6.EE.B.5Common CoreMathExpressions and EquationsGrade 6
6.EE.B.5: Solving Equations and Inequalities by Testing Values
In plain English: 6.EE.B.5 is the Common Core grade 6 math standard that asks students to see solving an equation or inequality as answering a question: which values from a given set, if any, make it true? Students substitute each value, decide true or false, and learn that a set can hold one solution, several or none. It prepares for solving one-step equations in 6.EE.B.7.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
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.5 · Official standard
Students learn that solving an equation or inequality means answering a question: which values make it true? An equation is a statement that two expressions are equal, such as x + 8 = 15. An inequality is a statement that compares two expressions with a symbol such as < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). A variable is a letter that stands for a number, and a solution is a value of the variable that makes the statement true.
In this lesson, the possible values always come from a specified set: a short list of numbers written inside braces, such as {5, 6, 7, 8}. Students use substitution (replacing the variable with a number) to test each value, compute both sides, and decide whether the statement is true or false. They see the three cases the standard's words "if any" point to: a set can contain exactly one solution, several solutions (common with inequalities), or no solution at all. All numbers are whole numbers, simple fractions or decimals, and students test values instead of using formal solving steps, which come next in 6.EE.B.7.
Learning Objectives
By the end of this lesson, students will be able to:
Explain that solving an equation or inequality means finding which values, if any, make it true
Use substitution to decide whether a given number makes an equation true
Use substitution to decide whether a given number makes an inequality true, including values that make both sides equal
Find every solution in a specified set, and recognize when a set contains no solution
Decide whether solutions found by substitution make sense in a real-world situation
Prior Knowledge Required
Students should already be comfortable with:
Evaluating expressions with parentheses and the order of operations 5.OA.A.1
Adding, subtracting, multiplying and dividing decimals 6.NS.B.3
Reading inequality statements as positions on a number line 6.NS.C.7a
Writing expressions with letters standing for numbers 6.EE.A.2a
Evaluating expressions at specific values of their variables 6.EE.A.2c
Write four numbers on the board: 4, 7, 9 and 12. Read the riddle aloud and give students one minute to think.
Warm-Up Prompt
"I am thinking of a number from the list 4, 7, 9, 12. When I add 6 to my number, I get 15. Which number is it? Now suppose the sum were 11 instead. Which number from the list works then?"
Collect answers and ask how students found them. Most will try the numbers one at a time: 4 + 6 = 10, 7 + 6 = 13, 9 + 6 = 15. So the first riddle's number is 9. For the second riddle, no number on the list works: the sums are 10, 13, 15 and 18, and none of them is 11. Tell students that trying each number is a real mathematical method called substitution, and that "none of them works" is a correct and complete answer when the list is fixed ahead of time.
Direct Instruction15-20 minutes
Solving as answering a question. Write n + 6 = 15 and say: an equation says two amounts are equal. The letter n is a variable, a symbol that stands for a number. To solve the equation is to answer the question "which values of n make this true?" When the values must come from a specified set, such as {4, 7, 9, 12}, we test each one. A value that makes the statement true is a solution. Show the four steps of substitution:
Copy the equation or inequality.
Substitute: replace the variable with the value you are testing. Write the value in parentheses when it is multiplied, for example 3y with y = 5 becomes 3(5).
Compute each side, following the order of operations.
Decide: is the statement true or false? Record the value as a solution only if it is true.
Equation with one solution in the set
Which value from the set {5, 6, 7, 8} makes x + 8 = 15 true?
Which values from the set {5, 6, 7}, if any, make 3y = 20 true?
Equation: 3(5) = 15, 3(6) = 18 and 3(7) = 21 are all false, so no value in the set is a solution.
Inequality with several solutions
Which values from the set {2, 4, 6, 8, 10} make 5w > 28 true?
Equation: 5(2) = 10 and 5(4) = 20 are not greater than 28; 5(6) = 30, 5(8) = 40 and 5(10) = 50 are. The solutions are 6, 8 and 10.
