Unleashing Children's Multiplicative Brilliance

 

What is the capital of Kansas?

If you are like me, you probably memorized in school that the answer is Topeka. But what else do you know about Topeka? 

Unlike many of you, I grew up about 40 minutes away, so Topeka is more than just a fact I memorized. I remember field trips to the zoo from my tiny hometown of 700 people. I remember eating lunch at a park with a giant wooden slide made of rolling logs that practically guaranteed you would leave with splinters. As I grew older, I learned about the famous Brown v. Board of Education case connected to Topeka. And I still remember standing beneath John Steuart Curry’s Tragic Prelude mural in the state capitol rotunda.

In other words, I do not just know Topeka is the capital of Kansas—I have experiences and connections tied to it.

I think we want the same thing for children when they encounter a multiplication fact like 8×6. It is one thing to know the answer is 48. It is another thing entirely to experience the relationship.

I am not opposed to flash cards. But initially, those flash cards should help students see the mathematics, not just recite it. Students need opportunities to visualize the multiplicative relationship and talk about what they notice.

When students see 8×6, I want them to see 8 groups of 6. But I also want them to see something deeper: the reorganization of those groups into our base-ten system. Forty-eight is not just “the answer.” It is 4 tens and 8 ones. Multiplication is about reorganizing groups of units into decimal units.

Just as importantly, I want students to describe what they see using conceptually consistent language such as “groups of.” That language matches the mathematics students are visualizing.

We often default to the word “times,” but I have found that “groups of” language is more productive for many learners. “Times” tends to sound comparative, while “groups of” emphasizes the actual structure of multiplication. With both children and adult learners, I consistently notice richer reasoning when students describe multiplication as groups of units rather than simply “times.”

A few weeks ago, I asked an adult learner to solve 8×6. They responded by skip counting:

6, 12, 18, 24, 30, 36, 42, 48.

That is certainly a conception of multiplication, but it is largely additive reasoning disguised as multiplication. The learner must keep track of both the running total and the number of counts. Many elementary students—especially those who struggle in mathematics—approach multiplication this same way.

The problem is not that repeated addition is “wrong.” The problem is that students often never move beyond it. When multiplication remains rooted only in additive reasoning, students struggle to develop the multiplicative reasoning needed later for proportional reasoning, fractions, and algebra.

My goal is that when students see 8×6, they connect the fact to a network of ideas. Of course I want them to know the answer fluently, but I also want flexibility.

I want students to know that 8 groups of 6 and 6 groups of 8 produce the same product. I want them to understand that rectangular arrays can be viewed as rows or columns depending on how we define the unit.

And I want them to see how multiplication facts are connected to the properties of algebra. In fact, they can use these strategies to move from known facts to unknown facts. 

For example:

• 5 groups of 6 plus 3 groups of 6 gives us 48. Welcome to the distributive property. 

• 2 groups of 4 groups of 6 also gives us 48. Welcome to the associative property. 

• 10 groups of 6 minus 2 groups of 6 gives us 48. Another use of the distributive property. 

I also want students to see how halving and doubling preserve products:

4×12=48

16×3=48

These are not isolated tricks. They are connected relationships within multiplication.

And with a strong understanding of place value, students can extend these same ideas naturally to problems such as:

8×600

8×0.6

8×6/5

In every case, students are still working with 48 of a particular unit:

48 hundreds 

48 tenths 

48 fifths 

The first two problems involving decimal units “place” nicely, 4 thousands and 8 hundreds, and 4 ones and 8 tenths because the regrouping happens at 10 of a particular unit (e.g., 10 hundreds = 1 thousand). The third problem, 8 x 6 fifths, can also be expressed as 48 of a unit,  fifths,  but since fifths regroup every 5 fifths, 48 fifths can also be expressed a 9 ones and 3 fifths. 

That is the kind of multiplication understanding I want for children—not isolated answers but connected mathematical relationships.

Every place we travel gives us opportunities to make connections and build understanding. Learning state capitals becomes more meaningful when we have experiences tied to those places. The same is true for multiplication facts.

Students deserve more than memorized answers. They deserve opportunities to explore the rich mathematical connections hidden inside those facts.

To learn more ways to connect mathematics ideas that may seem disconnected check out my book, See It, Say It, Symbolize It: Teaching the Big Ideas in Elementary Mathematics.

