multiplication

Place Value Part 4 — Multiplication

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by C. Elkins, OK Math and Reading Lady

In this post, I will focus on some strategies for multiplying double digit numbers (whole and decimal):  using manipulatives, the area model, partial products, and the bowtie method.  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 as well as 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 (or to use two strategies to check their answers). Try my “Choose 3 Ways” work mat as bell work or ticket in the door. Get it free here.

Using Manipulatives:  Using base ten pieces is helpful to see that multiplication means equal groups. Students build understanding at the concrete and pictorial levels first. The following free virutal manipulatives are from didax. The good thing about these is you don’t have to worry about having enough pieces for everyone!!:  https://www.didax.com/math/virtual-manipulatives.html

Area Model: This method can be illustrated with base ten manipulatives for a concrete experience. 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 using Ten Frames or Base Ten

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by C. Elkins, OK Math and Reading Lady

Yes, you can even use ten frames to teach multiplication concepts! Here are my mini ten-frames with dot cards from 1 – 10:  Click HERE to get a free copy. These are helpful to use, especially if you don’t have enough tens/ones blocks . . .  or you prefer manipulatives that are slightly easier to manage. These provide a strong connection to place value, and the commutiative and distributive properties.

I recommend two sets of the cards (1-9) per student. Each set has multiple copies of the same number. They can be laminated, cut, and placed in a baggie for ease in handing out and storage.

Multiplication Examples:

  1. Single digits (basic facts): 
    • For the problem 3 x 6, the ten frame is really helpful for the student to see 3 x 6 is almost like 3 x 5 with one more group of 3 added on (by being familiar with the fact that the top row on a ten frame is 5).
    • Because of the commutative property, I know these two facts will have the same answer. But which of these below do you think might be “easier” to solve? Students don’t often know they have a choice in how they can use the numbers to their advantage!
      6 x 3 (six groups of 3)
      3 x 6 (three groups of 6)
  2. Double digit x 1 digit:
    • Use of these also provides a strong connection of place value and multiplication. Notice how students can see the breakdown on the 4 x 12 problem (4 groups of 12 = 4 x 10 plus 4 x 2). Great introduction to the distributive property of multiplication!
    • Here is where application of the commutative property also comes in handy. Which of the methods below would you rather use to solve: count by 4’s or count by 12’s? Again, show students how to use their strengths to decide which way to think about solving the problem.
    • Even though the number of total pieces might seem to be a little overwhelming, it definitely is worth the effort for a few lessons so students get a visual picture of the magnitude of the products.
  3. Here are other ways to model multiplication problems with manipulatives like base ten rods or base ten disks.

After students get practice using these manipulatives (concrete), then proceed to pictorial models to draw them as simply as possible.  This will give them a good foundation to apply to the abstract (numbers only) problems.  I always pitch for the CPA progression whenever possible!!!

I will pause a while for the summer and just post once a month until school starts up again.  Take care, everyone!  But please don’t be shy.  Post your comments, ask your questions, etc.

Multiplication strategies — Equal groups

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by C. Elkins, OK Math and Reading Lady

Thanks for checking in on another multiplication strategy! The focus for this post 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. Be sure to see the links at the end for books and my free equal groups story cards.

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:  

  1. 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.
  2. 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.
  3. I bought 8 pair of earrings. How many earrings are there? (8 x 2). The pairs are the groups.
  4. 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:

  1. 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.)
  2. 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
  3. 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?
  4. PE Class activities:  If your PE teacher likes to help you with your learning objectives, let them know you are working on equal groups strategies. While I’ve not done this personally, I think having relay races related to this would work perfectly. For example, the teacher presents a problem and each team must use hula hoops and objects to show the problem (and the answer).
  5. Try these story books about multiplication:
  6. Equal groups story problems to solve:  Here are some story problem task cards and templates for solving multiplication and division problems using the equal groups strategy. Click to see the blog post on equal groups story problems and get my FREE set of story problem cards:  HERE

Enjoy!!  Many of you are now off for a well-deserved summer break. Use your summer time to catch up on my postings from this past year, or email me for more information. Also, feel free to comment on any article about your experience or additional tips. 

