The Arizona STEM Acceleration Project
“Scale Tales”: Exploring Systems of Equations
Scale Tales: Exploring Systems of Equations
A 9th grade STEM lesson
Julia Wright
1-6-2024
Notes for teachers
This lesson requires some setup but it’s worth it!
Students will conduct a hands-on experiment to experience the value of systems of equations.
Teach system of equations prior to conducting this activity.
List of Materials
Notes for teachers
The container should be opaque so that students cannot see the items inside. (could also be an opaque bag/paper envelope, etc)
A removable lid is recommended so students can verify their results at the end.
Walmart marbles come in bags of 50 - recommend weighing the marbles in advance and remove any outliers.
Fishing weights/sinkers are small, inexpensive, and come in a variety of weights.
Photos are hyperlinked.
Not as durable a the plastic jars, but budget friendly.
Notes for teachers
Scales
Any kitchen scale should work for this activity.
Before you buy - check with your schools science department and see if you can borrow a scale.
I liked this one because it could be recharged via USB rather than requiring batteries.
Math Standards
A1.A-CED.A.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. A1.A-REI.C Solve systems of equations.
A1.A-REI.C.5
5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions
A1.A-REI.C.6
Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables. Include problem solving opportunities utilizing real-world context.
ISTE Standards
1.1: Empowered Learner
1.3: Knowledge Constructor
1.6: Creative Communicator
1.7: Global Collaborator
Objectives:
Students will apply their understanding of systems of equations to solve a real-world problem. By using given weights of two different types of items and the total number of items in an opaque jar, students will determine the quantity of each item type. This activity aims to enhance students' problem-solving skills, reinforce their knowledge of linear equations, and demonstrate the practical application of systems of equations in a tangible, engaging manner.
Agenda (45 to 60 Minutes)
Intro/Driving Question/Opening
Imagine you have a mysterious jar in your hands. It's filled with two different types of items, but you can't see inside. What if I tell you the total number of items and the individual weights of each type of item? Can you figure out exactly how many of each item are in the jar without opening it?
Today, we're going to explore how the mathematical magic of systems of equations can turn us into detectives, uncovering the secrets hidden inside this jar. How can we use mathematics to reveal information that isn't immediately visible to our eyes? Let's dive into the world of equations and find out!
Hands-on Activity Instructions
Assign small groups.
Each group receives:
Demo video
Hands-on Activity Instructions
3. Read the scenario from your Mystery Jar. Use that information and the weight to create a system of equations.
4. Solve with your group.
5. Verify your answer by opening the jar.
6. Repeat with the next Mystery Jar.
Things to consider:
For example: because of slight variations in the item weights, you might get a result that says there are 2.1 marbles in the jar, when there are actually 2.
Marble Collector
You are a collector of rare and valuable marbles. At an antique store, they are selling sealed 'mystery jars' of balls. Each jar contains a mix of marbles and rubber balls and is labeled with the total number of balls. You want to purchase the jar with the highest number of marbles, but how can you tell without opening them? Luckily, you have your digital scale and a sharp mind trained in systems of equations. Each marble weighs 15 grams, and each rubber ball weighs 5 grams. With a limited budget, you need to decide wisely. Work with your group, apply your mathematical skills, and figure out which jar offers the best collection of marbles. This is not just about math; it's about strategy and teamwork. Are you ready to take on the challenge and add to your marble collection?
Emeralds and Rubies
You've just purchased a fascinating container from an antique shop, filled with a mix of emeralds and rubies. The shopkeeper tells you that the quantity of gemstones inside is written on the container, but they're all mixed up. He also shares that each emerald weighs 22 grams and each ruby weighs 14 grams. With this information and your digital scale, your challenge is to determine the number of emeralds and rubies in the container using your knowledge of systems of equations. Can you use the total weight of the gemstones to solve this intriguing puzzle without opening the container?
Lemon Drops and Chocolates
Today's activity is a delightful puzzle involving a jar filled with a mix of two types of candies: colorful lemon drops and creamy chocolate candies. The exact number of each type of candy in the jar is unknown, but it's a part of your challenge to find out. The total number of candies is written on the jar. Your mission is to figure out how many lemon drops and how many chocolate candies are inside the jar. But, there's a catch: you cannot open the jar or see its contents! However, you do know that each lemon drop weighs 4 grams, and each chocolate candy weighs 7 grams.
Armed with a digital scale, your task is to use your knowledge of systems of equations to determine the exact count of lemon drops and chocolate candies in the jar. Can you use the total weight of the candies and the weights of individual candies to solve this puzzle? Let the challenge begin!"
Piggy Bank
You've just discovered an old piggy bank in your attic. After shaking it, you realize it's filled with a mix of dimes and nickels. Someone has labeled it with the total number of coins. Unfortunately, the piggy bank is sealed, and you can't open it without breaking it. In this challenge, you have access to a digital scale and a similar empty piggy bank. There are also some loose dimes and nickels nearby. How can you use these resources to determine the number of dimes and nickels in the sealed piggy bank without breaking it open?
Consider the problem carefully. What information do you need? How can the weight of the coins and the piggy banks help you solve this mystery? Work with your classmates to devise a strategy, using your knowledge of systems of equations, to crack the code of the piggy bank. Remember, sometimes the key to solving a problem lies in observing and using the resources available in creative ways.
Marbles Examples: 6 jars - one is empty to use as a tare. The other containers have a variety of marbles and rubber balls. For example, the container could contain:
Container | Marbles | Rubber balls | Total |
1 | 9 | 0 | 9 |
2 | 8 | 1 | 9 |
3 | 6 | 1 | 7 |
4 | 5 | 2 | 7 |
5 | 3 | 2 | 5 |
You could make an answer key but there maybe be value in having the students check their own work by opening the container to count the contents.
Sample “candies” : fishing sinkers painted with nail polish.
Piggy Bank: Actual coins. Dimes and nickels work well
Assessment
Use the worksheet as an exit ticket and to assess where students are in their learning.
Differentiation
Provide students with the system of equations (or a worked problem on the board).
Remediation
Extension/Enrichment
Note: The Piggy Bank example has a greater degree of difficulty because the students have to weigh the nickel and dime on their own.
Have students create their own Scale Tale story problem using items around the classroom. (ie pencils and pens) Trade these with other students to uncover the mystery.
Have students solve by substitution, elimination, and/or graphing (perhaps using Desmos).
Have students start by predicting how many of each item will be in the jar. Calculate percent error.
Optional Resources