To learn more about the author's work and contact him about professional development opportunities  email him at patrick_e2e@outlook.com or check out his website  www.elevate-2-excellence.com.

See It, Say It, Symbolize It--Transformations of Functions

 The See It, Say It, Symbolize framework is about first making explicit the conceptual idea that you want students to see. This happens in conjunction with the relevant conceptual language with both ideas bound together by the symbolic representation of the idea. 

SEE IT 

Let's examine this idea through an example of transformations of functions. In this particular instance what we want to make explicit is the "activity" of the transformation which in this instance begins with a graphical representation of the parent function 

The prompt given to students is to "move each point in the plane 3 units to the right". For this particular  example we are going to focus attention on the transformation of five specific points. 

SAY IT 

As we ask students to describe what they are "seeing" we challenge them to use meaningful conceptual language to describe the transformation. A few examples of the conceptual language students shared is shown below. The first two example of Saying It is a restatement of the initial prompt while the third example describes the actual aspect of the functional relationship that is being transformed using terms (input and output) that speak to specific aspects of the functional relationship. 

1. Each point in the plane is shifted three units to the right.
2. The graph is formed by shifting each corresponding point on the original function relationship 3 units to the right.
3. Each input value of the original function is increased by 3 units while the output values remains the same. 

SYMBOLIZE IT

Then we challenge students to use algebraic symbols to describe the functional relationship. At the beginning we provide them with choices since the algebraic symbols are oftentimes very abstract ideas to a novice learner. Four possible options are shown below.

By doing this the cognitive load is lessened for the learner because they do not need to spend the energy on, at least initially, generating the functional relationship, but on making a connection between known input and output values from the graph of the transformation and selecting the function rule, that "fits" those inputs and outputs. For example, choosing a known input-output relationship of (3,0) there is only one function rule that "fits".

As learners gain more experience with these ideas they have the tools to  "look through" the abstract symbolic representation because they have visual images (seeing) and the necessary language (saying) to communicate the meaning. For example, consider the transformation shown below.  

The hope is that they can "see" through the graphical representation that the graph is shifted 3 units upward and "say" that the "input value remains the same, while the output value is increased by 3 units".

In summary, utilizing the See It, Say It, Symbolize It to organize instruction involves placing the algebraic symbolic representation on the back burner until students have made associations and starting with the "seeing it" utilizing the  representation that will best enable the learner to enable the main ideas of the concept to be seen explicitly. 

Imagining Possibilities: Algebraic Procedures and Algebraic Reasoning

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Is Mathematics Teaching in Need of "Retooling"?

 

In the movie Hidden Figures Dorothy Vaughn, played by Octavia Spencer, recognizes that with the advancements in technology her staff’s skills are going to become obsolete thus losing their jobs with NASA. She embarks on a journey to not only train herself but also to “retool” her employees so that when a different skillset was required, they were ready. "Retooling" is hard! Despite our best efforts  many of us have never been given the time, guidance and direction, or the resources to “retool” our teaching. In fact the constraints of assessment practices and the general public's reluctance to accept and embrace the need for change has left large swaths of the mathematics teaching workforce struggling to move beyond a 1970s mindset. This is in spite of the fact that many of us would agree that our students need a different experience.  Parenting and teaching are no different in a lot of ways. We tend to parent and teach the way that we were taught. If we are not “seeing” a different perspective modeled by our mathematics teachers, it is difficult to imagine ourselves doing something different when we have our own classrooms. 

What is a 1970s mindset? In 1970, Fortune 500 ranked computational skills as the 2nd “Most Valued Skill” by employers. Problem solving was ranked 11th. Computational skills are still important, but they are important in a different way and current mathematics teaching needs to reflect this difference. In the “Most Valued Skills” ranking over the past ten years, computational skills are no longer a top 15 required skill. Today’s workforce requires problem-solving, team-work, and quantitative reasoning skills, not computational skills. The reason that computation skills is no longer at the top of the list is that all of us have in our pockets the technology to perform these same skills accurately and efficiently. 

For example, if you need to know the answer to 2/3 * 9/10 all you have to say is “Hey, Siri!” or “Hey, Alexa”. Before you can write out the steps the technology replies with an answer of “point six”. [See! Even Siri and Alexa do not like fractions!!! More on that at another time!] The proliferation of technology enables us to shift our instructional focus away from a significant emphasis on efficient computational strategies such as “multiply straight across” to obtain correct answers because the technology can now do that work more efficiently. This strategy as well as others (e.g., “keep, flip, change”) were borne out of an era in which quick, accurate, and efficient calculations were required. Many of these strategies involve a set of steps to perform an operation without the student knowing the mathematical underpinnings (e.g., axioms, definitions, and properties). There was not as much concern for the “why” behind these "tricks" because it was not a focus of learning. I am not saying these tricks do not have a purpose later in solving more complex algebraic expressions, but they definitely do not have value in elementary and middle school classrooms when students are beginning to make sense of numerical relationships.

As was the case in the movie Hidden Figures, advancements in technology have enabled us to shift the focus of our teaching goals. Students can quickly get answers, so it is no longer the answer that matters as much as the thinking, questioning, and reasoning that we engage them in along their mathematical journey. Now we need students who can mentally reason as to whether the technology is providing a correct answer and, if not, how to adjust the technology so that it does provide a correct answer. Our students need to know the “why” behind what they are doing so that they can reason about the validity of the result and, if necessary, troubleshoot. For example, what our students need to know is that an answer of 0.6 or 6/10 to the problem 2/3 of 9/10 is a reasonable answer. Our students need to be empowered with the capacity to reason that the answer is less than 9/10 because taking 2/3 of something will result in less of that something. They also need to see that the problem 2/3 of 9 and 2/3 of 9/10 are similar because both involve the same actions. That is, the amount, 9, is partitioned into 3 equal groups (3), and the required amount is 2 of those groups is 6, or 6 tenths, respectively. 

After some consultation with some of my respected colleagues there is also a point to be made about efficient computational choices. The choice to partition 9 into 3 equal groups before selecting 2 of those groups was because I was able to easily determine the size of the groups. I could have just as easily "stretched" to create 2 groups of 9 (18) and then partition into 3 equal groups (6). It is also about the efficiency of those choices. I would not make the same choices if the problem was 2/3 of 5/7. I think that understanding is part of what we want our students to learn. It just about the "when" to apply an understanding as much as it is about the "why". My point again is a student may be able to obtain an answer of 10/21 to 2/3 of 5/7 but is that all we want them to know or is there something more we want from them? 

Another quick example to further illustrate this point. Much of the internet has been ablaze with problems like 8÷2(2+2). These problems claim to “break the internet”! Of course, there is an axiomatic need for an order of operations, but problems like this are not where the mathematical value lies. One of the values of order of operations lies in entering a formula into a spreadsheet. For example, assuming you have entered a numerical value in cell a2 you will get a different output if you enter =4 * a2 + 2 as compared to =4*(a2 + 2) into different cells.  That is because the parentheses create a different order of the operations. In the first example the order of operations is to multiply by 4 and add 2 while in the second the order is to add 2 and then multiply by 4. With quick access to spreadsheets, we can also quickly compute monthly payments using a complex formula that we can enter into a spreadsheet program.

This requires the correct placement of parentheses to perform the order of operations so that the output value correctly represents the formulaic computation. 

I think one of the most powerful thought experiment that all of us can engage is to reflect on the places in our own curriculum that might still be cultivating a 1970s mindset. We can then imagine what we can do to tweak it so that we can engage our students in the type of mathematical reasoning that will not only support current technology-driven workforce demands, but also give them the tools to meaningfully reason and think more deeply about mathematics. We have come so far with technology that we might even be able to have ChatGPT tell us where those areas in our curriculum might be! 😀

Please!!! No More Timed Multiplication Tests!

These past few weeks have been some of my most enjoyable in my over thirty years in education. This semester I have the privilege of re-imagining the content and teaching methods used in a developmental mathematics course at MSU. With failures rates hovering around or above 50% in this course change is needed. 

 I have watched my students as they have become accustomed to my methods and been vulnerable enough with me to share their personal math journeys. Many feel like math has damaged them and one theme that emerged as they shared their stories is that it started with learning multiplication facts by taking timed multiplication tests. Their stories include frustrated parents, missed recess time, missed sundae parties, teacher scorn, and downright embarrassment because "other kids got it and I didn't!"

