Homebody2 Posted February 22, 2018 Share Posted February 22, 2018 We can solve this, but not by using a number line. it's a 3rd grade problem, so no knowledge of common denominators or multiplying fractions: Liz and Jay each have a piece of string. Liz's piece of string is 4/6 of a yard, and Jay's is 6/7 off a yard long. Whose string is longer? Draw a number line to model the length of each string. Explain the comparison using pictures, numbers and words. Quote Link to comment Share on other sites More sharing options...
Petrichor Posted February 22, 2018 Share Posted February 22, 2018 draw two pieces of string(important that they are equal lengths), one above the other. use a ruler to divide one into 6 pieces, and the other into 7 pieces. like this: _ _ _ _ _ _ - - - - - - - now color in 4 of the 6 piece string, and 6 of the 7 piece string, and write numbers under each section of string, to show the "counting." _ _ _ _ _ _ 1 2 3 4 5 6 - - - - - - - 1234567 which one has more colored in part? it is apparent that 6/7 is more than 4/6 answer would be: _ _ _ _ _ _ 4/6 yard 1 2 3 4 5 6 - - - - - - - 6/7 yard 1234567 Jay's string is bigger because 6/7 is longer than 4/6. note: I assume what they mean by "number line" is that you number the lengths of string 4 Quote Link to comment Share on other sites More sharing options...
readinmom Posted February 22, 2018 Share Posted February 22, 2018 (edited) Here... Just scroll down the page. Edited February 22, 2018 by readinmom 2 Quote Link to comment Share on other sites More sharing options...
Petrichor Posted February 22, 2018 Share Posted February 22, 2018 We use singapore, so I do teach math like this, but I do appreciate that we can, at least, do this "thinking" orally rather than having to write it out! Quote Link to comment Share on other sites More sharing options...
Homebody2 Posted February 22, 2018 Author Share Posted February 22, 2018 Thank you for the responses. Can you explain the main concept the teacher is teaching or what understanding this shows? Quote Link to comment Share on other sites More sharing options...
freesia Posted February 22, 2018 Share Posted February 22, 2018 It visually reinforces the fractions and how they compare. It models drawing pictures to help solve problems. When the child has to explain what they've shown it solidifies learning ( you learn what you teach). You also can't do it by just memorizing a system, so it shows conceptual understanding. 8 Quote Link to comment Share on other sites More sharing options...
Mergath Posted February 22, 2018 Share Posted February 22, 2018 Drawing this stuff out also helps kids understand that if you take two identical somethings, when you divide it into a greater number of pieces, each piece is smaller than if you divide it into a smaller number of pieces. 6 Quote Link to comment Share on other sites More sharing options...
Homebody2 Posted February 22, 2018 Author Share Posted February 22, 2018 (edited) Thank you for your responses. I guess we are just having a hard time with a child accurately drawing a model that shows a line divided into sevenths. I do see the value in drawing a visual, but I see this example as being a bit tricky. The two fractions are so close that I could see how a child could draw them and see that they are the same length. Edited February 22, 2018 by Homebody2 Quote Link to comment Share on other sites More sharing options...
8filltheheart Posted February 22, 2018 Share Posted February 22, 2018 (edited) Thank you for your responses. I guess we are just having a hard time with a child accurately drawing a model that shows a line divided into sevenths. I do see the value in drawing a visual, but I see this example as being a bit tricky. The two fractions are so close that I could see how a child could draw them and see that they are the same length. You could have them draw an object (like rectangle/yard stick to represent a very fat string ;) ) above the number line so that they can visualize dividing the object into 7ths. Honestly, that is something that is very typical for 1st/2nd grade. The wording for the number line is confusing, but I am assuming that is the wording they are using in class. Learning to compare fractions by drawing pictures and shading in, etc is not "common core" but pretty standard math practice. (My kids have been doing it in Horizons math since 1994.) Edited February 22, 2018 by 8FillTheHeart 2 Quote Link to comment Share on other sites More sharing options...
Tsuga Posted February 22, 2018 Share Posted February 22, 2018 (edited) Obligatory notice that common core is an outcomes standard, not a curriculum. So homework that says "common core" is merely marketed as such. The government should have trademarked the phrase. Beast Academy and Singapore also use the number line method and also meet or exceed common core standard outcomes. Number lines are very useful in developing number sense though I agree that sevenths is not an easy demarcation. Good luck. A centimeter ruler can help as 7 cm is more likely to fit in a standard problem set. Edited February 22, 2018 by Tsuga 6 Quote Link to comment Share on other sites More sharing options...
Tsuga Posted February 22, 2018 Share Posted February 22, 2018 (I mean physically fit, whereas seven inches will take up the whole page.) Quote Link to comment Share on other sites More sharing options...
Petrichor Posted February 22, 2018 Share Posted February 22, 2018 Thank you for your responses. I guess we are just having a hard time with a child accurately drawing a model that shows a line divided into sevenths. I do see the value in drawing a visual, but I see this example as being a bit tricky. The two fractions are so close that I could see how a child could draw them and see that they are the same length. If the student(or parent even, really) is having trouble understanding how to draw it out, they could demonstrate it by taking two pieces of string or thread (2 different colors would be good for clarity) and fold and cut one so that there are 7 pieces and fold and cut the other so that there are 6 pieces. Line the pieces up to compare. Draw bars to show what it looks like. Can also print and cut out fraction bars to trace to assist with getting the right sizing until the student has practiced enough to be familiar with how to somewhat accurately draw the shapes. https://www.timvandevall.com/math/printable-fraction-strips-fraction-bars/ You can find fraction bars sold as magnets for more practice. I just remembered, I need to get myself some of those for my 3rd grader! 1 Quote Link to comment Share on other sites More sharing options...
