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The Impact of Use of Manipulatives on the Math Scores of Grade 2 Students

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The Impact of Use of Manipulatives on the Math Scores of Grade 2 Students

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Algebra Tiles: Learning Sequence

The following is a suggested teaching and learning sequence for using Algebra Tiles. These ideas can be used from Year 3 onwards.

We use concrete materials and manipulatives in the classroom in the lower primary years but often move away from them as students get older, giving students the impression that they are 'babyish' and that it is more 'grown-up' to manage without them. This is a misguided approach and manipulatives should be used all the way through school.

Students often experience difficulty with Algebra and the notation it employs. Algebra tiles are a manipulative that can help develop student's understanding and confidence with algebra, at many different levels.

The tiles employ the area model of multiplication and this needs to be explored and understood if the students are to get the maximum understanding and benefit from their use. See the references at the end of the document for more detail on the area model.

This lesson sequence is not meant to replace a textbook but is an introduction to the use of algebra tiles. Use your usual textbook and classroom resources for exercises and practice.

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The following is a suggested teaching and learning sequence for using Algebra Tiles. These ideas can be used from Year 3 onwards. We use concrete materials and manipulatives in the classroom in the lower primary years but often move away from them as students get older, giving students the impression that they are 'babyish' and that it is more 'grown-up' to manage without them.

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A good explanation of the CRA model

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Instead, I want to encourage you to view it more as this Venn diagram. These overlapping circles that if we can hit an activity in what I call the Sweet Spot of CRA. If you get an activity where you can help kids see that what they're doing concretely, with manipulatives, connects to this drawing that we're doing right over here, and that connects to these written symbols.

When you help kids make all three of those connections within the same lesson, it's those lessons where at the end of it we're like "that was so awesome!" But we don't really know why it was so awesome sometimes. Was it because kids could see all of these connections, things were flowing, or they started having those light bulb moments? When you're in that sweet spot, it just makes a world of difference for kids. And the cool part is that you allow kids to work wherever they are at in any activity.

So if I go back here to this visual of this lesson plan, basically, you have a whole class of 30 kids and you're doing a lesson using manipulatives to represent addition, what about those kids who are past it, who don't need the manipulatives? And then maybe in the next couple lessons you're jumping into the written method, but you still have half your kids who need the manipulatives.

Being able to do an activity where you allow all three of these phases to happen at the same time allows your students to be able to work where they are at within these phases.

Allow kids to be doing stuff with objects. Have them always out, it doesn't matter if your lesson plan calls for it or not. Letting kids always have objects available so that they could work the problem out concretely. Then have visuals, help kids be able to do drawings and models.

Now, here's one thing, change your wording from drawing to modeling because when a kid wants to draw a representation, especially in the young grades, they want to draw a very detailed drawing of it. If it's about butterflies they want to draw the butterflies and color it, so in the representational phase is a time where you can teach kids the difference between drawing and modeling the mathematics. There's a time and place for detailed drawings, but there's a time and place where we're just modeling what's happening in the problem. And then we can move them into attaching those symbols.

That's one of the downsides that I see is when kids are working hands on and they're doing manipulatives, we aren't connecting it to a drawing and we're not connecting the abstract symbols, and so when they get to those phases, they don't connect it back to the hands on stuff that we did five lessons ago.

So, here's a little example, just starting out with letting kids build a certain number. Here's a couple examples of moving them from building it concretely to representational to abstract. We want them to be able to see those visuals, and connect it to the abstract because there's so many more ideas around that number that they build besides just seeing it as a four and a three for 43. When they can build it hands on, they see the drawings, and they connect the representation to it, the abstract representation to it, so much more number sense gets built.

Then as they start moving into working with numbers, doing operations, we still want to do concrete, representational to abstract. We want kids to physically build and model the problem.

Here's an image of a rekenrek. They're building seven plus eight on a rekenrek. And then they can use the number path to represent what they saw on the rekenrek, so it's the same visual that these kids created. They both had seven on the top and eight on the bottom, but one kid saw the fives within there and so on the number path, they're circling the fives and then the group of two and three. While another kid saw the groups of sevens within, that seven plus eight is just like having seven plus seven. So on their number path, they modeled what they did on their rekenrek there. And then we attach the symbols to it.

