Foolproof method for finding factors

Knowing when you’ve found ALL the factors is the hard part.

Students have to learn how to find the factors of a number because several tasks in working with fractions require students to find the factors of numbers. Thinking of some of the factors of a number is not hard. What is hard is knowing when you have thought of ALL the factors. Here is a foolproof, systematic method I recommend: starting from 1 and working your way up the numbers. This is what student practice in the Rocket Math Factors program.

https://youtu.be/fDYMRfxtGIc

I have a white board type video lesson that explains this in 6 minutes. https://youtu.be/fDYMRfxtGIc

Bookmark this link so you can show it to your students.

How to find all the factors of numbers
Always begin with 1 and the number itself-those are the first two factors. You write 1 x the number.  Then go on to 2. Write that under the 1. If the number you are finding factors for is an even number then 2 will be a factor. Think to yourself “2 times what equals the number we are factoring?” The answer will be the other factor.
However, if the number you are finding factors for is an odd number, then 2 will not be a factor and so you cross it out and go on to 3. Think to yourself “3 times what equals the number we are factoring?” There’s no easy rule for 3s like there is for 2s. But if you know the multiplication facts you will know if there is something. Then you go on to four—and so on.

The numbers on the left start at 1 and go up in value.  The numbers on the right go down in value.  You know you are done when you come to a number on the left that you already have on the right.  Let’s try an example.

Factors Answers d

Let’s find the factors of 18.  (To the left you see a part of a page from the Rocket Math factoring program.)
We start with the first two factors, 1 and 18. We know that one times any number equals itself. We write those down.
Next we go to 2. 18 is an even number, so we know that 2 is a factor. We say to ourselves, “2 times what number equals 18?” The answer is 9. Two times 9 is 18, so 2 and 9 are factors of 18.
Next we go to 3. We say to ourselves, “3 times what number equals 18?” The answer is 6. Three times 6 is 18, so 3 and 6 are factors of 18.
Next we go to 4. We say to ourselves, “4 times what number equals 18?” There isn’t a number. We know that 4 times 4 is 16 and 4 times 5 is 20, so we have skipped over 18. We cross out the 4 because it is not a factor of 18.
Next we go to 5. We might say to ourselves, “5 times what number equals 18?” But we know that 5 is not a factor of 18 because 18 does not end in 5 or 0 and only numbers that end in 5 and 0 have 5 as a factor. So we cross out the five.
We would next go to 6, but we don’t have to. If we look up here on the right side we see that 6 is already identified as a factor. So we have identified all the factors there are for 18. Any more factors that are higher we have already found. So we are done.

Now let’s do another number.  Let’s find the factors of 48. 

We start with the first two factors, 1 and 48.  We know that one times any number equals itself.

Next we go to 2.  48 is an even number, so we know that 2 is a factor.  We say to ourselves, “2 times what number equals 48?”  We might have to divide 2 into 48 to find the answer is 24.  But yes 2 and 24 are factors of 48.

Next we go to 3.  We say to ourselves, “3 times what number equals 48?”   The answer is 16.  We might have to divide 3 into 48 to find the answer is 16.  But yes 3 and 16 are factors of 48.

Next we go to 4.  We say to ourselves, “4 times what number equals 48?”  If we know our 12s facts we know that 4 times 12 is 48.  So 4 and 12 are factors of 48.

Next we go to 5.  We might say to ourselves, “5 times what number equals 48?”   But we know that 5 is not a factor of 48 because 48 does not end in 5 or 0 and only numbers that end in 5 and 0 have 5 as a factor. So we cross out the five.

Next we go to 6. We say to ourselves, “6 times what number equals 48?”  If we know our multiplication facts we know that 6 times 8 is 48.  So 6 and 8 are factors of 48.

Next we go to 7.   We say to ourselves, “7 times what number equals 48?”   There isn’t a number.  We know that 7 times 6 is 42 and 7 times 7 is 49, so we have skipped over 48.  We cross out the 7 because it is not a factor of 48.

