I never liked word problems as a student. It was difficult for me to figure out which procedure to use, but I really didn’t like problems like this:
Robert is three times as old as his younger brother Mark. Mark is 7 years old. How old is Robert?
As I reflect on my experience, I realize I memorized everything for school. I was good at filling out my worksheets and getting the right answer, but I didn’t know the concept behind the procedures I used. Deep understanding happens with a web of connections in a students’ mathematical knowledge. It is so important for students to see how things are connected.
Today I will be blogging about the third and final situation for multiplication called Comparison. If you are just joining me, you may want to check out Part One and Part Two. You may recognize the title Comparison because it is also a situation for addition and subtraction.
If we look at a set of 3 and a set of 9, we can compare it two ways. In addition and subtraction we ask the question “What number when added to one set equals the other set?” (What number when added to 3 equals 9?) In multiplication, we ask the question “How many groups of one set do we need to equal the other set?” How many groups of 3 do we need to equal 9?
These questions have been adapted from the K-5 Operations and Algebraic Thinking Progression. The key for students is seeing the multiplicative relationship among the numbers. I will show you two ways for students to express this that I found in NC States Division and Multiplication Learning Trajectories. Click here to read a blog my colleague, Dr. Richard Theissen, wrote about this fabulous tool.
Lets look at a problem:
Zayla has 3 dolls. Her sister Rose has four times as many dolls. How many dolls does Rose have?
Students can use a Division/Multiplication Box to help them see the relationship. They may think about the relationship as multiplication or division so I have given you an example of both. Click on one of the examples to download a blank box.
Students can also use a number line to express the multiplicative relationship. Each jump is the iterated unit. In this example the unit is 3.
Both of these tools will help students look for patterns and build understanding of multiplicative comparison. How have you helped your students understand these word problems that illustrate the multiplicative comparison relationship among the numbers?