6th grade math activities

Lemonade ratios rebuilt so 6th graders rank recipes and defend the scale-up, five more activities by standard, a free printable, and the 7 standards the page teaches.

  • By Skyler Carr, co-founder and CEO, Mission.io
  • Last updated

Who we are. Mission.io makes Missions: teacher-led, whole-class simulations where students use what they’re learning to work through a real scenario together.

Everything on this page works without us. The 7 standards this page teaches are at the end, each with its Mission where one exists, free. (There are three. Lab Beneath the Waves is the underwater one, where a bad ratio means a long swim home.)

On this page

What 6th grade math looks like

Sixth grade math activities are problems where the class has to use the math to decide something, argue about whether the numbers are right, and leave each student answerable for a piece. This page teaches 7 of the grade’s 29 Common Core math standards, with an activity for five of them and a Mission for five.

The year has four big jobs. Ratio and rate, connected to multiplication and division (6.RP, new this year). Dividing fractions by fractions, and extending the number line to the negatives (6.NS). Writing, reading, and using expressions and equations with variables. And statistical thinking: questions that expect variability, and the numbers that describe a data set (6.SP). Geometry rounds it out with area, surface area, and volume (6.G).

You know the activity that usually stands in for ratios. Two cups of lemonade, a show of hands for which is more lemony, and a worksheet after. The hands settle it, and nobody writes a ratio down. Put the ranking on a sign with 60 cups riding on it, and the class needs a ratio table to get it right.

Six things that turn an exercise into math

In 2024, 28% of eighth graders scored at or above Proficient in math (NAEP). Sixth grade is where ratios, negatives, and variables arrive, and the way the class works on them counts: cooperative learning beat whole-class instruction on achievement, and the effect was stronger in science and math than in social sciences and languages, and stronger in elementary grades than in middle and high school (Kyndt and colleagues, 2013).

  1. 1. A real problem with stakes the class can feel.

    “Which is more lemony?” is a question. “The ranking goes on the sign at the fair, and 60 cups get mixed at the winner’s ratio” is a problem.

  2. 2. Apply, don't recall.

    A second round that only works with the tables from the first. If the ranking can be settled by a show of hands, it isn’t a math activity.

  3. 3. Group goals with individual accountability.

    The one most ratio lessons miss: one kid fills in the table, three kids copy the ranking. When the group shared a goal and each student answered for a part, 84% of studies found a positive effect; without both, 17% (Slavin, 1991).

  4. 4. The teacher leads the thinking.

    Three questions at the table (what did you scale to, do the table and the number line agree, how much concentrate for 60 cups) beat a worksheet of twenty at the desk.

  5. 5. A debrief where the class argues it out.

    The ranking is the setup. The argument afterward, about whose table has the error and why, is the practice the standards name: construct viable arguments and critique the reasoning of others (MP3).

  6. 6. A way to see who did what.

    If you can’t tell which student scaled the ratio and which one copied the ranking, you can’t teach the reasoning. You can only grade the sign.

Anyone can call a cup stronger. The table has to show it.

Everything the lemonade needs is on the sheet at the end of the next section: recipe cards, ratio tables, the number lines, and an answer key for you. Get the sheet →

Lemonade ratios, rebuilt for 6th grade (6.RP.A.3)

Mix lemonade from concentrate and water in different ratios, then answer “which is more lemony?” with ratio tables and double number lines. It’s the ratio lesson in program after program, because ratios are new in 6th grade (6.RP), and “which is stronger” is the question that makes a ratio feel like more than a fraction with a new name. The way it usually runs is a whole-class comparison and a worksheet, with nothing riding on the answer. Here’s the version that hits all six.

Time
50 minutes. The 45-minute cut: three recipes only, and skip the real mixing; colored water in clear cups shows strength you can see and steers clear of food rules.
Groups
Fours. With 30 students, seven groups of four and one of two, where the Table Builder also draws the line.
Supplies
Yellow food coloring in a pitcher as the “concentrate,” water, clear cups, a scoop per team, and the recipe cards on the sheet. About $3.
Standards
6.RP.A.3 (ratio and rate reasoning with tables, tape diagrams, and double number lines), with 6.RP.A.1 (ratio language), and the practices MP3 (construct viable arguments: “more lemony” is an argument) and MP2 (reason abstractly and quantitatively).

  1. Teach first, 10 minutesApply

    Show two cups: 2 scoops of lemon to 3 of water, and 3 of lemon to 5 of water. Ask which is more lemony, and let the wrong answer (“3 is more than 2”) stand for a minute. Then build both ratio tables out to the same water, 15, on the board. Name the tool: scale both ratios to the same water, or the same lemon, and compare.

