Learning Tracks

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How does this Learning Track teach?

Learning to Subtract Integers displays problems on a vertical number line and then teaches students two rules about how to solve problems in which you subtract positive and negative numbers.

Rule 1:  When you subtract a positive number, go DOWN.
Rule 2:  When you subtract a negative number, go UP.

Doing problems on the vertical number line is more intuitively appealing because UP is more and DOWN is always less.  This makes crossing zero a little easier to comprehend.

Total of four problem types to learn.

Students learn how these two rules play out with two types of problems: when starting with a positive number and when starting with a negative number. Students gradually learn all four types of problems.  On each worksheet they see how to solve each problem type using the number line working with their partner.  Then students learn to recognize the pattern of each problem type by orally answering several examples of each type with their partner (going around the outside of the page).  You will probably not be surprised that there is a one-minute test on each set.   Students are to be 100% accurate and to meet or beat their goal from the special writing speed test for Learning to Subtract integers (the fastest goal is only 28 problems in a minute).

Online Video lessons can teach students the process.

Here are links to the 4 online lessons teach students how each type of problem is solved and why it is correct. These are available in the virtual filing cabinet as well.

(1) Subtract Integers Set A Positive subtract a positive

(2) Subtract Integers Set B Positive subtract a negative

(3) Subtract Integers Set G Negative subtract a negative

(4) Subtract Integers Set L Negative subtract a positive

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What is Skip Counting?

Not counting while skipping!  Learning to skip count by fours is to learn this sequence by memory: “Four, 8, 12, 16, 20, 24, 28, 32, 36, 40.” Skip counting is also known as learning the “count-by” series.

Why teach skip counting?

Skip counting is the best way for students to prepare for multiplication. It teaches the concept of successive addition which is the root of multiplication.  In addition, skip counting helps prepare the ground for memorizing the multiplication facts.  Students who have learned to skip count find the chore of memorizing multiplication facts much easier. This Learning Track is excellent for second or third grade students after learning addition and subtraction facts and before learning Multiplication facts. Skip counting is not something older students past fourth grade really need to do, and of course, it’s no longer useful once multiplication facts have been committed to memory.

Each series is broken into easy-to-learn parts

Students learn part of each sequence on a page, then the next page they learn the rest. For example: in Set O students learn to count by 3s to 12, (3, 6, 9, 12) then in Set P they learn to count by 3s to 21 (3, 6, 9, 12, 15, 18, 21), and then in Set Q they learn to count by 3s to 30 (3, 6, 9, 12, 15, 18, 21, 24, 27, 30).

How does Skip Counting teach?

As in most Worksheet program Learning Tracks, students practice with a partner who has the answers. Because of the way the rockets go around the page (see above), students and their checkers will have to pick up the pages and turn them as they are working. You’ll be able to see if they are really engaged and they will have fun turning the page around. The test in the center has them write the count-by series they have learned for one-minute and they need to meet or beat their best–just like the rest of Rocket Math. Here is the sequence students will learn in this order: 2s, 5s, 10s, 9s, 4s, 25s (so they can count quarters), 3s, 8s, 7s, and 6s. Probably our most fun Learning Track.

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What does this Learning Track teach?

Learning to Add Integers teaches four types of problems to solve.  It displays problems on a vertical number line and then teaches students two rules about how to solve problems that add positive and negative numbers.
Rule 1:  When you add a positive number, go UP.
Rule 2:  When you add a negative number, go DOWN.

Doing problems on the vertical number line is more intuitively appealing because UP is more and DOWN is always less.  This makes crossing zero a little easier to comprehend.

Students learn how these two rules play out with two types of problems: when starting with a positive number and when starting with a negative number. Students gradually learn all four types of problems.  On each worksheet they see how to solve each problem type using the number line working with their partner.  Then students learn to recognize the pattern of each problem type by orally answering several examples of each type with their partner (going around the outside of the page).  You will probably not be surprised that there is a one-minute test on each set.   Students are to be 100% accurate and to meet or beat their goal from the special writing speed test for Learning to Add integers (the fastest goal is only 28 problems in a minute).

Video Lessons for students to learn the process.

4 online lessons teach students how each type of problem is solved and why it is correct.

(1) Add Integers Set A Positive add a positive

(2) Add Integers Set B Positive add a negative

(3) Add Integers Set G Negative add a negative

(4) Add Integers Set L Negative add a positive

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Add and Subtract Fact Families are a different way to learn facts.