Inequality in a real-world situation
School play tickets cost $6 each, and Maya has $20. Which numbers of tickets from the set {0, 1, 2, 3, 4} make 6t ≤ 20 true?
Equation: 6(0) = 0, 6(1) = 6, 6(2) = 12 and 6(3) = 18 are all at most 20, but 6(4) = 24 is not. Maya can buy 0, 1, 2 or 3 tickets.
Fractions in the set
Which value from the set {1/2, 3/4, 1, 1 1/2} makes 4k = 3 true?
Equation: 4 × 1/2 = 2, 4 × 3/4 = 3 (true), 4 × 1 = 4 and 4 × 1 1/2 = 6. The solution is 3/4.
Three possible answers. The standard asks which values make the statement true "if any". Point to the first three examples: a set can hold exactly one solution, no solution, or several solutions. Diagram 1 shows the third example as a table, one row per value. Inequalities often have several solutions, because many numbers can be greater than 28.
Inequality symbols. Read each symbol aloud: < is "is less than", > is "is greater than", ≤ is "is less than or equal to" and ≥ is "is greater than or equal to". Test a value that makes both sides equal: 12 ≥ 12 is true, but 12 > 12 is false. Diagram 2 shows the ticket example on a number line, with filled green dots for solutions and open red dots for values that do not work.
Guided Practice15 minutes
Pairs test every value in each set on mini whiteboards, writing one substitution per line. After each problem, one pair shows its work and the class agrees on the solutions.
Guided practice: find every solution in the set
Problem
Statement
Specified set
a
m ÷ 3 = 12
{4, 9, 36, 39}
b
y - 4 < 6
{8, 9, 10, 11, 12}
c
2.5 + d = 6
{3, 3.5, 4}
d
2n + 1 ≥ 11
{3, 4, 5, 6}
Answers: (a) 36, because 36 ÷ 3 = 12; a student who picks 4 has divided 12 by 3 instead of testing values. (b) 8 and 9, because 8 - 4 = 4 and 9 - 4 = 5 are less than 6, but 10 - 4 = 6 is not. (c) 3.5, because 2.5 + 3.5 = 6. (d) 5 and 6, because 2(5) + 1 = 11 and 11 ≥ 11 is true, and 2(6) + 1 = 13. Listen for students who drop the value that makes both sides equal in (d), or who keep it in (b).
Independent Practice10-15 minutes
Students work alone and show every substitution. (1) Which value from {10, 11, 12, 13} makes 9 + p = 21 true? (12.) (2) Which values from {5, 6, 7}, if any, make 7r = 40 true? (None: 35, 42 and 49.) (3) Which values from {6, 8, 10, 12} make h ÷ 2 > 4 true? (10 and 12; 8 ÷ 2 = 4 is not greater than 4.) (4) Each bag of apples weighs 3 pounds, and a shelf holds at most 25 pounds. Which numbers of bags from {6, 7, 8, 9} make 3b ≤ 25 true? (6, 7 and 8; 3(9) = 27 is too heavy.) (5) Which value from {4, 6, 8} makes 15 - q = 9 true? (6.)
Closure5 minutes
Exit ticket: (1) Does x = 4 make 5 + 3x = 17 true? Show the substitution. (Yes: 5 + 3(4) = 5 + 12 = 17.) (2) Which values from {1, 2, 3, 4} make 10 - z ≥ 7 true? (1, 2 and 3; 10 - 3 = 7 counts because of the "or equal to" part.) (3) In one sentence, what does it mean to solve an equation or inequality? (Find which values, if any, make it true.)
