Don’t argue, just prove it works!
How can we encourage the teacher who refuses rocket math and administration does not reinforce (or enforce) the program’s use?
Using a vertical number line can help provide certainty.
Adding and subtracting positive and negative numbers can be confusing for students. You can either start with a positive or a negative number and combine it with a positive or a negative number. That makes for four types or patterns of problems. Then when you consider addition and subtraction, the total is 8 problem types. Rocket Math has three learning tracks to help students learn how to deal with integers. Mixed Integers include all eight types, whereas Learning to Add Integers and Learning to Subtract Integers each just deal with four types. [Mixed Integers may be too hard for some or all of your students–meaning they can’t pass levels in 6 tries. In that case, put them through the Learning to Add Integers and Learning to Subtract Integers first.]
Part 1: Using the vertical number line to solve integers problems
The first issue for students is just to be certain of the answer. A vertical number line, where “up” is more and “down” is less helps provide certainty.
I have posted a series of free lessons online (links below) that use a vertical number line and a consistent procedure to take the confusion out of the process. Students can solve all eight types of problems with the same process on the vertical number line. Using the vertical number line there are two rules to learn. Rule 1: When you add a positive or subtract a negative, you go up on the number line. Rule 2: When you subtract a positive or add a negative, you go down on the number line.
So the first thing to figure out is what you are being asked to do (add or subtract a positive or a negative), and then use the rule to tell you whether you’re going up or down. Next step in the procedure is to circle the starting point on the number line. Once you circle the starting point, you show how far you’re being asked to go. You simply make the right number of “bumps” going either up or down from where you start. That gives you the answer without any uncertainty. These online lessons are quick (about 2 minutes) and identify a pattern of whether the answer is like the sum or the difference between the numbers. Once students can recognize the pattern they can begin to answer fluently and without a struggle
Part 2: Using the Rocket Math Integers learning track(s) to develop fluency in recognizing the type of problem
Here is a part of a page from the Mixed Integers learning track. The paired practice part of the program helps students learn to quickly and easily recognize each pattern. First, students use the vertical number line to work on a problem. In this example: -6 minus (-4). Then they have a set of problems with the same pattern (a negative subtracting a negative), which they should be able to answer orally without using the number line. Each worksheet includes all the types learned so far in the learning track.
As with all Rocket Math programs, there is a 2 to 3-minute practice session (at this level, I’d recommend 3 minutes), with a partner. Then the two switch roles. The practice is followed by a one-minute test. If the student can answer the problems in the test fluently (essentially without hesitations), the level is passed. As always, the student’s goals are individually determined by a Writing Speed Test. If a given level is still difficult, the student stays with that level a bit longer.
When a new pattern or type of problem is first introduced the one-minute tests will have a whole row of problems that are the same pattern. When the student passes the level, the next test will have two types of problems in each row. The next level has 3 types in a row, culminating in the fifth level, where the problem types are mixed. This way, the student develops fluency in recognizing the type of problem and how to derive the answer quickly. The Learning to Add Integers and Learning to Subtract Integers learning tracks take more time to learn the patterns, while Mixed Integers move more quickly.
Don’t forget that Rocket Math has a money-back guarantee. So if this doesn’t work for you and your students, we will refund your subscription price.
Many students find integers confusing. If you add a negative to a negative are you getting more or less??? Over the years different “rules” have been used to try to remember what should happen. Rules such as “two negatives make a plus” or “opposite signs subtract.” Whatever is used to try to remember, it interferes with a student’s ability to quickly and reliably get the answers without having to stop and puzzle it out.
I have posted a series of free lessons online (links below) that use a vertical number line to take some of the confusion out of the process. Turns out there are a total of eight types of problems but all of them can be solved with the same process on the vertical number line. Intuitively on a vertical number line, up is more and down is less.
Two rules to learn: when to go up and when to go down.
Using the vertical number line there are two rules to learn. Rule 1: When you add a positive or subtract a negative you go up on the number line. Rule 2: When you subtract a positive or add a negative you go down on the number line.
So the first thing to figure out is whether you’re going up or down. Once you do that you simply make “bumps” going either up or down from where you start. That gives you the answer without any uncertainty. These lessons are quick (about 2 minutes) and identify a pattern of whether the answer is like the sum or the difference between the numbers. Once students can recognize the pattern they can begin to answer fluently and without a struggle.
To help with the work of learning to quickly and easily recognize each pattern in Integers Rocket Math now includes three “Integers” Learning Tracks in our Worksheet Program Universal Level Subscription. (Click here to get a 60-day initial trial subscription for less than the standard full-year subscription.)
Learn adding and subtracting separately or mixed.