That's all it takes is when you're doing the concrete stuff, layer in the representational, layer in the abstract piece so that we can help kids start making those connections and go quicker throughout these phases. We want them to go from concrete and representational and finally get to abstract, but it doesn't happen in a linear path. We need to be doing it together so that when they eventually get to the point where they don't need the concrete, they can jump straight to the drawing piece and connect that to the written piece. And then when they get comfortable with that, they will naturally not want to do the concrete, they won't want to do the representation because they see it just takes way longer than that.

I see far too many worksheets that have kids draw out all of the 10's and ones when a kid could do it quickly. If a kid is at that abstract stage, and they've built that representation, they understand the representations, they understand the connections and now they're just able to do it abstractly, that's okay, even if your textbook says they have to do it with a drawing, they don't. And even if your textbook says they should do the written method and you've got kids who need to draw it out, let them draw it out. The concrete, representational and abstract, kids will be working in those phases in a messy way. It does not happen the way that our textbooks want it to happen.

Here's another quick example using multiplication. We do a lot with building area model when it comes to multi-digit multiplication and we use base 10 blocks to model that. So the concrete phase we're modeling with base 10 blocks. Then we move into the representational phase of drawing an area model and then we move kids into what's known as a partial products or even the traditional algorithm. But we don't let kids see the connections between those.

How often have you done a lesson where kids do all three of those within the same lesson and then, the key point here is, at the end, is you ask how these are connected?

If somebody just did the concrete, they did just the base 10 blocks and they didn't make it all the way to the partial products algorithm, can we help them see connections between those? Or do they see these as three separate algorithms to do? We want kids to see that you're doing the exact same things, it's just one was physical, you're doing the concrete. One was a drawing and one we're doing it without any pictures at all, but we're doing the exact same mathematics in every single one of those.

Now, take a look at your standards, because I can bet at almost every grade level, you have at least one standard in there that has it specifically called out that kids need all three of these stages.

This is one example from second grade where it's talking about adding and subtracting within a thousand and they need to have concrete models. They need to do drawings and they need a written method. It doesn't say these need to be separated though.

We can do all three of those at the same time. That's my reminder to you is the concrete, representational and abstract is a very powerful thing in mathematics, but it's not a linear progression. These things overlap and there are definitely times when you might just work in the concrete phase. Or you might just be doing drawings or you might be doing a drawing and the abstract, right?

That's the cool part about this Venn diagram, sometimes you're in that spot where just two of those overlap. Or sometimes you're at the spot where you're just doing abstract, you're only working on the written method. That's okay. They're all spots that work for kids, but when you get all three of them together, it helps build some magical connections for students that are just so, so, powerful.

I hope that this helped you build your math mind, so that you can go build the math minds of your students.

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What Is The Concrete Representational Abstract Approach - The Recovering Traditionalist

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Are you curious about what the Concrete Representational Abstract approach is and how to use it in your classroom wen teaching math to your students?

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Concrete - Representational - Abstract

Sequence of Instruction

Purpose

The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning. When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level.

What is it?

Each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, beans and bean sticks, pattern blocks).

Students are provided many opportunities to practice and demonstrate mastery using concrete materials

The math concept/skill is next modeled at the representational (semi-concrete) level which involves drawing pictures that represent the concrete objects previously used (e.g. tallies, dots, circles, stamps that imprint pictures for counting)

Students are provided many opportunities to practice and demonstrate mastery by drawing solutions

The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols)

Students are provided many opportunities to practice and demonstrate mastery at the abstract level before moving to a new math concept/skill.

As a teacher moves through a concrete-to-representational-to-abstract sequence of instruction, the abstract numbers and/or symbols should be used in conjunction with the concrete materials and representational drawings (promotes association of abstract symbols with concrete & representational understanding)

What are the critical elements of this strategy?

Use appropriate concrete objects to teach particular math concept/skill (see Concrete Level of Understanding/Understanding Manipulatives-Examples of manipulatives by math concept area). Teach concrete understanding first.