fralala Posted February 22, 2018 Share Posted February 22, 2018 Here... Just scroll down the page. I think this is probably what they're looking for. Not lines made with a ruler, but the conceptual understanding that 1/7 is smaller than 1/6, and therefore approximately where each tick will be on a number line in relation to each other. I assume this is why they call the fraction 4/6 and not 2/3. They are looking for understanding of what each jump on the number line will mean if you're starting with sixths vs. sevenths. First, I would ask my third grader how she would divide a line into sixths without a ruler. I'd have her make a good long line. That will make the difference more apparent.Then I'd say, "Okay, so where will 1/7 be on the line?" Make sure she gets that it's pretty close to 1/6, but smaller. Now, what about 2/7? Where is it going to be in relation to 2/6? Bigger or smaller? Now, how about 3/7? What is happening? Is each seventh going to be equally close to each sixth? Oops, what happens if you do that? Cool, now you see what you need to do differently next time! Let's draw another line." I think once you start pulling out a ruler and trying to measure precisely, and figure out exactly how you can put precise sixths and sevenths on a line, that's when this problem becomes confusing. Being able to free-hand a line approximately divided into sixths, though, and to know that 1/7 is just a little smaller than 1/6, but then the difference between 2/7 and 2/6 is going to be a little greater, and forth-- I think these are the skills the problem asks for kids to demonstrate. And yes, it does require thinking and very likely a little frustration as a kid realizes, okay, these were not sevenths because my last line is still not in the right place...but actually a pretty neat problem! 4 Quote Link to comment Share on other sites More sharing options...
wendyroo Posted February 22, 2018 Share Posted February 22, 2018 If it helps the child understand, you can also attach numbers to the number line. If we announce that each 1/7 is equal to 10 units, then we can draw a number line that goes from 0 to 70 with the sevenths marked at each 10. Okay, then how long is 1/6 of the number line? Well 6 * 12 is 72 which is just a bit longer than 70. So we can estimate that each 1/6 of the number line must be a bit less than 12. These can be roughly marked on the same number line. This emphasizes the idea that 1/6 is bigger than 1/7. Now we can see visually that 4/6 is less than 6/7 and we can confirm that with multiplication. 6/7 = 6 segments that are each 10 long, so it falls at 60 on our number line. 4/6 = 4 segments that are each ~12 long, so it must be a bit less than 48 on the number line. 48 is clearly less than 60. Wendy Quote Link to comment Share on other sites More sharing options...
letsplaymath Posted February 23, 2018 Share Posted February 23, 2018 (edited) Liz and Jay each have a piece of string. Liz's piece of string is 4/6 of a yard, and Jay's is 6/7 off a yard long. Whose string is longer? Draw a number line to model the length of each string. Explain the comparison using pictures, numbers and words. One of the best ways to help kids understand fractions is by comparing them to familiar numbers. I think that is the goal of this problem -- compare the fractions to one whole thing. If you draw a free-hand number line, that is plenty good enough. No need for accurate measurement. Third-graders can't measure that precisely anyway. Does your number line show that 4/6 needs TWO more pieces to get to one whole thing? And 6/7 only needs ONE piece. Now, one piece CAN be bigger than two pieces, if (for instance) the one is a half and the two are tenths. But in THIS case, not only does 6/7 need fewer pieces, it's also true that the pieces are smaller than the sixths. Because when we divide a whole thing into seven pieces, those pieces have to be smaller than if we had only split it into six. So even if we only had 5/7 (like in the image that readinmom linked to above), we would still be closer to one whole thing than when we have 4/6. Therefore, 6/7 is greater than 4/6, and Jay's string (in the original problem) is longer. Edited February 23, 2018 by letsplaymath 2 Quote Link to comment Share on other sites More sharing options...
Tanaqui Posted February 23, 2018 Share Posted February 23, 2018 Thank you for the responses. Can you explain the main concept the teacher is teaching or what understanding this shows? That is a wonderful question! I won't repeat what everybody else has said, but I want to thank you for asking it. Sad but true, most people don't bother :( 3 Quote Link to comment Share on other sites More sharing options...
TrustAndLove Posted February 23, 2018 Share Posted February 23, 2018 Am I the only one feeling this question is a bit too much to the point of meaningless? If the child just starts to learn fraction, such as comparing 1/2 and 1/3, using a pie or a number line makes sense. But comparing 4/6 and 5/7? At grade 3, kids should learn 4/6 is same as 2/3,14/21, and 5/7 equals to 15/21, therefore, get the answer. I know my grade 3 child will not be able to draw the difference of 1/21. 1 Quote Link to comment Share on other sites More sharing options...
8filltheheart Posted February 23, 2018 Share Posted February 23, 2018 Am I the only one feeling this question is a bit too much to the point of meaningless? If the child just starts to learn fraction, such as comparing 1/2 and 1/3, using a pie or a number line makes sense. But comparing 4/6 and 5/7? At grade 3, kids should learn 4/6 is same as 2/3,14/21, and 5/7 equals to 15/21, therefore, get the answer. I know my grade 3 child will not be able to draw the difference of 1/21. Except the problem was 6/7 vs. 4/6 which is a difference of 4/21 which is almost 1/5. It can be seen visually. I just asked my 2nd grader to divide 2 equal rectangles that I had drawn into 7ths and 6ths (she just drew the lines to divide the objects. No ruler. No precision. Just 2nd grader, not quite equally proportioned lines) Then I asked her to color in 6/7 and 4/6. I asked which was bigger and she answered easily bc she could see it. I am not sure there is anything more complicated about this problem than that. 1 Quote Link to comment Share on other sites More sharing options...
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