 # Multiplication/Division Facts

This program is designed to help students attain mastery of their multiplication and division facts.

Before using this program, it is recommended that students practice their math facts using the multiplication/division fact cards available for purchase in the Cards section. The fact cards are color coded by type and strategy to make it easier for a student to learn the math facts for each level. The app, which has no color coding, presents math facts in a mixed, random order within a level. The app also serves as a periodic review to keep math facts “shiny.”

### Why is this important?

Students must have mastery (automatic recall) of their math facts to free up their working memory so that they can focus on harder problem solving processes. In order for students to be confident and successful in math, they must learn their math facts.

### What is mastery or automatic recall of math facts?

A student has automatic recall of a fact when he or she has practiced enough, using effective strategies, to able to produce a correct answer immediately. The student knows the fact automatically (without having to think) and no longer needs a strategy to figure it out. The goal is for the student to be able to automatically recall every fact like he or she knows that “1 + 1 = 2”. Without thinking!

A student has mastery of an arithmetic level when he or she has automatic recall of all facts within the arithmetic level.

### How are these fact cards different than other fact or “flash” cards?

• Not just the answers on the back! Self-teaching, strategy-based solutions on the back of each card for students, parents, and teachers
• Based on the Jerome Bruner instructional approach of teaching math in sequence from the concrete (manipulatives) stage to the pictorial (drawing) stage to the abstract (symbols) stage for meaningful understanding
• Pictorial solutions to help the child visualize the process of “reading” a table and using table-based row and column strategies
• Two or three step problem solving processes that improve visualization skills, working memory capacity and sequential processing skills, all of which are critical for solving word problems and for overall achievement in all academic areas, not just math.
• Multiple strategies modeled so that the student can learn the easiest, fastest, and most accurate strategies for a given class of problems. For example, the “counting on” strategy is effective for adding 1, 2, or 3 to a number but is slow and inefficient when adding more than 3 to a number. The student will often use fingers because there are too many steps to hold in working memory!
• No more rote memorization without understanding patterns!
• No more guessing!
• No more fingers!
• No timed practice!

### Is the program results-based?

Yes! For several years it has been used successfully in schools for grade level, advanced, special needs and intervention instruction. It has also been used in math tutoring sessions and summer math programs to catch up students who have fallen behind in learning math facts as quickly as possible. It has been proven to work for ALL kinds of learners.

However, it is important that the parent or teacher understand that this application is based on the methodology of teaching from the concrete to pictures to abstract stages. Students must progress through all stages and cannot skip the concrete stage to go directly to these picture-based fact cards. It is highly recommended that the adult read the fact coaching tips included in this application.

Like all technology, this application is a teaching tool that requires adult supervision and instruction for it to be appropriate and effective.

### Where does the student start?

Students should have mastery of their addition and subtraction facts through 20 before starting multiplication or division fact practice. Students should progress through the arithmetic levels in the following sequence:

1. Multiplication Level 1
2. Division Level 1
3. Multiplication Level 2
4. Division Level 2

Student must have mastery (automatic recall of all facts) of prior arithmetic levels before progressing to the next arithmetic level.

### Example multiplication fact: 6 X 5 Before learning the 5x table in a picture, the student should model the 5x table in rows or columns of equal groups of 5 with concrete objects such as counting tiles or blocks.

The student should see the pattern of equal groups in the rows or columns. In this example, the groups of 5 are shown in the rows. The student should learn how to read a table from top to bottom, adding groups of 5, or bottom to top, subtracting groups of 5.

“Do you know 5 groups of 5? If 5 groups of 5 is 25, then how much is 6 groups of 5? Add 5 to 25 to get 30”. This strategy, shown in the picture above, is called using the nearest fact. The student should see that it would take much longer to figure out this fact by starting at the top of the table and counting (5, 10, 15, etc) than finding the nearest fact and adding a row.

### Example division fact: 42 ÷ 7 The student can use the multiplication fact that he or she has mastered. Additional prior knowledge comes from modeling division concepts with concrete objects. He or she can imagine splitting 42 blocks into groups of 7. This example shows the concept of pulling out equal groups of rather than dividing into 7 equal groups. Either concept can be used, but only one picture solution is shown on the card.

If the student can’t answer, “What times 7 is 42?”, then he or she should ask a different question. “How many groups of 7 blocks can I get out of 42 blocks?” The student should estimate. He or she can start with 5 groups of 7, a fact in the middle of the table. “5 groups of 7 is 35. 6 groups of 7 is 42 (7 more than 35). 7 groups of 7 is 49. 6 × 7 = 42.”