6.EE.B.7Common CoreMathExpressions and EquationsGrade 6
6.EE.B.7: Writing and Solving One-Step Equations
In plain English: 6.EE.B.7 is the Common Core grade 6 math standard that asks students to write and solve equations of the form x + p = q and px = q to answer real-world and math questions. Every number in these equations, including the answer, is zero or positive: a whole number, a fraction or a decimal. Students undo addition with subtraction and multiplication with division, which prepares them for two-step equations in grade 7.
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Reason about and solve one-variable equations and inequalities. Also written as 6.EE.7 · Official standard
Students write and solve one-step equations of two forms: x + p = q and px = q. In both, p and q are known numbers and x is the unknown. The standard limits every number, including the answer, to nonnegative rational numbers: zero and positive whole numbers, fractions and decimals. Students turn short stories about money, measurement and recipes into equations, then solve them with inverse operations (operations that undo each other) and tape diagrams (bars split into parts).
The lesson starts with number puzzles that students solve in their heads, then connects that thinking to tape diagrams and equations. Students practice choosing the right form from the words in a problem, solving with subtraction or division, checking by substitution (putting the answer back into the equation), and deciding whether an answer makes sense. Equations with two steps (such as 2x + 3 = 11) and negative numbers wait until grade 7.
Learning Objectives
By the end of this lesson, students will be able to:
Write an equation of the form x + p = q or px = q to represent a real-world or mathematical problem
Solve x + p = q by subtracting p from both sides, with whole numbers, fractions and decimals
Solve px = q by dividing both sides by p, including when p is a fraction or a decimal
Check a solution by substitution and explain whether it makes sense in the problem
Prior Knowledge Required
Students should already be comfortable with:
Finding an unknown number in a multiplication or division equation 3.OA.A.4
Adding and subtracting fractions and mixed numbers with unlike denominators 5.NF.A.1
Dividing fractions by fractions 6.NS.A.1
Adding, subtracting, multiplying and dividing decimals 6.NS.B.3
Using substitution to test whether a number makes an equation true 6.EE.B.5
Write two number puzzles on the board. Students answer them in their heads, then explain how they found the number.
Warm-Up Prompt
"(1) I think of a number and add 3.5. The result is 10. What is my number? (2) I think of a number and multiply it by 4. The result is 30. What is my number? How did you undo what I did?"
Collect answers: 6.5 and 7.5. Ask students how they found them. For the first puzzle, they took 3.5 away from 10. For the second, they split 30 into 4 equal parts. Some students may say 13.5 for the first puzzle because they added 3.5 again. Check it together: 13.5 + 3.5 = 17, not 10. Name the idea: subtraction undoes addition and division undoes multiplication. Pairs of operations that undo each other are called inverse operations.
Direct Instruction20 minutes
Key words. A variable is a letter, such as x, that stands for an unknown number. An equation is a statement that two amounts are equal, such as x + 3.5 = 10. A solution is a value of the variable that makes the equation true. In this standard, every number is nonnegative rational: zero or a positive whole number, fraction or decimal. Two forms appear. In x + p = q, a known amount p is added to the unknown. In px = q, the unknown is multiplied by a known number p, called the coefficient. The expression px means p times x.
A tape diagram is a bar split into parts to show how amounts fit together. Diagram 1 shows both forms. For x + p = q, the whole bar is q, and it is made of the unknown part x and the known part p. For px = q, the whole bar q is split into p equal boxes, and each box is x. Work the five examples, writing the equation first, then solving, then checking by substitution (putting the answer back into the equation).
Writing and solving x + p = q with decimals
Maya saved some money. She earned $12.75 walking a neighbor's dog, and now she has $40.00. How much did she have before?
Equation: x + 12.75 = 40, so x = 40 - 12.75 = 27.25. Maya had $27.25. Check: 27.25 + 12.75 = 40
Writing and solving x + p = q with fractions
A bean plant grew 3/4 inch this week. It is now 5 1/2 inches tall. How tall was it last week?
