Mixed Numbers and Improper Fractions

We introduced fractions greater than one by modeling them with pattern blocks and writing them as a mixed number, and also as an improper fraction. Then we discussed the similarity between the two numbers.

After lots of examples with pattern blocks, we moved to pictures in our journals. We started with the mixed number, drew the picture, then named it as an improper fraction. After several of those, we started with the improper fraction, drew the picture, then named it as a mixed number.

The next day, we talked about how we went between the two types of fractions using pictures to help us. After examining our pictures, we noticed some shortcuts to help us be more efficient when changing from one type of fraction to the other. Eventually, the kids discovered the multiplication and division connection to help speed them along.

Fractions Greater Than One Anchor Chart

Equivalent Fractions

Our 1/2 fraction flipbook

We begin talking about equivalent fractions after we've had a lengthy discussion regarding the relationship between 1/2, 1/4, 1/8, 1/16. etc. as well as the relationship between 1/3, 1/6, 1/12, etc. The kids have a really good grasp on this idea.

We begin by making our fraction flipbooks (which takes a bit!) and then lifting up a fraction to see its equivalent fractions. For example, when we lift up 1/2, we can see it is equivalent to 2/4, 4/8, and 8/16. After we write down lots of equivalent fractions we find, we discuss their relationship. Finally, we discuss the multiplication and division involved. This is a great visual way for kids to come up with the relationships on their own!

Flipbook showing fractions equivalent to 1/2

Our 1/3 Fraction Flipbook

Fractions equivalent to 2/8

Fractions equivalent to 1/3

Our Equivalent Fraction Anchor Chart

Our journal entry

Forms Of Energy Flip Book

 As we discussed different forms of energy, (we focused on Mechanical, Sound, Electrical, Light/Solar, and Heat/Thermal) we decided to make a flip book to hold all of our information. Inside we put the definition of the form of energy, and added picture examples as well. Here are two examples of our flip books - one picture is the outside view and the other picture is inside a flap.

Intro to Fractions

Fraction Basics

 

In math we have started to discuss fractions. By 4th grade, the kiddos have had a ton of practice naming fractions of a shape, as well as naming fractions of a set of objects. We continued that practice and discussed the definitions of the numerator and denominator. We've also extended our knowledge by figuring out how to find fractions of a number, such as 12 and 24.

We began by pulling out 12 squares and physically dividing them up into equal groups. It didn't take the kids long to figure out that to find fractions of a number, all they had to do was divide the number by the denominator of the fraction they were looking to find. For example, if they want to find 1/4 of 24, they could just divide 24 into 4 equal groups. Since 6 would be in each group, 1/4 of 24 is 6. Finally, we discussed how, if we found 1/4, we could also find 2/4, 3/4, etc.

Finding fractions of 12 with blocks

Finding fractions of 24

Finding fractions of 24

Division without Dividing???

Today we began division. I always suck my kids in by telling them I'm going to teach them how to divide without division. They never believe me, but it doesn't take long for them to figure out where I'm going. We use multiplication to divide! After all, multiplication is way more fun than division, right? ;)

We begin with numbers that have no remainder, such as  256 divided by 2. We start by taking out small groups of 2, perhaps 10 groups of two. 10 groups of 2 would be 20, so we subtract 20. Then we can take another 10 groups out. After we do this a few times, the kids make the connection they can take bigger groups out and save themselves a few steps. Instead of taking 10 groups of 2 out 5 times, they can just take 100 groups of 2 out from the get-go. Then we continue to take out groups until we can no longer take any more. I have two examples here, and I've tried to color-code them so they make sense.

Multiplication Matrix Box

Our class anchor chart

Close-up of the box itself

An example

We finally began 2-digit by 2-digit multiplication this week! The kids are absolutely loving the matrix box we use to introduce 2x2 multiplication!! They love it because it's super easy. I love is because it shows them the place value involved in 2x2 multiplication. They can actually see the process of multiplying the ones and tens of one number by the ones and tens of the second number. When they finally move to vertical multiplication, they'll actually understand why they put zeros here and there.Yea! There's nothing better than actually understanding WHY you do something!!!

The matrix box can actually be formatted to fit more than 2x2 multiplication. We'll be working on making our matrix boxes the size we need next week. More to come!!

