What is a part of a part? A story of fractions with pictures

Lesson Objective: Multiplication of Portions

Level 1 Goal - Goal specific to the lesson

Students can draw a picture to represent a fractional multiplication problem

Level 2 Goal - Goal specific to the unit

Students can use words, numbers, and diagrams to explain why the algorithm for the multiplication of fractions works

Level 3 Goal - Broad Subject-Matter Goals

Students use diagrams to support and develop mathematical understanding

Level 4 Goal - Long-Term Goals for student development

Students are confident in their reasoning and persevere through logical explanations

Today students will use an appropriate tool, the area model, to multiply portions. They will return to this tool many times in the future when they multiply quantities. Allow ample time for all students to create a representation, making sure that they make sense of it before you begin closure. By using a model, students will make sense of these problems and persevere in solving them.

Lesson duration: One day (45-50 minutes)

Core Problems:

Dance Problem

Chico Junior High School (you can replace with another school name for buy in) had a fall dance. Half of the students attended the school dance, but only three-fourths of those students came in costume. What fraction of students were in costume at the dance. Justify your solution.

Minecraft Problem

You and your 2 friends are building the floor of a castle in Minecraft, and you are responsible for building your third. You have decided that ¾ of your part will be built with red sandstone. If your friends build their portions with all quartz and wool, how much of the whole floor will be made of red sandstone? Use a picture to support your answer.

Lesson Overview:

It is expected that students are already familiar with the standard algorithms for multiplication of fractions and decimals, so this lesson serves as a review of those procedures through both linear and area models. By diagramming and solving problems within different contexts, students practice and make sense of procedures. Note: While this lesson stresses the use of a generic unit rectangle, do not discourage students who are correctly drawing different representations that also correctly depict the situations. The standard algorithm for multiplying fractions will be reviewed and investigated further in Lesson 2.2.6.

Suggested Lesson Activity:

Begin the lesson with the school dance problem. Project the question on the board and ask the students to think to themselves with no pencils for a minimum of 30 seconds. They should be attempting to come up with a strategy for starting and solving the problem. At this point they can turn and talk to their neighbors and discuss their strategies for solving the problem. After, have students share their strategies with the whole class. One of the goals of the class discussion is highlight students who have chosen to use a diagram, as this is one of our lesson goals.

Most students will be using pictures to multiply a fraction by a fraction. The problems do not actually say “multiply” as the connection to multiplication won’t be made explicit until Lesson 2.2.6. Typically students see the use of the word “of” as meaning multiplication. While this is often the case, avoid letting students latch on to meaningless code words as a way to define what operation to use. They must be able to justify why you multiply in order to use this strategy.

Watch one of Mr. DeLuna's teams tackle this problem.

Some potential student questions:

Do we need to know how many students attend CJHS/MJHS?

Students are given fifteen minutes to work as a team to solve the problem. Look for multiple representations for students to share with the whole class, especially those with pictures. During the class discussion ask students who are willing to share to walk you through their work. One question to emphasize when evaluating picture representations is why the whole school has been cut into eighths. Also encourage students to look for relationships between the algorithm and the pictorial representations.

Next, distribute the Minecraft problem and allow students 15 minutes to work through it as a team. If students are able to solve this quickly ask them if they can use a different strategy to solve it. Again, wander the room looking for multiple representations of the problem.

Samples of student work from Mrs. Cross' and Mr. D's classes


Created with images by Pexels - "close-up cogs gears" • JDmcginley - "masks colorful mask" • mrsdkrebs - "2013-10-18 Collaborative Video Editing"

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