Introducing Multiplication of Fractions to Whole Numbers: Building on Prior Knowledge
Lesson 2 of 8
Objective: SWBAT show two different ways of multiplying a fraction by a whole number.
In order to get students thinking about math facts, I asked students to count by 12's. I had them stand in a circle and every time the number had a "two" in it, they would say "buzz". This helped them think about place value and where the two would be. We counted around the room as I pushed them to be thinking about what number would be next by thinking ahead. So, if you heard the person two people away from you say "24" you already needed to be thinking about saying "48". I demanded complete silence except for the person who should be counting. This increased their listening and their concentration. If you missed or forgot to say "buzz', you needed to do a yoga pose until it was your turn again. This was better than sitting down, because you still had to listen and then concentrate on what you would be saying on your next turn. The kids loved it!
We went around the room once and then when I started again, I started them on 25. That way they had to add twelve to an odd number and look for patterns. When we were done, I asked if they saw a pattern with adding an even number to an odd? How many times did a 2 show up this time?I also asked them how they quickly added.
These questions were answered by two people who really were quick at their turns. One told us he adds ten and then two. I heard a lot of "ohhhh!"
We had so much fun with this!
Classroom Notes SB File: This portion of the lesson was taught whole group. I knew I needed to do this because the entire class did not understand the concept of showing and proving multiplication with a fraction and a whole number. I developed this SB that systematically develops their understanding using prior knowledge and connections.
On the first page I explained exactly what students needed to know to master the standard. I then turned to the second page and explained how we needed to activate our prior knowledge of multiplication and what it means to multiply. Students generated answers and I wrote them on the SB. Multiplying Fractions and Whole Numbers SB Classroom Notes. On the next page, we showed different ways to express 3x3. I connected whole numbers to whether or not it would work with fractions because I wanted students to understand that concepts of multiplication do not change, just the numbers can. I think this is important because then they can see that they already know something about multiplying fractions!
As we explored solving 1/3 x4. I asked a higher achieving student to show us another way of writing the equation. He wrote the addition equation instantly. I asked him to explain why his addition equation equaled the multiplication expression. He explained that it would be the same thing as saying 3 groups of 4 or adding three four times, except it was fractions instead. The next student drew fractional models without hesitation. She showed complete understanding of the concept and could show the improper fraction. She remembered to decompose it too. This is where I saw the standard come to life and the depth of understanding show. Having my students explain the concepts to others engaged the whole class as we were learning together.
I asked them what they thought would happen if we doubled the numerator and multiplied it by four? We solved the problem together as I wrote out the drawings and continued to show it two ways. They immediately saw the answer had doubled. One student asked if it happens with whole numbers and another student chimed in and explained. He said that if you double two and make four and times it to the same number, the answer doubles. He gave the example
"2x2 is 4 but 4 x 2 is 8. 8 is double four." Suddenly, I didn't need to teach and the number sense talk was going on around the room with kids giving kids examples of different products that showed the same idea.
I let them all discuss this idea for a minute and continued with the lesson. This idea transferred into the next concept that I was worried about on page 4. I realized I needn't have worried since they understood and saw the process because of the order in which I had taught it. We moved on to the next page as I taught them the concept of how any integer can be written as a fraction because the denominator is 1, for 1 whole. This clicked too!
I turned to a blank page and wrote a multiplication problem for them to try independently. They had to show two ways of solving it. I asked them to consult with their group after they finished and compare drawings and answers. I roved the classroom and saw that everyone was drawing their fractional models more than using repeated addition. I saw that one student was drawing pictures that did not represent multiplication and had lined up the numbers as if she were adding. "Not quite sure yet" shows how she had lined up numbers, used a comparison model and could not explain what she was doing yet. I needed to work with her one on one.
We worked on one more together before I assigned homework.
Aha! and Homework
With every new concept and idea, I try to close the class with an "Aha!" moment sharing time. I think this helps students understand and monitor their mastery of the standard. "Aha!" moment.
I continued asking them to share some more. (See: A whole number is written differently in order to multiply) One more student shared his concepts and we realized he needed to go one step further to prove his answers. (See: Show me you can combine them). I told that they would have time now to work on their assignment but that they needed to be sure to ask any questions if they were stuck. I also told them to be sure to show two ways to prove to me that they could master the standard.
I assigned 10 problems from IXL.com using their iPads. On this website, there are two leveled assignments on level F (4th grade). S.1 is mainly unit fractions times a whole number and S.2 uses larger fractions and whole numbers. I told them that if they found that S.1 was too easy, to switch to S.2. My students who master above grade level work were expected to do all 10 from S.2.
In this twenty minutes, I was able to work with a few students who were having difficulty. After the time was up, I felt confident that they could go home and finish their work. I emailed the pdf file to their emails so they had copies of the notes from the class.