Equation: h + 3/4 = 5 1/2, so h = 5 2/4 - 3/4 = 4 3/4 inches. Check: 4 3/4 + 3/4 = 5 1/2
Writing and solving px = q with a whole-number coefficient
Six movie tickets, all the same price, cost $49.50 in total. What does one ticket cost?
Equation: 6c = 49.50, so c = 49.50 ÷ 6 = 8.25. One ticket costs $8.25. Check: 6 × 8.25 = 49.50
Writing and solving px = q with a fraction coefficient
Two thirds of a hiking trail is 4 miles long. How long is the whole trail?
Equation: x = 1.8 ÷ 0.4 = 4.5 and y = 2.5 - 2.5 = 0. Zero is allowed: it is nonnegative
Choose the form: if a known amount is added to the unknown (earned, grew, more, total of two parts), write x + p = q. If the unknown is repeated p times or is a fraction of the whole (each, per, equal groups, 2/3 of), write px = q.
Undo the operation on both sides: subtract p from both sides of x + p = q, or divide both sides of px = q by p. Diagram 2 shows the first on a number line and the second with a fraction tape diagram.
Check and answer the question: substitute the solution into the equation, then write a sentence with units. Ask whether the answer makes sense: a price should be less than the total, and a whole trail should be longer than two thirds of it.
Guided Practice15 minutes
Pairs work through four problems, one at a time. For each one, they write an equation, draw a tape diagram, solve, and check. After each problem, one pair shows its diagram.
Guided practice problems (answers for the teacher)
Problem
Equation
Solution
1. A backpack with a lunch inside weighs 4.2 kg. The lunch weighs 0.6 kg. How much does the empty backpack weigh?
b + 0.6 = 4.2
b = 3.6 kg
2. Eight packs of stickers, all the same price, cost $14. What does one pack cost?
8p = 14
p = $1.75
3. A recipe uses 4 1/2 cups of flour, which is 1/4 of a bag. How many cups are in the whole bag?
(1/4)f = 4 1/2
f = 18 cups
4. Solve a number puzzle: a number plus 2 1/3 is 7.
n + 2 1/3 = 7
n = 4 2/3
Listen for students who add instead of subtract (writing 4.2 + 0.6) and for students who divide in the wrong order, such as 8 ÷ 14 instead of 14 ÷ 8. Ask: "Should one pack cost more or less than all eight?"
Independent Practice10 minutes
Students solve five problems on their own. For each word problem, they define the variable in words before writing the equation. (1) The rain gauge showed 1.15 inches before a storm and 2.8 inches after it. How much rain fell? (r + 1.15 = 2.8, so r = 1.65 inches.) (2) Solve 5x = 3.5. (x = 0.7.) (3) Jonah has swum 24 laps, which is 3/5 of his goal. What is his goal? ((3/5)g = 24, so g = 40 laps.) (4) Solve x + 5/8 = 5/8. (x = 0.) (5) A family paid $38.40 for 12 gallons of gas. What is the price per gallon? (12g = 38.40, so g = $3.20.)
Closure5 minutes
Exit ticket: (1) A water bottle holds 750 mL. After you drink some, 320 mL is left. Write and solve an equation for the amount you drank. (d + 320 = 750, so d = 430 mL.) (2) Solve 9x = 2.7. (x = 0.3.) (3) A classmate says x + 4 = 10 has the solution 14. Explain the error. (They added 4 instead of subtracting it; 14 + 4 = 18, not 10. The solution is 6.)
Differentiation Strategies
For Struggling Students
Give a tape diagram template with two parts for x + p = q and a row of equal boxes for px = q, and have students write the known numbers on it before writing the equation
Start with whole numbers, then change only one number to a decimal or fraction, so the method stays the same while the numbers get harder
Provide a sentence frame: "The unknown is ___. The known amount is ___. The total is ___."
