Fraction Equivalence

Fractions Day 14: To Be or Not to Be (Equal to 1/2)…

Jen Moffett, , Leave a comment

Good morning! This is the last lesson of this series of 15-20 min initial fraction explorations. I would have gone on for the duration of distance learning, but we have some measuring to do!

  1. We warmed up with some comparing of fractions. Using index cards, I just wrote a bunch of fractions (that he could check with the fraction strips). We flipped the cards face down and mixed them up. He would flip a card face up; I would do the same. We would compare our fractions, and whoever had the largest fraction got to keep the cards. Which one is bigger? How do you know? (Example: I know 1/2 is bigger than 3/8 because 4/8 is a half.) He often had to show me with his fraction strips to explain (See here? This one is bigger.). Note that some of the fractions are equivalent, which gave us some nice places to pause and discuss who won. I let him choose how to deal with the equivalent ones (He chose to put them back upside down and reshuffle.). See below for fraction cards for the game.
  2. I gave him the problems below. Each one compares 1/2 to other fractions. 1/2 is considered a “benchmark fraction”. It is a great one to use for estimation and relating to the size of other fractions. We looked at each pair individually. Is the fraction greater than, less than, or equal to half? How do you know? Notice the first 6 pairs start with the equivalent relationship, to push Chris into using half to reason about the very next pair of fractions with the same denominator. The last three pairs were to see if he would apply the “comparing to half” strategy. Below is a video of his thinking.
Comparing 6/8 to 1/2.

If I had another day, I would continue with this idea of comparing to half. It caused Chris to really consider how the numerator and denominator related to each other and to the fraction 1/2.

Comparing to 1/2.
comparing-fractionsDownload
Fractions Day 14: To Be or Not to Be (Equal to 1/2)…

Fractions Day 12: Renaming Mixed Numbers and Improper Fractions

Jen Moffett, , 2 Comments

Children often struggle with understanding how to rename an improper fraction to a mixed number (and vice versa). They learn ‘slick tricks’, such as Around the World or The Circle Method (I have no idea what that means). In fact, when I googled how to change a mixed number into an improper fraction, the first dozen (I stopped looking) used titles with “Easy”, “Trick”, “Fast”, and “The Neat Way”. Who are these ways easy for? Certainly they are easy to learn…and then forget.

My stance is to take time to work with small whole numbers (1 and 2, for instance) and concrete models (such as our fractions kits) and let children figure it out for themselves. It may take time, but in the end they will remember and will figure out a ‘slick trick’ that works for them.

  1. Played “Cover it Up”, first version (Please see Fractions Day 2 for directions) three times. He still wants to play, so we still play!
  2. Wrote 5/3 on his whiteboard. How many wholes could we make? (1) How many thirds would we have left? (2) So you would have 2-thirds? (Yeah. That is what I said, mom!) How could we write that amount? (1 + 2/3) So you have 1 AND 2/3 (That is what I wrote, mom!) Clearly I was annoying today. However, I wanted to make sure we restated the ideas clearly using appropriate language. He didn’t hear the difference, but I did.
  3. We continued with renaming improper fractions as a whole number + fraction. I tried to continue the questioning, but he worked so quickly I didn’t have time to question as he did some of them. He got stuck for a quick moment on 4/2. (It’s just 2 wholes, right mom? Because 2/2 = 1 and if I double that I get 2, right?) Otherwise, this progressed much faster than I thought it would!
  4. We then moved to renaming mixed numbers as improper fractions. I started with the same number as before: 1 2/3. Show me 1 whole and 2/3. (Mom, I can do this without the fractions (strips). Okay…Then if I want to rename this in all thirds, how many thirds would I have? (3/3 is a whole, so two more would make 5.) Five of what? (5 of the thirds) How would I write that? (Wrote 5/3 on the paper.)
  5. Again, I was surprised at how quickly he grasped this and was able to reason out each one without the fraction strips. Below is a clip of him reasoning out 7/3.
Renaming 2 1/3

It is important to note that Chris didn’t use the fraction strips, but could have. Each child is different and needs different representations to understand the math. If your child would like to continue using the fraction strips to work through the problems that is awesome! They are still gaining understanding regarding equivalence and renaming fractions greater than 1.

Below are the improper fractions and mixed numbers we explored.

Renaming Improper Fractions
Renaming Mixed Numbers
Fractions Day 12: Renaming Mixed Numbers and Improper Fractions

Fractions Day 11: ‘One More’ Than a Whole

Jen Moffett, , Leave a comment

Note: The title is not mathematically accurate, but since my tiny kept using the phrase I figured it was an appropriate way to express the meaning behind the lesson.

