Making Surface Area Visual

One of the students I tutor is a sixth grade girl.  When she came to me,
she was struggling with math anxiety.  When I gave her the Math Add+Vantage assessments it was apparent that she had very little number sense.  She struggled with composing and decomposing numbers with the help of structures such as ten frames and rekenreks and she had almost no concept of subtraction.  She was TERRIFIED of math and when I asked her a question she would just start throwing out answers and correcting herself.  It broke my heart.

To begin I had to get her feeling successful in math.  For the first couple of sessions, we just worked on working slowly and talking about strategies.  We played with fraction tiles, cuisenaire rods, rekenreks, ten frames, counters, dice, etc.  We played math.  We talked about what made her nervous and what she felt like she was good.    Most of all, we spent a lot of time working on structuring to 10, to 20 and to 100 and she has made great strides.  It was apparent with her from the beginning that she had not been provided concrete practice or examples in her early math experiences.

When she was working on fractions, she depended entirely on “tricks” and sayings to remember what to do.  The “why” was missing completely.

Last week, she failed a test on surface area and volume.  Her teacher was going to let her retake it the next week.  When we met to discuss it, it became clear that she needed 3 dimensional objects to connect back to.  When I asked her how many faces were on a rectangular prism she thought it might be 4.  We worked through some problems, but she continued to have difficulty visualizing the prism and was inconsistent in her answers.

This week I came prepared to provide those experiences.  She seemed to have a pretty good grasp on area, but volume and surface area were not sound.

I realized she needed to actually see me unwrap a rectangular prism into faces.

I provided this example:Drawing SA

I brought a rectangular prism and had her touch the faces.  I had her count them and we discussed how many numbers she might have for the area of each face.  She said 6.

I then provided her with a piece of graph paper and had her draw each of the faces (and shade the area).  We discussed that two faces are identical and found them on the prism as she drew.  She labeled them as we went with length and width.Concrete SA

After she had all of the faces drawn, I asked her how she would find the surface area of each.  She added the length and width.

I said, “Count the shaded squares inside and see if that works out.”

She replied, “Oh! It’s multiplication!”

I then asked, “How many surfaces are there? “


“So how many products will you have all together?”


“So what do you do with all of those products.”

“Add them together?”

“Does that make sense?”



“Because it will be the total area of all the faces.”drawing 4.png

I then had her try another problem using graph paper and asked her to try not shading the areas this time.

She was able to do that pretty easily, but she stopped drawing after four faces and I asked her how many faces she had so far.

She still struggled with the last set of faces.  We talked about which dimensions she had already used and she was able to draw the remaining two faces.  Visual.png

When she added the products I made note of some strategies we needed to look at for addition in a later lesson.

I asked her the units and she knew they were “squared” but we had to discuss how to show that in the answer.

The prior week we had talked about 2-dimensional shapes and how when we find area, the product is 2 factors which gives us units squared, but when we find volume it is a 3-dimensional shape and the product of 3 factors is units cubed.  She seemed to have retained that.

Drawing SA 3.png

Since she was having so much trouble with deciding which factors to multiply in the last two examples, I wanted to provide her an example that had two factors that were the same and see what she did with it.

This time I told her I was going to take away the graph paper, but that she could recreate it with a drawing.  I modeled the first one and then let her work on it.Representational.png

She did as predicted.  She just chose some numbers she saw and multiplied.  I asked her to identify the dimensions she was multiplying and she did, but still didn’t notice the problem of her 8 x 8 square.  I asked her to show me the two eights on the square she drew and she found her mistake.

She said, “Oh!  Now I get it!”

She continued with this example and then we moved on to a review on volume.

I asked her if she remembered the formula for volume and she said, “yes, length times width.”  I said, “isn’t that the formula we just used for area of a face?”

She said, “yeah.”volume.png

I pulled out my phone and showed her the video from the 3 Act Task, Stack Em’ Up and then drew this picture for her.  We talked about how it is three layers of 32 and she was able to tell me the formula was length times width times height.  I had her work a few problems and she did so with no problem.

I cannot stress enough the importance of making math visual for students.  This brilliant little girl has felt like a failure her whole life in mathematics because she was not given the tools to succeed.  This must not continue.  She can now solve simple subtraction problems with ease and mentally add and subtract 3-digit numbers using strategies such as compensation.  All because we took our time and connected the concrete -> visual -> abstract.

Most importantly, her attitude about math is changing.  She is more confident in her answers and she has strategies to prove it.  In my opinion, that is her greatest success!

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