Math 7 · 7.EE.A.2

Rewriting an expression to show what it means

Two expressions give the same number every time. Why would you ever trade one for the other?

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The distributive property with variablesMath 7

Where you are 12 in Expressions

Algebra9

Expressions12

▴ 6 above Distribute, then combine Multiplying two binomials

Rewriting an expression to show what it means

Equivalent expressions: the distributive property and like terms Factoring out the greatest common factor ▾ Show all 12 (1 more)

When you reach for it

Querida buys a lamp marked p dollars, and a sales tax of 9\% is added at the till. The receipt lists the price and then the tax, so the natural way to write what she pays is

p + 0.09p

The same amount can also be written 1.09p. The two expressions are equivalent: whatever number p is, they give the same result. What changes is what you can see. The first form says the price, plus a tax worth 9\% of the price. The second says the total is 1.09 times the price, that is, 109\% of it, found with one multiplication.

Reach for a rewrite when the form you have hides the thing you want to know: a percent change spread across a sum, a perimeter that is really two equal halves, a fixed fee tucked inside a bracket.

The moves are ones you already have: distributing a number over a bracket, factoring out a common factor, and combining like terms, often one after the other. This lesson is about which form to aim for, and how to read it once you have it.

The procedure

A rewrite never changes the value

Every rewrite uses properties that hold for all numbers: the distributive property, the commutative property (order does not matter in a sum or a product) and the associative property (grouping does not matter). So the new expression equals the old one for every value of the letter, and either one may be read as a description of the situation.

For Querida’s lamp, write p as 1p, which changes nothing, and use the distributive property from right to left to collect the two terms into one:

\begin{aligned} p + 0.09p &= 1p + 0.09p \\ &= (1 + 0.09)p \\ &= 1.09p \end{aligned}

The same step works for any rate: a price raised by a rate r is p + rp = (1 + r)p, and lowered by r it is p - rp = (1 - r)p. The number in the bracket is the multiplier for a percent change, which has its own lesson.

A number makes a useful check. For a lamp marked 40 dollars the tax is 0.09 \times 40 = 3.60 and the total is 43.60 dollars; and 1.09 \times 40 = 43.6, the same. A check catches an arithmetic slip, but it does not prove two forms equivalent, since different expressions can agree at one number by luck. The proof is the chain of properties, which holds for every p.

What each shape shows

A sum shows parts. Each term is one piece of the whole: a price and its tax, a fixed fee and the charges per person, the sides of a shape.

A single term shows a multiplier. In 1.09p the whole story is in the coefficient. A coefficient above 1 is an increase, a positive coefficient below 1 is a decrease, and its distance from 1 is the size of the change.

A product with a bracket shows equal groups. 2(l + w) is two copies of l + w, and 4(n + 1) is four copies of n + 1. The factor outside counts the copies; the bracket says what one copy is.

So first decide the question: what are the parts, how many times the original, or what is repeated. Then pick the move that produces that shape. To read the result, say what every term and every factor stands for in the units of the situation, a skill studied for its own sake in interpreting the parts of an expression.

Worked examples

Example 1

A price taken down, read two ways

A kayak marked k dollars is sold at 37\% off. Write the sale price as the marked price minus the discount, then as a single term. Say what each form tells you, and check both for a kayak marked 300 dollars.

Solution

The discount is 37\% of the marked price, which is 0.37k, so the sale price is

k - 0.37k

This form shows the two amounts involved: what was marked, and how much came off. To make it a single term, write k as 1k and use the distributive property from right to left:

\begin{aligned} k - 0.37k &= (1 - 0.37)k \\ &= 0.63k \end{aligned}

This form says the buyer pays 63\% of the marked price, found in one multiplication. Which form is more useful depends on the question. “How much do I save?” is the term 0.37k in the first form. “What do I pay?” is the whole of the second. Sale prices have their own lesson, discounts, sales tax, tips and commission.

Check with k = 300. The first form: 0.37 \times 300 = 111, and 300 - 111 = 189. The second: 0.63 \times 300 = 189. Both give a sale price of 189 dollars.

Example 2

Reading the change from a multiplier

Each expression gives a new amount in terms of an old amount t. Say what change happened.

(a) 1.13t \qquad (b) 0.74t \qquad (c) 2t

Solution

Split the coefficient into 1 and what is left over. The 1 stands for the old amount, kept as it was, and the leftover part is the change.

(a) 1.13 = 1 + 0.13, so by the distributive property

1.13t = (1 + 0.13)t = t + 0.13t

That is the old amount plus 13\% of it: a 13\% increase.