A number of math programs around the country introduce math facts in families.  Now Rocket Math does too!  A fact family includes both addition and subtraction facts. This program is Part 2 of Fact Families, coming after Fact Families 1 to 10. You can see above the 18 examples of fact families taught in this program starting with Set A; 11-2, 11-9, 9+2, & 2+9.  The sheet above shows the sequence of learning facts in the new Rocket Math  program Add-Subtract Fact Families from 11 to 18.  Each set that students learn from A to R adds just one fact family to be learned, so it isn’t too hard to remember.  (That’s the Rocket Math secret ingredient!) 

Learning math facts in families, is gaining in popularity these days.  Logic suggests that this would be an easier way to learn.  However, the research is not definitive that this is easier or a faster way to learn facts than separating the operations and learning all addition facts first and then learning all subtraction facts.  But learning in fact families is a viable option, and I wanted to have it available for Rocket Math customers.

Fact Families from 11 to 18 is part two and fit for second grade.

These facts come after the Add and Subtract Fact Families 10, which should be learned in first grade, so these are best for second grade.  The 25 fact families in 1s through 10s facts are just enough for one Rocket Math program.  It is a good and sufficient accomplishment for first grade.  With the 11 to 18 part for second grade there will be a lot of review.  In fact sets S through Z are all review. I have heard that some first grades prefer to keep the numbers small but to learn both addition and subtraction–so this program accomplishes that.

We  most sincerely want students to be successful and to enjoy (as much as possible) the necessary chore of learning math facts to automaticity. Please give us feedback when you use this program, Add-Subtract Fact Families 11 to 18,  letting us know how it goes for the students.

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A number of math programs around the country introduce math facts in families.  Now Rocket Math does too!  A fact family includes both addition and subtraction facts. You can see below the 25 examples of fact families taught in this program starting with Set A; 3+1, 1+3, 4-1 & 4-3.  The sheet shows the sequence of learning facts in the new Rocket Math  program Add-Subtract Fact Families to 10.  Each set that students learn from A to Y adds just one fact family to be learned, so it isn’t too hard to remember.  (That’s the Rocket Math secret ingredient!) 

Learning math facts in families, is gaining in popularity these days.  Logic suggests that this would be an easier way to learn.  However, the research is not definitive that this is easier or a faster way to learn facts than separating the operations and learning all addition facts first and then learning all subtraction facts.  But learning in fact families is a viable option, and we wanted to have it available for Rocket Math customers.

Flash news!! Someone looking for a master’s or doctoral thesis could do a comparative study of students using the fact families vs. the separated facts in Rocket Math. This could easily be a gold standard research study because you could randomly assign students to conditions within classrooms–the routine is the same for both–just the materials in their hands is different!  Just sayin’…

Best fit for first grade.  Dr. Don separated out the facts to 10 from the 11s-18s, because these 25 families are just enough for one Rocket Math program.  It is a good and sufficient accomplishment for first grade.  Some first grades prefer to keep the numbers small but to learn both addition and subtraction–so this program accomplishes that.

Add-Subtract Fact Families to 10 is the first half.  The rest of the addition and subtraction fact families, which students could learn in 2nd grade, are the Add-Subtract Fact Families 11 to 18.  We most sincerely want students to be successful and to enjoy (as much as possible) the necessary chore of learning math facts to automaticity.

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Why commit to memory equivalent fractions?

We expect students to reduce their answers to the lowest terms, but how can they if they don’t recognize them?  To do this easily, students need to know that six-eighths is equivalent to three-fourths and that four-twelfths is equivalent to one-third.  While they can calculate these, it is very helpful to know the most common equivalent fractions by memory.  Not knowing these by memory accounts for the most common problem students have in fractions: failing to “reduce their answers to simplest form.”

Each set (A through Z) has four fractions which are displayed on a fraction number line.  In addition to pairs of equivalent fractions, students frequently learn fractions equivalent to one, such as ten-tenths.  They also memorize fractions that can’t be reduced, for example three-fourths is only equivalent to three-fourths. Students use a number line and identify equivalent fractions by circling an equivalent pair of fractions.  They understand why they are equivalent as well as memorizing the pair.

Equivalent fractions, Factors, and Integers, are all pre-algebra programs that are recommended for upper elementary and middle school students who already know the basic facts.

5-minute Equivalent Fractions video lesson.

