New Learning Track: (LT 1.0) Conceptual Addition

Students have things to learn before memorizing Addition Facts

There are two Learning Tracks in the Rocket Math Online Tutor that can help students get ready for memorizing Addition Facts in the first grade.  Before doing addition students have to be able to count items and match the name of the number with its representation–that’s Counting and Subitizing.  The other skills they need to develop are the understanding of addition as adding two groups of objects along with the skill of “counting on” from a number other than one–that’s this one: Conceptual Addition.

 

We begin with adding two groups of objects

Children who understand the concept of addition know that it is counting all the items in two groups of objects.  Learning Track (1.0) Conceptual Addition begins with having the students provide the answer to simple addition problems with the objects displayed.  Each problem is first taught by the Online Tutor.  It shows a screen like this and says, “This shows 4 birds plus 2 birds equals 6 birds. Tap 6.”  Then it presents the problem without the answer showing for the student to practice.  If the student cannot answer in 3 seconds or answers incorrect, the Online Tutor teaches the problem again and then tests the student immediately.

 

“Counting on” from a number is the next skill

Student need to learn to “count on” from a number other than one in order to be able to add problems with sums above ten. (Without “counting on” you can’t add five and six on your fingers!)  The Conceptual Addition learning track teaches the skill of “counting on”  from early on in the sequence.  The teaching and correcting narration includes the counting on for each problem.  As illustrated in the example here, the Online Tutor teaches, “Start with four and count 3 more.  Four, five, six, seven.  Four plus three equals seven.  Tap seven.”  This instruction is presented the first time the student sees the screen and every time the student hesitates or makes an incorrect response. So the Online Tutor models “counting on” with every problem presented in the rest of the learning track.

 

Adding zero is taught as well

Students need to understand what to do when they encounter working with zero.  So there are some problems that include adding zero in the learning track. The concept of zero is taught in the Counting and Subitizing learning track.  Here students are taught that you “count zero more.”  Here is the teaching and correcting script for adding zero that goes with the screens on the right.  “Start with 4 and count zero more. Four. 4 plus 0 equals 4. Tap 4.”  This problem and another with adding zero continue to be reviewed throughout the learning track.

 

 

Problems go up to 9 plus 9.

Students encounter all of the basic facts up to 9 plus 9.  They learn that the procedure of starting with the larger number and “counting on” the number of times of the other number will lead to the right result.  We want them to understand the concept of addition.  They will understand the concept well before they finish the learning track.

Can individualized computer-delivered instruction be effective and easy to manage?

When instruction is below the student’s level (sometimes called zone of proximal development) they don’t learn anything they didn’t already know.  When instruction is over the student’s level they lack the prerequisite information to understand the lesson, and so they don’t learn as well. If you have the luxury of a homogenous classroom, thank your administration for making it possible to teach whole group. Most teachers don’t have that luxury and need to deliver instruction to many different levels because instruction needs to be individualized for the student.  It’s relatively easy to have individualized assignments that students can successfully complete and practice what they already have learned. What’s difficult is to have effective individualized instruction, where they are learning something new they did not know.  Computer delivered instruction can be effective, if there are five keys components to the software.

1. Testing for placement or promotion

The software must place students in the correct starting point for their instruction. That requires some kind of testing.  Either some kind of placement test is needed or the software must allow students to “test up” through and out of material they already know. This testing should be the same as the testing for corrections and for mastery.  Any teaching program that doesn’t have a testing component is not focused enough to be effective. Without testing you cannot know what students have learned.  Without testing the software cannot know that it has taught anything.

2. Good Quality Instruction

The software must impart the information to be learned to the students. Even in practice-type programs, such as math facts or spelling programs, the software should tell students what they need to learn.  It may seem simple, but it is critical for students to learn.

