Apply Two-Variable Relationships to Solve Problems Challenge

Grade 6 Math · Unit 9 Challenge Studio (6.AT.11) · Lesson 9-4
Progress: 0 / 31

I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.Puedo escribir una ecuación para representar la relación entre dos cantidades variables y usarla para resolver un problema real.

MissionMisión · Robot LabLaboratorio de robots · night shiftturno de noche

Test, measure, improve.Prueba, mide, mejora.

Budget review at the Robot Lab: three rental plans, two charging schedules and one tight deadline for a robot fair. Solve each equation, round the right way, and justify every recommendation with the math.

Revisión de presupuesto en el Laboratorio de Robots: tres planes de alquiler, dos horarios de carga y una fecha límite para una feria de robots. Resuelve cada ecuación, redondea en la dirección correcta y justifica cada recomendación con las matemáticas.

Word bankBanco de palabras 5 words5 palabrassolution, substitute, at least, …solución, sustituir, al menos, …
solutionsolución
A value of the variable that makes the equation true, like h = 8 in 6h = 48.Un valor de la variable que hace verdadera la ecuación, como h = 8 en 6h = 48.
substitutesustituir
To replace a variable with a number so you can compute.Reemplazar una variable por un número para poder calcular.
at leastal menos
That amount or more. The boundary number counts.Esa cantidad o más. El número límite cuenta.
predictpredecir
To use an equation or a pattern to find an amount you have not measured yet.Usar una ecuación o un patrón para hallar una cantidad que todavía no has medido.
justifyjustificar
To explain why your answer or recommendation makes sense, using the math.Explicar por qué tu respuesta o recomendación tiene sentido, usando las matemáticas.

Stage 1 of 5Etapa 1 de 5

Start StrongEmpieza fuerte

Each rung is harder than the last, and the numbers change every time.

Cada peldaño es más difícil que el anterior y los números cambian cada vez.

  1. 6 fresh problems with new numbers each time (interactive only).

Stage 2 of 5Etapa 2 de 5

Stretch Your ThinkingAmplía tu pensamiento

Plan, solve, then check that your answer makes sense.

Planea, resuelve y revisa que tu respuesta tenga sentido.

  1. Suppose the variable s counts 8-hour SHIFTS instead of hours. Which equation gives the cars produced?
    • c = 2.5s
    • c = 8s
    • c = 10.5s
    • c = 20s
  2. If the plant runs 24 hours a day, 7 days a week for the full 3 weeks, how many hours is that — and can it fill the 1,000-car order?
    • 72 hours — no, that is less than 400 hours
    • 168 hours — no, that is less than 400 hours
    • 504 hours — yes, because 504 hours is more than the 400 hours needed
    • 504 hours — no, because 504 is less than 1,000
  3. The solution to one problem was 41.7 cars. Would producing or washing a PARTIAL car make sense?
    • No — cars come in whole numbers, so round up to the next whole car
    • Yes — 0.7 of a car is a real amount you can wash
    • No — so round down and accept a smaller total
    • Yes — the equation is wrong if it gives a decimal
  4. A 3D printer makes 14 gears each hour, so g = 14h. Here h is the hours and g is the gears. The robotics club needs at least 200 gears. What is the fewest whole number of hours the printer must run?
  5. A robot rental costs c = 5h + 15, where h is the hours and c is the cost in dollars. The $15 is a setup fee. The club has $75. For how many hours can it rent the robot?
    • 12 hours
    • 18 hours
    • 15 hours
    • 60 hours
  6. Plan A costs c = 4h + 12 and Plan B costs c = 6h, where h is the hours and c is the cost in dollars. Fill in the cost of each plan.
  7. Plan A costs c = 4h + 12 and Plan B costs c = 6h. The club has a $90 budget and wants the plan that gives more whole hours of robot time. Which choice is correct?
    • Plan B, because it has no setup fee
    • Plan A, which allows 20 whole hours
    • Plan A: 19 hours; Plan B: 15.
    • Plan A, which allows 22 whole hours

Stage 3 of 5Etapa 3 de 5

InvestigateInvestiga

Use the model to test an idea. Then check it.

Usa el modelo para probar una idea. Luego compruébala.

  1. A robot's battery costs c = 6h + 14 dollars to run for h hours. The club has $62. Sort each step into the stage of solving it belongs to.
  2. Create two booth plans with different prices per item. Write both equations and a list of pairs for 1, 2 and 3 items. Choose a budget. Work backward to find how many items each booth sells for that budget, and justify which booth is the better deal.
  3. 4 fresh problems with new numbers each time (interactive only).

Stage 4 of 5Etapa 4 de 5

Find and FixEncuentra y corrige

Somebody made a mistake. Find the step and fix it.

Alguien cometió un error. Encuentra el paso y corrígelo.

  1. A welding robot makes 18 welds every hour, so w = 18h. Sam needs at least 100 welds and finds the hours.
  2. A robot charger uses c = 4t + 6, where t is the hours and c is the cost in dollars. Tariq has $46 and finds t.
  3. A bottling robot fills 25 bottles each minute, so b = 25m. An order needs 600 bottles. Mei finds the minutes.
  4. Robot X costs c = 3h + 20. Robot Y costs c = 5h. Here h is hours and c is dollars. For 8 hours of work, Dmitri says Robot X is cheaper because 3 is less than 5. Noor says to compute both totals first. Who is right?
    • Dmitri: the smaller hourly rate always means the lower cost.
    • Both: they use different information, so both are fine.
    • Neither: 8 hours is not enough to tell.
    • Noor: Y costs $40; X costs $44.
  5. What is the best next move?
    • Round up: 12 buses hold at least 400.
    • Report 11.4 buses as the answer.
    • Subtract 400 − 35 to get 365 buses.
    • Round down to 11 buses, because 0.4 is less than a half.
  6. 1 fresh problem with new numbers each time (interactive only).

Stage 5 of 5Etapa 5 de 5

Prove Your UnderstandingDemuestra tu comprensión

Say how sure you are, solve, then create a problem of your own.

Di qué tan seguro estás, resuelve y luego crea un problema tuyo.

  1. A painting robot paints 7 panels each hour, so n = 7h. What does the solution of 7h = 91 tell you?
    • The robot paints 91 panels in 7 hours.
    • The robot needs 13 hours to paint 91 panels.
    • The robot paints 13 panels every hour.
    • The robot needs 98 hours to paint 91 panels.
  2. A robot battery charges by p = 12.5h, where h is the hours and p is the percent of charge. The robot needs at least 80 percent to start. What is the fewest whole number of hours it must charge?
  3. Which check shows that h = 9 is the solution of 4h + 6 = 42?
    • 4 × 9 + 6 = 42
    • 4 + 9 + 6 = 19
    • 4 × 9 = 36
    • 42 ÷ 9 = 4.67
  4. The robotics club expects to sell 26 robot kits. It wants to raise at least $400. If p is the price of one kit in dollars, then 26p must reach 400. What is the least price, in dollars and whole cents, that reaches the goal?
  5. The table shows a battery that gains the same percent of charge every hour, so b = 8h. The battery holds at most 100 percent. After how many whole hours does the equation first give a charge above 100 percent?
  6. The Robot Lab must choose between two robot rental plans: Plan A, c = 4h + 12, and Plan B, c = 6h. The lab needs the robot for 10 hours. Write a recommendation. Use the words equation and solution, show your substitution, and justify your choice.