Dr. Don explains How the Equivalent Fractions program works.

Students will commit to memory 100 common equivalent fractions.

Click here for the full sequence of equivalent fractions that students will learn in this program.

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How to do (compute) multi-step division problems

After becoming fluent with division facts, the best way for students to retain the knowledge of those facts is by doing division computation.  If students have not been taught division computation, this program breaks it down into small, easy-to-learn steps that are numbered in a teaching sequence that leaves nothing to chance.

Assessment to find which skill to teach first

There is an assessment available so you can test and see where to begin instruction. Find where the student first starts having troubles and begin teaching there.

Skills in this Learning Track

Note that the number for each skill gives the grade level as well as indicating the teaching sequence.  Skill 3b is a 3^rd grade skill and after skill 3c is learned the next in the sequence is skill 4a.  The sequence of skills is drawn from M. Stein, D. Kinder, J. Silbert, and D. W. Carnine, (2006) Designing Effective Mathematics Instruction: A Direct Instruction Approach (4^th Edition) Pearson Education: Columbus, OH.

(3b) Dividing 1-digit divisor and quotient with remainder.

(3c) Division equation with ÷ sign; facts with no remainder

(4a) 1-digit divisor; 2- or 3-digit dividend, 2-digit quotient; no remainder.

(4b) 1-digit divisor; 2- or 3-digit dividend, 2-digit quotient; remainder.

(4c) 1-digit divisor; 2- or 3-digit dividend, 2-digit quotient with zero; remainder.

not yet completed levels–to be developed upon demand–ask Dr. Don

(4d) 1-digit divisor; 3- or 4-digit dividend, 3-digit quotient.

(4e) 1-digit divisor; 3- or 4-digit dividend, 3-digit quotient with zero.

(4f) 1-digit divisor; 4- or 5-digit dividend, 4-digit quotient.

(4g) Rounding to the nearest ten.

(4h) 2-digit divisor; 1- or 2-digit quotient, all estimation yields correct quotient.

(4i) 2-digit divisor with incorrect estimated quotients.

How to teach this Learning Track

For each skill there is a suggested Teaching Script giving the teacher/tutor/parent consistent (across all the skills we use the same explanation) language of instruction on how to do the skill.  The script helps walk the student through the computation process.  For the teacher, in addition to the script, there are answer keys for the five worksheets provided for each skill.

Each worksheet is composed of two parts.  The top has examples of the skill being learned that can be worked by following the script.  After working through those examples with the teacher the student is then asked to work some review problems that are already known.  The student is asked to do as many as possible in 3 minutes—a kind of sprint.  If all is well the student should be able to do all the problems or nearly all of them, but finishing is not required.  Three minutes of review is sufficient for one day.

There are five worksheets for each skill.  Gradually as the student learns the skill the teacher/tutor/parent can provide progressively less help and the student should be able to do the problems without any guidance by the end of the five worksheets.  There are suggestions for how to give less help in the teaching scripts.

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Teach how to do (compute) multi-step multiplication problems

After becoming fluent with multiplication facts, the best way for students to retain the knowledge of those facts is by doing multiplication computation.  If students have not been taught multiplication computation, this program breaks it down into small, easy-to-learn steps that are numbered in a teaching sequence that leaves nothing to chance.

Assess to find where to begin instruction

Included in the Learning Track is an assessment to help you find out where to start instruction in the sequence. Test the student and begin teaching with the first skill on which they have difficulty.

Skills taught in this Learning Track

Note that the number for each skill gives the grade level as well as indicating the teaching sequence.  Skill 3b is a 3^rd grade skill and after skill 3e is learned the next in the sequence is skill 4a.  The sequence of skills is drawn from M. Stein, D. Kinder, J. Silbert, and D. W. Carnine, (2006) Designing Effective Mathematics Instruction: A Direct Instruction Approach (4^th Edition) Pearson Education: Columbus, OH.

(3b) Multiplying 1-digit times 2-digit; no renaming

(3c) Multiplying 1-digit times 2-digit; carrying

(3d) Multiplying 1-digit times 2-digit, written horizontally.

(3e) Reading and writing thousands numbers, using commas.

(4a) Multiplying 1-digit times 3-digit

(4b) Multiplying 1-digit times 3-digit; zero in tens column

(4c) Multiplying 1 digit times 3 digit, written horizontally

(4d) Multiplying 2-digits times 2-digits.

