The purpose of this blog will be to introduce a grade 4 math standard from the Japanese Ministry of Education, Culture, Sports, Science and Technology [MEXT] and then to outline; three proficiencies for students to achieve, three learning activities to teach skills to accomplish the proficiencies and three assessments to gauge if the proficiencies were learned.
The Standard and Why?
(3) To help pupils deepen their understanding of division of integers, divide accurately, and extend their ability to use the calculation appropriately.
a. To explore ways of division in the cases where the divisor is a 1-digit or 2-digit number and the dividend is a 2-digit or 3-digit number, and to understand that these calculations are based on the basic calculations. Also, to understand the way of calculation using algorithms in column forms.
b. To divide accurately, and to use the calculation appropriately.
c. To investigate the relationships between dividend, divisor, quotient and remainder and to put them in the following formula: (dividend) = (divisor) × (quotient) + (remainder)
d. To explore properties of division and to make use of the properties in order to explore ways to calculate or check the results. (MEXT, 2005, p. 11-12)
I chose this standard because I feel the concept of division to be fundamental to both understanding and applying other mathematical functions and because the concept of division has many applications in the real world. For example; from the mundane calculating how many packages containing 100 napkins would be needed for a party of 237 people where you want to ensure that each guest has a lease 4 napkins, to calculations needed to manage a business i.e. dividing up a budget, distributing stock, pricing, calculating loss and profit, etc.
1. To be able to graphically represent division on a number line with the dividend represented as a collection of chunks (the divisor).
2. To correctly apply the long division algorithm to a 2 digit divisor and a 3 digit dividend.
3. To be able to check division results using (dividend) = (divisor) × (quotient) + (remainder)
As the standard states to deepen understanding an assumption of prior division knowledge is present. This assumption should be tested and that is what I will do. To begin the unit a short quiz on single digit and simple double digit division will be given to ensure that students are ready to proceed to 3 digit dividends. Those who are not can be brought up to speed. After the initial quiz, ongoing formative assessment will continue throughout the unit. Every activity’s worksheets will be checked and corrected, and returned to the student as an informal progress check. Those who show signs of struggling or who are falling behind can be identified and given additional support. Formative assessment will also be conducted in having students explain their reasoning behind answers in both small group discussions and in classroom discussions.
2. Summative – Long Division Poster
All students will complete an individual poster showing the steps of long division in an example using a 2 digit divisor and 3 digit dividend. The poster will include a mnemonic to remember the steps of long division; Divide, Multiply, Subtract, Bring down, Repeat/Remainder. Each divide example in the poster will be represented on a number line. Use of color to clarify the concepts will be encouraged.
3. Summative – Long Division Story
Students will write their own long division word problem and make a story. The story will have to include at least two problems encountered by the story’s characters that can be solved by long division (2 digit divisor and 3 digit dividend). In the first paragraph the students will have to set up the problem as a narritive, then in a separate paragraph write the solution as a narrative. Finally, the student will have to prove their solution by using the formula (dividend) = (divisor) × (quotient) + (remainder).
3 Learning Activities
1. Number chunking and, divisor, quotient, dividend poem.
- Objective: Students can draw a 3 digit number divided by a 2 digit number on a number line and identify all the elements by the names divisor, quotient and dividend.
- Materials: Blank A4 paper, 30cm ruler with mm increments, pencil and eraser, math Poem “Divisor, Quotient, Dividend” and worksheet (Greenburg, 2002, p. 13-14).
- Present the students an easy problem, what is 10 / 5 ?
- Tell the students 1 = 1 cm so can they draw a line to represent 10
- Then ask students to mark how many times the 5 fits on their line.
- Now present the students a more difficult problem, what is 252 / 12?
- Give them only a minute and then prompt with this question, “If 1 = 1 mm how would you draw the problem like we did with 10 / 5”
- Give the students time to do this, circulate and check what they are doing. Note who has correctly identified the way to do this using a number line. Call on those students to give the answer and explain their approach. Follow their solution on the main board.
- Give the students more problems to reinforce the concept.
- Read the poem “Divisor, Quotient, Dividend” together as a class. Then in small groups have them discuss which numbers from the number line activity would match these words. Circulate to check groups and have some of the groups who clearly understood this concept articulate it to the rest of the class.