Mixtures and Solutions

We began to investigate the difference between a mixture and a solution by making hot cocoa. First we examined the properties of the cocoa power - solid, brown, small granules, powdery - and the properties of the water - clear, liquid, tasteless. Then we combined them and talked about what happened. The two ingredients combined chemically to form a new substance. This is called a solution. Solutions cannot be separated physically, but require something more complex, such as evaporation. We'll get into separating solutions next week!

Then we listed out the properties of the popcorn and water. The kids filled a graduated cylinder almost to the top with water, then added as much popcorn as they could to raise the water level to the top. They were amazed by how much popcorn they could add to the water before the water level started to rise! That just added a little bit of fun to the investigation. When they finally achieved this, we discussed what they made was a mixture. The water's properties had remained the same, and they hadn't formed a new substance. Mixtures can be separated by physical means, such as filtration. We'll be looking at lots of physical ways to separate mixtures next week as well.

Popcorn Lab

Frozen "solid" of shortening and kernels

 The purpose of the popcorn lab was for the kids to have a visual example of what atoms look/behave like in all three states of matter. We began with a frozen "solid" of shortening with kernels (our atoms) packed in it.  The atoms are tightly packed in, and aren't moving. Then I challenged the kids to, without opening the bag, make their solid a liquid by adding thermal energy. They quickly figured out rubbing the solid was the fastest way to do this. After the solid became a liquid, we observed the atoms and noticed they were looser, not all touching, and moved around in our bag. Finally, I took a solid out of its bag and placed it in a beaker on a hot plate. We quickly got to see the solid changing states. As more and more thermal energy (heat) was added, the atoms finally became so energized they couldn't be contained in the beaker any longer and tried to escape. We kept this from happening with the foil.

After we finished our investigation, we discussed why these things happened to our atoms. The thermal energy gives the atoms so much energy, the result is the atoms becoming more and more active. This causes the change in states. We also discussed what this would look like if we removed thermal energy and went in reverse.

The solid beginning to melt

Liquid form

Student journal sheet page 1

Our "gas"

Student journal sheet page 2

Student journal sheet page 3

Balloon in a Bottle

Balloon without a hole

Balloon with a hole

Today we worked on properties of matter. After making predictions about what we thought would happen, we attempted to blow up a balloon in a 2-liter bottle. The balloon wouldn't blow up. So we removed the balloon to try to blow it up, and it worked just fine. Clearly, it wasn't a balloon issue!

We tried again, as unsuccessfully as before. Finally, we punched a hole in the bottle and tried again. This time it worked! We discussed why it worked this time, coming up with the idea that all matter, even gas, takes up space. When we blew up the balloon, we could feel the air coming out of the hole. The balloon was pushing the air out because matter cannot occupy the same space.

You should have seen all the red faces trying to blow up the balloons! It was priceless!

Breaking Apart Numbers in Multiplication

A student's list of combinations she's still working on

Breaking apart a number using arrays as a visual

Breaking apart a number using arrays as a visual

What we did this week was go through our basic facts and determine which ones we know quickly, and the ones with which we still need some practice. We made a list of those we're still working on, and came up with ways to "get" to that answer. For example, if 12 x 9 is still a struggle, we could think of it as (12 x 6) + (12 x 3). We made the arrays to go with the problem, which made visualizing the strategy a little easier.

If a student can visualize breaking apart a number for basic multiplication, it's going to make his life much easier when we get to 2-digit by 2-digit multiplication. It's so important for the kids to understand the place value involved in larger multiplication, and learn to make their numbers work for them. Flexibility with numbers is a HUGE concept to grasp, and it has arguably the greatest impact on the success of a mathematician!!!!!!

Divisibility Rules

Just a few divisibility rules

This week we listed out multiples of many numbers, and used these lists to help us come up with some divisibility rules. We thought knowing these rules might help us when determining factors of a number. We spread this process out over a couple of days, as it can be a bit overwhelming all at once! We made some great connections!!!

Making Arrays

Our introduction into finding all possible arrays

An example of our T-Chart to help keep us organized

As we continued our discussion on arrays, we tried to come up with a way to determine if we'd found all the possible arrays for a number. First we talked about what those arrays are actually modeling (factors). Then we came up with our T-Chart to list out the factor pairs in an organized way. We can determine we're done when we begin repeating numbers. All this discussion is leading up to 2x2 multiplication.

