Chapter 10: Basic Facts, Mental Computation, and Algorithms

Basic Facts, Mental Computation, and Algorithms

Introduction

Historically, math education focused on pen-and-paper practice of addition, subtraction, multiplication, and division.
Mental math was also emphasized, including memorization of times tables.
In the 1980s, the understanding of written calculation methods was questioned, as research indicated that proficiency in written computation didn't always mean understanding the procedures.
There were calls for math instruction that promoted both computational skill and conceptual knowledge.
In the technological age, the necessity of teaching standard written procedures is questioned due to the prevalence of calculators.
Some parents still value written computation skills and question math programs if their children can't perform them flawlessly.
Research emphasizes the importance of building number sense and understanding the operations being performed.

Working Flexibly with Number

An example of working flexibly with numbers is New Zealand's Number Framework.
This framework outlines nine stages for learning number, describing flexible thinking strategies at each stage.
The Number Framework suggests delaying standard written algorithms until children can use part-whole mental strategies.
Working flexibly with numbers in mental computation is possible with good number sense.
Students with good number sense can see numbers in various combinations and groupings.
For example, 6 + 7 can be seen as 6 + 4 + 3, enabling the student to add 4 to 6 to make 10 and then add the remaining 3.
In another context, 7 + 8 can be seen as 2 + 5 + 8.
Students need strong number sense and confidence to manipulate numbers in this way.
In Dutch research and curriculum development, the focus is on developing number sense and promoting intuitive strategies for mental computation in the first five years of schooling.
In Western classrooms, operations like 65 + 28 have been taught through a place value model.
The Dutch model encourages more flexible manipulation of numbers based on context.
For example, 28 can be broken into 20 + 5 + 3 to add to 65.
Students are encouraged to think flexibly with numbers, using strategies like jump and split.
An empty number line is a strategy to support this flexibility.
The number line can assist in tracking and thinking about numbers and operations.
Students will use the number line differently depending on their number sense.
The Number Line App is a useful tool for number line work with whole numbers, fractions, decimals, and negative numbers.

Fluency

In the Australian Curriculum: Mathematics, fluency is a key proficiency.
Other proficiencies are understanding, problem-solving, and reasoning.
The inclusion of proficiencies aims to shift math instruction away from algorithms, rules, and procedures.
Fluency is not about automatic recall of number facts or speed and accuracy in written computation.
Fluency involves choosing appropriate procedures and regrouping procedures flexibly, accurately, efficiently, and appropriately.
Students demonstrate fluency when they calculate efficiently, recognize robust ways of answering questions, and choose appropriate methods and approximations.
Fluency in computation is about using a range of strategies and being confident in doing so.
Fluency is rooted in a strong sense of number.

Basic Facts as the Foundation for Computation

'Basic facts' refers to simple addition, subtraction, multiplication, and division calculations expected to be automatic.
For example, 6 + 8 is a basic addition fact, and 4 \times 5 is a basic multiplication fact.
Multiplication facts are often referred to as 'the tables'.
Automatic recall of basic facts assists computation of larger numbers.
Fluent number fact knowledge goes beyond rote learning.
The approach suggested is to provide students with thinking strategies for basic fact calculation to assist in mental computation of larger numbers.
It's about building fluency through the development of number sense.
A strong sense of number provides students with a 'license to think' about numbers in different ways.
Fluency in number facts through a foundation of number sense grows out of discovering patterns and relationships.
Activities that focus on additive and multiplicative thinking, encourage number discussions, and engage with number sense have a profound effect on students' automaticity.
Practices that promote discussion of patterns, relationships, and connections outweigh blind memorization.
A strategies approach to teaching basic facts of addition and multiplication is elaborated, aiming to promote strong visual images through concrete materials.
The commutative principle (e.g., 3 + 5 = 5 + 3) is emphasized to reduce the number of basic facts to be learned.
Subtraction and division are addressed using the inverse principle (think addition and think multiplication, respectively).
Subtraction is the inverse of addition, and division is the inverse of multiplication.
This capitalizes on the part-part-whole composition of addition and the factor-factor-product of multiplication.
In addition, two parts are given, and the whole is determined (e.g., 3 + 4 = ?).
In subtraction, the whole and one part are given, and the other part is to be determined (e.g., 7 - 3 = ?).
If 3 + 4 is automatically recalled as 7, then to determine 7 - 3, thinking addition of '3 plus what makes 7' makes learning subtraction facts simple and draws on number sense.

