Here’s an activity I just drafted for my Algebra 1 class. I’m trying to help them get very comfortable working with the distributive property and fractions. Thoughts?
If one of them decides to combine the stuff in the brackets (parentheses) and then multiply, have they done it wrong? This was my first inclination.
The distributive law is flogged to death in arithmetic (is it called that nowadays?) but really comes into its own in algebra. I would start with 10(2+x) for example and see what you get for different x values, then look at the expansion of the brackets to get 20+10x and see that it gives the same results for the same values of x.
These type of calculations looks easy but we do silly mistakes while solving. These problems demands practice. The practice sets are really good for students. Really nice. Keep sharing. http://www.selectmytutor.co.uk/subject-maths.html
Math is alright, but is it useful?
Can you give a simple example for application of math to nature or to engineering?
This activity is good because your example can show the concepts of the distributive property and fractions. It is easy to mislead student when they do this type of problems, especially the second example. This activity can let students know how the questions with distributive property and fractions that can be asked trickily, and what concept they should completely understand on it. It makes students be more flexibility while they are doing the practices. Great!
I see that this can be a good example to teach children that how they multiply and add up fractions. I am pretty sure that children will get confused adding two fractions that has different denominator. It will be a great practice to find common denominator to add up two fractions or maybe more. Since children also know how to do multiplication with fractions, so teaching children step by step is important to help them understanding the concepts. I also agree that this is one of the common mistake when we learn this because many others and I also made same kind of mistake. I definitely would like to use this kind of practice if I teach children.
First of all the first solution was right, and the second solution was wrong. This is a common mistake that students will make when they did first learn addition/ subtraction on fraction. I remember when I first learn addition/ subtraction fraction, I also made the same mistake because I didn’t know the denominator need to be the same in order to add/ subtraction. While teaching fraction, I think teachers need to explain clearly what does the denominator and numerator, so students can understand why it is important to use same denominator on fraction. Anyway, this type of question is helpful because most of the students will get it wrong while they first learn, and teacher can use this opportunity to explain in depth.
For set 1 the solution is correct but the second problem of set one isn’t correct. I can understand how they got that answer because they just added the denominator and disregarded the numerator. When I was younger and when I first started working with fractions I was always confused about why we should make the bottom number a common denominator. I always thought to myself why can’t I just add straight across because its an addition problem. Fraction was always a confusing topic for me when I first started learning it. It didn’t occur to me that you have to make it a common denominator to add the fractions. It makes sense if there are pictures to show the problem but it is hard to just look at the numbers and try to figure out what is going on. The problem with the second problem was the child didn’t make it a common denominator. The rest of the problems are correct and its important to understand how a denominator works and how a numerator works.
The set that I decided to use was the first set. In the first set you can see how the student made the mistake on the second problem. The child forgot to add the top (numerator) and forgot about the bottom (denominator). To be honest fractions to this day are a little challenging for me and when I have a common denominator my brain automatically switches to thinking about the problem as a whole and either adding or subtracting what I need from a whole. When I have denominator that are not the same my brain starts trying to process the information and as an adult I know I have to convert, but the child in this case has always learned to add, subtract, multiply, divide from left to right and when you see this problem as a child your first response is to multiply straight across and seeing it as two problems (top row=1 problem, row two= 2nd problem). Then at the end they just combined them. For this particular child, they would need more practice with denominators and denominators for the sake of their mathematical experience and mathematical understanding to advance.
I thought this activity is very meaningful to examine if children understand how to apply concepts of the distributive property and if they truly understand how to solve the problem of fraction multiplication and addition because the questions involving multi-steps can be kind of tricky and confusing for children. For the solution of set 1 is correct that the child is multiplying the whole number 35 with each fraction in the parentheses and then add them up. The process shows he/she understands the distributive property and know the way to multiply whole number with fractions. However, he/she makes mistakes in set 2. He/she does it wrong by adding the denominators directly and disregarding the numerator. I think this might be the common challenging part for students who just touch on the fractions. For teachers, i think it’s important for us to know that it need process to lead them to understand why we need to make denominator same at first and then to add the fractions by visualized pictures. And then it also takes transition time for them to combine understanding on pictures to numbers and finally transit from pictures to solely numbers.
