Objective: Students will solve division problems by expanding the More Area Compare Strategy. / Notes
Materials:
1. One-Centimeter Grid Paper
2. Colored paper, scissors, rulers, and crayons/colored markers
3. Benchmark multiplication facts
Expanding the More Area Compare Strategy to Division
In this lesson students will use the More Area Compare Strategy used earlier to learn multiplication facts to now solve division problems. In 20.2, students will follow similar steps and procedures introduced in the 15.3 multiplication lesson, but now they will apply them to division operations. Once again, students will use a strategy they already know – the More Area Compare Strategy - to figure out more complex math problems.
The expansion of the More Area Compare Strategy to division leads to later concepts, such as: the division algorithm, repeated subtraction, factoring of polynomials, and solving one and two step equations.
Transitioning From Multiplication to Division Using the More Area Compare Strategy
In several RAMP units, students learned and practiced the More Area Compare Strategy. Students used the distributive property to find new products. For example, students used the known benchmark
5 X 6 = 30 to find 5 X 7. Numerically, 5 X 7 = (5 X 6) + (5 X 1) (one more group of 5). Specifically, students compared the rectangular area 5 by 6 to the rectangular area 5 by 7. Students focused on the idea that the side (factor) that remains unchanged is the amount added to find the larger product (area).
START by repeating the More Area Compare Strategy multiplication activities from earlier lessons, emphasizing these two attributes of a good benchmark:
A good benchmark is one that:
1) is less than or equal to the product you’re trying to find
2) has one factor that is the SAME
For example, 6 x 5 = 30 is a good benchmark for learning 6 x 6 = 36, because it is the closest benchmark that 1) is smaller than the product you’re trying to find (30 is less than 36), and 2) has one side that is the same (6).
Practicing Finding Good Benchmarks
When students used benchmarks and the More Area Compare Strategy to find values for new multiplication facts, they learned that the length of the sides was a determining factor in their process. For example, to find the product of 7 x 6, students used the benchmark 7 x 5 because the side (7) is the same. The benchmark 9 x 4 is closer to the product 42 and is less than the product, but it won’t work as a benchmark because they have no side in
common:
DO Have students practice identifying appropriate benchmarks, asking questions such as:
How do you know that … is an appropriate benchmark?
If we are looking for an appropriate benchmark with a side (divisor) of … which rectangle or area is appropriate?
Teachers should continue to model how to check the attributes that make an appropriate benchmark, emphasizing that the length of the side is the same and the product of the benchmark fact is less than (smaller in area) than the unknown fact’s product.
Transitioning to Division
DO To transition to division, teachers will change the language from multiplication language to division language, just as they did in Lesson 20.1, as they continue finding appropriate benchmarks.
DO Give students the division problem 42 ÷ 7. To figure this out, they will find an appropriate multiplication benchmark, only now the language changes to division language:
Please note that at this critical juncture, students will begin to integrate their knowledge of multiplication benchmarks into the process of solving division problems.
The whole rectangle is the dividend (instead of the product). Students need to find a benchmark that is less than or equal to the dividend (less than 42), because those are our attributes of an appropriate benchmark. The length of one side is the divisor, and the length of the other side is the quotient (instead of factors).
DO To figure out which is which, students need to find a benchmark that is less than or equal to the dividend and has a side in common with it. Then when students compare the two, the side that stays the same is the divisor. We always start knowing the length of the side the divisor, because it is given in a division problem. The length of the other side is the quotient. (Note: This is different than the Proportion Strategy, in which the area of the smaller rectangles became the quotient). Students need to compare an appropriate benchmark to the dividend in order to figure out the quotient.
SHOW students the rectangle for 42 ÷ 7 = 6 and ask them to find an appropriate multiplication benchmark for it, based on our two attributes. They will find 7 x 5 = 35 because it is the closest benchmark that is less or equal to 2 and has a side in common with the dividend.
ASK/SAY Which side do the dividend and the benchmark have in common? 7 is the divisor. So the side they have in common is 7. That makes the other side that is left the quotient.
