This week consisted of many number talks suggested by the Making Number Talks Matter book. Students also familiarized themselves with Algebra tiles, with the addition of x^2, y^2, y, and the xy tile. They also explored the perimeter of algebra tiles.
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:
Realized I whiffed on a teachable moment in number talk: S said answer was -35. Should of later asked when is that true? #MTBoS pic.twitter.com/0lGiKxh9Yk— Martin Joyce (@martinsean) September 20, 2016
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:
.@Dsrussosusan @rutherfordcasey me too. Saw it with top left. 1/4 minus 3/4 is -2/4 pic.twitter.com/gsp7RVdzLY— Martin Joyce (@martinsean) September 22, 2016
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.
Coached up S who went up to doc camera & explained perimeter of algebra tile figure! Great feeling for her to get that 3 second lap ! #mtbos— Martin Joyce (@martinsean) September 23, 2016
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?