first solvable part when the whole plan is unclear. Work forward then follows useful consequences from established facts, checking every needed input. Neither tool permits guessing a missing fact.

Two branches meet

A second supply holds 64 litres. After reserving 8 litres, the team sends three sevenths of the remaining water to North and all the rest to South.

Each North pot needs 4 litres; each South pot needs 8 litres. There are enough pots. Every allocated litre is used in full pots, with no loss or other transfer.

How many pots can be filled at North and at South, and how many is that altogether?

Shared starting facts

64 L supply
8 L reserved

North

3/7 of the remainder
4 L per pot

South

All the other water
8 L per pot

Both destinations depend on the amount left after reserving. Their different pot sizes matter before their counts can be combined.
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Show the answer and reasoning: both branches

Answer: 6 North pots and 4 South pots, making 10 pots altogether.

First find the amount being split: \(64-8=56\) litres. One seventh is \(56\div7=8\) litres, so North receives \(3\times8=24\) litres. South receives \(56-24=32\) litres.

8 LN
8 LN
8 LN
8 LS
8 LS
8 LS
8 LS
The fifty-six litres form seven equal parts. The first three labelled N go North; the four labelled S go South.

That is not yet a pot count. Carry each amount through the relationship for its own destination.

North

24 L ÷ 4 L per pot

6 pots
South

32 L ÷ 8 L per pot

4 pots

6 pots + 4 pots = 10 pots

Only combine the results after both branches count pots. The thirty-two litres in South is an intermediate amount, not a pot count.
\[\begin{aligned}24\div4&=6,\\32\div8&=4,\\6+4&=10.\end{aligned}\]

Rowan “I found six North pots. South still had thirty-two litres.”

CKSprout “How could you check whether your combined result answers the question?”

Amara “Pots. I need to turn South’s litres into four pots before adding.”

Check: \(6\times4+4\times8=24+32=56\) litres reach the pots. With the 8 reserved, that accounts for all 64. The branches used different amounts per pot, so dividing all 56 litres by one pot size would not fit the rules.

What do these facts decide?

A different model shares 36 litres between North and South. This time a North pot needs 5 litres and a South pot needs 3 litres.

There must be at least one whole pot at each destination. All 36 litres are used in full pots, with no loss or other transfer. There are enough pots. No rule tells us how the water is split.

Can you determine one unique total number of pots? If not, give two allowed water allocations and their pot counts that show why not.

CKSTEM Thinking CompassPlan and check your reasoning.

You don’t need the whole route yet. Start with one useful move, then notice what it tells you.

What do you know for certain? What are you trying to find?

The direction reflects your self-check. Use the lesson’s answer and reasoning to check further when you are ready.

Water to share

36 L in total

Split not specified
North

5 L per full pot

At least one pot

South

3 L per full pot

At least one pot

The total and pot sizes are given. A split rule is not.
Show the answer and reasoning: the missing split

Answer: No. One allowed allocation fills 10 pots; another fills 8. The total is not uniquely determined.

Allowed allocation 1

3 North pots + 7 South pots = 10 pots

Allowed allocation 2

6 North pots + 2 South pots = 8 pots

Both bars represent the same thirty-six litres on the same scale. N labels North and S labels South. Different valid allocations produce different pot totals.

For the first, North’s 15 litres fill \(15\div5=3\) pots and South’s 21 fill \(21\div3=7\): ten altogether. For the second, 30 litres fill six North pots and 6 litres fill two South pots: eight altogether.

Check: both allocations use exactly 36 litres, fill only whole pots and give each destination at least one. Two different valid totals are enough to show that a unique answer is impossible from these facts.

The earlier problem gave a split rule; this one does not. Knowing a total does not entitle us to assume an equal split. Forward reasoning must wait for enough information—or explain what remains undecided.

Will the team be ready?

On the same morning as the tour, setup begins at 9:00. Filling the display takes 12 minutes, and testing it takes 8 minutes after filling finishes.

Meanwhile, another person prepares the signs, starting at 9:00 and taking 15 minutes. This work is independent of filling and testing. A 4-minute team rehearsal can begin only when both the tested display and the signs are ready.