The idea of common denominator is not one that I expect children to understand quickly. It was definitely something I worked with for a long time, as refused to not just add the numerators and denominators separately and call it good. When I realized that was wrong was when I realized that each piece was a different amount of a whole, and that comparing them without making the “whole” be the same is not correct. So in the case of these math practices above, I can understand how the practice set 1 could be answered incorrectly. This is because the student just ignored the numerators and added the denominator together, as 4 plus 5 equals 9. The answer to this set is 9/20, so the student was somewhat on the right track. The student was probably confused why he or he could not just add straight across, and possibly though that the 1 could just be ignored. The teacher needs to remind his or her students that one can not assume that comparisons of parts of whole could be just done without modifications. I do think that bring the distributive property into things is just making everything else very confusing, so I am interested why that was even brought up in the first place.
I think that these are really great problems for children to practice the distributive property with fractions and also for teachers to check their understanding. As others have said, the first solution is correct but the addition in the second solution is incorrect. I understand why this child could make a mistake like this; it’s easy to think if we can just multiply fractions together, why can’t we do the same with addition? I would explain to this child more clearly how fractions work and how they are parts of a whole. Fractions must be part of the same whole in order to be able to add them together, or in other words, they must have the same denominator. The whole stays the same, but we add together how many parts of that whole there are; the denominator stays the same, but we add together the numerators. Explaining how to find common denominators and converting fractions might also be necessary. I think that these problems show a variety of different numbers with different denominators, so it would be great practice for a child still learning these concepts about fractions.
When looking at this I think that students are getting a really good practice on fractions and distributive properties. Clearly when learning fraction it can get really confusing and trying to remember all the different rules. Multiplying and dividing fractions have different rules from when adding and subtracting fractions. What I do like about these problems is that its repetition and when you do different problems but that have the same concept it teaches the student to really understand how to do these types of problems and also catch their mistake if they are also doing some part of a problem wrong.With these types of problems its really easy to get mislead by the way the question is being asked or trying to figure out what the question is asking especially of for the second one. Understanding that they need the same common denominators is upper level understanding in fraction and by this example the child think you should add the denominator and do nothing with the numerator. I think making that mistake is something that happens all the time and fraction is just one of those things you can not rush through and as teachers when teaching fractions we need to make sure students are following through with every step and really understanding and just take time with fractions and every level of it.
12 replies on “Mistake Analysis + Practice”
If one of them decides to combine the stuff in the brackets (parentheses) and then multiply, have they done it wrong? This was my first inclination.
The distributive law is flogged to death in arithmetic (is it called that nowadays?) but really comes into its own in algebra. I would start with 10(2+x) for example and see what you get for different x values, then look at the expansion of the brackets to get 20+10x and see that it gives the same results for the same values of x.
These type of calculations looks easy but we do silly mistakes while solving. These problems demands practice. The practice sets are really good for students. Really nice. Keep sharing.
http://www.selectmytutor.co.uk/subject-maths.html
Math is alright, but is it useful?
Can you give a simple example for application of math to nature or to engineering?
This activity is good because your example can show the concepts of the distributive property and fractions. It is easy to mislead student when they do this type of problems, especially the second example. This activity can let students know how the questions with distributive property and fractions that can be asked trickily, and what concept they should completely understand on it. It makes students be more flexibility while they are doing the practices. Great!
I see that this can be a good example to teach children that how they multiply and add up fractions. I am pretty sure that children will get confused adding two fractions that has different denominator. It will be a great practice to find common denominator to add up two fractions or maybe more. Since children also know how to do multiplication with fractions, so teaching children step by step is important to help them understanding the concepts. I also agree that this is one of the common mistake when we learn this because many others and I also made same kind of mistake. I definitely would like to use this kind of practice if I teach children.
First of all the first solution was right, and the second solution was wrong. This is a common mistake that students will make when they did first learn addition/ subtraction on fraction. I remember when I first learn addition/ subtraction fraction, I also made the same mistake because I didn’t know the denominator need to be the same in order to add/ subtraction. While teaching fraction, I think teachers need to explain clearly what does the denominator and numerator, so students can understand why it is important to use same denominator on fraction. Anyway, this type of question is helpful because most of the students will get it wrong while they first learn, and teacher can use this opportunity to explain in depth.
