I am not comfortable with fractions myself so these two problems took me a while to critique. The first problem I believe is correct because the student multiplied 35 by the numerator of both fractions getting (35/5 + 35/7) which is equivalent to 7+5= 12. This problem is correct. As for the second problem, the students strategy is incorrect because they multiplied the denominator and numerator of the outside number “1/4” with (4/3) and (4/5). With this being said, when I did the problem, I made 1/4 from the outside into a decimal. So then I get .25 on the outside and i multiple “.25” by the two numerators leaving me with (1/3+1/5) and I cannot add these two because they do not have a common denominator. So now I multiply the 1 from “1/3” by 5 and the 3 from “1/3” by 5 as well leaving me with 5/15. Now I multiply the 1 from “1/5” by 3 and the 5 from “1/5” by 3 as well and get 3/15. Now I can add these two fractions together because I have the common denominator 15. My new equation is 5/15+3/15=8/15. Ultimately after converting 1/4 to a decimal and finding a common denominator for 1/3 and 1/5 I get the answer to 1/4 x (4/3 + 4/5) = 8/15.
I am pretty sure that the first problem is correct. The student multiply 35 which is 35/1 to the numerators both 1/5 and 1/7 and got 35/5 and 35/7. The student knew that he can simplify this. 35/5 can be simplified as 7 and 35/7 can be simplified as 5. So, the student wrote 7+5=12 which is right answer. The second one, even I got confused at first. I concluded that this is incorrect. The student can do in two different ways. The first one is to multiply 1/4 and 4/3 first to get 1/3 because he needs to simplify denominator 4 with numerator 4. And then, he can multiply 1/4 and 4/5 to get 1/5. Now the student has 1/3 + 1/5 and need to find common denominator which is 15. Through this way, the student gets 5+3/15 which is 8/15. The second way to solve this is to calculate (4/3 + 4/5) first. The student can find the common denominator 15 and get (20 + 12/15) because those numerators should be increased at same rate with denominator. And then, the students can solve 1/4 x 32/25. In the case of product between two fractions, the student can reduce denominator 4 from 1/4 with numerator 32 from 32/25 and get 8/15.
Between the two problems we can see that the first problem is correct and the second problem is incorrect. The first problem is correct because it first distributes properly. when we multiply fractions we multiply straight across so in the first problem we see that 35 times 1/5 is 35 times 1 (numerator) and then 1 times 5 (denominator) which gives us 35/5. Then it is added to 35 times 1 (numerator) and then 1 times 7 (denominator) which is 35/7. Then when we see that the fractions can be reduced to a more simple number. 35/5 can be reduces to 7 and 35/7 can be reduced to 5 and when we look at the then simplified overall equation 7+5 the result is 12, which is exactly like the answer the student above gives.
The second equation is correctly done half of the way. the multiplication portion of the equation is done correctly. The 4/12 and 4/20 are the correct answers to when you distribute the 1/4 to (4/3 + 4/5). However this student makes a mistake when adding the 4/12 and 4/20. When adding fractions we need to remember that the whole stays the same so the denominator stays the same while the part (numerator) changes. In order to add the fractions we need to make the whole the same. in order to make 12 and 20 the same we need to make the fraction as simple as possible. 4/12 ca be reduced to 1/3 and 4/20 can be reduced to 1/5. then we can find the LCM for 3 and 5, which is 15. in order to make 15 the whole we need to multiply 1/3 by 5 and then 1/5 by 3 to make it 5/15 and 3/15. When we add these numbers it is 8/15 which is the correct answer to the second question.
We can see that the first solution is correct and the second one isn’t. In the first problem, we know that this student understands the distributive property of multiplication because they’re multiplying 35 to the fraction of 1/5 and then 35 to 1/7, which comes out to 7+5 = 12.
However, the second problem is a little bit more complicated. They still understand the distributive property when multiplying fractions with fractions, but they are not able to add the fractions correctly. I think they are confusing the rules with fraction addition when comparing the numerator with the denominator. We know that when you add numbers in the denominator, they have to be the same, which in this case would be 20×12 = 240. As for the numerator, you would multiply it by what you multiplied the denominator by.The 4/12 would be multiplied by 20/20 to become 80/240 and the 4/20 would be multiplied by 12 on both top and bottom to equal 48/240. After adding 80/240 to 48/240, you get 128/240, which you can divide by 8 on both top and bottom to simplify to 8/15.