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A Fact Family is an innovative way to group the learning of math facts. Some people are die-hard advocates of this way to learn facts. Yet students successfully learned math facts for decades without ever considering them as being composed of families. This blog discusses the pros and cons of learning with fact families. 

 What is a Fact Family

A fact family is a set of four math facts made with the same three numbers. The numbers 2, 3, and 5 can make a family of four facts: 2+3=5, 3+2=5, 5−3=2, 5−2=3. The numbers 2, 3, and 6 can make a different fact family: 2×3=6, 3×2=6, 6÷2=3, 6÷3=2.  

Why are Fact Families Important?

Fact families help children see that adding and subtracting are the opposite of each other. Multiplication and Division fact families do the same. Learning fact families may help students develop more flexibility in their number sense.   

Benefits of Thoroughly Learning a Single Operation at a Time

When it comes to memorizing math facts, students traditionally learn one operation at a time. They memorize only the Addition facts, typically in first grade. Not until later do students learn the concept of Subtraction, and then they begin memorizing Subtraction facts, usually in second grade. With Rocket Math, students will master Addition facts much better than most first graders. Students using Rocket Math will know Addition facts instantly without having to stop and think about them. When students have mastered Addition facts, an unusual thing happens when introduced to Subtraction. Students instantly recognize that the Subtraction facts are the opposite of those Addition facts they know so well, usually without even being told. This recognition does not typically happen in second grade with students who haven’t truly mastered Addition facts.  Therefore teachers think they need to teach in fact families to get students to recognize the reciprocal nature of Addition and Subtraction.   

Which Way is Better?

Only if students thoroughly master Addition facts, can they quickly recognize the reciprocal nature of Subtraction. Students find it easier when memorizing facts, to stick with Addition, rather than switching back and forth between adding and subtracting as is required by learning in fact families. They tend to memorize faster in single operations than they do in fact families.  However, if students memorize in fact families, they will learn the reciprocal nature of the fact families right from the beginning. They may learn it more thoroughly. That may help them develop more flexibility in their number sense. So that’s not a bad thing.

How to Learn Math with Fact Families and Single Operations

Rocket Math offers separate Addition and Subtraction sequences, as well as fact family sequences. Both the Online Game and Worksheet Program provides these. If teachers (or their regular math curriculum) are wedded to fact families, they are available to use. Usually, the first fact families learned are Addition and Subtraction Fact Families through 10, then Fact Families from 11.  

I would recommend beginning with learning the operations separately: learning Addition facts (1s to 9s) first, followed by Subtraction facts (1s to 9s). For some students, this may be all they have time to learn.  If students have time, use Fact Families as a review. If the student has time, it would be beneficial and relatively easy for them to do Fact Families (+,-) to 10 and then Fact Families (+,-) from 11 afterward. Students will learn the facts more thoroughly, and the reciprocal nature of Addition and Subtraction will be deeply ingrained. This is quite simple in the Online Game, as it only involves switching the student to another Learning Track, and doesn’t require a new filing crate, as in the Worksheet Program.  

Using Rocket Math to Teach Single Operations & Fact Families

Rocket Math offers separate Addition and Subtraction sequences, as well as fact family sequences through the Online Game and the Worksheet Program. For teachers looking to teach Addition and Subtraction math fact families, Rocket Math offers two sequences of Fact Families; the first, Fact Families to10 and the second, Fact Families from 11. For teachers looking to teach in separate operations, Rocket Math offers separate sequences for learning Addition facts (1s to 9s) first, followed by Subtraction facts (1s to 9s). If the operations are learned separately, fact families can be used as a review. 

Learn three keys to knowing how to find factors and primes

We have added Factors and Primes to the Online Game. Learning Track #15 Factors & Primes is a dream come true for Dr. Don.  Many middle grades students find that factoring and finding the GCF (Greatest Common Factor) is quite difficult.  This learning track will make it easy.  It accomplishes three key learning objectives at the same time.

(1) Recognizing the Primes.

Instead of just working on numbers with factors this Learning Track includes many prime numbers such as 7 in the illustration above.  When a number is prime, the student is shown 1 x the number and asked “What’s next?” Then the student is to answer with the check mark indicating that number only has 1 and itself as factors–there are no more. These prime numbers appear again and again, giving the student lots of practice in remembering the primes and in remembering their key characteristic.

