Symmetrical Names

If you look closely, you'll see Mrs.May on each section

Sarah

Taylor is written on each section

Connor

After our lesson on symmetry, we began making a symmetrical creation with our names. If you look closely, you'll see our names within the pictures. We folded a square sheet of paper into fourths, then folded that into a triangle. On each triangle, we wrote our names so they reflected the adjacent triangles. While they aren't perfect because they're drawn by hand, they turned out really cool! We had lots of discussions about the reflections as we went along...especially with the letter N. It's a confusing one! Some made their names go towards the middle, while others made theirs go towards the outside. Which ever direction they chose, they turned out great!

Comparing Decimals

Proving Zero the Hero (in orange)

As we think about decimals, we really try to make them look like money. After all, the kids have been working with money since about 1st grade, and will continue to use it their whole lives! We began our lesson on comparing decimals by simply coloring pictures of different decimals. When we got to forty hundredths and four tenths, the kids noticed they both cover the same amount of space. After we wrote the decimals in fraction form, the kids could really see why they are equivalent - because 4/10 and 40/100 are equivalent fractions. This lead into our discussion of Zero the Hero! When our decimals only go the the tenths place, we can add him in the hundredths place and it doesn't change the value. He can help us see the decimal as money, and he will also help us as we begin adding and subtracting decimals next week!

Adding Zero the Hero to make our decimal look like money

Decimal Introduction

Close-up of place value

As we introduced decimals, we talked about their relationship with fractions. Both are parts of a whole. We focused on our base 10 number system, and the pattern it follows. We always spend lots of time looking at picture examples, naming them in mathematical language, relating them to fractions, and labeling them. The biggest connection our kids make with decimals, is its relationship to money. We take lots of time to relate the tenths place to dimes, and the hundredths place to pennies.

Decimal Anchor Chart

Types of Triangles

Today we learned about different types of triangles - right, acute, and obtuse. We discussed that all triangles have two acute angles. The third angle is what determines its type. If the third angle is also acute, it's an acute triangle. If the third angle is right, it's a right triangle. If the third angle is obtuse, it's an obtuse triangle.

Naming and Measuring Angles

We began the week naming angles. Most of the students were familiar with the types of angles, but we reviewed the important characteristics of straight angles, obtuse angles, acute angles, and right angles.

Next we worked on measuring angles. Measurement is still a very abstract idea for 4th graders, so we began by discussing measurement as a whole. What do we use different measurement tools to measure? I posed the question, "Would a ruler work when trying to measure angles?" In the beginning of our discussion, many kids believed it would. After trying it out, they came to the conclusion that we needed something that measures in degrees. They understood the unit wasn't correct, but weren't sure what degrees actually are, though we had discussed degrees when classifying angles as right, acute, obtuse, and straight.

We went back to everything we already know about angles. Going in a complete circle means we have gone 360 degrees, while only going halfway around means we have gone 180 degrees. We already know that a right angle is 90 degrees. We already know a straight angle is 180 degrees. We know an acute angle measures between 1 degree and 89 degrees. We know an obtuse angle measures between 91 and 179 degrees. Once we illustrated that, we decided we could half each 90 degree piece and figure out where 45 degrees and 135 degrees are located. Next we moved to

Lines and Angles

We began lines and angles this week. We began with introductions of the different types of lines and angles, then moved to making them on dot paper. We moved on to looking at specific lines and angles within a large collection. This lead to the discussion of why it is important to have points on our lines - so we can name the ones we're specifically trying to reference. 

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.

Our Classroom Poster

Strip Diagrams to Model Computation

Most students at this point can read a word problem, figure out the operation(s) needed, and solve the problem(s). What might be a little more difficult for them is modeling their operation(s) with a strip diagram. This week we worked through all four operations, and discussed what the strip diagrams for each operation could look like. If a student truly understands what is happening with each operation, he/she should be able to model it.

Division and Subtraction Strip Diagrams

Multiplication and Addition Strip Diagrams

Fraction Review and Intro to Improper Fractions

This week we took a brief look at fractions. We started by reviewing what we already know about fractions such as definitions and pictures.

After we reviewed, we looked at pieces of fractions and discussed what happens when the numerator and denominator are the same number. We looked at pictures of pizzas cut in half. Once I have two halves, I have one whole. Finally, we explored what happens when we have more than one whole. We discussed how we would name these improper fractions (we'll learn about mixed numbers a little later this year.) We named them on a number line, as well as in picture form.

Moving to the Standard Multiplication Algorithm

The progression from matrix box to standard algorithm

This week we moved to the standard American algorithm for multiplication. Parents are always thrilled when we get to this, as this is the way they learned! We began by looking at our area model (in blue), and discuss what places were being multiplied during the process. The tens places of both numbers are multiplied together, then the tens of the first number is multiplied by the ones of the second number. Next the ones of the first number is multiplied by the tens of the second number. Finally, the ones of the first number is multiplied by the ones of the second number. Sounds confusing, right?! So we begin to discuss what this would look like vertically (in red). Then we discuss the traditional algorithm (in green), and make connections between all three strategies. This really helps your children understand the place value involved in the algorithm! Woohoo!

