Division Basics Part 3: Repeated Subtraction and # Line

by C. Elkins, OK Math and Reading Lady÷

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:

With the std. long division algorithm, students also must think of division as repeated subtraction. You wouldn’t believe the number of “ah-ha’s” I get from students when I show them this concept!!  Here’s a link to a previous post on long division. Making sense of division

So, how do we encourage this strategy with new learners using basic division facts? Check these out:

  1. Manipulatives:  If the problem is 20 ÷ 4, start with 20 objects.  Then take away 4 at a time by moving them to the side and forming a group. Repeat until all 20 objects have been taken from the original group. How many groups were made? This is a quick, meaningful any very efficient method to help students actually see equal groups being constructed. A must before going to drawings or paper pencil methods.
  2. Drawings:  If the problem is 15 ÷ 3, draw 20 objects.  Cross off 3 at a time and keep track (ex: tallies). Keep crossing off 3 at a time until all 15 objects have been crossed off. How many times was this done?
  3. Number line:  If the problem is 10 ÷ 2, draw an open # line and label with 10 points (0-10). Starting with 10, jump backward 2 at a time until 0 is reached. Count how many jumps were made. This is more effective with dividends 20 or less due to space and amount of time this takes. Consider a vertical # line as well.
  4. Subtraction:  If the problem is 56 ÷ 8, write the # 56, then subtract 8 repeatedly until there is zero left. Keep a running total. Common errors with this method are failure to follow regrouping steps and poor calculations.

I came across this FREE resource from Number Two Pencils: Free Multip. and Division templates showing a template for illustrating the 5 division strategies.  I completed two of them for you to see how these can be used effectively in the classroom.  They could also be used as anchor charts (and then student can create some of their own). There is also a companion set for multiplication strategies. Get your set here: Multip. and Div. strategy anchor charts

Enjoy your division journey!  Stay tuned for more!

Division Basics Part 2: Equal Groups

by C. Elkins, OK Math and Reading Lady

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.  

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:

  1. When using manipulatives or drawings, instead of randomly placing the objects being divided into equal groups, arrange them so it’s easy for the child as well as the teacher to see at a glance how many there are. In other words, if there are 5 in each group, are they randomly scattered?  If they are, the child wastes a lot of time recounting, which often invites error.  And the teacher has to spend time rechecking the child as well. Or, are the objects arranged in smaller arrays or groups making it very easy to see the total (like dice? by twos?).  This little requirement adds to a child’s understanding of number bonds and multiplication.

  2. Instead of placing individual objects, have the students try tally marks.  Again, these are counted more efficiently than a random organization – and it aids in multiplication.
  3. Instead of always using a one-at-a-time strategy as objects are being distributed, help them think that often they can try 2 at at time, or 5 at a time.  This aids with estimation and helps transfer this idea to future long division processes – especially partial quotients.
  4. Connect use of manipulatives and drawings with real life stories. What things come in equal groups?

    Refer to one of my previous blog posts showing this template for stories about equal groups (which can be multiplication or division): Equal Groups blog post. Help students notice each problem consists of these three components.

    • # of groups
    • # in each group, and
    • the total #
    • The division problem will usually provide the total and one of these (# of groups; # in each group). So the problem will be to determine the missing component by relating known multiplication facts and/or dividing.

Stay tuned!  Next week I will include some helpful basic division concepts resources.

Division Basics Part 1: Arrays and Area Model

by OK Math and Reading Lady

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?

Relate experience with arrays to determine area of a rectangle. This mostly just adds a measurement component to the problem.

  • Draw a rectangle and partition it into columns (length) and rows (width) to match the story.  Here are two sample stories:
    • I am making a rectangular shaped garden which I want to be 12 square yards in size.  If the length of the garden is 4 yards, how long does the side of the garden need to be?
    • I am using a rectangular piece of wood to cover a broken window that is 12 square feet.  One side of the wood is 3 feet. How long would the adjoining side be?

