Learning Tracks

Identifying Fractions Learning Track

Posted on by Dr. Don

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What do students learn in the Identifying Fractions Learning Track?

Identifying Fractions is a Learning Track to ensure that students have a firm and correct understanding of fractions.  This will prepare them well for all subsequent work in fractions.

Students will learn the essential rule about what the numerator and denominator mean, although they won’t be working with those terms.  They just learn through examples, practiced over and over.  The numerator, called simply the number on the top, tells how many parts are shaded.  The denominator, simply called the number on the bottom, indicates the number of parts in a whole.  If a whole is not divided into parts, it is a whole number.

Learning proper and improper fractions and mixed numbers.

Right from the beginning of Set A students will encounter improper fractions and mixed numbers. (See the illustration above).  They will see examples of every fraction first at the top of the page before they are asked to identify it on their own. You see that students see the fraction, see the words for how we say it and they see the fraction they are to write.

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

 

Halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths.

 

The fractions that students become familiar with include, halves, thirds, quarters, fifths, sixths, eighths, tenths and twelfths.  They see improper fractions and mixed number with every denominator.  In the beginning with the smaller denominators students see a variety of shapes for each denominator, so they learn that the identity of a fraction only has to do with the number of parts in a whole, not the shape of the display.  You can see thirds as cubes and as circles and as rectangles in the examples to the right.

When students are eventually introduced to eighths, tenths and twelfths we don’t want to slow them down by having to laboriously count the number of parts in each figure.

  • Eighths are always displayed as two sets of four rectangles on top of each other.
  • Tenths are displayed as two columns of five blocks with little numbers in them.
  • Twelfths are always displayed as three sets of four rectangles on each other.

Students should notice these conventions so they can quickly identify the number of parts in those figures without having to count them.

Be sure to do the Identifying Fractions writing speed test.

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

Division 0 through 9s Learning Track

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What do students learn?

These are the basic single digit Division facts 0 through 9s. Each of the 26 levels, A through Z, introduces two facts and their reverses.  You can see in the picture above of Set D, we have outlined the new facts in red.  Below you can see the sequence of facts that will be learned in the Division 0-9s program.  The program uses the four forms–that can be found in the forms and information drawer.

How do students learn?

Two students use Rocket Math Worksheets to practice their math facts.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.   The most succinct way to be introduced to this program are through these three videos.

A teacher sits with four kindergartners and is holding up her hand to count how many pencils they have on the desk.

Conceptual Addition (counting-on) Kindergarten Learning Track

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Conceptual Addition (counting-on) Kindergarten

Conceptual Addition is a beginning program for kindergarten students. It comes after they finish “Beginning Numerals.” You are teaching them the concept of addition.  They should also know how to write the numerals to 20 by memory or from dictation before starting.  We recommend using our “Rocket Writing for Numerals” to teach numeral writing.

Teacher-led instruction, not peer practice.

This is teacher-led instruction, rather than peer practice.  Each worksheet begins with an “I Do” portion that is a teacher modeled demonstration. That is followed by a “We Do” portion where the teacher leads the students in doing the problems together.  The last portion is called “You Do” which students should be able to do independently following the teacher modeling and leading on several items.

Begin by counting objects in both groups

For the first four worksheets you count all the objects in two groups to find the sum. When modeling slash the objects with a pencil as you count.  Teach the children to do the same.  The answer keys show the objects slashed.

“I can do this problem. I can add the butterflies by counting ALL the butterflies.
Watch me slash and count ALL the butterflies. [Slash while counting.]
One, two, three. There are three butterflies in all. One butterfly plus two butterflies equals three butterflies.
So they wrote three in the box. Touch here where the three is written as the answer. Good.
One butterfly plus two butterflies is how many butterflies, everybody? __________ Yes, three butterflies.
Now watch me do the next box.”

Later, begin with a number and count-on from there.

Starting in Set E you get the larger of the two numbers “going” and then slash and “count on” from there.