After rehearsal, everyone needs 5 minutes to walk to the entrance and then 3 minutes to welcome the visitors before the tour begins at 9:30. All tasks take exactly these times, may start as soon as their prerequisites finish, and have no other delays. Setup cannot begin before 9:00.

Can the tour begin at 9:30 under this plan? What is the earliest possible tour start, and how early or late is it compared with 9:30? For a future morning only, suppose the 9:00 start restriction is removed and both independent branches start together. What is their latest common start time that would let the tour begin at 9:30?

Start: 9:00

Display branch

Fill: 12 min

then test: 8 min

Signs branch

Signs: 15 min

Independent of display work

Both finished → rehearse 4 min

Then

Walk: 5 min

Then

Welcome: 3 min

Required tour

9:30 start

Independent branches may run together. The rehearsal depends on both, and the walk and welcome come after it.
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A question to help me begin

Which jobs can happen at the same time, and which jobs have to wait for another one to finish?

Show the answer and reasoning: the tour

Answer: No. The earliest tour start is 9:32, which is 2 minutes late. On the future morning, the latest common setup start is 8:58.

Work forward: filling finishes at 9:12, so testing finishes at 9:20. Signs are ready at 9:15. Rehearsal must wait until the later of those two finishes, 9:20; it ends at 9:24.

Work backwards

The future morning asks a different question: how late may setup begin while still meeting the fixed finish? Carry that required finish backwards through the dependencies.

The requested tour start gives us another useful fact. To begin at 9:30, the welcome must start by 9:27. The team must therefore leave the greenhouse, with rehearsal complete, by 9:22.

Work backwards · latest permitted times

9:30Tour begins
Subtract 3 min
9:27Welcome starts
Subtract 5 min
9:22Leave greenhouse
The team must be ready to leave by 9:22 to meet the requested 9:30 tour start.

Amara “Rehearsal must finish by 9:22, so it must begin by 9:18. Both branches must be ready then.”

Rowan “Signs could start at 9:03. But filling and testing need twenty minutes before 9:18, so they must start by 8:58. Starting both at 9:03 would leave the display late.”

CKSprout “Which of those limits controls a common start, and does your forward check confirm it?”

The common start must meet both branch deadlines, so choose 8:58, the earlier limit. From 8:58, signs finish at 9:13 and the tested display at 9:18; rehearsal, walk and welcome then reach exactly 9:30.

The earliest ready time, 9:24, misses that latest departure time by two minutes. Carry the actual plan forward: walk to the entrance by 9:29, then welcome visitors until 9:32.

9:009:089:169:249:32

Fill · 9:00–9:12

Test · 9:12–9:20

Signs · 9:00–9:15

Rehearse · 9:20–9:24

Walk · 9:24–9:29

Welcome · 9:29–9:32

Dashed line: requested 9:30 start. Every row uses the same clock scale.

Fill and test together finish at 9:20; signs finish earlier. Rehearsal ends at 9:24, and the walk plus welcome take the plan to 9:32.

Amara “I almost added fifteen minutes for the signs after testing.”

CKSprout “What does the plan say about when that work happens?”

Rowan “It happens alongside the display work. We wait for both branches, so rehearsal begins at 9:20, not 9:35.”

Check: the longest branch before rehearsal is \(12+8=20\) minutes. The signs need only 15. Then \(20+4+5+3=32\) minutes are needed from 9:00. The forward route and the backward deadline comparison agree. Starting any task later cannot improve this earliest time.

The tools meet at one question: Is our earliest ready time no later than the latest time we can leave?

Revisit Work backwards when a required finish helps you find an earlier deadline.

The last watering plan

A steady pump delivered 15 litres in 5 minutes during a calibration test. It keeps that same constant rate whichever outlet is used.

Now a supply holds 30 litres, and at least 8 litres must stay in it. North and South run one at a time. For each outlet, choose exactly 2, 3 or 4 minutes; it must run for its full chosen time.

Every North pot takes 2 litres; every South pot takes 3 litres. There are enough empty pots. All delivered water must go into whole, full pots at that destination: no partial pots, unused delivered water, rounding or other transfers.

