Basic Addition and Subtraction Strategies for Single-Digit Strategies

Foundations of Single-Digit Addition and Subtraction

• Scope of Study: The focus is exclusively on basic single-digit addition and subtraction facts. This foundational work is distinct from single-digit multiplication, which is typically addressed as a separate module (e.g., following a week of addition focus).

• The Addition Table: A standard addition table contains single-digit numbers ($0$ through $9$) along both the top row and side column. The goal is the mastery of combinations such as:

  • $4 + 3$

  • $6 + 3$

  • $8 + 5$

• Reusability of Facts: Mastery of these single-digit facts is critical because larger, more complex calculations are essentially the repeated application or reuse of these basic numbers.

Prerequisites for Effective Addition Strategies

Before implementing specific computational strategies, three fundamental prerequisites must be established to ensure accuracy and reduce cognitive load:

1. Identifying the Larger Number: When presented with an expression like $2 + 4$, the student must be able to recognize that $4$ is the greater value to use it as a starting point.

2. The Commutative Property of Addition: Also known as "Turnaround Facts," this principle states that the order of addends does not change the sum (e.g., $2 + 4$ is the same as $4 + 2$). Understanding this nearly halves the number of individual facts a student must memorize.

3. Proper Counting Start Point: Students must learn to start the count after the larger number. For instance, when adding $2$ to $4$, the student should hold "$4$" in their mind and count "$5, 6$." A common error is starting the count on the number itself (e.g., counting "$4, 5$" to add $2$).

Strategy 1: Counting On (The "Think Big, Count Small" Strategy)

• Inefficiency of "Count All": Younger children often use a "count all" strategy (e.g., for $4 + 2$, they count "$1, 2, 3, 4, 5, 6$"). This is laborious, contains more steps, and is significantly more error-prone.

• Definition of Counting On: Start with the larger number and count forward by the smaller number. This strategy is most effective when adding $1, 2, 3, \text{ or } 0$.

• Statistical Impact: Implementing the "Count On" strategy for the facts of $+0, +1, +2, \text{ and } +3$ (including their turnarounds) covers $64$ facts. This represents approximately two-thirds of all basic addition facts on a $10 \times 10$ table.

• Techniques for Teaching:

  • Trusting the Count: Students must be confident that the quantity of the larger set remains the same without needing to recount it every time.

  • Visual Aids: Use dominoes (e.g., a $6 \text{ and } 2$ domino results in $8$ regardless of orientation), bead strings, or joining cubes (like Lego bricks).

The Part-Part-Whole Model

• Conceptual Framework: Addition is the action of joining or combining. In this model, there are three elements: one part, another part, and the whole (the sum).

• Linking Addition and Subtraction: By covering different sections of a Part-Part-Whole diagram, students learn the relationship between the two operations:

  • Standard Addition: Part ($5$) + Part ($4$) = Whole ($?$).

  • Missing Part: Part ($5$) + Part ($?$) = Whole ($9$). This is algebraically identical to $9 - 5 = ?$.

• Linguistic and Practical Examples:

  • Scenario A: I have $5$ lollies, my friend gives me $4$; how many altogether? ($5 + 4 = 9$).

  • Scenario B: I have $5$ lollies, my friend gives me some more, now I have $9$; how many did they give me? ($5 + x = 9$).

  • Scenario C (Difficult): I had some lollies, my friend gave me $4$, and now I have $9$; how many did I start with? ($x + 4 = 9$).

Strategy 2: Facts to Ten

• Core Facts: These are pairs of numbers that sum to $10$. Many are already covered by the "Count On" strategy:

  • $9 + 1$

  • $8 + 2$

  • $7 + 3$

• New Facts Introduced: The remaining facts to master in this category are $6 + 4$, $4 + 6$, and the double $5 + 5$.

• Tools for Mastery:

  • Tens Frame: A $2 \times 5$ grid (which can be improvised with an egg carton) helps visualize the "hole" of $10$ and how many "parts" are needed to fill it. For example, placing $7$ counters on the frame clearly shows $3$ empty spaces.

  • Games: "Build to 10" or "Build to 20" involves using a spinner and identifying the complement needed to reach the target decade.

Strategy 3: Doubles Facts

• Definition: Adding a number to itself ($1+1, 2+2, 3+3, 4+4…$).

• Real-World Visuals: Pairs of shoes, packets of pears, or toilet paper rolls often come in doubles.

• Instructional Game (Doubles Spin): Use a spinner (made with a paperclip and pencil) to select a number, double it, and place a transparent counter on the result on a game board. The goal is typically to get three or four counters in a row (column, row, or diagonal) while blocking the opponent.

Strategy 4: Near Doubles

• Mechanism: If a student has recall of doubles, they can use them to solve nearby facts by making an adjustment of $1$.

• Examples:

  • To solve $6 + 7$: Use double $6$ ($12$) and add $1$ ($13$).

  • Alternative for $6 + 7$: Use double $7$ ($14$) and subtract $1$ ($13$).

• Prerequisite: This strategy requires some level of fluency with the baseline Doubles facts to be effective; it cannot be used if the doubles are not first memorized.

Strategy 5: Bridging Ten

• Application: This strategy is primary for numbers that "cross" the $10$ threshold, specifically involving addends like $8$ or $9$.

• Three-Step Process (Example: $9 + 6$):

  1. Know Facts to 10: Recognize that $9$ needs $1$ more to make $10$.

  2. Partition/Split: Split the second addend ($6$) into two parts ($1$ and $5$).

  3. Combine and Solve: Add the $1$ to the $9$ to reach $10$, then add the remaining $5$ to get $15$.

• Visualization: Using two Tens Frames side-by-side allows students to physically move counters from one frame to the other to "bridge" the first frame to a full $10$.

Questions & Discussion

• Host's Summary of Strategies: The host reviewed the five strategies mentioned: Counting on from the larger number, facts to 10, doubles, near doubles, and bridging ten.

• Resource Mention: Paul references "Milestones" documents available for download which summarize prerequisites, strategies, and order of instruction.

• At-Home Engagement: The host mentioned that her family (son and husband) plays "Ken Ken" at home, a math-based puzzle discussed in a previous webinar.

• Timeline for Competency: In the Australian curriculum, students typically experience and learn these facts by the end of Year 2. Year 3 is dedicated to increasing speed and fluency through games.

Practical Games and Instructional Tools

• Snap the Cubes: Start with a stick of $10$ cubes. Snap it behind the back and reveal one part; the student must identify the hidden part (e.g., reveal $7$, hidden is $3$). This can also be done using a "mystery bag."

• Snap to 10: Use a standard deck of cards (Aces to $10$). Instead of snapping on matching values, players "snap" when the two turned-over cards sum to $10$ (e.g., a $7$ and a $3$).

• Dice Variations: Use "Dot Dice" to encourage subitizing (recognizing quantity without counting) and transition to "Numeral Dice" to reinforce symbol recognition and "Think Big, Count Small."

• Bead Strings: Use strings of $20$ beads, often alternating colors every $5$ beads to help students subitize or "trust the count" of groups of five.