In this post, I will share some strategies for using concrete manipulatives and pictorial methods to solve **multiple digit multiplication** problems. By using these methods, students gain a better sense of **place value** as they work to **decompose** the problem into smaller units. Decomposing also allows a student to better perform mental calculations. Some helpful manipulatives: base ten materials (hundreds, tens, ones); place value disks; cups and pinto beans

What is the purpose of knowing multiple strategies? Some would argue that too many strategies are confusing for students. Some believe the only strategy needed is the standard algorithm. I believe teaching different strategies provides students with choices and improves analytical thinking. With only 1 strategy, if the “steps” are missed, the student has no other recourse. Student choice is a powerful motivator as well because they get a say-so in how they approach their own work.

I keep thinking about my past teaching when I only taught the standard algorithm (before I knew better). I recall saying: “Show all your work – because I said so.” This means I was not considering the students who were able to do some of the mental calculations in their head. I know I went through the steps in a robotic, don’t-question-me way: “Multiply the ones, carry to the ten’s place, multiply again and add the digit you carried. When multiplying the 2nd digit, be sure to watch the placement in the second row and scoot it over to the left one place.” None of this conversation (if you could even call it that) mentioned the place value relationship, what the carried digit represented, or why the second row of the answer should be scooted over one place.

**Here are some examples relating manipulative and pictorial methods with paper-pencil methods. I’ll use the problem 32 x 4. These methods help students use (30 + 2) x 4 to solve.**

- Base ten: Show 3 tens rods and 2 ones four times.
- Place value disks: Show three 10’s disks and two 1’s disks four times.
- Cups and beans: Each cup contains 10 beans. Ones are shown by individual beans. Show 3 cups and 2 beans four times.
- Pictorial drawings and decomposing models:
- Partial products: This is a great way to help student realize that the 3 represents 30.
- Area (box) model: Another ways to visualize and utilize place value knowledge to solve.

When it is time to introduce the standard algorithm, you can relate it to the partial products or area model. I always recommend showing both side by side so students now understand what the carried digit represents, and why the second row is scooted over to the left, etc. Then try some problems like this for your daily mental math number talks (show problem horizontally). I practically guarantee that students who can visualize the manipulatives or the partial products method will get the answer more quickly than those who are performing the std. algorithm “in the air.”

**I will take a break this summer and come back every now and then between now and August. Keep in touch! Enjoy your summer!!! Let me know if there are topics you’d like me to address on this blog.**

This is part 4 in a continuing series of posts about basic multiplication teaching concepts. Use them for beginning lessons or reteaching for struggling learners. Students could be struggling because they were not given enough exposure to concrete and pictorial models before going to the numbers only practices. **The focus in this post will be skip counting** to determine multiplication products. I will even focus on skip counting done in early grades (counting by 10’s, 5’s, and 2’s). Read on for **10 teaching strategies regarding skip counting**.

**I am going to give some of my opinions and misconceptions students have about skip counting.**

- Many students do not associate skip counting with multiplication, but just an exercise they started learning in KG and 1st (skip counting orally by 10’s, 5’s, and 2’s). This is often because they started with numbers only and did not have the chance to see what this looks like using concrete objects or pictorial representations.
- If you observe students skip counting, are they really just counting by 1’s over and over again? Or are they adding the number they are skip counting by repeatedly. You know the scenario. You tell a student to skip count by 3’s and they know 3, 6, 9, but then hold up their 3 fingers and count 10, 11,
**12,**13, 14,**15,**16, 17,**18,**and so on. Or are they truly counting like this: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30? - The main issue I have with skip counting is that if a student makes an error regarding just one of the numbers in the sequence, then the rest of the sequence is incorrect. So this should not be their only strategy. Do you recall a previous story I mentioned about the 5th grader who tried to solve 12 x 3 by skip counting on a timed facts test? He was unsuccessful because he kept losing track and didn’t have another strategy to use.
- Successful skip counting reinforces the concept that multiplication is repeated addition – do your students know this? I have witnessed many students who know the first 2-3 numbers in a skip counting sequence, but then don’t know how to get to the next numbers in the sequence.
- Students don’t often relate the commutative property to skip counting. Let’s say the problem is 5 x 8. The student tries skip counting by 8’s (because this problem means 5 groups of 8) and may have difficulty. Does the student try to skip count by 5’s eight times instead?