Use appropriate drawing techniques or appropriate picture representations of concrete objects (see Representational Level of Understanding/Examples of drawing solutions by math concept area). Teach representational understanding second.

Use appropriate strategies for assisting students to move to the abstract level of understanding for a particular math concept/skill (see Abstract Level of Understanding/Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers).

When teaching at each level of understanding, use explicit teaching methods (see the instruction strategy Explicit Teacher Modeling).

How do I implement the strategy?

1. When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding).

2. Provide students many practice opportunities using concrete objects.

3. When students demonstrate mastery of skill by using concrete objects, describe & model how to perform the skill by drawing or with pictures that represent concrete objects (representational level of understanding).

4. Provide many practice opportunities where students draw their solutions or use pictures to problem-solve.

5. When students demonstrate mastery drawing solutions, describe and model how to perform the skill using only numbers and math symbols (abstract level of understanding).

6. Provide many opportunities for students to practice performing the skill using only numbers and symbols.

7. After students master performing the skill at the abstract level of understanding, ensure students maintain their skill level by providing periodic practice opportunities for the math skills.

How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems?

Helps passive learner to make meaningful connections

Teaches conceptual understanding by connecting concrete understanding to abstract math process

By linking learning experiences from concrete-to-representational-to-abstract levels of understanding, the teacher provides a graduated framework for students to make meaningful connections.

Blends conceptual and procedural understanding in structured way

Additional Information

Research Support For The Instructional Features Of This Instructional Strategy: Allsopp (1999); Baroody (1987); Butler, Miller, Crehan, Babbit, & Pierce (2003); Harris, Miller, & Mercer (1993); Kennedy and Tips (1998); Mercer, Jordan, & Miller (1996); Mercer and Mercer (2005); Miller, Butler, & Lee (1998); Miller and Mercer, 1995; Miller, Mercer, & Dillon (1992); Peterson, Mercer, & O'Shea. (1988); Van De Walle (2005); Witzel, Mercer, & Miller (2003).

Create seperate pages
Concrete What is it?
Understanding manipulatives
Examples of manipulatives (concrete objects) by math concept level
Suggestions for using manipulatives (concrete objects) What is it?
The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills. Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed.
Understanding manipulatives (concrete objects)
To use math manipulatives effectively, it is important that you understand several basic characteristics of different types of math manipulatives and how these specific characteristics impact students who have learning problems. As you read about the different types of manipulatives, click on the numbers beside each description to view pictures of these different types of manipulatives.
General types of math manipulatives:

Discrete - those materials that can be counted (e.g. cookies, children, counting blocks, toy cars, etc.).

Continuous - materials that are not used for counting but are used for measurement (e.g. ruler, measuring cup, weight scale, trundle wheel). See example - 1

Suggestions for using Discrete & Continuous materials with students who have learning problems:

Students who have learning problems need to have abundant experiences using discrete materials before they will benefit from the use of continuous materials. This is because discrete materials have defining characteristics that students can easily discriminate through sight and touch. As students master an understanding of specific readiness concepts for specific measurement concepts/skills through the use of discrete materials (e.g. counting skills), then continuous materials can be used.
Types of manipulatives used to teach the Base-10 System/place-value (Smith, 1997):

Proportional - show relationships by size (e.g. ten counting blocks grouped together is ten times the size of one counting block; a beanstick with ten beans glued to a popsicle stick is ten times bigger than one bean).

- Non-linked proportional - single units are independent of each other, but can be "bundled together (e.g. popsicle sticks can be "bundled together in groups of 'tens' with rubber bands; individual unifix cubes can be attached in rows of ten unifix cubes each). See examples - 1 | 2 | 3 |

- Linked proportional - comes in single units as well as "already bundled" tens units, hundreds units, & thousands units (e.g. base ten cubes/blocks; beans & beansticks). Se examples - 1 | 2 | 3 | 4 | 5

Non-proportional - use units where size is not indicative of value while other characteristics indicate value (e.g. money, where one dime is worth ten times the value of one penny; poker chips where color indicates value of chip; an abacus where location of the row indicates value). A specified number of units representing one value are exchanged for one unit of greater value (e.g. ten pennies for one dime; ten white poker chips for one blue poker chip, ten beads in the first row of an abacus for one bead in the second row). See example - 1

