BTT Mathematics / Primary Mathematics Learning Hub / Guide 10
Early algebra begins when a learner treats a relationship as something that can stay true while the numbers or representations change. A pattern suggests a rule. An equation states that two expressions have the same value. An unknown stands for a quantity that must satisfy that relationship.
The equal sign does not mean “the answer comes next.” It means the value on the left is the same as the value on the right. This makes a sentence such as 8 + 4 = 7 + 5 just as valid as 8 + 4 = 12.
This guide develops patterns, function-like rules, missing-number sentences, equality, inverse operations and simple equations. It stays within Primary-style arithmetic reasoning while building a bridge towards Secondary algebra. Use sections according to current school scope in the MOE Primary Mathematics syllabus.
Patterns · Rules and tables · Equality · Unknowns · Simple equations · 24 questions · Worked answers
1. A pattern is evidence of a relationship
Consider 4, 7, 10, 13, 16. The difference between consecutive terms is consistently three. A simple rule is “start at four and add three each time.”
That rule predicts the next term nineteen. More importantly, it explains why nineteen belongs. Merely writing nineteen because it “looks right” is weaker evidence.
Describe what changes and what stays fixed
In 5, 10, 15, 20, the change is plus five each step. The step size stays fixed. In 3, 6, 12, 24, the rule is multiply by two each time. Here the operation is multiplicative rather than additive.
Do not assume every pattern uses addition because the numbers increase. The relationship between consecutive terms must be checked.
Several rules can fit a short beginning
The sequence 2, 4, 6 could continue 8 under the “add two” rule. But a more complicated rule could also produce the same first three terms and then change.
In school questions, the intended rule is usually the simplest regular pattern supported by the examples. State the rule rather than pretending the first few terms logically force one unique continuation in every imaginable sequence.
Visual patterns can be counted structurally
Suppose Figure 1 uses three tiles, Figure 2 uses five and Figure 3 uses seven. The increase is two tiles per new figure. Figure 10 therefore uses 3 + 9×2 = 21 tiles if the same construction rule continues.
Counting every imagined intermediate figure is possible but inefficient. A structural rule connects the figure number to the tile count.
Odd and even patterns
Odd numbers 1,3,5,7 grow by two. Even numbers 2,4,6,8 also grow by two. The two patterns have the same step but different starting values.
This shows why a rule needs both a starting condition and a change rule. “Add two” alone does not tell us whether the sequence begins with one, two or another number.
2. A rule can connect input to output
A table can show an input number and the output produced by a rule. For example:
Input 1 → Output 5
Input 2 → Output 8
Input 3 → Output 11
Input 4 → Output 14
The output increases by three when the input increases by one. A rule that fits every row is output = 3 × input + 2.
Check a proposed rule against every given pair
For input three, the rule gives 3×3+2=11. For input four, it gives fourteen. Testing only one row is not enough; many incorrect rules can match one example accidentally.
Worked example A: Use a rule forward
The rule is “multiply the input by four, then subtract one.” For input six, calculate 6×4−1 = 23.
Keep the order of operations tied to the words. “Subtract one, then multiply by four” would give twenty and describes a different machine.
Worked example B: Reverse a rule
A rule multiplies by five and then adds three. The output is thirty-eight. Work backwards in reverse order: subtract three to get thirty-five, then divide by five to get input seven.
Check forward: 7×5+3=38. Reversing operations in reverse order is the same logic used to undo a sequence of real actions.
Generalise a row pattern
A table gives n=1 → 4, n=2 → 7, n=3 → 10. The rule 3n+1 fits. At Primary level, n can be read as “the input number.” The letter is not mysterious; it is a placeholder for whichever input we are considering.
For n=20, 3×20+1 = 61. A rule allows a distant term to be found without listing nineteen earlier rows.
3. The equal sign states a balance
In 9 + 6 = 15, the left side and right side have equal value. The equal sign does not command the learner to calculate only the left side.
True equations can have operations on both sides
8 + 4 = 7 + 5 is true because both sides equal twelve. 8 + 4 = 8 + 5 is false because the sides are twelve and thirteen.
Ask “Are both sides worth the same amount?” This language makes equality a relationship rather than a signal.
Worked example C: Preserve equality with the same change
If 17 = 17, add five to both sides: 22 = 22. If two equal quantities are changed in the same way, equality is preserved.
