# Division Basics Part 3: Repeated Subtraction and # Line

In my opinion, the process of repeated subtraction is very important for students to practice. With repeated subtraction, we are actually asking this question:  “How many _____ in _______?”  If the problem was 20÷4, we can ask, “How many 4’s are in 20?”  The process is to keep subtracting 4 (using concrete, pictorial, and abstract methods) until zero is reached.  This would be done 5 times — thus, 20 ÷ 4 = 5.

Much like multiplication, there are different aspects of division children should become familiar with.

• Arrays
• Equal Groups
• Repeated Subtraction
• Number lines
• Skip counting

The focus today will be to help children understand how repeated subtraction can assist with the division process (using manipulatives, drawings, and paper-pencil methods). The template pictured here is FREE from: Multip. and Division templates FREE from Number Two Pencils @ TpT

The reason the repeated subtraction strategy is important is because this is what we are really asking students to do when they encounter long division or partial quotient problems. With the problem 100 ÷ 4, the question is, “How many 4’s are in 100?” If the repeated subtraction process is used, the answer is of course, 25.  But subtracting 4 twenty-five times is not very efficient.  So we want the student to get closer to 100 and subtract larger amounts than 4 at a time. The partial quotients method would allow the student to do this in chunks.  1 solution could be to subtract 40 (ten 4’s), subtract another 40 (ten more 4’s), subtract 20 (five 4’s).  See picture below: Continue reading

# Division Basics Part 2: Equal Groups

Last post featured division using arrays and the area model.  This post will focus on helping children see division as equal groups. Most of us have used the “plates of cookies” analogy to help kids see how to represent equal groups in a drawing.  I will just take that a few more steps to increase efficiency.

Much like multiplication, there are different aspects of division children should get familiar with:

• Arrays
• Equal Groups
• Repeated Subtraction
• Number lines
• Skip counting

In this post, I will break down the benefits of equal groups models to help children understand division (and how it is related to multiplication). Check out the freebies within this post.

If you haven’t utilized this book with your students, please try to find a copy!  It’s called The Doorbell Rang by Pat Hutchins.  In this story, Ma makes some cookies to be split between the kids.  Then the doorbell rings and more kids come, so the problem has to be refigured. This scenario repeats. As a class, you can duplicate the story with a different # of cookies and children.

Another great story emphasizing equal groups (as well as arrays) is the story One Hundred Hungry Ants by Elinor Pinczes.  In this story, 100 ants are on their way to raid a picnic. They start off in one straight line (1 x 100), but then rearrange into different equal groups to shorten the line (2 lines of 50, 4 lines of 25, etc.). A nice project after reading this book is to see how many ways a different given # of ants (or other animals / objects) can be divided into equal groups / rows.

By clicking on the links for each book above, you will be taken to Amazon for more details.

As I mentioned earlier, many children’s view of equal groups regarding division is to use manipulatives and/or draw circles / plates to match the divisor and then divide up the “cookies” equally in these groups.  Let’s say you had this problem: “There are 12 cookies to be divided onto 3 plates equally.  How many cookies would go on each plate?” As you observe the students:

• How are they dividing up the cookies? One at a time, two at a time, randomly, trial and error?
• Are the “cookies” scattered randomly on the plate / circle?  Or, are they arranged in an easy-to-see pattern so they are easily counted (by the student and yourself as you walk around the room)?
• Are the students able to verbally tell you how they divided them?
• Are the students making the connection to multiplication by noting that 3 x 4 = 12?
• Can they solve similar problems using language other than plates / cookies?
• Try shelves / books; trays / brownies; buildings / windows; flowers / petals; students / rows of desks, stars / # of points; aquariums / fish; boxes / donuts; etc.

Use of manipulatives of various types (cubes, tiles, counters) is important for children to have their hands on the objects being divided. This is how they work out their thinking. Then work toward paper/pencil drawings before going to the abstract use of numbers only.  This doesn’t have to be done in separate lessons, however. There is great value for children to see how the concrete, pictorial, and abstract representations all work together.

Also, help children list synonyms for the dividing process:  distribute, share, split, separate, halve, quarter, partition

Here are a few strategies I believe help make the equal groups process more efficient: Continue reading

# Division Basics Part 1: Arrays and Area Model

Division seems to be the hot topic with classes I have been visiting lately, so I thought I’d focus on that for now. Let’s look at some of the basics.  Students as young as first grade actually start thinking about division when working on fraction standards such as:  Determine fair share — equal parts. Most students have had practical experience with dividing sets of objects in their real life to share with friends, classmates, or family (cookies, pizza, crayons, money, pieces of paper). So now our job as teachers is to relate this real-life experience with the division algorithm.