(2) Learning factors in order. 

The game displays the factors with the lowest factor on the left and put them in order by that factor.  (see the Factors of 12 above). The factors are always in this order, so students learn them in order.  Notice that the last factors of 12 ends with 3 on the left and 4 on the right.  The factors on the right go up from there up to 12.  The key to knowing there are no more factors is when the next number down (in this case 4) is already showing in the list going up on the right.  By seeing the factors always in this order, student can more easily learn them. 

(3) Knowing when you’re done. 

Sometimes we require students to indicate that all the factors have been listed. The Online Game will show all the factors and ask, “What’s next?”  Students will know that there are no more factors because they have learned them in order.  Students will then put a check mark to say this is the complete list. When figuring GCF it is essential to know when you have listed all the factors.   

16 Learning Tracks in the Online Game

1. Addition                       5. Fact Families (+, -) to 10           9. Multiplication 10s-11s-12s
2. Subtraction                 6. Fact Families (+, -) from 11     10.  Division 10s-11s-12s
3. Multiplication             7. Add to 20, example 13+6        11.  Fact Families (x, ÷) to 20 
4. Division                        8. Subtract from 20, 18-5            12 Fact Families (x, ÷) from 21      13. Identifying Fractions                                                  14. Equivalent Fractions                   15. Primes & Factors              16. Fraction & Decimal Equivalents

Test drive the game as a student, right NOW!

Sign up for a free trial of the Online Game today!

These effective math teaching strategies will help your students be more successful in learning math in your classroom. These are not the “cool, pedagogically-correct things” you can brag about in the teacher’s lounge or your master’s program classes, but they will improve math for the kids.

1. Help Students Memorize Math Facts

Once students know how to count out or figure out math facts, they are ready to begin memorizing the facts. After they memorize math facts, students can do math assignments quickly and learn math easily. It’s cruel to make them continue to figure out facts over and over again for months or years. Find and use a systematic method of memorizing math facts, so they have this tool at hand. There’s no magic bullet. Whatever you use needs to be systematic because it has to build gradually as students learn more and more of the facts. Any fact is easy to learn, but there are a lot of them, so that takes time.

Helping your students memorize math facts is like buying math power tools for them. It makes everything go quicker and much better. More gets done. Attitudes will improve. You don’t have to use Rocket Math, but you should use something! You will change their trajectory in and attitude about math for years to come. Learn more about why memorization is one of the essential math teaching strategies.

2. Ensure Math Facts are Mastered Before Starting Computation

Kids love to go fast. They hate the drudgery of having to count out facts or look multiplication facts up in a table. Once they know facts instantly, they can rip through math computation–and enjoy math because they love to go fast.

Not knowing math facts interferes with learning computation.

Here’s another surprise. Memorizing math facts makes learning multi-digit computation easier. Students who’ve memorized math facts are not distracted by having to stop and figure out what is 14 minus 9. They know it, without having to stop and think about it. They can give more of their attention to the details of the computation process and learn them better and easier than before. Try getting all your students fluent on the operation and then teach computation AFTER they have acquired fluency. You will be astonished at how quickly they learn. Read more about how math fact memorization supports computation.

3. Teach Computation Procedures With Consistent Language

Learning the procedures to do computational math processes is just like learning to follow a recipe. Eventually, you learn the key ingredients and can do it without looking and can do variations. But in the beginning, you need the written recipe to follow. You also need to do it the same way a few times to learn it. Winging it and teaching math “improv style” will only frustrate students.

Use a script or a process chart to keep the instructions consistent.

No one can remember exactly the way you said it yesterday or last week, which is why you need a script or a process chart on the wall so you can show the students that you are doing it the same way as last time. Repetition helps, but only if you’re repeating the same steps in the same order with the same wording. 

Students will become flexible in their understanding at their own rate and in their own time. You just keep being consistent, and eventually, all the students will have mastered the procedure. I know that “thinking flexibly” is very much in fashion, but your students will thank you for providing them mastery of this one method of solving this type of problem. Flexibility can come later. Read more about how consistent language is key to teaching computation procedures.

4. Teach One Method and Only One Method at a Time

To solve a given type of problem, do NOT follow the advice to teach multiple solutions at one time. Teach only one way to go at a time, until that is learned, before introducing variations.  Multiple solution paths will confuse all but the most advanced students in your room. Just teach them one procedure that will work every time with that type of problem. Your students will thank you for it.

