The California Frog-Jumping Contest Fosnot- Algebra Day Two: Jumping Buddies Minilesson: Keeping the ratio constant in division Target: More frog-jumping problems as a context for examining common multiples of 8 and 12. Equivalent expressions are represented on the open number line.

Student Materials Needed Tables need their CFJC Packet Sticky notes- one per table Large piece of paper- one per pair of students Markers

Today’s learning target is to… Practice a range of strategies as students solve a string of division problems Represent your thinking on a number line Determine all possible points made by frog jumps Highlight the different ways to determining and representing equivalence

Mental Math Minilesson Keeping the Ratio Constant in Division Give a thumbs-up when you know the answer…. 36 ÷6= 72 ÷ 6 = 72 ÷ 12 = 144 ÷ 24 = 42 ÷ 6 = 126 ÷ 18 = 425 ÷ 25 = Explain your reasoning, and represent the strategy on an open number line.

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 Gallery Walk

Frog & Toad – Appendix D Frog’s Jump Problem: Frog jumps 8 times. Every time he jumps, he travels the same distance. After 8 jumps, he has traveled 96 steps. How long are his jumps? Toad’s Jump Problem: It takes Toad the same amount of time to get to 96, but he does it differently. Each of his jumps is equal to 8 of Frog’s steps. How many jumps does Toad make?

The Investigation Represent both problems on one diagram showing jumping amounts, and explain how they are different and how they are similar. Marking the Meeting Points! Where do Frog and Toad both land? Clearly, 96 is one answer. Are there other places where they both land?

Need Help? I encourage you to draw an open number line or double number line to represent your thinking At what points on the track do Frog and Toad both land? Try marking off all 96 steps on the number line Try skip-counting and identify common multiples Try using multiplication to derive common points Once you have solved the problems, can you determine all the possible points where Frog’s and Toad’s jumps meet?

Preparing for the Math Congress Gallery Walk Sticky notes are for comments or to ask questions about each other’s posters Conduct a gallery walk to give students a chance to review and comment on each other’s posters. Plan for a congress discussion that will highlight the different ways of determining and representing equivalence

Teacher Notes about the Math Congress The purpose of the Math Congress is to allow students to discuss their thinking about equivalence. Discussion should center on the idea of the various equivalences given by common multiples At the end, students should note that all the common multiples (24, 48, 72, 96) are multiples of the least common multiple, which is 24.

Teacher Notes about Facilitating the Math Congress Have students share the equivalence relationships they used in finding common landing points. Focus the conversation on the distance between those points to highlight how common multiples and common factors are related Have students consider the distance between the points Have students reflect on why they are equally spaced

Reflections on the Day The work today focused on equivalent expressions and their representation on the number line. In the minilesson, students explored division. The investigation of Frog’s and Toad’s jumps provided a context for students to think about common multiples and equivalent expressions. The number line used to represent unknown jump lengths became a tool for considering the relationships among different common multiples.