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Choose a letter for the unknown number
Task reminder
A letter stands for a number you do not know yet.
A notebook’s price is unknown. Let n mean the cost of one notebook in dollars. Every n card below names that same cost. The letter stands for a number, not for the object “notebook”.
The same cost repeats
The three notebook costs are equal. The delivery charge is a separate amount paid once.
Scroll across to read every label.
Figure data
This table is included in print while it is open.
| Set | Values |
|---|---|
| Three notebooks | n dollars; n dollars; n dollars |
| One delivery | 2 dollars |
Write an expression for the three notebooks alone, before delivery.
Connect the cards to the notationHide answer
The 3 is a coefficient: it counts the three equal lots of n. Including the one delivery charge gives 3n + 2 dollars. The notation 3n does not mean a two-digit number beginning with 3.
Check your understanding 1
Choose and use a letter
Check your understanding 1
Choose and use a letter
A ribbon has an unknown length in centimetres. A second ribbon is 6 cm longer. Choose a letter for the first length, say what it represents, and write an expression for the second length.
Keep working notes
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One choice is: let r be the first ribbon’s length in centimetres. The second length is r + 6 cm. A different letter works if its meaning and use agree. The expression 6r would describe six times the first length.
Read the operation and its order
A letter can take the place of a number in each operation:
- a + 3: add 3 to a.
- a-3: subtract 3 from a.
- 3a: multiply a by 3.
- a/3: divide a by 3.
Addition and multiplication let the two quantities swap: a + 3 = 3 + a and 3a = a × 3. For subtraction and division, which quantity comes first matters.
“Subtract a from 3” starts with 3, so it is 3-a. “Divide 3 by a” is 3/a, for a ≠ 0. If a = 6, then 3/a = 1/2 but a/3 = 2.
Check your understanding 2
Move between words and symbols
Check your understanding 2
Move between words and symbols
Write an expression for subtract b from 9.
Now write an expression for share b litres equally among 4 containers.
Finally, describe 7b by naming the two quantities and the operation.
Keep working notes
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The expressions are 9-b and b/4 litres per container. The expression 7b means seven times b, not 7 + b. For b = 8, sharing b litres among 4 containers gives 2 litres each. The reversed expression 4/b gives 1/2, so it describes a different division.
Read both quantities in a division
A fraction bar also means division. Read the numerator as the amount being divided and the denominator as the divisor. The expression 5/c requires c ≠ 0.
Check with a valueHide answer
If c = 10, then 5/c = 5/10 = 1/2. Reversing it gives 10/5 = 2. The same two quantities do not guarantee the same result.
Task reminder
Which terms count the same kind of amount?
Join terms of the same kind
Four lots of a and three lots of a make seven lots of a. The constant 5 counts ones, not more lots of a.
Group equal unknown values
Join the two groups of a cards. Keep the five ones as a different kind of term.
Scroll across to read every label.
Figure data
This table is included in print while it is open.
| Set | Values |
|---|---|
| First group | a; a; a; a |
| Second group | a; a; a |
| Constant | 5 ones |
Different letters need separate groups. For example,
An a alone means 1a, and a -b means subtract one lot of b. Keep every addition or subtraction attached to its term. The two letters may have different values, so the expression cannot become 8a or 8b.
Check your understanding 3
Repair a simplification
Check your understanding 3
Repair a simplification
A learner writes 6p-2p + 3q + 5 = 12p. Simplify the original expression and explain why the grouping does not work.
Keep working notes
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Only the two p terms combine. With p = 2 and q = 1, both the original and the collected expression give 16. The proposed 12p gives 24. Checking a value can expose a mistake; the like-term grouping explains why the correct expression works for every allowed value.
Fractions can count parts of the same unknown
Let a be a positive length. Each full strip below has length a. One third of a and two thirds of that same a make one whole a.
Keep the same unknown whole
Each full strip is one equal-sized whole.
Scroll across to read every label.
Figure data
This table is included in print while it is open.
| Whole | Parts in order |
|---|---|
| One third of a | marked; empty; empty |
| Two thirds of a | marked; marked; empty |
| Together: one whole a | marked; marked; marked |
Show workingHide answer
The picture uses a positive length to show the grouping. The same rule works when a is zero too.
Substitute a value for each letter
To evaluate an expression, replace each letter by its stated number, then calculate. For the notebook rule 3n + 2, let n = 4:
Multiply before adding. The delivery charge is added once; 3 × (4 + 2) would charge it three times.
Check your understanding 4
Use both given values
Check your understanding 4
Use both given values
Find 5a-2b + 1 when a = 3 and b = 4. Write the substitution before calculating.
Keep working notes
Show answerHide answer
The b term uses 4, not the value 3 shown for a. The subtraction remains subtraction.
Substitution need not give a whole number. Find a/2 when a = 3.
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Task reminder
What must you do to both sides to keep them equal?
Keep both sides equal
An expression, such as 3n + 2, gives a rule for a value. An equation, such as 3n + 2 = 17, says that two values are equal. Solving finds the number that makes that equality true.
Change both sides together
The two cards in each row name equal values. Subtracting two from both sides, then dividing both sides by three, preserves equality.
Scroll across to read every label.
Figure data
This table is included in print while it is open.
| Set | Values |
|---|---|
| Start: equal values | 3n + 2; 17 |
| Subtract 2 from both | 3n; 15 |
| Divide both by 3 | n; 5 |
Removing the same amount from both sides keeps equality, like removing equal loads from a balance. Dividing both sides into three equal groups also keeps equality. Check: 3 × 5 + 2 = 17.
Check your understanding 5
Explain the inverse steps
Check your understanding 5
Explain the inverse steps
A number is multiplied by 5 and then 7 is subtracted. The result is 28. Write an equation and solve it. Show how your steps keep equality, then check in the original equation.
Keep working notes
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Add 7 to both sides, then divide both sides by 5. Check 5 × 7-7 = 28. Subtracting another 7 would not undo the original subtraction.
Gather the unknown when it appears on both sides
For 5n + 2 = 2n + 23, remove the same 2n from each side:
Then remove 2 from both sides and divide both by 3. Substitution gives 37 on each side. You could subtract 2 before subtracting 2n; the steps need to preserve equality, not follow one fixed order.
Try a new equation: 6p + 1 = 2p + 17.
Show workingHide answer
Subtract 2p, subtract 1, then divide by 4 on both sides. Check: 6 × 4 + 1 = 25 and 2 × 4 + 17 = 25.
Find an answer that is a fraction
For 6x = 9, divide both sides by 6:
The coefficient 6 is a whole number, but the solution need not be whole. Check: 6 × 1 1/2 = 9.
Now solve 8u = 10. Give the exact value of u.
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Change the unknown, keep the story
A taxi-fare model is 4 + 2d dollars for a journey of d km. Using this stated model, find the fare for 6 km, then the distance for a 20-dollar fare. This is a model for this task, not an actual company’s pricing.
Keep working notes
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For the known distance, evaluate:
For the known fare, solve:
Check 4 + 2 × 8 = 20. The relationship stays the same; the question changes which quantity is unknown.
For a grown-up
Pip’s synthetic voice uses the University of Edinburgh’s Alba speech dataset, via Piper’s Alba voice, under CC BY 4.0. Audio is optional.
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