For Advanced Students
Ask students to write one story for x + 1.25 = 5 and one for 1.25x = 5, and explain why the solutions are different (3.75 and 4)
Give an equation with a fraction coefficient, such as (5/2)x = 10, and ask for two ways to solve it: dividing by 5/2 and multiplying by 2/5
Ask for a story whose equation has the solution 0, and explain what 0 means in the story
Assessment Guidance
What to Look For
Check that students define the variable in words and pick the form that matches the story: a known amount added to the unknown, or the unknown repeated in equal groups. Look for the inverse operation applied to both sides, with the division in the right order (q ÷ p, not p ÷ q). Students should check every solution by substitution and write the answer with units. Watch fraction work closely: dividing by 2/3 should give a larger number than 4, not a smaller one.
02
Classroom Activities
3 Activities
1
Story, Equation and Tape Diagram Match
20 minPairs
Each pair gets 8 story cards, 8 equation cards and 8 tape diagram cards. Pairs match each story to its equation and diagram, sort the matches into the two forms, and solve each equation.
Story Cards (8 cards)
A. Lena has read some pages of a book. After reading 18 more pages, she is on page 112. (p + 18 = 112; p = 94 pages)
B. Four friends split a $22 pizza bill equally. (4s = 22; s = $5.50 each)
C. A puppy gained 1.4 kg this month and now weighs 6.1 kg. (w + 1.4 = 6.1; w = 4.7 kg)
D. Three equal pieces of ribbon make 2 1/4 yards in all. (3r = 2 1/4; r = 3/4 yard)
E. The temperature rose 7.5 °F and is now 68 °F. (t + 7.5 = 68; t = 60.5 °F)
F. Half of the class, 14 students, bring lunch from home. ((1/2)c = 14; c = 28 students)
G. A jar has some marbles. After 35 more are added, it has 200. (m + 35 = 200; m = 165 marbles)
H. Ana earns $30 for 2.5 hours of babysitting. (2.5r = 30; r = $12 per hour)
Procedure
Deal the story cards face up. Match each story with one equation card and one tape diagram card
Sort the matches into two columns: x + p = q and px = q
Solve each equation on a sticky note and check it by substitution
Trade with another pair and check one of their answers
Discussion Questions
Which stories use the form x + p = q? (A, C, E and G) Which use px = q? (B, D, F and H)
Which words in the stories told you to use px = q? (split equally, equal pieces, half of, for 2.5 hours at the same pay)
In story D, why is each piece shorter than 1 yard?
Which solution is the largest, and does it make sense in its story? (G: 165 marbles)
Modification for Distance Learning
Put the cards on a shared slide and have pairs drag each story next to its equation and diagram. Pairs type their solutions in a text box and check another pair's slide.
2
Write the Story, Trade and Solve
15 minGroups of 3-4
Each group gets 4 equation cards. Students write a short real-world story for each equation, then trade stories with another group, which writes the equation back from the story and solves it.
Equation Cards (4 cards)
x + 0.75 = 3 (solution 2.25)
12x = 9 (solution 3/4)
x + 1 1/2 = 4 (solution 2 1/2)
(3/4)x = 15 (solution 20)
Procedure
Each student writes a story for one card and names what x stands for, with units
The group checks each story: does it really lead to that equation, and is the answer realistic?
Trade stories, not equations, with another group. The other group writes the equation from the story, solves it, and checks by substitution
Compare: did the other group write the same equation you started with?
Discussion Questions
The solution of 12x = 9 is less than 1. What kind of story fits that? (For example, 12 equal shares of 9 pounds of clay)
Could the story for x + 0.75 = 3 be about money? About time? What would each answer mean?
Why does (3/4)x = 15 have a solution greater than 15?
Challenge Variation
Groups write one story that needs both forms, for example a total cost split among equal items plus a known extra amount. They explain why it needs two steps and so belongs to grade 7 (going further than this standard).
