Conceptua Math content is initially displayed by domain, either Fractions or Multiplication and Division. Once a teacher is signed in, the content is also viewable by grade. When viewed or assigned by a domain, the content is shown by the "Big Idea" that the unit reflects. When viewed as a grade level scope and sequence, the content is shown as Units. Having two views for content provides an added measure of flexibility and adds value for accessing content. Teachers following the grade level scope and sequence can easily work from the grade level view. On the other hand, teachers working with students out of grade level can assign lessons that are not noted on the student screen by grade level but are instead noted by “Big Idea.”
Organization of Conceptua Math Units >In CM, Units and Big Ideas are organized into Topics and Lessons are a subset of Topics. Teachers can see the number and names for each lesson on the Teacher Dashboard. In addition to skills based lessons, each Unit or Big Idea contains Tool Investigations, Real World Investigations and content free Tools for teachers and students.
The Curriculum Library includes units organized by grade and domain. Each unit is broken down into topics with 2  5 lessons per topic. For information on how to navigate the Curriculum Library, watch the video below.
The videos and information in the lists below demonstrate how the lessons within each unit progress, explain the models and key ideas that should be reinforced, and provide teaching suggestions.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Student understandings. In Big Idea 1, the conceptual aspect of fractions is presented with a focus on partitioning or partwhole relationships. Beginning in grade 2 in the Common Core State Standards (CCSS), fractions are introduced. In the initial objective in grade 2 found in Geometry, students are partitioning circles and rectangles to show equal shares and develop the vocabulary of halves, thirds, and fourths. In grade 3, students continue the development of the conceptual understanding of fractions with the inclusion of a number line representation.
Demonstration of student understanding would include the ability to create equalsized parts, designate some of the parts to create a fractional amount, use multiple representations (including area and length models), and write a fraction. Across the topics in Big Idea 1, students are continually working with multiple representations that include area and number line models.
Teacher understandings. The introduction to fractions is essential in students’ understanding of what it means to have equalsized parts. Even though the software cannot demonstrate the idea that the parts do not have to be congruent (same size, same shape) in order to be equalsized, this is a concept that will need to be developed early in fraction instruction. In early stages of learning about fractions, students should be given the opportunity to use area models that may show, for example, fourths represented in different ways. (See figure 1.) Since the units are the same, students can use direct comparison methods where they impose one of the area regions on the other to determine that they represent the same area.
Figure 1. Same unit in area
There are two types of models used for fractions in Big Idea 1. One model is area, represented by the partwhole model of rectangles and circles. The other model is length, represented by the use of the number line or the strips. It is important as you are working with students to watch how they discuss the fractional models. It should be clear when they are using length models and when they are using area models, as the methods they use to compare the models to determine if fractional amounts are equivalent (or not) will be different depending on whether it represents length or area. For example, in figure 1, to determine that one part of either rectangle is the same sized area requires that students use a direct comparison method where they lay one area over another. In the case above, they may need to cut one of the parts into two or more pieces to see that the two parts are the same area. For length, students would lay the two lengths sidebyside to determine the relationship between the two quantities.
Notice that regardless of the model that is used, the size of the whole that is partitioned is consistent. This is an important idea related to determining equivalent fractions. Students’ use and understanding of the vocabulary is important. As they use words like numerator and denominator, students are also attaching meaning to those words when they create the multiple models that are used in the software.
Barbara Dougherty, Ph.D., is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
Students may approach fraction creation and identification tasks by applying what they’ve learned about whole numbers to these new numbers, called Fractions. Because of this, students may focus on only the numerator or denominator, not realizing that both are necessary in order to understand a fraction’s value. For example, students often think that 1/3 is greater than 1/2 because three is greater than two. Helping them to understand the relationship between numerators and denominators is an essential aspect of understanding fractions.
In addition, students may not realize that simply being one of three parts does not make a shaded area 1/3 of a larger area. As shown below, the large square is divided into three parts, and one part is shaded. The shaded area cannot be considered 1/3 of the larger square however, because it represents 1/4 of the large square, not 1/3.
(Adapted from Saxe, Taylor, McIntosh, & Gearhart, 2005.)
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
It is essential for students to have command of the manipulation of the models when working through these activities so they can develop conceptual understanding of the meaning of a fraction. As they manipulate the fraction models, they begin to understand and see important ideas such as the effect on the size of a fraction when more parts are used. Equally important is for students to verbalize their understanding, and for us, as teachers, to listen for that understanding and ask questions that guide students to be more explicit, use accurate terminology, and correct flaws in language that might foster or lead to misconceptions.
As students use different models, they should be encouraged to explain why one model is the same yet different from another model.
Example  
Context  Students count 1/2, 2/2, 3/2, 4/2, 5/2, . . . 
Student Talk  I am counting halves. No matter how many halves I count, I will still have halves. That’s why the denominator does not change. Every time I count one more half, I get one more in the numerator. That’s why it keeps changing. 
As students compare fractions, they need to be encouraged to use their understanding of a fraction to make comparisons.
Example  
Context  Students compare fractions 1/8 and 1/6 without the use of models. 
Student Talk  I know that 1/8 is smaller than 1/6 because when you cut one unit into 8 equal parts, the parts are going to be smaller than when you cut the same unit into 6 equal parts. 
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Student understandings. In Big Idea 2, the development of skill with computations (addition and subtraction) of fractions drives the lessons. In grade 4 in the Common Core State Standards, students begin their skill development with these computations by building on the conceptual understandings related to magnitude and equivalence of fractions. There is specific attention paid to developing the concepts of addition and subtraction by looking at addition as the joining of parts and subtraction as the separation of parts. Building on this idea, students decompose a fraction amount in a way similar to whole numbers. For example, 3/4 can be thought of as 1/8 and 5/8 or 1/2 and 1/4 much like they thought of decomposing 75 as 65 and 10 or 70 and 5 or 42 and 33. As students work through these standards, they also develop the use of common denominators for the operations of addition and subtraction. By grade 5, students can add and subtract fractions with a stronger focus on skill proficiency.
Demonstration of student understanding would include the ability to create equivalent fractions, add and subtract fractions, decompose a fractional amount in multiple ways, and apply addition and subtraction to word problem contexts. Across the topics in Big Idea 2, students are continually working with representing equivalent fractional amounts.
Teacher understandings. Understanding the foundations of addition and subtraction is critical. As students begin to work with addition and subtraction, the use of the decomposition of fractional quantities fits well. In the software, you will find the building of 1 with pattern blocks as one way to begin the computational development. For example, as you use actual pattern blocks in conjunction with the software, students can find that:
1 = 1/2 + 1/6 + 1/6 + 1/6 (hexagon is 1, trapezoid is 1/2, triangle is 1/6)
1 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 (hexagon is 1, triangle is 1/6)
1 = 1/3 + 1/3 + 1/3 (trapezoid is 1, triangle is 1/3)
It is important with tasks like this that students identify what the whole (1) is so that they notice that the area of the same piece, such as the area of the triangle, represents a different part of the whole, depending on what the whole is.
The use of decomposition helps to support students’ understanding of the concepts of addition and subtraction. Decomposition can be represented as an addition statement. For example, if ¾ is decomposed into two parts, we can write an equation such as 3/4 = 1/4 + 1/2. On the other hand, if we take 1/4 and 1/2 (both derived from the same whole or 1) and put them together, we can write the equation 1/4 + 1/2 = 3/4. Notice that the way in which the equation is written matches the action taken. If a fractional amount is decomposed, we start with the whole (3/4, in this case) and show the parts separated. If a fractional amount is composed, we start with the parts (1/4 and 1/2) and show the joining of the parts. Similarly, subtraction equations can be written. For example, if we start with 3/4 and take out or separate 1/4 from the quantity, we are left with 1/2. The corresponding equation that we might write is 3/4 – 1/4 = 1/2.
Notice that this helps to also illustrate the inverse relationship between addition and subtraction with fractions.
The vocabulary development for this component includes the use of decomposing to represent the taking apart and composing to indicate the joining together of fractional parts. This is also an opportunity to use addend, sum, difference, minuend, and subtrahend in the computation.
Barbara Dougherty, Ph.D., is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
When students first encounter addition and subtraction tasks involving fractions, they often view the numerators and denominators separately, instead of as two components of one number. Hence, they may perform calculations across numerators and denominators, solving an equation such 2/3 + 1/3 by writing 3/6. The partitioning, shading, and identification activities that students completed in Big Idea 1 can be built upon to help students understand that thirds plus thirds equal thirds, not sixths. Also, by comparing 2/3 to 3/6 using an area model or number line, students can see that 2/3 is already greater than 3/6, so adding any (positive) number to 2/3 would result in a sum even larger than 2/3.
