Summary
In this lesson, students will use questions, a book, and their favorite foods to explore multi-digit division with tools and strategies they already know. Students will also practice making conjectures (or reasonable estimated solutions). After students find a solution to the main problem, the teacher will use the students' strategies to facilitate a discussion that connects to and teaches a lesson on partial quotients.
Essential Question(s)
How are partial quotients similar to or different from partial products? How can partial products help us better understand and use partial quotients?
Snapshot
Engage
Students use Andrea Menotti's How Many Jelly Beans? to review rounding large numbers and multiplying those numbers to find approximate solutions.
Explore
Students work in small groups to estimate a solution to a multi-digit division problem. Then they find the exact solution.
Explain
Students share their groups' strategies and solutions. The class connects those strategies to division with partial quotients.
Extend
Students extend multi-digit division with partial quotients to problems where a quotient may contain a remainder.
Evaluate
Students answer questions to apply and reflect on their personal understanding of partial quotients.
Materials
How Many Jelly Beans? by Andrea Menotti
Paper and pencil
Engage
Begin by reading How Many Jelly Beans? by Andrea Menotti. Read to the page where the brother says, "In a whole year I could eat A THOUSAND JELLY BEANS!" and stop before the sister gives the answer. Ask students to use a Think-Pair-Share strategy and estimate how many jelly beans would be eaten per day.
If students offer more than one strategy and solution, ask students, Which strategy do you think would give us a closer estimate? Consider having students use the Think-Pair-Share strategy again if time allows.
Ask students to share their answers with the class. Then continue reading the story.
Throughout the rest of the story, periodically pause and ask students to estimate how many jelly beans someone would have to eat each day to hit 5,000 in a year. How about 10,000? 100,000? One million? Consider also asking students, How many per day is too many for you?
Explore
Ask students, What is your favorite thing to eat? and have students write their answers on notebook paper.
Now, ask students to imagine they eat that item six times a day, every day. How would they figure out how many days it would take for each student to eat that item 6,972 times?
Have students work in pairs of groups of three. Invite each group to make a conjecture or an estimate of how many days they think it would take them to eat 6,972 of their favorite food. Then, ask each group to decide whether the actual solution will be more or less than their estimated solution. After students have decided, invite them to determine exactly how many days it will take them to eat their item 6,972 times.
While students work, travel around the room and help groups who may be struggling. Use guiding and probing questions as needed.
Explain
Using a Strategy Harvest, ask each group to join with another group and share their solution strategy. Students could also share what did not work, but explain how their mistakes lead to finding their solution.
Ask each group what strategy they used. Have a volunteer from each group share the group's strategy. If possible, call on volunteers in an order selected to best lead into partial quotients. If any groups used a strategy involving partial products, connect this back to the lesson using its inverse, partial quotients.
Walk the class through another division problem using partial quotients. Examples include 721 divided by 3; 955 divided by 5; or 748 divided by 4.
Extend
Ask students to consider that none of the problems the class has worked on so far had anything left over—each dividend was divided equally by the divisor. What would they do if the dividend did not divide by the divisor equally?
Have students work with an Elbow Partner. Ask each group to select one division problem written on the board, write their own word problem, find a solution, and decide what to do with the remaining parts of the dividend. How will they represent those extra parts?
Depending on your classroom needs, either have students share solutions with the class or turn in their problems as a formative assessment. If choosing the latter option, consider selecting one or two problems for the class to evaluate and discuss.
Evaluate
Have students complete the reflection questions below as homework or as an Exit Ticket. Each answer should be detailed and require three or more sentences.
Why are estimating solutions helpful when working with large numbers?
How is the partial quotient strategy similar or different from partial products?
Describe a mistake or a misconception that you or a classmate had in class today. What did you learn from this mistake or misconception?
How could the ideas from today's lesson be used in life (Boaler, 2016, p. 109)? Give an example that could take place at home or in life.
Resources
Boaler, J. (2016). Mathematical mindsets unleashing students' potential through creative math, inspiring messages and innovative teaching. San Francisco: Jossey-Bass.
K20 Center. (n.d.). Bell ringers and exit tickets. Strategies. Retrieved from https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f505d6f2
K20 Center. (n.d.). Elbow partners. Strategies. Retrieved from https://learn.k20center.ou.edu/strategy/ccc07ea2d6099763c2dbc9d05b00c4b4
K20 Center. (n.d.). Strategy harvest. Strategies. Retrieved from https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f5062662
K20 Center. (n.d.). Think-pair-share. Strategies. Retrieved from https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f5064b49
Menotti, A. (2012). How many jelly beans? A giant book of giant numbers. San Francisco, CA: Chronicle Books LLC.