For more complex topics, the instruction, the telling of the information, should be as clear and simple as possible, and to the point.  It should be given in small bits, followed quickly by testing to see if the students learned the information (see correction procedures).  Too much information and not enough testing or checking for understanding and a program will be ineffective.  Unfortunately, sometimes “educational” software does not do a good job of teaching, the exposition is not clear enough.  The testing and correction procedures must be in place, so that the exposition will be seen to be ineffective and can be corrected.  Hopefully, the software developers used the feedback of initial trials to improve the lesson delivery.  In any case, the school or parents need to check the testing to see if the software is effective in teaching what it claims to be teaching.

3. Testing and Correction procedures

The software must include some immediate checking-for-understanding type testing after each bit of instruction. If that testing shows that a student has not learned the objective, the program must have a correction procedure in place.  It needs to be robust enough to ensure that the student learns the bit of information that was missed.  An effective correction will re-give the initial instruction and then immediately check for understanding.  A good program will repeat this step as many times as needed for the student to be able to correctly answer and show their understanding.  Then the program should intersperse something else and then re-test the student a couple of times more.  Then the information can be assumed to have been learned until mastery testing, which comes next.

4. Mastery-based promotion

Before students go on to learn another part, chapter, concept or unit the software must test them for mastery of all the objectives in this segment.  Mastery-based promotion means the student doesn’t go on to the next thing until the student has demonstrated mastery of the current material.  Instruction should, of course, be broken down into manageable size pieces, but each piece should be mastered before teaching additional content to be learned.  That requires a test of some sort.  Lack of mastery means the part needs to be repeated until it is mastered.  Demonstrating mastery on the test will result in the student being promoted on to the next segment of instruction.

 

5. Logical instructional sequences

Instruction must proceed through carefully thought-out, logical sequences.  Students must learn the prerequisites and learn them to mastery before they can take on a new skill.  Instruction must proceed carefully and thoughtfully so that students are never expected to know something that has not been previously taught and taught well.  Many topics that teacher and students find difficult to learn are simply a result of having several prerequisite skills that have not been brought to mastery ahead of time.  For example, long division requires that students have mastered subtraction, multiplication and division facts, as well as estimating and place value.  When several of those skills are not properly developed, long division become quite difficult to teach.  If the instructional sequence is sound, then each new step will not happen before the prerequisite skills are mastered.  When they are in a logical sequence and they are mastered, the new step will be relatively easy and will be learned quickly.

With these five components in place, computer delivered instruction can indeed be effective.  Students will be learning and the purchasers can see that students are learning based on the mastery testing in place.

Easy-to-manage requires that all the above are automatic

The above five components are necessary for instruction to be effective. They should be handled automatically by the software.  The testing and corrections, the re-teaching and the mastery testing, all should happen automatically and without fail.  When the instructional sequence is sound and students master each bit as they go along, then it will be easy-to-manage for the teacher.  For an example of an effective instructional program for math facts, see the Rocket Math Online Tutor.

Pre-algebra Learning Tracks for advanced students

Teachers can assign Rocket Math for all their students whether they are behind or advanced. Rocket Math has four pre-algebra topics for those students who already know their multiplication facts. Each of these topics will help them do pre-algebra processes more fluently and to quickly recognize relationships that they have memorized.

Learning Track 13: Identifying Fractions

Screenshot of Rocket Math Online Game for identifying fractions.

When students initially learn about fractions they are often only shown proper fractions. As a result, they have a limited understanding of fractions and can be confused by improper fractions or mixed numbers. Rocket Math prevents this problem. 

Students learn to name fractions shown in a picture as parts shaded.  Fractions are always shown with two or more whole units with a number of parts shaded. From the start, we teach students using examples of both proper and improper fractions as well as whole numbers and mixed numbers. Students learn from examples of proper fractions, improper fractions such as 5/3 that are equal to more than 1, and mixed numbers such as 2 and 1/2.

Students learn to identify over 90 different fractions quickly and easily by getting lots of practice. Their understanding of fractions will deepen and become more flexible as they learn to recognize many examples of fractions. Here’s half of the examples students learn from.