Much like multiplication, there are different aspects of division children should get familiar with:

• Arrays
• Equal Groups
• Repeated Subtraction
• Number lines
• Skip counting

In this post, I will break down the benefits and uses for arrays (and the related area model) to help children understand division (and how it is related to multiplication). There’s a fun FREE game (Block-It) at the end of the post.

What is an array?  An array is a rectangular model made up of rows and columns.  When an array is constructed, the factors are represented by the number of rows and columns.  So, do your students know the difference in a row and column?  (Rows go horizontally, while columns are vertical.)  These are important math terms students should be using.

• Give students experience constructing arrays with manipulative objects (tiles, chips, cubes, etc.):
• You can be specific, such as: “Build an array using a total of 12 tiles. Put them in 3 rows.  How many columns did you create?” In this scenario, there is only 1 way to show this array. Students would be modeling 12 ÷ 3 = 4. Twelve is the dividend (the total amount you started with). The # of rows is the divisor (how it was divided).  The quotient is the result (in this case the # of columns).
• You can also be a little more open ended such as:  “Build an array using 12 tiles. Is there more than one way to do this?” If students are given the opportunity to explore, they hopefully find arrays such as 3 x 4; 4 x 3; 2 x 6; 6 x 2; 1 x 12; or 12 x 1. Students would be modeling 12 ÷ 4; 12 ÷ 2; 12 ÷1, etc.
• Give students experience drawing arrays:
• You can be specific or open-ended (as above).
• Children can free-hand draw or use grid paper.  If using grid paper, then these can be cut out and displayed as “Different ways to divide 12.”
• Give students experience using pre-drawn arrays:
• Students should label the sides of the array with numbers.
• Use the numbers shown to determine the fact family.  Example:  3 x 4 = 12; 4 x 3 = 12; 12 ÷ 3 = 4; and 12 ÷ 4 = 3
• After the array is made, ask questions or explore more such as:
• How many 3’s are in 12? (count the columns)
• How many 4’s are in 12? (count the rows)
• Circle the rows and / or columns to see the groups more easily.
• Help children make up story problems to match the array:  “I have 12 desks that I need to arrange in 3 rows. How many desks will be in each row?” or “I need to put 12 books equally onto 3 shelves. How many books will go on each shelf?

# Math Problem Solving Part 5: Multiplication and Division Comparisons

As I promised, here is a post about another multiplication and division story structure.

The dinosaur is about twice as tall as the human. The dinosaur is about 12 times wider than the human.

While the previous structure I mentioned dealt with equal groups, this one deals with comparisons.  Like with addition / subtraction comparison problems, I will show some ways to use double bars to illustrate these. (I just added a FREEBIE – a 2-pg. pdf showing the pictures of the problems shown below – get it now Equal groups comparison problem pics)

Notice some of the questions associated with these stories. They sound a lot like questions used in other story structures — this is why I advise against statements like, “If it asks how many in all, add the numbers together.” Notice how this method guides a student as they go from pictorial to abstract. You can also represent these types of problems with cubes or money manipulatives if you need a concrete example.

This problem shows a multiplying process.

1. Read the problem. Ask who and what this story is about (Joe and Brent – Joe has more \$ than Brent). Notice  there is no additive or subtractive process, but one has more – the other has less. This is a signal to use comparison bars to help solve.
2. Make a vertical line (to help line up the bars evenly on the left side. Draw the bar with the known amount. (Joe has \$22.)  Label the second bar (Brent).

3. Since the story said Brent has 3 time more, draw 3 bars equal to the size of Joe’s bars.

4. Since Joe’s bar is \$22 and the bars by Brent’s name are equal in size, all of Brent’s bars should be labeled with \$22.

5.  To find out how much Brent has, solve by repeated addition or multiplication.

6.  IF the question was slightly different (How much do Joe and Brent have all together?), then the above steps would have to be followed by 1 more step (adding the two amounts together).

* If students had been taught earlier to “Add when the question asks How many in all” then the child would likely add \$22 + 3 to get their answer.  That signals no conceptual understanding of what the problem is all about:  One has more, the other has less.

