The divider (mixed number quotients)
This unit uses two of the series of four Divider learning objects to support the students division of whole numbers where the quotient is a mixed number (whole number and fraction). The Divider encourages the students to split a division problem into parts based on known division or multiplication facts. The divider is based on a rectangular representation of division where one of the side lengths is unknown.
AO elaboration and other teaching resources
- split division problems into parts using known multiplication and division facts.
- solve division problems where the quotient is a mixed number.
This unit is intended for students working at Stage 7 (Advanced Multiplicative) of the Number Framework. Students at this stage use a combination of known facts and mental strategies to derive answers to division problems.
The unit uses the following terms:
Dividend - the number that is being divided into.
Divisor - the number that the dividend is being divided by.
Quotient - the result of dividing the dividend by the divisor.
Mixed number - a number including a whole number and a fraction
The Learning Objects
The learning object, the divider, can be accessed from the link below, using your school's username and password:
Click for more information about the learning objects and who can register to use them.
Working with the learning object with students
In this teaching session the students use the learning object to solve problems where the quotient is a mixed number. We suggest that you use The Divider: solve your own problem however you could use the Divider: with and without remainders, which generates problems randomly.
- Show students the Divider and introduce them to David. Explain that the purpose of the learning object is to solve division problems by splitting them into easier to solve parts.
- Enter a problem into the learning object, for example: 217 ÷ 8
- Discuss what the numbers on the bar represent.
Why is 217 written in the bar?
What does the 8 on the side of the bar represent?
- Show that the bar represents 217 and that 8 is the length of the side. The length of the top or bottom of the rectangle is the amount they are trying to find. You may also like to represent the problem as 8 x ? = 217.
- Ask the students if any know the answer to the problem. If all the students know the answer show them where to enter it and how their response is checked. Select a new problem - this time choosing one where the problem is not a known fact.
- If at least some of the students did not know the answer to the problem show the students how to split the bar into parts. Each of these parts is represented by a division, which is shown in the boxes above the bar. Tell the students that the aim is to split the bar until they see a division that they know the answer to. For example, split the bar into 160 (160 ÷ 8) and 57 (57 ÷ 8).
- Show how to enter the answer into the first box (160 ÷ 8 = 20). Point out how this number appears along the bottom of the bar and also that it is entered in the equation at the foot of the learning object screen.
What does this [pointing to 20] represent?
Where is this number in the equation? Why is it there?
- Ask if they know the answer to the second box (57 ÷ 8). Enter "7" if suggested and discuss the feedback given in the learning object: "57 is not a multiple of 8. Move the slider until you see a number that is a multiple of 8."
- Move the slider to 56 showing the students that they can move the slider in steps of one by clicking on the + or - arrows.
- Enter the answer to 56 ÷ 8.
- Discuss how the 1 remainder has been represented in the learning object. What does this represent [pointing to the "1" which is now represented as 1/8].
- Discuss how the final answer requires them to sum the numbers along the bottom of the bar and show how this is represented in the equation.
What is the answer to 217 ÷ 8 ?
How did we work it out?
Why did we do it that way?
Could we have done it another way? If another way is suggested then click Reset and complete the problem again using different parts/splits.
- Repeat with other division problems where the quotient is a mixed number. We suggest that the problems are also posed in context so that the students are required to think which numbers from the problem need to be entered into the learning object. Also encourage the students to give the answer in context. Possible problems include:
A chocolate maker packs gourmet chocolates in boxes of 8. How many boxes can she fill with 245 chocolates. An adventure ride takes 7 people in each vehicle. How many vehicles can be filled with the 227 people in the queue?
Students working independently with the learning object
Pose problems set in context for the students to solve independently (individually or in pairs) using The Divider: solve your own problem learning object. Alternatively direct the students to use the Divider: with and without remainders learning object and ask them to create contexts for the problems generated.
Possible contexts for division problems:
Joe is paving the path. The path is 3 paving stones wide. How many rows can he tile if he has 487 paving stones.
Sione is packing biscuits into trays for the school fair. She can fit 9 biscuits on a tray. How many trays can she fill with 636 biscuits?
Once students have solved the problem encourage them to make a written record of the way that they solved the problem. For example the students may record the ways that they split the division problem into parts.
487 ÷ 3 = 17
Alternatively students can print the solution screen within the learning object. Provide opportunities to share solution strategies and written records with others.
How did you use the divider to solve the problem? What numbers did you use? Why did you use these numbers? How did you record your thinking?
Students working independently without the learning object
Independent activities that develop the same concepts as the learning object include:
- Figure It Out, Number, Book 5, Level 4, page 1, Revisiting Remainders
- Figure It Out, Number, Book 5, Level 4, page 2, Remainder Bingo
- Figure It Out, Number, Book 3, Level 4, page 10, Team Leaders
- Figure It Out, Number, Book 2, Level 3-4, page 5, Food for All
- Figure It Out, Number, Book 2, Link, page 6, Planting with the Whanau
Students could also work independently on division problems:
- Pose the students division problems (with and without remainders) in context to solve. Encourage them to make a written record of the way that they solved the problem. For example the students may record the ways that they split the division problem into parts (see above).
- Pose the students division problems as equations. First ask the students to create a context for the equations. Encourage them to make a written record of the way that they solved the problem. For example the students may record the ways that they split the division problem into parts (see above) or write a list of the splits.
Provide opportunities to share solution strategies and written records with others.
How did you use the divider to solve the problem? What numbers did you use? Why did you use these numbers? How did you record your thinking? How does that help you solve the problem?