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).
Knowing how to count fractions starts with a conceptual understanding of what a fraction represents. Before reading about interventions to support your student's ability to count fractions, read how a kindergarten and first-grade teacher, Ms. Murphy, exposes her students to the concept of fractions. As you read, consider: How is teaching a student to count fractions different than teaching a student to count whole numbers? Why is building this conceptual understanding so important before starting to intervene?
Conceptual Understanding of Fractions_0.pdf
Note: An Equal Sharing problem is one that allows a total number of items to be distributed to a given number of groups or people (Empson & Levi, 2011), such as the brownie example used in this reading.
As you've read, Ms. Murphy's students are able to grasp the concept of fractions without understanding the terminology, and without being able to read or write fractions themselves. Therefore, intervening to support a student's ability to count fractions (and, later, decimals) is different than intervening to support a student's ability to count whole numbers because a teacher will not start by teaching verbal, or rote, counting. Instead, she will develop a student's conceptual understanding of fractions, and then teach a student how to interpret, represent, and compare fractions. This page includes 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
Once your student has developed a conceptual understanding of what a fraction represents, you can explicitly teach what a fraction is. This sounds like:
Once a student has developed a conceptual understanding of what a fraction means, you can support his understanding of the terminology by implementing following interventions:
Activity A: Teach the Terms
If your student is struggling to conceptualize fractions, it's important that he becomes familiar with the terminology. Therefore, this intervention is a simple one: teach the student to describe the equal pieces that make up the whole (Empson & Levi, 2011). The teacher should present a visual representation of a whole and ask a question, "How many of these pieces fit into the whole?" The student should then look at a visual representation of a whole, count the number of smaller pieces that fit into the whole. Then, you should tell the student the term they would use to describe each fractional piece.Initially, encourage the student to write the name of the fraction in words instead of in symbolic notation.
Teach the Terms in Action
Teacher: Let's look at this brownie. It's divided into parts. How many of these parts fit into the whole brownie?
Student: 4
Teacher: Oh, so the brownie is divided into fourths. Into what?
Student: Fourths
Teacher: Since four of these little pieces make up the whole brownie, each little piece is called a fourth. What is this piece called?
Student: A fourth.
Teacher. Yes, one fourth. You can write that 'one fourth' or '1 fourth.'
Activity B: Teach the Symbols
As your student develops understanding of fractional terms, you should introduce the standard symbolic representation of a fraction (a/b). According to Empson and Levi (2011), symbolic notation is intuitive to students who understand fractional terms. They write:
For example, consider how the fraction symbol 2/3 can refer to an amount of a candy bar. The numerator, 2, refers to the number of pieces in the share. The denominator, 3, refers to the size of the piece relative to the whole. To represent 2/3, young children would write "2 thirds." When students begin to independently use correct fraction terminology to describe shares, you can introduce the standard fraction symbol (a/b).
Teach the Symbols in Action
Teacher: Let's look at this brownie. It's divided into parts. How many of these parts fit into the whole brownie?
Student: 4
Teacher: Oh, so the brownie is divided into fourths. Into what?
Student: Fourths
Teacher: Since four of these little pieces make up the whole brownie, what would we call two pieces of this brownie?
Student: Two fourths.
Teacher. Yes, two fourths. Show me how you could write that?
[Student writes 'two fourths']
Teacher: Another way to write two fourths is 2/4. Can you write your fraction the way that I did?
[Student writes '2/4']
Teacher: This means, were are talking about two out of four pieces that make up the whole brownie.
As a student learns to interpret fractions, he will follow a similar progression as he did for interpreting whole numbers. At first, the student might use more concrete objects to interpret fractions. Then, he'll develop the ability to interpret fractions using more abstract representations (such as drawing a circle cut into 5 pieces to represent a cookie cut into 5 equally-sized pieces). In order to support a student's ability to interpret fractions, you should support his understanding with the following interventions:
Activity C: Identifying Fractional Parts of Concrete Objects
If your student is struggling to understand fractions, specifically the concept that wholes are composed of many equal sized parts, teach him using concrete objects (e.g., a chocolate bar or a group of cubes) (Empson & Levi, 2011). Teach the student to start by examining the whole and then ask the student to describe how much each person would get if the object was divided in a certain way.
Identifying Fractional Parts of Concrete Objects in Action
Teacher: Here is a chocolate bar. Let's pretend that I wanted to split the chocolate bar with you. I'm going to divide the chocolate bar right here, so that we each get the same amount. How much chocolate do I get?
Student: One piece
Teacher: Yes. How many of these pieces make up the whole chocolate bar?
Student: Two
Teacher: Yes, so I get one half of a chocolate bar. How much do I get?
Student: One half of a chocolate bar.
Teacher: Yes. How much chocolate do you get?
Student: One half.
Teacher: One half of what?
Student: One half of a chocolate bar.