So often when students learn about improper fractions and mixed numbers, it is smooshed together as one lesson. I disagree. I think we should focus first on a little bit (unit-fraction) more than the whole to really understand how the whole number relates to our denominator. So though this lesson may seem unnecessary, I think it is a critical first move for children who are learning the relationship between fractions and whole numbers.

Thinking about 3-halves.
  1. Played the first version of “Cover it Up!” See Fractions Day 2 for directions on how to play the game. (We play 3 times.)
  2. Show me 4-thirds. (Chris laid them out side-by-side). How would we write that number? (He wrote ‘4/3’ on his whiteboard.) Is this more or less than a whole? How do you know? (It is more. He showed the whole strip, blue for us, and compared.) How much more? (1 more. ) One more of what? (1 more third.) So how can I write 4/3 in a way that tells me it is a whole and a third more? (He wrote ‘1+1/3’.)
  3. I showed him the very fancy worksheet. (Pencil and paper my friends are amazing!) The first one is 4/3. He wrote in ‘1 + 1/3’. I was going to start the same way with the next improper fraction, 9/8, but he said, “I can do it without the fractions, mom!” And he did! What amazed me was that he didn’t notice the pattern that they were ALLLLL a whole and 1/denominator more. He always said the whole number as a fraction (Example, 1 is the same as 8/8.) and added the unit fraction to it. (So 8/8 plus one more eighth is 9/8.)
  4. I flipped our fancy worksheet over and modeled for Chris the 1 whole strip and 1/3 more, laying them out as one long strip. How could I name this? (1 and a third more). If I want to name the length just in thirds, how many thirds would I need? (4-thirds.)Though I showed him the first two (Please see video below for 1 + 1/6.), he did the rest of them without needing the physical fraction strips.
Thinking about 1 1/6.

As I noted in Fractions Day 10, spending so much time at the beginning with the physical fraction strips and really exploring how the different size pieces relate was extremely important for this lesson. He ‘sees’ the pieces in his head and can mentally find relationships without needing procedures or sets of rules he may (or most likely may not) make sense of. This understanding will continue, as you will see tomorrow! Be well!

For the sequence of the fractions (both sides) on my fancy worksheet:

Renaming improper fractions as 1 + a unit fraction more.
Renaming 1 + unit fraction as improper fractions.
Fractions Day 11: ‘One More’ Than a Whole

Fractions Day 10: Show Me the Fraction!

Jen Moffett, , Leave a comment

Happy Thursday! I had thought about moving on to comparing, but a friend (You TOTALLY rock, Jen H.!) suggested thinking about fractions over a whole. Now that I have another week out of school, we reviewed with all of the denominators and moved slowly into mixed fractions.

A couple of things before we start…

  • Chris is in second grade, but even if he knew his multiplication facts I would never start there with fractions. Notice everything we have done has been built from the manipulatives (fraction kit) and not from rules or procedures. This is done on purpose, as you will see the power of it on Day 12.
  • We go slow to grow. You may think, “Could she GO any slower on these? Let’s move along now!” Remember, you get it. They do not. Let them process, let them get comfy, let them be successful. The slow to grow will pay off time and time again, versus speeding up and having to reteach time and time again. As the guide on this fraction journey, you choose your path, so choose wisely.
  • Though I have had Chris convince me with the fraction strips for most of these lessons, you will notice that I will slowly stop requiring it if he shows he is understanding. If I am unsure or not convinced, we can always go back to the physical models.
Fractions I read aloud for Chris to model and name symbolically.
  1. Played, “Cover it Up!” three times with the old dice (halves, fourths, eighths, and sixteenths).
  2. I read fractions aloud one by one and Chris modeled them with his fraction kit. He then wrote next to each what the symbolic fraction looked like. (This was a GOOD thing to do, as there were a few he wasn’t sure which was the numerator and which was the denominator.) Above are the fractions I read. Notice that each numerator is one of the pieces from our fraction kit, to ensure that he understood the numerator tells me the number of the size pieces. The denominator is the actual size of the pieces (How many of them make a whole). The last fraction was to get into our next part.
  3. I wrote 3/2 on the board and he modeled it with the fraction pieces. Is this longer or shorter than a whole? (Longer) How do you know? (Demonstrated with the blue 1 whole strip.) How much longer is it than a whole? (1/2 more). We did this several times using similar questioning until he could answer without using the fraction strips. Below are the fractions I wrote for him to model and discuss.
Fractions I wrote down for Chris to represent and rename as a whole and a unit fraction.

Notice the fractions I wrote were only a unit fraction more than the whole. This was intentional, as I really wanted him to understand that 1 = a/a. For example, 5/4 is 4/4 and 1/4 more. This will be very important for the next two lessons! Also, the fractions were done in the order we created them. I knew he had more familiarity with the halves, fourths, etc. Therefore I started at a place of comfort before moving to the new sets and to fractions we could not represent with our kit.

