Multiplicative Comparison Problems

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Objective

SWBAT solve multiplicative comparison problems and write multiplication and division equations for those problems.

Big Idea

Students solve multiplication comparison problems using models and equations.

Warm Up

5 minutes

I use this song, counting by 7's, and play it as students enter the classroom.  

They are used to these video and really like them. I sing along and they sing along as we all dance.  

 

 

This video also serve as great brain break and get kids up and moving.  This song helps students work towards mastering standard 4.OA.B.4Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

While students count by 7's, they are singing and saying the multiples of seven and thus working to determine whether a given whole number is a multiple of a given one-digit number.  


Number Hook - Magic Trick

4 minutes

This number trick I call Triple Triple because students think of a three digit number (triple) and end up seeing that number twice after multiplying that number. 

Directions:
Step1: Think of a 3 digit number.
Step2: Multiply it with x7x11x13.

Example: Enter 456 into a calculator and then multiply it by 7, then 11, then 13, and the answer is 456456

Note:  I also tell them that 7, 11, and 13 are prime numbers.  I play this up real dramatic as a little hint about tomorrows lesson. 

My students were very interested in finding out what prime numbers are. Some students reacted by saying they didn't know there were different kinds of numbers. I reminded them that they really do know this because they know that there are even and odd numbers.  Most of my students were very anxious for the next lesson to discover prime numbers. 

Concept development

40 minutes

In thinking about this lesson and the vocabulary, and fully reaching the depth of the CCSS 4.OA.2, I want to stress to my students about how multiplication comparison situations involve the reciprocal of a number.  For example, if one group is fives times as large as a smaller group, the smaller group is the reciprocal of 5, or 1/5 the size of the larger group.  I think this is very important for students to make connections and see relationships, especially with the tape diagram model. It is important to express the reciprocal vocabulary in this lesson in order for students to make connections between multiplication and division and whole numbers and fractions. 

I start this lesson by reviewing comparison language from third grade as well as from the additive comparison lessons in the previous unit: more than, less than.  I write on the board, Summer collected 15 apples while her little brother collected 3.  I then remind students that there are several different ways to compare the two quantities. For example, we could talk about how many more apples one has or how many less apples the other has. I then tell them they we can also use multiplication to compare the two quantities. I ask them to turn and talk with their learning partner and discuss how many times as many apples Summer has than her brother. In the past, most students are able to determine that Summer has 5 times as many apples. I ask learning partners to share with the class how they knew it was 5 times as many.  Some students say they know it because 5 x 3 is 15. Other students might say they know that 15 divided by 3 is 5.

I then draw a tape diagram to model this situation. Students draw a tape diagram in their math notebooks to model as well. (see photo)

At this time, is when I then tell students that we can also describe the smaller amount in terms of the larger amount. I express that we use a unit fraction to do so because we are fracturing, or equally dividing, something. So if we divide Summers apples equally into groups of three, we get five equal parts. Each of these equal parts is called one fifth written as 1/5.  I have them practice the sentences with me by describing the larger amount in terms of the smaller amount and then the smaller amount in terms of the larger amount. 

For the next part of the lesson, students will use their math notebooks.  I give various multiplicative comparison situations for students to model in their notebooks with tape diagrams, allowing students to use Math practice standard 4. I have students orally practice using the two different equations to compare the quantities.  From past experience, this is difficult for students when expressing the smaller amount in terms to the larger amount.  In the past my students have needed guidance for this. As students say the sentences, I write a matching equation on the board.  For example, in the apple situation, I would write 15 = 5 x 3 and 3 is 1/5 times as many as 15 which is 3 = 1/5 x 15 or 3 = 15/3. It is important for students to be comfortable and familiar with all equations and the different comparison language.

After several similar examples, I then ask students to write situation equations to match situations I tell them. For example,  Asher ate three times as many cookies as Kylee.  I then write A = 3 x K  I lead a class discussion regarding what the A stands for (Asher) and the K (Kylee).  I also reverse the situation and tell them that Kylee ate 1/3 times as many cookies as Asher and write K = 1/3 x A  I do several examples like this. After I sense that students are comfortable with writing situation equations, we put it all together to solve, model, and write an equation to match.  Students needs lots of practice with seeing and writing variables in order to solve word problems and master CCSS 4.OA.3.

Students work independently at solving the multiplication compare problems practice page.  For this assignment, I do not require students to write the situation equation comparing the small number in terms of the large number as a fraction mutlplied with a whole number. CCSS 4.NF.4 asks students to apply and extend previous understanding of multiplication to multiply a fraction by a whole number. At this point in the year, students have not had the lessons necessary to do this, thus I do not have them write that equation.  Many students write a division equation which is acceptable and anticipated for this time of year.

 

Student debrief - Wrap up

5 minutes

I end this lesson with a big roller coaster cheer.  From experience, this lesson is challenging for students as they think about equations for situations and make sense of multiplication comparisons.   I go over the answers to the independent practice page. I ask students to raise their hands for answers and then have others give a thumbs up if they agree of a thumbs down if they disagree or have a different answer.