In addition, by using the models from Big Idea 1 to visualize the fractions in an equation, students can estimate sums and differences before completing any calculations–this can help them to consider whether their answers make sense. For example, by visualizing a region divided into thirds with 2/3 shaded, students can see that the shaded portion is already more than 1/2 of the original region. Adding another third will result in a sum that is greater still. Finally, materials such as Pattern Blocks can be used to support students’ understanding of fraction addition and subtraction. By designating the hexagon as the whole, students can find that the blue rhombus is equivalent to onethird of the hexagon and the green triangle is equivalent to onesixth of the hexagon. Thus 2/3 (or two onethirds) + 1/3 (or one onethird) = 3/3 (three onethirds), not 3/6 (or three onesixths). In addition, students may not realize that simply being one of three parts does not make a shaded area 1/3 of a larger area. As shown below, the large square is divided into three parts and one part is shaded. The shaded area cannot be considered 1/3 of the larger square however, because it represents 1/4 of the large square, not 1/3.
Students may confuse 1/4 with 1/3.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
The conversations that occur in a classroom related to adding and subtracting fractions with like denominators should target some of the following understandings.
Encouraging students to use models and drawing are essential tools for having these conceptual conversations. They are important for all students and are essential for English language learners and students with special needs.
The following vignettes are examples of classroom conceptual conversations.
Example  
Context 
Teacher presents the problem: Use number lines to do the following problem, 5/6  2/6.
Bill used the following number lines to subtract 2/6 from 5/6. José used the following number lines to subtract 2/6 from 5/6. 
Dialog 
Teacher: What would you say about Bill and Jose’s work? Student: José’s number lines are the same length and Bill’s are not. Teacher: Why does that matter? Student: When you subtract things they have to be the same. Teacher: Aren’t Bill’s lines both cut up into sixths? Student: Yes, but the sixths of one is not the same as the sixths of the other. Teacher: Oh, you mean that Bill did not use the same unit of length for both of his fractions. Can you please correct Bill’s work? 
Example  
Context  Teacher presents the problem: 3/8 + 2/8. 
Dialog 
Teacher: Put 3/8 in one hand and 2/8 in the other hand. Now add them together and tell us what you have. [Students follow the instruction and state 5/8.] Teacher: How did you get 5/8? Where did the 5 come from? Where did the 8 come from? Student: I counted the number of eights and got 5. The 8 is because that’s what these pieces are. I just had more of the same pieces when I put them together. 
Example  
Context 
Problem: 3/5 + 1/5. Without using models students give the answers 4/5 and 4/10. The teacher writes both answers on the board. 
Dialog 
Teacher: I want you to think about what you know about the answer. 3/5 + 1/5, look at these two fractions and then decide if the sum if more than a half or less than a half. Student: It has to be more than 1/2 because 3/5 is more than 1/2 and I am adding 1/5 more to it. Teacher: How do you know that 3/5 is more than 1/2? Student: Because if I have fifths I need more than 2 but less than 3 to make a half. Teacher: What do the rest of you think about that? Can you use a diagram or a model to prove or disprove this thinking? [After some work the students indicate agreement.] Teacher: Okay. If you know that the sum must be more than 1/2, then which of your answers (pointing to the board) is correct. Please talk in your groups for 2 minutes and be prepared to share your decision and your reasoning. 
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
Student understandings. In Big Idea 3, the conceptual aspect of fractions is presented with a focus on partitioning or partwhole relationships, similar to Big Idea 1, but now extending fractions to mixed numbers. While students have worked with fractions that are less than or equal to 1 in grades 2, 3, and 4, there is an expectation in grade 4 that they will use mixed numbers in computations.
In order to compute with mixed numbers, students should be able to create a whole with equal sized parts and realize that if each part represents 1/8, for example, creating a whole would require 8 of those parts. Similarly, if there are 11 of the 1/8 parts, the composition of those parts would create a quantity that is represented by 11/8 or 1 3/8.
Demonstration of student understanding would include the ability to create equalsized parts, designate some of the parts to create a fractional amount, use multiple representations (including area and length models), and write a fraction that is less than, equal to, or greater than 1. Across the topics in Big Idea 3, students are continually working with multiple representations, including area and number line models.
Teacher understandings. It is essential that students’ understanding of what it means to have equalsized parts is well developed as they begin to work with mixed numbers. If a mixed number is represented as, say, 1 2/3, there must be the foundational understanding that:
There are two types of models used for fractions in Big Idea 3. One model is area, represented by the partwhole representations with rectangles and circles. The other model is length, represented by the use of the number line or the strips.
Notice that regardless of the model that is used, the size of the whole that is partitioned is consistent. This is an important idea related to all aspects of the students’ work in fractions.
While there are no specific objectives in grades 3 and 4 regarding an understanding of mixed numbers, this component cannot be omitted from instruction. If students are not given sufficient time to develop an understanding of mixed numbers and the representations that are linked to them, their ability to proficiently perform computations will be hindered.
Students' use and understanding of the vocabulary is important. Consistently using numerator and denominator continues to be an important component of communicating ideas about fractions.
Barbara Dougherty, Ph.D., is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
Students may not be aware that fractions can be greater than one. When working with mixed numbers, it is helpful for students to orient a number by considering the two whole numbers that it is between. For example, when viewing the number 2 1/4, encourage students to first note that it is between the numbers 2 and 3. They can then consider the fractional part of the mixed number to determine how much greater than 2 (or less than 3) 2 1/4 is.
In addition, when working with models for mixed numbers and improper fractions such as the one shown below, it is essential that students be aware of the unit, or whole. If the unit is one rectangle, then the shaded part of the model below shows 2 1/4 or 9/4. If the unit is all three rectangles however, then the shaded part shows 9/12.
It is not uncommon for students to confuse the unit, or whole, when working with fractions greater than one. Often answers that are considered incorrect are in fact correct answers to a slightly different question. Using the example above, a student who provides the answer of 9/12 for the shaded portion of the rectangles shows an understanding of part/whole relations, since the graphic does indeed show nine shaded parts out of twelve equalsized parts. What this answer indicates is that the student is considering all twelve of the parts to comprise the unit, instead of the graphic showing three units or wholes.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
The many activities in Big Idea 3 give ample opportunities for students to develop conceptual understanding of the relationship between mixed numbers and improper fractions. When these activities are presented and discussed, it is imperative that students are required to use language to explain what they see, draw, write, and think. This multimodal form of communication enhances their ability to understand and develop abstract processes and procedures.
As students write mixed numbers represented on a number line, they count the partitions of the unit from 0 to 1 to determine the denominator, then they count the total number of shaded parts to determine the numerator.
Example  
Context  Given a number line from 0 to 2 that is partitioned into 5ths and shaded to 6/5, students state an equivalent mixed number. 
Dialog 
Student: I know these are fifths because from 0 to 1 there are five equal parts. Teacher: Do you see fifths anywhere else? Student: Yes the whole line is cut into fifths. Teacher: So, how many fifths are shaded? Is there another way to express that number? 
As students convert improper fractions to mixed numbers they use many equivalent forms of 1 (e.g., 2/2, 3/3, 4/4, etc.).
Example  
Context 
Given 7/3 and asked to make a model and write a mixed number, the student creates three units divided into 3rds with appropriate ones shaded. 
Dialog 
Student: I know that 3/3 is one whole. Another 3/3 is another whole. That’s 6/3 so far. One more third makes 7/3 so I have 3/3 + 3/3 + 1/3 or 1 + 1 + 1/3 which is 2 1/3. Teacher: So, how many sets of 3 thirds are there in 7/3? (2). And those 2 sets equal how many thirds? (6) What other equations beyond 3/3 + 3/3 + 1/3 = 7/3 can you write? 
As students compare and place mixed numbers and improper fractions on a number line they explain the placements and locations of these numbers.
Example  
Context  Given a number line, from 0 to 3, partitioned into thirds, students are asked to place 1 4/8, 2 4/6, and 4/12 on the line. 
Dialog 
Teacher: How did you know where to place 1 4/8? Student: Well, 4/8 is the same as 1/2. So I put it at 1 1/2. Teacher: How did you decide where to place 2 4/6? Student: The line was already divided into 3 equal parts so I just thought about cutting each of those parts in half. Teacher: So you cut each third into 2 equal pieces and got sixths. 
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
Student understandings. In Big Idea 4, the development of skill with computations (addition and subtraction) of fractions is the focus. In grade 4 in the Common Core State Standards, students extend their skill development with these computations by building on the conceptual understandings and skills related to equivalence of fractions and computations with proper fractions. Using addition as the joining of parts and subtraction as the separation of parts continues to support student development. As students work through these standards, they also develop the use of common denominators for the operations of addition and subtraction. By grade 5, students can add and subtract fractions with a stronger focus on skill proficiency.