Multiplication Area Model

We finally got to begin 2-digit by 2-digit multiplication this week! I began by reviewing with them the area model we used when multiplying by 1 digit. Then the kids self-discovered (with my prompting and questioning) that they could create an area model for 2x2 multiplication. They referred to it as the "double-decker couch." Ha! The kids are absolutely loving the 2x2 area model!! 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 it works! There's nothing better than actually understanding WHY you do something!!!

The area model can actually be formatted to fit more than 2x2 multiplication. We played around with differently-sized numbers and made our matrix box fit those problems as well.

Our class anchor chart

Close-up of the box itself

An example

Breaking Apart Multiplication

As we move to multiplying larger numbers, I like to guide them into the algorithm. We start by investigating the pattern when multiplying by multiples of 10. We look at the problems 2 x 3 = 6, 2 x 30 =60, and 2 x 300 = 600. We discuss what is the same/different about each problem, then we look at several other similar examples. Eventually the kids see the pattern of just "adding a zero" each time. Patterns are a great shortcut, but it's important the kids understand why they work.
Next we move to breaking apart larger multiplication problems, such as 18 x 5. We can break the 18 into 10 and 8 to help us multiply easier.

We practice this with many numbers, also looking at the array model for each. We slowly get to larger numbers like this one...

This will lead us into the multiplication algorithm next week.

Finding the Factors of 12

We begin our lesson on finding the factors of a number by making all the arrays we can think of for the number 12. The kids usually do a really good job finding them all. When I question if we've found them all, however, the kids start to second-guess themselves. This moves us into the discussion of what we're actually finding when we're making these arrays. I lead them to discover they're finding the factors (number that are multiplied to make a number) of 12.

But how can we keep our factors organized so we can make sure we've found all the factors? This leads us to making our T-Chart. We always start with 1 and the number itself. Then we move to the next number in the number line, 2. We know 2 is a factor because 12 is even, so we then find the partner for 2 which is 6. Then we move to 3, which has a partner of 4. The next number in the number line is 4, which we've already used, so we know we're done. Once numbers start to repeat, we've found all our factors. This lesson is followed by lots of practice finding the factors of many different numbers. As we're making our factor lists, we discuss the definition of prime, composite, and square numbers. 

Elapsed Time

Elapsed time can be a challenge for kiddos because they like to try to find the difference between two times by lining them up vertically and subtracting for the difference. In a whole-group discussion, we talk about why this won't always work. Our number system is base-10, which means when we reach 10 in a place, we must move it over to the place to the left. For example, when we get 10 ones, we turn it into 1 ten. We also borrow in groups of ten. Time does not follow these rules, so we cannot always successfully borrow or carry vertically.

Introduction to Vertical Addition and Subtraction

When I introduce vertical addition and subtraction (done on different days because it's too much for one day!) I do it very concretely. We actually make the problem with base ten blocks and break apart our problem by place value. When we borrow a ten for the ones place, we actually take a rod out and replace it with ten units. Or when we carry a ten from the ones, we take ten units and turn them into a rod. It doesn't take long for the kids to understand why we're doing what we're doing. Soon we move to a written version with the place value broken apart. Finally, some students will move to the standard algorithm. They only make this move when they're successful with the broken-apart method. We continue practice with the vertical method until we've reached mastery. It's important to have these concepts mastered before we begin multiplication and addition! 

Tables and Measurement Conversions

This week we spent lots of time looking at sets of data in a table, and finding the relationship. It's often easy for students to find the pattern and fill in missing information. What can be difficult is trying to use number sentences to describe the pattern. Often our kids are shown a table, then given words that describe it to determine which descriptions are correct and which aren't. We spend time practicing actually plugging in the data to check the description.
This week your child was given a picture of a vehicle. They had to determine how many wheels the vehicle had, then create a table to prove how many of their vehicles were needed to reach 24 wheels. After we finished, we describe the table in many ways, then determined which of our descriptions were correct and which weren't.

Next we discussed how tables could help us with measurement conversions. The students figured out they are a great tool in keeping our conversion work organized!

Rounding

Rounding numbers is often taught in a very rule-based way, not conceptually.
"Just look at the next door neighbor and if it's 5 or higher, the number goes up. If it's 4 or less, the number goes down."
That probably sound really familiar to most adults. There are even a ton of cute sayings to help us remember this idea.
I like to focus on a couple of things when teaching rounding. Firstly, I ditch the phrase, "goes down" and replace it with "stays the same." The number in the position you're rounding never goes down. It either goes up or stays the same. This can be confusing for some. Secondly, and most importantly, we start rounding by placing numbers on a number line. This helps us determine if the number rounds up, or stays the same.
For example, if we're rounding 437,284 to the nearest hundred, we would make a number line. One end of the number line will be labeled with the hundred thousand the number is already in - 400,000. The high end of the number line will be labeled with the next hundred thousand - 500,000. Next we find the middle of the number line - 450,000. Finally we determine if the number falls to the left of 450,000 or to the right of 450,000. We do this by focusing on the ten thousands place (this is where the whole "look next door" idea originates.)

After we practiced this MANY times, rounding to MANY different places, we moved to the shortcut.

Relationship of Numbers in our Place Value System

One of the most important concepts a child can learn is number sense. This week we spent a lot of time investigating what happens as we move left or right in our number system, and comparing numbers in different places in our place value system. We related them to each other by describing their relationship. Using equations to describe the relationship is a pretty big concept for 4th graders to grasp. We spent lots of time writing equations to describe them. I tried to color code my journal examples so it makes sense.