We would next go to 8, but we don’t have to.  If we look up here on the right side we see that 8 is already identified as a factor.  So we have identified all the factors there are for 48.  Any more factors that are higher we have already found.  So we are done.

Can I upgrade from a Basic to a Universal subscription?

A teacher asks:

Can I upgrade from basic to universal? Will I just pay the difference?

Dr. Don answers:

Yes, indeed. If you login to your account you can see a link to upgrade (pictured above).   Yes you will pay the difference between your current subscription and the upgrade. Good decision to upgrade!  Once you upgrade you will be able to get into the drawers of the Universal subscription.  All of these are included:

  • Rocket Writing for Numerals,
  • Add to 20,
  • Fact families (+, -) 1-10s,
  • Subtract from 20,
  • Skip Counting,
  • 10s, 11s and 12s (Multiplication and Division),
  • Equivalent fractions,
  • Factors,
  • Integers (adding and subtracting positive and negative numbers, etc.

I also have completed and added to the Universal Subscription programs to teach computation.  The instruction proceeds in small steps from the beginning skills in an operation up through the highest levels. Each skill taught has a suggested teaching script to make it easy for tutoring.  So far Addition and Multiplication are done.  Each one has a placement test, so you can see where to begin.  Subtraction and Division are yet to be completed.

 

Challenge your students with the Race for the Stars Game Center!

race-for-the-stars-product-imagery-subtraction
How can you entice your students to play the Race for the Stars game in a center?

By letting them post their record/best time for completing the game boards. Race for the Stars is a great game for students to practice their math facts. The game provides 24 problems tiles students can race to put down next to the answers in the game board as fast as they can. There is now a Game Center Kit that give you everything you need to set up a center in your room that students will want to visit. The key is the poster with room for student labels with their names. Then next to their name is a place for them to write their best time at filling the game board. Well, actually their partner with the stop watch should probably write down their time. All the students in the class can rotate through the Game Center to record how fast they can fill either or both of the two game boards. One, for levels A-K are the first facts to be learned in Rocket Math. The second game board, L-Z are the facts learned in the last half of the Rocket Math levels.

But how can you motivate your students to play the game again and again to get the practice they need?

Here is the coolest part of the Race for the Stars Game Center. Students can go back and get someone to time them filling the Race for the Stars gameboard again. If the timer with the stopwatch sees that the student beat their previous best, they get to record the new record time, AND cover the old time up with a star sticker. Getting to put up a star sticker next to your name proves you were able to beat your own previous best–that is impressive! In fact, the savvy teacher will make a daily stop to see who in the class has been able to add a star sticker to the poster next to their name, showing that they beat their previous best time.

And that is how you motivate students to use the Race for the Stars Game Center (item #2112) to practice their math facts in their spare time!

Can you avoid summer losses with Rocket Math?

Take up where they left off before the summer!

Don’t think students have to start over in Rocket Math. They have learned the facts so well that with careful review they can take up where they left off!

There is a way to start off the year on the same set on which students left off at the end of last school year (providing you know where that was). You do need to do a slightly different procedure at the start of the year, however.

Notice on the Set L sheet above that only five of the facts on the One-Minute Test are new–the rest are review from the previous sets. That means that practice around the outside will help with the new facts, but won’t review those older facts from previous set. If you test the students on any set after the summer they might not pass because they need a little review of those older facts.

Here’s the way to beat the summer forgetting:
For the first week of school have the students add another practice session with the One-Minute Test each day. Give them the test answer keys, give them 2 to 3 minutes with their partner to orally practice the test problems with the same correction procedure as usual. Hint: have them take the sheets home and practice with a responsible sibling or a parent as well. A week to ten days of this extra review of the test problems and they will be successfully passing levels in Rocket Math–starting right from where they left off in June!

Can all 2nd graders finish subtraction?