Thinking about 5/4 as 4/4 and another 1/4.

Fractions Day 10: Show Me the Fraction!

Fractions Day 6: The Return of Equivalent Fractions

Jen Moffett, Leave a comment

Good morning! We needed more time with equivalent fractions and doubling/halving ideas before moving on to the family of thirds. Note: In two lessons, we will be making 3 more sets of colored fraction sets for our Fraction Kit. You will need 3 strips of equal size to the others in different colors.

There is no shame is repeating lessons. We do not master something in 15-20 min, and certainly not a new idea/concept. So this is going to sound a lot like Lesson 5. Trust me, it is helpful for later on!!! (And you really don’t have to prep anything new! BONUS!!!)

Trying to “Trick” his Mother!!! Commutative Property in Action!
  1. Play “Cover it Up!” twice. Please see Fractions Day 2 for how to play the game. We again focused on equivalence. How much more would I need to have the same amount as you? How much more would you need to be the same as me? How much more would we need to win? As always, I have Chris write his addition sentence once he covers the whole.
  2. We revisited the half and made equivalent fractions using only same-size pieces. See Fractions Day 4 for the lesson (the picture is above).
  3. Keeping with the half strip, I asked him to show 2 different ways to make a half without using ALLLLL of the same size pieces. I had to adapt and ask for 3 ways (As you will see why in the video above!). This was important to do, as it allowed him to really think about which pieces were equivalent and how to fill in the space to make half.
  4. My ink was out (and waiting for my Office Depot shipment!!!) so I hand wrote the fractions I wanted him to explore. See the picture below for the fractions and sequence we explored. This went much faster than I anticipated! I was especially excited to hear him stating things like, Well 2-eights is the same as 1-fourth or, If 2-sixteenths is an eighth then count by 2’s to get to 3-eighths. 2, 4, 6. So 6-sixtenths is the same as 3-eighths. I love making him tell me his thinking. It helps him learn how to explain and it helps me know what to do the next day!
Equivalent Fractions Work with Fraction Kit

Coming Next: Fractions and Cooking! We used the Visual Measuring Cups (I got mine on Amazon), but you can use whatever you have! We only used the 1-fourth measuring cup.

Fractions Day 6: The Return of Equivalent Fractions

Fractions Day 5: Which are Equivalent?

Jen Moffett, , , , 2 Comments

Equivalence is everything when it comes to fractions. Kids who understand which fractions are the same size and how to create fractions that are the same size are the ones in math class that say, “It’s easy!”. It’s not easy; it makes sense to them. Since I am at home with my tiny human for A-WHILE, I decided we would spend a good grip of time on equivalence. The next 3 lessons (and many more after introducing the family of thirds in Lesson 8) will focus on understanding equivalence with fractions.

Finding Equivalent Fractions to 1/2.
  1. Play “Cover it Up!” twice. Please see Fractions Day 2 for how to play the game. My questions to Chris focused on when our amounts were the same, or how to make them the same. How much more would I need to tie you? Which fraction would you need to have the same amount as me? I still had him write the addition sentence once he covered up the whole.
  2. I asked him to pull out one of his half pieces. How many eights would I need to cover up the half? (2). I then directed him to do the same with each fraction size. He had to cover up with only that size piece. We counted them out (One-eighth, two-eighths, three-eighths, 4-eighths…4/8 is the same as 1/2.) and wrote down each fraction amount that was equivalent to half. See image above for our work.
  3. On the whiteboard, I wrote 3/4. How many eighths would it take to cover 3/4? How many eighths are the same as 3/4?
  4. We continued looking at different fractions and finding the equivalent amount. The fraction sequence is below (See Equivalent Fractions Practice doc.), but looking at it now I would suggest less sixteenths and more fourths and eighths. Will need to do that tomorrow.
  5. I let him create two of his own. This was not easy. He just kind of stared at me. (Soooo tough not to just tell him, but the struggle is good for him!) I rephrased: Choose a fraction and find another that is the same size. Though this limited him at first, it helped get the ball rolling. He chose 1=2/2 for his first one. Though not what I was looking for, it is true! He second was more interesting (Wish I had asked for 3!) He chose 2/1=4/2. I was shocked, because we haven’t even talked about anything over a whole. However, when concepts make sense to kids, they can naturally apply them to unique situations.
Finding an Equivalent Fraction to 3/8.

Hope you have a great day exploring new ideas and making fractions fun!

Equivalent Fractions PracticeDownload
Fractions Day 5: Which are Equivalent?