Suggestions for using proportional and non-proportional manipulatives with students who have learning problems:

Students who have learning problems are more likely to learn place value when using proportional manipulatives because differences between ones units, tens units, & hundreds units are easy to see and feel. Due to the very nature of non-proportional manipulatives, students who have learning problems have more difficulty seeing and feeling the differences in unit values.
Examples of manipulatives (concrete objects)
Suggested manipulatives are listed according to math concept/skill area. Descriptions of manipulatives are provided as appropriate. A brief description of how each set of manipulatives may be used to teach the math concept/skill is provided at the bottom of the list for each math concept area. Picture examples of some of the manipulatives for each math concept area can be accessed by clicking on the numbers found underneath the title of each math concept area. This is not meant to be an exhaustive list, but this list does include a variety of common manipulatives. The list includes examples of "teacher-made" manipulatives as well "commercially-made" ones.

Counting/Basic Addition & Subtraction
Place Value
Multiplication/Division
Positive & Negative Integers
Fractions
Geometry
Beginning Algebra
Counting/Basic Addition & Subtraction Pictures

Colored chips

Beans

Unifix cubes

Golf tees

Skittles or other candy pieces

Packaging popcorn

Popsicle sticks/tongue depressors

Description of use: Students can use these concrete materials to count, to add, and to subtract. Students can count by pointing to objects and counting aloud. Students can add by counting objects, putting them in one group and then counting the total. Students can subtract by removing objects from a group and then counting how many are left.
Place Value Pictures

Base 10 cubes/blocks

Beans and bean sticks

Popsicle sticks & rubber bands for bundling

Unifix cubes (individual cubes can be combined to represent "tens")

Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values)

Description of use: Students are first taught to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g. ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side. Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds."
Multiplication/Division Pictures
Containers & counting objects (paper dessert plates & beans, paper or plastic cups and candy pieces, playing cards & chips, cutout tag board circles & golf tees, etc.). Containers represent the "groups" and counting objects represent the number of objects in each group. (e.g. 2 x 4 = 8: two containers with four counting objects on each container)
Counting objects arranged in arrays (arranged in rows and columns). Color-code the "outside" vertical column and horizontal row helps emphasize the multipliers

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Positive & Negative Integers Pictures

Counting objects, one set light colored and one set dark colored (e.g. light & dark colored beans; yellow & blue counting chips; circles cut out of tag board with one side colored, etc.).

Description of use: Light colored objects represent positive integers and dark colored objects represent negative integers. When adding positive and negative integers, the student matches pairs of dark and light colored objects. The color and number of objects remaining represent the solution.
Fractions Pictures

Fraction pieces (circles, half-circles, quarter-circles, etc.)

Fraction strips (strips of tag board one foot in length and one inch wide, divided into wholes, ½'s, 1/3's, ¼'s, etc.

Fraction blocks or stacks. Blocks/cubes that represent fractional parts by proportion (e.g. a "1/2" block is twice the height as a "1/4" block).

Description of use: Teacher models how to compare fractional parts using one type of manipulative. Students then compare fractional parts. As students gain understanding of fractional parts and their relationships with a variety of manipulatives, teacher models and then students begin to add, subtract, multiply, and divide using fraction pieces.
Geometry Pictures
Geoboards (square platforms that have raised notches or rods that are formed in a array). Rubber bands or string can be used to form various shapes around the raised notches or rods.

Description of use: Concepts such as area and perimeter can be demonstrated by counting the number of notch or rod "units" inside the shape or around the perimeter of the shape.
Beginning Algebra Pictures
Containers (representing the variable of "unknown") and counting objects (representing integers) -e.g. paper dessert plates & beans, small clear plastic beverage cups 7 counting chips, playing cards & candy pieces, etc.

Description of use: The algebraic expression, "4x = 8," can be represented with four plates ("4x"). Eight beans can be distributed evenly among the four plates. The number of beans on one plate represent the solution ("x" = 2).

