Interpreting the parts of an expression
A bank says your balance after t years will be 2500(1.04)^t dollars. Before you work anything out, what is that expression already telling you?
Where you are 12 in Expressions
Why this exists
A formula is usually met as a machine: put a number in, get a number out. But an expression written for a situation is also a description of it, and it can be read before any number goes in.
Zinovia’s savings account grows by 4\% a year, and the bank tells her that after t years her balance will be 2500(1.04)^t dollars. Read piece by piece, it says what she started with, what happens to the balance each year, and how many years have passed. Nothing was calculated, yet doubling the deposit visibly doubles the balance, and a better rate changes only the 1.04.
Reading this way checks that a formula fits its story and says what a change will do before any sums. The names for the pieces come from terms, coefficients and constants; this lesson asks what they mean, and how small pieces join into bigger ones.
Definition
To interpret an expression is to say, for each of its parts, what quantity that part stands for in the situation, in which units, and how the parts combine into the whole.
A part of an expression is any piece of it that gets worked out as a unit before being combined with the rest:
- a term, one of the pieces added or subtracted at the top level, with its sign;
- a factor, one of the pieces multiplied: in 3x(x - 4) the factors are 3, x and x - 4;
- a coefficient, the number factor of a term, and a constant, a term with no letter;
- a bracket, which stands for one number however much is written inside it;
- a base and an exponent: in (1.04)^t the exponent t says how many factors of the base 1.04 are multiplied together.
Which kind of part sits at the top is decided by the order of operations. Work the expression out in that order and look at the last step. If it is adding or subtracting, the expression is a sum and its top-level parts are terms. If it is multiplying, the expression is a product and its top-level parts are factors.
| Expression | Last step | Top-level parts |
|---|---|---|
| 2500(1.04)^t | multiply | 2500 and (1.04)^t |
| 45 - 4.9t^2 | subtract | 45 and -4.9t^2 |
| (x - 5)^2 + 2 | add | (x - 5)^2 and 2 |
| 1.06(24n + 5) | multiply | 1.06 and 24n + 5 |
Each part may have parts of its own. The term -4.9t^2 is a product of -4.9 and t^2, and t^2 is the base t with the exponent 2. Reading an expression means reading it at every level, from the top down.
Intuition
You can see that (x - 5)^2 + 2 is “something squared, plus 2” before you think about x at all. Treating a whole bracket as one quantity is the key move.
Casimir-Jo sews a fabric border b cm wide all round an embroidered panel 20 cm wide and 25 cm tall. The area of border fabric is
square centimeters. Read from the top, it is one thing minus another. Then read each thing.
The outer edge runs across the panel’s 20 cm and a border of b on each side, so it is 20 + 2b cm across, and in the same way 25 + 2b cm tall. The product of the two brackets is the area of the whole outer rectangle, panel and border together. The 500 is the panel, 20 \times 25. So the expression says “the whole, minus the hole”, which is the border. Nothing was multiplied out; the meaning came from the shape of the expression.
Results
The last operation decides the shape. Worked out in the order of operations, the last step says what kind of expression it is: a sum of terms, a product of factors, a quotient, or a power.
A bracket is one number. Whatever is written inside a bracket, the rest of the expression sees only its value. So 2500(1 + 0.04)^t is 2500 times a single number, the number the deposit is multiplied by over t years.
A square is never negative. For every real number u,
and u^2 = 0 only when u = 0. So A + u^2 is at least A, and A - u^2 is at most A.
Changing one part changes the whole predictably. Multiply one factor of a product by k, leaving the other factors alone, and the product is multiplied by k. Raise one term of a sum by d, leaving the other terms alone, and the sum rises by d.
Why it is true
The last operation, and brackets. Every step before the last produces a number, and the last step combines those numbers, so they are the top-level parts. A bracket is worked out first, so the rest of the expression has only its value to work with.
Squares. u^2 = u \cdot u. A positive times a positive is positive, a negative times a negative is positive, and 0 \cdot 0 = 0. So u^2 \ge 0, with 0 only at u = 0.
Changing one part. (kF)G = k(FG) by the associative property, and with more factors the commutative property first brings the changed one to the front. For a sum, (T + d) + S = (T + S) + d by the commutative and associative properties. The word “alone” matters: if one letter sits in two places, changing it changes two parts at once.