Demonstration of student understanding would include the ability to create equivalent fractions, add and subtract fractions, decompose a fractional amount in multiple ways, and apply addition and subtraction to word problem contexts. Across the topics in Big Idea 4, students are continually working with representing equivalent fractional amounts as part of the computational process.
Teacher understandings. Understanding the foundations of addition and subtraction is critical. As students begin to work with addition and subtraction, the use of the decomposition of fractional quantities fits well. In the software, you will find that the building of quantities that are greater than 1 supports this computational development.
There is a continued focus on the concept of the same whole. You can support students understanding this concept by drawing attention to the size of the wholes and noting that the parts must be from the same sized whole. Without that concept, the use of 2 3/8, for example, does not have any meaning. That is, if the whole from 3/8 is derived is a different size from the whole that constitutes each of the 2 wholes, then 2 3/8 does not make sense.
The use of decomposition continues to help to support students’ understanding of the concepts and skills of addition and subtraction. It is possible at this point to use fact families (or teams) to highlight the inverse relationship between addition and subtraction. For example, if we think of 3/8, 1 1/4, and 1 5/8, we can construct a fact team like this:
1 5/8 = 3/8 + 1 1/4
1 5/8 = 1 1/4 + 3/8
1 5/8 – 1 1/4 = 3/8
1 5/8 – 3/8 = 1 1/4
The use of fact teams or families also helps students consider the relationships among the parts and whole while accentuating the inverse relationships.
The vocabulary development for this component includes the use of decomposing to represent the taking apart and composing to indicate the joining together of fractional parts. This is also an opportunity to use addend, sum, difference, minuend, and subtrahend in the computation.
Barbara Dougherty, Ph.D., is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
As with addition and subtraction of the fractions encountered in Big Idea 2, students need to understand that when adding and subtracting fractions with like denominators, only the numerators are added or subtracted, and the denominators remain the same. Adding fractions that have a sum greater than 1, and subtracting fractions from a number greater than 1, presents another challenge for students. Helping them work up or down to 1 (or the closest whole number) and then complete the computation can be a very useful strategy for students. For example, when faced with a problem such as 7/8 + 3/8, students can be encouraged to approach the problem by breaking apart (or decomposing) 3/8 into 1/8 + 2/8. They can then see that 7/8 + 1/8 = 8/8 (or 1 whole); when they add the remaining 2/8 the sum is 10/8 or 1 2/8. Likewise, when subtracting 3/8 from 1 2/8, students can first subtract 2/8 to bring them to 1, and then subtract the final 1/8, resulting in a difference of 7/8.
When adding and subtracting mixed numbers, students can be encouraged to estimate the sum or difference by first looking at the whole numbers and determining a reasonable estimate for the answer. Once they’ve completed the actual computation, they can refer to their estimate to see if their answer makes sense. For example, when adding 3 1/6 + 2 5/6, students can first add 3 + 2 and determine that the final sum will definitely be greater than 5. Adding the fractional parts of the mixed numbers will tell them how much greater than 5 the final answer will be.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
Students build on the many understandings they have developed in Big Ideas 13 to add and subtract improper fractions and mixed numbers with like denominators. They can use models, pictures, language, and numerals to work problems and present solutions to the entire classroom. The many experiences they have had with the activities often lead to flexible thinking and more mental computation. It is important to continually ask students to expand on their answers, explain their thinking, and compare their strategies to those of others.
Students can use several methods to determine if the sum of 5/12 and 6/12 is closer to 1, 2, or 3.
When students compare sums such as 1/4 + 1/4 and 5/8 + 1/8, they might...
Using model to subtract mixed numbers such as 2 1/3 – 2/3, students may use different strategies such as...
Questions that Promote Conceptual Conversations
In order to have students reveal this kind of thinking, a teacher must ask questions that promote discourse and give students enough time and support explain their thinking.
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
Equivalent fractions name the same amount by using differentsized fractional parts.
This Big Idea focuses on developing a conceptual understanding of equivalence and the procedure for finding an equivalent for a given fraction. Students develop the understanding that any given fraction can be represented numerous ways but to be equivalent they must have the same value.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Student understandings. In Big Idea 5, students explore the concept and skills related to equivalent fractions. In grade 3 in the Common Core State Standards, students must demonstrate their understandings related to generating and explaining the equivalence of fractions. This continues in grade 4 where students move from a more conceptual approach to equivalency to formalizing an algorithm or rule about finding equivalent fractions. Critical to their understanding is the notion that equivalent fractions represent the same quantity.
Demonstration of student understanding would include the ability to not only apply an algorithm but to be able to model and justify why (or how) the fractions represent the same quantity. Generalizing the algorithm for finding equivalent fractions is essential in students’ work with computations.
Teacher understandings. To provide a strong foundation for students’ continued development of fractional concepts and skills, it is important to provide multiple experiences and models for students to identify, construct, and justify equivalent fractional quantities. Often, the approach to finding equivalent fractions is based solely on an algorithmic approach, but this is not sufficient for robust student understanding.
The software provides multiple models, including area and length, to use as the context for equivalent fractions. The use of the same sized whole continues to be an important component of this big idea. If the two wholes are not the same, the fractional quantities cannot necessarily be equivalent. In figure 1, the wholes are not the same and therefore we cannot say that 1/2 = 2/4.
Figure 1. The representations do not use the same whole and therefore 1/2 is not equal to 2/4.
It is possible to ask students to create instances where 2/4 does not equal 1/2 and compare them to instances where 2/4 does equal 1/2. As you ask students how these contexts are alike or different, you can develop the necessity of having the wholes be the same quantity.
The instructional component should include symbolic representations of the equivalencies that students find. For example, if they find that 6/8 and 3/4 can be used to represent the same length, then they should also represent it in equation form:
6/8 = 3/4
Student vocabulary for equivalent fractions should include "equivalency." An important idea to note is that equivalent fractions represent quantities that are equal. That is, an area of 2/4 is the same area as the one represented by 1/2 given that the whole is the same.
Barbara Dougherty, Ph.D., is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
Some students may be confused about the mathematical definition of "equivalent." While they may be able to generate multiple fractions that are equivalent to one another, they may not realize that these are all just different names for the same number. Having students show equivalent fractions on carefully partitioned number lines can help them to see that all fractions that are equivalent to 1/3, for example, are represented by the same point on the line. It is also important for students to understand that the relationship between the numerators and denominators of fractions that are equivalent remain constant, even though the actual numerators and denominators are quite different. For example, in the fraction 1/3, the numerator (1) is onethird of the denominator (3). Another way of saying this, is that if the denominator (3) is divided by 3 you’ll get the numerator (1). The inverse is also true – the denominator is three times the numerator. If you multiply the numerator (1) by three, you will get the denominator (3). This same relationship is true for all fractions that are equivalent to 1/3, regardless of the numbers comprising the numerators and denominators.
In addition, students can be encouraged to reflect on what they’ve learned about the relationship between the number of parts and the size of the parts, as discussed in the Common Misunderstandings section for Big Idea 1. If the region below is partitioned and shaded to show 1/3, that means that there are only 3 equalsized parts with one shaded.
If the same sized region is partitioned and shaded to show 5/15, each part must be much smaller (onefifth the size of the original), but there are five times as many parts.
Finally, when students begin using the multiplication and division procedure for finding equivalent fractions, i.e., multiplying or dividing fractions by n/n to find equivalent fractions, help them connect this to what they’ve already learned in previous grades about multiplying or dividing any number by 1. If students understand that all fractions that have the same numerator and denominator equal 1, you can build on this understanding to help them understand that multiplying any fraction by 2/2, or 3/3, or 10/10, or n/n, does not change the value of the fraction because you are simply multiplying the original number by 1. This is referred to as the Identity Property of Multiplication, which states that any number multiplied by 1 equals the original number (or n x 1 = n). The inverse is also true, any number divided by 1 equals the original number (or n ÷ 1 = n), however this is not generally considered one of the properties of numbers.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
When students use different models to find equivalent fractions they should be encouraged to explain how they “see” the equivalency in the model, and to make sense of why one fraction is equivalent to another. As they progress to finding equivalent fractions abstractly, i.e., using numbers only, they may rely on models or diagrams to explain their reasoning. Finally, they advance to determining equivalent fractions by multiplying by 1 (2/2, 3/3, 4/4…). To enable students to move through these stages from concrete to abstract, teachers plan on how to facilitate effective classroom discourse. The types of questions a teacher asks, to whom the questions are posed, and the length of time a student has to answer those questions are important considerations.
Asking the following types of questions will facilitate student discourse.
Example  
Context  Student is presented with models showing equivalent fractions. 
Dialog 
Teacher: When we use models to compare fractions we can decide if they are equivalent or not. What does that mean to say that fractions are equivalent? Student: They have the same amount of area shaded or have the same length on a number line. Teacher: Those two shaded areas do not look the same. How do you know that they are the same? Student: [The student should use a drawing or explain in words how he/she visualizes the areas to be the same.] Teacher: Why does it make sense that ___ is equivalent to ___? 