Suggestions for using manipulatives (Burns, 1996)

Talk with your students about how manipulatives help to learn math.

Set ground rules for using manipulatives.

Develop a system for storing manipulatives.

Allow time for your students to explore manipulatives before beginning instruction.

Encourage students to learn names of the manipulatives they use.

Provide students time to describe the manipulatives they use orally or in writing. Model this as appropriate.

Introduce manipulatives to parents

Representational
What is it?
Examples of drawing solutions by math concept level

What is it?

At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. It is appropriate for students to begin drawing solutions to problems as soon as they demonstrate they have mastered a particular math concept/skill at the concrete level. While not all students need to draw solutions to problems before moving from a concrete level of understanding to an abstract level of understanding, students who have learning problems in particular typically need practice solving problems through drawing. When they learn to draw solutions, students are provided an intermediate step where they begin transferring their concrete understanding toward an abstract level of understanding. When students learn to draw solutions, they gain the ability to solve problems independently. Through multiple independent problem-solving practice opportunities, students gain confidence as they experience success. Multiple practice opportunities also assist students to begin to "internalize" the particular problem-solving process. Additionally, students' concrete understanding of the concept/skill is reinforced because of the similarity of their drawings to the manipulatives they used previously at the concrete level.

Drawing is not a "crutch" for students that they will use forever. It simply provides students an effective way to practice problem solving independently until they develop fluency at the abstract level.

Examples of drawing solutions by math concept level

The following drawing examples are categorized by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes"). In each category there are a variety of examples demonstrating how to use these drawings to solve different types of computation problems. Click on the numbers below to view these examples.
Abstract
What is it?
Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers

What is it?

A student who problem-solves at the abstract level, does so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with numbers and math symbols only. Abstract understanding is often referred to as, "doing math in your head." Completing math problems where math problems are written and students solve these problems using paper and pencil is a common example of abstract level problem solving.

Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers

Students who are not successful solving problems at the abstract level may:

- Not understand the concept behind the skill.

Suggestions:

Re-teach the concept/skill at the concrete level using appropriate concrete objects (see Concrete Level of Understanding).

Re-teach concept/skill at representational level and provide opportunities for student to practice concept/skill by drawing solutions (see Representational Level of Understanding).

Provide opportunities for students to use language to explain their solutions and how they got them (see instructional strategy Structured Language Experiences).

- Have difficulty with basic facts/memory problems

Suggestions:

Regularly provide student with a variety of practice activities focusing on basic facts. Facilitate independent practice by encouraging students to draw solutions when needed (see the student practice strategies Instructional Games, Self-correcting Materials, Structured Cooperative Learning Groups, and Structured Peer Tutoring).

Conduct regular one-minute timings and chart student performance. Set goals with student and frequently review chart with student to emphasize progress. Focus on particular fact families that are most problematic first, then slowly incorporate a variety of facts as the student demonstrates competence (see evaluation strategy Continuous Monitoring & Charting of Student Performance).

Teach student regular patterns that occur throughout addition, subtraction, multiplication, & division facts (e.g. "doubles" in multiplication, 9's rule - add 10 & subtract one, etc.)

Provide student a calculator or table when they are solving multiple-step problems.

- Repeat procedural mistakes.

Suggestions:

Provide fewer #'s of problems per page.

Provide fewer numbers of problems when assigning paper & pencil practice/homework.

Provide ample space for student writing, cueing, & drawing.
Provide problems that are already written on learning sheets rather than requiring students to copy problems from board or textbook.

Provide structure: turn lined paper sideways to create straight columns; allow student to use dry-erase boards/lap chalkboards that allow mistakes to be wiped away cleanly; color cue symbols; for multi-step problems, draw color-cued lines that signal students where to write and what operation to use; provide boxes that represent where numerals should be placed; provide visual directional cues in a sample problem; provide a sample problem, completed step by step at top of learning sheet.

Provide strategy cue cards that student can use to recall the correct procedure for solving problem.

Provide a variety of practice activities that require modes of expression other than only writing

Student learning & mastery greatly depends on the number of opportunities a student has to respond!! The more opportunities for successful practice that you provide (i.e. practice that doesn't negatively impact student learning characteristics), the more likely it is that your student will develop mastery of that skill.

