Multiplication and Division Strategies

Below is a video clip from our math closing meeting yesterday. Check out these great strategies for multiplication and division!

Multiplication and Division Strategies from Melissa Ross on Vimeo.

Comparing Arrays

Last week in Math Workshop, we spent time arranging different amounts of chairs into rows and columns (arrays). Today, we are comparing the arrangement 16 and 17 chairs. Look at the different arrangements that we can make with each number below:

What do you notice about these arrays? Do any of the arrays have a special or unique shape? What do you notice about the number of arrays that can be made with 16 chairs compared to 17 chairs?

Hopefully, you notice that 16 chairs can be arranged in several different arrays. That is because it has many factors: 1, 2, 4, 8, 16. A number that has more than two factors is called a composite number. You probably noticed that 17 only has two arrays. This is because 17 is a prime number. Any number that has only two factors, one and itself, is a prime number. The factors of 17 are 17 and 1.

Also, you may have noticed that 16 can be arranged into a perfect square with 4 rows and 4 columns. Any number that results when another number is multiplied by itself is a square number. (ex: 3x3=9 Nine is a square number. 5x5=25 Twenty-five is a square number.) Sometimes math vocabulary can be confusing! For a reminder of the meaning of some math words that you may have forgotten, visit this great online math dictionary.

Thank you to our guest author this week, Miss Russell. :)

Closing Meeting Strategies

Check out this great thinking from a second grade closing meeting!

Oops! The factory put 101 paper clips in each box instead of 100! How many paper clips will you get if you buy 9 of these boxes?

Closing Meeting Paper Clips from Melissa Ross on Vimeo.

Take a look!

How do you add 258 + 392 ?

Yes, this clip is of a second grade student. Thank you for sharing, Miss Russell!

Modeling a Problem

We encourage students to visualize problems or create models for them if they do not understand what a problem is asking. Models are simply tools for thinking. We have problem solving strategy posters hanging in each second grade classroom to remind students of ways they can go about solving a problem if they do not understand it. Some of the posters include: make a chart or table, draw a diagram, guess and check, find a pattern, use logic, make it simpler, use real objects, or make a list. Let's take a look at how we can use some of these ideas to solve the problem below.

Miss Russell is getting ready for the Relay for Life. She needs 36 inches of tape to hang a sign over the Chets Creek booth. She heads to the store to buy the tape and realizes that they sell tape in feet, not inches. How many feet of tape will she need to buy to have the exact amount she needs to hang her sign? (She knows that 1 foot is equal to 12 inches.)

Your child may see problems similar to this one in Calendar Math. Students are discovering the relationship between units of measurement. (We have previously discussed the relationship in centimeters, decimeters, and meters.) As concepts are learned, we try to present them in a real-life context for students.

One thing that would help getting started with this problem would be to make it simpler. Find out what the problem is really asking. 12 inches = 1 foot so 36 inches = ? feet

Once students determine what the problem is asking, they can then use a strategy to help them. For this type of problem, students may want to make a t-chart or table to help show their thinking. Below are three student samples:

This student used three different strategies to model the problem. In strategy A, the student used a ratio table to show the relationship between inches and feet. In strategy B, the student used a t-chart to show the relationship between inches and feet. In strategy C, the student used a number line to hop by feet and write the inches beneath. All three are perfectly acceptable models to use to make sense of the problem.

Using Landmark Numbers on an Open Number Line

A landmark number is simply a number that is easy to work with. In second grade terms, the easiest numbers to work with are numbers we use when we "skip count by tens" (10, 20, 30). In using landmark numbers on the open number line, we want to find a landmark number that is easy to count on from.  There are many variations to this strategy. The example below shows one of the most common variations. 

Example: 28 + 17 

Students will think28 plus something is going to get me to my next landmark number of 30.

I know that 28 plus 2 will get me to the landmark 30. 

