Fosnot- Algebra Day Three: The Benches Students consider a new context of placing benches alongside two frog-jumping tracks.

Student Materials Needed Tables 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 & Responsibilities For 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 Context On a large piece of paper, draw the following diagram as I introduce today’s investigation (page 28):

Jumping Tracks Information The 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 Discussion What 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 E You 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 Suggestions Try adding up sixes and eights to see what they add up to Make 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 Walk Prepare 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 Day Several 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.