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Partial Quotients Method
Students will be able to divide using the partial quotients method.
- Display a simple multiplication chart with no coloring or notations.
- Conduct a whole-class discussion on which factors are easiest to estimate products (e.g., 2, 5, 10, 100, etc.) Highlight multiples and circle related factors on the multiplication chart so they can visually think about patterns they see in the chart. Students should understand that factors follow a pattern and that some numbers are easier to compute or estimate products.
- Tell students that today they'll use their understanding of factors and their multiples to use the partial quotients method to solve division problems.
Explicit Instruction/Teacher modeling(10 minutes)
- Define FactorAs the number multiplied by another number to get a Product,Or the answer to a multiplication problem. MultiplesAre a list of products specific to a factor (e.g., 4, 8, and 12 are all multiples of four). When discussing division, the QuotientIs the answer to a division problem. Review other key terms about division, such as the divisor and dividend if necessary.
- Display the teaching component from the worksheet Partial Quotients with Two-Digit Divisors. Label the products (i.e., totals that are subtracted), quotients (numbers that are added), divisor and dividend.
- Define the Partial quotients methodAs a way to solve division problems by repeatedly finding pieces of the quotient, or partial quotients, and then adding up all the quotients to determine the answer to the division problem.
- Model redoing the given model on the board, thinking aloud each step, and using the same colour-coding shown in the example. (Tip: use correct place value terminology, such as, "I need to borrow one 100 to make 110 so that I can subtract 70 from 110.")
- Discuss ideas that can make the partial quotients methods efficient, such as choosing to multiply the divisor by 10 because it was a familiar calculation. (Tip: it can be helpful for students to list the multiples of the divisor, or round the divisor up when deciding which partial quotient to use throughout the process.)
Guided practise(15 minutes)
- Ask students to use their whiteboards to redo the example problem again. Tell students to continue to use the same terminology you modeled to explain their answer to their elbow partners when they are done.
- Challenge a student to redo the problem again for the class but create different products to subtract from the dividend. For example, the student can choose to multiply 20 x 17Instead of 10 x 17.
- Distribute the worksheet Partial Quotients with Two-Digit Divisors and have students complete problem #1 and #2 in partnerships, making sure to write their answers on their own sheets.
- Review the answers as a class, correcting misconceptions about the strategy as necessary.
Independent working time(12 minutes)
- Pass out three dice to each partnership and ask students to roll the three dice to create the dividend and then roll only two dice to get the divisor. Then, ask students to use a sheet of copy paper to solve the division problem using the partial quotients method.
- Tell students to complete at least three problems. Challenge them to complete the problem on their own and then share it with their partner. Ask them to note which multiples they chose and how their choices differed from each other.
- Conduct a class discussion where students make generalizations about the partial quotients method with division. Ask students to make some generalizations about how they were able to find the quotient faster. Also have them think about which method was easier for them. Write their ideas on the board with tally marks next to each idea to represent the number of students who agree with the idea.
- Provide visuals and definitions for difficult vocabulary and sentence stems for ELs and students with disabilities.
- Allow students to use colour-coding for each of the partial quotients and products. Ask students to line up the partial products so they do not forget digits as they add them up to get the quotient.
- Have students model their thinking aloud by allowing them to share their explanations. Use some of their language and write it on the board for other students to consider.
- Challenge students to use more dice or 10-sided dice for their division problem creation.
- Ask students to connect their ideas about the partial quotients method to that of the partial products method. Allow them to share their ideas during the Review and Closing section if time allows.
- Have students consider other ways of solving the division problem for each of the problems on the worksheet.
- Hand out one large index card to each student and have them create a division problem either using their shared dice, or just writing a problem with Three-digits ÷ two-digits.
- Collect all the index cards and randomly pass out one card to each student. Have students complete the division problem with the partial quotients method. If time allows, ask them to solve the same problem on the back of the index card a different way (e.g., repeated subtraction, area model, standard algorithm, etc.).
Review and closing(5 minutes)
- Have students write their name on their index card and give it to another student. Ask the student to correct the problem by solving it themselves on their whiteboard.
- Ask students to consider the generalizations they created in the Independent section and add on more or eliminate some of them.
- Ask students to vote on whether they enjoyed using the method.