Another way to support your student's understanding of quantity is by teaching him to count. Knowing how to count includes the following: knowing how to verbally count, understanding cardinality (that numbers represent quantities), and having the ability to interpret, represent, and compare quantities. Counting ability develops in the following developmental stages:
Before you learn about interventions that support one of these stages, you'll learn about tools you can use to support student mastery.
The intervention pages to come refer to the following tools: Counters, Drawing and Labeling Objects, Ten-Frames, Number Lines, Base-Ten Blocks, and Area Models. Use this page as a resource as you learn more about each of the tools in this activity.
Tool # 1: Counters
Counters are the concrete objects that students can manipulate in order to count or build quantities. Counters can come in many different forms: gummy bears, pebbles, buttons, Unifix cubes, and so on. When students are first learning to count, counters are the best tool to use, since the student can physically hold and move this object.
Tool # 2: Drawing and Labeling Concrete Representations of Objects
As students become better at counting or building groups of objects using counters, you can link the counters to visual representations of each quantity. Forbringer and Fuchs (2014, pp. 92-93) explain why and how to do so:
Sometimes students initially model a concept or procedure with counters and then are asked to perform the skill using visual or symbolic representation but without practice that explicitly connects the various forms of presentation. While normally achieving students may be able to successfully transition from one form of representation to another without the need for scaffolded support, students who have difficulty with number sense often struggle to connect the various forms of mathematical representation (Hecht, Vogi, & Torgesen, 2007), and so benefit when these connections are made explicit. For example, if students initially used M&Ms to count, they could use pencil, crayon, or chalk to draw a model of their M&Ms. Recording the written numeral on their drawing helps students connect the three-dimensional objects with pictorial and symbolic representations.
Drawing and labeling objects is a great tool to use if your students does not need to physically hold an object to count it.
Tool # 3: Drawing and Labeling Abstract Representations of Objects
As students become more fluent with drawing and labeling pictures of objects, they are able to represent objects through symbolic representations of the same objects. Using this tool, a student would draw a circle to represent each object in the story, instead of a picture of the object itself, knowing that each circle represents the actual object. Drawing and labeling abstract objects is faster than drawing and labeling more detailed pictures of objects.
Tool # 4: Ten-Frames
A ten-frame is a graphic organizer with ten empty spaces on it, over which a student can place concrete objects as he models a quantity. Ten-frames are a particularly useful tool for students who are learning about place value. Ten-frames help students quickly organize their counters into groups of ten when modeling larger numbers.
One of the best tools to help students connect three-dimensional concrete representation to two-dimensional visual representation is the ten-frame. A ten-frame consists of an empty 2 x 5 grid onto which students place counters...
When [students] are first learning to model numbers with frames, students should lay the frame horizontally so that there are five boxes in the top row. They begin by placing the first counter in the upper-left corner and progress from left to right across the top row, then move to the bottom row and continue placing counters from left to right, just as the eyes move when reading. Placing counters on the ten-frame in this set order helps students organize their counting and develop a mental model of each quantity.
Click here for a ten-frame template.
Tool # 5: Hundred Chart
Another graphic organizer that a student can use as he learns to count is a hundred chart. A hundred chart lists all the numbers from 1-100, in order, in rows of ten. This tool helps a student visualize each quantity (for example, 23 represents all of the space on the 100 chart up to the number 23). It can also help a student recognize symbolic representations of numbers as he learns how to count (e.g.,, the student can touch each number and count up to 31 to find out that 31 is written "31").
Tool # 6: Number Lines
A number line is another tool you can use to support your student's ability to represent a number visually or recognize symbolic representations of numbers. Teach your student that the distance from zero to a given number on the number line represents the size of that number. Or, teach the student to touch each number as he counts to find out what the symbolic representation of a certain number looks like.
