Fifth Grade - November 10th - 14th, 2008

Fifth graders begin their study of fractions this week. They will make fraction strips in order to look at equivalent fractions. They will explore the meaning of numerators and denominators and look at how the size of the denominator affects the size of the pieces being looked at. For example, would you rather have 1/2 of a candy bar or 1/3 of a candy bar. At first, it may seem like 1/3 is bigger because the 3 is bigger than the 2. However, when you cut one candy bar into 2 equal pieces and another candy bar into 3 equal pieces, each of the 2 pieces are bigger than each of the 3 pieces.

Fourth Grade - November 10th - 14th, 2008

Fourth graders will begin their geometry unit by studying the characteristics of polygons. They will look at shapes that are polygons and shapes that are not polygons. From their observations, students will decide what is true and untrue about polygons. Some characteristics of polygons include:

• It is a closed figure - there are no openings in the shape.
• It only has straight sides - there are no curves.
• The lengths of the sides don't have to be equal.
• It doesn't have to be a typical shape like a square or rectangle

Students will then extend their study to polygons with 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 sides.

Third Grade - November 10th - 14th, 2008

We had an interesting discussion with one of the homework problems last week. The challenge was to find 70 less than 231. It is always important to put a problem into a story context so that the numbers make sense. Imagine this problem as a student having 231 stickers and they give 70 away to their best friend. One way to solve this problem is by making the numbers easier to work with. Instead of having 231 stickers, imagine that you only have 230 stickers and you give away 70 of them. Now the problem becomes much more manageable. It can be thought of as 23 tens minus 7 tens which is 16 tens or 160. The question is what happened to that extra sticker. Do we have 161 or 159 stickers left over. Well, think of our story. I started with 231 stickers but I made the problem easier by changing it to 230 stickers. It is like I put the extra sticker in my pocket. After I gave away 70 of my stickers, I can pull the extra sticker out of my pocket, so now I have 161 stickers.

This week, students will look at a problem like this, but the extra sticker is being subtracted. instead of 231 - 70, think of the problem 230 - 71. Again, lets make the problem easier by solving 230 - 70 which is 160. So, what happened to the extra sticker? In this case, the extra sticker is being given away, so after I give away the 70 stickers I still have to give away one more. Instead of having 160 left over, I now have 159.

The general idea is that if 120 - 50 = 70, then 120 - 54 will be 4 less or 66. If 120 - 50 = 70, then 124 - 50 will be 4 more or 74.

Second Grade - November 10th - 14th, 2008

Second graders are beginning to solve problems with bigger numbers. One problem we will look at this week is 12 + 24. Here are a few strategies we expect kids will discover as they solve this problem:

• Adding in Groups - Some students will start at 12 and count by ones 24 times. This method will work, but it becomes inefficient with large numbers. We would like to see kids who can start at 12 and add 10 at a time instead of 1 at a time. 12 + 24 can be thought of as 12 + 10 + 10 + 4.
• Breaking the Numbers Apart - Students may see that they can break the numbers 12 and 24 into (10 + 2) and (20 + 4). Since we just studied that order does matter when adding, we can rearrange our numbers to be (10 + 20) + (2 + 4). Now students are adding the tens together and the ones together. 10 + 20 = 30 and 2 + 4 = 6, so 30 + 6 = 36.

First Grade - November 10th - 14th, 2008

First graders are working on addition and subtraction story problems. Students are learning how to show their work using pictures, words, and numbers. It is very important for students to be able to show or explain how they got their answer. We often hear young children say, "I just did it my brain." We are trying to move kids away from that answer to actually showing their work.

Here are some of the strategies our students have learned to add and subtract:

• Counting All - Students add 3 + 7 by counting 3, then counting 7, and then starting over again with 1 and counting all the way to 10. We would like to see students move away from this strategy when they feel more comfortable with numbers.
• Counting On - Students add 3 + 7 by counting 3, then counting on from 3 an additional 7 so that they get to 10.
• Counting Back - Students solve 9 - 4 by starting with 9 and counting back 4 so that they arrive at the answer of 5.
• Using Known Combinations - Students solve 6 + 8 by reasoning that if 6 + 6 = 12, then 6 + 8 must be two more than 12 which 14.