More Density

View of our jars from the top

A fuzzy side view of our jars.

As we continued our exploration of density, we began another investigation using three jars of unknown liquids. The kids didn't know the contents of the jars until after the investigation was completed. Jar #1 contained 60 mL of dish soap (blue liquid). Jar #2 contained 60 mL of baby oil. Jar #3 was nothing but air, which is why I covered it in foil. I didn't want to spoil the surprise. Some kids knew right away there was nothing in it, but some were convinced I had put something in there! The groups all measured the mass of each jar using the triple beam balance. Then they added the jars to the tub of water to observe which was the most dense/least dense. All groups came to the conclusion the blue liquid had the most mass and was the and most dense, while the foil-covered jar had the least amount of mass and was the least dense.

We then discussed why it was so important both the liquids have the same volume. If the volumes are different, we wouldn't know if the blue liquid was, in fact, more dense, or if there was just more liquid in the jar.

Arrays

Arrays

Yesterday we began discussing arrays. This will eventually lead us into bigger multiplication. We just discussed what an array looks like (equal rows and columns), the dimensions, and how we find the number of objects in the array without counting each one individually.

This is never a hard concept for the kids to grasp. Usually the only challenge they have is forgetting to count "Mr. Popular." This is what I call that little guy in corner. We have to remember he's super popular, and both the rows and the columns want him in their groups. So he must be counted with the rows, and with the columns! 

Our Arrays Anchor Chart

Adding and Subtracting Money

Adding and Subtracting Money

This is our journal entry for adding and subtracting money. It was basically just a run-though of all our addition and subtraction strategies, but I threw in money instead of just whole numbers. Lots of discussion along the way!

Subtraction Strategies

Student Journal - Subtraction Strategies

As a class, we discussed different subtraction strategies. The strategy most of our kids are familiar with (at least if they've been in the district a while) is the number line. If the problem were 634-319, they would start their number line at 319, and count on by taking "jumps" until they reached 634. Then they add up their jumps, and there's the difference. It's like subtraction without subtracting!! For those kiddos who struggle with subtraction, this strategy is perfect until they're more comfortable!

The second strategy we discussed is subtracting by place value. With subtraction (unlike adding by place value) it's very important to only break up one number by place value. Otherwise, you're looking at major confusion for most kids. If we were to solve 634-319 again, we would start with 634-300 = 334, then 334-10=324, then finally 324-9=315.

Of course one of our strategies is vertical subtraction, which is my absolute LEAST favorite way for kids to subtract - at least until they have a major grasp on place value! For me to allow a child to subtract vertically, he/she must explain to me step-by-step how to do it, all the while talking in terms of place value, not numbers. For example, 634-319 would be explained something like this:

"I cannot take 9 from 4, so I borrow a group of 10 from the 30 so it now becomes 20. The 4 now becomes a 14. 14 minus 9 is 5. 20 minus 10 is 10, but I only write the 1 because I'm working in the tens place. 600 minus 300 is 300, so my answer is 315."

I know that may not sound too difficult, but when the numbers get bigger and there's more borrowing involved, it's gets pretty confusing. If they can explain the process to me in a way that shows me they know what they're doing and why they're doing it, they're free to do vertical subtraction all day long!

Our last strategy is shifting up/shifting down. I presented this strategy as a super-secret strategy that only they know. We even closed the blinds and door. Ha ha!! This strategy is for those kiddos who need the organization of the vertical strategy, but are still struggling with borrowing. We started by marking 8 and 16 on our number line and subtracting for a difference of 8. Then I added 1 to each number, making 9 and 17. Looking at our number line, we discovered it still had a difference of 8. Then I subtracted 3 from each number, making 5 and 13. Again, still a difference of 8. This led us into the discussion that if you change both numbers in the same way, you're not changing the difference between them. So we concluded that sometimes shifting the numbers up or down might help us in subtraction. For example, if I had the problem 2,002-412, I could shift both my numbers down 3, leaving me with 1,999-409. This is a much friendlier number to subtract! We did lots of examples using this strategy! Then we spent a little while coming up with reasons why this strategy wouldn't work the same for addition!! :)

Our Subtraction Strategies Anchor Chart

Another Subtraction Strategy