Basic Facts of Addition

Basic facts of addition involve adding two addends up to 10.
Basic facts of subtraction are the addition facts in reverse.
There are 100 basic facts of addition—or 121 if facts with zero are included.
The facts are categorized according to thinking strategies to assist in understanding and recall.
The thinking strategies for seven categories of facts are:
- Count-on facts
- Doubles
- Tens facts
- Adding 10 facts
- Bridging 10 (9, 8) facts
- Doubles + 1, + 2 facts
- Last facts
The table highlights overlaps of facts between categories (e.g., 6 + 4 is in both tens facts and doubles + 2).
The teacher's role is to encourage discussion of new fact categories and the different strategies that can be used to recall the same fact.
Students should express their opinions on particular strategies and why they might employ them.
The table also lists the spin-around for each fact to reflect the efficiency of learning facts by strategy and visualization.
Facts are not seen as isolated but as flexible, considering related spin-arounds at the time of learning.
This promotes understanding of the commutative principle.
The fact categories are not necessarily hierarchical, but some groups of facts must be in place before others can be learned.
The sequence presented is a suggestion, and the introduction of groups of facts according to strategy will depend on the needs of the students and the teacher's judgment.

Basic Addition Facts Sorted by Strategy

Count-on facts:
- Count-on 1: 2 + 1, 3 + 1, 4 + 1, 5 + 1, 6 + 1, 7 + 1, 8 + 1, 9 + 1
  - Spin-around: 1 + 2, 1 + 3, 1 + 4, 1 + 5, 1 + 6, 1 + 7, 1 + 8, 1 + 9
- Count-on 2: 3 + 2, 4 + 2, 5 + 2, 6 + 2, 7 + 2, 8 + 2, 9 + 2
  - Spin-around: 2 + 3, 2 + 4, 2 + 5, 2 + 6, 2 + 7, 2 + 8, 2 + 9
- Count-on 3: 4 + 3, 5 + 3, 6 + 3, 7 + 3, 8 + 3, 9 + 3
  - Spin-around: 3 + 4, 3 + 5, 3 + 6, 3 + 7, 3 + 8, 3 + 9
- Count-on 0: 0 + 1, 0 + 2, 0 + 3, 0 + 4, 0 + 5, 0 + 6, 0 + 7, 0 + 8, 0 + 9, 0 + 10
  - Spin-around: 7 + 0, 2 + 0, 3 + 0, 4 + 0, 5 + 0, 6 + 0, 7 + 0, 8 + 0, 9 + 0, 10 + 0
Doubles: 1 + 1, 2 + 2, 3 + 3, 4 + 4, 5 + 5, 6 + 6, 7 + 7, 8 + 8, 9 + 9, 10 + 10
Tens facts: 10 + 0, 9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5, 4 + 6, 3 + 7, 2 + 8, 1 + 9, 0 + 10
Adding 10: 10 + 0, 10 + 1, 10 + 2, 10 + 3, 10 + 4, 10 + 5, 10 + 6, 10 + 7, 10 + 8, 10 + 9, 10 + 10
- Spin-around: 0 + 10, 1 + 10, 2 + 10, 3 + 10, 4 + 10, 5 + 70, 6 + 70, 7 + 70, 8 + 70, 9 + 70
Bridging 10 (9): 9 + 2, 9 + 3, 9 + 4, 9 + 5, 9 + 6, 9 + 7, 9 + 8
- Spin-around: 2 + 9, 3 + 9, 4 + 9, 5 + 9, 6 + 9, 7 + 9, 8 + 9
Bridging 10 (8): 8 + 3, 8 + 4, 8 + 5, 8 + 6, 8 + 7
- Spin-around: 3 + 8, 4 + 8, 5 + 8, 6 + 8, 7 + 8
Doubles + 1: 2 + 3, 3 + 4, 4 + 5, 5 + 6, 6 + 7, 7 + 8, 8 + 9
- Spin-around: 3 + 2, 4 + 3, 5 + 4, 6 + 5, 7 + 6, 8 + 7, 9 + 8
Doubles + 2: 2 + 4, 3 + 5, 4 + 6, 5 + 7, 6 + 8, 7 + 9
- Spin-around: 4 + 2, 5 + 3, 6 + 4, 7 + 5, 8 + 6, 9 + 7
Last facts: 7 + 4
- Spin-around: 4 + 7