- Finish with the “Divisor, Quotient, Dividend” worksheet, collect for correction and return to the students.
2. Learn Long Division through Story and Mnemonic
- Objective: Students can identify and apply each of the procedures of long division: divide, multiply, subtract, bring down and repeat/remainder.
- Materials: Note book, pencil and eraser, math story “Johnnie Diviso, Division Detective” and worksheet (Greenburg, 2002, p. 24-26).
- Ask the students to draw 42 / 3 on a number line as in the previous activity. Secretly time the students and when the majority of them are finished report the time.
- Tell the students “We are now going to learn a faster way” (much cheering ensues)
- Read the story “Johnnie Diviso, Division Detective” together as a class.
- Break the class into small groups and tell them “In this type of division there are five steps. All the steps are in the story. In your group try to find them.” Circulate to monitor and prompt the discussion. Have some groups who did well share with the class.
- Introduce the mnemonic Dangerous, Monkeys, Swipe, Bananas, Rapidly for the five steps; divide, multiply, subtract, bring down, repeat/remainder. Have the small groups try to make their own interesting and easy to remember mnemonic.
- Have students finish by completing the long division worksheet independently and collect for correction and to give back to students.
3. Division House Activity
- Objective: Students draw a house with the number of details such as; doors, windows, flowers, trees, roofs etc. decided by solving division problems. Students use (dividend) = (divisor) × (quotient) + (remainder) to check each others work.
- Materials: Division House Worksheet, colored pencils/crayons/markers, blank paper, pencil and eraser.
- Show the students some examples of division houses. Point out the differences in the number of windows, doors, flowers, trees, etc. from picture to picture and ask the students why that is?
- Give the students the division house worksheet and work through the first few problems together. Then have the students complete the worksheet.
- As students complete the worksheet pair them up and have them check each others work using (dividend) = (divisor) × (quotient) + (remainder).
- Once the worksheet has been given the peer okay students proceed to draw their Houses. Complete houses are posted on the class bulletin board.
Division is one of the four fundamental operations in mathematics and developing deep understanding in division is important. The learning activities presented above certainly give the students the opportunity to deepen their division skills. Representing division on a number line allows students to form a visual spacial relationship with division which can aid later on in the more abstract applications of division. Learning long division through stories and discussion activates several learning types and correct use of long division will aid students in more complex problem solving. Turning a series of division questions into a pictorial house representation is another opportunity to explore division as more than just an operation on numbers. However, although deepening division skills is important it is also important to plant the seeds of the big ideas in mathematics and allow students to develop schema and skills which will serve them in years beyond long division. Big ideas in mathematics such as “Numbers, expressions, and measures can be compared by their relative values.” (Charles, 2005, p. 14) should always serve as a touch stone even at the fundamental skills training level. The number line activity promotes thinking about this big idea and as the students progress, if their teacher is diligent, they will encounter more instances where this idea holds true. Then once the students have encountered this idea enough they will be able to utilize it broadly as a tool for problem solving in novel situations. This more than anything is the key of backward instructional design and hopefully I have provided a small window into that process here at the grade 4 math level.
Charles, R. I. (2005). Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics. Journal of Mathematics Education Leadership, volume 7, number 3. [PDF file] Retrieved July 13, 2017, from https://www.google.co.jp/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&sqi=2&ved=0ahUKEwjJpLyVu4XVAhViCMAKHWnQAgYQFggkMAA&url=http%3A%2F%2Fwww.authenticeducation.org%2Fbigideas%2Fsample_units%2Fmath_samples%2FBigIdeas_NCSM_Spr05v7.pdf&usg=AFQjCNHKmtdKmLq_V0mw9NIqi0dW3U_zqw&cad=rja
Greenberg, D. (2002). Mega-funny division stories: 24 rib-tickling reproducible tales with companion practice sheets. New York: Scholastic Professional Books.
Ministry of Education, Culture, Sports, Science and Technology [MEXT]. (March, 2008). Improvement of Academic Abilities （Courses of Study）: Section 3 Arithmatic. [PDF file] Retrieved July 13, 2017, from http://www.mext.go.jp/en/policy/education/elsec/title02/detail02/1373859.htm