Exploration: Multiplying Fractions & Decimals Using an Area Model REPRESENT MULTIPLICATION SITUATIONS INVOLVING FRACTIONS & DECIMALS WITH MODELS INCLUDING PICTURES IDENTIFY FACTORS OF POSITIVE INTEGERS Prerequisites Some experience with fractions, decimals, & percents Preparation Worksheet # & # for each student Overhead # Color Tiles or Substitute Exploring Factors The first part of this lesson may seem unnecessary since it focuses on factors of a given whole number, and idea of factors does not apply to fraction or decimals. However, the factor portion of this lesson gets student comfortable representing multiplication problems using area models. This is an important connection to make before moving on to multiplying fractions or decimals. Give each student a handful of color tiles. You ll need exactly tiles. Please set the rest aside. I want you to find all the rectangles you can make with exactly tiles. Demonstrate by making one of the rectangles on the overhead and explain that, for instance a 3 x 4 rectangle counts the same as a 4 x 3 rectangle. Remember to use all tiles. Once student have found the three possibilities, pass out Worksheet # and show them how to record. the answer on page 3 under the section for tiles. Have students work through the rest of the numbers through, and then as a class work through question #-#6: Without building it, how many rectangles can be made with 3 tiles? How do you know? Which numbers had the most rectangles? Is that what you would have expected? Which numbers had the least rectangles? Is that what you would have expected? Do you see any patterns? Which number would you predict would make the most rectangles:, 3, 4? Why? Which would make the least rectangles? Why? by = 6 by = 4 by 3 = 3 Multiplying Fractions Now that students have some experience representing whole numbers using the area model, it s a good time to segway to multiplying fractions. Pass out Worksheet # and begin by asking: What would / by or / x look like? Draw in space # what you think that rectangle would look like. Discuss what students drew and record it on the overhead. : www.mathlessonbank.com pg. of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL LESSON PAGES # by or x Note it is important to label the / on the x or y-axis. Of course it doesn t matter if they make the rectangle / x or x /. I try to keep students in the habit of putting the x-axis number first, and the y-axis number second. Now the question is, how big is this rectangle? Drawing the whole will help. # OR # by = or x = Now let s do x /3. # by = or x = : www.mathlessonbank.com pg. of 4
It is important to mark both /3 and /3 on the axis. What size is this rectangle? The best way to make the answer clear is to draw the whole in a solid line and to divide the parts using dotted lines. #3 is / x / = Circulate as students work. Then discuss the answer, always beginning by drawing the shaded portion and then asking what size the rectangle is. #3 # x = by = or x = #3 x = ¼ EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL LESSON PAGES first then Continue working through the first 8 as a class. #4 #5 x = x = : www.mathlessonbank.com pg. 3 of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL LESSON PAGES #6 #8 x = /6 x = 4/5 #7 ¾ /4 ¼ x ¾ = 6/ Then have students work through #9-#6 on their own. Here are some suggested problems: 9. 3/5 x /3. /6 x 3/4. 3/4 x 3/4. /3 x 4/5 3. 3/5 x / 4. 3/5 x /4 Challenge Problems: 5. x 6. ¼ x /3 Multiplying Decimals This process of multiplying fractions using the area model easily transitions into multiplying decimals. I begin by passing out Worksheet #3. Show me what / x / will look like on graph #. The lines are already there to make it easier to determine that / x / is /0. : www.mathlessonbank.com pg. 4 of 4
I begin with the fraction number sentence, since that is what student were just working with. Beneath it I record the number sentence in words, and finally I record it using decimals. I think progression helps bridge from fractions to decimals. # 9 8 7 6 5 4 3 3 4 5 6 7 8 9 x = tenth x -tenth = -hundredth 0. x 0. = 0.0 0 EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL LESSON PAGES For #, work through 6/ x 5/ on your own. As we did with #, record problem in words and in decimals as well. # 6 x 5 = 30 0 6 tenths x 5-tenths = 30-hundredths 0.6 x 0.5 = 0.30 9 8 7 6 5 4 3 3 4 5 6 7 8 9 : www.mathlessonbank.com pg. 5 of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL LESSON PAGES Here are some suggestions for the rest of the problems: 3. 0. x 0.6 = 4. 0. x 0.7 = 5. 0.8 x 0.3 = 6. 0.9 x 0.9 = Challenge Problems: 7. 0.5 x. = 8. 0.7 x.3 = Here s what challenge problem #8 would look like: #8 9 8 7 6 5 4 3 0.7 x. = 0.84 3 4 5 6 7 8 9 : www.mathlessonbank.com pg. 6 of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL OVERHEAD # pg. 7 of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL OVERHEAD # pg. 8 of 4
Name: Multiplying Fractions # # #3 #4 Date: EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL WORKSHEET # #5 #6 #7 #8 pg. 9 of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL WORKSHEET # #9 # # # #3 #4 #5 #6 pg. of 4
Name: # Multiplying Using Area Model Date: EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL WORKSHEET #3 # pg. of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL WORKSHEET #3 #3 #4 pg. of 4
#5 EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL WORKSHEET #3 #6 pg. 3 of 4
EXPLORATION: MULTIPLYING FRACTIONS & DECIMALS USING AN AREA MODEL WORKSHEET #3 #7 #8 pg. 4 of 4