Place value understanding is critical when solving addition, subtraction, multiplication, and division problems. Add fractions and decimals to the mix too! This post will focus on building on parts 1 and 2 (counting and composing and decomposing numbers by place value) to add and subtract whole numbers and decimals.

**Virtual Manipulatives for Place Value:**

Here is one of my favorite free websites for all kinds of virtual manipulatives. You will defintely like the base ten and place value disks. Both of these include options for decimals. The 0-120 grid is also great! You can add different colors to the numbers using the paint can.: https://www.didax.com/math/virtual-manipulatives.html

With the above manipulatives, students can practice these basics concretely and pictorially; then transition to solving them mentally:

- 10 + single digit such as 10+7 = 17, 10 +3 = 13
- Multiple of 10 + single digit such as 20 + 4 = 24, 40 + 8 = 48
- Multiple of 100 + single or double digit such as 100 + 5 = 105, 200 + 30 = 230, 500 + 25 = 525
- 1 more, 10 more 100 more as well as 1 less, 10 less, 100 less
- Add to numbers with 9’s such as 90 + 10, 290 + 10, 1900 + 100

**Addition and Subtraction:**

- Decompose and then add or subtract
- Partial sums
- Rounding

Next post will be ways to emphasize place value when solving multiplication and division problems. If you have some ideas to share, please feel free to comment.

I appreciate all of my faithful followers the past 5 years! Thank you for viewing and passing this along to other teachers or parents. I hope you all have a restful holiday break!!

]]>Students know how to count orally, can subitize, can count on one more, know ordinal positions, and have cardinality (know that the last number counted **is** the number of objects). These are usually prerequisites with number sense *before* introducing the symbolic representation and base ten understanding. Our system is based on the repeated groupings of ten. This is a follow up from my last post (Place Value Part 1 — counting).

**There are two levels of understanding place value symbols**.

- Place value: In the number 23, the 2 has a place value representing the tens place.
- Face value: With 23, the 2’s value is 2 tens or 20.

This shows 23 and 32 are not the same amount even though they use the same digits. Conversely, this same understanding applies to decimals. For example, .4 is the same value as .40.

Here are some ways to help students develop place value competence. Below this section are activities to help teach and practice:

**Organize objects into groups when counting.**- If given a group of 72 objects, do students try to count them one at a time (and probably losing count somewhere along the way)? Or, can they make specific size groups such as 10’s to see they have 7 groups of 10 and 2 extra?
- If given a group larger than 100 (like 134), do they note that 13 groups of ten is the same as 130?

**Partition numbers into groups based on powers of ten (ones, tens, hundreds).**- Students learn that 52 = 5 tens, 2 ones = 50 + 2
- Students learn that 348 = 3 hundreds, 4 tens, 8 ones = 300 + 40 + 8
- Students learn that .45 = 4 tenths, 5 hundredths = .4 + .05

**Realize the relationship among the different places. Using the number 67 . . .**- Most frequently it is represented as 6 tens, 7 ones.
- But it can also be represented as 5 tens, 17 ones (which by the way is crucial to understanding the regrouping process for subtraction. Show your students a problem in which regrouping is needed for 67, with the result of 5 tens 17 ones. Ask them if 5 tens, 17 ones = 67? How many think no?).
- 67 can also be represented as 4 tens 27 ones, and so on.

A clear understanding of these three guiding concepts of place value are very important and you will do well to thoroughly help students experience them before embarking on number operations.

Some activities to help with the above:

**Organizing**:

- Provide objects for students to count: beans, cubes, tiles, candy, popcorn kernels, etc. Depending on the size of the objects, give directions such as “take a handful” or a spoonful, or a cupful. Make it a competition with a partner, who has more? who has less? Provide different containers for students to compare which holds more or less (and thus, working a little on measurement standards and conservation of space understanding).
- Use base ten ones units. Give an amount and after making piles of tens, have student trade each pile for a tens rod. How many tens? How many ones?