(b) 0.74 = 1 - 0.26, so

0.74t = (1 - 0.26)t = t - 0.26t

That is the old amount minus 26\% of it: a 26\% decrease.

(c) 2 = 1 + 1, so 2t = t + 1t. The whole of t has been added again, and the whole of an amount is 100\% of it. So doubling is a 100\% increase, which sounds odd until the rewrite shows it: the extra amount is as large as the original.

Example 3

Two ways round a rectangle

A rectangle has length l and width w.

(a) Show that 2l + 2w = 2(l + w), and say what each form shows.

(b) Ludovica has 34 meters of fencing and will use all of it to build a rectangular pen. What does the form 2(l + w) tell her about every pen she could build?

Solution

(a) Both terms of 2l + 2w have a factor of 2, and factoring it out is the distributive property read from right to left:

2l + 2w = 2(l + w)

The two forms count the Perimeter in two ways. 2l + 2w says two lengths and two widths: the four sides, taken in their two equal pairs. 2(l + w) says one length and one width, twice. Walking one length and then one width takes you from a corner to the opposite corner, half the way round, and the other half is the same trip again.

Figure 1One length and one width take you from a corner to the opposite corner, half the way round. Twice that trip is the whole perimeter, 2(l + w).

For a rectangle 12 by 5, the first form gives 2(12) + 2(5) = 24 + 10 = 34 and the second gives 2(12 + 5) = 2(17) = 34.

(b) Using all the fencing means the perimeter is 34 meters, so 2(l + w) = 34. Two copies of l + w make 34, so one copy is half of that:

l + w = 17

Every pen Ludovica could build has a length and width that add to 17 meters: 12 by 5, 10 by 7, the square 8.5 by 8.5, and so on. The form 2l + 2w holds this fact too, but the factored form puts l + w where it can be seen.

Example 4

A fixed fee hiding in a bracket

A climbing center quotes a group price of 15(n + 4) dollars for n climbers.

(a) Rewrite the price as a sum, and say what its two terms mean.

(b) What does the bracket form say?

(c) Sigrun’s group has 8 climbers and splits the bill equally. Find each share, and use the sum form to explain why it is more than 15 dollars.

Solution

(a) Distribute the 15 over both terms in the bracket:

15(n + 4) = 15n + 60

The term 15n grows with the group, 15 dollars for each climber: the coefficient of the letter is a rate. The term 60 has no n in it, so it is the same for any group: a fixed fee of 60 dollars.

(b) 15(n + 4) is what n + 4 climbers would pay at 15 dollars each with no fee at all. Read this way, the fixed fee costs the same as bringing 4 extra climbers who never climb.

(c) For n = 8 the bill is 15(8 + 4) = 15 \times 12 = 180 dollars, and the sum form agrees: 15 \times 8 + 60 = 180. Shared by 8, each climber pays 180 \div 8 = 22.50 dollars. The sum form explains the extra: each climber pays their own 15 dollars and also an eighth of the fixed fee, 60 \div 8 = 7.50, and 15 + 7.50 = 22.50.

The same split works for any group size. Dividing by n is multiplying by \displaystyle \frac{1}{n}, so by the distributive property each term is divided by n:

\frac{15n + 60}{n} = 15 + \frac{60}{n}

This form shows what the other two hide: each share is always more than 15 dollars, and the extra, \displaystyle \frac{60}{n}, shrinks as the group grows. A group of 20 pays 15 + 3 = 18 dollars each.

Example 5

Counting the tiles round a pool

A square pool is n meters on each side. A border of square tiles, each 1 meter on a side, runs round it, one tile wide. Thaddeus counts the tiles as 4n + 4, Octavian as 4(n + 1), and Peregrine as 2(n + 2) + 2n. Explain each count from the picture, and show that the three are equivalent.

Figure 2One border, cut three ways. Drawn for n = 5, where every count gives 24 tiles.

Solution

Thaddeus counts each side of the pool, which has a strip of n tiles along it, so four sides give 4n. The four corner tiles touch no side of the pool, so they are added at the end: 4n + 4.

Octavian cuts the border into four strips that turn like a pinwheel. Each strip is one side’s n tiles together with the corner just beyond it, so it holds n + 1 tiles, and there are four of them: 4(n + 1).

Peregrine takes the top and bottom rows across the full width of the border. Each holds the pool’s n tiles and a corner at each end, n + 2 tiles. The two side columns fit between those rows with n tiles each: 2(n + 2) + 2n.

