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Bar model for Multiplication

Multiplication Bar Models Worksheets & Teaching Resources

  1. Students will use bar models representing multiplication facts to find the product or missing factor on each card. They can check their own answers using the QR codes or turn in completed task sheet to be gr
  2. Singapore Math Bar Models are a very useful, visual method to teach simple word problems. Some weeks ago we learned how to use the Singapore Bar Models applied to fractions and today we are going to use it for multiplication and division problems. We have two quantities where one is a multiple of the other
  3. Big Idea: Bar models can be used to solve different kinds of multiplication and division word problems. Use bar models to solve one-step multiplication word problems. Use bar models to solve two- step word problems. Choose the correct operations in two-step word problems
Multiplication Division Worksheets 3rd - 4th Grade (Bar

A set of Easter themed NO PREP differentiated word problems using the bar model strategy, focussing on multiplication and division for grades 2 - 3. The bar model is great to help children visualise and solve problems and is becoming much more widely used across schools. This pack will guide your c Multiplication Bar Model. Multiplication Bar Model - Displaying top 8 worksheets found for this concept.. Some of the worksheets for this concept are Bar modeling, Student activity, Multiple step word problems and bar models grades 4 6, Sample work from, Math mammoth grade 5 a worktext, 1 mark, C h a pte using bar models r 9 multiplication and division, Measurement and data volume grade 5. Bar models of multiplication start with the same 'real' and 'representative counters' stages as addition and subtraction. Then moves to its final stage, drawing rectangular bars to represent each group: Each box contains 5 cookies. Lionel buys 4 boxes 3rd Grade Real World Problem3rd3rdBeginning in 2nd grade and continued in 3rd grade students learn to use the bar model for multiplication and division. These concepts are based on equal groups or the part-whole concept, where each equal group is one part of a whole. In 4th grade and 5th grade students will use the bar model as a helpful tool i In math, a bar model can be defined as a pictorial representation of a number in the form of bars or boxes used to solve number problems. Bar models help us to attain an understanding of how a problem needs to be solved and calculated. Here, for instance, one rectangle or box of the bar model represents a value of 6

Singapore Bar Models for Multiplication and Division

Chapter 9: Using Bar Models: Multiplication & Division

Bar Models For Multiplication And Division Worksheets

Multiplication Bar Model Worksheets - Kiddy Mat

Embedded in this unit of printable division worksheets are varied strategies for 3rd grade, 4th grade, and 5th grade kids to learn the basics of division with a range of hands-on tasks, such as cut-and-paste sharing activity, drawing objects for equal sharing, division by grouping objects, dividing by distributing objects in rows and columns of the array model, number line, bar model and area. Using bar models for addition and subtraction problems; Drawing bar models to represent word problems; Using bar models to represent addition and subtraction word problems; Using bar modelling to represent multiplication and division word problems; Drawing bar models to represent multiplication and division word problem

How to Teach the Bar Model Method in Maths to Ace

  1. Math in Focus Chapter 9: Bar Models: Multiplication & Division Trimester 2 *Tie into area Big Idea: Bar models can be used to solve different kinds of multiplication and division word problems. - Use bar models to solve one-step multiplication word problems. Use bar models to solve two- step word problems. - Choose the correct operations in two-step word problems
  2. Displaying top 8 worksheets found for - Multiplication Bar Model. Some of the worksheets for this concept are Bar modeling, Student activity, Multiple step word problems and bar models grades 4 6, Sample work from, Math mammoth grade 5 a worktext, 1 mark, C h a pte using bar models r 9 multiplication and division, Measurement and data volume grade 5 formative assessment
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  4. Given a model that represents two quantities that are compared through addition, the student will be able to solve for a sum or a missing addend with 85% accuracy over 5 consecutive sessions. 2. Given a model that represents two quantities that are compared through multiplication, the student will be able to solve for a product or a missing.
  5. Bar Model Multiplication And Division Worksheet is Important to Improve your Math Skills. You will find a number of things which both parents and teachers equally can do to aid the student become successful. On the part of their moms and dads, they must do a number of stuff. First of all, a due diligence system needs to be implemented in the.
  6. 1 Quiz. Outline of Session. Part-Whole Bar Model for Multiplication and Division Problems. Discussion Questions. Quiz Lesson 8: Part-Whole Bar Model for Multiplication and Division. Lesson 9: Comparison Bar Modeling for Addition and Subtraction. 3 Topics. |
  7. Bar Modelling Multiplication. Showing top 8 worksheets in the category - Bar Modelling Multiplication. Some of the worksheets displayed are C h a pte using bar models r 9 multiplication and division, Singapore math bar model strategy, Bar modeling, Using bar modelling to solve problems primary, Using bar models to find the fraction and percent of a number, The bar model work one catering, Math.

