Sunday, October 2, 2016

Week 6: Days 18-22 3rd week of number talks, participation quiz

Above is the overview of this week's and next week's number talks.
As you can see one answer was 5.6 with no bar over it. Henry explained the traditional way using long division with a decimal. Jennifer showed how the remainder is the numerator and the divisor was the denominator of the mixed number. Tommy multiplied by the reciprocal. I really like Patrick's way. He knew 18 / 3 is 6. So, 1 divided by 3 is 1/3. 6 minus 1/3 is 5 2/3. I also honored someone who counted up by 3's on their finger and got 5 groups of 3 being 15. I also illustrated how an array of dots would look when grouping them into groups of 3 and having 5 groups.

On Monday in all 3 of my 8th grade classes I did a participation quiz. If you have students work in groups and have never done this, you must do it. Notice how group 1 got off topic and the task manager didn't get the group back on track. I noticed group 2 had no one working ahead. Group 3 a teammate asked another if they were good. In group 4 I clarified shaded is positive and unshaded is negative. When it's unshaded the red side of the tile would be facing UP.

I absolutely love Justice's method here. He said 18 / 3 is 6. 15 /3 is 5. So, 17 / 3 must be 5 and 2/3, because 17 is 2/3 of the way from 15 to 18.

I like how group 5 verified what part of the problem they were on. Group 1 had to re read the problem to make sure they needed a negative sign. Group 2 was discussing how regions affect the sign of a tile. In group 7 I wanted them to speak up when they saw a group member not using the tiles as tools. I also recognized a person for being brave enough to tell their group they didn't get it.

We discussed 2 ways of representing an expression. -(y-2) is huge. In this class I was surprised they didn't have the shaded y  and the unshaded unit tiles both in the negative region, I always try to make sure I have that representation displayed.

I like how one student here just wrote the division problem as an improper fraction in their head then figured out it's mixed number. I like how this class said 5 remainder 2 and 5.2. This is a common misconception that the remainder is the decimal. I notice we didn't show the way the repeating decimal was found in that class period. I also brought in a strategy from 2nd period into the 5th period discussion because I liked it so much (Justice's in the bottom right).

5th period is right after lunch and is a class that I constantly have to remind to stay on topic. The participation quiz might be necessary more often with this group. Group 2 noticed one of their diagrams had all unshaded tiles. Someone from group 1 told their teammate to do their work! Group 1 asked a question when they were all stuck, which was good. A student in group 3 asked a question that started with "What's the difference between..?" to clarify how a group of algebra tiles were represented. As you can see group 8 was talking about their siblings when I walked by. When some students have tiles in their hand, they stop writing, so I reminded a student their to put the tiles down when drawing their diagram.

This was a great conversation in 6th grade math support. We talked about how it  didn't make sense to regroup when subtracting 18 from 21. One student explained they counted up 3 jumps from 18 to get to 21. Another said, no way that's not how I did it! They started at 21 in their head, and counted back until they got to 3 after 18 jumps back. I think by illustrating it on the board I hope to get to them that some strategies were better in certain situations.

We had no wrong answers to discuss in this class. I liked when a student added 1 tenth or 0.1 to 8.9 to get 9. Then added 7.48 to get 16.48. Then subtract 0.1 for 16.38.

2 students in this class really know their place value because one separated .48 into .4 and .08. Another way was decomposing the decimal.

Here we reviewed how to 3 situations represented zero and how to represent it with symbols. They also reviewed the 2 ways to make zero.

Here Vivian divided 600 by 30 to get 20. Then divided 43 by 30 to get 1 13/30. Then added those together. One student added 17 to 643 to get 660. Then 660 divided by 30 is 22. Then they subtracted the 17/30 to get 21 13/30. This is the one class where no one gave a decimal answer.

Here we showed how the negative sign is distributed to each term in the parentheses.