Count-On Facts

The count-on strategy is where, when given two numbers, one of the numbers is the starting point for counting on the amount of the second number.
The count-on strategy is used frequently, but is often the most difficult strategy to overcome once learned.
Many students use the count-on strategy to mentally add two numbers that are far too large to ensure accuracy or efficiency.
When used appropriately, the count-on strategy is very useful for mental computation, but students need to know when and how to use this strategy, and when it is more efficient to use other strategies.
The count-on strategy is only suitable for counting on small numbers such as 1, 2, 3 and zero.
In the early years, special instruction needs to be given for counting on zero, and this should be taught after the count-on strategy for numbers 1, 2 and 3 has been consolidated.
When teaching basic fact groups, selection of the examples used needs to be considered carefully so that the usefulness of the strategy will be demonstrated clearly.
To teach the count-on facts, select examples in which one of the two numbers in the pair is considerably greater than 1, 2 or 3.
The following facts match this description: 4 + 1, 5 + 1, 6 + 1, 7 + 1, 8 + 1, 5 + 2, 6 + 2, 7 + 2, 5 + 3 and 6 + 3.
These are the best facts for initial experiences with the strategy.
Visual, auditory and tactile experiences can also be used to introduce students to the count-on strategy:
- Using marbles in a tin is a useful teaching approach for developing the count-on strategy.
- Show students a group of five marbles, then place them in an empty tin.
- Let the students see and hear the group of five marbles rolling around in the bottom of the tin.
- Add one more marble and have the students count on from five to six as they hear the marble drop into the tin.
- Practise with other examples.
- Having the marbles in the tin discourages students from counting the first collection of objects, as they cannot see them.
- The noise created when a marble is added assists in exemplifying the counting-on action.
- Generate a group of counters on an Interactive White Board (IWB).
- Flash the screen to students and ask them to state how many counters are on the screen.
- This action capitalises on students' natural ability to subitize—that is, to recognize the size of a collection of objects without counting.
- Check by counting this collection if necessary.
- Cover the initial collection on the IWB, place two further counters on the screen and flash the screen again so that only the two additional counters are visible.
- Ask students to state how many counters there are altogether.
- This action discourages the urge to recount the first collection and to count on from this number.
- Flash sets of counters in various count-on fact representations on the IWB and ask students to state the solution.
- Ask the students to explain their strategy.
- Encourage students to use their subitizing skills to determine the amount of counters in each group, and then to select the largest group and count on from there.
- Provide students with the fact in symbolic form and ask them to show that fact with their counters.
- Ask students to cover the biggest amount with their hand and to count on the remaining amount.
- As students are presented with particular facts, encourage them to press their fingers on their skin, or to slap their thigh, as they count on each number to help keep track of the amount they are counting on.
- What can you see?

The Commutative Law (Spin-Arounds or Turn-arounds)

The count-on strategy is where the larger of the two numbers is the starting point for counting on the other number.
As children explore this strategy through use of concrete materials, they will naturally engage with one of the laws of arithmetic: commutativity—the concept that order does not matter (e.g. 7 + 2 results in the same total as 2 + 7).
The commutative law must be made explicit to students so they can see that, as they learn each basic fact, they are actually learning two facts: each fact and its spin-around.
To explicitly model commutativity, and therefore the spin-around or turn-around notion for the count-on facts, use a piece of paper or a card that is divided into two parts.
Place counters on each half of the card to model a count-on fact.
Ask students to state the fact being modelled.
Then physically turn the card around and ask the students to state the fact being modelled.
Similarly, students can use Unifix to represent the task and twist this in either direction to demonstrate the commutativity principle.
Reinforce the count-on strategy: select the larger of the two numbers and count on the other amount.
This strategy can be further reinforced by providing students with practice sheets of count-on facts.
Encourage students to circle the larger of the two numbers as the first step, before they count on, and then record the answer.

