Lesson 8.78.7 Small Group · Group 2
Start hereWords, worked example, and sentence starters
Learning target I can model and solve real-world problems using equations and inequalities, explain what each part stands for, and build one for a situation I have not seen before.
Why This Matters This is your challenge small group for Lesson 8.7. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| ModelSpanish: Modelar | To show a real-life situation with an equation or inequality. | 'Twice a number is 18' → 2n = 18 — the equation models the words |
| EquationSpanish: Ecuación | A math sentence with an equal sign showing both sides are the same. | x + 8 = 15 — exactly one answer: x = 7 |
| InequalitySpanish: Desigualdad | A math sentence that compares two sides with <, >, ≤, or ≥. | x + 5 > 12 — many answers: x = 8, 9, 10, ... |
| ReasonablenessSpanish: Razonabilidad | Checking if your answer makes sense. | If the problem asks for a number of people and you get x = −3, that is NOT reasonable |
| VariableSpanish: Variable | A letter that stands for a number that is unknown or can change. | “5 less than a number is 12” becomes x − 5 = 12, where x is the variable. |
| ConstraintSpanish: Restricción | A limit in a problem that tells what values are allowed. | “You can spend at most $20” is a constraint: cost ≤ 20. |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A van holds 8 students. A class has 30 students. Write and solve an inequality to find how many vans v the class needs.
- Decide: equation or inequality?: enough seats → inequality
- Write the inequality: 8v ≥ 30
- Divide both sides by 8: v ≥ 30 ÷ 8 → v ≥ 3.75
- Check if the answer makes sense: 3 · 8 = 24 ✗; 4 · 8 = 32 ✓
- v ≥ 3.75, so the class needs 4 vans.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Clue: "The lookout earned less than $50." Write a model for the earnings e. Then explain your choice.
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- I use an equation when the problem says , but I use an inequality when the problem says because .
Word bank ModelEquationInequalityReasonablenessVariableConstraintequationinequalityequalat leastat mostmore than
Watch out!A common mistake
A common mistake in Equations and Inequalities Problem Solving is matching the wrong model to a key phrase. For 'the getaway car held no more than 6 bags,' students often write the equation b = 6 instead of the inequality b ≤ 6, because they see one number and assume it must be exact. The same mistake happens in reverse: for 'the reward split equally among 5 people gave each $40,' students sometimes write the inequality 5p ≥ 40 instead of the equation 5p = 40, because 'split equally' sounds like it could mean a range. Before you submit, ask: does this phrase give one exact value (equation, =) or a limit/range (inequality, <, >, ≤, ≥)?
Lesson 8.78.7 Small Group · Group 2
Group 2Challenge practice
Learning target I can model and solve real-world problems using equations and inequalities, explain what each part stands for, and build one for a situation I have not seen before.
Why This Matters This is your challenge small group for Lesson 8.7. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.
-
1Complete each step, then write the final answer.
Buses Students they hold 1 40 b ? Each bus holds 40 students. A bus holds 40 students. 175 students are going on a trip. Write and solve an inequality for the number of buses b.
- 1Write the inequality from the story.
- 2Divide both sides by 40. What is the boundary number?
- 3How many whole buses are needed?
Final answer: -
2Complete each step, then write the final answer.
Pens Cost 1 $1.25 n $15 n pens cost $15 in all. Pens cost $1.25 each. Max spends exactly $15 on pens. Write and solve a model for the number of pens n. Then explain your choice of model.
- 1Equation or inequality? Write the model.
- 2Divide both sides by 1.25. What is n?
- 3Check: what is 1.25 × 12?
- 4Why is this an equation and not 1.25n ≤ 15?
Final answer: -
3Complete each step, then write the final answer.
Each shelf holds 18 books. A library has 250 books. Write and solve an inequality. How many shelves are needed?
Show your workFinal answer: -
4Complete each step, then write the final answer.
Tickets cost $7.50 each. Ray has $40. Write and solve an inequality. What is the greatest number of tickets he can buy?
Show your workFinal answer: -
5Complete each step, then write the final answer.
A rope is cut into 6 equal pieces. Each piece is 2.25 meters long. Write and solve a model. How long was the rope?
Show your workFinal answer: -
6Complete each step, then write the final answer.
A class needs at least $300 for a trip. They earn $12 for each car wash. Kim writes 12c = 300. Lee writes 12c ≥ 300. What is the least number of car washes they need?
Show your workFinal answer: