Math Final Lesson Plan
Distributive Property of Multiplication over Addition
SESSION TITLE: Distributive Property of Multiplication over Addition
Grade: 4th
GOALS OF THE SESSION:
۰The students should be able to understand the distributive property.
۰The students should be able to explain and justify the process of the distributive property.
۰The students should be able to explain the distributive property through the use of concrete and semi-concrete examples.
ENDURING UNDERSTANDING:
۰Students understand for all whole numbers a,b,c, a * (b+c) = (a*b) + (a*c).
MULTIPLICATION REPRESENTATION (TOOLS):
۰Manipulatives
۰Drawings – draw the blocks model
۰Worksheets
SESSION-RELATED QUESTIONS:
۰Why not leave the problem in a * b terms?
۰Will you get the same answer if you add the b+c , and then multiply by a?
۰When will we ever use this in real life?
RESEARCH/BEST PRACTICES:
When looking at research and best practices, we found several websites and texts that gave examples of the best methods to use when teaching children how to apply the distributive property of multiplication over addition. The following are examples of what we found:
The problem 8 x 7 is equal to 8 x (5+2) which is equal to (8 x 5) + (8 x 2). To represent this problem, the text suggests having a segmented rectangle (8 by 7) and cutting the model to be two rectangles, one that is 8 x 2 and the other 8 x 5. The students will then see that the two rectangles are equal to 8 x 7 and both 8 x 7 and (8 x 5) + (8 x 2) are equal to 56.
In addition to this example, we found numerous other resources that suggested a similar
model, but using grid paper rather than homemade segmented paper. We feel that the segmented paper would probably be more effective for 4th graders because it would be much larger and easier to visualize.
Give the students homemade tiles to represent the distributive property.
Show that 3(2+1) is equal to 6 + 3 by having tiles that are equal to designated units. Show how 3 two units and 3 one units can be arranged to equal both
3(2 + 1) and 6+3.
7 x 5 = (5 x 5) + (2 x 5) and this is represented by placing the cards like this…
* * * * * * * * * * ***** *****
* * * * * * * * * * ***** *****
* * * * * *****
* * * * * *****
* * * * * *****
(5 x 5) + (5 x 2) = 35
(25) (10)
SD K-12 CONTENT STANDARDS & NCTM STANDARDS & PRINCIPLES:
K-12 SD Standards
SD K-12 Algebra, Grade 4, Standard 1:
1. The student will relate the concepts of addition, subtraction, multiplication, and division to one another.
Rational and Justification: According to the NCTM, students should acquire further knowledge of multiplication as they work with different multiplication concepts and relate these concepts to prior mathematical knowledge. It is important to foster this relationship in multiplication by “showing how a product is related to its factors.” “The distributive property is particularly powerful as the basis of many efficient multiplication algorithms.” Our activity is an effective way to meet this standard because the students work with the distributive property and are able to relate the property to prior knowledge, since they are already competent in the area of addition and in the area of simple multiplication. As the students break down the problem from a two digit number into two numbers that add up to that number, they are able to see how addition and multiplication are directly related. When the students work with the semi-concrete model, they are again seeing how multiplying and then adding simple equations (using the distributive property), gives you the same answer as multiplying the original (more difficult) problem.
SD K-12 Algebra, Grade 4, Standard 3:
2. The student will relate the concept of addition, subtraction, multiplication, division, and parenthesis.
Rational and Justification: After searching the NCTM, and multiple texts, we were unable to find any rational that supported the standard. After much debate and thought, we decided that the standard definitely meets our activity and therefore must be included in the standards, even without the rational to support it. The reason we believe this standard meets our activity is because the distributive property of multiplication over addition is based on the concept that multiplication and addition are directly related. We are showing the students how multiplication and addition are related when we have them take a multiplication problem and break it down and add it to one another to get the same results. It is important for students to see that multiplication and addition are related to help students see that multiplication is not a foreign concept, and it can be related to information they already know (addition).
SD K-12 Algebra, Grade 4, Standard 7:
7. The student will describe given problem situations in multiple ways.
Rational and Justification: According to the NCTM, students need to be able to represent problems in numerous ways. Students should be able to understand problems so they “can generate equivalent representations for the same number.” In order to master this idea, “students need to have many experiences decomposing and composing numbers in order to solve problems…” They also need to model “multiplication problems with pictures, diagrams, or concrete materials” to help them learn about what factors represent. It is important to use multiple ways to solve problems because it helps show students that there is not just a single way to solve a problem. By showing the students how to solve the problems using the distributive property, both by drawing out blocks on paper (semi-concretely) and then solving it traditionally with just a written problem (symbolically), and by including an adaptation for those really struggling (having the student use concrete blocks), we are giving all the students an opportunity to see how to solve the problem using various methods. Students are then able to take this information and describe the problem in multiple ways. In addition to the different methods used to solve the problem, the students are decomposing the original problem using semi-concrete or concrete methods. By having the students solve problems using all the different models, we are ensuring that they understand that there are multiple ways of solving a multiplication problem involving the distributive property and multiple ways of describing that problem.
IMBEDDED ASSESSMENT OPPORTUNITIES:
Pre-assessment – observing the students as they solve the first example problem (2 * 12) and the explanation of the answer to this problem
Formative assessment – The example problem they work on in their groups, and the worksheets that are completed in the group
Formative observation – questions asked of the students throughout the session, and rubric of group participation
INSTRUCTOR MATERIALS:
۰Blackboard
۰Worksheets
PARTICIPANT MATERIALS:
۰Pencil
۰Paper
۰Worksheets
SESSION NOTES:
1) Have students pair up with a partner to figure out more than one way to solve the problem example 2 * 12. After the students have discussed their problem, regroup, and have the groups share their outcomes. Also, explain the process they used to come up with their outcome. As the students are explaining their procedures the teacher will be representing their outcomes on the board.
2) After the students have given their procedure and outcomes, discuss with the students the different ways people solved the problem. Ask the students if some of the procedures used were easier than others? Why? Then find an example that uses the distributive property or one that looks similar and focuses on this outcome. The teacher will then go back to the original problem and show how 2 * 12 is equivalent to 2 * (10 + 2), which equals (2 * 10) + (2 * 2). As you are explaining this concept to the students, be writing it on the chalkboard, including arrows to show how the 2 distributes to each the 10 and the 2. Explain to the students that this is the distributive property. Ask the students-what are some reasons that we would use the distributive property instead of just solving the original problem? Do you think it makes some multiplication problems easier?
3) Write another example problem, 4 * 13, on the board (appendix A). Rewrite the problem to be 4 * (10+3) and ask each the students to take out a piece of paper and individually solve the problem just as you had on the board (semi-concretely). As the students solve the problem, walk around and see their work and assist as needed. When each person is done, they will put their pencil down and you will check to see if they solved the problem correctly. (For the students that are struggling, see adaptation below.) When all students have had their work looked at, complete the problem on the board, solving semi-concretely and symbolically.
4) Break students into groups of four. One more example problem will be placed on the board, and the students will be asked to solve the problem using a semi-concrete example. The teacher will write the problem 6 * 18 on the board, and the students will work with their groups to show the problem solved semi-concretely, and also solved symbolically. When the group is done with their work, one group member will represent their answer on a designated spot on the board. Another group member will share their explanation with the class. This is to ensure that all of the students understand the distributive property of multiplication over addition both semi-concretely and symbolically. Groups will then be given a worksheet to finish in class (appendix B). When they have completed their worksheet, one group member will shared one of their answers by writing it on the board (both semi-concretely and symbolically) and another group member will explain how they came to that answer. Throughout the group work time, observe the students work while referring to the group work rubric and scoring the students appropriately (appendix C). When the students are done with the group worksheet, the students will be asked- could the problem 40 * 260 be solved using the distributive property? What would it look like? Is there anytime that it would not be beneficial to use the distributive property when solving multiplication problems? Why?
5) At the end of the lesson the teacher will give the students a worksheet over the distributive property of multiplication over addition, which involves solving problems both semi-concretely and symbolically (appendix C). This worksheet will be reviewed the following day, along with opportunities to ask any questions about the concept. Students will then be asked- can you think of examples of real life situations when the distributive property of multiplication over addition would be useful?
ADAPTATION:
For those students that are greatly struggling with using the semi-concrete and symbolic method of solving the problems, have blocks available so they are able to see how to solve problems involving the distributive property of multiplication over addition using concrete methods (an array model). As students grasp the concrete method of solving, scaffold their learning to the semi-concrete model on paper, and then to symbolically solving the problems.
ASSESSMENT:
Summative assessment – a homework worksheet that determines how well each student understands the concept of the distribution property of multiplication over addition. The upcoming chapter test will also include problems that need to be solved using the distributive property of multiplication over addition.
RATIONAL AND JUSTIFICATION FOR LESSON PLAN FOR UNDERSTANDING:
By researching and through a series of tedious discussion with mathematic educators, we came up with the belief that the property of distribution of multiplication over addition was a concept that most students struggle to fully understand. After researching in books of mathematic theory, in articles and discussing with teachers, we found numerous amounts of data that supported our beliefs to be accurate.
The property of distribution of multiplication over addition needs to be taught for “understanding to engage students in performance of understanding” (Wiske). However, these concepts need to address more than an “idea about the nature of understanding and its’ development” (Wiske). You must base your lesson on four key questions in order for students to have a relational understanding of a concept. A teacher should ask themselves when thinking of a lesson if “a topic is worth understanding, what about the topic needs to be understood, how can I foster understanding and how can we tell what students understand?” (Wiske).
When planning to collect evidence of understanding, teachers should consider a range of assessment methods (Wiggins & McTighe). Understanding develops as a result of ongoing inquiry and rethinking, the assessment of understanding should be thought in terms of a collection of evidence over time, instead of an event” (Wiggins & McTighe).
When approaching the concept of the property of distribution of multiplication over addition, one must utilize the six facets of understanding. By doing so, students can acquire a meaningful understanding for the concept, thus ensuring life long comprehension and application. Also, by designing a lesson that involves concrete, semi-concrete and symbolic thinking skills students can develop the “ability to explain their answers and how the developed it” (Wiggins & McTighe, 47).
When delivering this lesson we felt that group work of problem solving would allow the students to foster a deeper understanding. Then, individuals would have the capability to apply this concept of their own via a worksheet because “to understand is to be able to use knowledge” (Wiggins & McTighe, 51).
We feel that we have found a concept that is difficult for students to understand, and have strategically designed a lesson in which students enhance understanding. By utilizing different teaching strategies for the diverse learner the students can “grasp the concept, which can bring then to bear on new problems and situations” (Wiggins & McTighe, 51).
References:
Wiske, M.S. (1998). Teaching for understanding: linking research with practice. San
Francisco: Jossey-Bass.
Wiggins, G., & McTighe, J. (1998). What is backward design? In G. Wiggins & J. McTighe (Eds.), Understanding by Design (pp. 7-19). Alexandria, VA: Association of Supervision and Curriculum Development.
Wiggins, G., & McTighe, J. (1998). Understanding understanding. In G. Wiggins & J. McTighe (Eds.), Understanding by Design (pp. 7-19). Alexandria, VA: Association of Supervision and Curriculum Development.
Appendix A
Semi-Concrete example to be used on board
4 x (13)= ____
Show the students that 4 x (13) is equal to 4 x (10 + 3) is equal to (4 x 10) + (4 x 3)
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4 x 10 + 4 x 3 = 52 |
This way students can see that the 4 is distributed to both the 10 and the 3 to be the equation 4(10+3)=52.
Appendix B
Distributive property worksheet
Group Members:
Complete the following problems using the distributive property of multiplication over addition. Make sure to show all work using the space along the right side of the paper.
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Appendix C
Group Assessment Rubric
Group Members: ____________________________________________
Date: ___________ Subject: _____________________
Assignment: ________________________________________________
Rating Key: (low) 1-5 (high)
___ 1. Work Cooperatively:
___ 2. On Task:
___ 3. Board Work:
___ 4. Explanation:
Total ___/20
Distributive Property Homework Worksheet Appendix D
Name________________________
Date_________
Solve the following equations using the distributive property of multiplication over addition. Be sure to represent your answer in the box and show all work to the right of the problem.
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Solve the following problems using the distributive property of multiplication over addition. Be sure to show all work.
7. Do you think the using distributive property is a good way to solve some problems? Why? Is there a time when it wouldn’t be good to use the distributive property? Why?