Differentiation Strategies
For Struggling Students
Give a blank testing table with the columns "value", "substitute", "compute" and "true or false?", like Diagram 1, and fill in the first row together
Start with sets of three whole numbers and equations with one operation, then add inequalities once students test values reliably
Post a symbol card that reads each inequality symbol in words, with a number line example under each one
For Advanced Students
Ask students to write a set of five numbers that contains exactly two solutions of 3a > 14, and another set that contains none
Ask which whole numbers from 0 to 20 make both 2c > 9 and c + 3 < 12 true, and how they can test fewer than 21 values
Ask students to explain why the equation k + 4 = 10 can have only one solution, while k + 4 > 10 has many
Assessment Guidance
What to Look For
Check that students write each substitution on its own line and compute both sides before deciding true or false. Watch for three errors: reading 3y with y = 5 as the number 35 instead of 3 × 5, dropping or keeping the value that makes both sides equal for the wrong symbol (≤ and ≥ include it, < and > do not), and stopping after the first solution when an inequality has several. Students should be comfortable answering "no value in the set" and should explain it by showing every substitution.
02
Classroom Activities
3 Activities
1
True or False Card Sort
15 minPairs
Each pair gets 8 statement cards, each printed with its own set of four value cards. Pairs substitute each value and sort the value cards into two columns on the desk: "makes it true" and "makes it false". Every pair records the solutions for each card on a sticky note.
Statement Cards (8 cards)
Card A: n + 13 = 20, values {6, 7, 8, 9}
Card B: 6n = 54, values {7, 8, 9, 10}
Card C: n - 5 = 12, values {15, 16, 18, 19}
Card D: n ÷ 4 = 2.5, values {8, 9, 10, 12}
Card E: 2n < 13, values {5, 6, 7, 8}
Card F: n + 1.5 ≥ 4, values {2, 2.5, 3, 4}
Card G: 4n > 30, values {7, 7.5, 8, 9}
Card H: 18 = 3n, values {5, 6, 7, 8}
Answer Key
A: 7. B: 9. C: no value in the set (the solution, 17, is missing). D: 10. E: 5 and 6. F: 2.5, 3 and 4. G: 8 and 9. H: 6.
Discussion Questions
Which card has no solution in its set? (Card C) How did you know you were finished?
Which cards have more than one solution? (Cards E, F and G, the three inequalities) Why do inequalities often have more than one?
On Card G, 4 × 7.5 = 30. Why is 7.5 not a solution? (30 > 30 is false)
On Card F, why is 2.5 a solution? (2.5 + 1.5 = 4, and 4 ≥ 4 is true)
Modification for Distance Learning
Put the cards on a shared slide with two drop zones labeled "true" and "false". Pairs drag the value cards and type one substitution next to each card.
2
Stand Up If You Make It True
15 minWhole class
Hand out 13 number cards, 0 through 12, to 13 students (the rest of the class checks the work). The cards form the specified set {0, 1, 2, ..., 12}. The teacher reads a statement about k. Each card holder substitutes their number, and stands if it makes the statement true. The checkers confirm or challenge each person standing.
Statements to Read (6 rounds)
Round 1: k + 4 = 10 (6 stands)
Round 2: 3k > 20 (7, 8, 9, 10, 11 and 12 stand)
Round 3: k ÷ 2 = 5.5 (11 stands)
Round 4: 2k + 1 = 8 (nobody stands)
Round 5: 12 - k ≥ 9 (0, 1, 2 and 3 stand)
Round 6: k + k = 16 (8 stands)
Procedure
Before each round, card holders write their substitution on a mini whiteboard
After everyone stands or sits, the checkers test one standing and one sitting student out loud
Swap card holders and checkers after Round 3
Discussion Questions
In which round did nobody stand? (Round 4) Does that mean the equation is wrong?
Which round had the most students standing? (Round 2, with six students)
In Round 5, why did the student holding 3 stand even though 12 - 3 is not greater than 9?
Modification for Distance Learning
Assign each student a number and have them unmute, raise a virtual hand or type "true" in the chat when their number is a solution.
3
Does It Make Sense? Real-World Stations
20 minPairs
Pairs rotate through 4 stations. Each station card describes a situation, gives an equation or inequality and a specified set, and asks which values are solutions and what they mean in the situation.
