How to teach inequalities

Why this matters

A student who can solve 15 + 5v = 40 can still write the inequality pointing the wrong way, or count the tie as a win. Few students will ask out loud whether 5 belongs in the answer. On a worksheet the mistake looks like a right answer, and the teacher has to wait for quiz day to find out.

Writing inequalities from a word problem

To write an inequality from a word problem, students name what the variable counts, write an expression for each quantity being compared, and pick the sign. "At most" and "no more than" mean ≤, "at least" means ≥, and "more than" or "less than" mean a strict or . The solution is a range of values, so students also decide whether the boundary value belongs in it: where two plans cost the same, neither plan is cheaper. Algebra 1 students create inequalities in one variable and use them to solve problems (Common Core A-CED.A.1).

In the Flint activity, each student picks something they'd pay for every month, and Sparky, Flint's AI tutor, turns it into two plans and a spreadsheet that recalculates both costs. The student finds where the cheaper plan switches, writes and solves the inequality, then writes a budget inequality with ≤. When a student's sign points the wrong way, Sparky asks which plan the inequality says is cheaper, and a student who counts the tie gets sent back to the spreadsheet row where both plans cost the same.

Marcus picks a gym plan and writes the inequality that proves it

Every student in the class gets the same activity: compare two monthly plans, then write and solve the inequality that says when one is cheaper. The plans, the numbers and every question after the first come from the student's own chat. Marcus is a made-up 9th grader, and this chat is an example we wrote for this page.

Sparky uses what Marcus would pay forSparky answers what Marcus just wrote

M

Marcus

Interest: Club soccer, training at a gymStarting point: Solves two-step equations, treats < like =

Sparky

Today you'll pick between two monthly plans and write the math that proves which one is cheaper. First, what's something you'd pay for every month, like a gym, a game pass or music lessons?

Marcus

A gym. I play club soccer and I want to train more in the off season.

Sparky

Gym training for soccer it is. Plan A is $40 a month with unlimited visits. Plan B is $15 a month plus $5 a visit. Change any number in the Visits column and both costs update. Which plan is cheaper when you go only a few times, and which is cheaper when you go a lot?

Marcus

Plan B is cheaper if I go less. Like at 4 visits it's 35 bucks. At 6 visits Plan A is cheaper.

Understanding check

Exactly right. Can you write an inequality that shows when Plan B is cheaper than Plan A? Use v for the number of visits.

Marcus

15 + 5v > 40 so v < 5

Sparky

Check the sign in your first line. Your inequality says Plan B costs more than Plan A. Which plan did you want to show is cheaper?

Marcus

oh Plan B. so 15 + 5v < 40. from the table I think it's v ≤ 5

Sparky

15 + 5v < 40 is right. Now solve it. And look at the row for 5 visits in the spreadsheet. Are the two plans different there?

Marcus

5v < 25 so v < 5. at 5 visits they're both 40 so plan B isn't cheaper there. so it's v < 5 not ≤

Understanding check

That's right: Plan B is cheaper for fewer than 5 visits, and at 5 they tie. Here's a harder one. Your budget for the gym is at most $50 a month. Write an inequality for Plan B that shows how many visits you can afford.

Marcus

15 + 5v ≤ 50. 5v ≤ 35 so v ≤ 7. I can go 7 times a month

Sparky

Correct. With $50 a month, you can go to the gym up to 7 times on Plan B.

Fixed the sign himself, then solved the $50 budget with ≤

What Flint did for Marcus

• Sparky built both plans around the gym Marcus said he'd pay for, so the numbers meant something to him.
• Sparky made a spreadsheet whose formulas recalculate, so Marcus could check any number of visits himself.
• When Marcus wrote the inequality the wrong way, Sparky asked which plan it said was cheaper instead of fixing it.
• When Marcus guessed v ≤ 5 from the table, Sparky sent him back to the 5-visit row, where both plans cost $40.
• Once he got it right, Sparky gave him a harder question with a budget and ≤.

A teacher can't build 30 sets of plans and watch 30 inequalities as they're written. Sparky does it in every student's chat.

See which students still treat < like =

Each student's chat shows up on the activity page as they work. Activity analytics sums up what the class did well and which mistake to practice next.

Set up the activity for free

The plans activity is in Flint's public library, with Sparky's role, guidelines, greeting and greeting suggestions already written. Open it, select Use activity to copy it into a free Flint account, and move it to a class. Every student gets their own chat with Sparky, and the activity page fills in as they work.

In the activity's Preview tab, select Try session myself to test the activity as a student before you assign it. Add a rubric and Sparky grades each chat for you.

For another Algebra 1 activity, see how to give every student a personal tutor. For more math activities, see Flint for math.