This past week I was sitting with a student helping her review for our first test. Despite showing me great thinking she began to cry. I asked her what was a matter. She said, "I am worried that if I do not pass this test, I might not pass this class,  I won't make it through school, and I won't get a good job! I have just struggled with math for so long I just don't know if I will ever get it!" As tears flowed down her face I asked her how long she had felt this way. She said, "Since we learned times tables." She shared, in great detail,  that she had been slow to learn them, could never do them as fast as the other kids, and missed out on going to the sundae party with the rest of her class. She said her parents would try to help her by doing flash cards with her, but it did not help. She said it made her feel stupid, hate math, and she stopped trying because she believed she would never figure it out.

I am not opposed to fluency with multiplication facts and practicing those facts, but why do we have to time the process? Do there need to be winners and losers based on speed? 

As we talked more I asked her if she could tell me what 8 * 6 was? She nodded her head that she couldn't. I asked, "How about 4 * 6?" She nodded again that she couldn't. I asked, "What about 2 * 6?" She said it was "12". I wrote the problem on a sheet and asked, "study 2 * 6 and tell me what 4 * 6 would be?" She said "24". I asked what did you notice? She said it "doubled". Then I asked, "What about 8 * 6?" She took a moment but said "48".  I did not do my best work to uncover her thinking, but I pressed her to think about what 16 * 6 would be. She wrote out 16 + 16 + 16 = 48 and doubled that to get an answer of 96. I knew she was starting to see something. I finished by asking her if she could use all her work to tell me what 15 * 6 would be? She paused a bit, and finally said "90".  She smiled from ear to ear in a way that made might heart grow "10 times larger!"

If you ask a mathematician, in general, what their work entails they would tell you it is about "looking for patterns and finding relationships." I believe there is a mathematician in every child, but it is slow thinking, not fast thinking, that wins the prize because often patterns and relationships take time to see.  Are we giving our students an opportunity to do this?  Think for a moment the power of the following sequence of multiplication problems. Oh the places you could go!

2 * 6

4 * 6

8 * 6

16 * 6

15 * 6

I think we are all wired to think this way. We just need the right sequence of problems, a little time, and the opportunity to explore. I know we want to do what is best for our children, but as this story exemplifies, and is repeated many times as I talk to other students, timed multiplication tests seem to be doing more harm than good.  If that is the case, why does it continue? 

 

Drugs and "Keep, Change, Flip"--Just Say No!

I am a product of 80s schooling.  A popular public service announcement during that time was a man with a perturbed look on his face, arms crossed, asking the viewer, “Is there still anyone out there who still doesn’t understand what doing drugs does?” Holding an egg, he exclaims “this is your brain,” cracking it up against the side of a heated cast iron skillet he says, “This is your brain on drugs” as the sound of the sizzling egg plays in the background. The ad finishes by him asking, “Any questions?”. The motto we rehearsed in school, thank you Nancy Reagan, was “Just Say No!”.

I didn’t like eggs or drugs, so I am not sure the ad had much impact on me--I had already said “No”! However, I did get the gist. If you take drugs you are going to do bad things to your brain like impede mental growth, stunt brain development, lower cognitive functioning, etc. As a thirty-year teacher I often reflect on the question, “Are their current practices in math classrooms, as well as my own, that are deadening minds and stunting development?  Yeah, I know, drugs and mathematics teaching practices, a tough analogy. I believe one such of those practices that we need to “Just Say No!” is tricks to teach mathematics. A prevalent one is the use of the phrase “Keep, Change, Flip” to teach division of fractions. Okay, all I ask is that you hear me out.  

I witness on a regular basis the intellectual residue of students who have been raised on math tricks such as “Keep, Change, Flip” to learn mathematics. I teach mathematics content classes to future elementary, middle, and secondary mathematics teachers. I also teach the lowest college-level developmental mathematics course at our university--a course in which 95% of students have successfully completed Algebra 2 or above in high school. I also have conducted research in 4th through 7th grade classrooms related to understanding of fractions. I often ask these students, including preservice teachers,  to explain the meaning of 3/4÷1/3 or solve a word problem such as, “A cookie recipe calls for 3/4 of a cup of sugar and you have 6 cups of sugar, how many batches, including partial batches, of the cookie recipe, can you make?”.  Questions like these throw many students into a panic. They often reach for a calculator, or simply state, “Is this where I use, ‘Keep, Change, Flip’?” I have also had the opportunity to teach students from South Korea, Taiwan, China, Oman, Germany, Greece, as well as many others. The one constant is that most of the students from these countries have never heard of “Keep, Change, Flip”. Many U.S. students know “Keep, Change, Flip”, but have no idea mathematically why it makes sense to even perform that operation. This is an American-based shortcut that yields correct answers to division number sentences which are prevalent in our assessment-driven culture. However, a shortcut, like the 80s claim about drugs,  that promotes intellectual neuropathy. 