Counting on Zero

Special attention needs to be given to counting on with zero.
The count-on strategy is just that—counting on; that is, an action is expected and required.
However, when counting on zero, no action is immediately evident.
This may be a potential source of confusion to some students.
Through use of concrete materials and careful use of language, students will come to see that counting on with zero results in the original amount that they started with.

Doubles Facts

The doubles are usually the most easily learned facts for students.
Often, students know their doubles facts up to 5 + 5 before they commence formal schooling.
When introducing the doubles thinking strategy, encourage students to brainstorm things that come in a particular amount that can be doubled.
For example, the number of wheels on a bicycle is two—bicycle wheels come in twos.
So two bicycles is four wheels (double two is four).
The following is a list of suggestions that can be used to help children visualize a particular number and then double it:
- 1—person, stop sign, sun
- 2—drumsticks, eyes, ears, bicycle wheels
- 3—clover leaf, cricket stumps, tricycle wheels
- 4—car tires, legs on a dog, cat, horse, etc.
- 5—fingers
- 6—legs on an insect
- 7—calendar (2 weeks)
- 8—legs on an octopus/spider
- 9—Channel 9 TV symbol.
When teaching doubles, focus on the language and a visual image—'5 fingers on one hand; 10 fingers on two hands; 5 and 5 is 10; double 5 is 10'.
When students falter in thinking of the solution to 8 + 8, have them think of a spider: 'How many legs does a spider have? Now double it.'
For 7 + 7, think of a calendar: 'How many days is 2 weeks?'
For 6 + 6: 'How many legs do two insects have?'
The images students use to assist in learning the doubles should come from them.
They need to select the item that most appeals to them and then double it; however, too many images may be a source of confusion.

Doubles Plus One and Doubles Plus Two

Once the doubles are mastered, they can be used as a platform for facts that are almost doubles.
Facts that fall into this category are the doubles plus one facts (2 + 3, 3 + 4, 4 + 5, 5 + 6, 6 + 7, 7 + 8, 8 + 9) and the doubles plus two facts (2 + 4, 3 + 5, 4 + 6, 5 + 7, 6 + 8, 7 + 9).
As a category, these facts should be introduced once the doubles facts have been consolidated.
Recognizing doubles plus one and doubles plus two facts requires considerable cognitive effort, so this category should be delayed until students are familiar with easier categories of facts.
As students learn more and more facts, it may become apparent that particular facts have previously been learned through another strategy
This negates the need to relearn the same fact with a different and possibly more difficult strategy.
Through examination of the list of doubles plus one and doubles plus two facts, it can be seen that some facts fit within other fact categories.
The following facts have been covered through the count-on strategy: 2 + 3, 2 + 4, 3 + 4, 3 + 5.
The 4 + 6 fact is a simple tens fact.
The facts 7 + 8, 8 + 9 and 7 + 9 belong to the bridging 10 facts category, and may be simpler for some students to learn through that strategy than as an almost double fact.
The almost double facts have been presented here because of their link to the doubles facts.
To assist students to visualize facts that are almost doubles, use counters on 2 cm grids.
Use different colored counters to show the two numbers in the fact.
Display counters as a mirror image, with an extra counter clearly visible for doubles plus one facts and two extra counters for doubles plus two facts.
When the counters are displayed, describe the fact as being a particular double plus one or plus two.
From the examples given below, it can be seen that 5 + 6 gives the same total as double 5 + 1 more.
Similarly, 6 + 8 gives the same total as double 6 + 2 more.
- For some students, these facts may be more appropriately recognized as doubles less 1 or 2.
- That is, 5 + 6 may be seen more easily as double 6 less 1, 6 + 8 may be double 8 less 2, and so on.
- Promote this type of discussion around the visual images provoked by the concrete representations, and encourage students to use the strategy that most appeals to them.
- It is often better if students select and use one strategy until particular facts become consolidated; otherwise, confusion may occur.