**Ten teaching strategies for skip counting:**

- For
**young students**skip counting, use objects to show how to keep track:- Base 10 rods
- Rekenrek (easily slide 5 or 10 beads at a time)
- Hand prints (for counting 5’s or 10’s): Which do you think would give students a better understanding: Holding up one hand at a time and counting by 5’s or lining up several children and having them hold up their hands as you continue counting? The second scenario enables students to see the total of fingers as opposed to just 5 at a time.
- Use money: nickels and dimes
- Associate counting by 2’s with concepts of even and odd

- Use
**manipulatives**. Do it often and with a variety of materials. The arrangements should emphasize the other strategies (equal groups, arrays, repeated addition). - Draw and label
**pictures**. The labels for this strategy would show the cumulative totals instead of just the number in each group. **Arrange students**in line or groups to practice skip counting. Example if practicing 4’s: Every 4th student turns sideways, every 4th student holds up their hands, every 4th student sits down. every 4th student holds a card with the number representing their value in the counting sequence, etc.- Practice skip counting while bouncing or
**dribbling a ball**. Great for PE class! - Associate skip counting with
**sports**: 2 and 3 pointers in basketball, 6 points for touchdowns in football, etc. **Use a 0-100 chart to see patterns**made when skip counting. I love the 0-100 pocket chart and translucent inserts that allow you to model this whole group. Individual 100 charts are readily available in which students can mark or color the spaces. Here are links to the chart and the translucent inserts: 1-100 pocket chart and Translucent pocket chart inserts- Look for other
**patterns regarding skip counting**. Refer to my previous post on this for more details: Skip counting patterns - Relate skip counting to
**function charts and algebraic patterns**using growing patterns. - Practice skip counting using
**money**: by 5’s, 10’s, 25’s, 50’s

**What strategies do you like for multiplication? What misconceptions do you see with your students? **

**Next post will be part 5 of my multiplication posts – and the last one for this school year. I will focus on using these basic concepts with double-digit problems. Stay tuned!!**

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Thanks for checking in on part 3 of my multiplication posts. Focus will be on the **equal groups** strategy — looking at how students can efficiently use this strategy to help learn basic multiplication facts. My angle will be at the conceptual level by using concrete and pictorial methods.

**Basics:**

- Instead of in array or area format, equal groups are
*separate*groups. - The “x” means “
*groups of.*” So 3 x 4 means “3 groups of 4.”

**What things normally come in equal groups? Conduct a brainstorming session.** I love the book “What Comes in 2’s, 3’s, and 4’s” as a springboard. After reading the book, let students brainstorm other things that come in equal groups. See the pictures below for some more ideas. After some internet research, I also made this attached list to use (in case you or your students draw a blank): click here: Equal groups pictures and list template

**Use these lists to help students generate stories about equal groups. When students can create (and maybe illustrate) their own stories, they are much better at solving problems they must read on their own. This also helps students think carefully about what in the story constitutes a “group” and what the “groups of” represents: **

- There were 5 bowling balls on the rack. If you count all of the holes (3 per ball), how many holes are there all together? (5 x 3). The bowling balls are the groups. The holes are what is being counted in each group.
- How many numbers are shown on 3 clocks? (3 x 12). The clocks are the groups. The numbers are what is being counted in each group.
- I bought 8 pair of earrings. How many earrings are there? (8 x 2). The pairs are the groups.
- Seven ladybugs were crawling on the leaves. How many legs would there be? (7 x 6). The ladybugs are the groups. The legs are what is being counted in each group.

**Ways to show equal groups with objects and drawings:**

- Hula hoops (great to use these in PE class to emphasize multiplication)
- Embroidery hoops
- Circles of yarn
- Dishes: cup, bowl, plate, tray
- Baskets
- Shelves

**Objects to use to show equal groups:**

- people
- cubes
- tiles
- mini erasers
- teddy bear manipulatives
- base ten materials
- food: pinto beans, macaroni, cereal, candy
- practically anything you have an abundance of!!