“I can do this problem. I can add two chicks plus four by counting on from four. Watch me get four going and
then slash and count the two chicks. F-f-o-o-u-u-r-r, [Slash while counting.] five, six. Four plus two equals six.
So they wrote six in the box. Touch here where the six is written as the answer. Good.
Two plus four is how many, everybody? __________ Yes, six.
Now watch me do the next box.”

In each worksheet, the students are asked to add the number and the items in each box and write the correct number. The teacher models and students are required to count by slashing the items shown.  Slashing ensures the count is correct. Besides cute items there are also dice to count, fingers to count and hash marks to count–so students can learn multiple ways of keeping track of numbers.

Passing a level requires 100% accuracy.  Students who make any errors should be worked with until they can complete the worksheet independently and get all the items correct.

This will prepare kindergarteners for learning the concept of addition.  Combined with Rocket Writing for Numerals it will set students up for success in elementary math.

Addition 0 through 9s Learning Track

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What do students learn?

These are the basic single digit Addition facts 0 through 9s. Each of the 26 levels, A through Z, introduces two facts and their reverses.  You can see in the picture above of Set B, we have outlined the new facts in red. Student should already know how to add with fingers or pictures or manipulatives before beginning to memorize these facts.  See the Beginning Numbers and the Conceptual Addition Learning Tracks if you’re not sure.  Students should also be able to write numbers quickly and easily.  If not they need the Rocket Writing for Numerals Learning Track to learn how.

How do students learn?

Students practicing their math facts together.Students practice orally with a partner. They read and answer 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.  The test includes only the facts learned up to this level.  Each student has individual goals based on writing speed.  Students cannot pass a level if there are any errors.   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 Addition 1s-9s 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 is this 8-minute video.

 

Beginning Numerals–Kindergarten counting Learning Track

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What do students learn?

This is a beginning program for kindergarten students.  You are teaching them to count objects aloud and then match the word with the numeral.  Here is a link to a video explanation of how to teach this learning track.

In this Learning Track students learn through teacher-led instruction.

I Do portion–count objects and see the number circled (above picture).

Each worksheet begins with a teacher demonstration of counting objects and circling the numeral that matches.  On Worksheet A there are two and three only to learn.  The teacher demonstrates (best with a document camera so all students can see) how she counts the objects and then points out that the answer is circled.

We Do portion–counting stars together (also above picture).

In the “We do” portion of the worksheet the teacher counts the stars first as a demo and then with the students.  Worksheet A you all just count 3 stars.   By Worksheet S the teacher and the students are  counting 12 stars together.

 

You do portion (above picture).

In the “You do” portion of the worksheet after learning the numerals with the teacher, the students are asked to count the items in each box and circle the correct number.  They are not asked to form the numerals–that’s numeral writing skill.  They just identify the numeral and circle it. Besides cute items there are also dice to count, fingers to count and hash marks to count–so students can learn multiple ways of keeping track of numbers.

Passing a level requires 100% accuracy.  Students who make any errors should be worked with until they can complete the worksheet independently and get all the items correct.  Read an explanation of how to bring a small group to mastery in a program like this one.

This will build strong beginning math skills for kindergarteners learning the meaning of numerals.  Combined with Rocket Writing for Numerals it will set students up for success in elementary math.

 

Rocket Writing for Numerals Learning Track

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What do students learn?

Click here for the Rocket Writing for Numerals Teacher Directions.

Students learn to write numerals efficiently, quickly and legibly using the Rocket Writing for Numerals Learning Track.

There are 72 pages of practice divided into four chapters (each 18 worksheets long) which gradually increase in difficulty.

Read this blog on “How much practice is enough in Rocket Writing for Numerals?”

Chapter 1 Learn to trace numerals correctly and efficiently.

  • Chapter 1 has students tracing the numerals and learning to form them in the correct manner. The correct manner is from the top down which is the most efficient way to write numerals. Students work through each of the numerals individually.   Then they practice them in concert with other previously learned numerals.  Teacher directions suggest fun ways of teaching numerals by tracing them in the air. Students should stay in this chapter until they habitually form the numerals in the correct manner. (See the chapter 1 sample.)

Chapter 2 Learn to write numerals free-hand (without tracing) and smaller.