  2. The problemStakes

    A student-run lemonade stand opens at the school fair with three recipes, one from each of three classes. The stand manager has to publish one strength ranking on the sign and mix 60 cups of the strongest at the same ratio. The team gets the three recipes, written as ratios with different totals, and 10 units of concentrate to test with. Say the stakes out loud: a wrong ranking means the sign lies, and a wrong scale-up means 60 cups of watery lemonade.

  3. Four roles, one job eachAccountability

    On the sheet, and everyone signs their role’s box.

    • Mixer

      Measures the test cups with the scoop. The only hands on the concentrate.

    • Table Builder

      Builds the equivalent-ratio tables to a common amount. The only pencil on the tables.

    • Line Drawer

      Shows the same comparison on a double number line, and has to match the Table Builder or find the error.

    • Scaler

      Takes the strongest recipe and works out the 60-cup mix, then the concentrate it needs.

  4. You lead the thinkingTeacher leads

    Three questions, asked at the table: What did you scale to, and why that number? Do the table and the number line agree, and if not, who’s wrong? How much concentrate for 60 cups, and how do you know? The hardest part is not picking the common amount for them. Let them argue about it.

  5. Two rounds, so the first answer has to be rightApply

    Round one ranks the three recipes. Round two: the fair adds a fourth recipe written as a rate, 4 scoops per 10 cups, and the team has to place it in the ranking and redo the 60-cup mix if it wins. The Mixer and the Scaler swap.

  6. The debrief, 10 minutesDebrief

    Two teams’ tables on the board. Did every team get the same ranking, and if not, which table has the error? Why does scaling to the same water make “more lemony” fair? Where on the double number line is the recipe that fooled you? Who in your group found a mismatch between the table and the line? The second question is the lesson: a fair comparison holds one quantity still.

  7. See who did whatSee who did what

    The observation sheet, four behaviors per student: scaled a ratio correctly to a common amount, explained a comparison with a reason, caught a mismatch, changed their mind with a reason. Tally while you walk the eight tables; you’ll catch about half the room. The tables are quiet work, so loop the room twice.

(The rate card waits for round two on purpose. Teams that ranked by gut start over; teams with tables scale one more ratio.)

In Lab Beneath the Waves, the crew compares ratios again, this time to find the shortest tunnel to the lab. The lemonade tables are how they’ll get there.

The lemonade stand ratios, free

Emailed now, ready to copy: the teacher page, the Mixer, Table Builder, Line Drawer, and Scaler role cards, the three recipe cards with the rate card below a cut line, the ratio tables with the 60-cup mix box, the double number line strips, an answer key, and the four-behavior tally.

It’s on its way.

Check your inbox for the lemonade stand ratios.

Email me the sheet. It comes as a PDF, ready to print. Unsubscribe anytime.

Five more 6th grade math activities, by standard

Five more on every 6th grade list: what each one is, how it usually runs, and what it needs to be a math activity, in the order the standards run.

Ratios, integers, area, data

2. Paint mixing ratios (6.RP.A.3)

Needs red and blue paint, cups or a palette.

Students mix red and blue paint in different ratios, 2 scoops to 3, then 3 to 5, and decide which mix comes out more purple. It’s the lemonade’s structure with paint, so the ratio reasoning is real, and the argument over which cup is more purple makes a good debrief. As usually run, though, one student mixes while the others watch, and you can’t tell who compared the ratios and who went by the color.

The fix

  1. Borrow the lemonade’s jobs: one Mixer is the only pair of hands on the paint, and one student builds the ratio tables.
  2. Before anyone names the most purple cup, the group has to show it with equivalent ratios scaled to a common amount.
  3. Have each student sign the part of the sheet they did.

No page for this one; it’s the lemonade’s structure, so the lemonade above is the model.

3. Unit price shopping (6.RP.A.3.b)

Needs price tags or ads with sizes and prices. Individual, usually.

Students compare price tags or store ads, a 12-ounce cereal box against a 20-ounce one, work out each unit price, and decide which is the better deal. It already has a real problem and real ratio math. As usually run, though, it’s a solo worksheet, and a circled answer can’t tell you who found the unit price and who guessed the bigger box.

The fix

  1. Give the group a budget, so the better deal also has to fit what they can spend.
  2. The Pricer works out each unit price and writes it down.
  3. The Checker works each one again before it counts, and the group signs off only when both agree.

No page for this one; the three-part fix is the whole change.

4. The integer number line walk (6.NS.C.6)

Needs a tape number line on the floor.

A tape number line runs across the floor with zero in the middle, and students walk to place numbers and compare them: stand on -3, now find a number less than that. Standing on the line makes it plain that -5 is less than -2, which trips students up on paper, and you’re leading the questions. As usually run, though, one student walks while the class watches, and you can’t tell who could place -7 on their own.