**Teaching concepts regarding equal groups:**

- When students are placing objects or drawing inside, do they randomly place objects? Or do they organize them to enable ease in counting? Showing students how to organize the objects in each set contributes to their knowledge of equal groups — AND it’s a big help to you as the teacher as you check on students. If the dots are randomly placed, the teacher and student must count one at a time to check. If they are organized, teacher and student can tell at a glance if the amount in each group is correct. Notice the difference below: Which ones show a student’s understanding of 9? Which ones can a student or teacher check rapidly?
- When counting the objects or drawings to determine the product of these equal groups, are students counting one at a time? Or are they counting in equal groups (such as by 2’s, 5’s, 3’s, etc.)? If we allow students to just count by ones, then they are not practicing multiplication, just counting!!

**Activities to practice equal groups strategy:**

- Circles and Stars: Roll a dice once. This is the number of circles to draw. Roll a dice again. This is the number of stars to draw inside. If played with a partner, students can keep track of their totals to determine a winner. Dice can be varied depending on the facts that need to be practiced. A spinner can also be used. (See picture at beginning of this post.)
- Variation of above: Use other materials (such as those listed above).
- Dice roll #1 = # of cups. Dice roll #2 = number of cubes
- Dice roll #1 = # of hoops. Dice roll #2 = # of pinto beans
- Dice roll #1 = # of plates. Dice roll #2 = # of Cheerios

- Write and illustrate stories: Provide a problem for students to illustrate (example: 6 x 3 or 3 x 6). Then each student can decide how to form the story and illustrate. I always tell students to choose items they like to draw to make their story. Here are some examples. See some examples from former students.
- There were 6 monsters in the cave. Each monster had 3 eyeballs. How many eyeballs all together?
- Six princesses lived in the castle. They each had 3 ponies. How many ponies in all?
- There are 3 plants in the garden. They each have 6 flowers. How many flowers are in my garden?
- I made 3 pizzas. Each pizza had 6 slices. How many slices of pizza did I make?

- PE Class activities:
- Try these story books about multiplication:
- What Comes in 2’s, 3’s, and 4’s by Suzanne Aker: Amazon.com link
- Each Orange Had 8 Slices by Paul Giganti: Amazon.com link
- One Hundred Hungry Ants by Elinor Pinczes: Amazon.com link
- Teaching Multiplication with Children’s Books list: Click HERE

- Equal groups story problems to solve: See my previous post related to this. You will find some story problem task cards and templates for solving multiplication and division problems using the equal groups strategy. Click HERE

**Enjoy!! **

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Last week I posted my thoughts about multiplication strategies using the repeated addition strategy. This time I will focus on using **arrays**. Do you have some arrays in your classroom? Look for them with bookshelves, cubbies, windows, rows of desks, floor or ceiling tiles, bricks, pocket charts, etc. Students need to know arrays are everywhere! It is also very helpful for students to build arrays with objects as well as draw them. This assists students with moving from concrete to pictorial representations — then the abstract (numbers only) can be conceptualized and visualized more easily. Some good materials for arrays:

- cubes
- tiles
- circular disks
- flat stones
- pinto beans (dry)
- grid or graph paper
- bingo stamper (to stamp arrays inside grids)
- mini stickers
- candy (Skittles, M&Ms, jellybeans)

**Array Basics:**

- Arrays form rectangular shapes.
- Arrays are arranged in
**horizontal rows**and**vertical columns**. This vocabulary is very important! - The number of objects in each row (and column) in an array are equal.
- Arrays can be formed by objects, pictures, or numbers.
- Arrays can be described using numbers: If there are 4 rows and 3 columns, it is a 4 by 3 array.
- The number of rows and number in each row are the
**factors**. The**product**is the total. - When an array is rotated, this shows the
**commutative property**.