Jul 24, 2013 - WEBSITE: http://www.teachertube.com Multiplication Model Drawin A model that is great for problem solving with either adding, subtracting, multiplying, or dividing is a bar model. Bar models can be used starting as young as first grade but are also used well into algebra. A bar model is so helpful because it allows the student to decipher between the WHO and the WHAT in a word problem, and visually see. Bar Model Multiplication Worksheets Printable Worksheet Template. Article by Abby Kay | Preschool and Kindergarten Worksheets. Multiplication Anchor Charts Math Charts Teaching Multiplication Math Anchor Charts Multiplication Strategies Fractions Strip Diagram Bar Model Math Problem Solving Use an area model for multiplication of two-digit numbers by two-digit numbers. You'll gain access to interventions, extensions, task implementation guides, and more for this instructional video. In this lesson you will learn how to multiply a 2 digit number by another 2 digit number by applying your understanding of the area model for. The Singapore bar models are a linear representation of a problem, and would not be an effective strategy for array/area models. As you read these examples, notice how multiplication equations are often solved with division and vice versa

The bar model helps students visualize the relationships between the numbers given and the amount you are looking for in a word problem. What's great about this tool is that the model starts in elementary school with addition and subtraction problems and then can be expanded to work with multiplication and division problems, fractions, ratios, and percents In this lesson, Becca Sherman introduces the Singapore Bar Model to students who have never seen it in before. She uses the context of solving division story problems to introduce the model. The number talk connects the basic components of the Bar Model with students' original thinking, thus front-loading students with several applications of the Bar Model as a representation of equal. A bar model is a visual representation of known and unknown quantities. They are often used to help solve number problems including addition, subtraction, multiplication and division. Using bar models during maths lessons to help explain fractions, addition, subtraction, multiplication, and division is an excellent way to engage visual learners In addition and subtraction, we use a non-proportional bar to represent an amount. If it's a larger number, like 54, it would be a larger bar. For a smaller number like 24, it would be a smaller bar. When we move into part-whole multiplication and part-whole division, we are still using a non-proportional bar, but we're chopping it up into. To create bar models for multiplication & division. 22m video. Lesson . 12. Finding the whole and the parts when using a bar model. 17m video. Lesson . 13. Exploring how bar models represent word problems. 13m video. Lesson . 14. Solving word problems by using bar models. 13m video. Lesson . 15. Consolidating and applying: Using bar models and.

Multiplicative reasoning is central to the KS3 curriculum, but needs to begin at Primary. Representations in terms of the bar model and double number line are used to support reasoning and problem solving, which will in turn support problem solving Comparison Model: This model is good for addition and subtraction. It shows how big and small quantities are related to each other. Equal Parts of a Whole: This model is good for multiplication. It shows how a whole is broken down into separate parts. Comparison and Part-Part Whole: This model is good for addition, subtraction, multiplication. Bar Modelling your way through multi-step word problems is the latest silver bullet in primary school maths for good reason. Here I look at how you can use the bar model to solve the sorts of Year 6 word problems pupils will face in their KS2 SATs This is the first unit of the special topic The Bar Model Method.This is a pedagogical strategy widely used in Singapore to help students solve word problems. This unit Introduction to Bar Model Method introduces students to two types of bar models for solving word problems: part/whole models and comparison models, each applied to addition, subtraction, multiplication and division Bar models help us to understand what operations, whether addition, subtraction, multiplication or/and division should be used to solve a word problem. Any two factors and their product can be read as a comparison. In a multiplicative comparison problem, one quantity will always be smaller or larger than the other quantity