An example of a discussion when students are using the sliders to create models follows.
Example  
Context  The teacher asks the student to use the sliders to make equivalent fractions. 
Dialog 
Teacher: How did you know when to stop changing the slider? Student: I knew I needed the same area (or length) shaded so I “cut/divided” that area into more parts. I had to do that to all the parts. Teacher: What other fraction could you make that is equivalent to ___? 
An example of a discussion when students are comparing fractions follows.
Example  
Context  Teacher presents two fractions that are not equal, such as 3/5 and 3/4. 
Dialog 
Teacher: Is there another way you can say the same thing/describe the comparison? Student: 3/5 < 3/4 can be stated as 3/4 > 3/5. 
An example of a discussion when working with multiplication by 1 in the form of x/x follows.
Example  
Context  When completing a problem such as 1/3 = ? / 12. 
Dialog 
How did you know that the new numerator is 4? What were your thinking steps?

Considering whom to ask questions of is important in engaging all students in mathematical discourse. English language learners and other students who may need more processing time profit from having discussions in small groups before having whole classroom discussions. Asking questions to small groups and pairs enhances the classroom interactions and the engagement in the mathematical tasks for all students.
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
Student understandings. As students begin to develop stronger computational skills, the use of common denominators play a major role in their proficiency. Big Idea 6 builds on the common denominator and equivalency concepts that have been developed in the previous two Big Ideas. In the Common Core State Standards, at grade 4, students must demonstrate their understanding of adding and subtracting fractions with unlike denominators. There is an expectation that students will be proficient by grade 5.
To demonstrate their understandings, students should be able to explain why a common denominator is needed for addition and subtraction. They should be able to demonstrate a process for finding common denominators that can be justified with a rationale. In order to be proficient, the accuracy of their computations must also be present.
Additionally, it is important that students have the opportunity to perform addition and subtraction of fractions in problem contexts. That is, they should be given word problems that describe situations in which addition or subtraction computations would help them find a solution. These problems might involve joining and separation actions that are linked to addition and subtraction, respectively.
Teacher understandings. Using common denominators to add and subtract fractions involves understanding the relationships between equivalent fractions. When two fractions are rewritten with common denominators, students are actually finding fractions that are equivalent with respect to the same number of parts in a whole.
Common denominators are needed when adding and subtracting so that the sums and differences can be expressed in a way that makes sense. Without common denominators, we would have difficulty expressing the quantity that results when we join together two fractional amounts or when we separate two fractional amounts.
Common denominators are needed when adding and subtracting so that the sums and differences can be expressed in a way that makes sense. Without common denominators, we would have difficulty expressing the quantity that results when we join together two fractional amounts or when we separate two fractional amounts.
The concept of representing fractions with common denominators is based on the use of the same whole. If the two fractions that we are rewriting with common denominators were not created from the same sized whole, then our process would be faulty. Some exercises or tasks that students engage in often assume that it is understood that the fractions are taken from the same whole, but an explicit discussion of this with students is critical to their deep understanding.
Barbara Dougherty, Ph.D., is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
It is important for students to understand that when they are changing fractions in an equation, such as 3/5  1/3 = ?, to make two fractions with common denominators, they are not changing the values of the fractions in any way. Instead, they are simply finding equivalent fractions for 3/5 and 1/3 that have the same denominator and can then be added and/or subtracted. Reminding students of the strategies they used in Big Idea 5 to create equivalent fractions can support this understanding.
Another way to help students understand this important idea is to present the scenario of adding any two unlike groups together. For example, if your class has 12 boys and 11 girls, in order to add boys and girls together you need to find a common name for the two groups. In this example, one could use "children," "students," or "fifthgraders," to name a few. By finding a common name, you are not changing the values in any way merely changing the name so that the groups can be added. If students truly understand that equivalent fractions are merely different ways to name the same amount, or number, by finding a common denominator (or common name), they will be much less likely to struggle with this content.
As suggested in the Common Misunderstandings sections for Big Ideas 2 and 4, it is important to encourage students to estimate sums and differences before beginning any computation. This is particularly useful when working with fractions with uncommon denominators, as students' estimates can provide you (and them) with essential assessment information concerning what they do and don't understand. This information can then help you tailor your future instruction to best meet the needs of your students.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
Once students have developed understanding of the meaning of a fraction, the relationship of a fraction to benchmark quantities, how to find equivalent fractions, and how to add and subtract fractions with common denominators, they have the foundational skills to add and subtract fractions with different denominators. Students’ ability to recognize the relationship between the denominators is very important. Also important is requiring students to use the term "related fractions" in their talk. There are many conceptual conversations that will enhance students’ fluency with this idea.
As the class begins to use models and sliders for finding common denominators of related fractions, they should be asked to explain why the process is necessary.
Example  
Context  Given models of 1/3 and 1/6 
Dialog 
Teacher: The problem asks for us to move the slider for 1/3 to make it into sixths. Why do we need to do this? Student: We cannot add unless the denominators are the same. Teacher: Why do they have to be the same? Student: The number of equal parts must be the same so we can count the answer. 
As students begin to use a "one" fraction to find common denominators, they should be able to explain why they chose a particular "one" fraction since many "one" fractions can be used to find equivalent fractions.
Example  
Context  3/4  2/3 
Dialog 
I know that these fractions are not related so I need to multiply both of them by a "one" fraction. I know that I can change fourths to eighths by using 2/2, to twelfths by using 3/3, to sixteenths by using 4/4… I know that I can change thirds to sixths, to ninths, to twelfths… I can change them both to twelfths. So I need to multiply 3/4 by 3/3 and 2/3 by 4/4. 
To summarize the experiences with adding and subtracting fractions with common denominators, related fractions, and fractions with different denominators, students can be asked to describe the differences in fractions.
Example  
Context 
Students are given the three problems

Dialog 
Teacher: Look at these three problems. What do you notice is the same? What do you notice is different? Student: Possible answers: The all have fractions. You have to add in all three. The second one is easier. Teacher: Why do you think it is easier? Student: It has fewer steps because all you need to do is add the numerators. Teacher: What do you notice about the other two problems? Student: 3/4 and 5/12 are related fractions. Teacher: What does that mean? Student: That means that I only have to multiply one fraction by one. Teacher: Do all of you agree with what ____ said? Teacher: Talk to your partner about how that might be different from adding 2/5 and 1/3? Now work each of the three problems in your journal. But before working them, write in words what you know about the denominators of these fractions. Draw a diagram if that helps. 
Considering whom to ask questions of is important in engaging all students in mathematical discourse. English language learners and other students who may need more processing time profit from having discussions in small groups before having whole classroom discussions. Asking questions of small groups and pairs enhances the classroom interactions and the engagement in the mathematical tasks for all students.
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
Student understandings. Big Idea 7 continues with the development of addition and subtraction using common denominators. In the Common Core State Standards, at grade 4, students must demonstrate their understanding of adding and subtracting fractions with unlike denominators. There is an expectation that students will be proficient by grade 5 with all fractions, including mixed numbers.
As in Big Idea 6, to demonstrate their understandings, students should be able to explain why a common denominator is needed for addition and subtraction. They should be able to demonstrate a process for finding common denominators that can be justified with a rationale. In order to be proficient, the accuracy of their computations must also be present.
Additionally, it is important that students have the opportunity to perform addition and subtraction of fractions in problem contexts. That is, they should be given word problems that describe situations in which addition or subtraction computations would help them find a solution. These problems might involve joining and separation actions that are linked to addition and subtraction, respectively. Because this Big Idea includes mixed numbers, students should be engaged in problems that are situated in contexts that include amounts larger than 1.
Teacher understandings. Using common denominators to add and subtract fractions involves understanding the relationships between equivalent fractions. When two fractions are rewritten with common denominators, students are actually finding fractions that are equivalent with respect to the same number of parts in a whole. With the inclusion of mixed numbers, the whole components of a mixed number must also be the same size as the whole from which the fractional amounts were created. The consistency across these models is critical.
Common denominators are needed when adding and subtracting so that the sums and differences can be expressed in a way that makes sense. Without common denominators, we would have difficulty expressing the quantity that results when we join together two fractional amounts or when we separate two fractional amounts.
The concept of representing fractions with common denominators is based on the use of the same whole. If the two fractions that we are rewriting with common denominators were not created from the same sized whole, then our process would be faulty. Some exercises or tasks that students engage in often assume that it is understood that the fractions are taken from the same whole, but an explicit discussion of this with students is critical to their deep understanding.
