# Fractions Day 3: Which is Bigger?

Science and a Birthday gift of slime/putty jars kept us busy and at a very quick math lesson today. ‘Cuz you know…priorities!

1. 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 had more? or Who is winning? How do you know? to front load for today’s lesson.
2. Using a whiteboard (or scratch paper), I drew the equal symbol (=). What does this mean? (Chris said it meant they were the same size, which I was fine with.) Show me which pieces would be equal. He showed the 1 whole and 2/2. I then wrote 1=2/2.
3. Under that work, I drew the greater than symbol (>). What does this symbol mean? (Chris said it was an alligator. More on that another blog. I am not in the mood for that one!) We discussed that it meant the first number you write must be bigger than the second number. The math vocab wasn’t very sophisticated, as I just want him understanding the idea. I had him choose two fraction pieces from the Fraction Kit to compare, with the first being bigger than the second. We then wrote the number sentence that it represented.
4. I had Chris compare pairs of the unit fractions from the Fraction Kit and tell me which one was greater and why.  See the unit fractions to the right for sequence of the pairs we explored.
5. Once he had the idea of comparing we moved on to 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 greater, circling it on the whiteboard. I asked him to convince me why one fraction was larger than the other, and he verbally explained or showed me with his fraction pieces. See below for the sequence of pairs we explored (The circled ones are the greater fractions.).I often asked, How many more of ____ would you need to make them equal?, just to start the seed of equivalent fractions (Day 5). Then I threw a pair of equivalent fractions in our set to see what he would do. (He rolled his eyes and said they were equal. Duh, Mom!)

Overall, Chris did well with circling which fraction was bigger, so long as he could use the pieces to work through the pairs. This is appropriate, as he hasn’t learned any other strategies for comparing. We will move to which fraction is smaller tomorrow to continue comparing sizes of fractions and really understanding what the numerator and denominator mean with respect to the Fraction Kit pieces.

# Fractions Day 2: Cover It Up!

Good morning! So I thought I would get this to you prior to Monday in case you have to search for materials to play the game. I love this game! It is easy to play, yet emphasizes so many important fraction ideas that might go missing in a regular math book. We played 3 times and called it quits, because I knew we would continue to play it every day for the entire week and I didn’t want him to tire of it.

## Cover It Up!

Materials: The Fraction Kit (See Fractions Day 1 for how to make the Fraction Kit), one kit per person, sharpie, and ideally a blank wooden cube. (See below for alternatives for a cube.) We also used a whiteboard and dry erase pen, but those are totally optional.

How To Play:

1. Using the sharpie, label the 6 faces (one fraction on each face) of the cube as follows. See below for other options if you do not have a blank cube.
2. Place the 1 whole fraction strip in front of each player. This is your “game board”.
3. Player 1 rolls the die, and puts that fraction piece on his/her 1 whole to the far left.
4. 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.)
5. 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).
6. 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):
1. Who has more? How much more? (They can use their pieces to figure it out. No actual arithmetic!!!)
2. Who has less? How much less?
3. Do you need more or less than 1/2 to win the game? How do you know?
4. How much more do you need to win (get to 1 whole)?
7. Once a player has won, have him/her write the number sentence for his/her board. (We totally cheat and I let Chris roll as many times until his board was filled as well.)
8. Repeat the game 2 more times. Best out of 3 is the winner.

Alternatives to a Blank Number Cube: If you don’t have a blank wooden cube, below are some options so that you can still play the game. I have done all of these and they are all great options.

• Make your own die: See below for blank template and write the fractions we used on #1. Note: This is better printed on card stock or heavy paper.
• Make a spinner. See the PDF below and use Spinner #3. Label the sections as we did the cube above in #1. Using a paper clip and a pencil, place the pencil in the paper clip in the center of the spinner and spin the paper clip. Where it lands is your fraction. This can be on plain paper and it works great.
• Roll a regular die. See below for the fraction you get for each number on the die.