Julie writes,

In our district, we have data that shows students are struggling with subtraction. We really want to put emphasis on getting the subtraction facts memorized. What are your thoughts about 3rd grade starting with subtraction in the beginning of the year and switching to multiplication the second half of the year regardless of having completed Z in subtraction?

Thanks, Julie

Dear Julie,

Multiplication facts are VERY important and you want to be sure that all third graders have enough time to master them. The best solution would be to get all your second graders through subtraction during the second grade year. Here are some suggestions to get better results–where more students develop automaticity in subtraction facts in second grade.

1) Start Rocket Writing for Numerals in the second half of Kindergarten, so that students leave K able to write numerals quickly.

2) Start Rocket Math Addition in the first month of first grade (because most all students) are able to write numerals fast enough. If you have a few who are below 18 boxes in a minute–give them extra work on Rocket Writing for Numerals to get them up to speed and into Rocket Math Addition before the end of October.

3) Monitor closely students who don’t pass an operation in six days and make sure they are getting a bonus practice session each day at school or encourage their parents to practice with them at home. Some few students will need extra help to practice twice or three times a day to make progress as fast as others.

4) Save folders from grade to grade (over the summer) so that students don’t have to start an operation over from the beginning (just continue on from where they left off) and so can finish an operation in a timely manner.

5) If students get stuck for more than a week, and definitely when they come back from the summer in the middle of an operation, have them practice the test problems orally as well as the outside problems. If they are only practicing the outside, and they are slow or have forgotten some of the cumulative review problems in the test, they need to practice the test problems (as well as the newer problems on the outside) to bring them up to speed.

I think that continuing a student in subtraction at the beginning of 3rd grade from where they left off in 2nd grade is an OK policy, but it is far better to have as a clear goal to get students through subtraction during 2nd grade. Hope this helps. -Don

Five interventions for frustrated students

Erica writes:
My son currently is in second grade and uses Rocket Math in his school. He has been doing addition and is on Level “S”. Most of his class has moved onto subtraction and multiplication.
My concern with him is where does this leave him next year in 3rd grade?? Is he left behind? He knows the addition facts orally but fails to meet his goal on the 1minute drills due to his anxiety and frustration with being timed. He struggles to move forward even though he knows his addition facts!
With this scenario, how does Rocket Math help? How will he ever move on to learn subtraction and multiplication? He’s a smart kid but can’t seem to succeed with this method!
Please help me to see otherwise!

Dr. Don answers:
Erica, I can see why you are frustrated. Students should not take more than three to five days to pass a level in Rocket Math and no more than a year to pass an operation such as addition. The rule is: If you child is frustrated by Rocket Math–it isn’t being done right! The school should not be complacent and should not leave your son to fall way behind his classmates. If a student takes longer than six days to pass a level I recommend that the school or the teacher should intervene. Interventions should happen in a matter of days, rather than allowing students to stall for weeks or months. You are writing in response to my post, so you know you can read the directions, even get the program and work with your son at home to help him. What interventions should be tried with your son?

1. My first intervention would be to practice Level S test (inside the box) orally with your son in the evenings once or twice. If he has been on Level S for a long time, there are facts on the test that need review–through oral practice. Use the same correction procedure we recommend everywhere: if he makes any hesitation, give the correct problem and answer, have your son repeat it three times, then back up three problems and go again. A couple of days of that and he should pass handily at school.

2. If that didn’t help your son pass in two days, my second intervention would be to watch your son take a written test and see what is going on, see if there is evidence of frustration, or anxiety, or if there are behaviors during testing that are interfering with good test results. If you want to send me video of your son taking the test at home, I could probably tell what is the issue that is holding him back. There may be some test-taking behaviors he can learn, such as not stopping during the test, or not erasing sloppy answers, which would improve his test results.