Station Cards (4 cards)
Station 1, pencils: a box holds 12 pencils. Jin needs 60 pencils for the class. Which number of boxes from {4, 5, 6} makes 12p = 60 true? (5)
Station 2, fair ride: riders must be at least 48 inches tall. Four friends are 46, 48, 51 and 53 inches tall. Which heights make h ≥ 48 true? (48, 51 and 53)
Station 3, water cooler: each player drinks 2 cups from a 30-cup cooler during a game. Which team sizes from {12, 14, 16, 18} make 2n ≤ 30 true? (12 and 14)
Station 4, bake sale: cookies sell for $0.50 each, and the class wants to earn $40. Which numbers of cookies from {60, 75, 80, 90} make 0.50c = 40 true? (80)
Procedure
At each station, test every value in the set and write the substitutions on the recording sheet
Write one sentence that explains each solution in words, for example "48 inches is tall enough to ride"
Spend about 4 minutes at each station, then rotate
Discussion Questions
At Station 2, the friend who is exactly 48 inches tall can ride. Which part of the symbol ≥ shows that?
At Station 3, why do all the sets list only whole numbers?
Which station had exactly one solution in its set? (Stations 1 and 4)
Challenge Variation
Pairs write their own station card with a situation, a statement and a set of four values that contains exactly two solutions, then trade with another pair.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Testing Each Value in a Set
A testing table for the inequality 5w > 28 and the set {2, 4, 6, 8, 10}. Each row substitutes one value, computes 5w and compares the result with 28. Three values make the inequality true, so the set contains three solutions: 6, 8 and 10.
Diagram 2: Showing Solutions on a Number Line
A number line for the ticket example, 6t ≤ 20, with the set {0, 1, 2, 3, 4}. Filled green dots mark the values that make the inequality true (0, 1, 2 and 3 tickets). The open red dot at 4 marks a value that makes it false, because 4 tickets cost $24.
04
Homework Assignment
~30 min
6.EE.B.5 Homework: Which Values Make It True?
Directions: For every problem, test each value in the set by substitution. Write one line per value, compute both sides, and label it true or false. Then list the solutions, or write "no value in the set" if none work.
Part 1: Equations (Problems 1-3)
Which value from the set {6, 7, 8, 9} makes 8a = 56 true? Show each substitution.
Liam buys a sandwich for $4.75 and a drink, and he spends $6.00 in all. Which drink price d from the set {$1.00, $1.25, $1.50, $1.75} makes 4.75 + d = 6.00 true?
Which values from the set {10, 12, 14, 16}, if any, make g - 9 = 4 true? Explain what your answer means.
Part 2: Inequalities (Problems 4-6)
Which values from the set {3, 4, 5, 6, 7} make 9 + k > 14 true?
A school club has $50 to spend on pizzas that cost $11 each. Which numbers of pizzas from the set {2, 3, 4, 5} make 11p ≤ 50 true? What is the greatest number of pizzas the club can buy?
Ana says that 4 is the only solution of 5x ≥ 20 in the set {2, 4, 6, 8}. Is she right? Test every value and explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Substitution
Every value in every set is substituted and computed correctly
One or two values skipped or computed incorrectly
Values guessed without substitution
True or False
Each substitution labeled correctly, including values that make both sides equal
One symbol misread (for example ≤ treated as <)
Labels missing or mostly incorrect
Solutions
All solutions listed, and "no value in the set" given where no value works
Some solutions missing
Solutions missing or incorrect
Explanation
Problems 3, 5 and 6 explained in words that fit the situation
Explanation given for only some problems
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order and test values by substitution. 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
What does it mean to solve an equation such as n + 5 = 12?
Answer: B
Solving answers a question: which values of n make n + 5 = 12 true? Here the answer is 7, because 7 + 5 = 12. Choice A treats the numbers as something to combine, which ignores the variable. Choice C describes rewriting, not solving. Choice D substitutes a number from the equation itself instead of asking which value makes it true.
Question 2 of 20 · Multiple Choice
Does n = 6 make 4n - 5 = 19 true?