Barbara Dougherty, Ph.D., is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
As students encounter more advanced fraction topics, it is imperative that they continually build on and apply what they already know about part/whole relations, equivalent fractions, and fraction computation. When using the procedure to convert mixed numbers such as 2 1/4 and 3 2/3 so that the fractions in each addend have the same denominator, students may mistakenly multiply the whole numbers as well. This could result in students converting 2 1/4 to 6 3/12 (by multiplying 2 by 3 and 1/4 by 3/3) and changing 3 2/3 to 12 8/12 (by multiplying 3 by 4 and 2/3 by 4/4). Remind students that they have already learned that equivalent fractions are merely different names for the same number and that 2 1/4 is a lot less than 6 3/12. When they convert 2 1/4 to a fraction with a different denominator, it should still be a number somewhere between 2 and 3.
It is also not uncommon for students to focus so much on computing with the fractional parts of mixed numbers that they forget to add or subtract the whole numbers as well. This is another reason to have students estimate answers before doing any computation. By deciding ahead of time about how big or small the answer should be, they can selfcorrect when they see that their calculated answer is far different than their estimated answer.
It is important for students to remember that when using the procedure to convert improper fractions to find common denominators, they are merely multiplying the original number by 1, thus its value is not changed. As with mixed numbers, encouraging students to reflect on the whole numbers that an improper fraction is equal to (in the case of a number such as 15/3) or between (in the case of a number such as 16/3) will help them consider the answers to their calculations for reasonableness. Students may benefit from reviewing these ideas by partitioning regions and number lines as discussed in previous Common Misunderstandings sections.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
As students begin to abstractly add and subtract fractions and mixed numbers with unlike denominators, it is imperative that they can explain their thinking using models, drawings, or language. Their explanations should include the use of terms and phrases such as proper fractions, improper fractions, mixed numbers, related fractions, fractions with unlike denominators that are not related, equivalent fractions, convert, numerator, denominator, common denominators, and a “one fraction” (a fraction equivalent to one).
The following vignettes are examples of classroom conceptual conversations between a teacher and a student.
Example  
Context  Teacher asks student to convert 27/12 to a mixed number. 
Dialog 
Student: 27/12 is 2 and 3/12. Teacher: How did you get that answer? Student: I know that 12/12 is equal to 1 whole. Another 12/12 is another whole. That makes 2 wholes with 3 extra twelfths. Teacher: Where did you get the 3? Student: 12/12 and 12/12 is 24/12. That leaves 3/12 since I started with 27/12. Teacher: Can you show the 24/12 and the 3/12 using a number line? 
The teacher presents a task in which students work in groups to discuss related fractions.
Example  
Context  Teacher directions: In your groups, write three related fractions for 2/3. Also write three fractions that are not related fractions for 2/3. 
Dialog 
Student: Some fractions related to the fraction 2/3 are 1/6, 4/15 and 7/30. Teacher: How did you decide that they are related fractions? Student: We can easily make 3rds into 6ths and into 15ths and 30ths. Teacher: How? Can you use a diagram to explain how to do that? Students draw a model of 2/3, and cut each third into 2 equal parts. They follow this by using a model that shows thirds converted into 15ths and 30ths. Student: See the whole now has 6 equal parts so if we have a denominator of 3 we can change it to a denominator of 6, or 15 or 30. Teacher: ____, draw a diagram that shows how to convert thirds to 15ths and _____, draw a diagram that shows how to convert thirds to 30ths. After students have completed this the teacher asks, Teacher: Now, write a fraction equivalent to 2/3 that has a denominator of 15 and a second fraction that has a denominator of 30. After students display their fractions on the individual white boards, the teacher says, "In your groups complete the following statements on the chart paper provided,

Example  
Context  Teacher presents the problem 1/4  5/8 
Dialog 
Teacher: As I walk around, I see two different ways of finding the difference. ___, can you explain how you got your answer? Student1: I decided to change the mixed number to an improper fraction. So 1 1/4 equals 5/4. Teacher: How do you know that? Student1: One whole is 4/4 and 1 more 4th is 5/4. Teacher: What do the rest of you think about that? Can you use a diagram to show that? The student draws a diagram shading in 1 1/4. Then cuts the whole into 4/4. Teacher: Okay. Then what did you do? Student1: Since 5/4 and 5/8 are related fractions, I just had to convert 5/4 to eights. It equals 10/8. Teacher: Can you show that on your diagram? The student cuts each fourth into 2 equal groups forming eights and writes 5/4 x 2/2 = 10/8. Student1: Now I have 10/8 – 5/8 which leaves 5/8. Teacher: ___, you did yours a little bit different. Please share your work with the class. Student2: I just converted 1/4 to 2/8. Then I subtracted the 5/8 from the 1 whole and was left with 3/8. Teacher: What did you do with the 2/8? Student2: After I subtracted the 5/8 I added what I had left to the 2/8 and got 5/8. Teacher: Hmm! Do you think that might always work? Class, what do you think about that? Let’s try ___’s method on 2 1/2  1 3/5. Then try, [S1]’s method to see if it gives you the same answer. The students try both methods and determine that they both give the same answer. 
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
5.NF.5 Interpret multiplication as scaling (resizing), by:a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
Student understandings. The focus in Big Idea 8 is the development of conceptual understanding and skill with multiplication of fractions. In grade 4 of the Common Core State Standards, students build on their understanding of equivalence as they represent a fraction as a multiple of a unit fraction. This concept is extended to multiply a fraction and a whole number using visual models. In Grade 5, students apply previous understandings of multiplication to interpret the product of two fractions of any kind.
Demonstration of student understanding includes the ability to use repeated addition to represent multiplication of a fraction by a whole number, to describe multiplication by a fraction as finding part of a starting amount, to explain why the result of fraction multiplication may be less than or greater than the factors, and to model, solve, and create word problems involving fraction multiplication.
Teacher understandings. To provide a strong foundation for fraction multiplication, students need to understand the relationship between repeated addition and multiplication, as well as the concept of multiplication as the product of two factors. The approach to fraction multiplication is often based solely on an algorithmic procedure, which prevents students from grasping the logic of the product relative to the factors. There must be foundational understanding that:
• Multiplying a fraction by a whole number results in a product that is smaller than the whole number, because it is a [fractional] part of it.
• Multiplying a fraction by a fraction results in a product that is smaller than either factor, because it is a piece of a piece.
The software provides ample opportunities for students to create visual models that reinforce the meaning of fraction multiplication and provide a sensemaking explanation for the relative size of the factors and the product.
¾ of 8 is shown by identifying 8 wholes and finding the sum of three onefourths, or threefourths, of it. ¾ of 8 is also shown as three parts when 8 is partitioned into 4 parts.
¾ × 8 = ¼ (8) + ¼ (8) + ¼ (8) = 2 + 2 + 2 = 6
¾ × 8 = 3 × 8 ÷ 4 = 24 ÷ 4 = 6
¾ of ½ is shown by identifying one half of a whole and finding the sum of three onefourths, or threefourths, of it. ¾ of ½ is also shown as three parts when ½ is partitioned into 4 parts.
¾ × ½ = ¼ (½) + ¼ (½) + ¼ (½) = 1/8 + 1/8 + 1/8 = 3/8
¾ × ½ = 3 × ½ ÷ 4 = 3/2 ÷ 4 = 3/8
The vocabulary development for Big Idea 8 includes the use of terms learned in whole number work, such as repeated addition, sum, factor, and product, as well as terms relating to fractions, including numerator, denominator, equivalence, proper fraction, improper fraction, and mixed number.
Robyn Silbey is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
Multiplication with fractions can be challenging for students for a variety of reasons. Since the actual computation is generally not difficult, students may arrive at correct solutions with little understanding of the mathematics involved. The confusion is often related to a misapplication of the rules students have previously learned for adding and subtracting fractions Thus, it is especially important to encourage students to estimate answers before they carry out the computation and probing for understanding once students have arrived at a solution.
Providing students with meaningful contexts and visual models, such as area models or number lines, can support their understanding of fraction multiplication. It can also help them connect to their understanding of multiplication with whole numbers. When using contexts and models to find a solution to a multiplication problem involving fractions, one aspect that can be especially challenging however, is understanding the order in which to represent the values in the problem. For example, in the problem, "Ryan needed to plant 3 fields of corn by the end of the day. If he planted 2/3 of his fields before lunch, how many fields did he plant?" the expression for this problem is 2/3 x 3 (or 2/3 part of 3), not 3 x 2/3 (3 groups of 2/3). When representing this problem visually, however, students need to be aware that they should begin by representing Ryan’s three fields and find 2/3 of that amount, as opposed to beginning by representing 2/3 of one field and finding three groups of 2/3.