Tens Facts

The tens facts are those facts where the number pair combines to give a total of 10.
Recognizing combinations to 10 is, a frequently used strategy in mental computation.
Tens facts are often readily acquired, but it is important to ensure that all tens facts are equally familiar to students so that they have access to a valuable strategy for mental computation.
Orientation to the tens facts is best achieved through use of a tens frame.
A tens frame is a 2 \times 5 grid with grid cells large enough to accommodate a colored counter.
Before using the tens frame to represent tens fact combinations, students need to be able to instantly recognize numbers from 0 to 10, as represented with counters on the grid.
This is promoted by positioning counters in a particular, consistent arrangement and by drawing attention to the number of counters, and the number of counters missing, as follows (for the numbers 10, 9, 8, 5, 6 and 7):
- 10—no counters missing
- 9—1 missing
- 8—two missing
- 5—top line filled
- 6—five and one more
- 7—five and two more.
The numbers 0 to 4 are readily recognized by students.
A digital version, Number Frames, is available here: https://apps.apple.com/au/app/number- frames-by-the-math-learning-center/id873198123.
In addition to the tens frame, this resource includes options for a five frame, a 20 frame and a range of different numbered frames for later fraction work.
Once students can readily recognize numbers 0 to 10 on the tens frame without counting, link tens fact pairs by emphasizing the amount needed to fill the tens frame for the following five facts: 10 and 0, 9 and 1, 8 and 2, 5 and 5, 7 and 3, 6 and 4.
Once these five number combinations have been consolidated, the rest of the tens facts are the spin-arounds.
The sequence for presentation of the tens facts and accompanying language is as follows:
- 10 + # = 10
  - What goes with 10 to give 10?
- 9 + # =
  - What goes with 9 to give 10?
- 8 + # =
  - What goes with 8 to give 10?
- 5 + # =
  - What goes with 5 to give 10?
- 6 + # =
  - What goes with 6 to give 10?
- 7 + # =
  - What goes with 7 to give 10?
- 0 + # =
  - Zero and what gives 10?
- 1 + # =
  - One and what gives 10?
- 2 + # =
  - Two and what gives 10?
- 3 + # =
  - Three and what gives 10?
- 4 + # =
  - Four and what gives 10?

Adding Tens Facts

Understanding what happens, when a ten is added to a number relates to place value and numeration knowledge.
To explore this notion with students, counting activities with a calculator or spreadsheet are useful.
If a calculator or a spreadsheet has an inbuilt constant function, it can be programmed to count by entering a starting number, pressing the addition key and then entering the number by which you wish to count.

Teaching Idea Adding Tens Facts

To make your calculator count in tens:
- Enter any number (e.g. 6).
- Press the add key.
- Enter 10.
- Press the equals button repeatedly.
- Observe the changes occurring.
Alternatively:
- If that doesn't work, try entering the starting number, then press the add key twice before entering the counting number.
- Then press equals repeatedly.
In the example of counting in tens from 6, the idea is to make students focus on what is happening every time the equals button is pressed.
Students will see that when adding ten, the number in the ones place does not change, but the digits to the left of the entered number increase by one every time.
The pattern for 6 + 10 repeatedly is 16, 26, 36, 46, 56, 66, and so on.
Try other starting numbers and see which digits change and which stay the same.
Link this to a general principle of adding ten to a number—the digit in the ones place does not change, but a ten is added to the digit in the tens place.
From this generalization, students can quickly determine the solution to facts in which ten has been added.
This is an extremely useful strategy for mental computation of numbers with more than two digits.

Bridging Ten Facts

The bridging ten facts are when one number of the fact pair is close to 10.
Bridging ten is very useful when one digit is 9 (9 + 4, 9 + 5, 9 + 6, 9 + 7, 9 + 8) or 8 (8 + 4, 8 + 5, 8 + 6, 8 + 7), but for numbers smaller than 9 or 8 the strategy loses its usefulness and efficiency.
To show students how to use the bridging ten strategy and to provide them with a strong visual image of this strategy in action, use a double tens frame.
A double tens frame consists of two tens frames displayed one above the other.
Place in the top frame nine (or eight) counters in one color.
Note that 'filling' of ten frames is consistent with that suggested for developing tens facts.
On the bottom frame, display the second number in the fact pair with counters in a different color.
The top frame represents the number in relation to 10.
Clearly, when the number is 9, one more counter is required to fill the ten frame.
For 8, two more counters are required.
Take the required number of counters from the bottom frame to fill the top frame.
The thinking behind this strategy is to think of the first number as 10 and alter the second number accordingly.
For example, 9 + 4 is 10 + 3; 8 + 5 is 10 + 3.