Videos

There are no videos for this Strategy. See Explicit Teacher Modeling

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Concrete - Representational - Abstract Sequence of Instruction Purpose The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning.

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Here is another way to introduce directed integer operations

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A nice way to introduce Algebra Tiles

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Using algebra tiles for distributing and factoring expressions!

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Look for other DESMOS activities by Paul Jorgens

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Create Algebra Tiles in GeoGebra

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Use as a tool for students to explore patterns in products of binomials.

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Look for Algebra Tiles by Mathies in your app store. A free Algebra Tile interactive

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These are only in two colours - my preference

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Using Algebra Tiles in Middle School Math

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Various manipulatives to use with algebra

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Various manipulatives to use with algebra

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Introduction

Even though they cannot make algebra easy, manipulatives can play an important role in the transition to a new algebra course:

They provide access to symbol manipulation for students who had previously been frozen out of the course because of their weak number sense.

They provide a geometric interpretation of symbol manipulation, thereby enriching all students' understanding, and making a powerful connection to another part of mathematics.

They support cooperative learning, and help improve discourse in the algebra class by giving students "objects to think with" and talk about. It is in the context of such reflection and conversation that learning happens.

There are four main commercial versions of algebra manipulatives. In order of their appearance on the market, they are Algebra Tiles (Cuisenaire), the Lab Gear (Creative Publications, now Didax), Algeblocks (Southwestern Publishing), and Algebra Models (Classroom Products). All four provide a worthwhile model of the distributive law. However, note that students prefer blocks to tiles, and moreover, only blocks (the Lab Gear and Algeblocks) allow work in three dimensions.

The Lab Gear

My creation! Read about it in:

And/or see the applet representing the factoring of a trinomial.

For a full presentation of the Lab Gear, see the Algebra Lab Gear books (from Didax), the Algebra: Themes, Tools, Concepts textbook ( free download), or the video course I helped write for Dr. Ed Dickey of the University of South Carolina. This article is adapted in part from the print material that accompanied the video. ( How to get some of my publications.)

The Algebra Tiles

The Algebra Tiles are inexpensive and widespread. They make it possible to do most of the activities that are needed to introduce and explain the distributive law and factoring.

The Algebra Tiles do not provide a corner piece. Since the corner piece is a very important support for beginners, I recommend you add corner pieces to your algebra tiles, either by purchasing them separately, or by just drawing them on paper.

The absence of 3-D blocks is a limitation, but much can be done in two dimensions.

Likewise, the fact that there is only one variable is a limitation, but it is not fatal.

The lack of 5, 25, 5x, and 5y blocks does not really interfere with the key concepts, though it does make it inconvenient to represent expressions involving larger numbers.

Algeblocks
Algeblocks do have a variation on the corner piece (the quadrant mat) and they do support two variables and three dimensions. They lack the 5, 25, 5x, and 5y blocks, which is merely inconvenient. You can use them to do almost all the basic activities about the distributive law and factoring.
Algebra Models
Algebra Models do include two variables, a 5 block, and a variation on the corner piece (the Work Tray.) They are tiles, and thus they are less expensive, but cannot be used in three dimensions. Their main advantage is that they are incorporated into an excellent Algebra text: Algebra Connections, by the College Preparatory Mathematics program (CPM).

Representation of Minus

The representation of minus is the most controversial part of manipulative models of polynomial algebra. The four commercially available products each handle it in a different way. These ways have consequences in four areas:

the misconception that -x is negative, and x is positive

the integrity of the area model of multiplication

the representation of expressions such as 5x-(x-1)

the complexity of the model

The Algebra Tiles model of minus is based on color: one color corresponds to positive numbers, and another to negative numbers. This is then generalized to variables: blue x's represent x, and black x's represent -x.

Advantage: the model is simple

Disadvantages:

the model reinforces the misconception that -x is negative, and x is positive;

the area model of multiplication is not geometrically sound when minus is involved;

only the simplest expressions involving minus can be represented.