For set 1 the solution is correct but the second problem of set one isn’t correct. I can understand how they got that answer because they just added the denominator and disregarded the numerator. When I was younger and when I first started working with fractions I was always confused about why we should make the bottom number a common denominator. I always thought to myself why can’t I just add straight across because its an addition problem. Fraction was always a confusing topic for me when I first started learning it. It didn’t occur to me that you have to make it a common denominator to add the fractions. It makes sense if there are pictures to show the problem but it is hard to just look at the numbers and try to figure out what is going on. The problem with the second problem was the child didn’t make it a common denominator. The rest of the problems are correct and its important to understand how a denominator works and how a numerator works.
The set that I decided to use was the first set. In the first set you can see how the student made the mistake on the second problem. The child forgot to add the top (numerator) and forgot about the bottom (denominator). To be honest fractions to this day are a little challenging for me and when I have a common denominator my brain automatically switches to thinking about the problem as a whole and either adding or subtracting what I need from a whole. When I have denominator that are not the same my brain starts trying to process the information and as an adult I know I have to convert, but the child in this case has always learned to add, subtract, multiply, divide from left to right and when you see this problem as a child your first response is to multiply straight across and seeing it as two problems (top row=1 problem, row two= 2nd problem). Then at the end they just combined them. For this particular child, they would need more practice with denominators and denominators for the sake of their mathematical experience and mathematical understanding to advance.
I thought this activity is very meaningful to examine if children understand how to apply concepts of the distributive property and if they truly understand how to solve the problem of fraction multiplication and addition because the questions involving multi-steps can be kind of tricky and confusing for children. For the solution of set 1 is correct that the child is multiplying the whole number 35 with each fraction in the parentheses and then add them up. The process shows he/she understands the distributive property and know the way to multiply whole number with fractions. However, he/she makes mistakes in set 2. He/she does it wrong by adding the denominators directly and disregarding the numerator. I think this might be the common challenging part for students who just touch on the fractions. For teachers, i think it’s important for us to know that it need process to lead them to understand why we need to make denominator same at first and then to add the fractions by visualized pictures. And then it also takes transition time for them to combine understanding on pictures to numbers and finally transit from pictures to solely numbers.
The idea of common denominator is not one that I expect children to understand quickly. It was definitely something I worked with for a long time, as refused to not just add the numerators and denominators separately and call it good. When I realized that was wrong was when I realized that each piece was a different amount of a whole, and that comparing them without making the “whole” be the same is not correct. So in the case of these math practices above, I can understand how the practice set 1 could be answered incorrectly. This is because the student just ignored the numerators and added the denominator together, as 4 plus 5 equals 9. The answer to this set is 9/20, so the student was somewhat on the right track. The student was probably confused why he or he could not just add straight across, and possibly though that the 1 could just be ignored. The teacher needs to remind his or her students that one can not assume that comparisons of parts of whole could be just done without modifications. I do think that bring the distributive property into things is just making everything else very confusing, so I am interested why that was even brought up in the first place.
I think that these are really great problems for children to practice the distributive property with fractions and also for teachers to check their understanding. As others have said, the first solution is correct but the addition in the second solution is incorrect. I understand why this child could make a mistake like this; it’s easy to think if we can just multiply fractions together, why can’t we do the same with addition? I would explain to this child more clearly how fractions work and how they are parts of a whole. Fractions must be part of the same whole in order to be able to add them together, or in other words, they must have the same denominator. The whole stays the same, but we add together how many parts of that whole there are; the denominator stays the same, but we add together the numerators. Explaining how to find common denominators and converting fractions might also be necessary. I think that these problems show a variety of different numbers with different denominators, so it would be great practice for a child still learning these concepts about fractions.
When looking at this I think that students are getting a really good practice on fractions and distributive properties. Clearly when learning fraction it can get really confusing and trying to remember all the different rules. Multiplying and dividing fractions have different rules from when adding and subtracting fractions. What I do like about these problems is that its repetition and when you do different problems but that have the same concept it teaches the student to really understand how to do these types of problems and also catch their mistake if they are also doing some part of a problem wrong.With these types of problems its really easy to get mislead by the way the question is being asked or trying to figure out what the question is asking especially of for the second one. Understanding that they need the same common denominators is upper level understanding in fraction and by this example the child think you should add the denominator and do nothing with the numerator. I think making that mistake is something that happens all the time and fraction is just one of those things you can not rush through and as teachers when teaching fractions we need to make sure students are following through with every step and really understanding and just take time with fractions and every level of it.