Here’s a great game called “Block it” which utilizes arrays. It can have variations depending on the level of your students. Here is a FREE copy of the directions: Block-It Game Directions

Materials needed:

  1. 1 grid paper (1/2″ is great)
  2. 2 players
  3. Each player needs 1 crayon or colored pencil (light colored). Different color per player.
  4. 2 number cubes or dice (6 sided).

How to play:

  1. Player 1 rolls the dice.  Let’s say a 3 and 4 are rolled. The player makes a 3 x 4 “block” or array. Be sure to show them how to use the lines on the grid paper to make this (as I discovered it’s not always clear to some students). Color it in with crayon.  Inside the block, write the product (12).
  2. Player 2 then rolls the dice and uses their 2 numbers to create another block, colors it, labels it, etc.
  3. Repeat
  4. The goal is to create as many blocks / arrays as possible (more than the opponent). There is a strategy to maximize the use of the space. Repeated play helps children see they need to consider this so they don’t end up with little unusable spaces.
  5. As the board gets filled up, players may have to miss a turn or roll again if not enough space is available.

Variations:

  1. As the board gets filled up, students may need to start thinking of alternate ways to make their blocks to fit the available space. For example, if the player rolls a 6 and 4 but there is no room to fit a 6 by 4 array, they can think of other ways to make an array of 24 that might work (such as 8 x 3, 12 x 2).
  2. Students can keep track of their score by keeping a running total of each block / array they make.
  3. Use smaller size grid paper and use 9, 10, or 12 sided dice.
  4. Write the fact family members for each block created.

Enjoy! Have you / your students played Block It?  Let us know if you like it!  

Math Problem Solving Part 5: Multiplication and Division Comparisons

by C. Elkins, OK Math and Reading Lady

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

by C. Elkins, OK Math and Reading Lady

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)

by C. Elkins, OK Math and Reading Lady

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

Discovering Decimals Part 3: Multiplication and Division

by C. Elkins, OK Math and Reading Lady

This is the last part of number operations dealing with decimals: multiplication and division in a concrete and pictorial method. This is actually not part of the KG-5th grade OAS standards, but it is addressed in 6th grade and for those of you utilizing the CCSS, you will find multiplication and division with decimals starts in 5th grade.  Or you may have advanced students who are ready to explore this concept. There are a couple of freebies included in this post. Read on to find them!

Multiplying Decimals:

Typically we teach our students that when you multiply 2 numbers together, the product is larger than the 2 factors. And when we divide two numbers, the quotient is smaller than the dividend.

Be careful about stating this generalization: This remains true when multiplying whole numbers (or even a combination of whole and decimal such as 5.2 x 6.4 = 33.28 in which the product is larger than either factor), but NOT with decimals or fractions less than 1 (example: .7 x .2 = .14 in which .14 is less than either factor .7 or .2). With division of decimals less than 1, the answer is often a whole number larger than either the divisor or the dividend (such as 5 divided by 1/2 = 10). This is a difficult concept, but modeling and practicing with concrete and pictorial models helps to see the reasoning. So, be careful not to say, “When you multiply two numbers together you always get a bigger number.” because it’s not always true.

To further understand this principal, let’s look at the use of the times (x) sign with whole numbers. The problem 5 x 2 could be shown in an array as 5 rows with 2 in each row (phrased as 5 “rows of” 2).  It can be shown in a set model such as 5 groups and each group has 2 (phrased as 5 “groups of” 2) . It can also be shown in an area model (box) as a shape divided equally into 5 rows and 2 columns (phrased as 5 “by” 2). Continue reading

Multiplication Strategies Part 5: All Strategies Combined PDF

by C. Elkins, OK Math and Reading Lady

One of my subscribers really liked the multiplication strategies I posted in my 4 part series, but was finding it difficult to copy them for easy reference. So . . . I pooled pictures of the boards onto a single PDF (it’s 10 pages) and am providing it here (and also with the 4th part of the series). Click here for your copy: Multiplication Strategies PDF

Enjoy!!

