Wrapping Up Multiplication and Division with Multi-Digit Numbers
In this unit, students multiply and divide multi-digit whole numbers using place value understanding, properties of operations, and the relationship between multiplication and division. They use the standard algorithm to multiply multi-digit whole numbers and partial quotients algorithms to divide whole numbers up to four digits by two digits. They then apply these skills as they solve problems involving volume.
Section A: Multi-digit Multiplication Using the Standard Algorithm
Students begin this unit by estimating products and quotients in a real-world context. Students use their understanding of place value, and their understanding of powers of 10 to make reasonable estimates. Students connect multiplication strategies, like partial products, to the standard multiplication algorithm. This is the partial products area diagram for \(412 \times 32\).
They find partial products using area diagrams, and then translate that to a series of equations. These equations are compared against the steps in the standard algorithm to learn how the steps are based on place value reasoning and why the algorithm works. This table shows the connection between an algorithm using partial products and the standard algorithm.
Partial Products Area Diagram
Area Diagram Aligned to Standard Algorithm
Section B: Multi-digit Division Using Partial Quotients
Students begin the work on whole number division by deepening their understanding of division expressions and the effect that changing the divisor or dividend has on the value of the quotient. In a progression that leads to students engaging in algorithms using partial quotients, students estimate quotients and write partial quotient equations that match their own methods for finding the value of the quotient. Once students understand that they can find the value of the quotient by decomposing the dividend into multiples of the divisor, students learn to express this decomposition using equations and then an algorithm using partial quotients.
Decomposition of the Dividend
An Algorithm Using Partial Quotients
Section C: Let’s Put it to Work
Students practice their multiplication and division skills as they solve problems involving volume. Students are using the volume formulas (\(V = l \times w \times h\) and \(V = b \times h\)) to practice the multiplication and division work of the previous sections. Students engage with relatively large numbers to multiply and divide using these volume formulas, developing fluency with the standard algorithm for multiplication and the algorithm using partial quotients.
Try it at home!
Near the end of the unit, ask your student to solve the following problems:
- \(219 \times 52\)
- \(868 \div 14\)
Questions that may be helpful as they work:
- Can you draw a diagram to help you solve the problem?
- Can you explain the steps of your algorithm?