Activity D: Identifying Fractional Parts using Pictorals
Once a student has learned how to identify fractional parts of concrete objects, you should move on to teaching the student to describe fractional parts using pictorals (Empson & Levi, 2011). In this intervention, the teacher draws or references a fraction bar, and asks a student to identify the fraction as she shades different parts.
Identifying Fractional Parts using Pictorals in Action
Teacher: Here is a fraction bar. How many equal parts is it divided into?
Student: Three
Teacher: Yes. So if I asked you to color in the entire fraction bar, how many parts would you color?
Student: Three
Teacher: Yes. What if I asked you to color in one part of the fraction bar? How much is that?
Student: One out of three.
Teacher: Yes. We can say one out of three, or 1/3. What if I asked you to color in two parts?
Student: 2 out of 3, or 2/3.
Activity E: Identifying Fractions on a Number Line
If your student is able to name the fractional parts of a rectangle, you'll want to teach him to extend this understanding to identify fractional parts on a number line (Empson & Levi, 2011). In this intervention, the teacher gives the student a number line from zero to one. Then, he draws the student's attention to the number of parts that make up the whole distance between zero and one.
Identify Fractions on a Number Line in Action
Teacher: This is a number line from zero to one. Today, we are going to talk about what to call numbers that fall between zero and one on the number line. Each of the spaces between these tally marks represent parts of one. How many spaces fit into the distance between zero and one?
Student: Three.
Teacher: So, what can we call each of these parts?
Student: Thirds.
Teacher: Yes, I can label them on the number line like this: one third, two thirds OR like this: 1/3, 2/3.
Once a student is able to interpret fractions, he can start to represent (or build) fractions. You should support his understanding with the following interventions:
Activity F: Representing Fractional Parts using Concrete Objects
If your student is able to identify fractions with objects, but needs additional support to build fractions on his own, teach him Representing Fractional Parts using Objects. This strategy is similar to Identifying Fractional Parts of Concrete Objects but instead of having the student simply identify the fraction, the teacher asks the student to build (or represent) the fraction with objects (Empson & Levi, 2011). For example, the teacher might give the student a chocolate bar (such as a Hershey's bar that has 12 equal parts) ask the student to make 8/16. The teacher can then ask challenge the student to identify another fraction that amount makes (6/12 is the same as 1/2 of the bar).
Represent Fractions with Objects in Action
Teacher: Here is a chocolate bar. As you can see, it has 12 equal parts. I am going to show you how I use this bar to represent fractions. For example, if I wanted to show 6/12, I could break off 6 pieces, like this (teacher breaks bar in half). Now, I have 6/12. As I look at this piece, I also notice that I just broke the bar in half, so I also am representing the fraction of 1/2. So, that tells me that 6/12 is actually the same as 1/2. Now, I'm going to ask you to represent a fraction using this Hershey bar.
Activity G: Representing Fractional Parts using Pictorals
If your student has mastered interpreting fractions with pictorals, but needs additional support to build the fractions on his own, teach him Representing Fractional Parts using Pictorals (Empson & Levi, 2011). In this intervention, the teacher tells the student a fraction and asks the student to build it, using the model.
Represent Fractional Parts using Pictorals
Teacher: I'm going to show you how I can build 1/3 by drawing a fraction bar.
[Teacher draws a fraction bar and draws two lines, splitting the fraction bar into 3 equal pieces.]
Teacher: Ok, so I now have 3 parts total. Now, I need to represent 1 of them. I'm going to shade in this segment. (Teacher shades in segment.) Now, I have 1/3. Now, it's going to be your turn. I'll give you a piece of paper to represent the fraction 2/8.
Activity H : Represent Fractions on a Number Line
If your student is able to interpret fractions on a number line, but needs continued support to build them on his own, teach him to Represent Fractions on a Number Line (Empson & Levi, 2011). In this intervention, the teacher gives the student a blank number line and asks him to show a fraction (such as 1/3).
Represent Fractions on a Number Line in Action
Teacher: This is a number line. I can use it to represent a fraction. Let's say I had to represent the fraction 1/3. First, I would need to designate those marks on the number line by drawing tally marks. So, I know that I need to separate my unit interval into three equal parts, so that means I'll draw two tally marks. Now, I have to count them. 1/3, 2/3, 3/3, which is one. So, the space between 0 and this first tally mark is 1/3.
Activity I: Write the Fraction
If your student has been able to successfully interpret fractions, it's time for him to learn to Write the Fraction (Stein et al., 2006).This strategy helps a student's ability to represent fractions using numbers. In this intervention, the student will convert a fraction written as words (two-thirds) to its corresponding fraction (2/3).
Write the Fraction in Action
Teacher: You are going to convert a fraction written in words into numbers. Watch as I do this one: two-thirds. Hmmm....Well, 2 is the first number. Let me think: two-thirds. If I have two-thirds of something, that I know I have two parts out of three. That means that two must be my numerator, or the number of parts I have out of the whole. So, I'll write two on the top part of the fraction bar. Three represents how many pieces are in the whole, and I know that number goes on the bottom of the fraction bar as my denominator. So, two-thirds written as a fraction is 2/3. Now, you try!