It’s far better to know only one way to get there than to get lost every time!

Learning math procedures is no different than learning how to get to someone’s house (before GPS). You want to go the same way every time, until you have learned it. If you go a different way every time you’ll just be confused. You can learn another way later, once you can get there reliably using one route, but before that, you cannot learn another route, because you’ll mix up steps from each route. Read more about why teaching one procedure at a time is one of the effective math teaching strategies.

5. Separate the Introduction of Similar Concepts

The classic example is teaching parallel and perpendicular on the same day. The two concepts have to do with the orientation of lines, and the new vocabulary terms for them are similar. So teaching them at the same time means some or many students will have the two terms confused for a long time. That is known as a chronic confusion–possibly permanent. They will know that the orientation of lines is one of those two terms, but will be confused about which is which.

Parallel vs. Perpendicular

For example: For this picture, you would ask the students.

A “Are the two lines in item A parallel or not parallel?”   Ans: Not parallel.

B “Are the two lines in item B parallel or not parallel?”   Ans: Parallel.

C “Are the two lines in item C parallel or not parallel?”   Ans: Not parallel.

After a couple of weeks of this kind of practice, you could introduce perpendicular. Again teach it on its own and then contrast perpendicularly with non-examples until the vocabulary is clear.

Probably do that for a couple of weeks. Only then can you combine both terms in the same lesson.

Numerator vs. Denominators

Other examples abound in math. Teaching numerator and denominator in the same lesson is common. Teaching the terms proper fractions and improper fractions on the same day is another example. Acute and obtuse angles are yet another pair of chronically confusing concepts that are introduced simultaneously. Separate them in time, and you’ll be amazed at how much better students can learn these concepts. Learn more about how to successfully teach similar concepts separately. 

6. Teach  New Concepts Using Common Sense Names

Here’s an example of the problem. A brachistochrone (pictured here) is a curve between two points along which a body can move under gravity in a shorter time than for any other curve. Introducing this concept and the term brachistochrone at the same time would cause students difficulty learning it.

When Instruction is not working

If you watch instruction where the term and the concept have been taught simultaneously, confusion ensues when the teacher uses the term. Sometimes the teacher will notice students looking confused and give a thumbnail definition or example of the term, and the students will then remember. However, the teacher should then realize that the concept was not connected to the new vocabulary term. 

Students can quickly understand and use new concepts and ideas in math if they don’t have to learn a new word for it. Using a common sense term, the idea or concept can be applied to real-world problems almost immediately. Students can later quite easily learn proper vocabulary terms for concepts they understand and recognize. Here’s an example. 

The “shortest time curve.”

I would call a brachistochrone the “shortest time curve.”**  Instruction would proceed with the, “Do you remember that ‘shortest time curve’ we talked about last week?”  Students would easily be able to remember it. Instruction would go like this: “So ‘the shortest time curve’ has some other cool properties. What’s the primary thing we know about the ‘shortest time curve’?”  Students would easily be able to answer this question.

Then after students have worked with the concept of “the shortest time curve” for a couple of weeks, you can add the vocabulary term to it. “By the way, the proper mathematical name for “the shortest time curve” is called a brachistochrone. Isn’t that cool?”  Read more on how to teach new math concepts with common sense names.

**Actually, that’s what brachistochrone means in Greek: brakhistos, meaning shortest and khronos meaning time.

Learn More Effective Math Teaching Strategies with Rocket Math

Rocket Math Online Game and Rocket Math Worksheets Program are math teaching strategies that will work for any classroom. Use the Pre-Test worksheets to understand how fast your students can answer math facts and what level they are at. The Rocket Math Online Game will help students succeed in math by creating a fun and enjoyable way for them to learn.

The rapid speed required to answer our Online Game is a feature, not a bug.

You can change how fast the student has to answer, but you probably should not.  The main goal of Rocket Math is for students to commit facts to memory, to be able to answer them instantly, from recall. That is called “automaticity.”  It is the point of Rocket Math.

Recall is instantaneous, but “figuring out” is not.

The fast pace (3 seconds to input an answer) means students don’t have time to count on their fingers or to “figure out” a fact–they just have to remember it.  If they don’t remember, then the game gives them a LOT of practice on only a couple of facts, until they do remember them. That is exactly the point of the game. The game gives more practice through having students start parts over when they are not able to answer quickly.