Fifth Grade - November 3rd - 7th, 2008

Fifth grade students are finishing up their unit on addition and subtraction. They are using the strategies they have already explored (see blog entry on October 22nd) to solve problems with larger numbers like 69,703 - 55,675.

Fifth graders will also be assessed on how well they know their division facts. Students should be able to solve 30 problems in about two minutes. Students who are not this proficient with their facts should practice at home.

Fourth Grade - November 3rd - 7th, 2008

4th graders are beginning a unit on 2-D geometry and measurement. This week we will study linear measurement using both the U.S. system and the Metric system. Students will identify objects they can use as benchmarks to help them estimate lengths. For example the length of a piece of notebook paper is about 1 foot. Students will explore rulers, yardsticks, and metersticks and use these tools to measure the length of the classroom and other objects. They will also explore the idea of perimeter and measure different items in the room. Finally, students will design a path in the school that is 100 feet long. Most students (and many adults) find it difficult to picture 100 feet. This will be an eye-opening experience.

Third Grade - November 3rd - 7th, 2008

This week, third graders are focusing on how to find the difference between a 2-digit number and a number over 100 (128 - 36). Students have already worked on finding how far a number is from 100. Students should be able to reason that 36 is 4 away from 40 and then 60 away from 100 (36 + 4 = 40 and then 40 + 60 = 100). So, 36 is 64 away from 100 (36 + 4 + 60 = 100). Students are now reasoning that if 36 is 64 from 100 and 128 is 28 from 100, then the difference between 36 and 128 is 64 + 28, or 92.

Second Grade - November 3rd - 7th, 2008

Our second graders are working on adding three numbers at a time. They will explore the idea of whether order matters when adding a group of numbers. When adding 2 + 3 + 5, does it matter if we add 2 + 3 first? 2 + 5? 3 + 5? As students discover that order does not matter, they are really investigating the associative property of addition. a + (b + c) = (a + b) + c.

Students are also reviewing their doubles - 2 + 2, 3 + 3, 4 + 4, etc. Students use this knowledge to determine the near-doubles. 2 + 3, 3 + 4, 4 + 5, etc. Students should notice that the answer is just one more than the answer to the doubles combination.

By this point in the year, second graders should be fluent (not necessarily instantaneous recall) with the following combinations: plus 1 (4 + 1), plus 2 (4 + 2), combinations that make 10 (7 + 3), and doubles (6 + 6).

First Grade - November 3rd - November 7th, 2008

Our first graders are still working on addition problems. We're looking for students who can use the strategy of Counting On to add numbers together. To help students with this strategy, we are playing games with a number cube and a dot cube. When students add the two numbers together, they will not be able to Count All the dots because one of the cubes only has numbers. For example, imagine that you roll a 4 on the number cube and a 3 on the dot cube. A student has to start at 4 and count 3 more to get a total of 7. This idea of Counting On is much more efficent than Counting All the items in two groups.

Another strategy for addition is to use combinations already known to figure out new combinations. If a student knows that 5 + 5 = 10, they can reason that 6 + 4 = 10 because you just take one away from one of the 5's and give it to the other 5.

We are also working on the idea of finding all the addition combinations that equal a given number. Students are told that they have 7 items. Some of the items are blocks and some of the items are marbles. What are all the different combinations of blocks and marbles? We want students to develop a systematic way of finding all the combinations. Start with 1 block and 6 marbles, then move to 2 blocks and 5 marbles, then 3 blocks and 4 marbles, etc. Students should also realize that they can find opposites. If they have 3 blocks and 4 marbles, they can also have 4 blocks and 3 marbles.

Finally, we will start to study subtraction this week. It will be interesting to see the different types of strategies students use to solve subtraction problems.

5th Grade - Adding and Subtracting Large Numbers

Next week, our fifth graders will continue using different strategies to add and subtract large numbers.

Let's look at some different strategies to solve the problem 892 - 567.