**Ways to incorporate arrays into story problems:**

- Desks in a class (5 rows, 4 desks in each row)
- Chairs in a classroom or auditorium (10 rows of chairs, 8 chairs in each row)
- Plants in a garden (6 rows of corn, 8 corn plants in each row)
- Boxes in a warehouse (7 stacks, 5 boxes in each stack)
- Pancakes (3 stacks, 5 pancakes in each stack)
- Cars in a parking lot (4 rows, 5 cars in each row)
- Bottles of water in a crate (3 rows, 8 bottles in each row)
- Donuts or cupcakes in a box (how many rows? how many in each row)

**Activities to encourage concrete and pictorial construction of arrays:**

- Start off using manilla grid paper you probably have available with the construction paper supply at your school. This will help students keep their rows and columns even. Pose a problem and allow students to use manipulatives you have available to construct the array. If you say, “Build an array for this multiplication problem: 3 x 5,” do they know the 3 refers to # of rows and the 5 refers to the number in each row?
- Turn the paper after building the above array to see the commutative property. Now the picture shows 5 x 3 (5 rows with 3 in each row). The product is still 15.
- Use the manilla grid paper along with bingo dobbers to create the array. The grids can also be completed with mini stickers (I get them all the time in junk mail) or drawings.
- When using pictures of arrays, direct your students to always label 2 sides of the array (the rows and columns). Try to label different sides of the array so it’s not always presented in the same format.
- Find the product: The whole point of using an array as a multiplication strategy is to visualize the rows and columns to help calculate the product. If students create rows and columns and then just count the objects one-by-one, then this does
**not**accomplish the objective. Show students how to skip count using the # of objects in the rows or columns. Believe me, students don’t always know to do this without a hint from the teacher. Or better yet, before actually telling them to do this, ask students this question: “How did you get the total number of objects?” When you pose this question, you are honoring their strategy while secretly performing an informal assessment. Then when the student who skip counted to find the total shares their strategy, you give them the credit: “That is an efficient and fast way to count the objects, thank you for sharing! I’d be interested to see if more of you would try that with the next problem.” Plus now students have 2 strategies. - Use the distributive property to find the product: Let’s suppose the array was 6 x 7. Maybe your students are trying to count by 6’s or 7’s to be more efficient – but the problem is that counting by 6’s or 7’s is difficult for most students. Break up (decompose) the array into smaller sections in which the student
**can**use their multiplication skills. Decomposing into rows or columns of 2’s and 5’s would be a good place to start. This is the distributive property in action – and now the students have 3 strategies for using an array!! This is a great way to use known facts to help with those being learned.Here is a link to Math Coach’s Corner (image credited above) and a great array resource: Multiplication arrays activities from TPT $5.50. Here is my FREE guided teaching activity to help students decompose an array into 2 smaller rectangles. Click HERE for the free blank template. - Use the online geoboard I described a few posts back to create arrays using geobands. Click here for the link: Online geoboard Click here for the previous post: Geometry websites (blog post)
- Try these freebies: Free array activities from k-5mathteachingresources.com. Here’s a sample.
- Play this game I call “Block-It.” This is a competitive partner game in which students must create arrays on grid paper. Click here for a FREE copy of the directions: Block-It Game Directions
- Relate use of arrays when learning strategies for division and area.

**In a future post I will show some ways to use manipulatives and pictures arrays for double digit multiplication problems. Stay tuned!!**

The next few posts (until I take a break over the summer) will focus on the basic multiplication concepts one at a time. This will allow the opportunity to dig deeper into the concepts we want students to understand. This one will focus on the concept that **multiplication is repeated addition.** These posts will be helpful to teachers introducing multiplication to students in 2nd and 3rd grade as well as those in 4th, 5th, 6th and beyond who have missed some of these basic concepts. Future posts will focus on the area (array), set (equal groups), counting, and decomposing models as well as the associative and distributive properties.

Do your students know what the “times” sign means? They may hear it frequently, but not realize what it means. I like to interpret it as “**groups of**.” So a problem like 3 x 4 can be said as “3 **groups of** 4.”

To show repeated addition, that same problem would be 4 + 4 + 4 = 12.

**Repeated addition can be shown with numbers, and also with arrays and equal groups**. These pictorial models are great for developing multiplication concepts (and will be topics of future posts). However, when students are presented with these models they often count the individual pieces one at a time rather than adding the same amount repeatedly. Observe your students to see how they are counting.