Worked examples
A music bill in three pieces
Aurek buys a cello from his music school for 720 dollars, paid off in equal amounts over 24 months. His lesson package costs 120 dollars a month, and each hour he books a practice room costs 3 dollars. In a month when he books h hours of practice room, he pays
dollars. (a) Say what each term means. (b) Which term can change from month to month? (c) If the cello were paid off over 36 months instead, which part would change, and by how much?
Solution(a) The last step is adding, so the parts are the three terms. The term \displaystyle \frac{720}{24} is the cello’s price split into 24 equal parts: the monthly payment for the cello, 30 dollars. The term 120 is the lesson package. The term 3h is the charge for the practice room: the coefficient 3 is the price of one hour in dollars, and h is how many hours.
(b) Only 3h, the one term with a letter in it. The other two add to 30 + 120 = 150 dollars every month.
(c) Only the first term, which becomes \displaystyle \frac{720}{36} = 20. Each bill falls by 30 - 20 = 10 dollars, whatever h is, since one term fell by 10 and the others stayed. The cello still costs 720 dollars; it is spread over more months.
A savings account growing by 4\%
Zinovia deposits 2500 dollars, and her balance after t years is 2500(1.04)^t dollars. (a) What do 2500, 1.04 and t mean? (b) Why is the base 1.04 and not 0.04? (c) What does the factor (1.04)^t stand for?
Solution(a) The power is worked out first and the multiplication last, so this is a product of two factors, 2500 and (1.04)^t. The 2500 is the deposit in dollars, where the balance starts. The base 1.04 is what the balance is multiplied by in one year. The exponent t is the number of years, and so the number of times the balance has been multiplied by 1.04.
(b) Read 1.04 as 1 + 0.04. The 1 keeps the whole balance, and the 0.04 adds 4\% of it. Multiplying by 0.04 alone would keep only the interest. Check one year: 2500(1.04) = 2600, which is 2500 plus 4\% of 2500, that is, 2500 + 100. This is the multiplier for a percent change.
(c) (1.04)^t is t factors of 1.04, one for each year: how many times bigger the balance is after t years than at the start. After 2 years it is 1.04 \times 1.04 = 1.0816, and the balance is 2500 \times 1.0816 = 2704 dollars. Since the deposit is a separate factor, doubling it would double the balance at every time.
This is a first look at exponential growth, where the letter sits in the exponent.
Where the tax is charged
Briallen orders n concert tickets at 24 dollars each. The site adds a booking fee of 5 dollars to the order and then charges 6\% sales tax. Her total is
dollars. (a) Read the parts. (b) Another site’s total is 1.06(24n) + 5. What is different about how it charges? (c) Compare the two totals for 3 tickets.
Solution(a) The last step is multiplying, so this is a product of 1.06 and the bracket 24n + 5. The bracket is the bill before tax: 24n dollars for n tickets at 24 dollars each, and 5 dollars for the fee. The factor 1.06 multiplies the whole bracket, so the tax is charged on the tickets and on the fee.
(b) In 1.06(24n) + 5 the last step is adding, so it is a sum of two terms: the tickets with tax, and the fee, added after the tax and so not taxed. Where the bracket closes decides what gets taxed. The gap is the fee’s tax, 0.06 \times 5 = 0.30 dollars, whatever n is.
(c) For n = 3 the first total is 1.06(72 + 5) = 1.06 \times 77 = 81.62 dollars. The second is 1.06 \times 72 + 5 = 76.32 + 5 = 81.32 dollars. They differ by 0.30 dollars, as (b) said.
The fabric in a border
The panel in Figure 1 is 20 cm by 25 cm with a border b cm wide, and the area of border fabric is (20 + 2b)(25 + 2b) - 500 square centimeters. (a) Give the units of each part. (b) Find the area of fabric when b = 3. (c) If the border is made twice as wide, 6 cm, does the area of fabric double?
Solution(a) 2b, 20 + 2b and 25 + 2b are lengths, in centimeters. The product of the two brackets and the 500 are areas, in square centimeters. Both parts of the difference are areas, as they must be: a length is never taken away from an area.
(b) With b = 3 the outer rectangle is 26 by 31:
The area of fabric is 306 square centimeters.