The fix

  1. Tape a second and third line, so small groups walk at the same time.
  2. Work in pairs: one student places the number, the other explains why it goes there, and they swap.
  3. Have each student write two comparisons, like -6 is less than -2, on a card with their name.

No page for this one; the two misses are the fix.

5. Area of composite shapes on grid paper (6.G.A.1)

Needs grid paper and shape cards. Individual, usually.

Students break odd shapes on grid paper, an L, a house, an arrow, into rectangles and triangles, find each area, and add them up. Decomposing is the skill, and the worksheet gives it plenty of practice. As usually run, though, there’s no reason to find any shape’s area, and a total at the bottom can’t tell you whether a student split the shape or counted squares.

The fix

  1. Give the shape a reason, like a floor plan to carpet, so the area answers a question.
  2. Have students shade and label each piece they split off, so you can see how each one decomposed the shape.

No page for this one; row 9 below is the Mission on this standard.

6. Class data: mean, median, and the outlier (6.SP.B.5)

Needs something in the room to measure, rulers or tapes, a recording sheet.

The class measures something in the room, like hand spans, then works out the mean and median at the board and looks at what one outlier does to each. The data is real, you’re leading the thinking, and the math gets used. As usually run, though, the work happens at the board while a few students answer, and you can’t tell who could find the median alone.

The fix

  1. Put students in groups of three: the Measurer measures each person, and the Recorder writes every value down.
  2. The Plotter puts the group’s data on a dot plot, and the group finds the mean and median from it.
  3. Swap jobs for a second round, so each student does each part once.

No page for this one; the three roles are the fix.

  • Our part · We make Missions

    Then the unit has somewhere to go

    Plenty of teachers rebuild these by hand every unit, and it works. A Mission is a teacher-led, whole-class simulation built to the lemonade’s own standards (6.RP.A.1 and 6.RP.A.3): your class becomes the crew, the scenario is built from the ratio unit you just taught, and the only way through is using it together. After the lemonade, run Lab Beneath the Waves, and the unit has a day two.

    You're at the helm, not on the hook.

    Lab Beneath the Waves: the ratio unit, as a Mission

    Lab Beneath the Waves · 6th grade · About 25 minutes · One device per student, plus your projector · Free with a teacher account

    A full room of students on their feet with laptops in hand, while the wall screen shows the code to join the Mission.
    Laptops up and the join code on the wall: the first minute of a Mission.

    A hidden underwater laboratory has gone quiet. The crew has to compare the ratio of time to distance through several tunnels to find the shortest way to the lab, then weigh fuel against power across a grid of prototype batteries to find the ones that get them out of the collapsed tunnel. The story plays on the projector. The work happens at seven stations (Command, Engineering, Medical, Security, Science, I.T., Operations) and in the arguments between them.

    Every student is assigned to a team, and each team’s work is themed to it, from its own information and challenge to the same problem the rest of the crew is solving. The team gives a student a role in the story and people to lean on while you move between teams. You’re at the helm, asking what the rate is for this tunnel, and how they know without finishing the table.

    It’s built to the lemonade’s standards (6.RP.A.1 and 6.RP.A.3), with unit rates (6.RP.A.2) alongside, and the vocabulary is the unit’s: ratio, rate, number line, proportional, equivalent. Run the lemonade first, and tell the class why: the tunnel only gets picked by a crew that can put two ratios on the same footing. The lesson prepares them. The Mission is what they’re preparing for.

    “At the end of the year last year, every group in my classroom listed Mission IO in the ‘Top 10 Things about Sixth Grade.’”

    Sharla York, 6th grade teacher, Heritage Elementary School

    Will it work in my room?

    Students join with a class code: no student accounts, no student email, no passwords. Chromebooks, laptops, or tablets, plus your projector. Free for teachers, no credit card. Student data is covered by COPPA and FERPA, and we signed the Student Privacy Pledge. At about 25 minutes, Lab Beneath the Waves fits inside a 45-minute period with time left for the debrief, and it’s the shortest of the grade’s three.

    START FREE →

    Communication first, then two of the year's three

    What the report shows

    A Mission.io class report: the class score in each of six skills, from knowledge and application to collaboration, critical thinking, and resilience, with the class chart across the year below.
    A class report from a real Mission: the class score in each of six skills, then each skill across the year.

    Every Mission produces a report for every student, built from what that student did across the whole Mission, on the skills the Mission captured. The six it can capture are Knowledge, Application, Initiative, Collaboration, Critical Thinking, and Resilience. The first two are what a test already sees. The other four are the human skills every school promises. The test shows what they know. The report shows the rest. SEE THE EVIDENCE →

    You taught it. Now find out who can use it.