How many pairs of North and South run times meet every rule? Find every such pair that fills at least 4 North pots and at least 3 South pots while keeping the required reserve. For each valid pair, give the pot counts and litres left in the supply.

Calibration

15 L in 5 min

Rate: ? L/min
Supply

30 L at the start

Keep at least 8 L

Choices

North: 2, 3 or 4 min

South: 2, 3 or 4 min

Choose one full duration for each outlet. The pump rate has not yet been worked out.
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Show the answer and reasoning: all valid settings

Answer: There is one valid pair: North 4 minutes, South 3 minutes. It fills 6 North pots and 3 South pots, leaving 9 litres in the supply.

The constant calibration rate is \(15\div5=3\) litres per minute. Each duration therefore determines an amount, then a destination-specific pot count.

Start with what you know

Amara “I can settle North’s part before pairing it with South. Two minutes gives six litres—only three pots. Three minutes gives nine litres, which cannot fill whole two-litre pots. Four minutes gives twelve litres and six pots.”

Rowan “So every valid plan must run North for four minutes. That leaves only South to choose, then we check the reserve shared by both.”

CKSprout “Which rules let you settle that part independently, and which condition must still wait for the combined choice?”

North’s minimum and whole-pot rules concern North alone. Choosing that subproblem removes six impossible pairs before any repeated combined-reserve calculation.

Work forward: each chosen number of minutes now tells us its delivered litres. That amount tells us a pot count—or reveals that a whole-pot rule fails. Combining the two delivered amounts tells us what remains in the supply.

2 minutes

6 L delivered

North: 3 pots

South: 2 pots

3 minutes

9 L delivered

North: 4½ potloads · not whole

South: 3 pots

4 minutes

12 L delivered

North: 6 pots

South: 4 pots

The rate gives litres, then each outlet’s pot size gives its candidate count. A fractional potload cannot satisfy the full-pot rule.

Make an organized list

North is forced to four minutes, but a working South trial would not settle “every pair”. Hold that North choice fixed and cover South at two, three and four minutes. These are all the remaining possibilities. Read each card as “If we choose these times, then these amounts follow”.

N 4 min · S 2 min

North: 12 L → 6 pots

South: 6 L → 2 pots

Supply left: 12 L

× Reject
  • Too few South pots
N 4 min · S 3 min

North: 12 L → 6 pots

South: 9 L → 3 pots

Supply left: 9 L

✓ Fits every rule
N 4 min · S 4 min

North: 12 L → 6 pots

South: 12 L → 4 pots

Supply left: 6 L

× Reject
  • Reserve below 8 L
North’s separate rules have forced four minutes. All three remaining South choices are shown once; only N4/S3 meets every condition.

The valid pair sends \(3\times4=12\) litres North and \(3\times3=9\) litres South. North fills \(12\div2=6\) pots; South fills \(9\div3=3\). The supply retains \(30-12-9=9\) litres.

Check: six is at least four, three meets the South requirement, all pots are whole, both run times are allowed, and nine litres is at least the required eight. A three-minute North run delivers nine litres, which cannot all fit into whole two-litre pots. Four minutes at both outlets fills enough pots but leaves only six litres, so it fails the reserve.

The three tools share different jobs: Start with what you know chooses a solvable first part; Work forward carries its result through the other relationships; Make an organized list makes the search complete. A shorter argument can rule out some choices early; it still needs to cover every allowed choice.

Revisit Make an organized list when you need every possibility, rather than just one successful choice.

Rain today, water for later

Rainwater collection stores water for later use, such as watering outdoor plants. Keeping some water for another day also gives the team a reason to test its reserve, instead of using every litre simply because it can.

The model’s exact amounts and pump rate come from its stated rules. A plan for a real place would need real measurements too.

Your turn

Think through each problem before opening its answer. Choose a useful next fact and keep track of what it means. No writing is needed.

Clips for the music corner

A club has 96 minutes of audio. It makes 4-minute clips, using all the audio exactly once, then puts 3 clips in each set, using every clip exactly once. It keeps 2 complete sets for its own display.