3
Paper Strip Tape Diagrams
15 minPairs
Pairs build real tape diagrams from paper strips that are 24 cm long. They write an equation for each task, solve it, then measure with a centimeter ruler to check.
Tasks (3 strips)
Strip 1: fold the strip into 4 equal parts. Write an equation for the length of one part and solve it. (4x = 24; x = 6 cm)
Strip 2: fold a new strip into 3 equal parts. Write and solve the equation. (3x = 24; x = 8 cm)
Strip 3: measure 9.5 cm from one end of a new strip and cut there. Write an equation for the length of the other piece. (x + 9.5 = 24; x = 14.5 cm)
Procedure
Label each strip with the equation and the variable
Solve the equation first, then measure the piece with a ruler
If the measurement and the solution differ by more than 0.2 cm, check both the folding and the arithmetic
Discussion Questions
Why does folding into equal parts match the form px = q?
Which strip matches the form x + p = q, and why? (Strip 3: one known piece plus one unknown piece make the whole strip)
Your measurement may be a little off. Should you trust the equation or the ruler more? Why?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Tape Diagrams for x + p = q and px = q
Top: the whole bar is Maya's $40.00, made of the unknown amount x and the $12.75 she earned, so x = 40 - 12.75 = 27.25. Bottom: the whole bar is $49.50, split into 6 equal boxes, one for each ticket, so c = 49.50 ÷ 6 = 8.25. Both bars are drawn to scale.
Diagram 2: A Number Line and a Fraction Tape Diagram
Top: a number line marked in quarter inches. A jump of 3/4 from h lands on 5 1/2, so h is 3/4 to the left of 5 1/2, at 4 3/4. Bottom: the trail is split into 3 equal thirds. Two thirds are 4 miles, so one third is 2 miles and the whole trail is 6 miles, which solves (2/3)d = 4.
04
Homework Assignment
~30 min
6.EE.B.7 Homework: Writing and Solving One-Step Equations
Directions: For every word problem, say what the variable stands for, write an equation of the form x + p = q or px = q, solve it, and check your answer by substitution. Write units with every answer.
Part 1: Writing Equations (Problems 1-2)
A sunflower grew 14.5 cm this week. It is now 92 cm tall. Write an equation for its height last week, draw a tape diagram, and solve.
Seven identical bottles hold 3.5 liters of water in all. Write an equation for the amount in one bottle and solve it. Is your answer more or less than 1 liter? Explain why that makes sense.
Part 2: Solving Equations (Problems 3-4)
Solve each equation and check by substitution: (a) x + 3.08 = 10 (b) y + 2/5 = 1 1/2 (c) z + 4 = 4
Solve each equation and check by substitution: (a) 6x = 4.5 (b) (5/6)m = 10 (c) 0.25k = 7
Part 3: Real-World Problems (Problems 5-6)
A charity walk is 7 1/4 miles long. Dan has walked 2 5/8 miles of it. Write and solve an equation to find how far he still has to walk. Which operation did you use to solve it, and why?
The tomato bed in a school garden covers 3/8 of the garden. The tomato bed is 15 square meters. Write and solve an equation to find the area of the whole garden. A classmate got 5 5/8 square meters. Explain why that answer cannot be right.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Writing Equations
Variable defined in words, and the equation matches the story
Equation matches, but the variable is not defined
Equation does not match the story
Solving
Inverse operation used on both sides, and every solution is correct
Correct method with one arithmetic error
Wrong operation or order of division
Checking
Every solution checked by substitution
Some solutions checked
No checks shown
Making Sense
Answers have units and the reasoning questions are explained
Units or one explanation missing
No units and no explanations
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which equation represents "a number plus 4.6 equals 11"?