**Partitioning**:

- Build given numbers with base ten pieces. Example: “Build 47”
- Match pictures with expanded form using task cards.
- I recommend using place value pieces for KG and 1st graders because the pieces show the relationship (a tens rod is 10 ones units stuck together, the 100’s flat shows ten 10s and 100 units). For 2nd and up, it might be more feasible to use the place value disks because these allow students to work with larger amounts. I know I never had enough hundreds’ flats to build larger numbers. Here’s a link to place value disks on Amazon: Place value disks
- Use place value number strips that layer: 2000 + 500 + 30 + 8 when laid on top of each other shows the number 2,538. See sample here at Amazon: Place value layered strips
- Work on mental math thinking of adding tens and ones: 10 + 2 = 12, 30 + 5 = 35, 70 + 6 = 76. I often use this technique to show visually what we want our brain to do. These are tens and ones layered strips I made, available here for
**free: Digit cards 0-10 and 10-100**

**Relationships**:

- Practice making exchanges with place value pieces. Build a number the standard way (67 = 6 tens, 7 ones). Then exchange a ten for ten ones. Verify that 5 tens 17 ones = 67.
- How many ways can you show this number? (Give a number and provide manipulative for students to check and list.)
- Using base ten rods and flats, verify that 15 tens = 1 flat + 5 tens (100 + 50)
- Show how number changes by changing the ones or the tens.
Use a 100 chart. Select a number and find what ten more or ten less would be — also 1 more, 1 less. Use the chart to show ways to count by tens other than the traditional 10, 20, 30, etc. Try 27, 37, 47, 57. Practice counting forward as well as backwards.

**Important tip when using base ten manipulatives **(from personal experience):

When students are using base ten pieces, don’t allow random placement of the ones on their work mat. Why? This often means a student may not have the correct amount because they miscounted their “mess.” It also means the teacher has to spend more time when circulating the room to count their “mess” to check for correctness.

All it takes is noticing the student(s) who likes to organize their ones pieces. Their 5 pieces look like the dots on a dice or how it might look on a ten frame, or the 9 ones are in 3 rows of 3. As the teacher, you can show others how this is so helpful to you (and them) to arrange them in some kind of order. Knowing they get some choice in the order is key. And noticing how different students show the same amount in different ways is an ego booster to most students. Finally, it really does help boost students’ understanding of number bonds when they show 5 can be 4 and 1, or 2 and 3, or 2 and 2 and 1. That’s a two for one, right?

Enjoy your place value lessons — and share some you think would also be helpful!

Take care, stay safe!

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Place value is such an important math concept from KG and up. It starts with counting and recognizing amounts and in later grades plays a huge part in composing and decomposing numbers, multiplication, division, decimals, and yes . . . even fractions. Students need a solid understanding of it to estimate and compare numbers as well. Stay tuned for ideas and **freebies** below.

If you look up the definition of *place value* in a dictionary or math glossary it’s likely to refer to “the value of the place, or position, of a digit in a number.” But I think young students start off with a different understanding of numbers and may become confused. Does a child age 2 see their age and a quantity of 2 cookies in the same way?What about these numbers? – – Should I interpret these in terms of ones, tens, hundreds, tenths, hundredths, etc.? Does place value apply to these?

- telephone number: 123-456-7890
- address numbers: 1234 Happy Lane
- zip codes
- # on a sports jersey
- identification numbers (on badges, Social Security, etc.)
- # on a license plate

The examples above are actually referred to as *nominal* or *nonnumeric* because they are used for identification purposes and rarely have any meaning associated with place value. For example, the address above (1234) is most likely NOT the one thousandth plus house on Happy Lane.

So on the way to understanding place value, let’s look at **ways numbers are classified and the basic heirarchy even before we expect them to use the written notation for numbers:**

- Rote counting: saying numbers in sequence
- Counting objects: using a 1 to 1 correspondence between number and quantity. You may have to teach
*how*to keep track of counting objects like sliding them to the side when counting, or marking pictures with checks or circles as they are counted on paper. - Subitizing: recognizing a quantity without counting (accomplished using ten frames, dot cards, dice dots, a Rekenrek, tally marks). See my other blog posts on subitizing for more info and resources.
- Cardinality: associating the last number named when counting as the quantity of the set. After a child counts a set of objects, ask him/her this: “How many ___ are there?” Can they name the amount without recounting?
- Naming the next number in the sequence: Give a child a set to count. After announcing the amount, add one more object to see if they can name it — or do they start over and recount? Cardinality and naming the next number are needed in order to practice the skill of
*counting on.* - Concept of zero: To a young child this means “nothing.” With place value it can be a place holder within a larger number.
- Ordinal positions: learning terms such as first, second, third . . . which don’t even sound like the numbers one, two, three, . . .
- Part-Whole relationship: recognizing that quantities can be decomposed different ways. With 5 objects, can student show different combinations such as two and three, four and one, five and zero. I often refer to this as number bonds.

The message with today’s blog is to make sure young children have a firm understanding of the above *before* use of number symbols and teaching about “tens and ones.” I relate this to reading: Students need to develop phonological awareness about the *sounds* of letters and words before associating with the printed form (which is the study of phonics).