Since I am adding 17 altogether, I need to subtract 17-2 (the hop I already took) to  equal 15. This tells me I still need to take a hop of 15 on the number line. If students cannot add 30 + 15 in their head, they can break apart the 15 in a way that is helpful to them. (10 + 5 hops, 15 single hops, etc.)

Second Grade Math Night

Join us for second grade Math Night this Thursday, October 9, 2008 from 7-8:15 PM. During Math Night, the second grade teachers will be going over strategies that your child will be learning this year to increase number sense and bring their problem solving skills to the next level. If you are unable to join us, or would like a refresher when you get home, check out the clips below.  They have been filmed by Mr. Pinchot, a fourth grade teacher and Math Coach at our school.

Addition Strategies:

Addition Overview

Counting Up By Ones

Counting Up By Tens and Ones

Using a Hundreds Chart

Decomposing 

Left to Right Addition

Compensation

Regrouping (The "traditional" method)

Subtraction Strategies:

Subtraction Overview

Counting Down By Ones

Counting Down By Tens and Ones

Distance Between Two Numbers

Landmark Numbers

Decomposing into Tens and Ones

Compensation

Expanded Notation with Negative Numbers

Regrouping (The "traditional" method)

What is Combinations Club?

This year, we will be focusing on becoming fluent with the addition combinations up to 10 + 10. In second grade, students learn these combinations over the course of the year. Combination sets include:

Make Ten (combinations of 10 with two numbers like 8 + 2)
Plus 1 (any number plus one like 1 + 8)
Plus 2 (any number plus two like 3 + 2)
Doubles (any number plus itself like 5 + 5)
Near Doubles (any number close to a double like 5 + 6)
Plus 10 (any number added to 10 like 3 + 10)
Plus 9 (any number added to 9 like 3 + 9)

Students will learn these combinations through frequent and repeated use. In school, we will be playing several games that help students learn particular combinations. Students will also play some of these games for homework, such as Plus 1 or 2 Bingo and Make 10.

We will also be using “Combinations Cards” to help students practice. We are not encouraging students to memorize these combinations! We are encouraging students to find efficient ways to solve the combinations. For example, look at the sample cards below:

We want to encourage students to efficiently (quickly and accurately) solve basic addition combinations. We are doing this by having students think of clues to help them solve the combinations. They may write the clues on their practice cards to help them. For example, if they are familiar with the double combination of 6+6, they may choose to look at 6+7 as one more than 6+6. If they are more familiar with the combination 7+7, they may choose to look at the combination as 7+7-1. There are many different ways to look at these combinations. Again, we are stressing that we are NOT looking for memorization, but efficiency and problem solving.

When your child is comfortable with a set of Combinations Club cards, he or she will receive a new set. (Often students will be asked to explain their thinking aloud on certain cards.) If they are still having difficulty with a set of cards, you will see them sent home again. If we feel they have mastered the concepts presented in those cards, a new set will be sent home. Please add the new set of cards to the previous set of cards, and keep practicing. Write any clues on the front of the cards that may help remember the combinations.

What is compensation?

Compensation is the latest strategy some of my students have been exploring in class. This strategy requires students to have a complete understanding of place value and the number system. It is a little different for addition and subtraction, so I will break each section apart below.

Addition:
This strategy is based on maintaining the value of the whole. With addition, as long the whole is maintained the sum will be the same. Therefore, what ever is done to one addend, the opposite must be done to the other addend. Let's say that we are adding 36 + 45.

Step 1: Find the amount that can be added or subtracted from the addend (36) so that the digit in the ones place becomes a 0.

Step 2: Once you determine the number you are adding or subtracting from the addend (36), the opposite must be done to the addend (45).

Step 3: Find the sum of the two new numbers you created.

This probably sounds very confusing, so lets look at a picture for clarification.

Subtraction:
This strategy is based on the fact that subtraction is the distance between two numbers on the number line. As long as the distance is maintained, you can adjust the numbers to make an easier problem. Let's take a look at the problem 57-28.