The following example involves the division of a bar to help solve it. Continue reading

# Math Problem Solving Part 4: Equal Groups

Are you ready for a way to help students solve multiplication and division word problems? I have developed a template that will help students record the given facts and think through the process involving equal groups. There are 3 basic types of equal groups problems:

• # of groups and # in each group are known
• # of groups and total are known
• # in each group and total are known

Pictured here is an illustration of the strategy which can be used for multiplication and division problems. Get my FREE packet right here: Equal groups strategy with template.

One of the most important steps I recommend when working with equal groups problems is for students to brainstorm things that typically come in equal groups. There is a great book titled “What Comes in 2’s, 3’s, and 4’s” which can set the stage. It’s a picture book, but helpful to get kids’ brains warmed up.  Here are some visuals I made to illustrate the point. Click here for your FREE copy. They are included in this set of Equal groups pictures and list template

Kids may need your help to think of things to add to the list. See some hints below. This template and a full list is included in the above FREE download. Continue reading

# Making Sense of Division (3rd-5th)

Is division a dreaded topic on your list of objectives to teach? Like many math topics, students have a harder time understanding it most likely because it’s not something they use regularly in their lives. Students should understand why division is useful before they have to start solving division problems. In this post, I will focus on helping students see the relationship between subtraction, multiplication, and division both with concrete objects, pictures, and the partial quotients model. Freebies available below!!

Then let’s talk about what division really is — it is repeated subtraction; much the way multiplication is repeated addition. The issue is that repeated subtraction is not always very efficient. Here’s what I mean.

Let’s say I have the basic problem 25 ÷ 5.  I could start with 25 and then subtract 5, subtract another 5, another 5, another 5, and another 5 until I run out and reach zero.  I would have to do this 5 times. If I had 25 cookies that I wanted to share equally among 5 friends, I could do the “one for you, one for you, one for you, one for you, and one for you” process and still end up with 5 cookies for each. Or I could try “two for you, two for you,” etc. to make the action of passing out the cookies faster. When I get down to 5 cookies, I return to the “one for you . . .” to make it work.

With a larger problem such as 72 ÷ 6, I can again try subtracting 6 at a time until I reach zero. This would take 12 repetitions — not efficient, but still accurate. Could I subtract 12 at a time instead (2 groups of 6) to be more efficient? Or 18 at a time, or 24 at at time? This is the idea behind the partial quotients model I will refer to a little later. Continue reading

# Number Talks Part 3: Computational Strategies 3rd-5th grades

This is the Part 3 of Number Talks. If you are just tuning in, please refer to NT Parts 1 and 2. As I mentioned before, conducting a Number Talk session with your students is a chance for them to explain different ways to solve the same problem. This is meant to highlight strategies which have already been taught.

Click below to watch  2 videos of how to conduct a Number Talk session with intermediate students. You will see many strategies being used.

Number Talk 3rd grade 90-59 = ____

Number Talk 5th grade 12 x 15 = ___

Addition and Subtraction Strategies:  I like using the methods listed below before teaching the standard algorithm. This is because they build on a solid knowledge of place value (and number bonds 1-10). If your students are adding and subtracting using the standard algorithm and can’t adequately explain the meaning of the regrouping process in terms of place value, then try one of the following methods. In many cases, I will ask a student the meaning of the “1” that has been “carried” over in double-digit addition. About 85% of the time, the student cannot explain that the “1” represents a group of 10. When adding the tens’ column, they often forget they are adding groups of 10 and not single digits. So they get caught up in the steps and don’t always think about the magnitude of the number (which is part of number sense). You will notice teachers write the problems horizontally in order to elicit the most strategies possible.

• Partial Sums
• Place Value Decomposition
• Expanded Notation
• Compensation
• Open Number Line (to add or subtract)

Here are some possible Number Talk problems and solutions:

Multiplication and Division Strategies: I like using these methods before teaching the standard algorithms. Again, they build a solid understanding of place value, the use of the distributive property, and how knowledge of doubling and halving increases the ability to compute problems mentally. Once these methods have been learned, then it is easy to explain the steps in the standard algorithm.

• Area Model
• Partial Products
• Distributive Property
• Doubling and Halving
• Partial Quotients

Here are some possible Number Talk problems and solutions:

# Daily Math Meeting Part 1: Ways to Build Number Sense K-5

To build number sense, students need frequent exposure or review of concepts you have previously introduced. There are many ways to build number sense on an on-going, informal basis – especially when you can squeeze in 10-15 minutes daily:

• During morning meeting time
• During a Number Talks session
• At the beginning of your math lesson
• At the end of your math lesson
• End of day closure time

I have included several of my power point slides on this topic as a PDF file (daily-practice-to-build-number-sense-pdf). Continue reading