094i - Consolidation Formulas and Digital Tools

Lesson 94 — Problem Solving and Consolidation: Formulas and Digital Tools

Strand: Algebra | Descriptor: AC9M7A06 | Duration: 45 minutes

Learning Intentions

• To consolidate manipulating multi-variable formulas and exploring them systematically.
• To use a formula, a table and a digital tool as three views of one relationship.

Success Criteria

I can:

1. Find any variable in a multi-variable formula.
2. Build and read a table generated by systematic variation.
3. Describe an effect as additive or multiplicative, with a structural reason.
4. Reconstruct a formula from generated results.

Warmup

(6 minutes — three-way check, mini whiteboards)

For :

1. when .
2. when .
3. The effect of one extra hour.
4. Does doubling double ? Give the structural reason.
5. Write it as a spreadsheet formula with in A2.

Answers: 1. 13318h = 162 \Rightarrow h = 9$18$25$ never doubles; 5. =25+18*A2.

The five questions are the whole block: forward, backward, step, scaling, digital. Every task today asks some combination.

Activities

Activity 1 — Mixed Skills Circuit (16 min)

Stations or graded worksheet spanning Lessons 91–93.

Station A — Find any variable.

1. : find when , , .
2. : find when , , .
3. : find when , , .
4. : find when , and .

Station B — Describe the effect.

5. : effect on of adding to .
6. : effect on of doubling and doubling .
7. : effect of tripling from to — state the actual outputs.
8. : effect on of halving .

Station C — Spreadsheet translation.

9. Write as a spreadsheet formula (inputs A2, B2, C2).
10. A grid formula reads =12+3*$A2+5*B$1. What are the row-step and column-step, if column A steps by and row 1 steps by ?
11. Why would =12+3*$A$2+5*B$1 ruin the grid?

Station D — Reconstruct.

12. A formula gives , , . Name it.
13. A table for a two-variable formula shows and . Find the coefficient of the first variable.

(Answers: 1. ; 2. ; 3. years; 4. total ; ; 5. adds ; 6. ; 7. — not tripled (), the stands still; 8. ; 9. =A2*B2*C2/2; 10. row-step , column-step ; 11. $A$2 freezes the first input, so every row uses the same value; 12. ; 13. .)

Activity 2 — The Extended Task (16 min)

Pairs. One task, carried through all three representations.

The school camp costing.

The camp cost is modelled by

where is the number of students and the number of teachers. The school requires one teacher per students (rounded up).

1. Interpret each of the three numbers.
2. Find for students with the required teachers.
3. Build (or tabulate by hand) the cost for , computing correctly each time.
4. Describe the effect of each extra student. Is the effect steady across your table? Explain any jumps.
5. The budget is 7000$. What is the largest group that fits?
6. Write the spreadsheet formula you would use, given in A2 and in B2.

Socratic scaffolding for Q4:

Prompt	Purpose
From the formula alone, what does one extra student add?	85$.
Check your table: is the step per student always 85$?	No — every time ticks up, an extra 120$ lands.
Where do the jumps fall?	At — each new teacher.
So is the model additive or something else?	Additive in each variable, but depends on — the ceiling creates a staircase.
How would you describe it honestly in one sentence?	”Each student adds $85, plus $120 every time the group crosses a multiple of 12.”
Looking back	Real models often chain variables. The formula alone under-describes it — the table revealed the staircase.

Answers: 1. 450$85$120t = \lceil 48/12 \rceil = 4C = 450 + 4080 + 480 = $5010s=24t=2$2730s=36t=3$3870s=48t=4$5010s=60t=5$6150s=72t=6$7290s = 60$6150s = 72s = 66t = 6450 + 5610 + 720 = $6780s = 69$703568t=6450 + 5780 + 720 = $6950$ ✓. Largest group: 68 students; 6. =450+85*A2+120*B2.

Note on Q5: the answer requires trial across the staircase — a legitimate use of a table or sheet, and a good argument for building one.

Activity 3 — Inquiry: Break the Model (7 min)

Whole class, quick.

Our camp model says .

1. Name two costs a real camp has that this model ignores.
2. Would adding them change the structure or just the numbers?
3. The bus seats . What does that do to the model?

Discussion targets: ignored costs — transport, dietary variations, equipment hire, deposits. Most change only numbers (bigger constant, bigger per-student rate). But the bus is structural: at students a second bus appears — another staircase, like the teachers. Structural features are the ones a formula alone hides. (Bridges to L74–77’s modelling assumptions and L82’s constraints.)

Checks for Understanding

(5 minutes — exit ticket, collected)

1. : find when , , .
2. : state the effect of each extra , and whether doubling doubles .
3. Write as a spreadsheet formula.
4. A formula gives , , . Name it.
5. Reasoning. In the camp model, why isn’t the cost per extra student exactly 85$ at every group size?

Answers: 1. ; 2. Adds 35$8015 + 9a + 6b12$120$85$.

Common Misconceptions

Misconception	How to pre-empt it
Assuming the per-unit effect from the formula holds in every real table.	The camp staircase — the table contradicts the naive reading.
Rounding teachers to nearest rather than up.	The ceiling requirement, as in L82’s supervising adults.
Freezing an input with $A$2 in a grid.	Station C Q11, and L92’s demonstrated breakage.
Doubling an input and expecting a doubled output regardless of structure.	Station B Q7 gives the actual numbers.
Reconstructing a formula from too few results.	Station D needs one result per unknown — L93’s minimal-experiment finding.
Believing a formula fully describes a situation.	The break-the-model inquiry.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). with , , . Find .

Answer

.

E2 (AMC Junior style). A formula returns , , . Reconstruct it.

Answer

-step ; -step ; base : . (Check : ✓)

E3 (Challenge). In the camp model, at which group size does the average cost per student first fall below 100$?

Answer

Average . At : 113.75s = 48$104.38s = 72$101.25$450$ spreads ever thinner, but each new teacher nudges the average back up. Testing near the crossing:

average	100.33$	100.16$	100.00$	99.84$	99.69$

At the average is exactly 100s = 95$**. (A spreadsheet finds this in seconds — a strong argument for building one.)

E4 (Challenge). A box has . For a box, find ; then find if every dimension is tripled, and state the factor.

Answer

. Tripled (): — factor , since each term is a product of two lengths.

Homework

1. Find the requested variable: (a) , , , — find (b) , , , — find (c) , , , — find as a percentage.
2. For : (a) effect of each extra (b) at and (c) did doubling double ? Explain structurally.
3. For (fixed cross-section): (a) effect of each extra (b) effect of quadrupling (c) why do both descriptions agree here?
4. Write as spreadsheet formulas: (a) (b) (c) .
5. A grid uses =5+2*$A2+4*B$1 with column A stepping by and row 1 by . (a) The row-step. (b) The column-step. (c) The value when and .
6. Reconstruct the formula from , , .
7. The camp model with one teacher per students: find the cost for students and for students, and explain the size of the jump.
8. Reasoning. Explain why a model can be “additive in each variable” and still have a staircase in its table.
9. Challenge. For , a cube has . Find its side length, and the surface area if the side is doubled.

Answers: Q1 — (a) (b) (c) . Q2 — (a) 45$255$390$120+8\times 42125 + 8 + 36 = 49a= 21/3 = 7b= 12/2 = 6y = 32 + 7a + 6bs=30t=3450 + 2550 + 360 = $3360s=37t=4450 + 3145 + 480 = $4075$7157 \times 85$1206s^2 = 150 \Rightarrow s = 56(100) = 600$, four times as much.