Another aspect that can be challenging for students is understanding how the product of a multiplication problem can end up being a lesser value than either of the two other values in the equation. Many students have concluded, through experience or instruction, that multiplication always makes things bigger. For example, when working with fact families such as 2, 3, and 6, students are often encouraged to look for the largest number (6) and use it as the product, and use the two smaller numbers as the factors. While this works perfectly well when working with whole numbers, it is not necessarily the case when working with fractions. Helping students see patterns that occur when multiplying whole numbers, and extending those patterns to include fractions, can be very beneficial for students. For example, presenting students with a series of multiplication problems like 6 x 4, 6 x 2, and 6 x 1, and discussing what they notice about the product as the multiplicand (in this example 4, 2, and 1) gets smaller, can help them develop an understanding of the relationship between the numbers in the problem. You can then introduce the problem 6 x ½, and ask students to use the pattern to predict the answer.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
Multiplication of fractions presents the opportunity for students to either revise or extend their understanding of multiplication. Students who have conceptualized multiplication as repeated addition and not as equal groups may need to rethink these ideas and recognize that a problem such as 3 x 7 can be interpreted as 3 groups of 7, which will give the same answer as 7 + 7 + 7. In addition, prior experiences with multiplication defined only as repeated addition promote a misconception that a product must be larger than its factors. Multiplication of fractions will begin to address this misconception.
Classroom conversations that engage students in thinking, sharing their thinking, and using precise mathematical language can help address the concerns described above. The classroom dialogue should address the meaning of a problem, an estimation of the answer based on the meaning of the problem and the quantities involved, a description of how one can/did use the model to work the problem, and a connection to the symbolic recording of the process. In addition, multiplying by and with mixed numbers offers students a reason to use the distributive property. The models also extend students' conceptual understanding of the meaning of improper fractions and their equivalent forms. Often, teachers need to model precise mathematical language and mathematical notation for students.
The following vignettes are examples of classroom conceptual conversations between a teacher and a student.
Example  
Context  Problem: 2/3 of 3/4. 
Dialog 
Teacher: Talk to your partner and describe what this problem is asking you to do. What did your partner say? Student: This means that we need 2/3 part of 3/4. Teacher: Does everyone agree with that? Will the answer be more than 1 or less than 1? Student: It has to be less than 1. Teacher: Why do you think so? Student1: Because we are starting with 3/4 and taking a part of 3/4. Student2: Because we have two factors that are less than 1 and since 1 times 1 is 1, then the product of two factors less than one must be less than one. Teacher: So, is the answer going to be more or less than 3/4? Student: Less than 3/4. Teacher: If you were going to use a model or draw a picture, what fraction would you start with? Why? Student: I would start with 3/4. Teacher: Okay. Now draw that. Then talk to your partner and describe what you would do next. Student: We need to find a part of 3/4. Teacher: How much of a part? Student: We need to find 2/3 of that 3/4. Teacher: What does that mean? Student: Let's break up 3/4 in 3 parts. Then we will need 2 of the 3 parts. Oh, it's 2/4. 
The beginning of this vignette shows how a teacher provides opportunities for students to discuss and reason about the mathematics in the situation. This encourages all students to be engaged and gives processing time to all students.
Example  
Context  Problem: 1 1/2 x 4/5 
Dialog 
Teacher: What does this problem mean? Paraphrase this problem. Write the paraphrase in your journal. Teacher: Share what you wrote in your group If everyone in your group agrees, display a “thumbs up”. _____, please share your paraphrase. Teacher: Now look at the two factors and write an estimate of the answer. Don't work the problem. You will be asked to defend your estimate. Teacher: As soon as you have your estimate, write it on the board. If your estimate is already listed, record a checkmark next to the answer. (Students write the following on the board: 4/5, a little more than 1, a little less than 1 1/2, ) In your groups, look at these estimates and try to decide why these estimates might be reasonable. Then determine which one you think is the best estimate. Be prepared to defend your thinking. Student1: Our group thinks 4/5 is the best estimate because we have more than one group of 4/5. Student2: Our group decided that a little more than 1 was the best estimate because there is one group of 4/5 plus a half group of 4/5, which should be more than 5/5. Student3: Our group decided that whomever gave the answer of 1 1/2 might have thought of it as 4/5 of a part of 1 1/2 which would be a little less than 1 1/2. We still don't know if that is a better estimate than a little more than 1. Teacher: Now that you have heard the reasoning of your classmates, have a one –minute conversation with your group and determine if you have changed your mind. Teacher: Okay. Now draw or use manipulatives to represent the problem and find an exact answer and compare your exact answer to the estimate. Teacher: ____, I heard you say earlier that you could paraphrase 1 1/2 x 4/5 as one group of 4/5 plus a half of a group of 4/5. Why can you do that? Student: It's like when we multiply by 14, we can multiply by 10 and then by 4. Teacher: Oh, since you can break apart 14 or decompose 14 into 10 + 4, you thought of decomposing 1 1/2 as 1 + 1/2. So now your problem reads (1 + 1/2) x 4/5. Now let's write what you said: one group of 4/5 so we can write this as 1 x 4/5 plus a half group of 4/5 so we can write this as 1/2 x 4/5. Now what does this equal (Pointing to 1 x 4/5)? And what does this equal (Pointing to 1/2 x 4/5)? How did you decide that 1/2 x 4/5 is 2/5? Student: If I have four 1/5 then 1/2 of them is two 1/5 or 2/5. 
The class discussion ends with the students working on a similar problem, by first paraphrasing it, then estimating the answer, writing a defense for their estimate, working the problem and then comparing the solution to the estimate.
Nora Ramirez is a retired faculty member from Arizona State University and PastPresident of TODOS, an organization that advocates for equitable and high quality mathematics education for all students, particularly Hispanic/Latino students.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
Student understandings. The focus in Big Idea 9 is the development of conceptual understanding and skill with division involving fractions. In grade 5 in the Common Core State Standards, students interpret a fraction as division of the numerator by the denominator. This concept is extended to divide a whole number by a unit fraction. In Grade 6, students extend their knowledge to include finding the quotient of two fractions of any kind.
Demonstration of student understanding includes the ability to use visual fraction models and equations to model, solve, and create word problems involving division with fractions and to explain why the result of division involving fractions may be greater than or less than the dividend or the divisor.
Teacher understandings. To provide a strong foundation for division with fractions, students need to understand the relationship between multiplication and division. Although the “invert and multiply” procedure provides an accurate quotient, it does not promote deep conceptual understanding. Since division of a whole number by a whole number is the same as multiplying the number by its inversea unit fractionstudents can reason why dividing by a fraction is the same as multiplying by its inverse or reciprocal.
8 ÷ 4 = 8 × ¼
8 ÷ ¼ = 8 × 4
In order for students to grasp the concept of division with fractions, they must understand that:
• Dividing a fraction by a whole number results in a quotient smaller than the divisor or dividend, because a part is being separated into smaller, equal portions.
• Dividing a whole number by a fraction results in a quotient greater than the divisor, because the whole number is being partitioned into many more pieces.
• Dividing a fraction by a fraction involves partitioning a part into differentsize portions, with a quotient dependent on the size of the divisor and the dividend.
½ ÷ 6 = 1/ (2 × 6) = 1/12
6 ÷ ½ = 6 × 2 = 1
¾ ÷ ½ = ¾ × 2 = 6/4 = 1½
The vocabulary development for Big Idea 9 includes terms from whole number division: dividend, divisor, and quotient, as well as the fraction terms learned in multiplication plus reciprocal and inverse.
Robyn Silbey is the Director for the Center for Excellence in Science, Mathematics & Engineering Education at Iowa State University. Over the past 20 years, she has directed elementary mathematics pedagogy projects at the University of Hawaii and the University of Mississippi. She is widely published and is particularly known for her work on the Measure Up! project, a rational numbers program for elementary school students that originated in Russia, and which she administered in the United States.
Division with fractions can present substantial challenges for students and teachers alike. The difficulty is generally twofold. First, understanding what it means to divide when either the dividend, divisor, or both, are fractions can be very challenging to comprehend. In addition, the standard “invert and multiply” procedure is generally easy to execute but is rarely taught in a meaningful way. As a result, students may apply the procedure correctly but have no idea what their answer means, or, students may make a minor computation error that results in an incorrect solution but have no way of judging the reasonableness of their answer. Building on what students know about division with whole numbers, and the relationship between division and multiplication, can support their understanding of, and accuracy with, fraction division.
One strategy for helping students connect division with whole numbers to division with fractions is to present them with a series of equations such as those shown below. Ask students to notice what happens to the quotient as the value of the divisor changes. Help students see that as the divisor gets smaller, the quotient gets bigger. 6 ÷ 6 = 1 6 ÷ 3 = 2 6 ÷ 2 = 3 6 ÷ 1 = 6 6 ÷ ½ = ? 6 ÷ 1/3 = ?
By helping students consider the equations shown above using a quotative model of division (may also be referred to as “repeated subtraction” or “measurement division”), they can interpret the equation to mean “How many ______s (or groups of ______) are in ______?” Using the second equation as an example, the question becomes, “How many 3’s (or groups of 3) are in 6?”