Screenshot of Rocket Math Online Game of Equivalent Fractions practice.Students will memorize the most common equivalent fractions (by reducing them to lowest terms) with this Rocket Math Learning Track. They will also learn to identify a number of fractions, such as 2/9,  that do not “reduce” or for which there are no equivalent fractions in lower terms. Students also learn to recognize a fraction equal to 1 whole in its various forms. When students don’t instantly know the answer they are told the equivalent fraction and given practice on it. The computer gives help in the Online Game.  Their partner gives that help in the Worksheet Program. By the end of the program, students will learn over 90 equivalent fractions. This gives students an excellent start on being able to manipulate fractions quickly and easily.

Learning Track 15: Factors & Primes

factors and primes game screensStudents are required to “find the factors” when dealing with unlike fractions and reducing fractions. Rocket Math Worksheet and Online Game teach students how to find factor pairs. Students learn how to find all the factor pairs and what they all are for many common numbers. They also learn to identify prime numbers and their characteristic of having only one and themselves as factors.

Students learn Dr. Don’s foolproof method for finding the factor pairs in order. https://www.rocketmath.com/2020/12/29/foolproof-method-finding-factor-pairs/ By this method they know the “last” factor pair when they see it. When the game asks “What’s next?” students can provide the next pair of factors or click the checkmark to indicate there are no more factors. When students go through this Learning Track they will no longer hesitate when asked for the factors of common numbers.

Learning Track 16: Fraction & Decimal Equivalents

Screenshot of Rocket Math Online Game of Fraction & Decimal Equivalents practice.

Learning common fractions and their decimal equivalents, learning fractions and their fraction equivalents in 10ths or 100ths.

Common fraction and decimal equivalents should not require a laborious process to “figure out.” Students should just know these, so this Learning Track in the Online Game allows them to memorize a bunch of common decimal and fraction equivalents.

Having a facility with a lot of fraction and decimal equivalents means faster computation as well as a way to check their process when manipulating fractions and decimals. Students also learn another essential pre-algebra skill that often confuses them.  They learn to correctly and fluently translate a fraction into a division problem and vice-versa.

“Test Drive” any of the 16 Learning Tracks in our Online Game demo accounts. 

Register for a 30-day free trial of Online Game for all your students

Explore the options for subscribing to the Worksheet Program.

Assign re-take when results of a 1-minute race go down

As students take the scheduled 1-minute races we display the results of the last two races on the Tab (A) Review progress screen.  As they learn more, they should be improving, their fluency should be going up.  These “races” are really fluency tests of a random selection of problems from throughout the Learning Track they are practicing.  The student shown above answered 8 of those problems in a minute the day they finished the first set (A).  After working a while and getting up to set (i) they are then scheduled for another fluency test. Then, the day the student finished set (i) the student was able to answer 10 in a minute. That’s what we expect to see.  

Their score really should not go down (unless they happened to get more problems they didn’t know).  If their score does go down significantly, as this student did, we offer the teacher the opportunity to assign a re-take of that test.  Perhaps the student was distracted during the 1-minute race, or they weren’t really giving it their best effort.  We leave this up to the teacher or parent to decide, because they know the student.  If the student may not have made their best effort, a re-take can give them another opportunity. 

Once that is scheduled, the student will do the re-take the next time they login.  In the meantime, the assignment will show on that student’s row on the Review Progress screen.  If on the re-take, the student’s score improves, the improved score will replace the lower score.  If it does not improve, then the current score stands.

Students can also choose to re-take a poor showing.

Sometimes during the 1-minute race, students get distracted, upset, or confused and they are pretty sure they can do better.  We give them a chance to prove it.  At the end of the race, it displays their score and gives them the choice to re-take the race to try for a better score.  If they click on the green “I want to re-take it.” button they get another chance to take the 1-minute race. If their score is better, it will replace the score they had earned previously.  If it isn’t any better, they get to keep the original score.  On the other hand, if the student thinks they did their best, they can click the top red button and they don’t have to do a re-take.