Activity J: What Stays the Same
Once your student is able to write fractions, he is ready to write equivalent fractions (Empson & Levi, 2011). In this intervention, the teacher shows the student how to write equivalent fractions by visualizing an object being split multiple times, and writing a fraction every time the object is split. By using a visual (such as imagining splitting a chocolate bar or cutting a cake), the student can understand how different fractions can represent the same amount.
What Stays the Same in Action
Teacher: I'm going to show you different ways to write 1/3. The new fractions I write will look different, but they will still represent the same amount. In my head, I'm going to picture a cake split in thirds. Can you see that cake, split into thirds?
Student: Yes.
Teacher: If I have 1/3 of the cake, I have one piece. Now, I could split all of those pieces of cake again. So, in my mind, I'm going to cut each of those pieces in half. How many pieces of cake will I have now?
Student: 6.
Teacher: That's right. The denominator is now 6. But, if I still have 1/3 of the case, I have more than 1 piece out of the 6. I now have how many pieces of cake?
Student: 2.
Teacher: So, fraction is equivalent to, or the same size as 1/3?
Student: 2/6.
In order to support a student's ability to compare fractions, you should support his understanding with the following interventions:
Activity K: Compare the Fraction
Once a student is able to match, draw, and write fractions, he is ready to Compare the Fraction (Stein et al., 2006). In this intervention, the student determines whether a fraction is more than, equal to, or less than 1. Once a student can solve these types of problems with fluency, he can then practice comparing two fractions to identify which is greater. Empson and Levi (2011) write that when asked to compare two unit fractions such as 1/5 and 1/6, "children...often mistakenly conclude that 1/6 is bigger than 1/5 because 6 is bigger than 5. They make be thinking in terms of the number of total parts that is created when a whole is partitioned, or they may be simply looked only at the denominators." Therefore, interventions that support a student's ability to compare fractions are important to help a student better understand the relationship between the numerator and denominator.
Compare the Fraction in Action
Teacher: You are going to be reading a fraction and identifying if it is greater or less than one. Watch as I complete this problem: 2/3 is ______ 1. Well, I see that my numerator is 2, so I have 2 parts. I see that my denominator is 3, so there are 3 parts total. If I have 2 parts out a whole that is made up of 3 parts, do I have the whole thing? If my numerator were the same as my denominator, or 3/3, I would have the whole thing. I also know that 3/3 would equal 1. Do I have more than the whole thing? If my numerator were greater than my denominator, such as 4/3, it would be greater than 1. Hmm... I think maybe I have less than the whole thing. Let me check. If I have 2 parts of 3 parts total, I have less than the whole thing, since my numerator is smaller than my denominator, so 2/3 must be less than one. Now, you try.
Activity L: Ordering Fractions
Once a student can state whether a given fraction is greater than or less than one, he is ready to practice ordering fractions. In this intervention (Empson & Levi, 2011), the student learns principles that help him put fractions in order. For example, the teacher may teach the student one (or more) of the following principles of order:
At first, a student might only order fractions with one principle in mind. However, over time, a student should be able to use all four principles to put fractions in the correct order.
Ordering Fractions in Action
Teacher: We are going to order fractions, and I want you to decide which is greater based on the following principle: The more people sharing, the smaller each share will be. Let's think about this. Hmm...the bottom number, or the denominator, tells me how many sharing make up the whole, so if my denominator is small, what size are the shares? (They are big) What happens as my denominator gets bigger? (The shares get smaller)
Teacher: So if the top number, or numerator, is the same, we can look at the size of the denominators to figure out how big the shares are. My first problem is :Which is greater? 2/7 or 2/11. Well, there are 7 people sharing in the first, and 11 in the second. If there are 11 shares, each piece will be smaller than if there are 7 shares. The numerator is the same. So, 2/7 is greater than 2/11. Now, you try.
Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for counting fractions.
Teacher: "Can you write the fraction two-fourths?"
Student writes 24
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 to write that again?" |
| Medium Scaffold | Back it Up. If a student is struggling, back up your process. Show another fraction, and see if this model helps. | "I can see that you are stuck. One-fourth is written like this: 1/4. Can you write two-fourths?" |
| Highest Scaffold | Model. If the student continues to struggle, model how you write the fraction, explaining the terms numerator and denominator as you go. | "I can see that you are stuck. I'll show you how I write the fraction two-fourths. The two, my numerator goes on top." (Teacher writes 2.) "The fourths tells me how many parts I have total, and that goes on bottom. This number is called the denominator." |
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 when writing fractions, bring out a visual to help him develop his conceptual understanding. | "Here is a pie cut into four pieces. Two of the pieces are colored in, so I have 2/4. Let me show you how I write that fraction, using this visual." |