(c) With b = 6 the outer rectangle is 32 by 37:
That is more than 2 \times 306 = 612, so the fabric more than doubles. Scaling one factor does not apply here: b sits inside both brackets, so a wider border makes the outer rectangle both wider and taller, while the panel’s 500 is taken away unchanged.
A falling stone
Emlyn drops a stone from a bridge 45 meters above a river. After t seconds its height above the water is about 45 - 4.9t^2 meters. (a) What do 45 and 4.9t^2 measure? (b) How high is the stone after 1 second and after 2 seconds? (c) Does it fall as far in the second second as in the first?
Solution(a) The expression is a difference. The 45 is the starting height in meters. The term taken away, 4.9t^2, is the distance the stone has fallen after t seconds, also in meters. So it reads “starting height minus distance fallen”. The coefficient 4.9 is half of about 9.8, the pull of gravity near the Earth’s surface in meters per second squared.
(b) After 1 second it has fallen 4.9 meters and is 45 - 4.9 = 40.1 meters high. After 2 seconds it has fallen 4.9 \times 4 = 19.6 meters and is 45 - 19.6 = 25.4 meters high.
(c) No. In the second second it falls 19.6 - 4.9 = 14.7 meters, three times as far as in the first. The reason is the square: t^2 = t \cdot t has the time as two factors, so doubling the time multiplies the distance fallen by 2 \times 2 = 4. The stone keeps speeding up.
Reading a bound from the structure
Use the structure of each expression, not a table of values.
(a) Show that (x - 5)^2 + 2 is never less than 2, and find where it equals 2.
(b) Find the largest value of 9 - (2t + 1)^2, and where it happens.
(c) Find the largest value of \dfrac{12}{x^2 + 3}, and explain why it is never 0.
Solution(a) This is a square plus 2. Read x - 5 as one number u. Then the first term is u^2, which is never negative, so the sum is at least 0 + 2 = 2. It equals 2 exactly when the square is 0, when x - 5 = 0, at x = 5.
(b) This is 9 minus a square. The square is at least 0, so the most it can take away is nothing, and the largest value is 9, when 2t + 1 = 0, at \displaystyle t = -\frac{1}{2}.
(c) This is 12 divided by x^2 + 3, read as one number. Since x^2 \ge 0, that number is at least 3. Dividing 12 by a larger positive number gives a smaller result, so the quotient is largest when the denominator is smallest: \displaystyle \frac{12}{3} = 4, at x = 0. It is never 0, because 12 divided by a positive number is positive.
No value was tried. The bound came from knowing what kind of number each part is.
Boundaries
The meaning comes from the situation, not the symbols. 24n + 5 is a ticket order only because the story said so. Without a situation, all you can describe is the structure.
A formula holds only where the story does. 45 - 4.9t^2 describes the stone from t = 0 until it reaches the water; after that the expression gives negative numbers that describe no stone. A count such as n tickets is a whole number, so n = 2.5 has no meaning, though the expression has a value.
The parts belong to the written form. (20 + 2b)(25 + 2b) - 500 and 4b^2 + 90b are equal for every b, but only the first shows the outer rectangle and the panel. When the form you have hides what you need, rewrite it.
A bound is reached only if the bracket can reach the value that gives it. u^2 \ge 0 shows that (x^2 + 1)^2 + 2 is at least 2, which is true, but x^2 + 1 is never 0, so 2 is never reached. The smallest value is 3, at x = 0.
Squares are never negative for real numbers. Later there are numbers whose squares are negative.
Where this goes next
Reading 2500(1.04)^t as a starting amount times a repeated multiplier is exponential growth and decay, and the repeated number is the growth or decay factor. A fixed charge plus a rate times a count, as in Aurek’s music bill, is the starting value and rate of change of a linear function in context, and the numbers in a model are read the same way. The argument that a square plus 2 is at least 2 is how a quadratic in vertex form shows its lowest or highest point without a graph.
Exercises
Naming parts and their meanings
- A van rental costs 40 + 0.75d dollars for a day in which the van is driven d kilometers. Say what 40, 0.75 and 0.75d mean, and find the cost for 120 kilometers.
- A car’s value after t years is 16000(0.85)^t dollars. What does the base 0.85 say about the value each year? Find the value after 1 year and after 2 years.