Play this game to review Mathematics. What number sentence is represented in the bar model above? Preview this quiz on Quizizz. Quiz. Bar Model Multiplication. DRAFT. 3rd grade . Played 0 times. 0% average accuracy. Mathematics. 6 hours ago by. sbrazeau. 0. Save. Share. Edit. Edit. Bar Model Multiplication DRAFT. 6 hours ago by. sbrazeau. 3rd. Big Idea. There are two visual models that are very useful for division that will help us model every kind of division problem. Fluently divide multi-digit numbers using the standard algorithm. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics Solve addition word problems by building bar models. Launch. Model Builder - Subtraction within 100. Solve subtraction word problems by building bar models. Launch. Explore and identify patterns for multiplication tables to 10. Launch. Multiplication with Groups. Arrange groups of food on plates to visually represent the concept of. Let's Practise Bar Modelling (KS2) Bar Models worksheet for Year 5 and Year 6 to practice problem solving (includes answers) Bar modelling worksheet of 25 scaffolded word problems. Practice questions available with or without suggested bar models. Suitable for use in a mixed ability classroom and for mastery approach Using fraction strips or fraction bars, show students an example of a multiplication model with a fraction. Write the problem on the board (i.e., 3 x ⅕). Hold up one ⅕ bar and explain that 3 x ⅕ is three groups with ⅕ in each group. Draw three circles and place a ⅕ bar in each

Bar Model Word Problem For Grade 3. Bar Model Word Problem For Grade 3 - Displaying top 8 worksheets found for this concept.. Some of the worksheets for this concept are C h a pte using bar models r 9 multiplication and division, Bar modeling, Word problems and bar models, Introduction to singapore math, Multiple step word problems and bar models grades 4 6, Math mammoth grade 4 a, Name date c. Features of Singapore Math- Bar Models: -Instruction and Examples with a fun, engaging avatar to guide children at each level -Problems involving addition, subtraction, multiplication, and division that lay the foundation for algebra -Interactive bar model tool allows children to customize and personalize their problem solving experienc

Introducing New and Improved Thinking Blocks®. We redesigned Thinking Blocks and packed it full of new features! read aloud word problems - visual prompts - better models - engaging themes - mobile friendly. Thinking Blocks works well on all devices Building a strong foundation in mathematics is an important step in helping your child become proficient and confident. The worksheet gives students an opportunity to apply the concepts of comparison models to complete the sentences. Regular practice will help your students develop confidence in the classroom and in the real world.- SplashLear Using Bar Models: Multiplication and Division C H A P T E 9 R Worksheet 1 Real-World Problems: Multiplication Study the bar models. Then complete the sentence. Example BA A B is twice as many as . Twice means 2 times. 1. P Q is three times as many as . 2. A B is four times as many as . 3. P Q is fi ve times as many as The Bar Model is a teaching and learning strategy for mathematical problem solving which can be applied to all operations. Our Bar Model Poster Pack contains 16 display posters. The pack illustrates the five basic variations of the Bar Model and includes examples of how to apply them to different operations

Provide vocabulary cards with key terms that students can reference throughout the lesson, such as multiplication, division, bar model, clue words. Guided Practice (10 minutes) Tell students you would like to solve two more problems together as a class using the Steps for Solving Word Problems Chart The Bar Model . Identify the groups and each and then draw a bar to help you understand the problem. There are 3 plates so make a bar showing 3 groups and then each group has 4 cupcakes. Now multiply by knowing your facts or drawing a picture. 3 x 4 = 12 . Repeated Addition and skip countin Bar Model For Second Grade. Displaying top 8 worksheets found for - Bar Model For Second Grade. Some of the worksheets for this concept are Bar modeling, Multiple step word problems and bar models grades 4 6, Singapore math simple or complex, P t e r h a using bar models addition and subtraction, Grade 2 mathematics instructional focus documents, Summer math packets curr 2 grade 2, Mathematics.

