Tuesday, September 20, 2016

Week 4: Days 12-16. Dot talks, proportional relationships, division.

I saw the below image on Twitter and want to ask my 6th grade support class what they notice and wonder about it. We have played the game Fizz Buzz twice for 5 minutes each and this relates to that. Google the game if you want, but the rules are you count off by 1's in a circle. If you have a multiple of 3, you say fizz, if it's a multiple of 5 you say buzz. If it's a multiple of 3 and 5 you say Fizz Buzz. If it's not a multiple of either, you say the number. It's a great game, and when mistakes are made, I'm trying to get students to explain what they should have said and why.

This week was the first week of 4 straight weeks of number talks to improve fluency and flexibility of number sense. The first 3 days we did dot images, and the last 2 we did a worksheet with circles on it where they found as many ways as they could to count the circles without counting one by one.

Above you can see day 1. Most students saw it in rows or columns. Some saw the dots as diagonals, squares, or L shapes. This dot image came from Jo Boaler's week of inspirational math on youcubed.org.

In class I focused students attention on the above problem and asked which was correct. They reasoned that the one on the right was correct, because the problem says the puppy is 14 ounces at birth (which must be interpreted as Day 0). In the first 10 days, the puppies weight doubled. Therefore, day 10 the weight is 28 ounces. Then the problem says, assuming the puppy grows at a constant rate, find it's weight on other days. Some students continued doubling every 10 days, while others figured that you had to find the difference between 28 and 14, which was 14, so see how many ounces were added every 10 days. This is a great problem to focus students attention on the fact that just because a table grows at a constant rate, it is not always a proportional relationship. It's not because it doesn't have a weight of 0 ounces at 0 days, and if you double 10 days, you get 20 days. If you double 28, you get 56, but it should be 42 ounces because it grows at a constant rate.

Above in 6th grade math support we reviewed the traditional way of dividing as well as the partial quotient method.

Here we reviewed the terminology. Also, there is a NEED for partial quotient when you divide 84 by 12. Most students don't have the multiples of 12 memorized, so this is where partial quotient comes in handy.  So students reasoned 12 fits into 84 at least 4 times. So, 12 times 4 is 48, which you write below 84 and subtract it. That leaves 36. 12 fits into 36 3 times, so you put a 3 above the 4, multiply, subtract, and have a remainder of 0. Therefore, the quotient is the 3 groups plus the 4 groups which is 7 groups.

We also talked about the order of division. Students thought the big number always goes "in the house." This is not always true. The dividend always goes under the division symbol.

Above is Day 2's number talk. This is a popular one from the book I read called Making Number Talks Matter. I loved that the words trapezoid, hexagon, and triangle came up.

The homework provides an important opportunity to review concepts from previous years. Here we discussed how to find the 5 numbers for a box plot (minimum, maximum, median, lower and upper quartiles). We had to clarify that if you had a true median, you had an odd amount of numbers. When you find the quartiles, the median is NOT included. If you had an even number of numbers, you would include the median.

I liked that a student saw a smiley face, one saw a 3 dimensional cube, and another saw the 5 dots on a dice, with a dot on each side.

After students shared, I saw that I saw 3 squares with 4 dots each, but 2 were double counted so I had to subtract 2 after multiplying.

In this class we discussed the difference between a trapezoid and a parallelogram. Students reasoned a parallelogram has 2 pairs of parallel sides, while a trapezoid has only 1.

Reviewing the homework provided an opportunity to review integers as well as dividing by a fraction. When I asked students why we multiply by the reciprocal, they weren't too sure. I reminded them by writing it as a vertical division problem. Then I asked what you multiply 2/3 by to equal 1. Oh, 3/2. Well, 3/2 is the reciprocal which we can multiply the numerator and denominator by in a Super giant one (7th grade Course 2 CPM). I heard a lot of ohhhs and ohhh yeahhh.

I liked how this student saw 4 rectangles of 6 circles that overlapped on 4 circles, so they subtracted 4.

This blew my mind. Only one student in one class did it this way. Filling in "ghost circles" to make it a 6 by 6 square, and subtracting the ghost circles.

Assigning less homework allows for deeper discussions about the prior concepts. Here we reviewed finding the area of the border as well as how to construct a stem and leaf plot. They also analyzed how changing the 51 to a 33 would change the mean but not the median.

8th grade students investigated using a Giant One, Undoing Division, and Fraction Buster methods for solving a proportion (groundwork for later equation solving). Some students saw the connection between the Fraction Busters method and the shortcut of setting the cross products equal to reach other.

This was a very interesting 6th grade division problem. Basically, they had to find the quotient and the remainder. Since each vase has 21 flowers in it with 7 left over, you'd need 42 - 7 or 35 more flowers to be able to put 1 more in each vase to have 22 flowers in each vase with no flowers left over.

This was a much needed discussion of 2 8th grade homework problems. Students had to make the connections that a video game order had a 4.85 shipping cost PER ORDER. Not per item. Therefore, the unit rate was not equal AND the table did not increase at a constant rate of change.

Karin came back the next day with her another strategy to make a square out of the diagram. This time she moved the circles in a clockwise motion to form a square. You can see her expression 5 * 5 = 25. I crossed out the circles she moved so students could see her strategy better.

This is a Which One Doesn't Belong that I found on twitter than I want to save for later when we do 10 days of those warmups.

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