Multiplication Strategies Part 4: Doubling and Halving (and Lattice)

by C. Elkins, OK Math and Reading Lady

Doubling and . . .

I hope you have had a chance to look at Parts 1, 2, and 3 of my multiplication strategy posts. These strategies are especially helpful with 3rd – 5th grade students (and beyond). I have been reading a book by Dr. Nicki Newton called “Guided Math in Action.” She discusses five components to being mathematically proficient. One of them is strategic competence.  What is strategic competence? The National Research Council defines it as “the ability to formulate mathematical problems, represent them, and solve them.” The first process std. in Common Core (Make Sense of Problems and Persevere in Solving Them) emphasizes strategic competence in this way: “they try special cases and simpler

Halving Multip. Strategy

forms of the original problem in order to gain insight into its solution, . . . students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? . . . and, they can understand the approaches of others to solving complex problems and identify correspondences between different approaches.”

With all of that said, I would like to show two other multiplication strategies: 1) Doubling and Halving, and 2) the Lattice Method. All of the strategies picture boards I have shown are available on this free PDF (It’s 10 pages): Multiplication Strategies PDF

Doubling and Halving
Doubling and halving is supported by the associative property. It also enables the student to use mental math strategies. Here is an example:  Original Problem: 25 x 12.  When changed to 50 x 6, I can solve it mentally which equals 300.  From what I have researched and applied, here are some tips:

  • Use when a problem has one even and one odd factor (or two even factors).  It won’t work with 2 odd factors because you can’t break an odd number in half and still get a whole number.
  • Double the odd factor and halve the even factor. Do this 1-3 times until you get two factors which can be multiplied mentally.
  • Most useful when the odd factor has a 5 in the one’s place (because 5 doubled is 10, so the result will be a multiple of 10 which can usually be taken care of mentally).

How is the above problem connected to the associative property?

  • Think of 25 x12 as 25 x (2 x 6).
  • If I apply the associative property, I get (25 x 2) x 6 = 50 x 6 = 300.
  • I can also go another step further and show that 50 x 6 = 50 x (2 x 3) = (50 x 2) x 3 = 100 x 3 = 300.
  • Here is another one: Think of 15 x 24 as 15 x (2 x 12) = (15 x 2) x 12 = 30 x 12 = 360.
  • See my pictures below of how that looks in array form with the problems 3 x 4 and 5 x 12.

Continue reading

Multiplication Strategies Part 3: Connecting to Place Value

by C. Elkins, OK Math and Reading Lady

In  Multiplication, Part 3  I will focus on 3 strategies for double digit numbers:  area model, partial products, and the bowtie method. Please also refer back to my Dec. 6th post on Number Talks for 3rd-5th grade where I mentioned these and other basic strategies for multiplication. I highly recommend helping students learn these methods BEFORE the standard algorithm because it is highly linked to number sense and place value. With these methods, students should see the magnitude of the number and increase their understanding of estimation and the ability to determine the reasonableness of their answer. Then, when they are very versed with these methods, learn the standard algorithm and compare side by side to see how they all have the same information, but in different format. Students then have a choice of how to solve. Try my “Choose 3 Ways” work mat as bell work or ticket in the door. Get it free here.

Area Model: This method can be illustrated with base ten manipulatives for a concrete experience. Remember the best methods for student learning (CPA) progresses from concrete (manipulatives) to pictorial (drawings, templates, pictures) to abstract (numbers only). Using a frame for a multiplication table, show the two factors on each corner (see examples below for 60 x 5 and 12 x 13). Then fill in the inside of the frame with base ten pieces that match the size of the factors. You must end up making a complete square or rectangle. This makes it relatively easy to see and count the parts: 60 x 3 and 5 x 3 for the first problem and (10 x 10) + (3 x 10) + (2 x 10) + (2 x 3) for the second. I’ve included a larger problem (65 x 34) in case you are curious what that looks like. The first 2 could be managed by students with materials you have in class, but I doubt you want to tackle the last one with individual students – nor do you probably have that many base ten pieces. A drawing or model would be preferred in that case. The point of the visual example is then to connect to the boxed method of the area model, which I have shown in blank form in the examples . . . and with pictures below. I also included a photo from another good strategy I saw on google images (sorry, I don’t know the author) which also shows 12 x 13 using graph paper. Continue reading

