Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.8

Lesson 8.1Understand Equations and Their Solutions

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can write an equation for a real situation and check whether a given value is a solution by substituting it.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
EquationSpanish: Ecuación A math sentence with an equal sign showing both sides are the same. x + 3 = 10
VariableSpanish: Variable A letter that stands for a number that is unknown or can change. the x in x + 3 = 10
Equal signSpanish: Signo igual The = sign, showing both sides are the same. =
ExpressionSpanish: Expresión A math phrase with numbers and letters, but no equal sign. 2x + 5 is an expression; it becomes an equation when you write 2x + 5 = 15
ConstantSpanish: Constante A fixed number in an expression or equation that does not change. In x + 7 = 12, the numbers 7 and 12 are constants; only x can change.
UnknownSpanish: Incógnita The value you are trying to find, usually shown with a variable. In n + 5 = 12, n is the unknown; you solve to find n = 7.

How it worksWorked examplechecking a solution by substituting

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Jade wrote the equation m + 23 = 41 for a number puzzle. Is m = 18 a solution?

  1. Substitute: replace m with 18. The equation becomes 18 + 23 = 41.
  2. Do the arithmetic on the left side: 18 + 23 makes 41.
  3. Compare the two sides: 41 and 41. Both sides balance.
  4. The equation is true, so m = 18 is a solution.

Answer: Yes — m = 18 makes both sides equal 41, so it is a solution.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

A club wrote t + 14 = 31 for its total members. Is t = 17 a solution?

  1. Substitute: replace t with .
  2. Left side: + 14 =
  3. Compare the sides: left , right . Do they match?
  4. So t = 17 (is / is not) a solution.
Answer:Is t = 17 a solution? yes / no

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • I substituted for the variable and got .
  • The value (is / is not) a solution because .
  • An equation is true when both sides .
  • The variable stands for in this problem.

Word bank equationvariablesolutionsubstitutetruefalseequal signboth sidesbalancecheck

Watch outA common mistake

A common mistake in Write Equations is writing subtraction and division clues in the wrong order — putting the number from the clue first instead of the variable first. For 'a number decreased by 8 equals 5,' students often write 8 − n = 5 instead of the correct n − 8 = 5; for 'a number divided by 4 equals 6,' they write 4 ÷ n (or 4n) instead of n / 4 = 6. Always start with the variable, then attach the operation in the order the words describe it, and check by substituting your answer back in.

Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.8

Lesson 8.1Understand Equations and Their Solutions

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can write an equation for a real situation and check whether a given value is a solution by substituting it.

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Sort each phrase into the correct operation.

    Groups: Addition (+) Subtraction (−) Multiplication (×) Division (÷)
    • more than
    • increased by
    • sum of
    • less than
    • decreased by
    • fewer than
    • times
    • product of
    • twice
    • divided by
    • split equally
    • quotient of

    Hint Read each phrase and think: does it mean combining, taking away, making groups, or sharing equally?

Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    Which equation represents: 'A number plus 12 equals 30'?

    1. An + 12 = 30
    2. Bn − 12 = 30
    3. C12n = 30
    4. Dn / 12 = 30

    Hint Find the operation word in the sentence. What does 'plus' tell you to do with the number and 12?

    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    Which equation represents: 'Seven less than a number is 18'?

    1. An − 7 = 18
    2. B7 − n = 18
    3. Cn + 7 = 18
    4. Dn / 7 = 18

    Hint 'Seven less than a number' is tricky — the number comes FIRST, and then 7 is taken away from it.

    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    Which equation represents: 'Three times a number equals 21'?

    1. A3n = 21
    2. Bn + 3 = 21
    3. Cn − 3 = 21
    4. Dn / 3 = 21

    Hint Circle the operation word: 'times'. What operation does 'times' always signal?

    Show your work
  4. 5Circle the letter of the best answer. Show how you know.

    Is n = 3 a solution of n + 5 = 9?

    1. AYes, because 3 is smaller than 9
    2. BNo, because 3 + 5 = 8, which does not equal 9
    3. CYes, because 3 + 5 is close to 9
    4. DYou cannot tell without solving first

    Hint Replace n with 3.

    Show your work
  5. 6Circle the letter of the best answer. Show how you know.

    Which equation represents: 'A number divided by 4 equals 8'?

    1. An / 4 = 8
    2. B4n = 8
    3. Cn − 4 = 8
    4. Dn + 4 = 8

    Hint The phrase 'divided by 4' names the operation. Which symbol shows division?

    Show your work

Explain your thinkingHow did you decide what expression to put on each side of the balance?

Sentence starter I put ___ on the left side because ___, and ___ on the right side because ___. The equal sign means ___.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.8

Lesson 8.1Understand Equations and Their Solutions

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can write an equation for a real situation and check whether a given value is a solution by substituting it.

Part 1Understand the ideamodels and meaning
  1. 1Write the letter of the matching item on each line.

    Match each situation to its equation

    1. Item 1
    2. Item 2
    3. Item 3
    4. Item 4
    • A
    • B
    • C
    • D
  2. 2Complete the table. Show how you found each value.

    For each situation, identify what the variable represents and write the equation.

    SituationVariable RepresentsEquation
    A box holds some files. After removing 5, there are 13 left.
    A detective split 48 photos equally into folders. Each folder has 6 photos.
    Show your work
  3. 3Decide whether each pair balances. Check Yes or No, then explain.

    Build the equation 2n + 3 = 11 on the balance scale and find n.

    Left sideRight sideBalanced?
    Explain your choice
Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    A detective found some fingerprints on Monday and 14 more on Tuesday, for a total of 30. Which equation represents this?

    1. Af + 14 = 30
    2. Bf − 14 = 30
    3. C14f = 30
    4. Df / 14 = 30
    Show your work
Part 3Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Crack the Equation Error

    1. 1ClueA number of case files divided by 3 equals 6.
    2. 2Choose a variableLet n be the number of case files.
    3. 3Write the equation3n = 6

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingHow did you decide what expression to put on each side of the balance?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.8

Lesson 8.1Understand Equations and Their Solutions

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can write an equation for a real situation and check whether a given value is a solution by substituting it.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    For each equation, explain why one situation matches and the other does not.

    EquationSituation That MATCHESSituation That Does NOT Match
    5n = 35Five identical bags hold 35 clues total.
    n − 8 = 12After giving away 8 items, a detective has 12 left.
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Equation Error

    1. 1SituationTwice a number is 16
    2. 2Identify operationTwice means addition
    3. 3Write equationn + 2 = 16

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 3Answer in complete sentences.

    Write a real-world situation that can be modeled by the equation n / 5 = 9. Explain what n represents and verify your equation makes sense.

    Write your answer

Explain your thinkingHow did you decide what expression to put on each side of the balance?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help