Here we honored the mistake of multiplying 27 by 2 and getting 51. Then dividing 51 by 3 to get 16. The process is correct, except they should have gotten 54 when they multiplied. One student said 27 divided by 3 is 9. So, that's 1/3. Double 9 to get 18, which would be 2/3. Another student discussed cross cancelling or simplifying when multiplying fractions.

I liked how this student quickly illustrated the 3 legal moves. It was pretty simple and to the point.

I liked how someone here thought of 2/3 as an equivalent fraction with a denominator of 27. Therefore they thought of Giant One in their head to multiply 2/3 by 9/9 to get 18/27, so 2/3 of 27 must be 18, the numerator.

One student was able to do improper fractions in their head. In this case it's quite a laborsome process. I liked when a student rounded 7 2/3 to 8. Then added 3 and 5/9. Then subtracted 1/3 which is 3/9 from that to get 11 2/9.

In 6th grade math support we did a number line activity. At first, students thought that the single digit decimals were the smallest and closer to 0. Double digit decimals were closer to 1 here. One student correct 1.0 so it was right under 1, because they said you can add a point zero on the end and not change it.

Here you can see a misconception of 4/5 being between .5 and .6. Students couldn't explain why that was not correct. Students correctly identified .90 as being close to 1.

Then a student moved the .01 to the left of .1. They said if you compare the tenths place, 0 is less than 1, so it has to be smaller. This was a huge jump and turning point.

I told students they could use extra clothespins to pin any decimals that were equal or equivalent. Eventually a student pinned .20 to .2. They said that they had the same number in the tenths place and the zero at the end didn't change it. I reiterated that 2 is 2/10 or two tenths as they said. .20 is 20/100 or twenty hundredths, which simplifies to 2/10.

This student below really understands proportional relationships, but I ended up taking a bit off for the explanation because passing through the origin is not sufficient to be a proportional relationship and their 2nd reason didn't make it clear to me.

Wednesday, September 28, 2016

Week 5: Days 17-21 Subtraction, Mixed #, Decimal, Multiplication Number Talks

As you can see, Mario's method was keeping track of subtract 3 groups of 10, and then adding 2 on because it was subtracting 28, not 30. When Patrick subtracted 3 from 68 and 3 from 28 to get 65-25 I illustrated this as equal distances on a number line. We also of course acknowledged the traditional algorithm of regrouping vertically.

Students saw the algebra tiles were 2 types of shapes: squares and rectangles. They noticed different sizes, side lengths, widths, and color. A student noticed the teal rectangle fits on 1 side of a teal square and same for the purple. Then students traced 1 of each tile and we color coded with 3 different colors after making a key.

In this class you can see Cameron added 2 to 28 to get 30. Then added 30 more to get to 60, then 3 more to get to 63. Then adding 2, 30, and 3. Jonathon did something interesting: He did 60-20 first. Then added 3 because it was 63 not 60. Then he subtracted 8 more from the 28, which got him 43-8 which was 35. I illustrated that in the upper left with backwards and forward jumps.

Here is some work from 7th and 8th grade Math support students. Finding the area of figures, computing decimal integers, fractions, as well as drawing pictures of tiles and combining like terms.

When given the "Jumbled Tiles" problem, Michael organized his expression by categorizing each type of tile. I thought it was interesting.

Day 2 was more subtraction, but with decimals. Here you can see a student said add 3 after subtracting 2, and I wrote it as 3 to make 1.97 into 2. He later added onto his idea to say it was 3 hundredths after hearing other students reasoning. Damon added .03 to both numbers to make 4.37 - 2. Some students came up with 3.37 after not regrouping the ones place.

I liked Justice's method. Subtract 1. Then .07. Then 0.9. It showed true understanding of place value. EMac had an interesting method. She added 0.37 to 1.97 to get 2.34. Then she subtracted 2.34 from 4.34 to get 2. Then she added on the 0.37.
Here is the tweet that sparked the revisit:

This is the class where I revisited the wrong answer yesterday of -35 to 63-28. I asked students how someone could have come up with -35 because it shows some understanding of something. Students saw that -35 was the difference if you did 28-63. This could have been another additional opportunity to show that the difference of 2 numbers is the same if you take the absolute value of both.

