Linear Equations

A tall colored tile stands for a number nobody has named yet; a small gray square is one unit. Tiles and squares in a bin make an expression, and the palette builds it token by token: one blue tile and four squares is k plus 4, two green tiles and five squares is 2m plus 5, three purple tiles and six squares is 3r plus 6. Then the hidden number is named and the tiles show it: k is 3, so k plus 4 is 7; m is 10, so 2m plus 5 is 25; r is 4, so 3r plus 6 is 18. Two teaching boards name the parts, the variable term and the constant term, and the three skill checks evaluate 2y plus 3, 3r plus 6 and 5q plus 2 at named values.

In the first lesson the tiles hid their numbers and the total came out of them; here the total is known first and the hidden number comes out of it. A pink tile and three gray squares total 8, so the tile is dragged taller until d plus 3 reaches 8 and d is 5; three orange tiles and two squares total 14, so w is 4; four teal tiles and five squares total 17, so g is 3. Then the value goes into the equation itself: 3u plus 4 is 22 gives u equal to 6, 2e plus 6 is 20 gives e equal to 7, and with no tiles at all 3u plus 8 is 20 gives 4 and 4w plus 12 is 40 gives 7. Two teaching boards name the two moves, subtract the constant and divide by the coefficient, and the three skill checks find j in 2j plus 3 is 13, d in 3d plus 2 is 20 and v in 4v plus 6 is 30.

Four teal tiles of n and a gray tile marked 3 sit in one bin, and the four matching tiles gather into a single term, 4n; the 4 in front counts the tiles and is called the coefficient. Three tiles of h with a 2 and a 4 gather into 3h plus 6; two purple tiles already marked 3s and 4s, with a 2 and a 3, gather into 7s plus 5; a mixed bin of 2t, 3, 4t, a lone t and 5 gathers into 7t plus 8, because order changes nothing. Then the tiles go and the expression stands alone: 3f plus 4f plus 2 plus 6 is 7f plus 8, and 5 plus 3m plus 4 plus m is 4m plus 9. Two teaching boards name the two moves, gather the letter terms and gather the plain numbers, and the three skill checks collect 3k plus 3k plus 1 plus 4, 3p plus p plus 2 plus 4 plus p, and 2r plus 3 plus r plus 6.

A sentence can hide an expression. Adding 6 to an unknown number k is k plus 6; doubling a number r and adding 15 is 2r plus 15; four more than triple a number h is 3h plus 4. Each description is built from tokens into its expression, and two expressions go the other way: 3m plus 8 finds its description among three, and 6y plus 5 is only found once 2y and 4y are collected into 6y. Four added to the sum of g and 7g collects into 8g plus 4, two teaching boards name the moves, doubling and tripling set the coefficient and more than sets the constant, and the three skill checks write 5d plus 7, 3w plus 11 and, from the sum of 2q and 4q plus the sum of 5 and 4, 6q plus 9.

Stories written as expressions, and expressions read back as stories. 8 coaches travel with t teams of 11 players: 11t + 8 people. 15 loaves at b dollars and 24 muffins at m dollars, with 6 dollars for delivery: 15b + 24m + 6. c trays of 24 cookies with 17 eaten: 24c − 17 left, and the family of 5 is not counted. Then the other way: 5s + 7 lanterns are s stalls of 5 lanterns and 7 more over the fountain, and a profit of 4f − 15 dollars is f faces painted at 4 dollars each, less 15 dollars of paint.

Evaluate, solve, combine and write linear expressions, each answer typed into its slot, from every lesson of the level.

A merchant's bill has three lines: the count of items, the price of each, and the total. Three reed pens at four dollars each come to twelve dollars, and six ink pots at three dollars each to eighteen; when q sheets cost three dollars each, the total is 3 times q. Then the bill is read backward: a total of 36 dollars at four dollars each hides a count of nine, because 4k equals 36 and dividing each side by 4 leaves k equal to 9. Six m equals 30 is undone by dividing by six, so m is 5; eight r equals 56 gives 7. Five lamps at six dollars each come to 30, and the last bills give w equal to 8, y equal to 6 and z equal to 4. Ten boards, and one move under all of them: when a number stands in front of the unknown, divide every side by it.

An invoice with a delivery fee: 4 items at $3.00 and an $8.00 fee come to $20.00, and when the count is a letter the total reads T = 3 · k + 8, the price times the count and then the fee. Then the total is printed first. 8 + 3m = 50 gives up m in two moves, the fee off both sides and the price divided out, so m is 14; 4b + 5 = 29 gives 6; 1.5q + 6 = 18 gives 8; 7 + 6w = 61 gives 9. Two operations built each total, and the same two, undone in reverse order, hand back the count.