Multiplication strategies — using arrays

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by C. Elkins, OK Math and Reading Lady

In the last post, I shared 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:

  1. Arrays form rectangular shapes.
  2. Arrays are arranged in horizontal rows and vertical columns.  This vocabulary is very important!
  3. The number of objects in each row (and column) in an array are equal.
  4. Arrays can be formed by objects, pictures, or numbers.
  5. Arrays can be described using numbers:  If there are 4 rows and 3 columns, it is a 4 by 3 array.
  6. The number of rows and number in each row are the factors. The product is the total.
  7. 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?  Starting off using this graph paper may help when students freehand their own array (to help keep rows and columns lined up and not going astray).
  • 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 in a previous post 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!!

Multiplication: Repeated addtion

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by C. Elkins, OK Math and Reading Lady

The next few posts will continue to 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, decomposing, and doubling/halving models as well as the associative and distributive properties.  Freebies below!!

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. . . and encourage counting in equal groups to promote a growth of the multiplication mindset.

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 (using his fingers). 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.

Here are a few resources (FREE) that might help with this strategy:

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”).

Multiplication — Developing an understanding

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by C. Elkins, OK Math and Reading lady

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 – and I will focus on all of these in future articles:

  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 facts have interesting relationships — stay tuned for future posts or check out my blog archives for more
    • 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
    • 2’s, 4’s, and 8’s are related
    • 5’s and 10’s are related
    • 3’s, 6’s, and 9’s are related
  6. Multiplication can be shown by skip counting.  Aside from my comments above about errors students make with skip counting, there are some ways to arrange skip counted numbers in distinctive organized groupings so the patterns become more noticeable, perhaps leading more to memorization than just a horizontal list of numbers, or using fingers.
    • 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). Visualizing and explaining the patterns is a good exercise for the brain.  Can your students come up with another way to visualize the patterns with these numbers?

 

Stay tuned for more blog entries about multiplication!

Interactive math lessons and activities on NCTM

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Review by C. Elkins, OK Math and Reading Lady

Resource – http://illuminations.nctm.org

This is a math resource I absolutely love! It is a product of the National Council for the Teachers of Mathematics (NCTM).

This site includes lesson plans and interactive activities. Search in several ways: by topic, by standard, or by grade level. Need some strategy games? Check out “Calculation Nation” (some of which can be played against other players), and “Brain Teasers.” I have just added this link to my Resources page (on my blog home page).  Pass this along to parents for them to use with their children at home!

Many of the lessons connect to exploration projects and literature. The interactive features are outstanding!! These are perfect for the smartboard, on laptops, or tablets. Doing Zoom lessons? Then these are also wonderful for sharing the screen to introduce or review concepts. Once you are on the home screen, click the Interactives box (right side) and then the desired grade level. There are dozens of great applets, but here are a few you might really like. I have linked them for easy reference, so just click on the  title and you’ll be there:

Dynamic Paper: Customize graph paper, number lines, spinners, nets, number grids, shapes (to include pattern blocks, color tiles, and attribute blocks), and tessellations. You can also choose inches or cm. These can be customized, saved and printed as jpeg or pdf. I created the spinner shown here from this application.

Five Frame and Ten Frame tools: Geat activities to build number sense using five or ten frames. These may take 1-2 minutes to load.

Cubes: Build a rectangular prism one cube, or row, or layer at a time and then compute the volume or surface area.

Coin Box: Drag and exchange coins. There is also a feature I like (the grid at the bottom right corner), which puts coins in blocks (by 1s for pennies, 5s for nickels, 10s for dimes, and 25s for quarters). This really helps see the value of the coins. Want more info about coin blocks? Once on the Coin Box page, click on the “Related Resources” tab.

Try these for fractions: Fraction Models (which includes decimal and percent equivalencies) and the Fraction Game.

Geometric Solids: Create a shape (either transparent or solid) and swivel it around to see all of the faces, vertices, and edges. It has a cube and pyramid as far as basic 3D shapes are concerned. I wish it had more that 3rd-5th students would encounter.

Here’s a nice multiplication game:  Product Game  Two players (or a player vs. the computer) choose factors from the bottom bar to create products shown on the game board grid to get 4 in a row (and try to block your opponent from getting 4 in a row).  Be sure to see the directions included.

Some of the interactives require an NCTM subscription.  The ones I have listed above should be okay to access without membership. I have subscribed for years and just paid $94 for this coming year. Well worth it if you plan on using their site extensively.  This subscription also entitles you to a print and and online journal, blog capabilities, and more.

Enjoy these and so many more!!! Let us know if there are others you recommend.  I’ll highlight more on my next post.

 

Multiplication facts: What happens when students don’t or can’t memorize them?

