**Rationale:**

Another important math skill is to be able to count by numbers other than 1’s. Some count-bys are relatively more useful than others. Counting by 5’s has application to money and time-telling applications, while counting by 8’s has few directly applied uses, but is still useful when estimating. Being able to do fluent count-bys is helpful with multiplication and division computation where the student is dealing with the same number again and again. Using recursive patterns to teach count-bys illustrates the relationships between numbers and makes them easier to learn. In recursive patterns numbers are presented to show the minimal change in the pattern.

**Materials:**

The numbers as printed from a particular number to some other

number (forward). A second blank sheet of paper.

**Â Â 2 Â Â 4 Â Â 6 Â 8 Â 10 **Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (5)**
12 14 16 18 Â 20 **Â Â Â Â Â Â Â Â Â (10)

**Â Â Â Â Â Â Â (15)**

22 24 26 28 Â 30

22 24 26 28 Â 30

**Â Â Â Â Â Â Â (20)**

32 34 36 38 Â 40

32 34 36 38 Â 40

**Â Â Â Â Â Â Â Â Â Â Â Â Â (25)**

42 44 46 48 Â 50

42 44 46 48 Â 50

**(30)**

52 54 56 58 Â 60 Â Â Â Â Â Â Â Â Â Â Â Â Â

52 54 56 58 Â 60 Â Â Â Â Â Â Â Â Â Â Â Â Â

**Â Â Â Â Â Â Â Â Â Â Â Â Â (35)**

62 64 66 68 Â 70

62 64 66 68 Â 70

**Â Â Â Â Â Â Â Â (40)**

72 74 76 78 Â 80

72 74 76 78 Â 80

**Â Â Â Â Â Â Â (45)**

82 84 86 88 Â 90

82 84 86 88 Â 90

**Â Â Â Â Â Â Â Â Â Â Â Â (50)**

92 94 96 98 100

92 94 96 98 100

**Rote Counting by 2’s. See and Say Numbers**

This task should be done for a couple of minutes each day until the student counts quickly and easily without errors from any number to any other number.

**Task 1: Counting by 2s**

Say to the student, “Now **we are going to learn to count by 2s. I am going to count by 2s to 20. I will touch each number and say it. Listen. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.**

Touch each number from 2 to 20 and say it**. Listen again, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.**

**Count from 2 to 20 with me. I will touch the numbers. When I touch the numbers we will both say them. Ready.** (Signal) Touch each number and count from 2 to 20 with the student.

**Say, “Good job. You did it!”**

**Now try that by yourself. Touch each number and count from 2 to 20 and say it. Ready.**(Signal)

Watch the student touch each number and listen to the student count from 2 to 20.

Say, **You did it. Good counting. Do that for me one more time. Ready.** (Signal) Repeat task.

Say, **Excellent. You are learning to count well.**

**Correction Procedure:**

If the child cannot count from 2 to 20 without making errors, change the task. Review counting to 100 until the student is fluent at 150 counts/minute and try the task again.

**Exercise #2 â Short form recursive pattern for 2’s**

On a blank sheet write of paper Â Â 2, 4, 6, 8, 0.

Say to the student, **Â Here is an easy way to count by 2’s. Any count will always end with these 5 numbers. If you want to count further, use the numbers again.**

**Look at these numbers and count by 2s to 10. Ready**. Signal.

Students count Â Â Â Â Â Â Â Â *2, 4, 6, 8, 10.*

**Now count by 2âs from 10 to 20. Ready. **Signal.

Students countÂ Â Â Â Â Â Â Â *12, 14, 16, 18, 20.*

**Look at the underlined numbers. They are 2, 4, 6, 8, 0. If you remember these five numbers, you can count by 2’s to any number.**