Now there might be a place for “Keep, Change, Flip” in simplifying an algebraic expression such as 

 but, I am becoming more and more convinced that it does not have a place in elementary classrooms. You are welcome to try to convince me otherwise, but at least give me a chance to make my argument.  I myself have done this and I agree that “Keep, Change, Flip” is an easy, efficient way to get an answer to a school-based problem. I am now sitting on the other end witnessing the unintended consequences of this narrow view of mathematics. Our educational goals should push students beyond the “quick high” of just getting a correct answer.

What is the alternative to “Keep, Change, Flip”? Provide students with the tools that they will need to reason about division problems in the real-world, the language necessary to make meaningful interpretations about any division problem and learn an algorithm that connects to whole number division. The book Extending Children’s Mathematics: Fractions and Decimals by Sue Empson and Linda Levi is a great resource for anyone desiring to learn more about a different view of division of fractions. 

Take a moment to think about the solving the problem, Gwen has 6 yards of ribbon to make bows. If each bow takes 3/4 of a yard of ribbon, how many bows can Gwen makes before she runs out of ribbon. In fact, can you create a picture that would represent the situation? Before we formalize the notation that cues dividing fractions it is important that students intuitively experience the actions associated with the process of division. I would venture that in modeling this solution you created 6 whole objects to represent the 6 yards of ribbon, partitioned each whole object into 4 equal pieces (i.e., fourths), and counted sets of 3 of those pieces (i.e., 3 fourths). In general, we want students to connect the process of partitioning a whole unit into n equal parts with the size of a part, 1/n, that results. And, that they can combine unit fractions to make fractions that are multiples of unit fractions (p. 74, Empson & Levi, 2011). In the ribbon problem, the answer of 8 means there are 8 groups of 3/4 of a yard in 6 whole yards.

Students must also develop the language associated with a symbolic division problem. For example, the above problem 6÷3/4 is asking, consistent with the visual model, “How many groups of 3/4 can be created from 6 whole units?” The first number in a division number sentence signifies the whole and the second number, using a measurement perspective, signifies the size of the group. Many of the college level students I teach do not have a language beyond “Keep, Change, Flip”. This is very limiting to their intellectual development because it does not provide them with the tools to use mental arithmetic to solve problems nor the means to estimate or check the reasonableness of their answer. Consider how a student empowered with this language might reason about problems such as 4 1/2÷1/4  or 6÷0.01.  For example, 4 1/2÷1/4 is asking how many 1/4s are in 41/2?, and, 6÷0.01 could be interpreted as, how many pennies in $6?  And, how they might estimate the answer to a problem such as 1/4÷1/3. That is, that the answer is less than one because the whole is smaller than the size of the group. 

If students are to learn an algorithm it should be an algorithm that connects with what they already know about whole number division. That algorithm is called the Common Denominator Algorithm. The visual action of the earlier ribbon problem illustrates this algorithm. The 6 whole units was recomposed into 24 fourths which was used to form groups of 3 fourths resulting in an answer of 8. In other words, the fraction division problem 6÷3/4 is equivalent to the fraction division problem 24/4÷3/4  which is equivalent to the whole number division problem 24÷3 because it is asking, “How many groups of 3 fourths can be made from 24 fourths?”.

Consider another problem introduced earlier 3/4÷1/3. The problem is asking, “How many 1/3s can be created from 3/4?” or making the units explicit, “How many 1 thirds can be created from 3 fourths?” Using what we know about equivalent fractions, an equivalent problem can be created, “How many 4 twelfths can be created from 9 twelfths?”. In other words, the fraction division problem 3/4÷1/3 is equivalent to another fraction division problem  9/12÷4/12 which is equivalent to the whole number division problem 9÷4. 