How many complete sets are left for visitors? Explain the useful intermediate counts.

Known recording · requested visitor sets

96 minWhole recording
4 min per clip
?Clips
3 clips per set
?Complete sets
Keep 2 sets
?Visitor sets
Each step changes what is being counted. Two complete sets, not two clips, are kept.
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Show the answer and reasoning: the music sets

Answer: 6 complete sets are left for visitors.

First \(96\div4=24\) clips; then \(24\div3=8\) sets. Keeping two leaves \(8-2=6\).

Work forward · keep the unit meanings

96 minWhole recording
÷ 4 min per clip
24Clips
÷ 3 clips per set
8Complete sets
− 2 sets
6Visitor sets
The 24 clips form 8 sets. Keeping 2 sets leaves 6 sets for visitors.

Check: the six visitor sets and two kept sets contain \(8\times3=24\) clips, accounting for \(24\times4=96\) minutes. Subtracting two from 24 would confuse sets with clips.

Is the green lamp certain?

In a model, a green lamp is on exactly when both conditions hold: the water level is at least 10 litres and the lid is closed. Nothing else controls the lamp.

The level is 12 litres, but we do not know whether the lid is open or closed.

Can we be certain the lamp is on? Explain what can happen under each allowed lid state and what information is still needed.

CKSTEM Thinking CompassPlan and check your reasoning.

You don’t need the whole route yet. Start with one useful move, then notice what it tells you.

What do you know for certain? What are you trying to find?

The direction reflects your self-check. Use the lesson’s answer and reasoning to check further when you are ready.

Level

12 L

Rule needs at least 10 L

LidState unknown
Lamp rule

Both level and lid conditions must hold

The level is given. Whether the lid is closed has not been established.
Show the answer and reasoning: the lamp

Answer: No. With the lid closed, the lamp is on; with it open, the lamp is off. We still need to know the lid state.

Lid closed

12 L meets the level rule

Lamp on
Lid open

Lid condition fails

Lamp off
Both cases fit the known level. A conclusion requiring two facts cannot be certified from only one.

Twelve meets the level condition. It does not settle the lid condition. The rule needs both before its “on” conclusion follows.

Check: both lid states are consistent with the given 12 litres. Forward reasoning preserves that uncertainty; it does not turn one satisfied condition into two.

One printer, four badges

One morning at 10:00, one printer begins making 4 badges. Each takes 6 minutes, and the printer makes only one badge at a time.

After all four are printed, decorating the whole batch takes 10 minutes. Packing then takes 8 minutes after decorating finishes. There are no other delays, and each stage starts as soon as possible.

What is the earliest time the packed badges are ready? Can they be ready for a 10:40 pickup that morning, and if not, how late are they?

Print

One printer

4 badges · 6 min each

Then decorate

10 min after all printing

Then pack

8 min after decoration

The four print jobs share one printer. Decoration waits for all four; packing waits for decoration.
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Show the answer and reasoning: the badges

Answer: 10:42. No, they cannot meet the 10:40 pickup; they are 2 minutes late.

The one printer needs \(4\times6=24\) minutes, reaching 10:24. Decorating ends at 10:34, and packing ends at 10:42.

Work forward · one printer, successive stages

10:00Printing starts
4 × 6 min
10:24All 4 printed
+ 10 min
10:34Decoration finished
+ 8 min
10:42Packed and ready
The equipment permits only one badge at a time. The earliest final time is 42 minutes after 10:00.

Check: \(24+10+8=42\) minutes after 10:00 is 10:42. Unlike independent signs and display work, the four badges cannot all print at once. The equipment rule changes which steps may overlap.

Does every step follow from the facts?

Each new fact should have a reason and a meaning. Ask what it unlocks, whether all the needed information is present and whether it brings you closer to the actual question.

At a fork, follow the useful branches. At a meeting point, wait for every needed input and use compatible units. If a link is missing, say what remains unknown.

Our next focus is Start with what you know: choose a part you can settle even when the whole problem still looks difficult.

CKSTEM Thinking Framework

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