Two ways to buy pizza for your party. Figure out which deal saves money and exactly when the winner changes.
The number of slices I eat is the because I get to choose it.
The total cost is the because it changes based on how many slices I eat.
In y = kx, k = $3 per slice is the — it tells me how much cost goes up per slice.
x = because that is what I choose.
y = because it depends on how many slices I eat.
For Deal A, as x goes up by 1 slice, y goes up by $.
For Deal B, as x goes up, y because the price is always $.
| Slices — x | Deal A Cost ($) | Deal B Cost ($) |
|---|---|---|
| 1 | ||
| 2 | ||
| 4 | ||
| 6 | ||
| 8 |
Both deals cost the same at slices, where both equal $.
At 2 slices, Deal is cheaper by $.
k = $3 means for every 1 slice I add, the total cost goes up by $.
Deal B's cost is always $18, so Deal B is — cost doesn't change with x.
Use Deal A's equation to find the cost for 5 slices:
y = 3 × 5 = $. So 5 slices costs $ with Deal A.
What shape does Deal A's line make, and why?
Deal A makes a line because its cost
What shape does Deal B's line make, and what does that tell us?
Deal B makes a line that goes because each extra slices adds $
Do the lines cross? If yes, at approximately what slices amount?
Yes / No — the lines look like they cross near slices
What does the crossing point mean in this situation?
At that crossing point, both plans cost the same ($). Before it, is cheaper. After it, is cheaper.
If eating fewer than slices, choose Deal because it costs less.
If eating more than slices, choose Deal because it costs less.
My table shows: at 6 slices, Deal A = $ and Deal B = $, so at exactly 6 slices both deals cost the same.
My graph shows: Deal A's line goes as x increases, which means more slices = cost.
x = ___ because ___ . y = ___ because it depends on x.
Deal A's equation is y = ___ x. The k value of ___ means that for every slice, the cost goes up by $___ .
For a party of 5 people eating 2 slices each (10 slices total), I recommend Deal ___ because my table shows ___ and my graph shows ___ .
Two ways to buy pizza for your party. Figure out which deal saves money and exactly when the winner changes.
What is the independent variable? Write a complete explanation.
The independent variable is because I get to choose how many slices to eat.
What is the dependent variable? Explain why it depends on x.
The dependent variable is because it changes based on how many slices I eat.
If x goes up by 1 slice, y goes up by $ for Deal A because .
| Slices — x | Deal A ($) | Deal B ($) | Cheaper Deal |
|---|---|---|---|
| 1 | |||
| 3 | |||
| 5 | |||
| 6 | |||
| 8 | |||
Describe the pattern as slices increase. What happens to each deal's cost and why?
As slices increase, Deal A's cost because .
Deal B's cost because .
They behave differently because Deal A is while Deal B has a cost.
What is Deal A's k value and what does it mean in the context of pizza?
Deal A's k = $. This means for every slice I add, the total cost goes up by $.
So k is the — it is the rate of change because for every 1 extra slice, y increases by $.
Why doesn't Deal B's equation include x? Is Deal B proportional?
Deal B's equation is y = 18 with no x because the cost no matter how many slices.
Deal B is proportional because a proportional relationship requires y to when x changes.
Use Deal A's equation to find the cost for 7 slices:
y = 3 × 7 = $. So 7 slices costs $ with Deal A.
Give the approximate coordinate where the two lines cross:
The lines cross at approximately ( slices , $ )
What does that crossing point mean for someone choosing between Deal A and Deal B?
At that point, both options cost $. If you need fewer than slices, use because it's cheaper.
If you need more than slices, switch to because
Describe Deal A's line shape and explain it using the equation:
Deal A's line is because its equation is y = , which means the cost
never / always changes as x increases.
What does the steepness of Deal B's line tell you about its k value?
The steeper the line, the the k value. {plan_b}'s k = , so every unit of x adds $ to the cost.
My table shows the Cheaper Deal column switches from Deal to Deal at slices.
At 5 slices, Deal A = $ and Deal B = $, a difference of $.
Deal A's equation y = 3x shows k = $3 per slice. Deal B is always $18, so Deal B is cheaper when x is than slices.
After slices, Deal is cheaper because at that point 3 × x exceeds $18.
On my graph, the lines cross at ( slices, $ ).
To the left of that point, Deal is cheaper. To the right, Deal is cheaper.
For a party eating about slices, I recommend Deal because
Deal A's k = $3 per slice. Deal B's cost never changes with x. This tells me Deal A grows ___ expensive faster. For small slice counts, Deal ___ is cheaper because ___ .