Note: A common misunderstanding when using a number line is thinking that the number of ticks on the line represents the size of the number (when actually the quantity represented is the distance between the number and zero). Teach your student to focus on the spaces between ticks and not the ticks themselves, when identifying a quantity. You can do this by helping him build a number line, placing equal-sized units (e.g., blocks or squares) next to each other. Or, you might incorporate movement to draw your student's attention to the spaces. Forbringer and Fuchs (2014, pp. 99-100) write "...students can walk along a large number line taped on the floor, counting their steps as they move. The large muscle motion helps them focus on counting space. They can then progress to demonstrating bunny hops... on a small number line taped on the desk."
Tool # 7: Base-Ten Blocks & Place Value Organization Mats
Base-ten blocks are physical counters that are scaled to represent ones, tens, hundreds, and thousands. There are four types of base ten blocks, including a cube thousand (representing 100 units), a flat hundred (representing 100 units), a 10-rod (representing 10 units), and a single cube (representing one unit). These blocks allow for more efficient modeling of multi-digit whole numbers. To use base-ten blocks, a student will represent a given number using the lowest number of blocks. For example, if a student wanted to represent the number 23, he would use two 10-rods and 3 single cubes (instead of counting out 23 individual cubes). Using base ten blocks allows a student to quickly glance at the blocks and understand how many there are.
Tool # 8: Area Models
There are numerous concrete objects that students can manipulate in order to identify or build fractional parts of a whole. These include: fraction tiles, pattern blocks, legos, or strips of paper. Each of these tools can be used to visually model the number of equal parts a whole is divided into (as represented by the denominator). The student can then identify the number of equal parts represented by a particular fraction (as reflected in the numerator).
Learning about decimals is a natural extension of what a student already understands about base ten and fractions (Empson & Levi, 2011). The progression that a student goes through as he learns to count decimals is similar to the progression of counting fractions, and includes: deepening conceptual knowledge, interpreting, representing, and comparing decimals. This page includes intervention strategies that you can use to support your students in this area. As you read, consider which of these interventions best aligns with your student's strengths and needs in the whole learner domains.
Explicit Instruction
If you are intervening to support your students' ability to count decimals, you should start by explicitly teaching how a decimal is a different way to represent a fraction. This sounds like:
Typically, a student is not taught to verbally count decimals: rather, he learns that the position of a number behind a decimal (similar to numbers in front of a decimal) is related to it's value (0.1 is in the tenths place, 0.01 is in the hundredths place, 0.001 in in the thousandths place). This concept is akin to the concept of place value when learning whole numbers. Therefore, in order to teach a student to understand this concept, you should use the following intervention:
Activity A: Read the Place!
If your student is struggling to conceptualize decimals, it's important that he understands that the position of its number is related to its value (Empson & Levi, 2011). Therefore, this intervention is a simple one: teach the student to "read" the place in which a number sits. In this strategy, a teacher teaches a student to read tenths, hundredths, and thousandths place, simply by practicing reading this numbers until the student understands the concept.
Read the Place in Action
Teacher: In order to read and understand fractions, we need to understand that the position a number is in, behind the decimal point, tells us about it's value. So, we are going to practice reading these. The number directly behind the decimal is a tenth. What is it?
Student:A tenth.
Teacher: When we read this number, we say the name of the number, and then the word tenth. So, if I were reading this number 0.4, I would say four-tenths. What would I say?
Student: Four-tenths.
Teacher: Now you try.
Empson, S., & Levi, S. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH: Heineman.
In order to support a student's ability to interpret decimals, you should support his understanding with the following interventions:
Activity B: Convert Fractions to Decimals
If your student has developed fluency with fraction, but struggles to understand and identify decimals, teach him that any fraction with 10, 100, or 1000 in the denominator can be written as a decimal (Empson & Levi, 2011). In this intervention, the teacher asks the student to convert fractions to decimals.
Convert Fractions to Decimals in Action
Teacher: We've talked about how fractions are similar to decimals, and today we are going to practice converting fractions to decimals. This is called writing fractions in decimal notation. Now, I know that when I have a ten in the denominator, I have tenths. If I have 7/10, I can write this as a decimal by writing the numerator, or the number of tenths I have, after the decimal point in the tenths place, so let's try it. 7/10 would be what decimal?
Student: 0.7?
Teacher: That's correct! How did you figure that out?
Student: I thought about how the tenths place is the first place behind the decimal and I wrote the number of tenths that I have in that place.