Answer: A
Substitute 6 for n: 4(6) - 5 = 24 - 5 = 19, and 19 = 19 is true. Choice B reads 4n as the two-digit number 46 instead of 4 × 6. Choice C forgets to subtract 5 after multiplying. Choice D subtracts before multiplying, which breaks the order of operations.
Question 3 of 20 · Multiple Choice
Which value from the set {6, 7, 36, 48} makes 6c = 42 true?
Answer: B
Test the values: 6(6) = 36, 6(7) = 42, 6(36) = 216 and 6(48) = 288. Only 7 makes 6c equal 42. Choice C, 36, is what you get by subtracting 6 from 42 instead of testing values, and choice D, 48, comes from adding 6 to 42. Choice A is close, but 6(6) = 36, not 42.
Question 4 of 20 · Multiple Choice
Which values from the set {4, 5, 6}, if any, make 8m = 44 true?
Answer: D
Test each value: 8(4) = 32, 8(5) = 40 and 8(6) = 48. None of them equals 44, so no value in the set is a solution. (The value that works, 5.5, is not in the set: 8(5.5) = 44.) Choice A drops the remainder of 44 ÷ 8 and keeps 5. Choice B rounds 5.5 up to 6. Choice C picks the two values that 5.5 falls between.
Question 5 of 20 · Multiple Choice
Which values from the set {3, 5, 7, 9} make 4q > 20 true?
Answer: C
4(3) = 12 and 4(5) = 20 are not greater than 20, while 4(7) = 28 and 4(9) = 36 are. The solutions are 7 and 9. Choice A lists the values that make 4q ≤ 20 true, so it reverses the symbol. Choice B treats > as ≥ and keeps 5, but 20 > 20 is false. Choice D stops too early and misses 7.
Question 6 of 20 · Multiple Choice
A backpack can safely hold at most 12 pounds of books. Each textbook weighs 3 pounds. Which numbers of textbooks b from the set {2, 3, 4, 5} make 3b ≤ 12 true?
Answer: A
3(2) = 6, 3(3) = 9 and 3(4) = 12 are all at most 12, but 3(5) = 15 is too heavy. The solutions are 2, 3 and 4. Choice B treats ≤ as < and drops 4, even though 12 ≤ 12 is true. Choice C reverses the symbol and finds the values that make 3b ≥ 12 true. Choice D keeps only the value that makes both sides equal.
Question 7 of 20 · Multiple Choice
How many values from the set {1, 2, 3, 4, 5, 6} make y + 2 ≥ 5 true?
Answer: C
Substitute: 1 + 2 = 3 and 2 + 2 = 4 are less than 5, while 3 + 2 = 5, 4 + 2 = 6, 5 + 2 = 7 and 6 + 2 = 8 are at least 5. So 3, 4, 5 and 6 work: 4 values. Choice A counts only 3, the value that makes both sides equal. Choice B treats ≥ as > and leaves out 3. Choice D counts every value in the set without testing.
Question 8 of 20 · Multiple Choice
Which value from the set {1/2, 2, 3, 18} makes 6 × f = 3 true?
Answer: A
6 × 1/2 = 3, so 1/2 is the solution. Choice B divides 6 ÷ 3 = 2, which puts the numbers in the wrong order: 6 × 2 = 12. Choice C copies the right side of the equation: 6 × 3 = 18. Choice D multiplies 6 × 3 instead of testing values.
Question 9 of 20 · Multiple Choice
Which value from the set {2.4, 3.6, 4.4, 8.4} makes w + 2.4 = 6 true?
Answer: B
3.6 + 2.4 = 6.0, so 3.6 is the solution. Choice C comes from subtracting only the whole numbers (6 - 2 = 4) and keeping the .4, but 4.4 + 2.4 = 6.8. Choice D adds 6 + 2.4 instead of testing values. Choice A copies the number already in the equation: 2.4 + 2.4 = 4.8.
Question 10 of 20 · Multiple Choice
Mia tested every value in the set {10, 20, 30} for the inequality 5 + p < 10 and found that none of them made it true. What can she conclude?