Step 1: Find the amount that can be added or subtracted from the minuend (57) so that the digit in the ones place becomes a 9. (We want to eliminate any kind of regrouping.)

Step 2: Once you determine the number you are adding or subtracting from the minuend (57), you must add or subtract the same number from the subtrahend (28) to keep the distance the same on the number line.

Step 3: Find the difference between the two new numbers you created.

Again, let's break it down a little more. Take a look at the number line below:

Keep practicing your strategies. Remember your child will not be asked at any time to only use a certain strategy (like compensation) on any assessments or classwork. Students choose the strategy that works best for them on the problem they are working on.

Story Problems Galore!

We are beginning our new math book, Putting Together, Taking Apart. This unit supports students in developing strategies for solving addition and subtraction problems based on an understanding of numbers, number relationships, and the operations of addition and subtraction. Students continue to work on counting with an emphasis on developing efficient addition and subtraction strategies.

In order to solve these problems completely, students are asked to follow several steps. These steps are meant to teach students how to organize their work. We teach students that mathematicians are organized and can clearly explain their thinking.

1. Close my eyes and visualize. Students will picture what they think is happening in the story problem.

2. Combining or separating. The next step is to decide if the story problem is describing a combining situation or a separating situation. We do not teach students to search for “key words” when reading the story problem.

3. Write the equation. After students have decided the type of problem, they will need to write out the equation, leaving a blank spot where the answer should be until the equation is solved.

4. Solve it. Now, students will solve the equation using a strategy of their choice. Students are never told which strategy to use to solve a problem. Blog postings thus far have explained how to use tally marks, open number line, and decomposing. I will be posting more strategies soon.

5. Check with a second strategy. Students are to check their work using a second strategy they know. Again, students are never told which strategy to use as long as they use a strategy different from the one used in the previous step.

6. Circle my answer. If the answer matched on both strategies tried, the answer is most likely correct. Students are then asked to circle their answer or write it in the blank space in the original equation.

7. Complete the sentence. The story problem asked an original question that the student was trying to solve. This final step asks students to write in a complete sentence the answer to the problem they were solving.

Let’s look below for a complete example. The only step not visible in the problem below is “close your eyes and visualize," for obvious reasons.

Ms. Groves has 24 pieces of chocolate. Mrs. Ross ate 13 pieces.

How many pieces of chocolate does Ms. Groves have left?

Story Problem Practice online. You will need your own paper to solve the problems on the websites below. Happy solving!

1. Math Playground http://www.actionmath.com/GSM1/GSMwp1.html

2. More Math Playground http://www.actionmath.com/Katie1/Katiewp1.html

3. Still More Math Playground http://www.mathplayground.com/wpdatabase/wpindex.html

We're Decomposing!

Our math strategies are in full swing. Math Night was a HUGE success! Thank you for coming out and spending a night of problem solving with us!

We have been through using tally marks to solve problems and using an open number line. Now, on to our next strategy: decomposing.

In decomposing, we are going to be solving combining and separating problems by decomposing, or breaking them down, into groups of tens and ones. In order to work on this strategy with your child, he or she must have a good grasp of place value. He or she will also continue to explore this strategy in multiplication problems in higher grades by using clusters of simpler problems.

Let's look at a combining example for the problem 24 + 12.

1. Decompose both addends into groups of tens and ones, making sure to line up work neatly. The neatly part is harder for most kids at this point than the actual decomposing. ;-)

2. Before finding the partial sums of the tens and ones, I have each student make sure that every number has a "buddy" to combine together. (10+10, 10+2) If not, I have them hold the place with a zero. (4+0) (This step may be skipped, but I stress it in the beginning as they are learning this strategy so they don't get confused as to which numbers to combine.)

3. Find the partial sums of the tens and ones.

4. Combine the partial sums to get the final sum.

Decomposing for subtraction is similar. Let's use the sample problem 24-12.