Empson, S., & Levi, S. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH: Heineman.
Activity C: Identify Decimals using Pictorals
If your student is struggling to understand decimals, specifically the concept that the numbers behind the decimal point are parts of wholes, teach him to Identify Decimals using Pictorals. This intervention is similar to Identifying Fractional Parts using Pictorals, but the student practices saying the name of the fraction, and then converting it to a decimal. In order to intervene with this strategy, a student should have fluency interpreting and representing fractions, so that he can rely on his understanding of fractions to develop his understanding of decimals. In this intervention, the teacher gives the student a model and asks the student to interpret the decimal.
Note: If the student has limited understanding of fractions (and struggles to convert the fraction into a decimal), the teacher has two options: revisit fractions until the student can rely on this knowledge to convert fractions to decimals, or start teaching decimals with objects that have ten parts (such as a paper folded into 10 squares). This will help a student understand the concept of tenths.
Identify Decimals using Pictorals in Action
Teacher: Here I have a pie divided into 10 pieces. Two pieces are shaded. Since we know that decimals use the base ten system, I can identify the fraction or the decimal that this picture represents. If I were to write it as a fraction, what would I write?
Student: There are 2 parts shaded out of 10 shares, so 2/10.
Teacher: Well done. Now, I know how many tenths I have, I can write this number as a decimal: 0.2 (two tenths).
Jabal, R. F., & Rosjanuardi, R. (2019). Identifying the secondary school students’ misconceptions about number. Journal of Physics. Conference Series, 1157(4)https://doi.org/10.1088/1742-6596/1157/4/042052
Activity D : Identify Decimals using an Area Model or Grid
If your student is able to interpret decimals up to 10 using a pictoral, but needs continued support to identify decimals in the hundredths place, teach him to Identify Decimals using an Area Model or Grid. In this intervention, the teacher gives the student a partially filled in area model or grid and asks them to skip count to figure out the decimal that is represented.
Identify Decimals using an Area Model or Grid in Action
Teacher: This is an area model. It has some boxes shaded in, so we are going to identify what decimal is being represented. This area model has 100 boxes, so I'm going to skip count by 10 to figure out how many boxes are shaded. 10, 20. Now, I see that the third column has every box shaded but one, so I know there are 9 boxes shaded in that column. So, that makes 29/100 (twenty-nine hundreds) shaded. My decimal would be .29.
Wilson, A. (2017). Area model. New York, NY: Relay Graduate School of Education.
Jabal, R. F., & Rosjanuardi, R. (2019). Identifying the secondary school students’ misconceptions about number. Journal of Physics. Conference Series, 1157(4)https://doi.org/10.1088/1742-6596/1157/4/042052
Once a student is able to interpret decimals, he can start to represent (or build) decimals. You should support his understanding with the following interventions:
Activity E : Represent Decimals using an Area Model or Grid
If your student is able to interpret decimals, but needs continued support to build them on his own, teach him to Represent Decimals using an Area or Grid. In this intervention, the teacher gives the student a blank grid with 100 boxes on it and asks him to shade a given decimal, such as .29.
Represent Decimals using an Area Model or Grid
Teacher: Here is a grid that contains 100 boxes. I'd like to represent the decimal .29 by shading the appropriate amount of hundredths.
Writing decimal numbers shown in grids (video). (n.d.). Khan Academy. Retrieved February 1, 2023, from https://www.khanacademy.org/math/arithmetic-home/arith-review-decimals/decimals-intro/v/decimal-intuition-with-grids
Activity F : Represent Decimals on a Number Line
If your student is able to interpret decimals on a number line, but needs continued support to build them on his own, teach him to Represent Decimals on a Number Line. In this intervention, the teacher gives the student a blank number line and asks him to show a decimal, such as 0.25, and explain why he put it there.