Answer: B
"A number plus 4.6" is x + 4.6, and "equals 11" sets it equal to 11, so the equation is x + 4.6 = 11. Choice A multiplies the number by 4.6 instead of adding. Choice C swaps the sum and the added number. Choice D multiplies and also swaps the numbers.
Question 2 of 20 · Multiple Choice
Solve x + 7.25 = 12.
Answer: A
Subtract 7.25 from both sides: x = 12 - 7.25 = 4.75. Check: 4.75 + 7.25 = 12. Choice B adds 7.25 instead of subtracting it. Choice C subtracts the 7 but then adds the 0.25 (12 - 7 + 0.25). Choice D divides 12 by 7.25, which undoes multiplication, not addition.
Question 3 of 20 · Multiple Choice
Solve 8x = 6.
Answer: C
Divide both sides by 8: x = 6 ÷ 8 = 6/8 = 3/4. Check: 8 × 3/4 = 6. Choice A divides in the wrong order (8 ÷ 6). Choice B multiplies by 8 instead of dividing. Choice D subtracts 8 from 6, which undoes addition, not multiplication, and gives a negative number, which this standard does not use.
Question 4 of 20 · Multiple Choice
Which situation can be represented by the equation 5x = 17.50?
Answer: A
5x means 5 equal amounts of x, and they add up to 17.50, so five identical notebooks costing $17.50 in all fits: x = 3.50. Choice B adds $5 to an unknown, which is x + 5 = 17.50. Choice C also describes adding two amounts. Choice D asks for the total, 5 × 17.50, so the unknown is not x in 5x = 17.50.
Question 5 of 20 · Multiple Choice
A tomato seedling grew 1 2/3 inches and is now 4 1/6 inches tall. The equation h + 1 2/3 = 4 1/6 gives its height before. What is h?
Answer: D
Subtract 1 2/3 from both sides. Rename with sixths: 4 1/6 - 1 4/6 = 3 7/6 - 1 4/6 = 2 3/6 = 2 1/2 inches. Check: 2 1/2 + 1 2/3 = 4 1/6. Choice A adds instead of subtracting. Choice B subtracts the whole numbers (4 - 1 = 3) and then the fractions in the wrong order (2/3 - 1/6 = 1/2). Choice C subtracts only the whole number 1 and forgets the 2/3.
Question 6 of 20 · Multiple Choice
Solve (2/5)x = 8.
Answer: B
Divide both sides by 2/5, which is the same as multiplying by 5/2: x = 8 × 5/2 = 20. Check: (2/5) × 20 = 8. Choice A multiplies 8 by 2/5 instead of dividing. Choice C subtracts 2/5 from 8. Choice D divides by 2 and forgets the 5 in the denominator.
Question 7 of 20 · Multiple Choice
Carlos bought 4 identical bags of ice for $11.00 in total. Which equation and solution give the price of one bag, b?
Answer: D
Four equal bags make $11.00, so 4b = 11 and b = 11 ÷ 4 = 2.75. One bag costs $2.75. Choice A adds 4 instead of using 4 equal groups. Choice B has the right equation but multiplies instead of dividing. Choice C swaps the count and the total.
Question 8 of 20 · Multiple Choice
Which value of y makes y + 0.9 = 3.2 true?
Answer: A
Subtract 0.9 from both sides: y = 3.2 - 0.9 = 2.3. Check: 2.3 + 0.9 = 3.2. Choice B adds 0.9. Choice C lines up the digits wrong and subtracts 0.09 instead of 0.9. Choice D divides 3.2 by 0.9.
Question 9 of 20 · Multiple Choice
Solve x + 3/8 = 3/8.
Answer: C
Subtract 3/8 from both sides: x = 3/8 - 3/8 = 0. Check: 0 + 3/8 = 3/8. Zero is a nonnegative number, so it is an allowed solution. Choice A adds 3/8 to 3/8. Choice B divides 3/8 by 3/8, which undoes multiplication, not addition. Choice D multiplies 3/8 by 3/8.