We want them to stop having to “figure out” facts and just remember the answer.   If the students are not used to “recalling” facts they will think that the game is just “too fast” for them.  If they keep playing and learning, after a bunch of repetitions, they will be able to remember the fact.   These days students are NOT asked to memorize anywhere else. Today’s students are unaccustomed to having to repeat things over and over to commit them to memory.  Almost everyone can do it, but it takes more practice than many students are used to doing. Consider the Toughness Certificate [located in the Online Game’s main navigation bar] for those who can overcome their frustration at having to start over.

The danger in slowing the game down for most students.  If you let students play at the slower speeds they may never use “recall” and instead may figure out the facts over and over.  Until I realized the difference, I allowed my students to take their time to figure out facts.  Many of my students never committed facts to memory all year long!  If, as we do in Rocket Math, you only ask them to remember two facts and their reverses at a time, everyone can remember two facts.  It takes just a few minutes to realize that they can, in fact, remember that answer instantaneously.  Once they use recall, they remember the answer in less than a second, and then three seconds to input it, is quite doable.  If you slow down the game speed, they may NEVER realize they can remember the fact, instead of figuring it out each time.

Only if their difficulty score is over 3.0 do they need an adjustment made.  Their difficulty score, shown on the Review Progress screen [Top of Main Navigation bar] tells you whether or not the game is too fast for them.

• We expect students have to start over at least once per phase, which gives them a difficulty score of 1.0.  Even having to start over, on average, twice per phase is not too much–giving a difficulty score of 2.0.
• [If you sort your class based on their difficulty scores you can see a display like the one in this picture.]
• Any difficulty score under 3.0 means the student has to start over on average fewer than 3 times for each part passed.  That is not too difficult.  Students may not be accustomed to repeating anything, but it is not a real burden. On the other hand, you may find some students have difficulty scores under 1.0 and Rocket Math is very easy for them.
• Only students with difficulty scores over 3.0 should be considered for a speed change–and then only if you know they require some kind of accommodation.  On the other hand, students with difficulty scores under 0.1 should be challenged to take on the Faster speed!

Here’s where you can make the change.  Go to the green “Individual Action” button at the right end of each student’s row. Click on “Change Game Speed.”  The popup gives you the choices of  Fast, Normal, Slow and Slowest.

The options for speed are:

• Normal, at 3 seconds to answer (double that for two digit answers)
• Slow, at 4.5 seconds to answer per digit
• Slowest, at 6 seconds to answer per digit
• Fast, at 2.25 seconds to answer per digit

Here you can see different students showing different speeds of play.

Fractions are unnecessarily hard for students to understand.  The reason?  Not enough practice with identifying a wide enough variety of fractions and ways of displaying them.  Rocket Math has introduced a great solution as part of the Universal Level Worksheet subscription–a new program called Identifying Fractions.

Identifying Fractions: Not just proper fractions anymore.

One of the most powerful aspects of the Identifying Fractions program is that it is not limited to showing proper fractions.  Right from the beginning of Set A, students are introduced to improper fractions and mixed numbers.  They are taught the fundamental understanding that the bottom number tells you how many parts each whole is divided into.  At the same time, if the whole is not divided into parts, then we represent it as a whole number.  Finally, the top number tells how many parts are shaded (or used) regardless of whether that is more than 1 whole or less than one whole.  Identifying fractions that include improper fractions and mixed numbers from the beginning insures that students really understand fractions and don’t accidentally acquire the misrule that fractions are always less than one.

Identifying Fractions: Learn to speak their names also

Without a lot of oral practice students do not always know how to say the names of fractions.  Identifying fractions introduces three fractions in each set and includes the words for how to say them.  In the example here one half, three halves, and one and one half are written out at the top of the page.  This is all that is practiced as part of this first set.  This way, orally practicing with a partner means saying the names of the fractions, which are shown at the top of the page.  Students are not asked to say any fractions they haven’t seen written out first.

The fractions that students become familiar with include, halves, thirds, quarters, fifths, sixths, eighths, tenths and twelfths.  They see improper fractions and mixed number with every denominator.