**How do you accomplish the above?**

- Lots of exposure to classroom manipulatives
- Oral counting practice (even in poems and songs)
- Match objects one to one. Place objects on top of dots on dot cards and count as you go, or Match # of objects from one picture card to objects of another picture card.
- Make designs. Example: “Using your color tiles, what design can you make with ten pieces?”
- Use ten frames and dot cards during Number Talk sessions (flash quickly and discuss how the quantity is seen). Example — If you show a dot card with 4 which forms a square shape, do you get a variety of responses such as, “I saw two and two.” or “If it makes a square, there are 4.” See some of my Number Talk blog posts for resources.
- Use class scenarios to help children name the next number. “There are 3 of you sitting on the carpet with me. If Megan comes to join us, how many would there be then?”
- Practice counting on with ten frames and Rekenreks. Ex: Show a ten frame like this. The top is full so it is 5. Then count on 6, 7. How many dots? 7
- Notice ordinal positions regarding lines of students or arranging manipulative objects. Ex. “Put the blue bear first, the yellow bear second, and the red bear third.”
- Experience part-whole counting by provide number bond activities such as my favorite, On and Off
- Share stories about counting. Check out this link from The Measured Mom: The Ultimate List of Counting Books
- Develop an observation-type informal assessment checklist to track each child’s ability to do the above. Assess while they are using math centers or during inside recess opportunities. Here’s a FREEBIE checklist you are welcomed to edit, so I kept it in Word format. Counting Fluency Observation Checklist

Enjoy your counting lessons with your children or students. Use these to identify gaps in students’ concepts. Stay tuned for more development toward understanding of place value.

]]>As I mentioned in my last post (More Number Talk Ideas – Part 1), there are many ways to conduct Number Talks with your students. The last post focused on Picture Talks and Which One Doesn’t Belong (WODB). This week I am focusing on Estimation Mysteries and Data Talks.

**Esti-Mysteries**

Steve Wyborney has been super gracious to share his math estimation mysteries with educators via the nctm.org blog and through his website: https://stevewyborney.com/

What are they? Each esti-mystery features a clear container with identical small objects (cubes, dice, marbles, manipulatives, etc.). Students estimate how many are in the container, then proceed through 4-5 clues revealed one at a time in a ppt format. Clues give information dealing with number concepts such as even/odd, less than/greater than, place value, multiples, prime, composite, etc. With each clue, students can then revise their estimate to try to eventually match the actual amount revealed on the last slide. Different clear containers are used with each mystery to include ones with irregular shapes.

It’s really interesting for students to share how they arrived at their estimate, to list possible answers, then defend their choice. And of course, the rejoicing when/if their estimate matches the revealed amount!

**Data Talks**

You may have heard of the youcubed website (https://www.youcubed.org) which is partially commanded by well-known Stanford mathemetician Jo Boaler and her co-hort Cathy Williams. You will be amazed at all of the math resources and task ideas for all grades at this site. I witnessed another webinar hosted by these scholars regarding the increased need for data science. While professionals surveyed rarely actually use the algebra, geometry, and calculus learned from high school or college courses, there is definitely an increase in the need for data science / statistics as noted in almost everything we do. So youcubed has made it their mission to ramp up data science resources. One already in the works is “Data Talks” which provides some real-world, interesting, thought-provoking data presentations ready for class discussion. The link is right here: https://www.youcubed.org/resource/data-talks/

You will find graphs and tables of all types (some very creative ones), with topics such as these:

- Steph Curry’s shooting and scoring % shown on a basketball court diagram
- Social media use
- Paper towel hoard in 2020
- Dice combinations

Before diving into the data presented, get students to notice first . . . “I noticed . . .” and follow analysis with “I wonder . . .” The “I wonder” questions promote ideas about trends and change in data. Here’s a sample graph regarding possible outcomes when adding 2 dice (graphic from google, not youcubed.org):

Possible noticing and wondering:

- I noticed the graph goes up and then down symmetrically.
- I noticed there are 11 possible sums using 2 dice.
- I noticed the bar for 7 is the highest.
- I noticed numbers on the left side go up by .02 each increment.
- I wonder why 7 is the highest? What are ways to roll a sum of 7?
- I wonder what a graph would look like when actually rolling 2 dice numerous times? Will it be similar to this one?

I highly encourage you to check these out! I will add the sites to my resource list (top bar of my blog) for easy access.

