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# Painting a Barn

Alignments to Content Standards: 6.G.A 6.RP.A.3

Alexis needs to paint the four exterior walls of a large rectangular barn. The length of the barn is 80 feet, the width is 50 feet, and the height is 30 feet. The paint costs \$28 per gallon, and each gallon covers 420 square feet. How much will it cost Alexis to buy enough paint to paint the barn? Explain your work. ## IM Commentary The purpose of this task is to provide students an opportunity to use mathematics addressed in different standards in the same problem. The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail. This task helps illustrate Mathematical Practice Standard 4 and engages students in grappling with aspects of the problem to determine how to use mathematics to solve it. Students must apply multiple mathematics standards in this real life problem situation and discuss what needs to be painted, how the area should be calculated, and how many cans of paint are necessary. Also, this problem connects very nicely to work on area, surface area, and volume (6.G.A.1). Students discuss assumptions and approximations used to solve the problem (MP.4). They share how they were able to model the situation and their subsequent results. Through the sharing of models and results, incorrect and correct assumptions will be uncovered. Questions that students may ask each other include: “How many cans of paint are really needed?” “Will there be left over paint, how do you know?” “How did you decide which measurements to use to figure out what need to be painted?” This problem can be seen as a stepping stone problem towards more complex or abstract modeling problems students might tackle in the future. While the student is given everything they need to solve the problem within the stem of the task, and also they do not have to sort through additional material that is not necessary for solving the problem in the task, they do have to apply skills from different standards that they may have learned at different times to a real life situation. So while the task is more straight forward than some modeling tasks, it is still along the continuum of modeling with mathematics and can help our student get ready for less well defined tasks in the future. ## Solution First Alexis needs to find the area she needs to paint. Alexis will need to paint two 30 foot - by - 50 foot walls and two 30 foot - by - 80 foot walls. $$2 \times 30 \textrm{ feet } \times 50 \textrm{ feet } = 3000 \textrm { square feet }$$ $$2 \times 30 \textrm{ feet } \times 80 \textrm{ feet } = 4800 \textrm { square feet }$$ Alexis will need to paint 3000 + 4800 = 7800 square feet. Next, the table below shows how many square feet she can cover with different quantities of paint. Number of gallons of paintArea covered 1420 52100 104200 156300 208400 20 gallons is a little more than she needs, so she can check 19 gallons and 18 gallons: Number of gallons of paintArea covered 1420 52100 104200 156300 208400 197980 187560 18 gallons isn't quite enough and 19 gallons is a bit more than she needs. Since paint is usually sold in whole gallons, it makes sense for Alexis to buy 19 gallons of paint. Finally, since paint costs \$28 per gallon, the total cost will be $$19 \textrm{ gallons } \times \28 \textrm{ per gallon } = \532$$ It will cost Alexis \\$532 to paint the barn.

#### Glenn says:

I would ask some other questions like: Explain the steps you would take to solve this problem. What do you know about rectangles that will help you? What are two things that you need to calculate?

#### Jason says:

4 months

I would say something like: 'How much will it cost Alexis to buy enough paint to paint the barn?" Only the cost of paint is being considered.

#### Bill says:

4 months

Good point. We'll change this as soon as we get the task editing bug fixed.

#### Paul Jacobs says:

almost 4 years

I believe this problem is way off base. The purpose of the problem is to make use of finding areas of simple polygons. What do 6th graders know about buying paint? I guarantee that not one 6th grader will get this problem right. I got it wrong and I was an engineer and a math major in college. A friend got it wrong and he's a retired math professor from U of Arizona. If I were a parent and my kid brought this problem home, having had it marked wrong I'd be furious and would be writing this letter to the school instead of to you. It's things like this that make kids hate and fear math, as well as not learning it well.

#### Kristin says:

almost 4 years

Is your objection the fact that the solution assumes you have to buy whole gallons of paint? Or do you have another concern? I don't consider getting a different answer than the posted solution as necessarily indicating you got the problem wrong if you think of this task as having a modeling component (MP4); part of mathematical modeling is making and defending assumptions. I can imagine using this task in an instructional context where students could discuss the assumptions they make as they solve the problem. If I understand your concern better I think we can add some commentary and possibly another solution approach to make the task a better illustration.

#### Michael says:

8 months

I agree totally. MP1, MP2, and MP4 are well within the realm of this problem. On a more practical note, having painted my son's rooms a few years ago, I found it nice to have more paint than what I needed. There will be some cases where you need to have more than what is called for. In 12 years of teaching, I have noticed that students have regressed in being able to think through problems critically and reason outside what plainly in front of them.

#### Paul Jacobs says:

And after a long vacation ....
If you want the kid to round up to the next higher gallon size, then state (very early in the problem) that "paint is only available in one gallon cans".) But the fact is, that you can buy paint in 1/2 gallon cans, or quart cans, or pint size cans. On the other hand, why throw in a red herring into the problem? Make the answer work out to be an integral number of gallons! And my original complaint still stands: this is why kids learn to hate math, when they should love it! It should be taught as a fun game, not a series of traps that they can get caught up in.

#### Todd Caulfield says:

Love this problem! So many opportunities for small group problem solving and discussion. I use this problem every year in my 6th grade classes. As a teacher, I believe that my job is to build mathematicians rather than just teach algorithms.

#### donna says:

over 3 years

love this, perfect for a mathematical modeling problem

#### holly says:

over 1 year

I used this task to start our geometry unit. I posted the task on the board and then we started with the questions.. Where is the math? and What questions do you have about the problem that you might need to solve this. Each student recorded each of their individual questions on a separate post it. The students then sorted their post-its by topic with a partner and then we organized them as a class, creating a heading for each set of questions. Our biggest revelation was that there were some questions we had to INFER, but really may never be able to answer- such as how many coats of paint, can you buy it in smaller gallons. It led to a great discussion of you can only use the information given in this case and that sometimes your background knowledge cannot not impact your solving plan.

Many students related this problem to volume and as they began to solve, realized that volume was unreasonable and then refocused on area. Which then led to questions about surface area and how the roof came into play.

This was a wonderful task to piggy back on the fact we just learned expressions and equations and students were able to use their knowledge of area and then create an expression to solve this problem. Some students created it before solving while others created it after finding the solution.