Level G Lesson 6
Add Fractions
In lesson 6 the objective is, the student will work with adding fractions and mixed numbers with like and unlike denominators.
There are three essential questions that will be guiding the lesson. Number 1, how does it help our understanding of fractions to build with concrete materials? Number 2, how can we add fractions and mixed numbers with like denominators? Number 3, how can we add fractions and mixed numbers with unlike denominators?
Our SOLVE problem for this lesson is, James is working on a packet of math review problems. The teacher gives him the packet on Wednesday and it is due the next Monday. James completes one fourth of the problems on Wednesday, one third of the problems on Thursday and one eighth of the problems on Friday. How many of the problems did he complete on Wednesday and Thursday?
When we S the problem or study the problem the first thing we’re going to do is underline the question. How many of the problems did he complete on Wednesday and Thursday? The second part of our S step is to complete the statement. This problem is asking me to find, how many problems James completed on Wednesday and Thursday.
We’re going to modeling adding like fractionsat the concrete level. What we’re going to do is use our fraction strips our manipulatives to model this problem one third plus one third. We are going to place those underneath our whole unit. Each of our addends is green. This indicates that our denominators are the same or common so we can add. The way we add our fractions is by pushing them together. Our sum is now two thirds, 2 groups of one third. We’re going to write that sum at the top of our page. What happened to the denominators in our problem? They remained the same. What happened to the numerators in the problem? We added the numerators 1 plus 1 for a sum of 2. The numerators changed when added. Could the sum be legally traded for fewer fraction strips in another color? No. That completes our model.
The next problem we are going to model is one fourth unit plus one fourth unit. We’re going to find the fraction strips for one fourth, that would be 2 yellows, so each of our addends are yellow. This indicates because they’re the same color that we have a common denominator. We can add our two fraction strips by pushing them together. The answer when we add our one fourth and one fourth is two fourths. What does this mean to us? Our denominators are the same. What happened to the denominators in the problem? It remained the same in the sum. What happened to the numerators? They changed when they were added. Could this sum be legally traded for fewer fraction strips in another color? And the answer is yes. We can do a legal trade of two fourths for our one half unit. Which ones? Two fourths can be trades for one half, so my final answer is one half.
We’re now going to model adding unlike fractions at the concrete level. We have a
One half unit which is represented by our brown and a one fourth unit, which is represented by our yellow. The color of our two addends are brown and yellow.What does this mean? This means that our denominators are different. We cannot add before finding a common denominator or having our fraction strips all in one color. What we need to do is see if we can legally trade our fraction strips so that they are all in one color before we add. We are going to look at our half unit and see if we can legally trade our one fourth for one half. And when we look we see that one half can be legally traded for two fourths.
We are going to legally trade those. Now we have all of our fraction bars in one color and we can push them together. Our answer then is three fourths. How did you change the denominators in the problem? We changed the 2 in our one half unit to 4 in our two fourths units. What happened to the numerators in the problem? When I changed the one half to two fourths the numerator doubled, just like the denominator, 2 doubled to 4 and 1 doubled to 2. Could this sum be legally traded for fewer fraction strips in another color? And the answer is no.
The next problem we’re going to model is one third plus three fourths. The color of our 2 addends are green and yellow. This indicates that our fractions have different or not common denominators. What we are going to do is legally trade for a different color so that both of our addends are all in one color. I’m going to I look at my 3 and my 4 and I know that both factors of 12 so I’m going to try my twelfths for my legal trade. I see that I can legally trade my one third for 4 twelfths, And I can legally trade my one fourth for 3 twelfths. I noted that my denominators are different, I completed the legal trade. What happened to the denominators in the problem was the one third changed to four twelfths, the 3 changed to 12, and the three fourths changed to nine twelfths for a new denominator of 12. What happened to my numerators. They both became larger as the denominators became larger. Now I’m going to find the sum of one third and three fourths or four twelfths and nine twelfths. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. I now have thirteen twelfths as my sum. Thirteen twelfths is an improper fraction, because the numerator is greater than or equal to the denominator. But we can write that improper fraction as a mixed number. A mixed number is a combination of a whole number and a fraction. What we are going to do is we are going to legally trade our thirteen twelfths to see if we can make fewer pieces in one color. And if you look at the whole unit we’ll see that 12 of those are equivalent to 1 whole unit so we’re going to do a legal trade for our 12, twelfths. I still have one twelfths left after I do my legal trade. I have 1 whole unit and one twelfth. Could my sum be legally traded for fewer fraction strips in another color? Yes, thirteen twelfths can be traded for 1 whole unit and one twelfth .