This principle later supports equation solving. At Primary level, use balances, number bonds and matched transformations rather than formal symbolic manipulation if that is more appropriate.
Missing-number equality
Find the box in 8 + 7 = □ + 6. The left side is fifteen. Therefore the right side must also be fifteen. The missing number is nine because 9 + 6 = 15.
A learner who writes fifteen in the box may be treating the equal sign as “write the answer next.” The repair is to evaluate both sides as complete quantities.
Compensation preserves a sum
48 + 27 can be rewritten as 50 + 25. Two was transferred from the second addend to the first. The total did not change.
This is an algebraic habit in arithmetic form: change the representation while preserving the invariant quantity.
4. An unknown is a quantity constrained by a relationship
In □ + 18 = 47, the box represents the number which, when increased by eighteen, gives forty-seven. Subtract eighteen from forty-seven to obtain twenty-nine.
The box does not mean “do subtraction” by itself. The relationship determines the inverse operation.
Unknown addend
For 35 + □ = 82, subtract the known part from the whole: 82 − 35 = 47. Check 35 + 47 = 82.
Unknown starting amount
For □ − 16 = 29, the starting amount must contain the remaining twenty-nine and the sixteen removed. Add them: □ = 45.
This differs from 45 − □ = 29, where the unknown removed amount is sixteen. Position and meaning matter.
Unknown factor
For 7 × □ = 56, divide the product by the known factor: 56 ÷ 7 = 8.
Use the equal-group interpretation. Fifty-six contains seven equal groups; one group has eight.
Unknown divisor and quotient
For 72 ÷ □ = 9, ask what number multiplied by nine gives seventy-two. The divisor is eight. For 72 ÷ 8 = □, the quotient is nine.
The same three numbers can occupy different roles. Reading the equation prevents automatic use of the wrong inverse fact.
5. Simple equations can be solved by undoing operations
Worked example D: One-step equation
Let x represent an unknown number. Solve x + 14 = 39. Subtract fourteen from the total: x = 25.
Check by substitution: 25 + 14 = 39. The check confirms that the found value satisfies the original equation.
Worked example E: Multiplicative equation
Solve 6x = 54. Six equal copies of x total fifty-four, so one copy is 54 ÷ 6 = 9.
The notation 6x means six times x. It does not mean sixty-something or the two-digit number formed by writing 6 beside x.
Worked example F: Two-step equation through a story
Three identical boxes and two loose counters contain twenty-three counters altogether. Let x be the number in each box. Then 3x + 2 = 23.
Remove the two loose counters: three equal boxes contain twenty-one. Divide by three: x = 7.
Check: 3×7+2=23. The bar-model route and the equation route express the same structure.
Worked example G: Keep the order of undoing
Solve 4x + 5 = 29. The last operation applied to 4x was adding five, so undo that first: 4x = 24. Then divide by four: x = 6.
Dividing twenty-nine by four first would not undo the structure correctly because the added five was not multiplied by four in the original expression.
Worked example H: A simple inequality idea
If a box can hold at most twelve books, and x books are inside, then x cannot exceed twelve. We can write x ≤ 12.
Primary learners may meet this idea verbally before formal inequality notation. The important reasoning is that a condition can restrict an unknown to a range rather than determine one exact value.
6. Practice: 24 questions
Keep the worked answers covered. State the pattern rule or relationship before calculating where possible.
Questions 1–8: Patterns and rules
1. Continue 4, 7, 10, 13, __, __.
2. Continue 3, 6, 12, 24, __.
3. A pattern starts at 11 and adds 5 each time. Write the first five terms.
4. A tile pattern uses 3, 5, 7, 9 tiles in Figures 1–4. How many tiles are in Figure 10 if the same rule continues?
5. The rule is “multiply by 4, then subtract 1.” Find the output for input 6.
6. A rule multiplies by 5 and adds 3. If the output is 38, find the input.
7. Inputs 1,2,3 give outputs 4,7,10. Give a rule that fits all three.
8. Using the rule from question 7, find the output for input 20.
Questions 9–16: Equality and unknowns
9. Is 8 + 4 = 7 + 5 true or false?
10. Find □: 8 + 7 = □ + 6.
11. Find □: 35 + □ = 82.
12. Find □: □ − 16 = 29.
13. Find □: 45 − □ = 29.
14. Find □: 7 × □ = 56.
15. Find □: 72 ÷ □ = 9.
16. Rewrite 48 + 27 as a sum using 50 while preserving the total.
Questions 17–24: Equations and generalisation
17. Solve x + 14 = 39.
18. Solve 6x = 54.
19. Solve 3x + 2 = 23.
20. Solve 4x + 5 = 29.
21. A number is multiplied by 3 and then increased by 7 to give 40. Find the number.
22. Three identical bags contain the same number of marbles. Two extra marbles make a total of 35. How many marbles are in each bag?
