This is the online discussion of teachers as we move through the lesson study process. Lesson Study provides an efficient organization for teachers to collaborate as we learn more about our content and how students learn.
Thursday, January 22, 2009
Please put comments here after January 22
Almost everyone is in so let's start a new page with just comments about our topic. Heather can you post the summary of our discussion? Thanks, Becky
Becky, Sue, Heather,Ken, What about percentages do you think we need to do a better job of teaching, or what your students find the most confusing? I think that is what perhaps we should tackle.
Housekeeping: New Blog Site coming soon! Please check everyday or every other day. Next meeting is Jan. 28th at 1:30. Feb. 11th at 2:30 Feb 25th at 1:30 March 11th at 2:30 March 25th (90 minute meetings during school) March 26th (90 minute meetings during school) March 30th (Teacher Duty Day 9:30-11:30) April 1st (Lesson Study to be taught) April 2nd (Math Lesson Study to be taught ?? time??)
Group introductions for Jennifer from American Educator. Will be back in April for final lesson!!
I. Percent Research How do we teach for understanding…not just an operation??
II. Brainstorm:
Definition(s) of percent: out of hundred relationship to a whole based on 10% 20% = 2 * 10% how do you get the “whole” to 100% (3/24 to ?/100 link is challenge standard unit to compare quantities **a ratio comparing a part to a whole using the number 100
Properties of Percents: Number (add, sub,<,>) Part to whole Ratio Shows relationship between 2 ratios The comparison to a standard allows understanding of very small or very Large quantities (estimation)
Usages of Percents: Fractional comparison…part whole(25% of the students in this class…) tax, dollars and cents Comparison percents (25% v. 50%) Probability Partition and iteration Discounts, interest rates
DRAFT Developmental Story 100 concept – visual- bundle -shapes physical 100’s chart comparing size of numbers place value groups of 10 part to whole amount (fair shares)…need to emphasize that the whole has changed (10 ones is one 10) 10 tens = 100 (bundles) bundled units given-proved earlier whole changes depending on the situation concrete/visual representations of ratios found in story situations (1 cow, 4 legs) “each, per” repeating patterns to T-chart w/numbers pattern unit (set) repeated addition multiplication (3 sets of…) ratio equivalent ratios decimals- concrete flats .1, .01, .001 -money -measurement(meter sticks) identify the whole – denominator identify the part- numerator (concrete, semi-concrete, semi-abstract, abstract) fractional part of a set equivalent fractions renaming fractions to a decimal ½ = .5 (concrete to money to division) renaming fractions to 100ths renaming fractions and decimals to “percents” **determining a purpose for percent** standard operations - 30% of 50 - 5 is 20% of what quantity - increase or decrease in %
Goal for blog: Firm up developmental story Where in the developmental story should our Lesson Study go?
Thanks Heather for posting such detailed notes for us!! In teaching percents in the past, where do the lessons seem to break down, or where is it in your lessons that you feel the children begin to lose their understanding of the topic..??
I'll try again...it booted my last post! I think kids break down when the "whole" is not 100. Also, they seem comfortable with fraction/percent circles, but as soon as it's something new (bar model for example) they have a hard time.
Remember when we discussed fractions a few years ago, we spoke of the power of introducing fractions with linear measurement rather than the area model. Wonder if this would be valuable? Perhaps even beginning with a meter and doing 70% of a meter then moving to half a meter or something similar. I know we need to determine our goal before thinking activity but I was just pondering. I need to reorganize the developmental story on a piece of paper to more clearly see connections and flow. I agree with Heather. Lessons fall apart at the what's 5% of 35 stage.
This may sound silly, but I am just trying to understand this. In the developmental story we have "renaming fractions and decimals to percents". I wanted to know if this is done side by side or individually. And if individually, which comes first?
In one of the TM articles Parker and Leinhardt states that percent "Always defines a relationship- comparison: sometimes fractionally and sometimes as a ratio." The part that got my attention was when they talk about students must understand BOTH interpretations to grasp percents BEYOND 100. Could students be relying more on 1 or the other interpretation and missing the connections?