In the quotative model of division, the size of the groups is known (this is the divisor) and the quotient refers to the number of groups. Another type of division context is referred to as partitive division. In a partitive model, the number of groups is known (this is the divisor) and the quotient refers to the amount in each group. Using the same equation as above, the question becomes, “If you divide 6 into 3 equal groups, how many are in each group?"
As with whole numbers, providing students with contexts can help them think more deeply about division with fractions. Given a problem in which both the dividend and divisor are fractions such as, ¾ ÷ ¼, any of the following examples of quotative division may help students make sense of the problem and understand the quotient (3):
• If you have ¾ pound of ground beef, how many ¼ pound patties can you make?
• If you need ¾ cup of flour, and you only have a ¼ cup measuring cup, how many scoops do you need?
• If you want to run ¾ of a mile, how many ¼ mile laps do you need to run?
Partitive division problems are much more difficult to consider when fractions are involved, and not all division problems with fractions are easy to interpret using a context, but student understanding can be greatly enhanced when initial instruction focuses on sensemaking rather than blind application of procedures. As with addition, subtraction, and multiplication of fractions, students should be asked to estimate quotients before performing any calculations and reflect on the reasonableness of answers once they are attained.
The “invert and multiply” procedure can add an entirely new layer of confusion to fraction division. One way to help students understand this algorithm is to reflect on what they know about the relationship between multiplication and division. Although they may not be able to articulate it, most students in fifth grade and above can be guided to the understanding that half of 24 and 24 divided by 2 both equal 12. Likewise, onefourth of 100 and 100 divided by 4 both equal 25. Providing students with many opportunities to reason about and discuss this important mathematical idea can help them come to the generalization that multiplying by a number is the same as dividing by its reciprocal (or a x b = a ÷ 1/b).
Another aspect of the invert and multiply procedure that can be confusing is that students may find that in some cases you actually can divide straight across numerators and denominators and arrive at the correct answer even though they’ve been told this is strictly forbidden^{1}! For example, given the problem from before, ¾ ÷ ¼, if I divide across the numerators I get 3 ÷ 1 = 3; divide across the denominators I get 4 ÷ 4 = 1; so my final answer is 3/1, or 3. We run into problems, however in situations where the numerator and denominator of the divisor are not factors of the numerator and denominator of the dividend, respectively. Using the problem ¾ ÷ 2/3, dividing straight across the numerators and denominators results in a quotient of (3/2)/(4/3) [Fig. a].
The invert and multiply procedure prevents having a quotient that is a complex fraction (a fraction with a fraction as the numerator and/or denominator) and can actually be explained by starting with this complex fraction and multiplying it by (3/4)/(3/4), a form of n/n, which is equivalent to 1 [Fig. b]. Our denominator is now ¾ x 4/3, which equals 1, and our numerator becomes ¾ x 3/2, instead of the original problem ¾ ÷ 2/3 [Fig. c].
Even before performing any calculations, however, students should be asked to consider what they know about the numbers in the problem to help them determine a reasonable estimate for the answer. Given the problem ¾ ÷ 2/3 students may say the answer will be close to one since ¾ and 2/3 are pretty close together, or that the answer will be a little greater than one since ¾ is a little larger than 2/3. Helping students reason about the numbers involved, connect to what they know about division with whole numbers, and build on the relationship between multiplication and division can help students make sense of fraction division.
^{1 }Actually, one can always divide across straight numerators and denominators, but depending on the values of the numerators and denominators, may end up with complex fractions.
Julie McNamara, Ph.D., is a mathematics specialist with TeachingWorks at the University of Michigan. She is the coauthor of Beyond Pizzas and Pies . . ., a book about fractions featured on this website. She has conducted extensive research focused on elementary and middle school students' challenges with fractions.
Organizing and chunking concepts and skills around big ideas facilitate greater retention, recall and transfer of learning when key concepts and big ideas are explicitly part of daily instruction. There are a number of strategies that teachers can employ to facilitate an understanding of the Big Ideas, some of which are noted below.
There are a number of researchers who have been studying what is referred to as the pedagogical content knowledge of teachers; that is their knowledge of the instructional methods for teaching mathematical content. Research that focuses on instructional approaches of teachers indicates that many elementary teachers struggle identifying which concepts require the most focus and which methods are most effective. Having content organized around big ideas and key concepts, with researched best practices readily available, supports teachers as they plan and deliver lessons.
The Class Dashboard is where teachers spend the most time in Conceptua Math. It gives teachers the ability to preview any or all parts of a lesson sequence, assign lessons and track student progress. The decisions made in the Assignments tab impact what is displayed in the "Topic/Lesson" and "Investigations and Tools" tabs. Students must be assigned either a topic or lesson in order to begin working in Conceptua Math.
Teachers can assign topics or individual lessons. Assigning by topic ensures that students receive approximately one week of carefully sequenced instruction that moves students from models to procedures. Assigning a single lesson may be helpful for review work or as a strategy for managing pacing. Information presented under the remaining tabs is determined by the selected assignment.
How to Assign a Topic >The Dashboard has three onscreen reports and a link to create customized downloadable reports.
Progress View >Coming Soon!
Leading mathematical discussions can be challenging. It can be difficult to meaningfully engage all students while introducing key vocabulary and linking new content to previous learning. The Conceptua Math Opener assists the teacher in delivering an effective opener by providing a set of carefully selected examples that supports the introduction of new content, skills, and vocabulary. This premade online opener engages students through the use of interactive visual models, while supporting teachers in facilitating conversation using rich mathematical language. Upon completion, students know what they are going to learn and teachers are confident that they have facilitated student use of the Standards for Mathematical Practice.
The following inclass videos show Conceptua Math Openers in reallife situations. The first four videos are spontaneous, and demonstrate the vital nature of classroom discourse while revealing some student misconceptions.
Remarkable interaction between two students that demonstrates "adding on." The topic is skip counting in multiplication. [3:05]
Students make a human number line of 1/4 and 1/2. Misconceptions are revealed and resolved. [2:32]
See a studenttostudent discussion regarding the relationship between 1/8 and 1/4. [1:36]
A helpful list of Talk Moves can be accessed by teachers at any time. Simply click the Talk Moves icon in the lower right of any Opener. Watch the video below to learn more.
Click here to view a printable Productive Talk Rubric that can help teachers analyze and improve their presentation of openers and talk moves.
The Lesson Openers have builtin teaching prompts called “sticky notes.” Each screen has one to five sticky notes with prompts. The prompts provide a framework for discourse aligned with the Common Core Standards for Mathematical Practice (SMP).
To conduct a Lesson Opener:
A good lesson opener is vital to a successful lesson because it immediately engages students in rich discussion and discovery as the learning intention is laid out. The Lesson Opener is an opportunity for teachers to build a high level of mathematical practice and discourse into the classroom. Upon completion of the Opener students should be primed for new learning.
Research tells us that classroom discussions are critical to learning mathematics. Teacher discourse or “talk moves” can assist teachers in facilitating discussions in which all students think deeply about content, question their own thinking, share their solutions, question the thinking of their peers and problem solve. Chapin and O’Connor define five styles of conversational prompts that teachers can use to implement classroom discussions. These “talk moves” refer to the ways that teachers can encourage their students to listen carefully, think deeply and communicate their reasoning using mathematical language.
More About Productive Talk Moves >
Rich teacherstudent discussion also supports the development of students’ reasoning and supports their abilities to express their reasoning.
1) Revoicing: This talk move allows teachers to interact with a student who is unclear or if the teacher is not sure that the other students in the class understand.
The Guided Lesson is an online 815 minute session in which students work individually, receiving instruction and practice utilizing interactive visual models, numbers and operations while the teacher circulates and provides support. Guided instruction is a structure for teaching in which the student is supported through increasing levels of difficulty as they develop proficiency in a concept or skill. Often guided instruction is conducted in small flexible groupings, based on performance data, in which the teacher focuses on a particular concept, strategy or skill. Guided lessons are typically short, contain scaffolding and provide immediate feedback so students gain intended learning outcomes.
Goals of the Guided Lesson >
Conceptua Math lessons use a variety of visual models that build conceptual understanding through a deliberate instructional sequence. Students are supported as they move from virtual manipulatives to number sentences and equations, building conceptual understanding and procedural fluency. The teacher engages students in mathematical discourse during the Guided Lesson. The lesson plan provides sample questions. These discussions between teacher and students during the Guided Lesson reinforce the Standards for Mathematical Practice. They emphasize strategies, reasoning, modeling, and perseverance to understand and solve problems.