My graph's crossing point is at ( ___ slices, $___ ). To the left use Deal ___ . To the right use Deal ___ because ___ .
For 4 slices: Deal A = 3 × 4 = $___ . Deal B = $___ . The cheaper deal is ___ , saving $___ .
Deal C: $5 delivery fee + $2.50 per slice. Equation: y = 2.5x + 5. What is Deal C's k value? Find the cost for 4 slices. Add Deal C to your graph.
Deal C's k = $ per slice. At 4 slices: y = 2.5 × 4 + 5 = $ + $5 = $.
On my graph, Deal C starts at y = $ and rises than Deal A.
Three ways to buy pizza. Compare all three and find out which deal is cheapest — and when the winner changes.
Before calculating: predict which deal is cheapest at 2 slices and at 8 slices. Use vocabulary terms.
At 2 slices I predict Deal is cheapest because its k (rate of change) = $ per slice, which means .
At 8 slices I predict Deal wins because .
Do you think one deal is always cheapest, or does the winner change?
I think the winner because Deal A charges per slice while Deal B has a fee.
Deal C starts with a $5 fee but has a lower k ($2.50 vs $3), which means it might win .
| Slices — x | Deal A ($) | Deal B ($) | Deal C ($) | Winner |
|---|---|---|---|---|
| 1 | ||||
| 3 | ||||
| 5 | ||||
| 6 | ||||
| 8 |
My prediction was because .
I was surprised that at slices, Deal was cheaper than I expected because .
The Winner column changes times, creating different zones.
Compare Deal A's k and Deal C's k. What does a smaller k value mean?
Deal A: k = $ per slice. Deal C: k = $ per slice.
Deal C has a k, which means its cost grows as slices increase.
A smaller k = lower rate of change, so for large slice counts, Deal will be less expensive.
Which deals are proportional? Which are not? Explain using the equation form.
Deal A is proportional because its equation is y = 3x — it's in form. If x = 0, y = $.
Deal C is proportional because of the + 5 — even at x = 0 slices, you'd still pay $.
Deal B is proportional because y = 18 on x at all.
Use Deal C's equation to find the cost for 6 slices:
y = 2.5 × 6 + 5 = $ + $5 = $. So 6 slices with Deal C costs $.
Deal C charges $12.50. How many slices was that? Steps set up for you:
Start: 12.50 = 2.5x + 5. Subtract 5: = 2.5x. Divide by 2.5: x = slices.
Where do Deal A and Deal B cross? What does that mean?
Deal A and Deal B cross at about ( slices , $ ).
At that point both cost exactly $. Before it is cheaper; after it is cheaper.
Where do Deal A and Deal C cross? What does that mean?
Deal A and Deal C cross at about ( slices , $ ).
This means at that point both cost exactly $. Before it is cheaper; after it is cheaper.
Describe the three zones your graph creates — which plan is cheapest in each?
Zone 1 (0 to slices): is cheapest because its line is the lowest here.
Zone 2 ( to slices): wins because
Zone 3 (above slices): is cheapest because
Why is the graph more useful than just the table for finding these zones?
The graph shows all three lines at once, so I can see the crossing points at a glance. With only the table, I would have to
The Winner column changes times, creating zones.
Deal wins at small slice counts (under slices) because .
Deal A: k = $3/slice. Deal C: k = $2.50/slice. Deal B: constant $18.
Because Deal A's k is than Deal C's k, Deal A becomes more expensive as slices increase.
Deal C's $5 starting fee means it starts higher, but its lower k means it eventually .
My graph has crossing points, creating zones.
Zone 1 (1 to slices): Deal is cheapest.
Zone 2 ( to slices): Deal wins.
Zone 3 (above slices): Deal is cheapest because .
Small party (1– slices): use Deal .
Medium party (– slices): use Deal .
Large party (more than slices): use Deal because .
Deal A's k = $3/slice and Deal C's k = $2.50/slice. This means Deal ___ grows more expensive faster because ___ . For large parties, Deal ___ becomes a better choice because ___ .
At 5 slices: Deal A = 3 × 5 = $___ . Deal B = $___ . Deal C = 2.5 × 5 + 5 = $___ . The cheapest at 5 slices is Deal ___ .
My graph has ___ zones. Zone 1 (1 to ___ slices): Deal ___ cheapest. Zone 2 ( ___ to ___ slices): Deal ___ cheapest. Zone 3 (above ___ slices): Deal ___ cheapest.