The Algeblocks model of minus is based on position: blocks in the shaded area of the various Algeblocks mats are taken to be preceded by a minus.

Advantage: the model is simple

Disadvantages:

the area model of multiplication is not geometrically sound when minus is involved;

only the simplest expressions involving minus can be represented.

The Lab Gear model involves two different representations, both based on position: blocks in the minus area of the workmat, and blocks sitting on top of other blocks ("upstairs") are taken to be preceded by a minus.

Advantages:

the upstairs representation allows for a geometrically sound representation of minus in multiplication;

the two representations can be combined to represent expressions such as 5x-(x-1).

Disadvantage: it is initially harder to learn than the other models.

The Algebra Models model is based on color. However, at least in the CPM implementation, this is combined with a workmat-like minus area, which makes it possible to do much work with reading and simplifying expressions, equation-solving, and the like -- very much like what can be done with the Lab Gear. Moreover, CPM avoids using minus in the area model of multiplication, and thus the geometric integrity of the model is not undermined. (See next section.)

Multiplication with Minus

With the Algebra Tiles, to multiply (y+3)(y-2), you do the same thing that you would do with (y+3)(y+2), except that you use the "minus" colored tiles for -2, and for -2y and -6 in the product.

Similarly for (y-3)(y-2), you do the same thing that you would do with (y+3)(y+2), except that you use the "minus" colored blocks for -2, and for -2y and -3y in the product.

Note that in this model, the products (y+3)(y+2), (y+3)(y-2), (y-3)(y+2), and (y-3)(y-2), are all congruent, which is geometrically incorrect, since for example y+3 clearly should be longer than y-3.

The Algebra Models have exactly the same problem.

With Algeblocks, factor blocks are placed on the right or above the center of the quadrant mat if they are preceded by a plus, and to the left or below, if they are preceded by a minus. The resulting product subrectangles are considered to be preceded by a plus for the 1st and 3rd quadrant, or a minus for the 2nd and 4th quadrant.

In this model also, the area model loses its geometric integrity when minus is used:

The factors are sometimes embedded in the product rectangle, distorting its dimensions.

Once again, the products (y+3)(y+2), (y+3)(y-2), (y-3)(y+2), and (y-3)(y-2), are essentially all congruent, which is geometrically incorrect.

With the Lab Gear, the upstairs representation of minus accurately represents y-2 as 2 units shorter than y. It is then possible to multiply in the corner piece, and use upstairs blocks in the product, to obtain rectangles with geometrically correct dimensions. However how to do this is not easy to learn in a case such as (y-3)(y-2), and many Lab Gear users stop short of teaching this to their students. This is not a big problem, because the Lab Gear materials encourage a transition from blocks to symbols with the help of the "multiplication table" format for polynomial multiplication ("the box"). That format is visually related to the basic Lab Gear multiplication format, it works fine with minus, and it guarantees that the student does not remain dependent on the blocks.

Curriculum Materials

I encourage you to compare the curriculum materials provided with the four models. In my opinion, the Algebra Tiles manual is confusing and essentially unusable. There is much of value in the Algeblocks binder, which seems in large part inspired by the original Lab Gear books. Moreover, their ingenious (if mathematically flawed) approach to minus does make the Algeblocks easier to use.

But for mathematical correctness, depth, and range, check out my Algebra Lab Gear: Basic Algebra and Algebra Lab Gear: Algebra 1 ( Where to get them.) They are the result of many years of work with algebra manipulatives, with both teachers and students, and include all the basic lessons on the distributive law, a rich introduction to equation solving, plus three important features:

explicit attention to the transition from blocks to symbols

connections to the graphing of linear and quadratic functions

a remarkably accessible approach to completing the square and the quadratic formula

For extended work on integer arithmetic, plus a general introduction to Lab Gear at a more basic level, see The Algebra Lab: Basic Algebra. ( Where to get it.) For many lessons using algebra manipulatives, including some seminal ones, and some available absolutely nowhere else (e.g. a model of polynomial long division), seeThe Algebra Lab: High School, a whole book in PDFs available for free download on this site.