Answer: A
5 + 10 = 15, 5 + 20 = 25 and 5 + 30 = 35, and none of them is less than 10. When no value in the set works, the correct answer is that the set contains no solution. This is the "if any" part of the standard. Choice B is false: a set can contain no solution. Choice C confuses the values with the right side of the inequality. Choice D picks a value without testing it.
Question 11 of 20 · Multiple Choice
Tickets to a science museum cost $9 each. A group paid $63 in all. Which number of tickets t from the set {4, 5, 6, 7} makes 9t = 63 true?
Answer: D
9(7) = 63, so the group bought 7 tickets. The other values give 9(4) = 36, 9(5) = 45 and 9(6) = 54. A student who chooses 6 may have stopped at 54 because it is the closest total below 63, without checking the next value.
Question 12 of 20 · Multiple Choice
Which value from the set {0.5, 1, 1.5, 2} does NOT make 3.5 + r ≤ 5 true?
Answer: D
3.5 + 2 = 5.5, which is greater than 5, so 2 is the only value that makes the inequality false. The other sums are 4, 4.5 and 5, and each is at most 5. Choice C is a trap: 3.5 + 1.5 = 5, and 5 ≤ 5 is true, so 1.5 is a solution. A student who treats ≤ as < would wrongly choose it.
Question 13 of 20 · Multiple Choice
Kai has 28 points in a reading game and earns p more points. He wins a badge when 28 + p > 40. Which values from the set {10, 12, 14} win the badge?
Answer: C
28 + 10 = 38 and 28 + 12 = 40 are not greater than 40, but 28 + 14 = 42 is. Only 14 wins the badge. Choice B treats > as ≥: 12 points brings Kai to exactly 40, which is not more than 40. Choice A accepts every value without testing. Choice D forgets to test 14 after finding that 12 does not work.
Question 14 of 20 · Multiple Choice
Which value from the set {3, 4, 5, 6} makes 20 ÷ n = 5 true?
Answer: B
20 ÷ 4 = 5, so 4 is the solution. Choice C copies the right side of the equation, but 20 ÷ 5 = 4, not 5. The other values do not divide 20 into 5: 20 ÷ 3 = 6 2/3 and 20 ÷ 6 = 3 1/3.
Question 15 of 20 · Short Answer
Test each value in the set {12, 14, 16, 18} for the equation k ÷ 2 = 8. Which value is the solution?
12 ÷ 2 = 6, 14 ÷ 2 = 7, 16 ÷ 2 = 8 and 18 ÷ 2 = 9. The solution is 16, the only value that makes both sides equal.
Question 16 of 20 · Short Answer
Which values from the set {0, 1, 2, 3, 4, 5}, if any, make 4 + 2h < 10 true? Show your substitutions.
4 + 2(0) = 4, 4 + 2(1) = 6 and 4 + 2(2) = 8 are less than 10. 4 + 2(3) = 10 is not less than 10, and 4 + 2(4) = 12 and 4 + 2(5) = 14 are greater. The solutions are 0, 1 and 2. A common error is to include 3, but 10 < 10 is false.
Question 17 of 20 · Short Answer
A phone plan includes 5 GB of data each week. Sam uses 0.75 GB each day. Which numbers of days d from the set {5, 6, 7, 8} make 0.75d ≤ 5 true? What does your answer mean?
0.75(5) = 3.75 and 0.75(6) = 4.5 are at most 5, but 0.75(7) = 5.25 and 0.75(8) = 6 are more than 5. The solutions are 5 and 6. At this rate, Sam's weekly data lasts 5 or 6 days, but not a full 7 days.
Question 18 of 20 · Short Answer
Write a set of four whole numbers that contains no solution of x + 6 = 10. Explain how you know.
Any set without 4 works, for example {1, 2, 3, 5}. The only number that makes x + 6 = 10 true is 4, because 4 + 6 = 10. Substituting 1, 2, 3 or 5 gives 7, 8, 9 or 11, so none of them is a solution, and the set contains no solution.
Question 19 of 20 · Short Answer
Does y = 2.5 make 4y = 10 true? Does it make 4y > 10 true? Explain.