Tens of thousands of students have taken these 1-minute races and demonstrated improved fluency as a result of doing Rocket Math.  See this link to see the real-time data as more students improve their fluency with Rocket Math.

 

 

 

Rocket Math pioneer program–pilot it for your school or district

Are you a Rocket Math pioneer?

If you’re the first to try Rocket Math in your school, you’re a pioneer.  You are blazing the trail for others to follow. Congratulations!  We have an offer for you–a pilot test of Rocket Math. 

Pilot test offer

Are you willing to run a pilot test of Rocket Math?   Would you share your success with an administrator who could choose to purchase Rocket Math next year?  
Contact Lisa@rocketmath.com to tell us you want to run a pilot.  Then tell us who is the decision-maker who needs to see the success of Rocket Math. 

If the decision maker will write an email to Lisa expressing interest in the results you get with Rocket Math, she can set you up for a free til-the-end-of-the-school-year pilot subscription. We’ll add the decision-maker as a co-owner to your account so they can see the student success as it happens.  

Things-to-look-for (any time of day) for Rocket Math implementation

Evaluating a Rocket Math implementation when you aren’t observing Rocket Math in action.

Most of the time when you go into classrooms, something other than Rocket Math® will be going on. These are the things you can check on even when there are no students in the room. There are eight indicators you can see by looking at student folders.  There are four indicators while looking at the Rocket Math filing crate.  There are additional indicators to look for if there is a Wall Chart being used or if there are Race for the Stars games in the room.  Here’s a link to the checklist.

Look at several Student Folders

(1). Students all have folders that appear to be used daily. The folders are the heart of the organizational system. Students should keep their materials in the folders and keep track of their progress on the folders. Whether the students keep folders in their desks, cubbies, or are collected each day, there should be some signs of wear and tear.

(2). Rocket Charts on student folders show dates of each attempt to pass a level. Each day when students take a 1-minute timing test to try to pass a set of facts, they should write the date of the “try” on the Rocket Chart on the front of their folder. Without this record you cannot tell if a student is stuck because he or she has missed two weeks of school, or if students are only doing Rocket Math® twice a week (not recommended!), or if a student has exceeded six tries without intervention.

(3). Rocket Charts on student folders are colored in when passed. Coloring in the row on the Rocket Chart for the fact set that was just passed is the primary reinforcer of all that hard work. It is essential that students are given the time (and the colored pens, pencils, or crayons) to celebrate their success. Don’t get fooled by the older students or the students who are “too cool” to color in the chart. Even if they only want to color in the row with their regular pencil, students need to be told that they have accomplished something important, and giving them the time to color in their chart is a critical component of the program. This is way more important than you might think. You can also praise students who have accomplished a lot or who have just passed a level. Hearing from an administrator or coach about progress in math facts sends a huge message regarding the importance of the task.

(4). Student folders include packets of answer keys on colored paper. In order to practice correctly, each student’s partner needs to have an answer key in front of them when practicing. Each student needs their own answer key packet (so they can practice with someone who doesn’t have that answer key or with a volunteer who has no answer key). All the answer sheets for their operation should be copied and stapled into a booklet so students don’t have to go hunting for answer keys. Having the answer keys copied onto a distinctive color is important for teachers to be able to monitor paired practice. When students are practicing, each pair should have one student with answers (in that distinctive color) and the partner without the answers (on white paper). Any variation of this means the students are not practicing correctly—and that should be easy for the teacher to spot. Additionally, if a teacher is ready to begin testing and sees a hot pink paper on a desk, the teacher knows someone has answers in front of him or her.

(5) Student folders have the next sheet ready before starting practice time. Some system needs to be put in place so that the limited amount of time available for students to practice is NOT taken up with all students trooping up to the crate to get the next practice sheet each day. The recommended system in the Teacher Directions is to refill student folders when they pass a level, after school, with a packet of six sheets. That way the only time teachers have to handle folders is when students pass and they check the “pass” for errors and refill with a new packet. Many other ways of refilling student folders are possible, but no matter the process, students should have a blank practice sheet or set of practice sheets in their folder—which you would see when you check folders.