To show they agree, rewrite each as a sum of like terms. Octavian’s needs one step of the distributive property: 4(n + 1) = 4n + 4. Peregrine’s needs three: distribute the 2, use the commutative property to bring the two n terms together, then combine them.

\begin{aligned} 2(n + 2) + 2n &= 2n + 4 + 2n \\ &= 2n + 2n + 4 \\ &= 4n + 4 \end{aligned}

All three counts equal 4n + 4 for every n, as they must, since they count the same tiles. For the pool in the figure, n = 5: 4(5) + 4 = 24, 4(6) = 24 and 2(7) + 2(5) = 24. A fourth count agrees: the whole square of pool and border is 7 by 7, which is 49 squares, and taking away the pool’s 25 leaves 24.

The three expressions are equal, but each one remembers how its count was made, and that reading comes from the picture, not from the algebra. The whole-minus-pool count written with letters is (n + 2)^2 - n^2, and rewriting it needs multiplying two binomials.

Example 6

Three numbers in a row

(a) Show that the sum of three consecutive whole numbers is always three times the middle one.

(b) Can three consecutive whole numbers add to 100?

Solution

(a) Call the smallest number n. The next two are n + 1 and n + 2, so the sum is

n + (n + 1) + (n + 2)

The associative and commutative properties let the terms be regrouped and reordered, so collect the three n terms and the two numbers: the sum is 3n + 3. That form is correct but says little. Factor out the 3:

3n + 3 = 3(n + 1)

Now it speaks: n + 1 is the middle number, so the sum is three times the middle number. The same form shows that the sum is always a multiple of 3.

(b) No. Every such sum is a multiple of 3, and 100 is not, since 100 = 3 \times 33 + 1.

Boundaries

Only an equivalent form may be read in place of the original. A rewrite counts only if it was made with properties that hold for every value. Agreeing at one number is not enough: 2x + 2 and 4x both give 4 when x = 1, but at x = 2 they give 6 and 8.

Terms join into one only when they are multiples of the same thing. p + 0.09p becomes 1.09p because both terms are multiples of p. A price plus a flat delivery charge of 9 dollars, p + 9, is not any single multiple of p: the 9 dollars is a different percent of each price, so there is no one multiplier to read off. In the same way 2l + 2w factors as 2(l + w), but it does not collapse into one term, because l and w are different lengths.

A decrease is at most 100\%. (1 - r)p is a sensible price only when r is at most 1. A rate of 1 gives 0p = 0, the whole price gone, and a larger rate, as in p - 1.2p = -0.2p, is correct algebra that describes no price.

The meaning comes from the situation, not from the algebra. 15(n + 4) reads as “the price of 4 extra climbers” only because 15 is a price per climber. Some correct forms, such as 4(n + 2) - 4 for the pool border, make sense only alongside the story of how they were counted.

Dividing by the group size needs a group. \displaystyle 15 + \frac{60}{n} is each climber’s share only when n is at least 1; with n = 0 there is no one to share the bill and the division has no value.

Where this goes next

The shape of 15n + 60, a fixed part plus a rate times a number, returns as the starting value and rate of change in linear functions in context. Two percent changes made one after the other are two multipliers multiplied together, which is its own lesson.

Later a quadratic written as a product shows where it equals zero, the idea behind solving quadratics by factoring, while the same quadratic in vertex form shows its lowest or highest point.

Exercises

Percent changes read two ways

A meal costs m dollars, and a service charge of 23\% is added. Write the total as a sum of two terms, then as a single term, and say what the single term shows.
A tent marked q dollars is sold at 27\% off. Write the sale price both ways.
A new amount is 0.79v, where v is the old amount. What change happened?
A new amount is 1.24s, where s is the old amount. What change happened, and how large is the change when s = 450?
Peregrine says that 1.4p shows a 4\% increase on p. Is he right?

Perimeters

A rectangle is twice as long as it is wide, and its width is w centimeters. Write its perimeter as the sum of its four sides, then as a single term. What does the single term say, and what is the perimeter when w = 4.5?
A rectangular garden has a perimeter of 50 meters. What must its length and width add to? If its length is 15 meters, what is its width?

Groups and fixed costs

A bowling alley charges 12(n + 3) dollars for a group of n. Write the charge as a sum and say what the fixed fee and the price per person are.
A coach company charges 20n + 100 dollars for n passengers. Rewrite the charge with 20 factored out, and say what the number in the bracket means.
The bowling group in Exercise 8 has 9 people and splits the bill equally. How much does each pay? Show the share as the price per person plus a share of the fixed fee.
For a group of n, one club charges 18(n + 2) dollars and another charges 18n + 30 dollars. Which is dearer, and by how much?