Kristen used improper fractions to have 13/4 - 7/4 to get 6/4. In the top left Lawrence showed how he regrouped 3 1/4 to 2 and 5/4. I showed how 3 is 2+1 and 1 is equal to 4/4, then he added it on. 5/4 - 3/4 is 2/4 so 1 2/4 or 1 and 1/2. Julianna used decimals and visually lined it up in her head. Vivan rounded 1 3/4 to 2, subtracted it, then added the 1/4 back on that was added to 1 3/4 to make it 2.

I liked Justice's method of repeated subtraction. He subtracted 1 to get 2 1/4. Then subtracted 1/4 from both 2 1/4 and 3/4 to get 2 - 2/4. Then he got 1 1/2.

Love the method by Maria, featured in a tweet below. She subtracted 1 from 3 to get 2. Then 1/4 minus 3/4 is -2/4. Then 2 and (-2/4) is 1 2/4 or 1 1/2:

I loved drawing 2 fingers and Joe's method of skip counting by 5 18 times. 5 times on each hand. Then 5 more on the left hand. Then 3 more to make 18.

Karin decomposed 18 into 9*2 then multiplied 2*5 to get 10. Then 10 times 9. EMac realized 5 was 10/2, so she multipled 18 by 10, then divded the product by 2. 180/2 is 90. Jonathon split 18 into 9+9 and multiplied each by 5, then added it together like the area model at the top. I also love rounding 18 to 20, multiplying by 5, then subtracting 2 groups of 5, or 10 from 100.

Here I intentionally had students raise their hands if they got 28 for the above problem. Then half the class raised their hand for 10. Students reasoned in the order of operations the exponents must be done first before multiplying by 2. I related this to algebra tiles, because 2x^2 is x^2+x^2.

The blank setup, before students discussed how to label.

And here is what we went over in each class.

The student above had the principal, Mr. Hophan, happen to pop in for an informal observation and observe Cameron label the sides of the algebra tiles that were on a scratch piece of paper. It was great with the pressure!

In each class students also came up with the alternative to the above, when you label the right side of the x tile x, and not labeling the right side of the unit tile and consider it part of X.

A highlight to the week was when I discussed a perimeter of algebra tile problem with a student who engaged with it and got it. I encouraged her to go to the document camera and show the class and she nailed it. Great moment. Also she didn't get a 3 second lap, she got a 3 second clap, a quick way to give props to someone in class.

I had to rock my CPM shirt on Friday to represent.

Vivian decomposed 16 into 2 times 8. Then 12 times 8 is 96. 96 times 2 is 192. Mario reasoned 12 times 12 is 144. Then 12 times 4 is 48. Add those together and you get 192.

Emma had a great way of breaking up 12 and 16 into their place values. I illustrated this with a generic rectangle. Sargent thought of it as 16 groups of 12. 5 groups of 12 is 60. 3 groups of 60 is 180, or 15 groups of 12. Then add 12 more to make 1 group of 12 to make 16 groups of 12.

Students brainstormed the meaning of minus.

I substituted Mr. Deabler's 7th grade class and we discussed multiple ways to know how 5/11 was less than 1/2. Students came up with converting them to common denominators of 22, showing 10/22 is less than 11/22. One student also looked at common numerators. Since 11 is a bigger denominator it's smaller pieces. Another student said to compare the cross prodcuts of 10 and 11.

In 6th grade math support we worked on a MARS task as a culmination of decimal activities. I think it opened my eyes to how the students were comparing decimal numbers to see if they were less than or greater than 5.5. Many studens saw the multiple digits after the decimal as bigger than 5 tenths.

Anyone know an easier way to solve the above problem from a high school entrance exam test prep book?

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

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.