Expressions using Letter-Numbers · Lesson 1 of 11
The Notion of Letter-Numbers
“Use letters to describe number relationships that remain useful when the numbers change.”
• Explain what a letter-number represents in a stated situation. • Translate an age relationship into an algebraic expression. • Replace a letter by a given number and evaluate the expression. • Describe repeated equal contributions using a formula. • Distinguish the changing quantities from the relationship connecting them.
One Relationship, Many Possible Numbers
Shabnam is three years older than Aftab. If Aftab is 10 years old, Shabnam is 13; if Aftab is 18, Shabnam is 21. The actual ages change, but the instruction for finding Shabnam’s age stays the same: add three to Aftab’s age. Mathematics needs a way to record this relationship without choosing just one pair of ages.
Let a represent Aftab’s age in years and let s represent Shabnam’s age in years. The expression a + 3 gives Shabnam’s age. Writing s = a + 3 says that the number represented by s equals the number obtained by adding 3 to a. Read the equal sign as “has the same value as”. It links two descriptions of the same age.
A letter used to represent a number. Its meaning must be stated: for example, a may represent an age in years or n may represent a number of objects.
A mathematical expression containing letters that represent numbers, together with numbers and operations. For example, a + 3 and 2 × n are algebraic expressions.
A letter is useful even when its value is not yet known. It also allows us to describe many allowed values at once. In this story, a and s are connected; we cannot select any two numbers independently and still keep the age relationship. The letter 3 is not needed because the gap is already known and fixed.
Problem
Aftab is 23 years old. Find Shabnam’s age using s = a + 3.
- 1.The letter a represents Aftab’s age, so replace a by 23.
- 2.The expression becomes 23 + 3.
- 3.Calculate 26. Shabnam is 26 years old, and 26 − 23 = 3 checks the given gap.
Reading the Relationship in Reverse
The same story can answer a different question. If we know Shabnam’s age, we find Aftab’s age by taking away the three extra years. The operation changes because we are now starting from the older person’s age.
Problem
Shabnam is 20 years old. How old is Aftab?
- 1.Aftab is younger by 3 years, so use a = s − 3.
- 2.Replace s by 20: a = 20 − 3.
- 3.Aftab is 17. Check in the original direction: 17 + 3 = 20.
We could use different letters, provided we define them clearly. For example, x for Aftab’s age and y for Shabnam’s age would give y = x + 3. Changing the letter names does not change the relationship. Within one problem, however, keep each letter’s meaning consistent so that the reader knows which quantity is being used.
Equal Contributions Give Multiplication
Letters can also describe repeated quantities. Imagine making several separate L shapes, each using one vertical stick and one horizontal stick. Before writing an expression, explain the counting in words: each L needs two sticks, and no stick belongs to two Ls.
If n is the number of Ls, the total number of sticks is 2 × n. For five Ls, there are five groups of two; for forty-five Ls, there are forty-five groups of two. The single expression records the same counting method for every allowed number of Ls. Here n is a whole-number count, rather than a length or an age.
Problem
How many sticks are needed for 7 separate Ls? How many for 45?
- 1.Each L contributes 2 sticks. No sticks are shared.
- 2.For 7 Ls, replace n by 7: 2 × 7 = 14.
- 3.For 45 Ls, replace n by 45: 2 × 45 = 90. The rule is unchanged even though the drawings would be very different in size.
A Formula Records a General Relationship
A square has four equal sides. To walk once around it, add its side length four times. If q represents the side length, the perimeter is 4 × q. This relationship is useful for small and large squares alike.
A mathematical statement that records a general relationship between quantities. A formula often gives one quantity in terms of others.
Problem
A square has side length 7 cm. Find its perimeter.
- 1.All four sides measure 7 cm.
- 2.Use P = 4 × q and replace q by 7: P = 4 × 7.
- 3.The perimeter is 28 cm. Adding 7 + 7 + 7 + 7 gives the same result.
Notice the common method in the age, L-shape, and square problems. First identify what is known and what can change. Next describe the relationship in ordinary words. Finally choose letters and write the operations that match those words. The letters make the statement shorter, while the reasoning gives it meaning.
A letter-number represents a number, not an object’s name. If q is a side length, 4 × q means four times that length. Also, defining a as Aftab’s age does not mean a has one permanent value across every problem. Read the definition supplied in the current situation.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
If a represents Aftab’s age and Shabnam is 3 years older, which expression gives Shabnam’s age?
Shabnam is 29 years old. Aftab is 3 years younger. What is Aftab’s age?
What does n represent in the rule “2 × n sticks for separate Ls”?
A square has side length q cm. Which formula gives its perimeter in centimetres?
Why is s = a + 3 useful for different ages?
Start with Aftab’s age and add the three-year gap.
Practice Problems
- Complete Shabnam’s ages when Aftab is 4, 10, 18, and 23 years old.
- If Shabnam’s age is s years, write Aftab’s age and evaluate it for s = 35.
- Write a formula for the sticks in n separate Ls. Use it for n = 5 and n = 12.
- Write a square-perimeter formula using a letter of your choice. Find the perimeter for side lengths 3 cm and 11 cm.
- A student defines a as Aftab’s age, then uses a for Shabnam’s age in the next line. Why is this confusing?
- Does the rule 2 × n still count sticks if neighbouring Ls share sticks? Explain.
Add 3 each time: 7, 13, 21, and 26 years.
Key Takeaways
• Define every letter before using it. • Letter-numbers let us discuss unknown or changing numerical values. • An algebraic expression describes operations on letter-numbers. • Substitution replaces a letter by a stated number. • Translate a relationship into words before symbols. • A formula can describe many cases with one statement.
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Next · Lesson 2
Writing Expressions for Everyday Relationships