A sentence about an unknown number is written as an equation: four times p is 212 becomes 4p = 212, and h decreased by 67 with the result 58 becomes h − 67 = 58. Then the algebra moves, the same operation on both sides: 4p = 212 divided by 4 on both sides leaves p = 53; h − 67 = 58 with 67 added to both sides leaves h = 125. 385 = 6c + 43 gives up c in two moves, 43 off both sides and then both sides divided by 6, so c is 57; 3g + 21 = 48 gives 9. The opposite operation, on both sides, until the letter stands alone.

An equation can arrive with parentheses: 2(u + 6) = 26 says two times the whole sum is twenty-six. The multiplier was applied last, so it is undone first: both sides divided by 2 leave u + 6 = 13, and then the 6 comes off, u = 7. Typed on a keypad, 5(g + 3) = 40 gives g = 5 the same way. Then the moves are chosen one at a time: 3(d + 4) = 30 divided by 3 and then less 4 gives 6; 4(h − 5) = 36 gives 14; half of s + 5 equal to 8, multiplied by 2 and then less 5, gives 11; 6(b + 1) = 54, 3(t − 4) = 27 and 7(r + 1) = 21 give 8, 13 and 2. The multiplier outside comes off first, the number inside second.

A letter wrapped in several operations. a divided by 5 is 30, so a is 150; h divided by 6, less 3, is 9, so h is 72; k plus 4, all over 3, is 9, so k is 23. Each layer comes off from the outside in, one line at a time: (3c − 6)/4 = 3 peels to c = 6, and 2(d + 3)/5 + 4 = 8 peels to d = 7. On 4(3g + 2) + 6 = 62 the moves are made by hand, each on both sides, the outermost layer first, until g stands alone at 4. The last operation put on is the first one taken off.

A story holds an equation. 52 sheets, 4 set aside for covers, the rest sewn into k books of 6: 6k + 4 = 52, solved by moves on both sides to k = 8, and read back as 8 books. A basket of m dates shared among 3 students, each going from 4 dates to 9: m/3 + 4 = 9, so m = 15 dates. Four travellers each buying a $3 loaf and a $y jar of honey, $36 in all: 4(y + 3) = 36, so a jar of honey costs $6.

Solve one-step, two-step and bracketed equations, from invoices and sentences as well as bare equations.

An inequality holds for many values. k is more than 5, so 6, 7 and 10 all work, and every value to the right of 5 is shaded as a ray from an open circle; w is less than 1, so the ray runs left from 1. The solution to an inequality is every value that makes it true, drawn from its boundary out to the arrow.

Graph the solution of an inequality on a number line, with an open or closed circle on its boundary.

Read an inequality with its letter on either side, graph its solution, and write a graph back as an inequality.

Solve the associated equation to find an inequality's boundary, then graph the side that makes it true.

Undo two steps to find an inequality's boundary, then graph the side that makes it true.

Write a story as an inequality, solve it, and read the answer back into the story as a whole number.

Solve, graph and write inequalities, and complete one so it has a given solution.

Two orders, each a rate times an amount, add into one total. When the same unknown amount goes to both shops, the two charges carry the same letter and join into one term, and the two fees join into one number. With the like terms joined, the equation is short enough to solve in a step or two.

A sentence about several terms becomes one equation: the sum of 2y and 7y is 81 is written 2y + 7y = 81. Combining like terms is a move of its own: 2y + 7y becomes 9y, and one division leaves y = 9. With a plain number beside the letter, 5c + 3c + 12 = 60 combines to 8c + 12 = 60; the 12 comes off both sides and the 8 divides out, so c = 6. With two plain numbers, 4m + 65 + 7m + 38 = 257 combines to 11m + 103 = 257, and m = 14. Fewer terms first, then the moves.

Letters on both sides of the equals sign. Four times k is k plus 12 is written 4k = k + 12; k taken from both sides leaves 3k = 12, so k is 4. 5p − 14 = 2p + 7 gathers to 3p − 14 = 7, so p is 7, and r − 6 = 0.8r gathers to 0.2r − 6 = 0, so r is 30. On 3(y + 4) = 9y the moves are made by hand, each on both sides: the 3 divides out, y comes off both sides, and y is 2. A letter term moves across the equals sign exactly as a number does.

Combine like terms, gather the letter on one side, then undo the rest until the letter stands alone.

Write two linked amounts from a story as one equation, solve it, and say what the solution means.

Combine like terms, gather a letter onto one side, and write equal amounts from a sentence as one equation.

A letter can stand for a number below zero. Its tiles hang below a waterline while the squares stand above it, and the recipe never changes: the value takes the letter's place inside brackets, the product comes first and the constant is added last. A total can land above zero, at zero or below it.

A missing value can sit below zero. Four k-tiles and ten squares make 4k + 10; when that comes to 2, the 10 comes off both sides to leave 4k = −8, and the 4 divides out to leave k = −2. At 4k + 10 = −6 the same two steps give 4k = −16, then k = −4. Writing the constant first changes nothing: 10 + 5n = −5 leaves 5n = −15, and n = −3. The constant is undone first, then the coefficient, and the sign rides along through both steps.