Question 10 of 20 · Multiple Choice
Solve 0.8x = 5.6.
Answer: D
Divide both sides by 0.8: x = 5.6 ÷ 0.8 = 56 ÷ 8 = 7. Check: 0.8 × 7 = 5.6. Choice A misplaces the decimal point in the quotient. Choice B multiplies 5.6 by 0.8 instead of dividing. Choice C subtracts 0.8 from 5.6.
Question 11 of 20 · Multiple Choice
A 2.5-pound bag of apples costs $4.75. The equation 2.5p = 4.75 gives the price per pound, p. What is p?
Answer: A
Divide both sides by 2.5: p = 4.75 ÷ 2.5 = 1.90. The apples cost $1.90 per pound. Check: 2.5 × 1.90 = 4.75. Choice B multiplies 4.75 by 2.5 (11.875, rounded). Choice C subtracts 2.5 from 4.75. Choice D divides 2.5 by 4.75, which gives pounds per dollar, not dollars per pound.
Question 12 of 20 · Multiple Choice
Ana has saved $38.50. A jacket costs $65. The equation m + 38.50 = 65 gives the amount she still needs, m. What is m?
Answer: B
Subtract 38.50 from both sides: m = 65 - 38.50 = 26.50. Ana needs $26.50 more. Check: 26.50 + 38.50 = 65. Choice A adds the two amounts. Choice C divides 65 by 38.50. Choice D subtracts only the $38 and ignores the 50 cents.
Question 13 of 20 · Multiple Choice
To solve 9x = 3.6, what should you do to both sides of the equation?
Answer: C
In 9x, x is multiplied by 9. Division undoes multiplication, so divide both sides by 9: x = 3.6 ÷ 9 = 0.4. Choice A undoes addition, but nothing is added to x. Choice B multiplies again instead of undoing. Choice D divides by the total instead of by the coefficient.
Question 14 of 20 · Multiple Choice
Three quarters of a cup of sugar is 1/3 of the sugar a recipe needs. The equation (1/3)s = 3/4 gives the total sugar, s, in cups. What is s?
Answer: D
Divide both sides by 1/3, which is the same as multiplying by 3: s = 3/4 × 3 = 9/4 = 2 1/4 cups. Check: (1/3) × 9/4 = 3/4. Choice A multiplies 3/4 by 1/3 instead of dividing. Choice B subtracts 1/3 from 3/4. Choice C adds 1/3 to 3/4.
Question 15 of 20 · Short Answer
A bike trail is 12.4 km long. Mei has ridden 7.85 km. Define a variable, write an equation of the form x + p = q, and find how far she has left to ride.
Let x be the distance left in km. x + 7.85 = 12.4. Subtract 7.85 from both sides: x = 12.4 - 7.85 = 4.55 km. Check: 4.55 + 7.85 = 12.4.
Question 16 of 20 · Short Answer
A box of 12 identical muffins costs $15.00. Write an equation of the form px = q and find the cost of one muffin.
Let m be the cost of one muffin in dollars. 12m = 15. Divide both sides by 12: m = 15 ÷ 12 = $1.25. Check: 12 × 1.25 = 15.
Question 17 of 20 · Short Answer
Solve (7/8)x = 21 and check your answer.
Divide both sides by 7/8, which is the same as multiplying by 8/7: x = 21 × 8/7 = 24. Check: (7/8) × 24 = 21. The answer is greater than 21, which makes sense because 7/8 of it is 21.
Question 18 of 20 · Short Answer
Solve x + 5.06 = 9.1 and check your answer.
Subtract 5.06 from both sides. Write 9.1 as 9.10 so the digits line up: x = 9.10 - 5.06 = 4.04. Check: 4.04 + 5.06 = 9.1.
Question 19 of 20 · Short Answer
A student solved 3x = 1.5 and wrote x = 4.5. Explain the error and give the correct solution.