(6). Students have clear goals indicated on goal sheet. After students complete the Writing Speed Test, they are to have goals set for their daily 1-minute timing. The goal sheet should be stapled to the inside left of the student folder, the goal line circled, and the 1-minute goal written at the bottom of the sheet. The goal may be crossed out and a higher goal written in if the student has consistently demonstrated the ability to write faster than the original goal. Sometimes, teachers also write the goal on the front on the Rocket Chart, but the student’s goal should be clearly indicated. If not, it may be arbitrary or inappropriate (the same for all students, for example).

(7). Individual graphs are filled in because 2-minute timings are happening. Every week or two, students should be taking the 2-minute timings. These timings are a progress monitoring measure. They could be used for RTI or for IEP goals, or for any other time when a curriculum-based measurement is useful. At least they can demonstrate to us (and to the students) whether they are making progress in learning math facts in a given operation. As students learn more and more facts in the operation to a level of fluency and automaticity, they will be able to write answers to more facts in the operation on the 2-minute timing. Each time they take a 2-minute test, they should count the number correct and graph that on the graph stapled on the inside right of their folder. Each test is graphed in the correct column for whichever week of the month the test was taken.

(8). Individual graphs show upward trends as students are learning facts. Once students are taking the 2-minute timings regularly, it should be easy to see a trend. It should be going up, even if somewhat unevenly. For example, scores might go down after the long December break, but they should recover after a couple of weeks. If these graphs do NOT show an upward trend, something is wrong. Practice may not be being done for long enough (less than 2 minutes a day), or frequently enough (only three times a week), or students may not be practicing correctly (not fixing hesitations and errors). If only one or two students have flat graphs, those students will need something more. The individual graphs will be your indication that there is something amiss. You will just have to figure out what could be wrong. This should lead you to do some observations during Rocket Math® practice in that classroom.

Look at the Rocket Math file crate

(1). There is a crate or set of files for each operation practiced in the room. Each operation fills a crate and requires a different set of files. In any classroom where not all students are working on the same operation, there will need to be more than one set of files. Sometimes, teachers who have only one or two students in an operation may use the files of a neighbor teacher, but that should be only a temporary fix. The rule is that there must be a crate for every operation being practiced in that class.

(2). Rocket Math® crate is filled and organized from A–Z, complete with tabs. As of the 2013 version of Rocket Math®, every operation goes up to the letter Z. So each crate should have hanging folders with tabs showing the letters A though Z. Tabs are important to save time finding sheets and filling folders. If the files are a mess, out of order, no labels, or some letters are empty, valuable practice time will be used up trying to find the right sheets. If everything is labeled, and there are sheets in each file, then efficiency is a possibility. The Rocket Math store has tabs for sale if you need them.

(3). Rocket Math® crate has 2-minute timings numbered 1–5. In order to make sure that teachers do the 2-minute timing and monitor progress readily, they need to have class sets of the 2-minute timings (1 through 5) available in the crate. This is easy for you to check. If they are not there, it is likely that the 2-minute timings won’t be done as regularly as they should be. It is important for those timings to be done so you can see if all the students are making good progress.

(4). Teacher has a hard copy of the directions available for reference. The best place to keep the directions is right in the crate, so they are handy at any time. We have found that most of the time, when teachers are not doing things as they should in their Rocket Math® implementations, they don’t have a copy of the directions. When teachers don’t have the directions handy, they will ask a colleague how to do things. Unfortunately, this is like a game of telephone and typically doesn’t end well. Being sure that every teacher has the directions available for easy reference goes a long way toward proper implementation. It also allows you to pick up the directions when you are in the room and point something out to the teacher or to reference an appropriate page number in the directions in your notes to the teacher.

You can print the Teacher Directions from the virtual filing cabinet, in the Forms and Information drawer, under Rocket Math Teacher Directions.  You can buy printed copies from RocketMath.com/shop.  There are additional things to look for on the form but they are optional and go with supplemental parts of the curriculum.