Represent Decimals on a Number Line in Action
Teacher: This is a number line. I can use it to represent a decimal. Let's say I had to represent the decimal 0.25. First, I would need to designate those marks on the number line by drawing tally marks in between 0 and 1. I need to figure out how many tally marks I need to make in between the 0 and 1. To do this, I'm going to think about how many times 0.25 goes into 1. Well, 0.25 + 0.25 is 0.5, and 0.5 plus 0.5 is 1.0. So, that means there are four 0.25 in 1. That means, I need to make sure there are four equal-sized components, which means I'll draw three lines in between 0 and 1. Now, I have to count them. 0.25, 0.50. 0.75, 1. So, I know the first one is 0.25.
Turito. (2022, May 23). Decimals on Number Line: Representation, Examples - Turito. US Learn. https://www.turito.com/learn/math/decimals-on-number-line
Activity G: Write the Decimal
If your student has been able to successfully interpret decimals, it's time for him to learn to Write the Decimal.This strategy helps a student's ability to represent decimals using numbers. In this intervention, the student will convert a fraction written as words (five-tenths) to its corresponding decimal (0.5).
Note: An alternate activity is to have the student convert a fraction written in standard form into a decimal (such as 5/10 = 0.5).
Write the Decimal in Action
Teacher: We are going to practice converting written decimals to their numeric counterparts. Watch as I do one first. This says five-tenths. Ok, so I know the digit 5 is in the tenth place, which is the first position behind the decimal. It must be 0.5. Now, you try one!
Writing and Naming Decimals | Prealgebra. (n.d.). Courses.lumenlearning.com. Retrieved February 1, 2023, from https://courses.lumenlearning.com/monroecc-prealgebra/chapter/decimals/#:~:text=Write%20a%20decimal%20number%20from%20its%20name.%201
In order to support a student's ability to compare decimals, you should support his understanding with the following intervention:
Activity H: Which is Greater?
Once a student is able to interpret and represent decimals, he is ready learn compare decimals to decimals, and decimals to fractions. In this intervention, Which is Greater?, the student determines whether two decimals are greater, less than, or equal to each other. When teaching this strategy, it is helpful to have one number compared written as words (five hundredths) and one number in numerical form (0.5). This way, students learn that although the written word "hundredths" may sound bigger, it actually refers to a smaller position.
Which is Greater? in Action
Teacher: We are going to practice comparing decimals. My problem is, which is greater: five-hundredths or 0.5? I notice that one of the decimals is written as a number and the other is in numerical form. First, I have to read the first number and turn it into a decimal. Five-hundredths...well, it sounds bigger, but I know that hundredths are smaller than tenths. I also know that the 100ths place is too spots behind the decimal, so it must be written as 0.05. Now, I'm going to compare the two decimals: 0.05 and 0.5. The second, 0.5 is greater! I know this because five tenths is bigger than 5 hundredths.
Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for counting decimals. The example refers to the following problem.
Teacher: "Can you write the decimal two-tenths?"
Student writes 210.
In such a case, what might you do?
When you are planning your lessons, you should anticipate that your student will make errors throughout. Here are a series of prompts that you can use to respond to errors. Keep in mind that all students are different, and that some students might respond better to some types of feedback than others.
| Level of Support | Description of Scaffold | Script |
|---|---|---|
| Smallest Scaffold | Try Again. Give the student an additional opportunity to demonstrate his understanding by asking him to try again. | "Can you try that again?" |
| Medium Scaffold | Back it Up. If a student is struggling, back up your process. Show how to write a difference decimal, and see if this model helps. | "I can see that you are stuck. One-tenth is written like this: 0.1. Can you write two-tenths?" |
| Highest Scaffold | Model. If the student continues to struggle, model how you would write the decimal, explaining place value after the decimal point as you go. | "I can see that you are stuck. I'll show you how I write the decimal two-tenths. The two tells me how many tenths I have. I know that the tenths place is the spot right after the decimal point, so I will put a two in the tenths place, like this 0.2." |
If your student struggles to meet your objective, there are various techniques that you might try to adjust the activity to your student's needs.
| Activity | Description of Strategy | Script |
|---|---|---|
| All Activities | Make it Visual. If a student struggles to write decimals, revisit place value for decimals. Refer to or create a place value chart that shows the tenths, hundredths, and thousandths places. | "Let's review what we call the places after the decimal point." |