With my kids, we don't typically get to questions like 5% of 35 (for those who don't know I'm working with kids in grades 3 - 5 who are often two or more years below grade level in math and reading). We teach basic fractions as parts of a whole, with an emphasis on trying to get them to understand that a fraction is just a part or piece of a whole thing. We work toward understanding that the "whole thing," whatever that may be, is 100%. So if you have a fraction or piece of the whole you have a percentage of the whole that is smaller than 100%. We start basic with examples like 1/4th being 25% of something. (like 1/4th of a dollar is 25 cents - think quarters). Money and food examples help as the kids can relate easily. If you eat 1/4th of a granola bar you ate 25% of it. We can break it into 4ths and count 25%, 50%, 75%, 100% seeing that it's equal to 1/4th, 2/4ths, 3/4ths, 4/4ths (100%) of the granola bar. Money is the best way we've found to tie fractions, decimals, and percents together. It's concrete enough for my kids to begin to see that in $1.25, the .25 is 25/100 cents or 25% of $1.00 (100 cents). As I think more about the example of 5% of 35, If we did try to tackle that we would discuss that the 35 in this case is the whole thing - 100%. If we had 35 cookies for the class, 35 would be 100% of the cookies we have. We would start with friendly numbers, like 10. What's 10% of 10? 1. Can we do 10% of 5 then? It would be half of 1, or .5. So 10% of 30? 3. Plus 10% of the remaining 5? 3.5. Now, since we want 5% (not the 10% we just figured) that's 1/2 of 10% so we must take half of 3.5. What's 1/2 of 3? 1.5. 1/2 of .5? .25. So 1.5 plus .25 is 1.75 or 5% of 35. For me, I think helping the kids to see that even if they are trying to find a % of something other than 100, we need to help them understand that whatever it is, it's still a whole which is still 100%.
Be sure to sign in before typing your blog. If you type first, then go to sign in, you lose what you typed. If you forget, click preview (it won't let you) but then you can copy and paste. As I have learned, it's not much fun to type in a dissertation only to lose it because you forgot to first sign in!
One more thought -- my students are always asking me, "What'd I get?" when we check over their work. They want to know their percentage grade to see if it falls as an A, B, etc. I tell them it depends on the number of questions there were on the assignment and how many they missed/how many they got correct as to what the percentage grade will be. This could be another way to show that the number isn't always based on 100, because they rarely have assignments or tests with exactly 100 questions. This came up today in math class and it hit me as another thought on a way to make a lesson problem meaningful to them. I really think a big part of the problem in trying to teach this is that the numbers by themselves do not have meaning unless we attach it to something they can relate to, like money, food, grades, etc.
Becky, Sue, Heather,Ken,
ReplyDeleteWhat about percentages do you think we need to do a better job of teaching, or what your students find the most confusing? I think that is what perhaps we should tackle.
Lesson Study Meeting 1
ReplyDelete1/16/09
Housekeeping:
New Blog Site coming soon! Please check everyday or every other day.
Next meeting is Jan. 28th at 1:30.
Feb. 11th at 2:30
Feb 25th at 1:30
March 11th at 2:30
March 25th (90 minute meetings during school)
March 26th (90 minute meetings during school)
March 30th (Teacher Duty Day 9:30-11:30)
April 1st (Lesson Study to be taught)
April 2nd (Math Lesson Study to be taught ?? time??)
Group introductions for Jennifer from American Educator. Will be back in April for final lesson!!
I. Percent Research
How do we teach for understanding…not just an operation??
II. Brainstorm:
Definition(s) of percent:
out of hundred
relationship to a whole
based on 10% 20% = 2 * 10%
how do you get the “whole” to 100% (3/24 to ?/100 link is challenge
standard unit to compare quantities
**a ratio comparing a part to a whole using the number 100
Properties of Percents:
Number (add, sub,<,>)
Part to whole
Ratio
Shows relationship between 2 ratios
The comparison to a standard allows understanding of very small or very
Large quantities (estimation)
Usages of Percents:
Fractional comparison…part whole(25% of the students in this class…) tax, dollars and cents
Comparison percents (25% v. 50%)
Probability
Partition and iteration
Discounts, interest rates
DRAFT Developmental Story
100 concept – visual- bundle -shapes
physical
100’s chart
comparing size of numbers
place value
groups of 10
part to whole amount (fair shares)…need to emphasize that the whole has changed (10 ones is one 10)
10 tens = 100 (bundles)
bundled units given-proved earlier
whole changes depending on the situation
concrete/visual representations of ratios found in story situations (1 cow, 4 legs) “each, per”
repeating patterns
to T-chart w/numbers
pattern unit (set)
repeated addition
multiplication (3 sets of…)
ratio
equivalent ratios
decimals- concrete flats .1, .01, .001 -money -measurement(meter sticks)
identify the whole – denominator
identify the part- numerator (concrete, semi-concrete, semi-abstract, abstract)
fractional part of a set
equivalent fractions
renaming fractions to a decimal ½ = .5 (concrete to money to division)
renaming fractions to 100ths
renaming fractions and decimals to “percents”
**determining a purpose for percent** standard
operations - 30% of 50
- 5 is 20% of what quantity
- increase or decrease in %
Goal for blog:
Firm up developmental story
Where in the developmental story should our Lesson Study go?