For link to die template: https://www.printableboardgames.net/preview/Blank_Die

# Number Choices Matter

I had the privilege to work with fourth and fifth grade teachers this week. We explored multiplicative comparison problems in fourth and division with fifth (more on those in another blog). What I came away with is this: NUMBER CHOICES MATTER.

We don’t get much choice on the concepts we teach, nor often on the program we have to use. But we do get to choose what numbers we use with children. This could make all the difference for kiddos. If we choose our numbers wisely, we can build understanding through the patterns they see, the differences that appear, and talk about why those differences happened.

For example, consider the following set of problems:What is the same about each? If I am thinking about this as a partitive model and using base 10 blocks, I am sorting the amount I am given into 3 equal groups each time.

What is different about each? The amount I am giving to each of the 3 groups.

In the first example, 36 can be created by using 3-tens and 6-ones, and each can be fairly shared without any problems. Each would get a ten and two ones, or 12.

In the second example, I can still give out a ten to each of the 3 groups, but now I have to figure out what to do with the leftover ten and eight ones. This one builds off the first example, but pushes students to think about exchanging (regrouping).

The final example builds off the 48, but leaves 2 left that students have to consider. This allows to have a conversation about remainders.

These three build understanding of division, regrouping and remainders through strategically chosen problems to build from one to the next. Students have something to grasp on to when negotiating meaning with this tough tough subject.

So where can you build understanding through your number choices? I challenge you to think about what you want your students to learn next week and how your number choices can contribute to students understanding those goals!!!!

# Why Distribute in Third Grade?

I am blessed to work with dedicated teachers who care deeply for their students and are working hard to understand the conceptual shifts CCSS brings to the table in math. One such teacher emailed me this weekend distraught, not knowing how to respond to a frustrated parent. The premise was the mother did not understand why her third grader was being asked to learn the distributive property, when she herself hadn’t learned it until Algebra I.

The arithmetic properties (commutative, associative, identities, etc.) were not created for Algebra I, though many of us didn’t learn them until then. I remember thinking that a bunch of old guys must have made them up for the sheer joy of torturing me into memorizing random stuff. No, the properties are the rules that give us the freedom to simplify math problems to make them easier to calculate while keeping the value the same.

An example. Which would be easier to solve in your head, 15 + 29 then add 5  OR  15 + 5 then add 29? The second, because we can make tens (15+5 = 20) and easily add 29 to it rather than have to “carry the 1” on the first example. This illustrates the commutative property: when adding or multiplying, I can perform that operation with any numbers in the problem first. I can switch the numbers around to make the problem easier to add (or multiply).

These properties should be celebrated as early as Kindergarten. Students do not necessarily need to know the names, but should realize through exploration that they exist and help them find their values.

Back to the distributive property. We old-timers saw it used like this: 6(5x + 2) = 30x + 12. This is not what we are asking third graders to do! Since the CCSSM standards require single digit multiplication fluency in third grade (1×1 through 9×9), it is natural to teach the distributive property at this level. This property allows me to break up one of my bigger numbers into parts. I can then multiply those smaller parts by the other factor to make it easier.

Example: 6 x 7   This is always a toughie. Is it 48? 42? 56? I don’t know! Even if I draw an array (the above pic), that is a whole lot of dots to count!

But if I know my 5’s and 2’s, I can figure it out using the distributive property!             Break up the 7 into 5 and 2.   So now 6 x 7 becomes 6 x (5 + 2).

I know 6 x 5 = 30.  (This is illustrated in yellow.)

I know 6 x 2 = 12 (This is illustrated in red.)

So 6 x (5 + 2) = 30 + 12 = 42.

You may also see the distributive property used as a “number bond”. This is when it is broken up into parts, either using boxes or circles to show the parts. In this case, 7 is broken up into two parts:  5 and 2.

Utilizing the distributive property is an amazing strategy for students who struggle with memorization! They can use the distributive property to break up larger values they don’t know the multiplication facts for, until they have time to build fluency. More important, it teaches kids the value of knowing that math isn’t just memorization. There are structures and patterns that I can use. If I know the rules that govern those patterns (the properties), I can change the structure to find the value in an easier way.