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by C. Elkins, OK Math and Reading Lady

If you teach 3rd and above, I am positive you have students who have not memorized their multiplication facts. So what do they do to try to get the answer? From my experience, most students seem to know that repeated addition, drawing equal groups or arrays, and skip counting are strategies to try. I do believe those are very helpful for students to conceptualize what multiplication is all about. But here is what is frustrating:

Let’s say the problem is 6 x 7:

  • Do they write 7 + 7 + 7 + 7 + 7 + 7 and then add each part at a time? Or a little better, do they add 7 + 7 three times?
  • Do they draw a picture such as 6 circles with 7 items inside each one? The main difficulty with this is most students using this method count each object one at a time, making this a counting practice, not multiplication practice.
  • Do they draw an array? If so, do they correctly line up the rows and columns? Do they count each item in the array one at a time? Or do they group some together (which is a little better because they are at least thinking of equal groups)?
  • Do they skip count by fingers or write the sequence on paper? And what happens then? They may start off okay with 7, 14, 21 and then repeatedly count 7 fingers to get to the next number (21 + 7 = 28, then 28 + 7 = 35,  then 35 + 7 = 42, etc.).

With all of these strategies, students can get the correct answer, but they are often not really even using multiplication. Their method is often counting the objects in each group one at a time.  And when skip counting, if just one number is missed in the sequence then the total is obviously off. In addition, students often spend so much time with each of these that they get frustrated and give up.

In previous posts, I mentioned different ways for students to skip count while focusing on the patterns numbers make (Click HERE) and ways to use arrays to break it down into smaller equal groups (Click HERE).  So those methods are a little more productive toward using multiplication than the above. Today, though, I will steer you toward a unique strategy which does the following:

  • Allows students to use readily known facts (especially the 5s and 2s)
  • Adds a pictorial component which builds on subitizing, number sense, and decomposing of numbers
  • Applies the distributive property so students are using multiplication and addition together

The strategy modeled here is based on facts students already know. This is likely to be different among your students. Some will say they are great with their 4s or 3s. But most students I work with are proficient with their 5s and 2s (and can skip count quickly and accurately if they haven’t memorized these). So a lot of the problems shown will focus on use of 5s and/or 2s.

Again, let’s look at 6 x 7.  The student doesn’t know their 6’s and doesn’t know their 7’s. So we will decompose 6 or 7 to include a group of 5’s, which is known (I’ll show both ways).

  1. Decompose 6:  Six is made up of a group of 5 and a group of 1.  This is a pictorial method to build on subitizing using a dot pattern to show 5 and 1 (similar to a domino piece).
  2. See how the connection to the familiar ten frame can illustrate 7 x 6 (7 groups of 6) in this manner.
  3. Condense this concept to this representation which shows 7 x 5 plus 7 x 1 (35 + 7 = 42)

To see 7 decomposed instead of 6: Seven is made up of a group of 5 and a group of 2.

  1. See what this looks like on a ten frame to illustrate 6 x 7 (6 groups of 7):
  2. Condense to the “domino piece.” This shows 6 x 5 plus 6 x 2 (30 + 12 = 42):

Click on this link Multiplication Strategy pictorial CE for a FREE copy of the pictures above and below which are used in this post (for easy reference later). Here are a few more examples. Some use 5s and 2s, while others will show other combinations using 3s or 4s. The use of dots instead of numbers inside the “domino” is suggested to keep it a little more pictorial and less abstract. Plus, it builds on knowledge of subitizing (which is recognizing quantity without physically counting). Numbers alone can certainly be used, but the quantity of numbers might frustrate some students.

 

Practice activity:

  • Use a set of dominoes and digit cards 1-9. Turn over 1 domino and 1 digit card. Write the problem and then the decomposed version. See photo for example. Click on this link Digit cards 0-9 for a FREE copy of the digit cards.I’d love to hear if you are able to try this with your students. Let me know if it helps. I have worked with a couple of classes so far with this and they have loved it.  It opened a lot of eyes!!

Have a great week!

C. Elkins, OK Math and Reading Lady

Ten Frames Part 4: Multiplication

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by C. Elkins, OK Math and Reading Lady

Yes, you can even use ten frames to teach multiplication concepts! Here are my mini ten-frames with dot cards from 1 – 10:  Click HERE to get a free copy. These are helpful to use, especially if you don’t have enough tens/ones blocks . . .  or you prefer manipulatives that are slightly easier to manage. These provide a strong connection to place value, and the commutiative and distributive properties.