Thanks Heather for posting such detailed notes for us!!
ReplyDeleteIn teaching percents in the past, where do the lessons seem to break down, or where is it in your lessons that you feel the children begin to lose their understanding of the topic..??
I'll try again...it booted my last post! I think kids break down when the "whole" is not 100. Also, they seem comfortable with fraction/percent circles, but as soon as it's something new (bar model for example) they have a hard time.
ReplyDeleteAh, there you are! I'm finally on the right blog. I'll read this more carefully after SAC and respond. Thanks!
ReplyDeleteAm I in?
ReplyDeleteHeather, what did you mean by bar model? example?
ReplyDeleteRemember when we discussed fractions a few years ago, we spoke of the power of introducing fractions with linear measurement rather than the area model. Wonder if this would be valuable? Perhaps even beginning with a meter and doing 70% of a meter then moving to half a meter or something similar.
I know we need to determine our goal before thinking activity but I was just pondering. I need to reorganize the developmental story on a piece of paper to more clearly see connections and flow.
I agree with Heather. Lessons fall apart at the what's 5% of 35 stage.
This may sound silly, but I am just trying to understand this. In the developmental story we have "renaming fractions and decimals to percents". I wanted to know if this is done side by side or individually. And if individually, which comes first?
ReplyDeleteIn one of the TM articles Parker and Leinhardt states that percent "Always defines a relationship- comparison:
sometimes fractionally and sometimes as a ratio." The part that got my attention was when they talk about students must understand BOTH interpretations to grasp percents BEYOND 100. Could students be relying more on 1 or the other interpretation and missing the connections?
With my kids, we don't typically get to questions like 5% of 35 (for those who don't know I'm working with kids in grades 3 - 5 who are often two or more years below grade level in math and reading). We teach basic fractions as parts of a whole, with an emphasis on trying to get them to understand that a fraction is just a part or piece of a whole thing. We work toward understanding that the "whole thing," whatever that may be, is 100%. So if you have a fraction or piece of the whole you have a percentage of the whole that is smaller than 100%. We start basic with examples like 1/4th being 25% of something. (like 1/4th of a dollar is 25 cents - think quarters). Money and food examples help as the kids can relate easily. If you eat 1/4th of a granola bar you ate 25% of it. We can break it into 4ths and count 25%, 50%, 75%, 100% seeing that it's equal to 1/4th, 2/4ths, 3/4ths, 4/4ths (100%) of the granola bar. Money is the best way we've found to tie fractions, decimals, and percents together. It's concrete enough for my kids to begin to see that in $1.25, the .25 is 25/100 cents or 25% of $1.00 (100 cents). As I think more about the example of 5% of 35, If we did try to tackle that we would discuss that the 35 in this case is the whole thing - 100%. If we had 35 cookies for the class, 35 would be 100% of the cookies we have. We would start with friendly numbers, like 10. What's 10% of 10? 1. Can we do 10% of 5 then? It would be half of 1, or .5. So 10% of 30? 3. Plus 10% of the remaining 5? 3.5. Now, since we want 5% (not the 10% we just figured) that's 1/2 of 10% so we must take half of 3.5. What's 1/2 of 3? 1.5. 1/2 of .5? .25. So 1.5 plus .25 is 1.75 or 5% of 35. For me, I think helping the kids to see that even if they are trying to find a % of something other than 100, we need to help them understand that whatever it is, it's still a whole which is still 100%.
ReplyDeleteBe sure to sign in before typing your blog. If you type first, then go to sign in, you lose what you typed. If you forget, click preview (it won't let you) but then you can copy and paste. As I have learned, it's not much fun to type in a dissertation only to lose it because you forgot to first sign in!
ReplyDeleteOne more thought -- my students are always asking me, "What'd I get?" when we check over their work. They want to know their percentage grade to see if it falls as an A, B, etc. I tell them it depends on the number of questions there were on the assignment and how many they missed/how many they got correct as to what the percentage grade will be. This could be another way to show that the number isn't always based on 100, because they rarely have assignments or tests with exactly 100 questions. This came up today in math class and it hit me as another thought on a way to make a lesson problem meaningful to them. I really think a big part of the problem in trying to teach this is that the numbers by themselves do not have meaning unless we attach it to something they can relate to, like money, food, grades, etc.
ReplyDelete