In Mathematics, a bar model is used to represent problems or concepts by the means of pictures. Each and every mathematical operation, such as addition, multiplication, division, and subtraction can be easily depicted by the means of bar models. Bar models are highly beneficial when it comes to explaining mathematical concepts to children We'll also look at how to represent multiplication and division using discrete objects (counters) and using a more continuous model (bar diagrams/Cuisenaire rods). Multiplication as repeated groups: The first way we understand and learn about multiplication is as repeated addition: the amount in several sets, each of which are the same size Math in Focus: Resources for Families / Bar Models for Problem Solving. Click on the link below to receive an overview of how bar models progress from Kindergarten to 5th grade. Bar Models for Problem Solving (K-5th Grades) Click on the links below for examples of part-whole and comparison bar models for addition/subtraction and multiplication. The following diagram shows how to use bar models for multiplication and division. Scroll down the page for more examples and solutions on using the multiplication and division part-whole model. Division Bar Modeling Examples: 1. A baker made 72 muffins. He places 8 baskets on the counter. How many muffins will the baker put in each basket? 2

What is Bar Model? - Definition, Facts & Exampl

Bar Models to Solve Word Problems. For Teachers 9th - 10th. Students model inequalities as they solve for the missing variable. In this algebra lesson, students model their inequalities and use it to identify less than, greater than and applying it to solving for the missing value. Get Free Access See Review Common Core Connection for Grades 3 and 4. Solve word problems involving equal groups, arrays and measurement quantities. Use equations with a symbol for the unknown number to represent the problem. Solve two-step word problems using the four operations. Represent problems using equations with a letter standing for the unknown quantity

Multiplication and Division Models and Strategies

Bar Modelling Explored. White Rose Maths (WRM) is proud to offer its Bar Modelling Explored online training, a compilation of three separate series of professional development videos. Each series comprises about ten short videos (around 2-3 minutes each) which lead you through the learning in small steps. Sessions are interactive, effective and. using the bar model or area/grid method. They may solve division problems by: sharing or grouping objects equally; using a number line to count backwards in steps of 2, 5 or 10; using their understanding of inverse to use multiplication to work out division problems. For example, when solving 8 ÷ 2, children may count in steps of two to find. Apr 27, 2016 - Third Grade / Special Education Math Anchor Chart: Intro to Fractions, Circle Model, Bar Model, Definition, Fraction Names. Pinterest. Today. Explore. When autocomplete results are available use up and down arrows to review and enter to select. Touch device users, explore by touch or with swipe gestures Content Skill: Multiplication Common Core State Standards: CCSS.Math.Content.3.OA.C.7 - Fluently multiply within 100, using strategies such as the relationship between multiplication and division or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers

Bar Model Worksheets by Once a Teacher Always a Teacher | TpT

Video Lesson on Using Bar Models for Multiplication and

In the last post, we looked at the use of part-whole bar models for multiplication word problems. Here's a similar application of the part-whole bar model method - this time for a division word problem. As with previous bar modelling videos, this video was produced using Matholia's 'Bar Model Maker' - just one of more than 50 primary maths. For over 10 years Conceptua Math has offered its innovative math curriculum to K-5 students and classrooms, in addition to a free online version of its math tools. However, with the end of support for Adobe Flash on December 31, 2020, most of Conceptua Math no longer functioned and with great disappointment the free tools were retired on March. Solve multiplicative comparison word problems by using bar models. You'll gain access to interventions, extensions, task implementation guides, and more for this instructional video. In this lesson you will learn how to solve word problems with multiplicative comparison by using bar models The necessity of multiplication practice should not be emphasized ample for today's elementary school college students. Understanding one's simple Bar Model Multiplication Worksheets information is a vital building block for later on mathematics classes where by this essential knowledge will likely be expected to be not just comprehended, but perfected Chapter 9 - Using Bar Models:Multiplication and Division. Helpful Videos. bar model for multiplication and division. bar models examples. more bar model examples. two step bar model problem. Worksheet Videos. pg. 159. pg. 160

Using bar models for multiplication and division word

Practice 1 Real-World Problems: Multiplication Solve. Use bar models to help you. Using Bar Models: Multiplication and Division C h a p t e r 1. Mrs. Bently pays $95 for a new bed. Mrs. Lee pays three times as much as Mrs. Bently for a new bed. How much does Mrs. Lee pay for the new bed? 1 unit $ 3 units $ w 3 $ Mrs. Lee pays $ for a new bed File: 082911-multiplication bar model.doc. Josh Alfred AF Endeavor Middle School Brooklyn, NY 69 Views. 3 Downloads. 0 Favorites 8th Grade Math - 2011-2012 8th Grade Math - 2011-2012 8th Grade Math - 2012-2013 8th grade math 8th Grade Math - 2013-2014 8th Grade Math - 2014-201

