Lesson 14.4 — Build Sum Pyramids and Work Backward
第14讲 · 填数阵图 · 和数塔
Build every lower circle from the two circles above it, discover why the bottom uses the weights 1, 4, 6, 4, 1, and work backward from a target without assuming the answer is unique.
Expand the pattern → use the weights → solve the remaining total → audit every circle.
Read the sum-pyramid rule
Every lower circle is the sum of the two adjacent circles directly above it. Begin with a three-number pyramid before moving to the five-number practice problem.
Warm-up controls
Use whole numbers from 1 to 20. Changing a control changes the diagram; the questions below still use 2, 3, 4.
Check the fixed example 2, 3, 4
Need a next step?
Hint
Each lower circle has exactly two parents.
Reasoning and worked explanation
For top a,b,c, the bottom is (a+b)+(b+c)=a+2b+c. The middle appears twice.
Discover the five top-number weights
The middle top values travel to the bottom through more addition paths. Their influence is therefore larger.
Build the coefficient pattern
The last row tells how many addition paths connect each of the five top circles to the final bottom circle.
Bottom as a weighted sum
The center top number has six routes to the bottom; each endpoint has only one.
Enter the five coefficients
Need a next step?
Hint
Track one top value down all addition paths.
Reasoning and worked explanation
Neighboring path counts add: 1; 1,1; 1,2,1; 1,3,3,1; 1,4,6,4,1. These are the five weights.
Build the worked example target of 50
The worked example chooses three small middle values, then turns the remaining work into a much simpler endpoint-sum problem.
Five different positive whole numbers on top
Every lower circle is the sum of the adjacent pair above. The final circle is 50, and the other fourteen circles must all contain different numbers.
The worked example makes the strategic choice:
Weighted middle contribution
What remains for the endpoints?
The whole pyramid is not solved yet. We still have to test endpoint pairs and check all fourteen values for repeats.
Complete the arithmetic ledger
Need a next step?
Hint
Subtract the weighted middle contribution from 50.
Reasoning and worked explanation
4×2+6×1+4×4=30, leaving 20 for the two endpoints. This equation is necessary but does not check repeated values.
Test every endpoint orientation
The endpoints must sum to 20 and must avoid the already used values 1, 2, and 4. Left–right order can change the lower rows, so both orientations must be checked.
Tested 0 of 12 orientations
Need a next step?
Hint
Predict a repeated value before revealing the audit.
Reasoning and worked explanation
Generate all four rows above 50. A repeat anywhere in those fourteen values rejects the row, even when its weighted total is 50.
Your independent check
You may study a model. To complete this mission, also answer this new prediction and press the mission’s Check button.
Explain this prediction
The first addition is 1+2=3. Since 3 already occurs on top, the pyramid fails distinctness.
Audit the three original models
This worked example displays three valid pyramids for its chosen middle values. View each one and verify the bottom and distinctness conditions.
Current top row
Viewed 0 of 3 original models.
What do the worked example models prove?
Need a next step?
Hint
A model demonstrates existence; two different models disprove uniqueness.
Reasoning and worked explanation
Each displayed pyramid has bottom 50 and fourteen distinct upper values. Left–right order matters; check the reversed row separately.
Work backward when one top value is missing
Use the weights before generating the whole pyramid. The missing top value may have weight 1, 4, or 6 depending on its position. This mission only requires the sum rule and positive whole numbers; repeated values are allowed. Enter a target and four known values from 1 to 1000000.
One-unknown laboratory
Known weighted contribution: 50. Remaining: 14. Divide by weight 1.
Solve two fixed challenges
Need a next step?
Hint
Subtract known weighted terms, then divide by the missing weight.
Reasoning and worked explanation
A missing endpoint has weight 1. Positions 2 and 4 have weight 4; the center has weight 6. A whole-number result still needs any extra distinctness checks the problem requires.
Build a different valid pyramid
Build a solution different from the three worked examples in Mission 5. The checker accepts any positive whole-number top row that creates bottom 50, keeps all fourteen upper values different, and is not one of the three displayed original rows.
Live audit
Enter five top values.
Need a next step?
Hint
Choose small middle values so there is room left for the endpoints.
Reasoning and worked explanation
For middle 1,3,2, the contribution is 4+18+8=30. Endpoints must total 20. Test each orientation and reject any repeated upper value.
Explain a complete search; explore the full table
This added exploration turns “the answer is not unique” into a complete case audit. Mirror-image top rows count as different ordered rows, but they form pairs.
| # | a₁ | a₂ | a₃ | a₄ | a₅ | Row above bottom | original? |
|---|
Read the complete search
Need a next step?
Hint
First understand why the search has a finite endpoint.
Reasoning and worked explanation
Positive weighted terms must fit in 50. List middle triples, compute the remaining endpoint sum, test ordered endpoint pairs, then audit all fourteen values. The optional table checks that none were missed.
Fresh strategy workshop
Try these new situations before opening the explanations. Use only the sum rule unless a question explicitly asks about distinct values; repeated values are allowed in the other questions. Correct all eight answers to complete this workshop.
Worked explanation — try first
The next row is 5,7; bottom 12. Equivalently 4+2×1+6=12.
Worked explanation — try first
The center has weight 6: 6×2=12.
Worked explanation — try first
An endpoint has weight 1.
Worked explanation — try first
5+4×1+6×2+4×3+7=40.
Worked explanation — try first
The middle contribution is 8+18+16=42; 62−42=20.
Worked explanation — try first
Known contribution: 6+8+24+12=50. The endpoint is 10.
Worked explanation — try first
Known contribution is 5+18+16+7=46. (58−46)÷4=3.
Worked explanation — try first
It appears on the top and again as 1+2. Checking the top alone is not enough.
0 / 8
Independent exit ticket
Try these new situations before opening the explanations. Use only the sum rule unless a question explicitly asks about distinct values; repeated values are allowed in the other questions. All five correct completes this mission. The certificate requires all ten missions.
Worked explanation — try first
3+8+6+20+8=45.
Worked explanation — try first
54−(7+4+18+8)=17.
Worked explanation — try first
The fourth position has weight 4: 4×2=8.
Worked explanation — try first
Known terms total 39. (63−39)÷6=4.
Worked explanation — try first
All five top, four next, three next, and two next values must differ: 5+4+3+2=14.
0 / 5
Certificate of Completion
Sum-Pyramid Reverse Engineer
This certifies that
Grade 5 Mathematician
used influence weights, endpoint totals, systematic case testing, and complete distinctness audits to build and solve sum pyramids.
Teaching notes
The original derives the bottom expression a₁+4a₂+6a₃+4a₄+a₅, chooses a₂=2, a₃=1, a₄=4, and displays three valid pyramids with bottom 50.It explicitly says the answer is not unique.
Guided Practice 2 asks students to fill a similar pyramid with a solution different from Example 4. The live builder therefore validates the mathematical conditions rather than matching one hard-coded picture. The warm-up, one-unknown laboratory, complete enumeration, workshop, and exit ticket are added teaching scaffolds.
For this lesson, “natural number” is stated as a positive whole number 1,2,3,… to match the displayed original constructions.