4(2.5) = 10, so y = 2.5 makes 4y = 10 true. It does not make 4y > 10 true, because 10 > 10 is false: 10 is equal to 10, not greater than it. It would make 4y ≥ 10 true.
Question 20 of 20 · Short Answer
Leo has 3 hours of free time on Saturday afternoon. Soccer practice takes 1.25 hours. Which amounts of time t from the set {1.5, 1.75, 2} for a trip to the library make 1.25 + t ≤ 3 true?
1.25 + 1.5 = 2.75 and 1.25 + 1.75 = 3 are at most 3, but 1.25 + 2 = 3.25 is more than 3. The solutions are 1.5 and 1.75 hours. The value 1.75 counts because 3 ≤ 3 is true.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.EE.B.5 mean?
6.EE.B.5 means students understand solving as answering the question "which values make this equation or inequality true?" Students are given a specified set of numbers, such as {3, 5, 7, 9}, and use substitution to test each one. A value that makes the statement true is a solution. The set may contain one solution, several, or none.
Is 6.EE.B.5 about solving equations with inverse operations?
No, not yet. In 6.EE.B.5, students solve by testing values from a set, which builds the meaning of a solution. Writing and solving equations such as x + p = q and px = q with inverse operations comes next in 6.EE.B.7. Many teachers use substitution again later as a way to check answers.
What is a specified set in 6.EE.B.5?
A specified set is the list of values students are allowed to test. It is usually written in braces, such as {0, 1, 2, 3, 4}, and it can include whole numbers, fractions or decimals. In real-world problems the set often comes from the situation, such as the number of tickets someone might buy.
What does substitution mean in math?
Substitution means replacing a variable with a number and then computing. To test whether 5 makes 2b + 3 = 13 true, write 2(5) + 3 = 10 + 3 = 13. Both sides are 13, so 5 is a solution. Writing the number in parentheses keeps students from reading 2b as a two-digit number.
Can an equation or inequality have no solution in a set?
Yes. If no value in the set makes the statement true, the answer is "no value in the set". For example, no value in {1, 2, 3} makes a + 10 = 15 true, because the only solution, 5, is not in the set. The words "if any" in the standard point to this case.
Why can an inequality have more than one solution?
An inequality compares amounts instead of saying they are equal, so many numbers can make it true. For c + 2 > 6 with the set {1, 4, 6, 10}, the values 6 and 10 both work, but 4 does not, because 4 + 2 = 6 is not greater than 6. Students should test every value in the set and not stop at the first solution.
What are common mistakes with 6.EE.B.5?
A common mistake is misreading ≤ or ≥: students forget that a value making both sides equal is a solution, or they include it for < and >. Other errors include reading 3y with y = 5 as 35, stopping after the first solution, and guessing instead of showing each substitution. A testing table with one row per value prevents most of these.
How is 6.EE.B.5 tested?
Test items usually give an equation or inequality and a set, then ask which values make it true, sometimes with more than one correct answer to select. Other items ask whether a single number is a solution, or ask students to explain what a solution means in a situation. Students should show or be ready to explain their substitutions.
How does 6.EE.B.5 connect to later math?
It builds the idea that a solution is a value that makes a statement true. That idea runs through 6.EE.B.7 and 6.EE.B.8 in grade 6, multi-step equations and inequalities in 7.EE.B.4, and solving equations in Algebra I (HSA.REI.B.3). Substituting to check an answer stays useful in every later course.
How can parents help with 6.EE.B.5 at home?
Turn everyday limits into quick tests. If a car holds at most 5 passengers, ask whether 4, 5 or 6 friends fit, and why 5 counts. At the store, give a budget and ask which numbers of items stay within it. Ask your child to say each test out loud, such as "3 times $4 is $12, and $12 is less than $15, so 3 works".
07
Related Standards
6 standards
These standards connect to 6.EE.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.EE.A.2Prerequisite
Write, read and evaluate expressions in which letters stand for numbers
