Unit 7 · Standard 6.EE.7
Equations and Inequalities Problem Solving
Key Vocabulary Level 1 support
Picture first, then the word, then a plain-language meaning. Say each word out loud.
'Twice a number is 18' → 2n = 18 — the equation models the words
Model
To show a real-life situation with an equation or inequality.
x + 5 = 12 — exactly one answer: x = 7
Equation
A math sentence with an equal sign showing both sides are the same.
x + 5 > 12 — many answers: x = 8, 9, 10, ...
Inequality
A math sentence that compares two sides with <, >, ≤, or ≥.
If the problem asks for a number of people and you get x = −3, that is NOT reasonable
Reasonableness
Checking if your answer makes sense.
Key Ideas & Notes
- Detective Santos is closing a complex case.
- She has collected equations and inequalities from different parts of the investigation.
- One clue says: 'The total value of stolen items divided equally among 4 accomplices gave each person $85.' Another says: 'The lookout earned less than $50.' She must model both clues mathematically and check if the answers are reasonable.
- Can you help?
- Read each word problem. Decide if it needs an equation or an inequality. Write the mathematical model, solve it, and check your answer for reasonableness.
Think About It
- Which clue leads to an equation and which leads to an inequality?
- What operation does 'divided equally among 4' suggest?
- What does 'less than $50' tell you about the inequality symbol?
My Notes
Guided Examples
Example 1
A museum has some paintings. After adding 12, they have 45. Which equation models this?
Solution: 'Adding 12' to the original number gives 45: p + 12 = 45. Solve: p = 33.
Answer: A. p + 12 = 45
Example 2
A detective needs more than 8 hours to finish the investigation. She has already worked 3 hours. Which inequality represents the additional hours h she needs?
Solution: 3 hours plus additional hours h must be more than 8: 3 + h > 8. Solve: h > 5.
Answer: A. 3 + h > 8
Example 3
Which model is correct for: 'A number divided by 6 equals 7'?
Solution: Divided by 6 is n / 6. Equals 7 means = 7. So n / 6 = 7.
Answer: A. n / 6 = 7
Write About the Math The Writing Revolution
I can explain my reasoning using the words model, equation, inequality, and reasonableness.
1. Kernel Sentence subject + verb
Model: Equation is a math sentence with an equal sign showing both sides are the same.Ecuación es una oración matemática con un signo igual que muestra que ambos lados son iguales.
Write a kernel sentence about equation. Use a subject and a verb.Escribe una oración base sobre ecuación. Usa un sujeto y un verbo.
2. Sentence Expansion because · but · so
Kernel: Equation matters in mathEcuación importa en matemáticas
Expand the kernel three ways. Add a reason, a contrast, and a result.
Equation matters in math because ___.Ecuación importa en matemáticas porque ___.
Equation matters in math, but ___.Ecuación importa en matemáticas, pero ___.
Equation matters in math, so ___.Ecuación importa en matemáticas, entonces ___.
3. Sentence Types 4 ways to write a math idea
Tell one true fact about equation.Di un hecho verdadero sobre equation.
Equation ___.
Ask a question about equation.Haz una pregunta sobre equation.
How does ___ ?¿Cómo ___ ?
Show excitement about equation.Muestra entusiasmo sobre equation.
Wow, ___ !¡Guau, ___ !
Tell a partner what to do with equation.Dile a un compañero qué hacer con equation.
First, ___ .Primero, ___ .
4. Explain Your Reasoning use a sentence starter
My plan was to ___.Mi plan fue ___.
I knew to ___ because ___.Supe ___ porque ___.
This is like when ___.Esto es como cuando ___.
Try It
Solve on your own. Check the answer key when you are done.
1. A bus can carry at most 48 passengers. There are already 31 on board. Which inequality shows how many more can board?
- 31 + p ≤ 48
- 31 + p > 48
- 31 + p = 48
- p − 31 ≤ 48
2. A student solved 4n = 52 and got n = 13. Is the answer reasonable if n represents the number of notebooks in a box?
- Yes — 13 notebooks per box is reasonable
- No — 13 is too many notebooks
- No — the answer should be 48
- Yes — but only if n is a fraction
Stretch Your Thinking Level 2 enrichment
Challenge task — explain your reasoning in full sentences.
Create two word problems about the same situation: one that requires an equation and one that requires an inequality. Solve both and explain why the models are different.
Sentence starter: Equation problem: ___. Model: ___. Solution: ___. Inequality problem: ___. Model: ___. Solution: ___. The models are different because ___.
Reflect — Exit Ticket
A box of donuts has some donuts. After giving away 7, there are fewer than 5 left. Which inequality represents the starting number of donuts d?
- d − 7 < 5
- d + 7 < 5
- d − 7 > 5
- d − 7 = 5
Answer Key & Teacher Guide
- Try It 1: A. 31 + p ≤ 48 — 'At most 48' means the total must be ≤ 48. So 31 + p ≤ 48. Solve: p ≤ 17.
- Try It 2: A. Yes — 13 notebooks per box is reasonable — 4 × 13 = 52 ✓. 13 notebooks per box is a reasonable whole number answer.
- Exit Ticket: A. d − 7 < 5 — Starting with d donuts and giving away 7 leaves fewer than 5: d − 7 < 5. Solve: d < 12. ✓
Writing (TWR) — what to look for
- Kernel sentence: A complete sentence needs a subject and a verb. Example: Equation is a math sentence with an equal sign showing both sides are the same.
- Expansion: because gives a reason, but shows a contrast or exception, so shows a result. Answers vary; each must keep the kernel idea and add the correct kind of detail.
- Sentence types: Statement ends with a period, question with "?", exclamation with "!", and a command starts with an action verb (a "bossy" verb).