Adaptive teaching draws upon the teacher’s insightfulness and judgment. Each oneminute visit begins when a student raises their hand after failing a skills check. The formative assessment screen allows the teacher to easily view the student’s actual answers (see graphic) and gain a window into the student’s thought process. The teacher has an opportunity to engage the student in a discussion to identify what is impeding the student’s success and prescribe the appropriate remediation lesson.
For each and every lesson, Conceptua Math provides three remediation options, on the spot:
The “More” lesson gives the student another try with different examples.  
The “Remedial” lesson covers the same content but with more visual and verbal support.  
The “Preskill” lesson is appropriate if the student has a misconception and needs to revisit prior content before returning to try this lesson again. 
By choosing from these three different remediation options, teachers are able to differentiate and personalize with ease.
Every lesson sequence in Conceptua Math has a Guided Lesson, which is used individually by students while the teacher circulates, engages and supports learning. Since students cannot progress in the lesson until a problem is correct, it is important that the teacher have a signal for students to indicate when they are in need of assistance.
Every Guided Lesson ends with a “Skills Check.” The teacher uses the results to check individual understanding, review student errors, and determine the student’s next learning experience or lesson.
The teacher can evaluate students with the Indicators of Understanding found in the lesson plan. Some ways students can demonstrate these indicators is through the use of tools or lesson examples in Conceptua Math, with manipulatives or through journaling. These indicators, used with the formative assessment, can give the teacher a strong sense of what students know and can do.
Teachers have a vital role during the Guided Lesson to engage with students in mathematical discussions and check for student understanding. A teacher is highly active while students are working in the lesson.
Before class, the teacher has the option to preview the Guided Lesson online. While students are working in the Guided Lesson, the teacher circulates among the students. The teacher supports students in multiple ways:
Numerous research studies show the benefits for using concrete materials to make mathematics meaningful for students. Experiences with handson manipulatives and visual models allow students to understand how abstract equations operate. While concrete manipulatives can be grasped and physically moved by students to build conceptual understanding, similar understanding can be gained through virtual manipulatives.
Virtual manipulatives are interactive representations that can provide students with the same opportunity to build conceptual understanding if they can be manipulated to show concepts, processes and relationships. Some virtual manipulatives are static representations that resemble concrete models and therefore can only provide pictorial support, as students cannot use them to perform the same actions as concrete objects. Interactive visual representations can be manipulated in a way that is similar to concrete manipulatives. Students can control the actions by sliding, moving, flipping, stacking, grouping and performing other actions for sense making.
The Lesson Closer is a sequence of online screens displayed to a class or group in which the teacher uses onscreen prompts to guide a 5 minute discussion to summarize the lesson. A good lesson ends with a teacherdirected closer. A good closer is not simply stating what has been taught in a lesson, but rather, it requires the student to reflect, summarize, articulate and demonstrate what has been learned. Effective lesson closers are studentcentered, requiring students to share thinking through additional practice, discussion or problem solving while permitting the teacher to identify if students have mastered the learning outcome of the lesson. The teacher can then determine if students require reteaching or if they are ready to move on to the next lesson.
Goals of the Lesson Closer >Every lesson sequence has a Lesson Closer, which can be conducted using an interactive white board or projector with the entire class or using a smaller device with a small group of students. There are three types of Lesson Closers indicated on the first screen of the online Closer and listed in the Lesson Plan.
Lesson Closers consist of one example presented on 1 or 2 screens and have builtin teaching prompts called “sticky notes.” Each screen has one to five sticky notes with prompts. The prompts help facilitate a discussion in which students demonstrate an understanding of the lesson and summarize their thinking.
Each Lesson Closer contains a Journal Prompt located on the last screen. It is one of the two Journal Prompts available in each lesson. Both prompts are available on the last page of the lesson plan. Types of Journal Prompts:
Instructional strategies regarding the use of closers can be used before the lesson, during the Closer and after the Closer.
Prior to the Lesson >Just as research indicates the importance of reflective teaching, summarizing and reflecting are important for student learning. Researchers such as Robert Marzano note that that summarizing strategies have a substantial effect on student understanding of academic content. Summarizing in a group setting allows students an opportunity to hear the thinking of their peers and solidify their own understanding of a particular concept or skill.
Sometimes during a discussion in which students share strategies or reflect on a problem as a group, a student can transition from using one strategy to another that is more useful or efficient. Allowing students to summarize the important features of the lesson and reflect on ways they addressed problems during guided practice facilitates their ability to organize their thinking and internalize the skills. The Lesson Closer is always relevant. It can be used as an effective means of developing communication, providing important feedback for students and assisting the teacher in gauging the success of the instruction. In addition to summarizing within a group setting, journaling is a great way for students to individually reflect on their learning and consolidate new learning with previous understandings.
Real World Investigations are powerful and engaging learning experiences where students learn about the mathematics in their world. These activities consist of 3 parts, each requiring a more indepth application of learning as students move from analysis of provided data to integration of their own data from the real world. All Investigations are freely available at the bottom of each topic page within the Curriculum Library.
Providing students with divergent problems that can be solved in different ways encourages creative reasoning, and offers an opportunity for students to challenge assumptions while explaining and connecting mathematical concepts. Students use their own data, linking mathematics to experiences that are relevant and real.
There are two ways to access Real World Investigations:
1. In the Curriculum Library, at the bottom of each topic page. (See image below)
2. Via the Real World Investigations page, which contains a selection of Investigations to get you started.
To view or print the Investigation student worksheets and lesson plan, click the Lesson Plan link next to or under the Investigation title. Each Lesson Plan document contains a lesson plan for all 3 parts of the Investigation, an optional Unit Launcher, the student worksheets (and solutions), and stepbystep instructions on presenting the Introduction and facilitating the Investigation all in an easytouse interactive PDF.
Before the lesson, review the Lesson Plan and print the student worksheets you will use from the Lesson Plan PDF. You may also want to View the interactive online discussion guide that you will use with a projector or interactive whiteboard to facilitate the wholeclass introduction at the beginning of the lesson.
Each of the three data tiers in Real World Investigations lends itself to specific instructional approaches. Generally, you are helping your class to matriculate from wholeclass, clearlydirected instruction in the first tier to group work with great latitude for individual inventiveness in the third tier.
Tier I Instruction: All Data is Provided >
George Polya developed a widely adopted inquiry model for problem solving which includes the following four stages:
Teaching students these stages and providing a variety of problemsolving opportunities is critical in learning mathematics. Students need opportunities to think critically and to apply learned skills in a meaningful way.
The first stage, Understand the Problem, is often the one that creates the most difficulty for students. It involves not only reading the problem, but also determining what information is necessary, if all necessary information is present, and what information is extraneous. It is important to provide sufficient support to ensure students understand the problem. Students should be asked to restate, paraphrase or simplify the problem prior to developing a plan.
There are many strategies for solving problems. For example, students can make models and drawings; make a list; work backward; or use interactive manipulatives. Teaching a heuristic can assist students in learning how to plan and solve problems. For example, one powerful strategy is to assist students in reading tables, graphs, and/or charts and using these formats to display solutions.
Once students carry out their plan, sharing their solutions as a group offers opportunities to explain and justify thinking as well as learn alternate strategies used by peers. Polya’s inquiry model and the Conceptua Math Real World Investigations provide an excellent vehicle for students to develop their problem solving skills and meet the Standards for Mathematical Practice.
Differentiated instruction is a philosophy of teaching based on the belief that all students have the capacity to learn and that it is the teacher’s role to adapt instruction to match students’ differences so learning occurs. Similar to the philosophy of universal design for learning (UDL), the design of curricula considers the needs of a wide range of learners rather than remediating or reteaching after instruction proves unsuccessful. There are many barriers to learning that can be mitigated when the instructional design attends to readiness levels, language needs, learning and accessibility needs.
Conceptua Math uses methods of instruction shown to be effective for students with special needs. With appropriate supports, many students with disabilities can participate in the general education environment. Conceptua Math supports inclusion in the least restrictive environment:
Teachers are able to create classes, in effect subgroups of students, and enroll students into the class as a way to reorganize a class into ability groupings. Students who are in multiple classes will have a dropdown menu on their dashboard to select the class for which lessons are to be completed.
To Create a Class >The Student Dashboard gives the student access to assigned lessons, unassigned lessons (teacher must allow access), tools and investigations.
To Access Differentiation Lessons >The tool investigation associated with the assigned lesson and all other tool investigations are readily accessible to students. In the student dashboard, clicking the Tool Investigation button to the right of the Assignments button launches the Investigation associated with the assigned topic.
Accessing other Tool Investigations >Differentiated instruction is an approach to teaching that requires the teacher to actively plan for student differences in classrooms. Actively planning for student differences prior to launching a lesson can eliminate much of the need for remediation and individual support during the guided lesson. However, at times it is necessary to differentiate during a lesson or upon completion of the lesson according to student performance and formative assessment data.
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