(4e) Multiplying 2-digits times 3-digits.

(5a) Multiplying 3-digits times 3-digits.

(5b) Multiplying 3-digits times 3-digits; zero in tens column of multiplier.

How to teach this Learning Track

For each skill there is a suggested Teaching Script giving the teacher/tutor/parent consistent (across all the skills we use the same explanation) language of instruction on how to do the skill.  The script helps walk the student through the computation process.  For the teacher, in addition to the script, there are answer keys for the five worksheets provided for each skill.

Each worksheet is composed of two parts.  The top has examples of the skill being learned that can be worked by following the script.  After working through those examples with the teacher the student is then asked to work some review problems of addition problems that are already known.  The student is asked to do as many as possible in 3 minutes—a kind of sprint.  If all is well the student should be able to do all the problems or nearly all of them, but finishing is not required.  Three minutes of review is sufficient for one day.

There are five worksheets for each skill.  Gradually as the student learns the skill the teacher/tutor/parent can provide progressively less help and the student should be able to do the problems without any guidance by the end of the five worksheets.  There are suggestions for how to give less help in the teaching scripts.

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Teach how to do (compute) multi-step subtraction problems

After becoming fluent with subtraction facts, the best way for students to retain the knowledge of those facts is by doing subtraction computation.  If students have not been taught subtraction computation, Subtraction–Learning Computation breaks it down into 18 small, easy-to-learn steps that are numbered in a teaching sequence that leaves nothing to chance. 

Find out where to begin instruction

Even better the instructional materials include an assessment of all the skills in subtraction computation in order, so you can test the knowledge of the student(s) before beginning instruction to see where to start. You can use this assessment to find very specific “holes” in student skills and then have the exact problems and explanation to fill that hole.

Skills taught in this Learning Track

Note that the number for each skill gives the grade level as well as indicating the teaching sequence.  Skill 3b is a 3^rd grade skill and after skill 3g is learned the next in the sequence, skill 4a is best taught in fourth grade.  Minor changes have been made, but for the most part, the sequence of skills is drawn from M. Stein, D. Kinder, J. Silbert, and D. W. Carnine, (2006) Designing Effective Mathematics Instruction: A Direct Instruction Approach (4^th Edition) Pearson Education: Columbus, OH.

(1b) Subtract from 2 digits; no renaming.

(2a) Subtract from 2digits; renaming required.

(2b) Subtract from 3 digits; borrow from 10s.

(3a) Subtract from 3 digits; borrow from 100s.

(3b) Subtract from 3 digits; borrow either place.

(3c) Subtract tens minus one facts.

(3d) Subtract from 3 digits; zero in 10s; borrow 10s or 100s.

(3e) Read and write thousands numbers, use commas.

(3f) Subtract from 4 digits; borrow from 1000s.

(3g) Subtract from 4 digits; borrow once or more.

(4a) Subtract from 4 digits; zero in 10s or 100s column

(4b) Subtract from 4 digits; zero in 10s column, 1 in 100s.

(4c) Subtract hundreds minus one facts.

(4d) Subtract from 4 digits; zero in 10s and 100s column.

(4e) Subtract 1, 2, or 3 digits from 1,000.

(4f) Subtract 5 and 6 digits with borrowing.

(5a) Subtract thousands minus one facts.

(5b) Subtract from a number with four zeroes.

How to teach this Learning Track

For each skill there is a suggested Teaching Script giving the teacher/tutor/parent consistent (across all the skills we use the same explanation) language of instruction on how to do the skill.  My favorite part is the rule students are taught for when to borrow (often confusing for students): Bigger bottom borrows.  Simple, easy-to-remember and consistently correct.  The script helps walk the student through the computation process.  For the teacher, in addition to the script, there are answer keys for the five worksheets provided for each skill.

Each worksheet is composed of two parts.  The top has examples of the skill being learned that can be worked by following the script.  After working through those examples with the teacher the student is then asked to work some review problems of addition problems that are already known.  The student is asked to do as many as possible in 3 minutes—a kind of sprint.  If all is well the student should be able to do all the problems or nearly all of them, but finishing is not required.  Three minutes of review is sufficient for one day.

There are five worksheets for each skill.  Gradually as the student learns the skill the teacher/tutor/parent can provide progressively less help and the student should be able to do the problems without any guidance by the end of the five worksheets.  There are suggestions for how to give less help in the teaching scripts.