3. The third intervention I would do is give the Level S test orally. There are a number of reasons a student might not be passing and I have blogged and have video clips on YouTube that address what to do with students who are “stuck.” You write that you know that he “knows his facts.” Probably because when you ask him a fact, he can answer it immediately, without having to stop and figure it out. If that is true he should be able to verbally tell you the answer (not read the problem, just say the answer) to 40 facts in a minute at Level S. You could test him at home to find out if he does know the facts at Level S. If he can orally answer 40 facts on the Level S test, he knows the facts well enough and should have passed. If he is not passing the written test even though he can verbally say the answer to 40 facts in a minute, then his writing goal is off for some reason. Another piece of data that would suggest his writing goal is off, is if he has been stalled at some rate of problems and hasn’t improved his rate for a week. That suggests that that number of facts (whatever it is) is all he can actually do, and his writing goals need to be revised down to the number he can write when taking a one-minute written test (assuming he is on a level on which he can verbally answer 40 problems in a minute). In a couple of places in the directions (FAQs), I explain that, so you can share with the teacher.

4. What if, as you suggest, he is freezing up during the written test due to “anxiety and frustration with being timed?” The best way to overcome anxiety is to keep doing the thing that makes you anxious, which is why most students stop being anxious about Rocket Math after a couple of weeks. A fourth intervention would be to practice taking the test in writing–but untimed. If he completed all the items on the test several times at home, untimed, he would stop being so anxious about doing it under timed conditions. Most students also understand why they are being timed (to see if they know the facts without hesitation). He will not get unduly frustrated if you explain to him this is just a race and if he doesn’t give up he will keep getting better until he wins. Of course, if his writing goal is too high and he can’t possibly meet the goal, he will become frustrated.

5. If your son cannot already orally say the answers to 40 problems on the Level S test in one minute, he needs some more practice. My fifth intervention would to practice with your son–be his partner. He may not have a conscientious partner at school and may not be getting the most out of his 2 minute practice time. I routinely find that when I practice with students (the right way with correcting hesitations as well as errors) even once, they suddenly pass or come very close. The quality of the practice is critical to learning to answer these facts without hesitation. If practice allows students to stop and figure out the fact every time they will take a very long time to get to knowing those facts instantly. If that is the case, if you practice orally with your son once or twice an evening at home, the right way, he will begin to pass every few days at school. He will finish addition this school year.

In third grade I recommend that all students start multiplication at the same time even if they have not “finished” subtraction. Multiplication facts are far more important, so subtraction facts can wait. If you closely follow your son’s progress in learning multiplication facts with Rocket Math, you can intervene in time to make sure he does not get frustrated or fall behind. It should take him three to five days to pass a level, but no longer than that. With some extra practice at home you can be sure he will be successful. Knowing basic facts instantly will be very important for him, so don’t give up!

Can we use Rocket Math worksheets at home?

A parent asks:

I am a parent of a second grader who struggles mightily with her math facts. Her school does not do Rocket Math, although other buildings in our district use your program. I would like to know if your math facts program is appropriate for me to buy to use at home with my daughter. Also, does the Rocket Math basic subscription contain the complete program that a classroom would get?

Dr. Don answers:

Yes, the Rocket Math worksheet program is appropriate to buy to use at home with your child. The basic subscription has everything a parent or classroom teacher needs to run the program. But…

That being said, a parent at home may want to consider using Rocket Math flashcards instead of the worksheets in the original Rocket Math program. Flashcards are designed for one-on-one where the worksheets are designed to run an entire class at the same time. You can download the Flashcard Directions for free–and I highly recommend you doing that, so you know exactly how to work with your child effectively to learn math facts from flashcards. I really like the watch-your-favorite-TV-show-together-and-do-flashcards-during-all-the-commercials plan.

If you are teaching your child math facts at home, you definitely wouldn’t want to work on more than one operation at a time; addition in first grade, subtraction next in second grade, multiplication in third, and division in fourth grade.

The practice procedures are very similar between the flashcards and the original worksheet program. In both cases the student is to read aloud the problems and say the answers from memory without hesitation. The person listening (tutoring) provides the same correction procedure–saying the correct fact and answer, having the student repeat the fact and the answer three times, then doing two more problems before revisiting the target fact (the one on which there was an error or hesitation). The difference is that in the worksheet program students are reading facts from the worksheet, while in the flashcard program the student is reading the facts off the flashcards.