Lesson: To create bar models for multiplication & division

Using Bar Models: Multiplication and Division C H A P T E 9 R Thinking Skills PROBLEM SOLVING Solve. Use bar models to help you. 1. The cost of 5 apples is $4. Mrs. Shulter has $28. How many apples can she buy? $4 $28? 5 apples 13(M)MIF2015CC_EnG3A_Ch09.indd 73 4/22/13 10:57 A Bar Model Online Course Chapter 2.5 Multiplication Chapter 2.5 Multiplication In Progress Your name How [

The Bar Model NCET

Interactive bar modelling to aid the use of manipulatives in the classroom I'm delighted to say that I can add another model of multiplication to your list: prime factor concatenation. For example, 6 = 2 * 3 and 10 = 2 * 5, so 6* 10 = 2*2*3*5 = 60. This was inspired by the game Prime Climb, so I really can't take much credit for it There are a variety to think of multiplication such as skip counting, area model, and repeated addition. With the chip model, however, we will rely on thinking of multiplication as equal groups. 34 x 7 means 34 groups of 7 and also 7 groups of 34. Most people would think it is easier to model 34 x 7 as 7 groups of 34 • Follows Area Model for Multiplication of Whole Numbers • A visual model that assists in learning the standard algorithm How can we model the following fraction multiplication problem with the Kit Kat bar? First split the candy bar lengthwise into 4 equal pieces as shown. This gives us pieces of ¼ each Let's Practise Using the Bar Model Multiplication Year 3. Practise the bar model method with a worksheet of 11 worded questions. Includes optional bar model scaffolding, and a challenge question. A great way to help your Year 3 pupils approach and represent a worded Maths problem. Includes 11 multiplication word problems, available with or.

Bar Model Multiplication. Showing top 8 worksheets in the category - Bar Model Multiplication. Some of the worksheets displayed are Bar modeling, Multiple step word problems and bar models grades 4 6, Student activity, Sample work from, Math mammoth grade 5 a worktext, C h a pte using bar models r 9 multiplication and division, Notes and guidance yearly overview, New bar modelling content This model is often used to help kids see the different qualities of addition and multiplication. Bar modeling (also known as a tape diagram) A bar model uses bars to visually represent numbers and unknowns in a word problem In today's post, we're going to talk about multiplying fractions using the Singapore Math Bar Model. Let's take a look at an example: Let's take a look at an example: Amy drew a rectangle and colored in 2/5 of the parts purple and after, she striped 1/2 of them yellow

Using Bar Models: Multiplication and Division C H A P T E R 9 Worksheet 1 Real-World Problems: Multiplication Study the bar models. Then complete the sentence. Example B A A B is twice as many as. Twice means 2 times. 1. P Q is three times as many as . 2. A B is four times as many as . 3. P Q is five times as many as Hi, At page below we present you several amazing photos that we've gathered so they might helpful, for this time we are more concern about Bar Model Multiplication Worksheets. When we talk about Bar Model Multiplication Worksheets, we already collected particular related images to inform you more. fixed and variable cost template, fixed and variable cost template and fixed and variable cost.

Examples, Solutions, videos, worksheets, games and activities to help students learn how to solve 2-step word problems using bar models in Singapore Math. Bar models are similar to tape diagrams used in Common Core Math. The following diagrams show the Part-Part-Whole Models and Comparison Models. Scroll down the page for examples and solutions This candy store math problem is a fun way to introduce 2nd and 3rd graders to the geometric model for multiplication, and kids really don't need to know much about multiplication to benefit from this Review: Multiplication and Division 1. Write four equations for each bar model (a fact family). Then solve for w. Multiplication and division have to do with things or groups of the same size. When you multiply the number of groups by the amount in each group (or the other way around), you get the total. When you divide the total by the number o The bar model method is a great way to visualise tricky mathematical concepts and solve problems.. They are a quick and easy way to represent a situation in a word problem. Bar modelling can be.