Last Facts

By highlighting facts on an addition grid as each fact category is introduced, all of the 121 facts are covered by the preceding strategies—except one.
That fact is 4 + 7, or 7 + 4, the only one that does not fit simply into any category.
In this light, this fact is special, and it should be indicated to students as, such.
With this fact, strategies for solution are a discussion point.
Some students may see it as an almost ten fact, some as a bridging ten fact, and some as double four add three; other students may simply see it as a special fact that gives 11.

Basic Facts of Subtraction

Once all addition facts have been consolidated, subtraction facts should follow easily.
Each fact should be considered as a set of four facts—two addition and two subtraction facts—generated through rearrangement of the three numbers in the fact.
For each fact encountered, students learn two facts using the commutative law—for example, 4 + 6 + 10; therefore, 6 + 4 + 10.
Two subtraction facts are generated from the addition facts: 10 - 6 + 4, 10 - 4 + 6.
Within each fact, the three numbers share a special relationship.
These sets of four facts are sometimes referred to as fact families.
The most efficient strategy for subtraction facts is to think addition.
The think addition strategy is best applied once the addition facts have been fully consolidated—that is, when students can rapidly determine the solution to the addition fact.
When they are presented with a subtraction fact, attention is drawn to the related, and previously learned, addition fact and the number combinations within that fact.
For example, when presented with 10 - 6, to think addition is to think of the number that, when combined with 6, makes 10.

Addition and Subtraction: Mental Computation

Having ready recall of basic facts assists in mental computation.
The strategies outlined in the previous section for promoting recall of basic addition facts were presented as ways in which learning basic facts can be facilitated through provision of visual images and thinking strategies.
The thinking strategies are also useful for mental computation.
Common strategies for mentally computing solutions to these examples will include the strategies outlined for learning the basic facts.
In the first example, it is possible to see that the bridging ten strategy can be used when adding 90 and 50; in the second example, the adding ten strategy can be used; for the third example, knowledge of tens facts is useful; and in the fourth example, doubles are presented.
For the last example, another strategy may be useful: the count-back strategy for subtraction.
Like the count-on strategy, this strategy is useful for counting back 1, 2 or 3, but becomes inefficient and prone to error when used to count back larger amounts.
The suggested methods for mental computation are not the only ways in which solutions may be determined, and mental methods are diverse and often quite idiosyncratic.
The important thing is that learners are given support and encouragement to determine their own solution methods, and that they have the confidence to do so.
Armed with automatic recall of basic addition and subtraction facts, as well as thinking strategies for computation, learners are in a strong position to invent their own procedures for computation that will be inherently meaningful to them.

Basic Facts of Multiplication

Automatic recall of basic multiplication facts is an important objective of primary-school mathematics, and is vital for mental computation and estimation.
There is no question about the value of students learning these basic facts, but the approach taken must be considered.
Why students need to learn their tables beyond the ten times must also be questioned.
For efficient mental computation, students need automatic recall of all multiplication facts up to ten times.
From this, they can derive solutions to eleven and twelve times tables and beyond.

Foundations for Basic Multiplication Facts

Counting patterns lay the foundation for multiplication facts.
Skip counting in various amounts encourages students to look for patterns in counting, and this builds number sense.
Early skip counting activities should provide students with some form of concrete reference, so that the size of numbers being counted can be conceptualized.
The following are some ideas for early counting in tens, fives, and twos where the students are standing in a line and count progressively in a specified amount:
- For counting in tens, the first student in line brings her/his hands forward, stretching out the fingers while shouting 'ten'.
- The second child in line does the same, but shouts 'twenty'.
- This continues down the line until all students have raised their hands.
- For counting in fives, the first student brings one hand forward and shouts 'five', then the other hand forward and shouts 'ten'; the second student brings one hand forward and shouts 'fifteen', then 'twenty', and so on down the line.
- For counting in twos, the students stand in line and, at the count of 'two', 'four', 'six', etc., each student blinks in turn (i.e. using two eyes), with a nod of the head.