- Is 3x(x - 4) + 7 a sum or a product at the top level? Name its top-level parts, and the factors of the first one.
- Is 5(2y + 1)^2 a sum or a product at the top level? Name its top-level parts, and say what is being squared.
Reading a bound from the structure
- Find the smallest value of (x + 4)^2 + 6, and where it happens.
- Find the largest value of 15 - (y - 2)^2, and where it happens.
- Can (3t - 1)^2 + 8 ever equal 5? Explain from the structure.
- Find the largest value of \dfrac{20}{x^2 + 4}, and where it happens.
- Find the smallest value of (x^2 + 2)^2, and where it happens.
Changing one part
- In Exercise 2, the car had cost 32000 dollars instead. How would its value compare at every time? Write the new expression.
- In Briallen’s ticket total, 1.06(24n + 5), the booking fee rises from 5 to 7 dollars. How much more does every order cost?
- In Aurek’s music bill, \displaystyle \frac{720}{24} + 120 + 3h, the cello costs 960 dollars instead. How much more is each monthly bill?
- A box with a square base x by x and height h has volume x^2h. How does the volume change if x is doubled? If h is doubled instead?
- A carpet cleaner charges 40 dollars to come out plus 25 dollars a room, and 8\% tax is added to the whole bill. Write an expression for r rooms, and find the bill for 4 rooms.
Reasoning
- Someone says the balance 2500(1.04)^t rises by 100 dollars every year, because 4\% of 2500 is 100. Explain, using the parts of the expression, why the rise in the second year is more than 100 dollars, and find it.
- Explain why (x - 1)^2 + (x + 1)^2 is never 0.
Answers
- 40 dollars is the fixed charge for the day, 0.75 the price per kilometer, and 0.75d the charge for the distance. For 120 kilometers: 40 + 0.75 \times 120 = 40 + 90 = 130 dollars.
- 0.85 = 1 - 0.15: each year the car keeps 85\% of its value, losing 15\%. After 1 year: 16000 \times 0.85 = 13600 dollars. After 2 years: 13600 \times 0.85 = 11560 dollars.
- A sum: the last step is adding 7. Its terms are 3x(x - 4) and 7, and the first has factors 3, x and x - 4.
- A product: the power comes before multiplying by 5. Its factors are 5 and (2y + 1)^2, and the number being squared is the bracket 2y + 1.
- 6, at x = -4. The square is at least 0, so the sum is at least 6, and the square is 0 when x + 4 = 0.
- 15, at y = 2. The square taken away is at least 0, so the result is at most 15, reached when y - 2 = 0.
- No. (3t - 1)^2 \ge 0, so the expression is at least 8, and 5 is less than 8.
- 5, at x = 0. The denominator x^2 + 4 is at least 4, and 20 divided by a larger positive number is smaller, so the largest value is \displaystyle \frac{20}{4} = 5.
- 4, at x = 0. The bracket x^2 + 2 is at least 2, and it is positive, so its square is at least 2^2 = 4, reached when x^2 = 0.
- Twice as large at every time, since the price is one factor and (0.85)^t does not depend on it. The new value is 32000(0.85)^t dollars.
- 2.12 dollars more, whatever n is. The bracket rises by 2 and is a factor, so the total rises by 1.06 \times 2 = 2.12.
- 10 dollars more. Only the first term changes, from \displaystyle \frac{720}{24} = 30 to \displaystyle \frac{960}{24} = 40.
- Doubling x multiplies the volume by 4, because x^2 = x \cdot x has x as two factors: (2x)^2h = 4x^2h. Doubling h doubles it, since h is one factor: x^2(2h) = 2x^2h.
- 1.08(40 + 25r) dollars: the bracket is the bill before tax, and 1.08 adds 8\% to all of it. For 4 rooms: 1.08 \times 140 = 151.20 dollars.
- 104 dollars. The 1.04 multiplies the whole balance, not the deposit, and after one year the balance is 2600. The second year adds 0.04 \times 2600 = 104, so the balance becomes 2704 dollars, not 2700.
- Each term is a square, so each is at least 0, and the sum is 0 only if both are 0. That needs x - 1 = 0 and x + 1 = 0 at once, so x = 1 and x = -1, which is impossible.