We are now going to be modeling adding like fractions moving to the pictorial level. I have my 1 whole unit here and I’m representing my problem two fourths plusone fourth. Underneath my problem I have three fraction strips represented. The first strip is going to be my addend. The second strip will be my second addend and the third strip will represent my sum. My first addend is two fourths, so I’m going to shade in two fourths of my strip, which is 2 out of the 4 sections. On the second strip I’m going to shade in one fourth or 1 out of the 4 sections to model my second addend. My third strip represents my sum. I’m going to push my two addends together and I’m going to have a sum of three fourths or 3 one fourth units. I’m going to model that on my last strip, and I’ve shaded in my sum of three fourths. The question I want to ask is, can this fraction be traded for fewer pieces, and the answer is no.
In my next problem, I’m going to be modeling moving to pictorial and we’re going to rename one of the addends. I’ve modeled with fraction strips up here my manipulatives. I’m going to add three, one fourth units plus one eighth unit. In the problem we did before the denominators were common or the same. This time we have different denominators, so we’re going to have to find a common denominator. We’re going to do that by finding an equivalent fraction. Again, our first two fraction strips are going to represent our addends. The first one is three fourths. I have that divided into four parts. I’m going to shade in 3 of the 4. My second addend I’m going to represent by shading 1 of the 8 pieces. My question I want to ask now is, can I legally trade fourths for eighths? How many eighths are in each group of one fourth? There are two groups of one eighth in each one fourth. I’m going to legally trade those, so three fourths is equal to six eighths. I’m going to model that on my fraction bar, my one fourth is now two eighths. My three fourths is now 1, 2, 3, 4, 5, 6, eighths. I have created a equivalent fraction. Now I can combine or add by pushing together my fraction strips. Because my common denominator is 8, I’ve divided my sum into 8 sections. And I’m going to model the sum by shading in seven eighths.
Three fourths plus one eighth is seven eighths. My last question is can this fraction be traded for fewer pieces in one color and the answer is no. This is my simplified fraction.
The next problem I’m going to be modeling is one third plus one fourth. We are going to be moving to the pictorial but we’re renaming both addends this time. We’re going to have to legally trade the sum of one third and one fourth for fraction strips of one color. So what we’re going to have to do is we’re going to be finding a fraction strip that we can legally trade the one third. What I’m going to do is try the one twelfth. I can legally trade one third for four twelfths and I can legally trade one fourth for three twelfths. I can then push those two together and I’m going to have a sum of seven twelfths. What happened to my numerators and denominator when I was finding equivalent fractions is that the numerators changed when the denominators were changed. I cannot simplify seven twelfths so this is my final answer. We’re now going to model the same problem we just did pictorially. In these problems we’re going to have to find a common denominator for the 3 and the 4. The first strip represents our one third. The second strip represents one fourth, and the third strip represents our sum. The fractions we have one third and
One fourth cannot be added. They must have a common or the same denominator. So what we’re going to do is we’re going to find the common denominator by listing the multiples of 3 and 4 and finding the least common multiple. I’m going to list the multiples of 3: 3, 6, 9, 12. I’m going to stop there because I know from using my manipulatives that my denominator will be 12. I list my multiples of 4: 4, 8 and 12. I’m going to circle my least common multiple, which is going to become by common denominator. We’re going back to our concrete representation again and when we have the fraction of one third, we did a legal trade for four twelfths. When we had ourone fourth we did a legal trade for three twelfths. One third was equivalent to three twelfths. We had to multiply the denominator 3 times 4 to get our new denominator of 12. We also had to multiply our numerator 1 times 4 to get our new numerator of 4. When we had one fourth and we’re finding an equivalent fraction, 4 times 3 is 12, we also multiplied our numerator 1 times 3 for our new numerator. When we add those two numbers our sum is seven twelfths. What we’re going to is we’re going to write the pictorial representation and we’re going to first of all model our first fraction fourth twelfths by dividing our first fraction strip into 12’s. We’re going to shade in 4 of those with my pink to represent my one twelfths. We’re now going to represent the second addend by dividing our second strip into twelfths. Our second addend is 3/ 12. We are going to shade in 3 of the 12 sections. Our last fraction strip, remember is our sum. We’re going to divide that into twelfths, because our sum also has a denominator of 12, 4 plus 3 is seven twelfths. The last question we want to ask ourselves is, can this fraction be traded for fewer pieces. The answer is no,seven twelfths is in simplest form.
We are now going to be adding unlike fractions, and we’re moving to the abstract representation. We’re adding three fifths plus one fifth. I’ve modeled that at the top with my fraction strips. And in the first box you’re going to draw the fraction bars and model shading in the three fifths and the one fifth to represent each addend. When we add we’re going to push together and because they have a common denominator and are all one color, we can push together or combine those fraction strips which is represented pictorially in our bottom strip here as four fifths. In the last column we’re going to write numerically what we have done. We’ve taken three fifths we’ve added one fifth for a sum of four fifths.
