The California Frog-Jumping Contest

Fosnot- Algebra

Day Three: The BenchesStudents consider a new context of placing benches alongside two frog-jumping tracks.

Student Materials NeededTables need their CFJC Packet

Length of 6 and 8 connecting cubes

Large pieces of paper- a few per team

Markers

Today’s learning target is to…Introduce and discuss the bench investigation, highlighting the requirement that the bench lengths must add up to the length of the track

Team Roles & ResponsibilitiesFor each team, select a role and responsibility for each person at your table.The Recorder’s job: to write the solutions for each

problem in the Team packet and to listen to the Speaker’s to check the accuracy of the team’s solutions

The Artist’s job: to create a visual model or poster of the solutions and clearly display the team’s knowledge of the investigation

The Speaker’s job: to present the poster created by Artist to the entire class and explain the team’s reasoning for each solution during the Math Congress

Developing the ContextOn a large piece of paper, draw the following

diagram as I introduce today’s investigation (page 28):

Jumping Tracks InformationThe frogs are planning a jumping contest. They have two jumping tracks; one is 28 feet long and the other is 42 feet long. They have decided to bring in benches from their storeroom and place them along both sides of each track end to end so the benches line the track lengths exactly. The benches in the storeroom are of two different lengths. One size is six feel long and the other size is eight feet long. How many six-foot benches should they get and how many eight-foot benches should they get in order to line both sides of both tracks?

Initial Thoughts and DiscussionWhat are the requirements for the bench

lengths?Can the benches be cut? NO!What does your team thinking about the

following:How many six-foot benches and how many

eight-foot benches are needed in order to line both tracks?

Are there other possible choices of six- and eight-foot benches that could be used?

The Investigation- Appendix EYou need to combine 6-foot and 8-foot

benches to line both sides of two jumping tracks. The total bench lengths must line the track lengths exactly. One track is 28 feet and the other track is 42 feet. How many 6-foot benches and how many 8-

foot benches are needed to line both tracks?Are there other possible choices of 6-foot and

8-foot benches that could be used?

Need Help?Try to find all possible solutions and I

encourage you to think about the relationships between the solutions

Are you able to use a combination of both bench sizes for one side of the track?

Think about the relationships between the solutions

Strategy SuggestionsTry adding up sixes and eights to see what

they add up toMake a list of all the multiples of six and a list

of all the multiples of eight and then adding various combinations of these multiples together to see which add to 28 and 42

Exchanging of equivalent pieces is a more algebraic strategy

What did they find in the storeroom?While you are working, let me tell you one more part of the story. When the frogs (frogs?, I think they meant people) went to the storeroom, they found that they had only 17 six-foot benches and 9 eight-foot benches. Will these amounts work?What should they do?

Preparing for the Math Congress Gallery WalkPrepare for a math congress by making posters

addressing the following questions and justifications for your answers:1. The frogs decide they want new benches so

they go out to price benches at different stores. What are their options for buying six-foot and eight-foot benches to place along both sides of both their 28- and 42- foot tracks?

2. Explain how you know you have found all the possibilities.

Plan for a congress discussion on equivalence and how equivalence can be represented using a double number line

Reflections on the DaySeveral big ideas about equivalence were explored

today as you investigated the benches. Some teams used trial and error to find the

possibilities while other teams may have adjusted their initial trials to find solutions.

By using organized lists or by making exchanges, you were able to find the different possible solutions to the investigation.

The investigation enabled you who started with arithmetical strategies to use more algebraic approaches.

The algebraic approaches, using equivalence and substitution, enabled you to show why your collection of possibilities was complete as you worked to develop proofs.