Do your students apply the **commutative property** of multiplication? This means if the problem is 3 x 4, it can also be solved by thinking of 4 x 3 (which is 4 groups of 3 OR 3 + 3 + 3 + 3). I want students to know even though the answers are the same, the way the factors are grouped is different. When used in a story, 3 x 4 is a different scenario than 4 x 3.

Do your students practice repeated addition, by combining 2 or more numbers? See the following for an illustration of **15 x 6**:

Do your students apply the concept of repeated addition to **multiple digit multiplication** problems as well? I have witnessed students numerous times who only try a problem one way and struggle. For example, on a timed test I witnessed a 5th grader attempt the problem 12 x 3. I observed him counting by 3’s. He was trying to keep track of this by skip counting by 3’s twelve times. I could tell he had to start over frequently, thus spending a lot of time on this one problem. It became obvious he had no other strategy to try. He finally left it blank and went on. Just think if he had thought of 12 + 12 + 12. This should have been relatively easy for a 5th grader. He also could have decomposed it to this: (3 x 2) + (3 x 10).

Do your students always go to the **standard algorithm** when they could perhaps mentally solve the problem by repeated addition? If the problem was **50 x 3**, are they thinking 50 + 50 + 50? Or are they using paper-pencil and following the steps?

What about a problem such as** 45 x 4**? Using repeated addition, is your student thinking of 40 + 40 + 40 + 40 combined with 5 + 5 + 5 + 5? This is then solved as 160 + 20 = 180.

**Students who are able to use repeated addition skillfully are showing a healthy understanding of place value and multiplication. This strategy also enhances mental math capabilities. Conducting daily number talks are highly advised as a way to discuss multiple ways to solve a given problem such as those mentioned above. Check out “Number Talks” in my category list for more information on this. Also check out some recommended videos about conducting number talks (above black bar “Instructional Resources”).**

There are several great math websites which might help you and your students with geometry and measurement standards such as area, perimeter, volume, surface area, angles, etc. The ones I am recommending are interactive and often customizable. Check them out!! (Each title can be clicked to take you directly to the linked website.)

- Geoboard by The Math Learning Center: I love the concept of geoboards to help children create polygons and measure area and perimeter. However, most teachers have ditched their physical geoboards. They are often in boxes relegated to the basement storage areas. I get it, though. They take up a lot of shelf space in the class, there aren’t enough rubber bands to go around (aka geobands), the kids misuse them or break them, they don’t stretch far enough, the pegs get broken, etc.

I think you will LOVE this app. Check out the little “i” on how to get the most use out of it, but it has 2 variations for the board size and you can show it with/without gridlines or numbers. There are different colored bands which you drag to the board and stretch to whichever pegs you need. You can shade in areas, copy, and rotate (which is helpful to see if 2 similar shapes are equivalent). There is also a drawing palette in case you want to freehand something or draw lines (and with different colors as well).

**What are the possibilities with this?**

- Use with primary students to
**create**squares, rectangles, and other**polygons**. The teacher can elicit different responses with directions such as: Make a square. Make a different size square. Make a trapezoid. Are any of our trapezoids the same? - Creations can sometimes be recorded on dot paper – although I would reserve this for less-complicated shapes.
- Count the pegs around the shape to determine
**perimeter**. The teacher might ask students to create a rectangle with a perimeter of 10 (or 12, or another amount). How many different ways are there? Be cautious with diagonal connections because they are not equivalent to vertical or horizontal connections. Think of how you can get students to discover this without just telling them. - Show the gridlines to help students determine
**area.**Initially, students may just count the squares inside the shape. Guide students to more efficient ways to figure this (multiplying, decomposing into smaller sections, etc.). - This app is also great for creating irregular shapes in which students may decompose into smaller rectangles or triangles. Then check them with the standard formulas.