The student multiplied 1.5 by 3 instead of dividing. Multiplication is undone by division, so divide both sides by 3: x = 1.5 ÷ 3 = 0.5. Check: 3 × 0.5 = 1.5. Substituting 4.5 gives 3 × 4.5 = 13.5, not 1.5.
Question 20 of 20 · Short Answer
Tomas walks at a constant speed and covers 1 1/2 miles in 1/2 hour. The equation (1/2)s = 1 1/2 gives his speed, s, in miles per hour. Solve it and explain whether the answer is realistic.
Divide both sides by 1/2, which is the same as multiplying by 2: s = 1 1/2 × 2 = 3 miles per hour. Check: (1/2) × 3 = 1 1/2. This is realistic: 3 miles per hour is a normal walking speed.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.EE.B.7 mean?
6.EE.B.7 means students write and solve one-step equations of two forms: x + p = q and px = q. Here p and q are known numbers and x is the unknown. All of the numbers, including the answer, are zero or positive, and they can be whole numbers, fractions or decimals. Students use these equations to answer real-world and math questions.
Is 6.EE.B.7 a grade 6 or grade 7 standard?
It is a grade 6 standard. In grade 7, students move on to two-step equations such as px + q = r and p(x + q) = r, with negative numbers too (7.EE.B.4). Grade 6 keeps each equation to one operation and uses only nonnegative numbers.
What does "nonnegative rational number" mean?
It means a number that is zero or positive and can be written as a fraction. Whole numbers such as 12, fractions such as 3/4, mixed numbers such as 2 1/2 and decimals such as 0.35 all count. Negative numbers, such as -3, are left out of this standard.
How do you know whether to subtract or divide?
Look at what is done to the unknown. If a number is added to it, as in x + 2.5 = 8, subtract that number from both sides. If the unknown is multiplied by a number, as in 2.5x = 8, divide both sides by that number. The two examples have different solutions: 5.5 and 3.2.
How do you solve an equation with a fraction in front of the variable?
Divide both sides by the fraction, which is the same as multiplying by its reciprocal (the fraction flipped over). For example, (3/4)x = 12 gives x = 12 × 4/3 = 16. A tape diagram helps: if 3 fourths are 12, one fourth is 4, and 4 fourths are 16.
Why should students check their solutions?
Checking catches mistakes before they count. Students substitute the solution back into the equation and see whether both sides are equal. This is the idea of 6.EE.B.5, the standard just before this one: a solution is a value that makes the equation true.
What are common mistakes with one-step equations?
A common mistake is using the same operation instead of the inverse one, such as adding 4 to both sides of x + 4 = 10. Another is dividing in the wrong order: for 5x = 4, the solution is 4 ÷ 5, not 5 ÷ 4. With decimals, many students line up the last digits instead of the decimal points when they subtract.
How do students pick the right form from a word problem?
They ask how the unknown and the known number are related. An amount added to the unknown (more, grew, earned, total of two parts) gives x + p = q. Equal groups, a price for each item or a fraction of a whole gives px = q. Drawing a tape diagram first often makes the choice clear.
Why are equations like 2x + 3 = 11 not part of this standard?
They take two steps to solve, so they belong to grade 7 (7.EE.B.4). In grade 6, students build the meaning of an equation and learn one inverse operation at a time. Teachers can show a two-step equation as a challenge, labeled as going further than this standard.
How does 6.EE.B.7 connect to Algebra I?
It is the first step toward solving linear equations. In Algebra I, students solve equations with variables on both sides and with letters as coefficients (HSA.REI.B.3). The same two ideas are used there: undo operations with inverse operations, and do the same thing to both sides.
07
Related Standards
6 standards
These standards connect to 6.EE.B.7: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.NF.A.1Prerequisite
Add and subtract fractions with unlike denominators, including mixed numbers
Lesson coming soon
6.NS.A.1Prerequisite
Interpret and compute quotients of fractions and solve related word problems