**Till next time . . . Cindy**

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I’m back after taking a couple of months off from blogging! I know some of you are already back at school, while others will be starting this coming week. I wish the best with all the uncertainties that still lie ahead. BUT most of you are back in the classroom this year, which is a good thing, right?

I am a big advocate of implementing **Number Talks** as part of a short daily math routine. Most of my previous number talk posts have focused on students sharing strategies for solving problems involving number strings and using known problems to stretch for new problems (such as 3 x 4 and then 30 x 4 or 10 + 8 and then 9 + 8). Today I would like to start a two-part post about other good quality number talk options which are also designed to elicit critical thinking.

- Picture Talks
- Which One Doesn’t Belong (WODB)

Next post will be these two:

- Esti-Mysteries
- Data Talks

**Tips for Implementing:**

- There are multiple ways to interpret, so students can participate at different levels.
- Project them on a large screen, and allow writing on it to capture the thinking process.
- A great question to start with is, “What do you notice?”
- These are great to share with a partner before discussing with the whole group.
- You may need to assist students with verbally explaining their thinking. Summarize so everyone understands.
- Relish the chance to introduce or review new vocabulary.
- Design your own, and have students create some as well.
- Be amazed at the many different ways to interpret these!

**Picture Talks**

This involves the use of pictures of objects with the purpose of telling how many and how they were counted (simiar to dot cards, but actual photos of objects arranged in rows, arrays, groups, etc.). A terrific way to practice subitizing, doubles, near doubles, equal groups for multiplication and division, fractions, as well as create story problems with them. Great questions for these picture talks: How many? How did you see them?

Many of them can be found on google images, but a good resource is via Kristen Acosta. I participated with her on a recent webinar and was hooked. I have tried many of these with my Zoom online students and they enjoy them because there are multiple ways to analyze a picture to determine how many.

- This is Kristen Acosta’s website. She has posted her photo images free, although you may need to subscribe to access them. She also has other math treasures on her website! She has a few using egg cartons, which inspired me to go crazy and make my own photos. Feel free to use these
**below**, or take your own! https://kristenacosta.com/number-talk-images/ - Char Forsten is well known in the Singapore Math world. I have had this book for many years and love it! It is great for PreK-2nd grade. What’s inside? Nursery rhymes with pictures that are full of math content. Suggestions for questions to help students notice the pictures to find number bonds. Other photographs you can place under your document camera to project as you discuss. The book is rather expensive, but I found the digit version which is $15.
- Math Talk by Char Forsten (Digital copy for sale by sis4teachers.org)
- Math Talk by Char Forsten & Torri Richards (Amazon)

Example of different ideas students might have on how to count this:

**Which One Doesn’t Belong?**

Inspired by the book (or vice-versa), you will see 4 images, numbers, letters, shapes, graphs, etc. To elicit critical thinking, the goal is to have your students select one of the images and tell why it **doesn’t** belong with the others. BUT, there are many possible responses — as long as the student can explain their reasoning. Follow some of the tips above and have fun exploring all of these free ready-made WODB images!

Image 1 thoughts to get you started:

- Top right because it’s the only one with no holes.
- Top left because it’s the only one with no icing.
- Bottom right: It’s pink and the others all have chocolate

Image 2 thoughts to get you started:

- 9: because it’s the only single digit
- 9: because the other numbers have digits that add up to 7
- 43: because it’s the only prime number
- 16: because it’s the only even number

WODB designs: Submissions by many, but website created by Mary Bourassa

Which One Doesn’t Belong: 2D shapes from Miss Laidlaw’s Classroom (FREE on TPT)

Which One Doesn’t Belong: 3D shapes from Miss Laidlaw’s Classroom (FREE on TPT)

Which One Doesn’t Belong: 2D shapes for 2nd-7th grades from Jeannie’s Store (FREE on TPT)

**Here are more of my egg carton images to get you started! Please share your experiences with these!**

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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:**

**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!

**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.

- 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
**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.

]]>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: **

- 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 t**o 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. **

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:**

- 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? 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!!**

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:

- Here’s one I developed to practice double digit x single digit problems: Multiplication is repeated addition double digit x single digit problems CE
- Chart/worksheet to show multiplication as picture, repeated addtion, equal groups (Free on TPT from Dotty Fobes)
- Repeated addition for mult. games (Free on TPT by Games 4 Gains)

**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”).**

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:

- Multiplication is repeated addition. For example: 3 x 4 means 3 groups of 4 or 4 + 4 + 4 = 12
- 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.
- 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)?
- 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.
- 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

- 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?

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**Stay tuned for more blog entries about multiplication!**