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Painting a Wall


Alignments to Content Standards: 5.NF.B

Task

Nicolas is helping to paint a wall at a park near his house as part of a community service project. He had painted half of the wall yellow when the park director walked by and said,

This wall is supposed to be painted red.

Nicolas immediately started painting over the yellow portion of the wall. By the end of the day, he had repainted $\frac56$ of the yellow portion red.

What fraction of the entire wall is painted red at the end of the day?

IM Commentary

The purpose of this task is for students to find the answer to a question in context that can be represented by fraction multiplication. This task is appropriate for either instruction or assessment depending on how it is used and where students are in their understanding of fraction multiplication. If used in instruction, it can provide a lead-in to the meaning of fraction multiplication. If used for assessment, it can help teachers see whether students readily see that this is can be solved by multiplying $\frac56\times \frac12$ or not, which can help diagnose their comfort level with the meaning of fraction multiplication.

The teacher might need to emphasize that the task is asking for what portion of the total wall is red, it is not asking what portion of the yellow has been repainted.

Solutions

Solution: Solution 1

In order to see what fraction of the wall is red we need to find out what $\frac56$ of $\frac12$ is. To do this we can multiply the fractions together like so:

$\frac56 \times \frac12 = \frac{5 \times 1}{6 \times 2} = \frac{5}{12}$

So we can see that $\frac{5}{12}$ of the wall is red.

Solution: Solution 2

The solution can also be represented with pictures. Here we see the wall right before the park director walks by:

Wall_painting_1_4740940e7f77654be15a5e4a8329e07c

And now we can break up the yellow portion into 6 equally sized parts:

Wall_painting_2_249d9952c6dcc948719bbbf61c38291f

Now we can show what the wall looked like at the end of the day by shading 5 out of those 6 parts red.

Wall_painting_3_3dc4f1d2c6a523ee49586290489a400e

And finally, we can see that if we had broken up the wall into 12 equally sized pieces from the beginning, that finding the fraction of the wall that is red would be just a matter of counting the number of red pieces and comparing them to the total.

Wall_painting_4_cdc4d525af7164d2436a8d51e40ece80

And so, since 5 pieces of the total 12 are red, we can see that $\frac{5}{12}$ of the wall is red at the end of the day.

Jennifer Harris says:

over 4 years

This problem was first introduced to us during our Summer Math/Science Partnership (MSP) workshop at Valdosta State University (VSU) for the southern RESA District in Georgia. The problem in video format with the solution aviailble will prove to be useful for our students who have special needs, need remediation, may have been absent, ect... We are very ecited about using this technology or that similar to it, to help bolster our students' knowledge/understanding of mathematical concepts and thier learning environment.

Please follow the link below for a video on how we were able to find a solution to this problem.

https://www.dropbox.com/s/in2ct3azs8sl4ck/VID_20130618_151341_175.mp4?m

Kristin says:

over 4 years

Dear Jenn Harr,

Thank you so much for sharing your video with us. It is wonderful to see all the ways that people are adapting the tasks for their own purposes.