Multiplication Strategies Part 2: Decomposing and distributive property to learn facts

by C. Elkins, OK Math and Reading Lady

In part 2, I will show you some ways to help students decompose a multiplication problem into 2 (or more) easier multiplication problems. Most students know problems with factors of 2, 5, and 10. The decomposing will allow students to use what they know to work on the unknown / unmemorized fact.

I frequently see students struggle with solving an unknown multiplication problem. Often they choose skip counting, but if they miss just one number in the sequence, the answer comes out wrong. I also see them use their multiplication chart, but this doesn’t do much to help them apply number sense. Other times I see students draw circles with dots inside, but this is time consuming and it often becomes just a counting practice. This method is using the distributive property. Students can break apart one of the factors into “friendly” addends. I usually advise making one of the addends a 2, 5, or 10 since those are usually easier to compute or are already memorized. Here are some examples:

 

 

I have also attached a class activity sheet in which students cut out grids, glue them on the worksheet and then decompose them. Get it free here: Distributive property teaching chart  Another resource for teachers is my multiplication strategies guide which shows some ways to break down each factor’s family. Get it here free: Multiplication fact strategies chart  Finally, here is a link to a TPT source with a freebie for using the distributive property with arrays: Distrib. Property of Multip. freebie by Tonya’s Treats for Teachers

Have a great Spring Break for Oklahoma teachers!! I will be back in 2 weeks with more multiplication strategies.

Multiplication Strategies Part 1: Basic Strategies and skip counting

Are you looking for some ways to help your students learn the multiplication facts? Or ways to help them solve multiplication problems while they are in the process of learning the facts? One way is to skip count or repeatedly add the number over and over again. While this is one acceptable strategy, I see many students skip count using their fingers, often starting over numerous times. And if the child miscounts just one number in the sequence, then all of the remaining multiples/products are incorrect. Sometimes the student will write down the sequence in a horizontal row (better than using fingers in my opinion), but again – if they miss one number . . . all the rest of the numbers in their list are wrong.

What I want to show you today (in Part I of my series about multiplication strategies) are ways to relate the multiples/products in recognizable patterns which may facilitate recall and help with committing the facts to memory. Yes, students should also know the following about multiplication:

  1. Multiplication is repeated addition. For example: 3 x 4 means 3 groups of 4 or 4 + 4 + 4 = 12
  2. Multiplication is equal groups. 3 x 4 might be shown with 3 circles and 4 dots in each one. Be cautious about continued use of this one. Students are good at drawing this out, but then are they actually adding repeated groups or just counting one dot at a time. Observe students to see what they are doing. Transition to showing 3 circles with the number 4 in each one.
  3. Multiplication is commutative. If solving 7 x 2 (7 groups of 2), does the student count by 2’s seven times, or perhaps make it more efficient by changing it around to make it 2 x 7 (2 groups of 7 — and adding 7 + 7)?
  4. Multiplication can be shown with arrays. If students are drawing arrays to help solve, watch how they are computing the product. Are they counting one dot at a time? Or are they grouping some rows or columns together to make this method more efficient. I will focus on this one in Part 2.
  5. Multiplication can be shown by skip counting.

I have included 2 visuals to see some of my favorite ways to relate skip counting to unique patterns (for the 2’s, 3’s, 4’s, 6’s, 8’s, and 9’s). And another pattern below:

  • An even number x an even number = an even number
  • An odd number x an even number = an even number
  • An odd number x an odd number = an odd number

 

Next post will be Part 2 of Multiplication Strategies. Have a great week!

 

 

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

by Cindy Elkins, OK Math and Reading Lady

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.

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

Here are some possible Number Talk problems and solutions:

Enjoy your Number Talks!!