I recommend two sets of the cards 1-9 per student. Each set has multiple copies of the same number. They can be laminated, cut, and placed in a baggie for ease in handing out and storage.

Multiplication Examples:

  1. Single digits (basic facts): 
    • For the problem 3 x 6, the ten frame is really helpful for the student to see 3 x 6 is almost like 3 x 5 with one more group of 3 added on (by being familiar with the fact that the top row on a ten frame is 5).
    • Because of the commutative property, I know these two facts will have the same answer. But which of these below do you think might be “easier” to solve? Students don’t often know they have a choice in how they can use the numbers to their advantage!
      6 x 3 (six groups of 3)
      3 x 6 (three groups of 6)
  2. Double digit x 1 digit:
    • Use of these also provides a strong connection of place value and multiplication. Notice how students can see the breakdown on the 4 x 12 problem (4 groups of 12 = 4 x 10 plus 4 x 2). Great introduction to the distributive property of multiplication!
    • Here is where application of the commutative property also comes in handy. Which of the methods below would you rather use to solve: count by 4’s or count by 12’s? Again, show students how to use their strengths to decide which way to think about solving the problem.
    • Even though the number of total pieces might seem to be a little overwhelming, it definitely is worth the effort for a few lessons so students get a visual picture of the magnitude of the products.
  3. Here are other ways to model multiplication problems with manipulatives like base ten rods or base ten disks.

Continue reading →

Multiplication Concepts Part 5: Multiple digit strategies

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by C. Elkins, OK Math and Reading Lady

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.

  1. Base ten: Show 3 tens rods and 2 ones four times.
  2. Place value disks: Show three 10’s disks and two 1’s disks four times.
  3. Cups and beans: Each cup contains 10 beans. Ones are shown by individual beans. Show 3 cups and 2 beans four times.
  4. Pictorial drawings and decomposing models:
  5. Partial products: This is a great way to help student realize that the 3 represents 30.
  6. 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.

Multiplication Concepts Part 4: Skip Counting

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by C. Elkins, OK Math and Reading Lady

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:

  1. 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
  2. Use manipulatives.  Do it often and with a variety of materials. The arrangements should emphasize the other strategies (equal groups, arrays, repeated addition).
  3. Draw and label pictures. The labels for this strategy would show the cumulative totals instead of just the number in each group.
  4. 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.
  5. Practice skip counting while bouncing or dribbling a ball. Great for PE class!
  6. Associate skip counting with sports:  2 and 3 pointers in basketball, 6 points for touchdowns in football, etc.
  7. 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

     

  8. Look for other patterns regarding skip counting. Refer to my previous post on this for more details: Skip counting patterns

     

  9. Relate skip counting to function charts and algebraic patterns using growing patterns.
  10. 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!!

 

Multiplication Concepts Part 3: Equal Groups

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by C. Elkins, OK Math and Reading Lady

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:  

  1. 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.
  2. 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.
  3. I bought 8 pair of earrings. How many earrings are there? (8 x 2). The pairs are the groups.
  4. 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:

  1. 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.)
  2. 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
  3. 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?
  4. PE Class activities:  If your PE teacher likes to help you with your learning objectives, let them know you are working on equal groups strategies. While I’ve not done this personally, I think having relay races related to this would work perfectly. For example, the teacher presents a problem and each team must use hula hoops and objects to show the problem (and the answer).
  5. Try these story books about multiplication:
  6. 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!!  

 

Multiplication Concepts Part 2: Arrays

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by C. Elkins, OK Math and Reading Lady

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:

  1. Arrays form rectangular shapes.
  2. Arrays are arranged in horizontal rows and vertical columns.  This vocabulary is very important!
  3. The number of objects in each row (and column) in an array are equal.
  4. Arrays can be formed by objects, pictures, or numbers.
  5. Arrays can be described using numbers:  If there are 4 rows and 3 columns, it is a 4 by 3 array.
  6. The number of rows and number in each row are the factors. The product is the total.
  7. 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!!

Multiplication Concepts Part 1: Repeated Addition

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by C. Elkins, OK Math and Reading Lady

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”).

Math Problem Solving Part 5: Multiplication and Division Comparisons

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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 Story Problems

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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 →

Multiplication Strategies Part 3: Connecting to Place Value

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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

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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

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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). Visualizing and explaining the patterns is a good exercise for the brain.  Can your students come up with another way to visualize the patterns with these numbers? Here’s 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

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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!!