• Subtracting in Parts - Start with 892 and subtract 500 (892 - 500 = 392). Next, subtract the tens place (392 - 60 = 332). Finally, subtract the ones place (332 - 7 = 325).
• Adding Up - For this strategy, start with 567 and add up until you get to 892 (567 + 33 = 600 and 600 + 292 = 892). Finally, add your numbers to get the final answer (33 + 292 = 325).
• Subtracting Back - For this strategy, start with 892 and subtract back until you get 567 (892 - 92 = 800 and 800 - 200 = 600 and 600 - 33 = 567). Finally, add your numbers to get the final answer (92 + 200 + 33 = 325).
• Changing the numbers to make an easier problem. For this strategy, it is important to remember that if you change one number you must also change the second number. Imagine you have 84 pennies and your brother has 55 pennies and you want to find the difference. What would happen if both of you got an additional 6 pennies? The difference would still be the same. Using this idea, we can change 892 into 900, but we must also change 567 into 575 (we added 8 to each number). Now we can use one of our first three strategies to solve this problem (575 + 25 = 600 and 600 + 300 = 900, so 25 + 300 = 325).

4th Grade - Multiplication

Next week, our fourth graders will continue their study of multiplication. We are espcially using the idea that you can multiply by 10 to solve a 2-digit problem. For example, to solve 15 x 13, students can think of the number 13 as 10 + 3. So, instead of 13 groups of 15, there are 10 groups of 15 and 3 groups of 15. 10 groups of 15 = 150 and 3 groups of 15 = 45. Add 150 + 45 to get your answer of 195.

4th Graders will also take a test next week on thier multiplication combinations. It is expected that students should be able to solve 30 problems (1-digit x 1-digit) in 3 minutes. Students who do not meet this expectation should continue to study thier combination cards at home.

3rd Grade - Addition and Subtraction

Next week, our third graders will continue working on addition strategies and begin studying their subtraction facts. By now students should be comfortable with at least one strategy for addition.

One strategy they have learned is Adding By Place Value. For example, to solve 184 + 235, students can add the hundreds first (100 + 200 = 300), then the tens (80 + 30 = 110) and finally the ones (4 + 5 = 9). Add your numbers to find the final answer - 300 + 110 + 9 = 419.

Another strategy is Breaking One Number Apart. To solve the same problem, 184 + 235, first add 184 + 200 = 384. Then add 384 + 30 = 414. Our students have been studying how to add groups of 10, so this step should be fairly easy. Finally, add 414 + 5 = 419.

Students will bring home some subtraction fact cards later in the week. It is expected that students are fluent with their subtraction facts by the end of third grade.

2nd Grade - Rectangles

Next week, our second graders will continue exploring rectangles. Students will build rectangles with tiles and write descriptions of what they build so that other students can build the same shape. Students should realize that it is important to include the number of tiles that are in each row and the number of rows. Students should also understand that a rectangle has 4 sides and 4 right angles.

After studying rectangles, we will move our study into rectangular prisms. Students will build boxes in the shape of a rectangular prism and discover that all the faces are rectangles and that opposite sides are congruent (equal).

1st Grade - Counting On and Finding All Possible Solutions

Next week, first graders will be working on the strategy of Counting On. Young students typically count two groups by counting all the items. For example, if one group has 5 buttons and another group has 6 buttons, students will count the 5 in the first group, the 6 in the second group, and then start with 1 again to count all the buttons. We are working with our students to learn the strategy of Counting On instead of Counting All. Once you count the number of items in one group, keep that number in your head and then continue counting the items in the second group. Students will play a game next week to reinforce this idea. The game is like Bingo where students are trying to get 5 numbers in a row on a game board. Earlier in the year students played this game by rolling two dot cubes and adding the numbers together to find a number on the board. This time students will roll one dot cube and one number cube. This will encourage students to Count On.

Another big idea we are studying next week is finding all possible solutions. Students are given a number and asked to find all the different ways to make that number. For example, we can make the number 5 by adding 0 + 5, 1 + 4, 2 + 3, 3 + 2, 4 + 1, and 5 + 0. First graders may not organize thier work in a systematic way, but we hope they will discover a method for making sure they have every possible answer. Another idea students will discover is that if I have 1 + 4 I can also have 4 + 1.

3rd Grade Measurement

Students in third grade are using measuring tools to collect data. Today they used rulers to measure the length of their classroom.

