Model and Solve Two-Step Multiplicative Comparison Problems
Two-step multiplicative comparison problems require drawing a tape diagram to model the relationship between quantities, writing an equation with a variable, and solving in two steps, as taught in Grade 4 Eureka Math. A typical problem might state that one quantity is a multiple of another, then ask for a combined or remaining value. For example, if Sam has 4 times as many stickers as Jo’s 12, find Sam’s count then add both. The tape diagram makes the multiplicative relationship explicit before equations are written.
Key Concepts
To solve a two step word problem involving multiplicative comparison, first draw and label a tape diagram to model the relationship between the quantities. Then, write an equation with a variable for the unknown value and solve it. The structure often involves finding a larger quantity through multiplication and then performing a second step of addition or subtraction: $Total = (n \times \text{base amount}) \pm \text{other amount}$.
Common Questions
What is a multiplicative comparison problem?
A problem where one quantity is described as a multiple of another. Example: ‘Ana has 3 times as many books as Ben.’ The comparison uses multiplication rather than addition or subtraction.
How do you model a multiplicative comparison with a tape diagram?
Draw one bar for the smaller quantity. Draw another bar that is n times as long for the larger quantity. Label known values and mark the unknown with ?.
How do you set up the equation from the tape diagram?
Let the unknown = variable. Write the equation based on the multiplicative relationship shown. Example: if larger = 4 × smaller, and smaller = 12, then larger = 4 × 12 = 48.
What makes a problem ‘two-step’?
Two-step means you solve an intermediate equation first, then use that result to answer the final question. Example: find the larger amount (step 1), then find the sum of both amounts (step 2).
What are common errors in multiplicative comparison problems?
Confusing ‘times as many’ (multiplication) with ‘more than’ (addition). Also forgetting the second step after finding the multiplied value. The tape diagram prevents both errors.