In our next example we’re moving again to the abstract representation. We’re going to be modeling three fourths plus one twelfth. One of our addends will need to be renamed. What we’re going to do here is we’re going to shade in three fourths in my first fraction strip, and one twelfth in the second to represent each of the addends. Because we’re going to need to find a common denominator, we’re going to do that by finding the multiples of 4, and the multiples of 12. The multiples of 4 are 4, 8, 12, and we’re going to be able to stop there because we found that 12 is a multiple of 4 and that’s our other denominator, so we are actually going to stop right there. We know our new denominator is going to be 12. What we’ve done in the second box is a legal trade from fourths to twelfths, three fourths is equivalent to nine twelfths.
When we found our equivalent fraction, we multiplied 4 times 3 to get 12, so we also have to multiply our numerator 3 times 3 for our new numerator of 9. Our second fraction one twelfth stays the same, because we already have a denominator of 12. Our last fraction bar we’re using to represent the sum. We have our 9 plus 1 which is ten twelfths. When we model that with our fraction strips, we can show the legal trade of our three fourths for nine twelfths. Push those together and our sum is ten twelfths. The last question we need to ask ourselves is if we can legally trade fewer pieces of the same color to simplify our fractions. We’re using the one sixth and we find that we can simplify to five sixths. We can legally trade our ten twelfths for 5 one sixth pieces, simplifying our final answer to five sixth.
The next problem we’re going to model is one half plus two thirds. This is our final modeling from moving from pictorially to abstract. And we’re going to represent our two fractions and this time we’re going to have to rename both denominators. The first thing we’re going to do is legally trade fraction strips so that we can have a common denominator. We’re trying the one sixth there and there. One sixth is a legal trade. One can be legally traded for one half, three one sixth units, one half and my two thirds can be legally traded for 4 of the one sixth units. I can find my common denominator by using the common multiples also. I’m going to list the multiples of 2 up through 10, then I’m going to list the multiples of 3 and I find I have a common multiple in 6. I have represented here my one half being equivalent to three sixths. When I look over here at my fraction one half being equivalent to three sixths. I see that I had to multiply the denominator 2 by 3 in order to get my new denominator 6. I also multiply my numerator 1 times 3 to get a new numerator 3. I have my two thirds which is equivalent to four sixths. We look over here at our two fractions in order to change our two thirds to an equivalent fraction we had to multiply 3 times 2 to get 6. We also want to multiply our numerator by the same number 2 times 2 is 4. I’m now going to push together my one half plus two thirds which I have found equivalent fractions three sixths plus four sixths. When I add those together I have, 1, 2, 3, 4, 5, 6, 7, seven sixths, and I have that written here. Seven sixths is an improper fraction and it needs to be converted to a mixed number. Can I legally trade sixth sixths for anything? The answer is yes, we can trade sixth sixths for 1 whole. My final answer is 1 and one sixth.
We’re now moving to adding fractions without models. We’re going to look at problems number 1, three tenths plus four tenths. What is this problem asking us? It’s asking us it is asking us three tenths plus four tenths. Do our fractions have common denominators? Yes they do. We do not have to find a common denominator We’ll add the numerators together and the denominators will remain the same. We don’t need to find the least common multiple because we already have the common denominators. We’re going to rewrite our number sentence, three tenths plus four tenths our denominator stays the same, and we add our numerators for a sum of seven tenths. Do we need to simplify the fraction? No, seven tenths is a simplified fraction.
In the second problem we’re modeling, one third plus three sixths. Do the fractions have a common denominator? No, they do not. We’re going to find a common denominator by finding the multiples that are common for 3 and 6. We’re going to list the multiples of 3, and we see right away that 6 is a multiple of 3 so we know that’s going to be our common denominator. We do have two common multiples but the least common multiple is 6 and we want that to be our denominator. We’re going to change one third to an equivalent fraction with the denominator of 6. In order to do that we have to multiply the denominator by 2, because 3 times 2 is 6. Whatever we multiply the denominator by we also multiply the numerator by. The reason we can do that is because 2 over 2 is equivalent to 1, and multiplying any number by 1 is the same value, 1 times 2 is 2. One third and two sixths are equivalent fractions. We don’t need to do anything to the 3 / 6 because it already has a denominator of 6. We found that our denominators were not common, are least common multiple is 6, and we made an equivalent fraction of one third and two sixths. The last thing we do is rewrite our number sentence: Two sixths plus three sixths equals five sixths. Do we need to simplify this fraction? No, it’s in simplest form.