One important idea during this investigation is how to use measuring tools, specifically 12-inch rulers and yard sticks. We discussed that 1 yard is equal to 3 feet and that 1 foot is equal to 12 inches. I found it interesting that as students measured the room, they came up with answers like 21 feet, 29 inches. They were able to determine that 7 yardsticks was the same as 21 feet. What they did not think about was that 29 inches could be broken down into feet and inches.

After some discussion, we decided that 21 feet, 29 inches could be written as 21 feet + 12 inches + 12 inches + 5 inches. We then changed this to read 21 feet + 1 foot + 1 foot + 5 inches which is the same as 23 feet, 5 inches.

As you talk withyour child, talk about the idea of converting measurements. Ask questions like:

• What is another way to say 17 inches? (12 inches + 5 inches or 1 foot, 5 inches)
• How many inches are in 3 feet, 4 inches? (12 inches + 12 inches + 12 inches + 4 inches = 40 inches)

When should kids know their combinations?

I have had many parents ask me when their child should know their addition or multiplication facts. I would first like to say that Investigations prefers to call them combinations. Calling them facts implies that you can't learn them through reasoning. However, if I know 8 x 3 = 24, then I should be able to reason that 8 x 6 is double 8 x 3 or 48.

What does it mean to know your combinations? Fluency means that you can immediately recall an answer or quickly perform a calculation to get the answer. If it takes longer than two seconds to figure a combination, a student is not fluent. This includes single digit addition and multiplication pairs and their counterparts for subtraction and division. We encourage students to work on combinations that they don't immediately know. This can be done during downtime in class or at home. I also encourage 3rd, 4th, and 5th graders to use arrays when working on their combinations.

The following are guidelines for learning combinations through the grades:

Addition - fluent by the end of Grade 2, with review and practice in Grade 3.

Subtraction - fluent by the end of Grade 3, with review and practice in Grade 4.

Multiplication - fluent with multiplication combinations with products to 50 by the end of Grade 3; up to 12 x 12 by the middle of Grade 4, with continued review and practice.

Division - fluent by the end of Grade 5.

Adding by Place Value

How would you solve the problem 254 + 763?

Many of us would line the numbers up like this:

However, there are other ways to solve this problem. One way is to break the numbers down by place value.

• We can start with the hundreds place and solve 200 + 700 to get 900.
  
• Moving to the tens place, we can solve 50 + 60 to get 110. If you have trouble adding these two numbers, think of the problem as 50 + 50 + 10.

• Finally, move to the ones place and solve 4 + 3 to get 7.

• Now we have the numbers 900 + 110 + 7. Another way to think of this problem is 900 + 100 + 10 + 7 which is 1,017.
  

Why does this strategy make more sense than lining up the numbers like above? Let's talk our way through solving that problem. The first step is pretty easy. 4 + 3 = 7. Now have a student explain the second step. You will probably hear something like "5 + 6 is 11, so put down a 1 and carry the other 1." What does this mean? The actual step is adding 50 + 60 to get 110. So, you are putting 10 down and carrying the 100. This explanation get a little confusing for kids.

While lining up your numbers and following the steps will work, it doesn't give students any meaning about what they are really doing. Adding, or subtracting, by place value makes sense of the problem. As our students move through Investigations, they will become more efficient at solving problems like this and may even begin doing them in their head.

New Math Program at Las Colinas

Welcome to a new school year at Las Colinas Elementary. My name is Craig Rich and I am the math instructional facilitator for our school. I am very excited because this year our district is adopting a new math program called Investigations. Many of our teachers had an opportunity this summer to work with other teachers from around the country who have already used Investigations for over ten years. I was very impressed with what I saw. This program helps kids to understand the meaning behind mathematics instead of just memorizing rules. As we learn throughout the year I will post tips, suggestions, and helpful hints for parents so that you will better understand how Investigations work. I will warn you in advance, this math program will not look like the math you had when you were a kid. You won't see many homework sheets with lots of addition, subtraction, multiplication, and division problems. Instead, many homework assignments will ask students to explain how they arrived at a certain answer. Don't get me wrong, I still think it is important for students to know their basic facts. I would love for parents to work with their child at home with these facts. However, time in the classroom will be spent exploring math and discovering the meaning behind the methods they use to solve problems.

Here is a problem to get you thinking. How would you solve 73 - 49? Can you solve it more than one way? Can you solve it in your head? These are some questions we would like our students to think about.