Write a subtracted constant as an added negative, and solve an equation written either way.

The number in front of a letter can carry a minus sign. Five −p tiles combine to −5p; −3w, −3w, 5 and −9 combine to −6w + (−4); −6z, 2z, −8 and 3 combine to −4z − 5. With a value in the letter's place, the sign rides into the product: with g = 3, −4g + 5 is −12 + 5, which is −7; with r = −3, −5r − 6 is 15 − 6, which is 9, because a negative times a negative is positive. The same care finds k in −4k + 13 = −7: −4k = −20, so k is 5.

Undo the constant, then divide by a negative coefficient to find a missing value, tracking every sign.

Evaluate and solve linear equations whose values, constants and coefficients sit below zero.

Solve an equation whose answer is a number below zero, taking each layer off both sides until the letter stands alone.

Read a subtracted term as adding its opposite, then solve by taking each layer off both sides.

Read a minus sign on a letter as a coefficient of negative one, and flip the sign on both sides to set the letter free.

Combine like terms with negative coefficients, gather the letter on one side, and solve the equation.

Write a story that falls by a steady amount as an equation, solve it, and read the answer back into the story.

Solve linear equations whose solutions, constants and coefficients sit below zero.

Test one value to find which side of the boundary holds every solution of an inequality.

Turn the symbol around when each side of an inequality is multiplied or divided by a negative number.

Solve an inequality that divides by a negative, then graph every solution as a ray on the number line.

Combine like terms and solve an inequality step by step, turning the symbol for a negative multiplier or divisor.

Turn a story about keeping at least an amount into an inequality, solve it, and round the answer down.

Solve and test linear inequalities where dividing by a negative number turns the sign around.

Multiply a number across a sum and read the product off the rectangle it builds.

Multiply the factor outside parentheses by each term inside to expand a product into a sum.

Multiply a signed number across a sum and keep track of the sign of every product.

Clear the parentheses by distributing, then undo each operation until the letter stands alone.

Solve a multiple of a sum by dividing first or distributing first, whichever keeps the numbers whole.

Distribute on both sides, gather the letter on one side, and undo the rest to solve.

Combine groups of the same expression into one group, then solve the simpler equation that is left.

Turn a story about identical groups into a bracket equation, then solve it by distributing or by dividing.

Distribute, combine groups and solve linear equations with parentheses.

Pull a shared factor out of a sum and write it as a product.

Write a sum both ways, as a product of its shared factor and as the terms it spreads into.

Pull the greatest factor every term shares out to the front of the parentheses.

Pull a shared number out in front of a side, divide it away, and finish the solve.

Factor out the greatest common factor and solve linear equations by factoring.

Some equations have no solution, because the letter cancels from both sides and the work ends on a false statement.

Show that an equation has infinitely many solutions by working each side until the two sides read the same.

Carry an equation down to its last line to tell whether it has one solution, infinitely many, or none.

Write a story about two prices as one equation and tell whether it has one solution, none, or infinitely many.

Decide whether a linear equation has one solution, no solution or infinitely many, and build equations of each kind.

Undo a fraction in front of a letter by multiplying both sides by its reciprocal.

Combine fraction terms that share a denominator by adding their numerators, then solve for the letter.

Rewrite fractions with different denominators over one common denominator, combine them, and solve.

Clear every fraction from an equation by multiplying both sides by the least common denominator, then solve.

Clear fractions outside and inside a group by multiplying both sides by the least common denominator.

Solve linear equations with fractional coefficients by combining or clearing the fractions.

Find a percent of a price and add it on as tax, first in dollars, then as one combined term.

A discount takes a percent of the price away. A $50 lamp with 10% off sells for $45, and an $80 rug with 25% off sells for $60, written S = 80 − 0.25 · 80. A scarf at p dollars with 20% off sells for p − 0.2 · p, and p − 0.2p is 0.8p: taking 20% off a price is the same as multiplying it by 0.8.

Add the sale prices of two discounted items, then combine like terms with decimal coefficients.

Weigh two deals by their sale prices, then solve an inequality for the prices at which one deal is cheaper.

Write, combine and compare decimal expressions for percent changes and sale prices.

Find the values that make two inequalities true at once by reading where their graphs overlap.

Read a three-part inequality as a pair joined by and, graph its values, and write one from its graph.

Graph two inequalities joined by or as one picture, and read an or graph back into its two halves.

Combine the graphs of two inequalities into one graph, and decide how many boundary points it needs.

A compound inequality joins two inequalities. With and, a value must make both true: 1 < a + 4 < 9 becomes −3 < a < 5, a stretch between two circles. With or, one is enough: 3m ≤ −6 or 3m ≥ 12 becomes m ≤ −2 or m ≥ 4, two rays pointing apart. Every move is applied to every part, and dividing by a negative number turns every sign around.

Turn a real rule with limits into a compound inequality, solve it, and read what it allows in whole numbers.

Solve, graph and write compound inequalities joined by and or or.