Identifying Fractions: It’s not the shape that matters

When students don’t have a lot of practice with identifying fractions they may not see different shapes being divided into the same number of parts.  In Identifying Fractions we make a point of showing each fraction with at least three different shapes.  In this example you see thirds in a circle, in a cube shape, and as upright rectangles making a larger rectangle.  All of those are equally “thirds” because each whole figure is divided into three parts.  So there are three different shapes for halves, fourths, fifths and sixths.

By the time students are introduced to eighths, tenths and twelfths, they have already learned the rule that the shape doesn’t matter.

When students are eventually introduced to eighths, tenths and twelfths we don’t want to slow them down by having to laboriously count the number of parts in each figure.  As you can see to the left, the eighths are displayed as two sets of four rectangles on top of each other.  Eighths are always displayed that way, so they are easy to identify quickly.  Tenths are consistently displayed as two columns of five blocks with little numbers in them. (I know it’s a little weird, but it works to make them easily identifiable.) Twelfths are always shown as three sets of four rectangles on each other.  Students should notice these conventions so they can quickly identify the number of parts in those figures without having to count them.

Identifying Fractions:  Advanced students can fit in during Rocket Math routine

Identifying Fractions follows the standard Rocket Math routine.  Each student practices orally with the partner for a couple of minutes.  Then the two switch roles.  Finally everyone takes a 1 minute Daily Test.  A student in Identifying Fractions can be paired with any student in any other Rocket Math program as long as the student has an answer key to hand to their checker. Hopefully you remembered to print the answers on colored paper.

Unlike other Rocket Math programs, the test and the practice items are the same.  Of course, the students have a page without the answers, while their checker holds the answer key. Students practice by saying aloud to their partner the fractions shown in the test.  Then they take the test on those same items, but write the answer.

Identifying fractions has its own writing speed test, to be sure that student goals are individualized to their writing speed.  By the time students complete Set Z in this program they will have a strong understanding of fractions that will be fluent.  There are even 2-minute timings you can give every week or two for them to chart their progress as they get faster.  This is a great program for students of any grade from second grade on up who have finished the basics for their grade level.  It will really put them in good shape when dealing with fractions in later years.

Rocket Math now has added an Online Game to its tried-and-true Worksheet Program.  Customers ask, “Which should I use?  Should I use both?”

Dr. Don’s answer is “Yes, I do recommend using both.  As that opinion may appear self-serving, here’s why.”

1) Online Game is an easier route to math fact fluency.

Most students begin passing levels in the Online Game right away.  They find it quicker and easier and can sometimes pass more than one level in a day.  This gives the students a taste of success.  The Online Game helps them realize they can learn facts and make progress almost from the first day.  Students are then more willing to do the Worksheet Program as well.  Rarely, there are a few younger students who cannot input answers within 3 seconds.  They won’t be able to pass levels and will have to start over many times on the Online Game.  When monitoring them in the Online Game, such students will have difficulty scores over 3.  If that’s the case, the Worksheet Program is more flexible and they may prefer that.  But for most students, with difficulty scores below 2 in the Online Game, they will require a lot less practice to pass levels with the Online Game than in the Worksheet Program.

2) Start with the easier implementation of the Online Game.

The Online Game is easier for teachers to get started using.  Teachers don’t have to print out worksheets, maintain files and organize student pairs so they actually practice with the Online Game.  It is therefore easier to implement.  Less than enthusiastic teachers, who might not start Worksheet Program, will at least start doing the Online Game. After they see the success of the Online Game and students’ enthusiasm, they will then be more willing to do Worksheet Program.

3) Online Game is easier for parents to support.

4) Worksheet Program more rigorously develops math fact fluency.

Compared to the Online Game, the Worksheet Program is a bit harder to pass a level.  Students have to practice with their partner more time before they pass, so students learn facts better with it. They are more solid in their knowledge of facts when they are done with an operation like multiplication in the Worksheet Program than they are if they just run through multiplication in the Online Game.  Which is a reason not to do the Online Game only.  Of course, students are even stronger in their facts when they practice with both.

5) Worksheet Program will generalize to computation more readily.

The purpose of learning math facts is to make it easier for students to learn and do basic computation.  Math work is written, so the Worksheet Program (which is also written) is closer to how math facts will be used.  That means the Worksheet Program will generalize better to computation assignments.  You will see a bigger benefit to students doing math assignments when they finish the Worksheet Program than with the Online Game.  Which is yet another reason to do both programs.