MthEd 377 Lesson Plan
Cover Sheet
Name: Krista Garner / Date: September 12, 2005Section Title: Capacity and Mixed Number Multiplication
Big Mathematical Idea:
Multiplying m x n involves adding groups of m (or n) units together n (or m) times whether m and n are whole numbers or whether they are mixed numbers
Why is this topic important?
Because mixed number multiplication is a skill that commonly arises in solving other mathematical problems
Because mixed number multiplication (in its comparison to whole number multiplication) enables students to recognize that the field of mathematics is consistent and orderly
How does this lesson fit in to the overall unit? (i.e., How does this lesson build on the previous lessons and how do subsequent lessons build on it?)
This lesson builds on previous lessons that introduced mixed numbers and mixed number addition (and subtraction) by providing students with an opportunity to experience the other basic component of mixed number arithmetic—namely, mixed number multiplication. It enables students to realize that these basic operations that they first learned in elementary school are at the heart of a lot of mathematics and that the field of mathematics obeys predetermined rules and procedures.
Learning mixed number multiplication now also prepares students for discovering mixed number division in subsequent lessons. These two operations are closely related for two reasons: first, mixed number division is obviously the inverse operation of mixed number multiplication; and second, mixed number division, which involves that mysterious invert-and-multiply rule, consequently uses mixed number multiplication practically by definition.
Grading rubric (for Keith’s use)
5 The Big Mathematical Idea addresses core mathematical concepts and is clearly articulated
5 Description of the importance of the topic is well thought out and relevant
5 There is a clear, insightful discussion of how this lesson fits in to the mathematical content of the overall unit
5 Lesson sequence is well thought out and detailed
5 Students' thinking is anticipated with forethought and detail
5 Reactions to students' thinking is mathematically oriented, insightful and detailed
10 3-5 reflection paragraphs demonstrate thoughtful reflection and are clearly articulated / 10 Met with Dr. Leatham and made appropriate revisions based on this discussion
30 3-5 page reflection paper demonstrates thoughtful reflection and is clearly articulated
Lesson Sequence: Learning activities, tasks and key questions (what you will do and say, what you will ask the students to do) / Time / Anticipated Student Thinking and Responses / Your response to student responses and thinking / Formative Assessment, Miscellaneous things to remember /
Launching the Lesson
Review what it means to multiply two whole numbers
Review definition of mixed numbers by asking the students to compare whole and mixed numbers
Ask, “In what kinds of situations might we have to multiply mixed numbers instead of whole numbers to solve a problem?”
Explain that we will be engaging in an activity today that explores the idea of multiplying mixed numbers / 5:00 minutes / “m x n means taking m/n groups of n/m”; “m x n means taking a group of m/n n/m times”; “It is like having m/n groups of n/m and adding them all together”
“A whole number and a fraction squeezed together”; an incorrect definition (like the definition of an improper fraction)
Various situations (including area calculation, recipe adjustment, etc)
Student response not required / Positive reinforce-ment of good ideas, with an emphasis on the idea that it is like having m/n groups of n/m and adding them all together
Write “whole number and fraction” on the board with an exam-ple; provide an identi-fication of what the incorrect definition really defines and a reminder of what a mixed number is
Positive reinforce-ment of good ideas; other questions that will enable the stu-dents to tweak slight-ly incorrect ideas
Teacher response not required
Orchestrating the Task
Organize the students into five groups; explain the roles each group member will play; instruct the students in each group to number off
Distribute pattern blocks and a worksheet to each group; instruct the students to begin
Move throughout the classroom as the students engage in the task and complete the worksheet / 10:00
minutes
If the students do not need this much time for the task, then we will spend more time in the discus-sion / Timely and appropriate response; slow or rowdy response
Same response as above
Students may define one block to be a whole cup and another to be ⅓ of a cup; they may only define one block to be ⅓ of a cup and transform the mixed number into a proper fraction; they may define every block and then make a number of transformations or realize that they do not need every block; they may define the blocks differently; they may perform the multiplication operation incorrectly / Positive reinforcement of good behavior; courteous reminder of what proper behavior is
Same response as above
Look for groups that are solving the problem in these different ways and ask them to write their series of steps on the board; ask questions that will enable students who are solving the problem incorrectly to readjust their thinking / The first solution described in the “Anticipated Student Thinking and Responses” category exemplifies the distributive property method
The second solution described in the “Anticipated Student Thinking and Responses” category exemplifies the improper fraction method
Facilitating the Discussion
Draw the attention of the class back to the teacher
Invite a representative from a group that used the distributive property method to explain their series of steps; lead the class in a discussion about the benefits and disadvantages of this type of solution
Repeat the step above for a group that used the improper fraction method and for groups with other solution methods / 10:00
minutes / Timely and appropriate response; slow or rowdy response
“This method works well when multiplying a whole number and a mixed number”; “This method may not work as well when multiplying two mixed numbers”; “Is there an algorithm [although students may not use that word] that we can use for this method?”
“This method works well when multiplying two mixed numbers”; “This method is not as easy as the previous method when multiplying a whole number and a mixed number”; “Is there an algorithm [although students may not use that word] that we can use for this method?” / Positive reinforcement of good behavior; courteous reminder of what proper behavior is
Positive reinforce-ment of good ideas; probe for explana-tions as to why these benefits and disad-vantages exist for this method; explain that we will formalize the algorithm in a subsequent lesson
Same response as above / If either the distributive property method or the improper fraction method is not described by any group, the teacher may consider presenting at least one of the methods and then following the pattern outlined in the “Lesson Sequence” category
Debriefing the Lesson
Ask, “How did we define whole number multiplication at the beginning of the lesson—say,
m x n?”
Ask, “Can we define mixed number multiplication in the same way?”
Explain that in a subsequent lesson, we will learn how to perform mixed number multiplication / 5:00
minutes / “It is like having m/n groups of n/m and adding them all together,” or something similar
“Yes, but the size and number of the groups is not a whole number anymore”; “No, because we had to use the distributive property method”; “No, because we had to use the improper fraction method”
Student response not required / Positive reinforce-ment of good idea
Positive reinforcement of good ideas; explain that a group of (1+⅔) is just like saying a group of one and one-third, and so on; explain that even though we may have to perform an extra step, it does not change the definition
Teacher response not required
Names ______Date ______
______
______
______
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Exploring Mixed Number Multiplication
Person 1: Scribe
Person 2: Pattern Block Manipulator
Person 3: Pattern Block Manipulator
Person 4: Quality Controller / Presenter (if necessary)
Person 5: Quality Controller
Each problem* addresses a problem that can be solved using mixed number multiplication. Follow the steps and use the pattern blocks to help you solve each problem.
1. You have been asked to prepare 5 batches of chocolate chips cookies for a community bake sale. The recipe for a single batch calls for 1 ⅔ cups of chocolate chips. How many cups of chocolate chips do you need to make all 3 batches?
(a) First, decide which pattern blocks will represent which measurements. Do not forget to write down these relationships.
(b) Next, use the pattern blocks to determine how many cups of chocolate chips you need to make all 5 batches, making sure to express your answer in simplest terms.
______
(c) Now that you know the answer, think about the individual steps you took with the pattern blocks to find this answer and write them down in order.
*Obviously a real worksheet would be longer than this one.