The first integers learning track, Learning to Add Integers, is limited to adding integers. In the above list of lessons, the adding lessons are numbers 1, 3, 7, and 8. The second integers learning track, Learning to Subtract Integers, is limited to subtracting integers and teaches the processes shown in lessons 2, 4, 5, and 6. The third learning track, Mixed Integers Drawer combines all 8 processes into one learning track. If students are likely to have issues or begin to find the Mixed Integers lessons difficult, have them do the separate operation first.
In the Rocket Math Integers lessons, students use the vertical number line to work a problem. In this example: -2 minus (-3). They use a rule and then apply it to the number line to find the answer. Then they have a set of problems with the same pattern they can orally answer without having to use the number line.
Students practice with a peer partner
As with all Rocket Math programs, there is a 3-minute practice session, with a partner. The partner who is checking has the answer key and corrects all errors immediately. Then the two switch roles. Then the practice is followed by a one-minute test. If the student can answer the problems without hesitation the level is passed. If it is still difficult the student stays at that level a bit longer. When a new pattern is introduced the tests will have a whole row of problems that are the same pattern. When that level is passed the next test will have two types of problems in each row. The next level has 3 types, then 4 types in each row. Then the problem types are mixed. This way the student develops fluency in recognizing the type of problem and how to derive the answer quickly.
Rocket Math has a money-back satisfaction guarantee. If you try this and find it isn’t everything you hoped, for in terms of helping your students become fluent with integers, I’ll gladly refund your money. I’m betting they’re going to love it.
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 Worksheet Program Factors Learning Track. Students also learn the pairs of factors in this sequence in the Online Game.
Dr Don has a white board type video lesson that explains this in 6 minutes.
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.
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.
Most people, for example, know their name, by memory.
In a previous blog I discussed What does CCSS mean by “know from memory?”
A reader asked the following question:
This topic of “know from memory” is something I have been digging into as a special educator. I wonder what your thoughts are about whether certain accommodations from these “know from memory” standards would actually be modifying the curriculum?
For example, if we used “extra time to respond” and the student had to use their fingers or some other method to count, would they then not be doing the standard?
This relates to where I’m at in middle school math, but I think that it’s reflected in the continuum of the common core maths.
Dr. Don’s response:
Actually, your example is very clear that it is not “knowing from memory.” You are describing “deriving from a strategy” or what I call, “figuring it out.” When you know it from memory, when you recall the answer, then you stop having to “figure it out.”
Knowing from memory and figuring something out are two very different things. I used to ask workshop participants to imagine sitting next to me in a bar and asking me for my name. What if, instead of saying, “Hi, my name is Don,” something different happened? What if, like the man pictured above, I was puzzled and said, “Wait a second, I have it here on my driver’s license.” Most people would likely turn their attention elsewhere while wondering what kind of traumatic brain injury I had sustained! They would very likely say to themselves, “OMG, that man doesn’t know his own name.”
The purpose of the verbal rehearsal that is a daily part of Rocket Math is to cement these basic facts in memory. Then when a student says to themselves, “8 times 7 is,” the answer pops into their mind with no effort. It takes quite a bit of practice to achieve that. However, the ability to instantly recall the answers to basic math facts makes doing mathematical computation a relative breeze. It make seeing relationships among numbers very obvious. It makes reducing fractions and finding common denominators easy. That’s why the Common Core thinks “knowing from memory” is so worthwhile. It’s why I began promoting Rocket Math in the first place.
Timed tests are not the important part of Rocket Math.
The “active ingredient” in the Rocket Math prescription, the thing that makes it work, is not timed tests. Timed tests don’t actually teach and often don’t really help students develop fluency. The usual timed tests of a random selection of all the facts can assess fluency in math–but they don’t work to develop it!
The “active ingredient,” the thing that makes Rocket Math effective, is verbal rehearsal. When students practice with their partner the students read the facts and RECALL the answers from memory and say them aloud. That verbal rehearsal is what cements them into memory. Reading the fact and recalling the answer from memory strengthens the neural connection.
Why do we give the daily tests in Rocket Math? Not to teach, but only to assess whether the facts introduced thus far have been learned well enough for the student to have new facts added to what they are learning. Individual students learn at different rates. Some students need only a couple of days of practice to memorize two new facts while others may need several days. The purpose of the daily tests is just to see if the student needs more practice time, or is ready to “swallow” some more facts.
As I note in my basic training presentation, “It’s like feeding mush to a baby. You have to make sure they have swallowed the last mouthful before you give them more.” See an explanation in this You Tube video in our Rocket Math channel: https://youtu.be/J8cWSDG0Di8
Do you want your students to learn OR are you just keeping them busy?
It’s critical to keep some of your students occupied in order for you to have the peace and quiet you need to teach other students. Those free math worksheets of random facts are fine for busywork, provided students already know the facts.