Counting tiles

A rectangular pool is a tiles long and b tiles wide, with a border one tile wide. Explain why the border has 2a + 2b + 4 tiles, and also 2(a + 2) + 2b tiles. Show the two forms are equivalent, and count the tiles when a = 6 and b = 3.
Sigrun counts the border of a square pool n tiles across as 4(n + 2) - 4. Explain where the -4 comes from, and show that her count equals 4n + 4.
A square pool n tiles across has a border two tiles wide. Count it as four strips of 2n tiles along the sides and four 2 by 2 blocks at the corners, and show this count equals 8(n + 2). Find the number of tiles when n = 5, and describe a way of cutting the border that gives 8(n + 2) directly.

Patterns

Show that the sum of four consecutive whole numbers is always even. Which four consecutive whole numbers add to 50?
Show that the sum of five consecutive whole numbers is five times the middle one. Which five add to 115?

Reasoning

Explain why p + 0.09p can be written as a single multiple of p but p + 9 cannot.
Without using letters, explain why three consecutive whole numbers always add to three times the middle one.
Give two expressions in x that are equal when x = 2 but are not equivalent, and show a value of x where they differ.

Answers

m + 0.23m = 1.23m. The sum shows the meal and the charge; the single term, by the distributive property (1 + 0.23)m, says the total is 123\% of the meal.
q - 0.27q = 0.73q. The first form shows the marked price and the 27\% taken off; the second says the buyer pays 73\% of the marked price.
A 21\% decrease. 0.79v = (1 - 0.21)v = v - 0.21v.
A 24\% increase, since 1.24s = s + 0.24s. When s = 450 the change is 0.24 \times 450 = 108, and the new amount is 1.24 \times 450 = 558.
No. 1.4p = (1 + 0.4)p = p + 0.4p, a 40\% increase. A 4\% increase is 1.04p.
w + 2w + w + 2w = 6w centimeters: the perimeter is six widths. When w = 4.5 it is 6 \times 4.5 = 27 centimeters.
2(l + w) = 50, so l + w = 25 meters. With a length of 15 meters the width is 10 meters. Check: 2(15) + 2(10) = 50.
12(n + 3) = 12n + 36. The fixed fee is 36 dollars and the price per person is 12 dollars. The bracket form says the fee costs the same as 3 extra players.
20n + 100 = 20(n + 5). The fixed fee of 100 dollars costs the same as 5 extra passengers at 20 dollars each.
16 dollars each. The bill is 12 \times 9 + 36 = 144, and 144 \div 9 = 16. Split: 12 + 36 \div 9 = 12 + 4 = 16.
The first club is dearer by 6 dollars for every group, since 18(n + 2) = 18n + 36 and 36 - 30 = 6.
22 tiles. Two long sides of a and two short sides of b, plus 4 corners, give 2a + 2b + 4. Taking the two long rows across the full width, each a + 2, with the two short sides of b between them, gives 2(a + 2) + 2b. Distributing, 2(a + 2) + 2b = 2a + 4 + 2b, which is 2a + 2b + 4. For a = 6, b = 3: 12 + 6 + 4 = 22; and 8 \times 5 - 6 \times 3 = 22.
Sigrun takes four full sides of n + 2 tiles each, which counts every corner tile twice, once in each side it ends, so she takes away the 4 extra. 4(n + 2) - 4 = 4n + 8 - 4 = 4n + 4.
56 tiles when n = 5. 4(2n) + 4(4) = 8n + 16 = 8(n + 2). For n = 5: 8(7) = 56, and the whole square, 9 by 9, minus the pool gives 81 - 25 = 56. Directly: cut the border into four pinwheel strips, each 2 tiles deep and n + 2 tiles long, which is 4 \times 2(n + 2).
11, 12, 13, 14. The sum is n + (n + 1) + (n + 2) + (n + 3), which is 4n + 6 = 2(2n + 3), two times a whole number, so always even. 4n + 6 = 50 gives n = 11, and 11 + 12 + 13 + 14 = 50.
21, 22, 23, 24, 25. The sum is 5n + 10 = 5(n + 2), and n + 2 is the middle number. 115 = 5 \times 23, so the middle number is 23, and 21 + 22 + 23 + 24 + 25 = 115.
Both terms of p + 0.09p are multiples of p, so the distributive property joins their coefficients. The 9 in p + 9 is not a multiple of p. If p + 9 equaled some kp for every p, then putting p = 0 would give 9 = 0, which is false. In the situation, a fee of 9 is 90\% of a price of 10 but only 9\% of a price of 100, so no single percent fits.
The largest number is one more than the middle one, and the smallest is one less. Move 1 from the largest to the smallest: the total is unchanged, and now all three numbers equal the middle one. Three equal numbers add to three times that number.
For example 3x and x + 4. At x = 2 both are 6. At x = 0 the first is 0 and the second is 4, so they are not equivalent.