Question
scenario: the stage
instructions:
view the video found on page 1 of this journal activity.
using the information provided in the video, answer the questions below.
show your work for all calculations.
analyze the students' conjectures
malcolm wants to build an outdoor stage with a total area of 350 square feet. the length of the stage should be 3 feet shorter than the width. he calculated the equation to be w2 − 3w = 350
1. complete the table to summarize each student's suggestion for figuring out the equation: (2 points: 1 point for each row of the chart)
student conjecture
malcolm's bandmate
malcolm
2. malcolm and his bandmate have different ideas for figuring out the equation. which one do you think will make it easiest to solve the equation? why? (2 points)
3. malcom's bandmate starts by making a factor table for w2 – 3w – 350 = 0.
he is looking for the factors (w + p)(w + q) = 0, where p • q = –350 and p + q = –3.
fill in the last row of the table with different factors, p and q, so that p • q = –350. (2 points).
p q p + q
10 –35 –25
–10 35 25
50 –7 43
–50 7 –43

4. the bandmate’s factor table is not complete. it does not contain all the factors of –350.
does the factoring table contain the factors that can be used to solve w2 – 3w – 350 = 0? explain your reasoning. (1 point)
5. to make a perfect square trinomial, malcolm said the rule for figuring out the number to add is . is he correct? if not, what is the rule? (1 point)
6. calculate the number that you need to add to each side of the equation w2 – 3w = 350 to create a perfect square trinomial. add this number to each side of the equation. show your work. (3 points)
7. factor the trinomial. (3 points)
8. now that you've factored the equation, find the square root of each side and solve for w. show your work and both solutions. (2 points)
9. do both of these solutions make sense in terms of the problem? why or why not? (2 points)
10. what are the maximum dimensions of the stage that malcolm can build? (round to the nearest foot.) (2 points)
w = feet
l = feet
prompt
audio:
o. k., everybody, it's "do-it-yourself" time! they say all it takes is the right tools, a little determination, and a good plan. well, i've got the lumber, i've got the power saw — i've even got the elbow grease! the only thing i don't have is a plan. [the images change to show a young man carrying a plank of lumber, then a circle saw, then of an up close shot of a nail with a hammer about to hit it and a person in the background driving the hammer.]
here's the situation: my rock band got permission from the community center to build an outdoor wooden stage so we can perform some concerts this summer. [a park-like area with lots of trees is shown with benches and a small wooden stage.]
i used all my powers of charm and persuasion to get a local lumberyard to donate some lumber. [a large red truck is shown filled with lumber.]
we have enough materials to build a stage with a total area of 350 square feet. but because of the way our band does our setup, the length of the stage should be three feet shorter than the width. [a grass area is animated with a dashed-line outline of a square with "350 ft2." wood is then shown inside the square but the right side elongates and "l = w – 3" is shown on the right side of the rectangle and "w" on the bottom of the rectangle.]
based on that info, i want to build the biggest platform i can. i think i can figure out the dimensions using a quadratic equation. (i know what you're thinking: he's musically talented and good at algebra? wow! )
here's what i've come up with so far:
we know that width times length equals total area. ["w • l = area" appears on-screen.] in this case, the length equals the width minus 3. so, width times width minus 3 should equal 350. [the following text appears on-screen: "l = w – 3; w(w – 3) = 350; w2 – 3w = 350"]
that means: w squared minus 3w equals 350. i've been arguing with my bandmate about what to do next.
[spoken by a different voice: ] hey, it's not an argument, it's just a lively discussion!
student: so what's your suggestion?
[spoken by a different voice: ] i think we need to subtract 350 from both sides, and then factor the equation. [w2 – 3w – 350 = 0 appears on-screen.]
student: i think it would be easier to factor if we change the equation so that it has a perfect square trinomial.
[spoken by a different voice: ] that sounds too complicated.
student: we just have to figure out what number will complete the square on the left side of the equation, and then add that number to both sides of the equation. i think the rule for figuring out the number is b divided by 2, squared. ["" is shown on screen.]