With the worksheet program you will have to print out the worksheets, the writing speed test, the goal sheet and the rocket chart. Each time you give the student the one minute test (to see if they are ready to move on to the next sheet) you’ll use up that sheet and have to print a new one. When your student passes the set of facts on that sheet, you’ll need to print the worksheet for the next set. With the flashcards, no additional printing is required. That alone is reason to use flashcards in my mind.

There is one very special circumstance in which it might be important to use the original Rocket Math worksheets at home. If your child is using Rocket Math in school, AND if the program is not being run correctly, AND if your child is being frustrated–then you might want to get a subscription. Watch our YouTube video on how to tutor Rocket Math.

If you read the Rocket Math Directions FAQs, you will be able to discover what is wrong at school. It may be that not enough time is spent practicing, or practicing the right way. It may be that your child’s handwriting speed was not taken into account when setting their goals. It may be that your child’s student partner in school is not correcting errors or hesitations in the right way. In any case it would be very important to show your child that he or she CAN in fact learn math facts successfully (all children can) and to overcome the frustration that improper use of the program is causing.

So you can buy and use the original Rocket Math worksheet program at home, but think about whether flashcards would be easier than the worksheet program. Teachers can’t effectively use flashcards in their classrooms because they can’t monitor the learning of that many students at once without the testing procedure. But you can when you are home alone with one child at a time–so flashcards can work for you.

Is Rocket Math frustrating your students?

If students (and parents) are really frustrated, Rocket Math is not being done the right way.

How should Rocket Math be done?

  • * Students should be practicing orally two or three minutes each day in school .
  • * Students should be practicing again at home for another two or three minutes.
  • * SOME students who need it, should be getting a second practice session during the day at school.
  • * When practicing the students should be saying the facts aloud and the answers.
  • * Students should be practicing with a partner who has an answer key.
  • * Partners should do the correction procedure if the student hesitates on any of the facts they are practicing.
  • * This practice should occur every day–not just once or twice a week.

With good practice several days running any child can learn those two new facts to automaticity and should be able to write the answers to those facts without hesitation–as fast as he or she can read the facts and write the answers. This is the point of Rocket Math and it works when done properly. How could this go wrong? Here are some things to look for that are WRONG!

  • * Testing only without the daily oral practice. Teachers sometimes prefer just giving tests and think this will accomplish the same thing, but it doesn’t. The learning occurs during the practice sessions with the partner. Without orally practicing students are not all going to progress as well as they should, and some will become very frustrated.
  • * Students who have bad habits that interfere with their ability to write quickly, such as erasing answers, counting on their fingers, looking at the clock, skipping around or writing answers in complicated patterns.
  • * Setting goals faster than students can actually write. (How this happens I haven’t a clue, but it does.) Students know the facts without hesitation but can’t write as fast as their goals demand. If they have practiced well for a few days and they can orally answer the facts without hesitation–giving 40 or more answers orally in one minute–reset their goals to what they have been doing and let them move on. Students don’t have to pass every day, but they should pass within six days.

Remember, the point is for students to practice the two new math facts on the sheet and add them to the ones they already know. As long as students can answer facts without hesitation (after reading the fact aloud they have the answer already in mind) then they know their facts well enough. This should not be driving anyone crazy and if we do it right it is fun and enjoyable–even though it is work.

Why should it take months to learn addition facts?

Because you need to remember these facts your whole life!

Learning all the addition facts well should take a while. What’s more, it is important to spread this learning task out over weeks and months. Why? Because the longer you spend learning something the better you learn it and the longer you remember it. Conversely, when you cram learning into a short period of time you will likely forget it soon. Remember, cramming for exams in high school or college? Remember, what you learned? Probably not.

