Last week, Pamela worked through the odd problems in Math-U-See's Lesson 1 according to plan. We transitioned from concrete work through the games I blogged in the plan to number lines (click the picture on the right to enlarge it), which was where I first introduced negative and positive signs. When we started working on word problems, I made a list of actions that usually go with negative and positive, grouped in +/- opposite pairs (sell/buy, have/owe, earn/spend, find/lose, etc.). Whenever we do one of these transactions in real life, I spotlight it for Pamela. Finally, last week, I introduced her to abstract symbols for problems like (+7) + (-10) = _____. She shifted very smoothly from concrete to pictures to words to symbols in the process of understanding the concept of negative.
Last Friday, Pamela ACED her first test straight from the book with no help from me. I was busy typing something on the computer while Pamela took her test. She must have thought about the time when her brother took his first Math-U-See tests because she grabbed the calculator and used it herself. Pamela proudly commented on using the calculator, probably a sign of growing up to her. You can see how she completed most of the test unassisted because she showed her work up until Problem 15. I am more delighted about Pamela reflecting on how David took tests and altering her behavior than the test score!
If you examined the test closely, you will see some fraction problems. Like Math-U-See suggests, we have been reviewing fractions for the past five weeks, falling back on manipulatives, drawings, real-life activities, etc. for concepts that are murky. We examined concepts like whole versus fractions, equal versus unequal parts, terms (numerator, denominator, proper, improper, equivalent, etc.), adding and subtracting fractions, and reducing fractions. I have assessed her understanding of fractions from concrete to pictures to words to symbols, and Pamela is recovering what she did not retain. One of these days, I might get around to blogging our plan for algebra/arithmetic, the third track, which focuses on the review of fractions, decimals, and percents until the seeds are sown for doing algebra.
We are making steady progress in our plan for geometry. After blogging our first week of geometry, I left you hanging!
Area of Triangles
Since Pamela caught onto the relationship between the area of a rectangle and the area of a triangle, we reinforced what she knew, NOT through kill and drill, but through new ways of exploring the same idea. She cut out rectangles on a grid and found the area of a rectangle. Then, she cut the triangles out of the rectangle and matched up the pieces to prove to herself that the triangle has the exact area as the remaining pieces of the rectangle. Since the triangle plus the group of remaining pieces form two equal parts of the rectangle, then you know the area of the triangle is half the area of the rectangle.
Then, we transitioned from the hands-on to pictures and she learned to draw a rectangle around the triangle, find the area of the rectangle, and divide by 2 to get the area of the triangle.
Pamela spent some time learning to measure the base and height of physical triangles or pictures of them to avoid the trap of thinking that half of one side times another side will work. I helped her get far enough to be able to handle finding the surface area of a pyramid which was the ultimate goal. Down the road, we will revisit the area of triangles and delve into why and how to extend the base to find the height.
Surface Area of Rectangular Prisms
We began by counting the number of surfaces of various polyhedrons to introduce the idea of a solid having faces. I found an awesome resource for making my own out of cardstock paper (you must check out the link--some of the models are fantastical). The first item we chose was the paper clip container, which perfectly matched the sticky notes (top and bottom) and only required minor adjustments for the sides. We labeled each side with letters and identified their position (top, bottom, front, back, left and right) to prepare her for transitioning to pictures later. Then, we made up a table of the areas of all the faces and added them up. Her eagle eye spotted a pattern right away (top = bottom, front = back, left = right).
I showed her how to open the box and flatten it out and draw a two-dimensional picture of it. Given the flat picture, Pamela easily found the surface area. Given a box in real life, she had no problem! The tricky part was, when shown a 3-D drawing of a box, drawing the flattened picture and correctly transcribing the height, width, and length. We continued to practice one a day, looking at the picture, and pointing to the parts of a real box. After consistent and slow work, everything fell into place suddenly and she figured out the logic in a pattern that made sense to her:
- Every number gets copied four times.
- Figure out the dimensions of the bottom and copy them to the top.
- Logically, you have to copy the width of the top and bottom to the front and back.
- The front and back involve the height, so transcribe that.
- Logically, you have to copy the length of the top and bottom to the left and right sides.
- The only thing remaining to transcribe is the height, which gets assigned to the left and right sides.
By now, your eyes are probably bugging or your head is spinning. Here is the thing to keep in mind. Unlike reading words on a page or taking notes in a lecture, working through these problems, day by day, develops understanding. Pamela thought of this way of solving the problems by knowing what makes sense and what does not make sense. That kind of knowledge comes from going from concrete to picture to words to abstract numbers and symbols.
Surface Area of Triangular Prisms
We applied the same process to a pyramid with a square based. She had no problems making the connections for this solid, drawing what she calls "the star", and transcribing numbers correctly.
To assess whether or not Pamela truly understood calculating surface area from a flat drawing, I gave her a slightly different problem. The last thing I want her to do is work math problems on automatic pilot. The only way to understand math is to think and make logical connections. I gave her dimensions for a pyramid with a triangular base and the star suited for that solid. Pamela had no problems adjusting her strategy for this problem.