Principal Free Space: Monitoring implementation

The Wall Chart is invaluable for monitoring an implementation

Principals and coaches:  Please read below, if you don’t already know that it is a great motivation for the students to be able to walk up and put their star on the Wall Chart and help towards meeting that class goal.


How can you easily monitor to know if classes are doing Rocket Math consistently? It can be hard to monitor, especially if you (wisely) scheduled a time each day for Rocket Math.  You can’t be everywhere at once!   

 
The Wall Chart is to be filled with the stickers from the bottom up and left to right.  You can walk through and quickly see what’s happening. The more stars on the chart, the more students are learning and achieving.  If more stickers are on the chart than the last walk-through, then you know Rocket Math is being done.

How the Principal Free Space works

 
How can you tell, when looking at a Wall Chart, whether this class has more stars than the last time?  When you walk through, praise the class and add a principal free space –a red dot or star to the next empty spot on their Wall Chart.  It would be a gift to the kids–one more spot towards their goal.  More importantly, it would show you what they have achieved up to today. 

When you come back a week or two later and there aren’t very many more star stickers after your principal free space, you can tell Rocket Math isn’t being done!   On the other hand, if they have filled in a bunch of star stickers you can really praise them again, congratulate the teacher and class on how hard they are working (and add another principal free space)!

How does the Wall Chart work?


The Rocket Math Wall Chart is designed to produce high levels of positive motivation in your students! Each time a student passes a level in Rocket Math they earn a star sticker and the right to parade up and put it on the Rocket Math Wall Chart.

As the star stickers begin to fill the 24 x 36-inch chart, the success of the whole class is demonstrated graphically, building excitement and pride. The goal arrows help the teacher graphically set intermediate goals and rewards, such as filling four rows in a month, and then celebrate success with a popcorn party or some other group reward! The chart has room for the over 700 star-stickers (that come with the poster), a year’s worth of Rocket Math success.  Includes directions and four (4) Goal Arrows.

Subtract from 20 (e.g., 18-15, 15-5, 19-8)

Back to Comparison Continue to Checkout 

Why teach Subtract from 20 facts?

The Common Core suggests that students be able to compute mentally facts such as 18-15, 15-5, and 19-8.  We call these the Subtract from 20 facts. These obviously build on the basic single digit facts such as 8-5, 5-5, and 9-8. Students should find these fairly easy to master but they still need some practice to commit them to memory.

Please give the special Writing Speed Test.

LOOK OUT! Because all the answers are two digits, the number of problems students can be expected to answer will go down!  You must give the special Subtract from 20 Writing Speed Test to set new, lower, goals for your students.  Above you can see the sequence of facts that will be learned in the Subtract from 20 program.  Otherwise, the program is exactly the same as the basic Subtraction Rocket Math program and uses the same forms–that can be found in the forms and information drawer.

 

Mult-Division Fact Families from 21 Learning Track


List of Learning Tracks Continue to purchase

Why learn in Fact Families?

Fact Families are another way to learn multiplication and division facts, or to review them once learned. 7 x 3, 3 x 7, 21 ÷ 7, and 21 ÷ 3 make up such a family.  Fact families are divided into two Learning Tracks.  This Learning Track is the second one, which goes on from 21 with 7 x 3 = 21 and larger facts.

How do students learn?

Students practice orally with a partner, reading and answering the facts going around the outside of the sheet.  The partner has the answer key.  Then the two students switch roles.  After practice everyone takes a one-minute test on the facts in the box–which are only the facts learned up to this level.  Each student has individual goals based on writing speed, but no one can pass a level if there are any errors.   You must give the special Writing Speed Test to set individual goals for your students.

Students should be able to pass a level in a week, if they practice the right way.   Below you can see the sequence of facts that will be learned in the Mult & Division Fact Families from 21 program.  The program uses the four forms–that can be found in the forms and information drawer.

The most succinct way to be introduced to this program are these three videos.