I begin this lesson by explaining to students that a multiplicative comparison is when one quantity is multiplied by a specific number to get another quantity. I use multiplication charts to make it easier for students to visualize this lesson. For example, If I say that there are 7 girls in this grade, but 3rd grade has 5 times as many girls as we have. i know to multiply 7 by 5 to get 35. I can also use the multiplication chart and count down five times to get to 35. Now to read this I can say that 35 is 7 times as many as 5 or 5 is 7 times as many as 35. I give other scenarios where students have to use their knowledge of multiplication to solve the problems.
I explain that another way to work this problem out would be to use arrays. I know that there are 7 girls in our grade. Now If I draw 7 girls in one row and make 5 rows of 7. It will show me how many girls there are in 3rd grade. I draw an image on the board for students to visualize.
I ask the following questions to see exactly what students are thinking:
What do the numbers used in the problem represent?
What is the relationship of the quantities?
How is _______ related to ________?
What is the relationship between ______and ______?
Can you draw an illustration to represent this problem?
It is important for students to see a visual interpretation of a problem. It help them to determine the structure of the problem. In other words it helps them determine which algorithm is being used to solve the problem. I go over additional samples using area models.
We will focus on the following Mathematical Practices in this lesson:
MP.2. Reason abstractly and quantitatively.
MP.4. Model with mathematics.
In this portion of the lesson, I want students to look at multiplication problems and come up with word problems that justifies the solution. I put students in groups of 4. I write 5 X 3 and 6 X 2 on the board. I ask them to use arrays to show how this can be done and to also add additional assistance when writing their problems. I encourage struggling student to draw area model, or use graphing paper to shade in the size of the model to represent the given set of equations. graph paper.pdf
I monitor students as they are working. I ask; Read me your problem. I have three dollars. Mom gave me 3 more. I see that there is some confusion therefore I get cubes to make the arrays. I have five yellow cubes in this row. What should I do next? Make two more rows of cubes with 5 in each row. How many will I have then? 15. Ok, so now let's look back at your problem, how much money will you have when your mom gives you 3 more dollars? 8 Is that the same problem? No. Can you think of another problem? I had $5. My mom gave me $5 two more times. Great! Now how much do you have? 15. What is the keyword in what you just said that let's us know that we should be multiplying? times.
I change the lesson around and show students that they can also find missing factors when given one factor and the product. For example 6X_=42. Now, if we know the multiples of 6, we can easily determine that 6X7=42. Therefore, we know that 42 is 6 times as many as 7 and vice versa. I also show students how to use the multiplication sheet and explain that if we put our finger on the 6 and count how many rows we cross to get to 42, we will find our missing product. I give several problems with missing products and ask volunteers to come up and demonstrate how to use the multiplication chart.
I probe students as they are working and only assist them as needed. I gradually release them to work on their own. For instance, I may ask, What do the numbers used in the problem represent? What is the relationship of the quantities? How is _______ related to ________? What is the relationship between ______and ______? Can you represent your problem using an illustration?
Most students are able to answer at ease. For instance, students pointed out the number that represented how many verse the number they were to add or multiply. They also noted the relationship among the different algorithm, and how they worked together to help solve another math situation.
TSW independently find missing factors for problems when given one factor and the product. TSW also be given two problems and create a number problem along with arrays to show understanding. Independent work (1).docx As students work, I circle the room to check for understanding. For instance, I ask, What do the numbers used in the problem represent? ; What is the relationship of the quantities?; How is _______ related to ________? ;What is the relationship between ______and ______? Can you represent your problem using an illustration? How do one operation relate to the other one? Explain?
I make note of students strong and weak areas. I will address these when students time is up.
If time allows I may ask a couple of students to share out what they have learned so far. Having students explain their answers build their ability to justify their answers, as well as, critique the reasoning of others.