Rocket Math is designed to motivate students to work through a long task of learning nearly 100 facts in addition. It is broken down in bite-sized pieces for a couple of reasons. One, so each piece is not too much to memorize (nobody can memorize ten similar things at once). Two, so students can experience success along the way. Little successes keep them motivated for the long haul, which is a key point.

Rocket Math is purposely demanding. We want student to learn the facts to the point of instant recall, without any hesitation. So we expect them to be able to write the answers to the facts as fast as their little fingers can carry them–without any having to stop and think about on the way. That is called the level of automaticity. One way to do this would be to simply require all students to practice for a week on each set. That wouldn’t be terrible, but it wouldn’t be motivating and it wouldn’t take into account learning differences–it wouldn’t differentiate properly. Some students can learn facts to that level in 3 or 4 practice sessions while others may take 10 or more practice sessions to get to the level of automaticity.

Practice must be focused on learning. It is very important that practice has to be focused on learning, rather than just “going through the motions.” It is critical that students realize they can move on as soon as they learn these facts, and not until then. If everyone moved on every week regardless of learning differences, it would be too soon (moving on too fast) for some students and too slow for others.

Students need to meet the rigorous tests of Rocket Math and it is optimal that they spend several days on each set. With 26 sets to master and 90 days in a semester, we should not expect students to master an operation in less than a semester. It is also acceptable for a student to take up to a school year to learn all the addition facts. If schools routinely taught addition in first grade, subtraction in second, multiplication in third and division in fourth grade, their students would find math computation a breeze. Even better, they would remember those facts, that took them a school year to learn, for life. Isn’t that really the point?

Is a goal over 40 fair or necessary?

A parent asks:
Our daughter has a goal of 50 problems in a minute. She is finding that hard as more of the answers in multiplication have two digit answers. Is this normal, and if so, should her target be lowered as a result? I don’t want to challenge her below her abilities, but is she actually learning less if she is only required to answer 40 to pass instead of 50?

Dr. Don answers:
You are right that a student who can answer 40 problems in a minute knows their facts well enough. Initially we did not have any goals over 40, but we discovered that a lot of older students (4th grade and up) can write a lot faster and are capable of having higher goals than 40 and can achieve higher goals with 3 or 4 days of practice.

There is a caveat on goals over 40, that is in the FAQs/Teacher directions in Part K. Here it is:
Please note: There is a special exception for students who are such fast writers that they have goals OVER 40 in a minute. On the bottom of the Goal Sheet, please notice the exception for fast writers. Students who have goals over 40 should try to meet those goals, but only for up to six days. As long as they are answering over 40 problems per minute without errors, they should be passed after six days. It is nice for those who can write faster to have higher goals, but we don’t want it to slow them down too much.

I have had people suggest to me that because of more two digit answers later in the sequence in Multiplication it slows students down and therefore goals should be lowered.  I would say, there is a case for keeping goals the same rather than raising them.  Here’s what the data show in terms the number of two-digit answers out of the 63 answers on a test:
Set B zero, Set D 23, Set G 37, Set K 43, Set N 46, Set Q 49, Set T 50
So there is a bit of a case to be made for the extra digits, especially if goals are raised really high during Set A or B.  The writing speed test is 50% two digit answers, but by Set G in Multiplication the test is a higher percentage of two digit answers than that.  So keeping the goals the “same” in Multiplication in terms of problems still means the challenge is increasing in terms of digits.  Watch out for any students who cannot pass within six tries/days with good practice.

The next caveat is that being able to answer orally 40 problems in a minute (just saying the answers) is fast enough to indicate no hesitations. So either orally or in writing to do 40 problems in a minute indicates a student is at mastery. In either case we would want the student to take six tries to meet a higher than 40 goal if they can. As I said in another post, we want them to practice 3 or 4 days on a set before passing, so we don’t need to pass them along before that.

The question is whether she can weather three or four days of practice without feeling like a failure. She shouldn’t, but compared to the passing every day she was doing before, it might seem like it.  Building up her stamina and getting her to take 3 or 4 days of practice on each set is optimal.