Multiplication Word Problems. Choose a problem type using the buttons above. Solve each problem by converting the given expresion of the problem in words to a pictorial representation using bars, by dragging the blue dividing and ending lines. Once the correct model has been constructed, the answer may be calculated and entered Chapter 5 - Using Bar Models: Addition and Subtraction. Helpful Videos. Bar model part 1. Bar model part 2. bar model part 3. Bar model problems. Workbook Videos. pg. 73. pg. 74 The array model, used throughout Bridges, is the subject of Arrays, Multiplication and Division by Jennie Pennant from nrich.maths.org. The piece outlines how the array model supports development of a sense of multiplicative relationships and describes how to move students from building multiplication facts and tables to exploring division as the inverse operation of multiplication

Lights, Camera, Action Learn!: Interactive Notebooks

Multiplication Bar Model - Displaying top 8 worksheets found for this concept. Some of the worksheets for this concept are Bar modeling, Student activity, Multiple step word problems and bar models grades 4 6, Sample work from, Math mammoth grade 5 a worktext, 1 mark, C h a pte using bar Multiplication with Bar Models. Complete the bar models (aka bar diagrams or tape diagrams) and write the multiplication fact for each. 3rd Grade. View PDF. Multiply, Cut, Glue (0-10) Solve these basic multiplication facts. Then cut out and glue the correct answer to the boxes Create. Make social videos in an instant: use custom templates to tell the right story for your business. Live Streaming. Broadcast your events with reliable, high-quality live streaming Multiplication; They're particularly useful for time and measurement problems as children often find these problems challenging. Solving addition and subtraction problems. Part-whole bar models are a simple way to show word problems with a missing number element Area Model Multiplication Activities. Once students get the hang of area method multiplication, they can use it as a primary problem-solving method or a way to check their answers. Here are some activities for introducing this method and using it at a variety of levels. 1. Start with arrays. Multiplication arrays are the basis for the area.

Chocomatic (Multiplication, Arrays, and Area) Launch Gizmo. Use the Chocomatic to design candy bars made out of chocolate squares. Use multiplication to find the number of squares in each chocolate bar. Build collections of chocolate bars that all have the same number of squares. Solve multiplication problems by joining two smaller chocolate. Bar modeling multiplication and division. Saved by Jodi Dobbs. 324. Multiplication Anchor Charts Math Charts Teaching Multiplication Math Anchor Charts Multiplication And Division Teaching Math Fractions Teaching Ideas Strip Diagram Children who are following the Asian maths mastery method are taught the foundation of bar models, for example using linking cubes, in the first term of Year 1. In Year 2 children might use the bar model method as a visual way of solving calculations for addition, subtraction, multiplication and division. They are particularly useful when solving multi-step problems involving missing numbers Thinking Blocks is a great resource for multiplicative comparison problems as well. They have a multiplication/division app that matches comparison bars for multiplication and division problems. They have some with two parts, some with three parts, some that involve more operations. Giving kids consistent, targeted practice within each of the. A collection of math resources based on Singapore Math, including video lessons, examples and step-by-step solutions of Singapore Math Word Problems, worksheets for Singapore Math from Grade 1 to Grade 6, What is Singapore Math, How to explain Singapore Math? What are number bonds, How to use number bonds, Model drawings, bar models, tape diagrams, block models

4th Grade Number Activities

Here is a collection of our printable worksheets for topic Model Multiplication with Arrays of chapter Understand Multiplication in section Multiplication. A brief description of the worksheets is on each of the worksheet widgets. Click on the images to view, download, or print them. All worksheets are free for individual and non-commercial use Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables. Write and calculate mathematical statements for multiplication and division using the multiplication tables they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods

Part-Part-Whole Word Problems (solutions, examples, videos)Bar Model: addition and subtraction facts - Reasoning

Teach ratio using ratio model, solve complex questions with the model method, learn different ways to draw such models and boost your kids' confidence in math Part-Whole Concept....Teach Kids Math Using The Model Method From Singapore Math Part-Whole Concept is one of the 3 main pillars of the Model Method widely used to teach Singapore Math The following bar model uses a repeated addition view of multiplication to visualize the problem. 12 + 12 + 12 + 12 + 12 + 12 = 72 (or 6 x 12 = 72) so Alana has 72 pieces of gum Arrays are useful models for multiplication which can be used in a variety of ways, ranging from highly structured lessons to games and open investigations. An array is formed by arranging a set of objects into rows and columns. Each column must contain the same number of objects as the other columns, and each row must have the same number as.