2. “Cubes” at NCTM’s site (Illuminations): This one is perfect for **volume and surface area.**

**Volume**: You can use the gear symbol to select the size (l, w, and h) of the rectangular prism, or use the default ones shown. Then there are 3 tools used to fill the rectangular prism: individual cubes, rows of cubes, or layers of cubes. I prefer using the layer tool to support the formula for volume as: area of the base x height. The base is the bottom layer (which can be determined by looking at the length x the width). The height is the number of layers needed to fill the prism. Once you compute the volume, enter it and check to see if it is correct.**Surface Area of Rectangular Prism**:- Area of the front and area of the back are the same
- Area of the top and area of the bottom are the same
- Area of each side is the same
- Be sure to explore what happens when the prism is a cube.

3.Surface area with Desmos: This link provides an interactive experience with surface area, using a net. This time, the three visible faces of the prism are color coded, which helps with identifying top / bottom; front / back; and side / side. The prisms on this site are also able to be changed so students can see how altering one dimension affects the surface area.

6. “Plane Figures” on GeoGebra

These three may be more relevant to middle school math standards. Check them out!! Also take a look at the “Resources” link (left side of web page). There are plenty of other good links for arithmetic standards as well – too many to list here. You may have to create a log-in, but it’s FREE!

**Enjoy! Do you have other websites to recommend? Let us know.**

Here are some cool graphic organizers for your math files! Make sets of them, laminate or put in plastic sleeves, and use them over and over again! Graphic organizers help students stay organized and teach them how to complete problems neatly. They are also a great way for students to show different strategies for the same problem. While primary students may need an already-made graphic organizer, intermediate students should be taught how to duplicate them on their own to use whenever the need arises – so the simpler, the better! With repeated use, students are more likely to utilize them regularly in their daily work (and on their scratch paper with assessments).

This one has **ten frames and part-part-whole models**. In my opinion, these are essential when working with K-2 students because they help children with subitizing, number bonds, and addition / subtraction facts. If you are using Saxon, you are missing these important strategies!!:

Here’s one to show **fractions** (area, set, length models)

Need a template for students to make **arrays**? This one is ready! I love showing students how to break an array into smaller parts to see how multiplication (or division) facts can be decomposed. Example: Make a 6 x 7 array. Section off a 6 x 5 part. Then you have a 6 x 2 part left over. This proves: 6 x 7 = (6 x 5) + (6 x 2). Or — 6 x 7 = 30 + 12 = 42

This graphic organizer shows **5 different multiplication strategies** using 2 digit numbers, and a blank one for students to record their thinking. Very handy!! One of my favorite strategies is partial products. I highly recommend this one before going to the std. algorithm because students decompose the problem by place value and must think about the whole number and not just the parts.

Do your students need something to help them see the different **models for a decimal**? Try out this graphic organizer. Students will utilize the pictorial forms as well as the abstract.

Do your students know that .7 (or 7/10) is the same as .70 (or 70/100)? Using this dual set of **tenths and hundredths grids** will help them see why this is true!

**Be sure to check out my FREE templates and organizers (see black bar above “links . . .”) Please share your favorite graphic organizers for math! Enjoy!!**

I highly recommend the use of graphic organizers. The purpose is to help students organize information with regard to different text structures:

- Compare and contrast
- Cause and effect
- Details / Descriptive
- Problem and solution
- Sequence

Graphic organizers are also helpful with standards such as:

- Main idea
- Summarizing
- Character analysis
- Story elements

Graphic organizers help organize the student’s thinking, and assist with note-taking. The visual pictures created help the student “see” the text structure, recall details, state the main idea, and summarize the selection.

**Here are links to some sites I think provide good quality graphic organizers which can be utilized with a variety of situations:**

- This one is more primary oriented: https://www.eduplace.com/graphicorganizer/
- This one is oriented more for 3rd and above: http://www.educationoasis.com/printables/graphic-organizers/
- This one is a FREE resource at TPT (as pictured above) that supports each of the 5 text structures: https://www.teacherspayteachers.com/Product/Non-Fiction-Text-Structures-Flip-Flap-and-Graphic-Organizers-Freebie-1777102

I have also linked these in “Instructional Resources” and in the categories list on my blog. ** Enjoy!!**

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:

**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.**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?**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.**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!**

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:**

- 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. - 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.
- 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.
- 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.**