DO/ASK Continue to integrate division language into this process, asking questions such as What is the closest benchmark to the dividend with the same divisor?
DO Continue to model and practice this process until students can fluently identify the dividend, divisor, and quotient:
ð First, give students a division problem that does not have a remainder.
ð Then, have students find an appropriate benchmark that is less than the dividend, and has a side the same.
ð Next, have students show you the divisor, and explain how they know.
ð Finally,have them show you the quotient, and explain how they know.
DO Students need extensive practice identifying appropriate benchmarks. During this practice time, the teacher continues to emphasize division language – dividend, divisor, quotient.
Teachers need to explicitly label both the number sentence and the diagram with these labels. In labeling the rectangle, teachers should rely on fact families (turn around facts) and the missing factor. The missing factor is the length of one side which is unknown. For example, given the area designated by 6 X 4, the dividend would be the area 24 squares. The divisor would be the length of the side that makes the benchmark appropriate. The quotient is the length of the side the student is trying to identify.
Note: Teachers need to avoid language such as “How many times does it go in?” This language implies a proportion model (two 3 x 4 rectangles placed over a 6 x 6 rectangle), which does not match the division algorithm. Instead, teachers use language such as, “What is the closest benchmark with the right length side?” When using stories, it would be appropriate to say, “How many times can….be subtracted from…..” but this is only appropriate after students know how to solve division problems using partial products rules (next segment), and not before.
Using Benchmarks for Division Problems with Remainders
DO give students the division problem 27 ÷ 4, leaving off the quotient.
ASK What is the closest benchmark to the dividend that is smaller with the same divisor? 4 X 6 = 24.
SAY That’s right, 4 X 6 = 24
.
SHOW students a rectangle representing 4 X 6 = 24.
SAY/SHOW So let’s compare the benchmark 4 x 6 = 24 to our dividend of 27.How many less than 27 is 24? 3
SHOW students the subtraction problem 27 – 24 = 3, written as illustrated below, to show the difference:
SAY So when we do division problems that have remainders, this is one way to figure out the remainder.
SAY That’s right, 24 is 3 less than 27. So if we add 3 to our benchmark 4 x 6 = 24, we have 27. So let’s add these three squares to the top of the benchmark, and compare them:
SHOW/DO Show students the two models, laying the benchmark over the dividend to show the remainder, 3. Also have students shading in this representation on their own paper to illustrate the difference.
DO Continue to have students figure out division problems with quotients less than 10 this way, reinforcing division language and referencing benchmarks to illustrate division problems.
Transitioning to the Coordinate Grid
SHOW students the division problem 48 ÷ 6, leaving the quotient off.
SHOW students how to diagram the problem on a coordinate grid, noting that you always put the divisor on the x axis, and since we are looking for an unknown height (y axis), we place a dotted vertical line on the x axis at the end of the divisor line. For example, in the problem 48 ÷ 6 the x axis value would be 6 and a vertical dotted line is placed at x = 6.
DO After students find an appropriate benchmark (6 x 8), have them lay that benchmark on top of the coordinate grid, drawing a line across the top to mark the top of the benchmark. Identify the quotient once they get the problem diagramed. When the problem is on the grid, students should easily recognize that the quotient will be 8, because they already know the multiplication benchmark 6 X 8 = 48.
DO Continue having students practice diagramming division problems without remainders on the coordinate grid this way, identifying the dividend, the divisor, and finally the quotient based on their knowledge of multiplication benchmarks.
Partial Products Strategy for Division and the Division Algorithm
Transferring to the Place Value Mat
Last revised 07/29/2010
Ó2010 University Place School District. All rights reserved. The Math: Getting It Project is a Mathematics and Science (MSP) Partnership funded by the Department of Education. Partners: University Place School District (lead partner), Peninsula School District, and Fife School District; the University of Washington/Tacoma; and the Pierce County Staff Development Consortium, Pierce County, Washington.
For more information, contact the Math: Getting Project Co-Directors, Jeff Loupas or Annette Holmstrom ,
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