    Who's behind it

    Mission.io is a research and design organization with one job: make collaborative, active learning something a teacher can run every unit instead of once a semester, and put a report in your hands when it’s over. Over a million students have run a Mission. The research behind them is funded by the National Science Foundation and the U.S. Department of Education’s Institute of Education Sciences, and Mission.io meets ESSA Level II standards, as validated by Instructure.

    6th grade math standards and the Missions on them

    Lab Beneath the Waves is one of three. Below are 7 of the 29 6th grade Common Core math standards, each with the Mission or the activity on this page that covers it. Three Missions, numbered 7 to 9, carry 5 of them. Each code links to its standard on thecorestandards.org.

    Ratios and proportional relationships (6.RP)

    New this year. Lab Beneath the Waves carries all three standards, and the lemonade, the paint, and the unit prices teach them by hand.

    6th grade ratio standards, with the Mission that carries all three and the activities on this page.
    NumberThe standardMissionMinutes
    76.RP.A.1, 6.RP.A.2, 6.RP.A.3 Ratio language, unit rates, and ratio and rate reasoning with tables and double number linesLab Beneath the WavesOn this page: 1. The lemonade, 2. Paint mixing, and 3. Unit prices25

    The number system (6.NS)

    Rational numbers, negatives included, on the number line and the coordinate plane. The number line walk teaches the first, and Brace for Impact runs on the four-quadrant plane.

    6th grade number system standards, with the Mission for the four-quadrant plane and the activity on this page.
    NumberThe standardMissionMinutes
    6.NS.C.6 Rational numbers as points on the number line and the coordinate planeOn this page: 4. The number line walk
    86.NS.C.8 Graph points in all four quadrants, and find distances along a grid line with absolute value (grades 6 to 8)Brace for Impact30

    Geometry (6.G)

    The area of polygons, by composing and decomposing. Infection is built on it.

    6th grade geometry standards, with the Mission for area of polygons.
    NumberThe standardMissionMinutes
    96.G.A.1 Area of triangles, special quadrilaterals, and polygons, by composing and decomposing (grades 6 to 9)InfectionOn this page: 5. Composite area34

    Statistics and probability (6.SP)

    Summarizing a data set in its context. The class data activity teaches it.

    6th grade statistics standards, with the activity on this page for summarizing data.
    NumberThe standardMissionMinutes
    6.SP.B.5 Summarize a data set in its contextOn this page: 6. Class data
    START FREEMISSION BRIEF →

    Free teacher account, no card. Start with the Communication Mission, marked Start here in your library, then the Mission for your unit, each after its 5-minute Teacher Preview.

    Also at this grade

    The 5th grade math activities page rebuilds the cereal box, with the 5th grade standards it teaches; the 7th grade math activities page is the year after. The 6th grade problem solving activities page rebuilds Open Middle, with Brace for Impact as its Mission, and the 6th grade STEM activities page rebuilds the straw rocket and lists every 6th grade Mission by subject.

Questions 6th grade math teachers ask

Which of these fits a 45-minute period?

All three 6th grade Missions do: Lab Beneath the Waves at 25 minutes, Brace for Impact at 30, and Infection at 34. The lemonade needs 50, so use the 45-minute cut: three recipes, and colored water instead of mixing.

Which 6th grade math standards does this page teach?

Seven of the grade’s 29 Common Core math standards, across four domains: 6.RP, 6.NS, 6.G, and 6.SP. Three Missions carry five of them: 6.RP.A.1 to 6.RP.A.3, 6.NS.C.8, and 6.G.A.1. Two more are taught by an activity on this page, 6.NS.C.6 and 6.SP.B.5.

Is this a math page or a problem solving page?

Both pages exist on purpose. This one teaches seven of the 6th grade math standards and rebuilds the lemonade. The 6th grade problem solving activities page rebuilds Open Middle and runs Brace for Impact as its Mission. Plan the unit from this page, and reach for that one when the reasoning itself is the lesson.

Do I need real lemonade?

No. Yellow food coloring in water, in clear cups, shows strength you can see and steers clear of food rules. The math doesn’t change: the Mixer still scoops, and the tables still decide the sign.

Is this aligned to my state's standards, not just Common Core?

Each Mission page lists its Common Core codes and the state sets it maps to. Lab Beneath the Waves also maps to Texas (TEKS), Florida (B.E.S.T.), and North Dakota’s math standards.

A lemonade sign that has to tell the truth

Day one, a problem by hand with the sheet. Day two, Lab Beneath the Waves after the lemonade, on the same standards, once you've watched its 5-minute Teacher Preview. Both free.

GET THE PRINTABLESTART FREETALK TO US →
CONTENTS ↑GET THE PRINTABLE