No pun intended. We are at the end of our exploration of initial fraction ideas. Even though I am a mathematics educator, and have been for over 20 years, this experience has taught me a few things.
NEVER underestimate a child. Though I knew what answers or ideas I wanted, they were not always what I got. And often, his answers and ideas were simpler and quite frankly, better.
Time, Time, TIME! Give lots of time to build these critical foundational concepts. Just when I thought, “He’s got this!”, the next day I would question and he would hesitate. Children need lots of time to practice, to play, to explore, and to find how to say and think about the ideas themselves.
Wait Time is not only necessary; it is CRITICAL. There were so many videos I cringe when hearing it again. Those times when I should have just sat quiet, and waited for him to work it out. The wait time isn’t awkward for him; it’s awkward for me. When I did just take a swig of coffee (instead of butting into his thinking) the thoughts he figured out were amazing.
“Mommy, this was FUN!” We often have a running joke in middle and high school where we say fractions are our friends (and not the other f-word). So many students are traumatized by the lack of understanding they have with fractions. Yet here is my 8 yo asking to play games, to cook, and to learn about fractions. Because it made sense to him. Because he got time to process and practice. Because he understood the relationships and how to utilize those relationships to make meaning.
We are on to measurement and I hope to revisit equivalence, addition and subtraction of fractions in May. I hope you were able to enjoy this journey with your little(s) as much as we did. 🙂
Take care of each other, stay home, and stay healthy.
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!
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.
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.
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.
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.
Played “Cover it Up”, first version (Please see Fractions Day 2 for directions) three times. He still wants to play, so we still play!
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.
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!
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.)
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.
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.
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.
Played the first version of “Cover it Up!” See Fractions Day 2 for directions on how to play the game. (We play 3 times.)
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’.)
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.)
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.
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:
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.
Played, “Cover it Up!” three times with the old dice (halves, fourths, eighths, and sixteenths).
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.
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.
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.
The one thing I love about trying this out with my little is that mistakes help us either laugh or learn (or both). Here is a laughable moment that I would totally do differently next time.
Materials: The Fraction Kit (See Fractions Day 1 and Day 8 for how to make the Fraction Kit), one kit per person, sharpie, and ideally a blank wooden cube. (See Fractions Day 2 alternatives for a cube.) We also used a whiteboard and dry erase pen, but those are totally optional.
How To Play:
Using the sharpie, label the 6 faces (one fraction on each face) of the cube as follows: 1/2, 1/3, 1/4, 1/6, 1/8 and 1/16. Reflecting back, I would have swapped the 1/16 out for 1/12. May have to try this next week…
Place the 1 whole fraction strip in front of each player. This is your “game board”.
Player 1 rolls the die, and puts that fraction piece on his/her 1 whole to the far left.
Player 2 rolls the die, and puts that fraction piece on his/her 1 whole to the far left. Who has covered up more of their 1 whole? (In our game, Chris had.) How do you know? (Chris originally said, “Because purple is bigger than pink.” I restated, “Oh, so you mean 1/4 is bigger than 1/16?” This helps them start visualizing the size of pieces and prepare for comparing.)
Player 1 rolls the die again and puts that fraction piece right next to the first one so they are touching, but there are no gaps or overlaps (as best as they can). Player 2 does the same on his/her board. Who has covered up more? Who has covered up less? If the two rolls were the same (e.g. I rolled two of the 1/16) How many sixteenths do I have? (2/16).
Play continues until a player covers exactly 1 whole. If a player rolls a fraction that is too big to fit, he/she loses that turn. Some questions to ask (as appropriate):
Who has more? How much more? (They can use their pieces to figure it out. No actual arithmetic!!!)
Who has less? How much less?
Do you need more or less than 1/2 to win the game? How do you know?
How much more do you need to win (get to 1 whole)?
Once a player has won, have him/her write the number sentence for his/her board.
Repeat the game 2 more times. Best out of 3 is the winner.
So the first two games went fine. Then the third game I was left with this:
Needless to say I lost, since I did not have the opportunity to roll 1/24! Chris thought this was great, since he then won 2 out of the 3 games. Next time, I am going to allow the player to remove a piece of his/her choice in exchange for a roll. THEN they may roll the turn after to continue playing the game. This sets us up for the Uncover It! game coming up in a few lessons…
Hope these lessons are going well for you! We love our fraction time and look forward to it daily!
Now that Chris is comfy with the denominators 2, 4, 8, and 16 (and halving/doubling to make equivalent pieces) we tried denominators of 3, 6 and 12. Why 12? I just figured it was easier to work with 2, 4, 3, and 6 once we move to equivalent fractions. Also, it was easier to fold (no joke).
Note: I would suggest you fold for your kiddo. Or have a set already folded for them in case they need it. I did not, and then had to spend time cutting more pieces to make new folds (He laughed…I did not.)
Show the 1 whole again. Explain we are making new fraction pieces for our kit (YAY!).
Let your child choose a color (Chris chose green). Fold into thirds. I cheated and measured it out, then divided by 3 and made marks with a ruler. He then folded on the marks. How many equal sized pieces do we have? (3) What do we call each piece? (1-third). Write 1/3 on each piece and cut them out. Count them out as you cover 1-whole strip: 1-third, 2-thirds, 3-thirds or 1 whole. You can write 3/3 on the whole strip if you would like.
Have your child choose a different color (Chris chose orange.) For the sixths, fold again into thirds, then in half. How many equal-sized pieces do you think we have? (5???) Open it up and see. (Oh-6! Well yeah, because 3 doubled is 6.) What do we name each of these pieces? (1/6) Write 1/6 on each piece and cut them out. Again, name the amount as you cover up the 1 whole strip with 1/6 pieces. 1-sixth, 2-sixths, 3-sixths, …, 6-sixths or 1 whole.
Take one final color and fold into thirds, fold in half once, then fold in half a second time. How many equal-sized pieces do you think we have? (I don’t know…ummmm…10?) He opened it up and saw 12 pieces. What is the name of each piece? 1/12. Write 1/12 on each, cut them out, and count them as you cover the 1 whole.
I asked Chris to show me different values.
Show me 3/12.
Show me 5/6.
Show me 2/3. (I soo wanted to start comparing, but that is not the focus for today. Keep it focused so they get what you want out of the lesson.)
Show me 7/6. (HA!) He just grabbed one of my pieces to make 7/6.
Show me 5/3. Same. Grabbed my pieces to figure it out.
If you can’t grab my pieces, how could you show 5/3 with your pieces? I wish I had taped it. I thought he would show 1 whole and 2/3. But no. He used the 3/3 then 4/6! I love when we let them do their own thing. They always seem to surprise us!
Today was our first day of Distance Learning from our school. Though I am grateful for all of the lessons now being planned out and having some awesome challenges to engage in (Shout Out to Coach SMITH and Mrs. NEAL!!!!), it messed up the routine we had been rolling with. I spent so much time navigating the different sites and trouble shooting that I didn’t have time to prep the new colored strips I needed to start thirds, sixths and ninths for the Fraction Kit.
I honestly thought Chris wouldn’t ask about the math, as he had so much to do for his other classes. (I was ready to ditch out on the fractions today.) But he asked at lunch, When are we having our fraction time, Mom? How can you say no to that?!!! (Well, okay lots of us can, but anyone who knows me knows I can’t. Ask my good friend who had to wait for me to write on receipt paper math tutorial links for the cashier at Total Wine the other day!)
Fortunately, a good friend had posted on Facebook a recipe for making salt-dough ornaments. See below for the recipe, but what I loved about it was there are only 3 ingredients (And I had them all!!!!) and the amounts were cups and halves. Perfect!
I pulled out all of my measuring cups and asked Chris how he would name them. He called the full circle 1 whole. He took the others and rotated them within the cup to figure out how many fit inside. Pretty similar to the Fraction Kit, but with circles. We named them verbally and I asked him to find the fourth. We put the rest away.
I asked him to measure 1 cup of flour with the 1/4. How many fourths would you need to make a whole cup? 4How do you know?Because it takes four of them to make a fourth. (He rotated the 1/4 in the air to show the whole circle.) 1-fourth, 2-fourths, 3-fourths, one-whole.
We needed to measure 1/2 cup of salt. How can you use the 1-fourth to find 1-half? Well, since 4 of them made a whole, I need only 2.
We needed to measure 1/2 cup of water. I didn’t bother asking; I knew he got the gist of it.
The recipe is super easy to make and fun to create different ornaments to paint and share with friends during this isolation time. I am having Chris write a letter with each to send out next week so we can get some Pen-Pal action going!
Note: We did not put parchment paper down the first time we rolled out the dough and it was a huge mistake! Place parchment or wax paper down and sprinkle some flour before rolling out the dough.
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.
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.
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.
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?
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.
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.
Hope you have a great day exploring new ideas and making fractions fun!
I am a fan of repurposing (NEW WoRd!!!) problems. I figure, if you can read a book for different purposes, why not math problems? Plus, it makes planning that much easier for you! And right now we have enough on our plates!
Played “Cover it Up!” twice. Please see Fractions Day 2 post for how to play the game. The questions I used were focused on were: Who has less? or Who is losing? How do you know? to front load for today’s lesson. NOTE: I still make him write the addition sentence at the end of each game to work on notation and such.
Using a whiteboard (or scratch paper), I drew the equal symbol (=). What does this mean? Today Chris said they were equal, or the same. We moved on.
Under that work, I drew the greater than symbol (>). What does this symbol mean? (The first number is bigger or greater than the second number.) Again, I had him choose two fraction pieces from the Fraction Kit to compare, with the first being bigger than the second. He actually did not choose the same ones as yesterday. We then wrote the number sentence that it represented.
I drew the greater than symbol (<). What does this symbol mean? (The first number is smaller or less than the second number.) I had him choose two fraction pieces from the Fraction Kit to compare, with the first being smaller than the second. He actually did not choose the same ones as yesterday. My little Sassy Sam just reversed the ones he had. We wrote the inequality and moved on.
I had Chris compare pairs of the unit fractions from the Fraction Kit and tell me which one was smaller and why. They were the same ones I used yesterday. He didn’t notice!
We moved on to the fractions with different numerators (still only using the denominators from the Fraction Kit). I gave him (one at a time) pairs of fractions to compare. He could use the Fraction Kit pieces to determine which was smaller, circling it on the whiteboard. I asked him to convince me why one fraction was smaller than the other, and he verbally explained or showed me with his fraction pieces. See below for the sequence of pairs we explored (The red fractions are the smaller fractions.)I often asked, How many ________ would you need to make them equal?, just to start the seed of equivalent fractions (Day 5). I threw at him two unit fractions (1/2 and 1/7) to see if he could apply his understanding without always using the Fraction Kit pieces.