23. The nth term of a pattern is 2n + 5. Find the terms for n=1, 2, 3 and 10.
24. A learner says 9 + 6 = 15 + 4 because “15 is the answer to 9 + 6.” Explain the error and correct the statement by changing only the number 4.
7. Worked answers
Answers 1–8
1. 16, 19. The pattern adds three each time.
2. 48. Each term doubles the previous term.
3. 11, 16, 21, 26, 31. Start at eleven and repeatedly add five.
4. 21 tiles. Figure 1 has three tiles and each new figure adds two. Figure 10 is nine increases beyond Figure 1: 3 + 9×2 = 21.
5. 23. 6×4−1 = 24−1 = 23.
6. 7. Reverse the rule: 38−3=35, then 35÷5=7.
7. Output = 3×input + 1. This gives 4, 7 and 10 for inputs one, two and three.
8. 61. 3×20+1 = 61.
Answers 9–16
9. True. Both sides equal twelve.
10. 9. The left side equals fifteen, so the right side must also equal fifteen. Nine plus six gives fifteen.
11. 47. Subtract thirty-five from eighty-two.
12. 45. Add sixteen back to the remaining twenty-nine.
13. 16. Forty-five minus sixteen leaves twenty-nine.
14. 8. Fifty-six divided into seven equal groups gives eight in each.
15. 8. Since 72÷8=9, the missing divisor is eight.
16. 50 + 25. Transfer two from twenty-seven to forty-eight, preserving the sum seventy-five.
Answers 17–24
17. x=25. Subtract fourteen from thirty-nine. Check 25+14=39.
18. x=9. Divide fifty-four by six. Check 6×9=54.
19. x=7. Remove two to get three equal groups totaling twenty-one, then divide by three.
20. x=6. Subtract five to obtain 4x=24, then divide by four.
21. 11. Reverse the operations: 40−7=33, then 33÷3=11.
22. 11 marbles per bag. Remove the two extras: 35−2=33. Divide by three equal bags.
23. 7, 9, 11 and 25. Substitute n=1,2,3,10 into 2n+5.
24. The equal sign requires both sides to have the same value. The left side is fifteen. To keep the form 15+□, the missing number must be zero. Correct statement: 9+6=15+0.
8. Diagnose the relationship before introducing more symbols
If a learner can continue a pattern but cannot state the rule, ask what changes from one term to the next. If a distant term is needed, ask whether that repeated change can be connected directly to the term number.
If equality errors appear, temporarily remove letters. Work with number sentences such as 12=7+5 and 8+4=7+5 until the equal sign is understood as balance.
Use boxes before letters when helpful
The box in □+18=47 and the x in x+18=47 have the same mathematical job. A letter adds compact notation, not a new kind of unknown.
Move to letters when the learner is comfortable with the relationship. Do not make unfamiliar notation the obstacle when the real goal is inverse reasoning.
Check equations by substitution
After finding x, put the value back into the original statement. If both sides match, the value satisfies the equation. This creates an independent return path.
Connect arithmetic transformations to algebraic thinking
Compensation, regrouping and equivalent fractions all change a representation while preserving value. Equation solving extends the same discipline: transform without breaking equality.
Continue through the Primary Mathematics series
For number structure, use Factors, Multiples and Divisibility. For equality inside word problems, use Word Problems, Bar Models and Checking. For broader reasoning habits, continue to Estimation, Reasoning and Checking.
Return to the BTT Primary Mathematics Learning Hub. The later conceptual owner is Algebra.
Original learning guide. Curriculum reference checked 6 September 2026. Early algebra examples are educational bridges and not a claim that every formal algebraic notation shown is required at every Primary level.