  • Chapter 2 (see the sample) gives more practice tracing but also requires students to learn to copy smaller examples.  The big challenge is to write the numerals free-hand (without tracing them) and to do so the appropriate size to fit 20 on a line.

Chapter 3 Learn to form numerals fluently (20 per minute).

  • Chapter 3 (see the sample) gives more practice tracing and more practice writing the numerals freehand and to put 20 on a line.  The big challenge is for students to do a one-minute timing to see if they can write 20 digits on a line and in one-minute. There are 18 worksheets in this chapter, so the goal is by the end of the chapter for students to be able to write 20 numerals in one minute. These one-minute timings can be done at the same time as other students who are doing Rocket Math worksheets.

Chapter 4 Learn to form numerals more fluently (40 digits per minute).

  • Chapter 4  gives more practice tracing and writing free-hand with 20 on  a row, but also has students aim for writing 40 digits in one minute. Once they achieve this milestone they are fast enough for Rocket Math. The last page of the program (the end of Chapter 4) is shown at the top of this article.

 

 

Foolproof method for finding factors

Posted on by Dr. Don

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.

https://www.educreations.com/lesson/view/how-to-find-all-the-factors-of-a-number/46790401/

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.

How can you improve writing speed?

Posted on by Dr. Don

Tina asks:
Hello Don,
Do you have any recommendations for improving writing speed? My son’s school does not use Rocket Math, but we use it at home. He knows his addition facts rather quickly orally but is stuck at a much lower level at school because he cannot write them fast enough.
Thanks, Tina

Dr. Don answers:
Tina,
That is a very good question. Yes, you can improve writing speed. Increasing writing speed will come with practice, but a special kind of practice. The biggest problem slow writers have is that they “draw” the numerals. That is to say, they decide how to make the numerals look like they should and then draw them, rather than having a set way of doing the numbers. Step 1 is for them to learn how to most efficiently write the numerals using strokes that consistently go down and from left to write. Students need to learn the right way to form the numerals and then practice it exactly the same way over and over until it becomes habit. In Step 2 the students need to practice writing the numerals small enough to fit on the line, while still forming them the right way. In Step 3 and 4 students need to practice writing the numerals until they are fluent (speedy and still form them correctly and legibly).

A student can practice each page of Rocket Writing several times. How many times you ask? See my blog on the topic of How much practice is enough in Rocket Writing, because it is interesting to see that you can arrange it so that you trust your son to know how much practice he needs.

Rocket Writing for Numerals is part of the Universal Level of the Worksheet Program. If you only have a basic level subscription, you can upgrade to that.

How much practice is enough in Rocket Writing for Numerals?

Posted on by Dr. Don

Students balance a desire for comfortable mastery against a desire for novelty.

A home-schooling mom asks:
After having read the Rocket Writing for Numerals teacher’s directions, I have a question about implementation: Should I have her do the same page twice in one day (at separate times) to help her get more practice? After re-reading the teacher directions again today, I also think I need to go back and do more demonstration and air writing.

Dr. Don answers:
Regarding Rocket Writing for Numerals, the focus of the air writing and demonstrations is to achieve accuracy and consistency in the way to form the numeral. Once she consistently knows how to form the letter (starting in the right place, making the strokes in the right direction, etc) then the rest is developing the motor skills. More air writing is not needed once formation is consistently correct.

Yes, you can have her do a page twice in a day. How many days in a row is needed before you can move on to a new page is not established by research. It would be different for each student anyway. If you watch her, then you can decide, or you can encourage her to decide.

You want a page to become easy or routine for her. She doesn’t have to do it perfectly, but don’t move on if she still seems to be struggling or having to go very slowly. You should move on if she seems to be unchallenged by the page. You can also engage her in deciding if she feels she is ready to go on to the next sheet or wants to practice on the same page some more. Generally, once children get the idea of what it feels like to master a performance, they want to do so and students balance that desire for comfortable mastery against a desire for novelty. My favorite image is of skateboarders in the park who practice and practice until they have a particular move down–but then move on to try something new when they think they have it.