Many of the best ideas I have come up with in this domain have been incorporated (unfortunately without attribution) in CPM's Algebra Connections, a rather wonderful Algebra 1 textbook. This includes everything from the engaging perimeter problems, to using "make a rectangle" puzzles prior to the introduction of the distributive law, to "which is greater?" as a lead-in to equation solving, to the step-by-step introduction to completing the square. Hats off to the authors for their ability to recognize a good activity when they see its effectiveness with students, and many thanks to them for bringing these approaches to a much larger population than I could.

History of Algebra Manipulatives

The first use of manipulatives to illustrate algebraic ideas was by math educator Zoltan Dienes, who used base-10 blocks. Using the "rod" (10) as x, and the "flat" (100) as x ², he showed how to use base 10 blocks to illustrate the distributive law. For example, (x+5)(x+2)=x ²+7x+10 can be seen by making a 12 by 15 rectangle, and seeing that its area is 100+7·10+10. The idea was powerful, and launched the idea of algebra manipulatives, but this model falls short when trying to factor: x ²+7x+10, represented by 100+7·10+10 can be arranged into two rectangles: 12 by 15, or 10 by 18. The first one is the correct factoring (x+2)(x+5), but the second is not: x(x+8) does not equal x ²+7x+10.

Mary Laycock improved on Dienes' model, by using multi-base blocks. Instead of just working with base 10 blocks, a trinomial factoring had to work in all bases. In other words, the same layout should work whether x=3, 5, 6, or 10. She also introduced the "upstairs" representation of minus, which made it possible to represent a product involving minus like (x-1)(x+1) in a geometrically correct way.

Peter Rasmussen used base ten tiles, plus 5-, 25-, and 50-tiles for convenience. More importantly, he created the non-commensurable x, which solved the problem of false factorings encountered when using arithmetic blocks for variables. He also laid out multiplications in a tray that was a precursor to the corner piece. His MathTiles were placed along the tray's frame in such a way that the student saw them edge-on, thereby suggesting the one-dimensionality of linear measurements of the dimensions of the two-dimensional rectangles inside the tray. His model of minus combined Laycock's upstairs method with a color scheme. The tiles were only painted on one side, so that if a tile is turned over to its unpainted side, it is considered negative. (Number tiles, x-tiles, and x ²-tiles each had their own color.)

The Algebra Tiles were based on Rasmussen's ground-breaking model, without the upstairs representation of minus. Unfortunately, they dropped the multiplication tray. Fortunately, they did keep the non-commensurable x.

The Lab Gear was based on Rasmussen's and Laycock's models, keeping the non-commensurable x, and adding a non-commensurable y, and blocks for 5x, 5y, and 25, for convenience. (Blocks for 10, 100, and 1000 can be readily added to the Lab Gear by using base 10 blocks, which could be quite useful in younger grades.) The corner piece extends the uses of the tray for multiplication into the third dimension, and 3D multiplication is supported by x ³, x ²y, y ²x, and y ³ blocks. The minus model is extended beyond the upstairs model by the use of the workmat, with its minus area. The workmat is an environment for working with polynomials involving minus signs, as well as for equation solving and inequalities.

Algeblocks are based on the Lab Gear, including the x and y variables, and variations on the workmat. However, the 5, 25, 5x, 5y, are missing. The corner piece is replaced by the quadrant mat, and the upstairs representation of minus is abandoned.

Finally, the Algebra Models combine ingredients from all these sources: tiles rather than blocks, x and y variables, minus indicated by color, the Work Tray instead of the corner piece, a 5 block, but no 25, 5x, or 5y.

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Even though they cannot make algebra easy, manipulatives can play an important role in the transition to a new algebra course: They provide access to symbol manipulation for students who had previously been frozen out of the course because of their weak number sense.

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Today's math curriculum is teaching students to expect -- and excel at -- paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.

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Algebra Tiles - From concrete to abstract

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Algebra Tiles are a simple manipulative that can help develop students' understanding and confidence with algebra, at many different levels.

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This is an amazing resource for teaching algebra and many other topics. The Lab Gear manipulatives are similar to algebra tiles.

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Links and information about the Lab Gear, a comprehensive manipulative environment for algebra designed by Henri Picciotto.

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