If we START with, “Keep, Change, Flip”, we are promoting intellectual neuropathy, a numbing of the brain, because our students feel they have all they need to get answers. As a colleague of mine said, “If we choose to share with students ‘Keep, Change, Flip’ we are making the decision that our students no longer need to think about ideas related to division of fractions.” It has been a challenge to develop a deeper understanding of division of fractions because students believe they have all they need if they can get correct answers. I have seen this first-hand in my college level classes. Recently, I asked a class of 50 students why, “Keep, Change, Flip” works. Not one student was able to provide a mathematically valid response (e.g., dividing is the same operation as multiplying by the reciprocal). There is so much more that we want them to understand about division of fractions than getting a correct answer. Let’s, “Just Say No” to “Keep, Change, Flip” and hope that what we replace it with is more effective than those 80s drug ads! I love to engage other teachers in ways to improve mathematics teaching. Feel free to reach out for further dialogue at patrick_e2e@outlook.com.

Adding Fractions: Making the Connection to Whole Numbers

One of the five recommendations put forth by the Institute for Educational Studies in Developing Effective Fractions Instruction for Kindergarten Through 8^th Grade (2010) is, “to help students understand why procedures for computations with fractions makes sense” (p. 5). One way to help students make sense of procedures for computations with fractions is to leverage procedures they may already understanding for computations with whole numbers. The argument is that fractions represent numbers and the reasoning to solve problems involving operations with fractions is closely related to the reasoning to solve problems involving operations with whole numbers because both problems involve numbers. For example, “addition is addition” whether the problem involves whole numbers or fractions. That is, addition is a process of defining numerically the amount of objects in two or more sets of like objects with same unit (e.g 9 apples and 4 apples is 13 apples). The purpose of this first blog is to make a clear link with addition. The necessary understandings of fractions as well as important terminology will be highlighted.

 Depending on the operation and the specific situation it is important to understand the different possible meanings of a fraction. In some instances it is important to interpret the fraction 3/4 as a scalar. For example, is 3 copies of 1/4 or 3(1/4), in which the 1/4, or fourth, is the iterating unit—that is 1/4 is repeated 3 times. In other instances the fraction 3/4 is interpreted as an operation. For example, in the abstract problem 3/4 * 16, the fraction 3/4 can be thought of through the lens of partitioning 16 into 4 equal groups {4} and choosing 3 of those equal groups                                     {3 equal groups of 4 = 12}.

Consider the following addition problems.

Jenny has 4 whole apples and her friend Sue has 9 whole apples. Between the two of them                 how many total whole apples do they have?

 Jenny has 1/3 of a whole pound of cheese and her friend Sally has  3/4 of a whole pound of cheese. Between the two of them how much cheese do they have?

The operation of addition is about describing numerically the sum of the number of elements in two or more sets of the same objects sharing the same iterative unit. For example, to answer the question “How many total whole apples do they have?,” the iterative unit is 1 whole and apple is the object. In performing the computation the action of addition involves combining 4 copies of 1 whole apple with 9 copies of 1 whole apple for a total of 13 copies of 1 whole apple or 13 apples.

 The critical aspect of addition in contextual situations is that not only is the object is the same, but so is the iterative unit. These ideas are lost sometimes when discussing de-contextualized problems such as 1 + 1/2.  The answer 1 and 1/2  is describing the numerically the combining action of 1 copy of 1 whole unit with 1/2 copy of 1 whole unit yielding 1 and 1/2 copies of 1 whole unit or 1 1/2 whole units. It seems important to mention that 1 + 1/2 can also be thought of as the sum of 2 copies of 1/2 whole unit and 1 copy of 1/2 whole unit which is 3 copies of 1/2 whole units or 3(1/2) or 3 halves.

  The same argument holds when adding fractions. In order to perform the combining action associated with addition we must have the same iterative unit and like objects or referent units.  We cannot combine 1/3 of a whole pound of cheese and 3/4 of a whole pound of cheese because although their referent unit (1 pound of cheese) is the same the iterative units are not. The fraction 1/3 can be interpreted as “1 third” or “1 copy of the iterative unit 1/3.” Similarly, the fraction 3/4 can be interpreted as “3 fourths” or 3 copies of the iterative unit 1/4.”

While thirds and fourths are different iterative unit preventing us from completing the combining action of addition it is possible to re-define each of those fractions in terms of the same iterative unit. That is, 1 third is the same as 4 twelfths or 1 copy of 1/3 is the same as 4 copies of 1/12 . Likewise 3 fourths is the same as 9 twelfths or 3 copies of 1/4 is the same as 9 copies of 1/12. 

Now that both fractions are expressed in terms of the same iterative unit the combining action of addition showing the sum of 4 twelfths and 9 twelfths as 13 twelfths i.e. --An analogous solution to the whole number addition problem.