Addition/Subtraction Facts

Addition/Subtraction Facts

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This program is designed to help students attain mastery of their addition and subtraction facts for sums through 10 and 20.

Before using this program, it is recommended that students practice their math facts using the addition/subtraction 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 by breaking the problem into easier steps
  • 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 the process!
  • 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 progress through the arithmetic levels in the following sequence.

  1. Addition through 10
  2. Subtraction through 10
  3. Addition through 20
  4. Subtraction through 20

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

It is important to understand that students master their math facts when taught effectively using the Concrete to Pictures to Abstract (CPA) approach. Students must progress through each stage in sequence. They cannot skip the concrete stage and go directly to these picture-based fact cards.

The concrete, pictorial, and abstract stages are overlapping and continuous; they should not be treated as discrete units to be passed through once and then abandoned. In order to attain math fact mastery, do not skip stages!

Know that some students need to stay in the concrete and pictorial stages for a long time before they internalize the picture. It is internalized when they can “see” it after the concrete and pictures are removed. If the student forgets a fact or is rusty after a period of no practice, he or she should be able to recall the pattern and corresponding strategy to figure out the fact. If the student cannot retrieve the pattern or accurately use the strategy, he or she must go back to the concrete and/or pictorial stages since the fact was never truly mastered.

First, model the strategy in a concrete way.

The student should model the fact by building the numbers with counters like counting bears or tiles, or with linking cubes to make “number trains”. It is recommended that the student build number trains for numbers 2 through 10. Make two trains of 10 for both the addition through 20 and subtraction through 20 fact card levels. If the student cannot “see” and use the number line in his or her head to count on and down effectively, he or she will need a picture, specifically a number line, 1-10 or 1-20, to see and touch until he or she has practiced counting enough that they have internalized the number line picture and no longer need the number line scaffolding.

Next, draw out the strategy in a special picture, a number bond.

After modeling concretely, ask the student to write the math fact on a dry erase board and draw out the number bond solution. Coach the student on which number to break apart and why. He or she will need to “read” the picture and, from the number bond break apart solution, write the two mental math steps to answer the question. It is not mental math yet! A student must stay in the drawing/writing stage until the number bond picture has been internalized which means he or she can “see” it and recall it without drawing it.

First, use concrete objects like number trains when drawing it out.

It’s important to overlap the concrete and pictorial stages for conceptual understanding. Remove the concrete objects once the student can “see” the pattern and corresponding strategy in the picture and write the steps without coaching. After drawing out the solution, the student should check his or her solution with the solution on the card.

Remember that a student’s solution may be different than the fact card. As long this solution is just as fast and easy (not too many steps or slower, harder steps), the student’s strategy may be used.

Important: If the student is confused, he or she should pull out the concrete number trains so that he or she is concretely modeling a two step strategy that leads to the correct answer.

Finally, use the strategy abstractly.

After enough concrete and pictorial practice, a student is able to “see” the strategy in his or head and articulate the steps out loud. After enough practice saying the steps out loud, he or she can just think the two steps. The student is in the abstract stage when he or she can visualize the solution without the use of concrete objects or pictures. If the student does not have a clear mental picture and is confused, he or she must draw the number bond picture. If this picture does not clear up the confusion, he or she must pull out the concrete again. Can he or she retrieve the pattern and corresponding strategy? Coach the student through the confusion.

Coach your student!

Use guiding questions to help the student arrive at a correct answer. If the student is stuck on a fact, do not allow guessing or rushing to an incorrect step or answer. Ask the student to pause and think, “What’s my strategy?” You can use the strategy on the card to coach the student.

Good coaching questions: “Which number should you break apart? How would that help you? Does it make the problem easier? Can we solve the problem in just two steps? What’s the first step? Say the step out loud. What’s the next step? Say the step out loud.”

If the student is still confused, he or she should return to the picture stage: draw out the number bond solution and steps. If he or she is still confused, coach the student through drawing the solution while modeling it concretely with number trains again. Use a number line, as needed.

There are many ways to find the answer! The student may solve a problem with a strategy that is different than the one presented on the back of the card. As a student develops good number sense, he or she may learn new patterns and strategies that are not shown in the card solution. What’s important is that the student uses the easiest strategy for him or her so that no more than two or three steps are taken.

A student should not be practicing in silence! If the student knows the answer, he or she should say the answer out loud. If the student does not automatically know the answer, he or she should say the “steps” out loud to sort through the confusion. Once the steps have been articulated clearly, then, he or she should say the entire fact out loud, not just the answer.

The goal is automatic fact recall, which means that the student no longer needs to practice using effective strategies, but knows the fact automatically, without thinking, just as he or she knows “1 + 1 = 2” automatically, without thinking.

Example addition fact:


If you knew your doubles fact, 4 + 4 = 8, would this be an easier problem?

“Can you pull 4 out of the number 6 and add it to 4 to get 8? Then, add the rest of the number 6, which is 2, to 8 to get 10”. This strategy, demonstrated with a number bond picture, is called doubles fact plus 2.

By using this two-step mental process, students not only master math facts but improve working memory and sequential processing skills.

Example subtraction fact:


“Can we break 14 into two parts, 10 and 4? Can we subtract 8 from one of these parts? 10 take away 8 leaves 2. What else is left in the number 14? Can we add the rest, which is 4, to 2? 4 and 2 makes 6.” This strategy, demonstrated with a number bond picture, is called subtracting from the 10.

By using this two-step mental process, students not only master math facts but improves working memory and sequential processing skills.