1. Decompose the minuend (24) and subtrahend (12) into groups of tens and ones.

2. Check for "buddies."

3. Find the difference of each problem. (subtract)

4. Finally, find the sum of the differences. This step (for obvious reasons) confuses children. I explain it to them by saying that we decomposed or broke apart the numbers 24 and 12 so that we could subtract them. Now, we have to put the numbers back together or "recompose" them to get our final answer.

Still confused?Below you will find a quick video clip of Mr. Pinchot (a fifth grade math teacher and coach from our school) explaining the strategy. Thanks Mr. P!

http://www.schultzcenter.org/mathvideos.shtml

Place Value

As part of our Calendar Math every morning, we review place value. We review it by creating groups of tens and ones for the amount of days that we have been in school. Place value is a very in-depth concept that students began learning in kindergarten and first grade. In order to fully understand the addition and subtraction strategies we will be starting soon, students should have a complete grasp of the concept of place value.

Simply put, place value is the value of where the digit is in the number. In second grade, we do not venture farther than the hundreds place often. Here is an example of what students should be able to explain:

124

-The number can be written as one hundred twenty-four.

-The number can be written as 100 + 20 + 4

-The 1 is in the hundreds place. This means that there is 1 group of 100 in this number. One group of 100 looks like this:

-The 2 is in the tens place. This means there are 2 groups of 10 in this number. Two groups of 10 looks like this:

-The 4 is in the ones place. This means there are 4 ones in this number. Four ones look like this:

- The entire number 124 can be represented as:

Great Place Value Practice Games:

**My favorite!!** Place Value Golf http://www.toonuniversity.com/flash.asp?err=496&engine=5

Dino Place Value Game http://www.ictgames.com/dinoplacevalue.html

Shark Place Value Game http://www.ictgames.com/sharknumbers.html

Introduction to Place Value Movie:

http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/3_Place_Value/

Using an Open Number Line in "Single Hops"

An open number line is another strategy that students have learned to use. In this strategy, students are recreating only the part of the number line that is useful to them to solve the problem they are working on. We begin by taking single "hops" on the number line. Students will eventually "hop" in groups of 10 or to landmark numbers.

Sample problem: 12 + 10 = ___

1. First, begin the open number line by starting with one of the numbers (usually the higher number) from the equation.

2. Next, make "hops" on the number line to show what you are counting to. In this problem, we will make 10 "hops." Just like in the tally mark strategy, students are asked to show all their work.

3. Fill in the numbers on the number line that correspond with each "hop."

Using Tally Marks to Solve Problems

In second grade, we really focus on increasing number sense with our students. One of the ways that we do this is to encourage student to have a bank of strategies to help them solve problems. I am going to go through each strategy as we go over them in class. This first strategy tally marks in groups of 5 was introduced in first grade. We review it at the beginning of second grade for those students who did learn it in first and introduce it to those students who have not learned it yet.

Sample problem: 12 + 10 = ___

1. Make a representation of the problem using tally marks in groups of 5. It is helpful to some students to circle the two groups to make sure that they did not leave anything out.

2. Count all of the groups of 5. Students are always encouraged in class to show all work.

3. Count any single tally marks that are left after the groups of 5 are counted. The hardest part for students in this strategy is to go from counting in groups of 5 to counting singles. (For example, instead of counting 5, 10, 15, 20, 21, 22 students may try to count the single tally marks as groups of 5 like 5, 10, 15, 20, 25, 30.)

I have to draw legs on a math problem?

You may have heard your child say "I have to show my work! I have to put legs on my math problem!" If so, they aren't decorating their math homework. This is a strategy we talked about in class to solve number strings to make them easier.

Here are some helpful steps:

1. Check for doubles or combinations of ten.
2. Combine them using "legs."
3. Check for any missing numbers and write a new number string.
4. Solve your number string.
5. Circle your answer.


Here are some samples of what these number strings could look like solved.