Wednesday, May 4, 2016

Day 152: Patterns in Prague MARS Re-Engagement & Transformations

Today's estimation was from Ontario, Canada to New York, New York. It was about 1/3 of the trip within Canada.

I passed out Chapter 10 homework packets and the checked and marked Patterns in Prague MARS task. Overall students were not very successful initially. Some students that did understand the area and got 80 square units forgot to multiply it by the area of 1 5 cm by 5 cm square. A few students got the correct answer of 2000.

For the perimeter, only a couple students got the correct answer the first try, and I showed that on yesterday's blog. I used the following Google Slides to re-engage students in a whole class discussion.

To begin I focused students on what the question was asking. I also had them cross out sentences that were not helpful and highlight or underline information that was important. They underlined squares that were 5 cm by 5 cm. They also underlined that some triangular blocks are made from cutting the square blocks diagonally in half. 

I had students use color and highlighters after they noticed the 8 by 8 perfect square in the middle. Then they reasoned that the triangles were 2 full squares and 4 half squares, so they were 4 total blocks. There were 4 sides, so 4 * 4 is 16. Then they added 16 to 80.

The final step is multiplying that number of blocks by the area of 1 5 by 5 cm block, therefore 25 square centimeters.

In every class I halted class for a number talk on how to multiply 80 by 25 without a calculator or pencil and paper. Student reasoned 25 times 8, then adding a zero at the end. I asked students how to do 25 times 8 in your head. Most students mentioned money. 25 cents is a quarter, so 4 quarters in a dollar, and 8 quarters in 2 dollars so 200 cents. Some students just said 25 times 4 is 100, then double that for 25 times 8.

One student in 5th period mentioned piecing together the 16 square units in the triangle as an array of 2 by 8 to add 2 rows to the perfect square, making the design a rectangle that had dimensions of 10 and 8. Only 1 person mentioned that all day. One more method is shown in the photo below.

I want students to think about what these cuisenaire rod arrangement represents.
Color coding the area. On the right one student suggested to cut all of the triangles off and re arrange them into a square. Students instantly saw that it was a 4 by 4 square therefore 16 squares.
Next time I would go into more depth on what diagonal actually means and why it's different than the side length.

Only in one class did I remember to relate the side length of a triangle to the distance from 1st to 3rd base in baseball. It's obvious that the distance from 1st to 2nd (a leg) is shorter than the distance across the diamond (the hypotenuse).

Students had a tough time verbalizing the part of the perimeter we knew without calculating anything. That was the 4 corners. Each corner had 4 side lengths of a square. Each side length of the square was 5 centimeters, so each corner had 4 * 5 or 20 centimeters of perimeter. With 4 corners, that made 80 centimeters.

The task even suggests using the Pythagoras' Rule to find the perimeter below the title of the task. Then students realized that half a square makes a right triangle. I reinforced this by asking how many right angles a square had (4). So when you cut it in half you have a right triangle and know the 2 side lengths or legs that are 5 each. Students said to use the Pythagorean Theorem. They stated it, substituted, and realized the c value or hypotenuse was the square root of 50.

Some students jumped to a calculator, but I asked students to approximate it. Before that I asked them what type of number it was. They said irrational. Some said the decimal goes on forever in a non repeating pattern but I pushed them to tell me how they knew by looking at it. Some said that (the radicand) is not a perfect square. The closest is 49, so the square root of 50 must be close to the square root of 49, which is 7. 

Then they multiplied that by 16 and got 112. Add that to 80 and you get your final answer. A challenging task for a majority of my students, but fulfilling. They corrected it and I am going to assess them on their growth, reflection, and how well they showed their work.

In accelerated students started their posters on function transformations. I checked in on all tables to discuss the differences between f(x)+k, f(kx), k * f(x), and f(x+k). They will be working on that for the next 2 days, possibly spill over to Monday for the post it note feedback because there is an assessment on Friday.

Below is some material I am tutoring an Algebra 2 student, showing how important it is to distribute negative signs when using the distributive property. It's good to see the 8th grade skill of fraction busters shows up in rational equations in Algebra 2.

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