Get a 60 day trial for only $13
(and this is a big but) if you want students to actually learn facts, you need math worksheets that are more systematic than the usual fact practice worksheets. A random mix of problems (on those free math worksheets) is fine for practicing what you already know, but it is USELESS for learning new facts.
Students who don’t know their facts are left painfully counting on their fingers to do their “work.” This just wastes their time and makes them come to HATE math.
I know, because I made my students do it for years. 🙁
I discovered that with systematic practice students can actually learn math facts!
In order to learn new facts students must concentrate on a few they don’t know and practice those particular facts until they know them “by memory” without having to figure them out. After students have learned those they can then tackle a few more. That’s the only way to learn a bunch of facts. That’s what Rocket Math does. Watch this video to see it in action.
Each sheet (A-Z) adds two new facts and their reverses, making the process of learning them painless. By the time students have worked their way through the A-Z worksheets of an operation they know the facts “by heart” or as the Common Core calls it “by memory.”
If LEARNING is your goal, you’ll need something more effective than the free math worksheets.
Rocket Math has a MONEY-BACK guarantee.
If you spend the $13 to get a trial subscription and you decide Rocket Math doesn’t work or you don’t want to use the program, we’ll gladly refund your money.
Students have more fun and learn better when they are practicing orally, with a partner so they can get corrections and extra teaching on any facts they don’t know well. That is part of how Rocket Math works. So it won’t just be busywork. Your students will actually learn the facts and be proud of it.
If you can afford $13,
We want your students to have fun during math.
At Rocket Math we believe that students should enjoy math. And we know that what students enjoy is going fast! They enjoy being able to slam through a page of math facts or even a page of computation quickly and easily. We know that students are motivated by a sense of accomplishment and a sense of competence. They love getting “good” at math.
Practicing math facts until they are fully memorized is NOT a quick fix. It takes time and dedication on the part of the teacher and the student. But done right, within a matter of weeks both parties begin to see a difference. Students say things like, “I can do this!” or “I’m good at math!” when they see themselves succeeding and working through the multi-month process of learning all the facts in an operation.
As one of our teacher friends said, “I always start my math class with facts practice now, because it gives my students a sense of accomplishment, of success right off the bat.” Let’s face it, practicing math facts, even with a partner, is not intrinsically interesting. It is true, that if it is not done right, students will not make progress. But if it is done the right way, students learn, get good, pass small milestones and can begin to see progress. Seeing progress gives students a sense of accomplishment and they love it.
Too many educators suggest that the way to get students to enjoy math is to avoid dull topics like math facts or computation. Instead they want to immediately dive into complex, real-world, authentic, head-scratching-type problems that take even a committee hours to figure out. For most students that is not enjoyable. It is painful. And those students tend to avoid math or say they aren’t any good at it.
Done the right way, students can learn and become proficient with math facts and computation. I know it seems counter-intuitive that developing skill and fluency with basic math facts and computation would help students come to enjoy math more. But maybe you ought to consider it, because for decades we’ve been doing the opposite. The results show that very few American-educated students major in math in college. Maybe if we helped them feel like they were “good” at the beginning levels of math they might stick with it. Just sayin’
Three important test-taking strategies that Rocket Math will turn into habits.
Students really need perseverance to get through today’s tests. You want your students to really work hard and do their best! To have that kind of perseverance students need to KNOW that it pays off. Sticking with learning and testing over and over until they win is a central lesson of Rocket Math’s daily practice and tests. Most days, most students do NOT pass the One-Minute Daily Test. They have to practice some more and try again the next day. If they try hard and do their best on each day’s test, eventually they do pass. This teaches perseverance like nothing else in the curriculum!
Students doing Rocket Math learn that to be successful you have to work as fast as you can. Their individualized goals require that they write answers as fast as they can write. Students who pause to look at the clock or look around the room during the one-minute test simply do not pass. This may be the only time of the day that students experience the need to work quickly and they get immediate feedback based on whether or not they do work quickly–and it is something they care about! So they are motivated to work quickly. It is important for students to have that kind of experience if they are to learn the general rule that you are supposed to work as fast as you can when taking a test.
Have you ever watched a student waste valuable time working on a test item you knew the student wouldn’t be able to answer? Nothing more painful. Students need to learn to skip the items to which they don’t know the answer readily. How are they going to learn that without practice? Rocket Math has a progress-monitoring component–a weekly 2-minute timing you can see to the right. These weekly tests sample all the facts in the operation, including ones they haven’t memorized yet. Therefore the strategy they should use is to skip the ones they don’t know yet, so as to answer quickly all the ones they do know. If you explain this to the students, and they can develop this strategy while taking these weekly tests.
If you aren’t sure that your implementation is developing these habits please feel free to download the Teacher Directions. If you have a school wide implementation of Rocket Math be sure you have the Administrator and Coach Handbook.