Partial Products in Algorithms
Warm-up: Which One Doesn't Belong: Multiplying Large Numbers (10 minutes)
This warm-up prompts students to compare four representations of multiplication. Students compare diagrams and equations that represent multi-digit multiplication. This prepares them for the work of the lesson where they compare different ways to represent products as sums of partial products.
- Groups of 2
- Display the image.
- “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
- 1 minute: quiet think time
- “Discuss your thinking with your partner.”
- 2–3 minutes: partner discussion
- Share and record responses.
Which one doesn't belong?
- “Why doesn't B belong?” (It's not a diagram. It's an expression.)
- “Does the value of expression B match the value represented in any of the diagrams?" (Yes, diagrams A and C both represent the product \(4 \times 5,\!342\) and that's the same as B.)
Activity 1: Partial Products Everywhere (20 minutes)
The goal of this activity is for students to examine different ways to write the product of a three-digit number and a two-digit number as a sum of partial products. Students match sets of partial products which can be put together to make the full product. Students are provided blank diagrams, familiar from the previous lesson, that they may choose to use to support their reasoning. In the activity synthesis, students relate the expressions and diagrams to equations to prepare them to analyze symbolic notation for partial products in the next activity.
When students relate partial products and diagrams to the product \(245 \times 35\) they look for and identify structure (MP7).
Advances: Speaking, Conversing
Materials to Copy
- Partial Product Expressions
- Create a set of cards from the blackline master for each group of 2.
- Groups of 2
- Display first image from student book.
- “What product does this rectangle represent?” (\(245 \times 35\))
- “Today, you are going to take turns with your partner picking expressions that can be added together to give the product \(245 \times 35\). You can use the diagrams to explain your reasoning, if they are helpful.”
- 10 minutes: partner work time
- Monitor for students who:
- use the diagram to determine which expressions they will use.
- look at the expressions and think about how they could be used to find the full product.
- compute the full product in different ways.
- Take turns picking out a set of expressions that are equal to \(245 \times 35\) when added together. Use the diagrams if they are helpful.
- Explain how you know the sum of your expressions is equal to \(245 \times 35\).
- What is the value of \(245 \times 35\)? Explain or show your reasoning.
Advancing Student Thinking
If students do not choose correct expressions to represent a sum that is equal to \(245 \times 35\), refer to one of the empty boxes in the diagram and ask, “Which multiplication expression represents this partial product?”
- Invite previously selected students to share their strategies. As students share, record their reasoning with equations.
- Display: \(245 \times 30 + 245 \times 5 = 245 \times 35\)
- “How do you know this equation is true?” (I can put the 30 and 5 together since they are both multiplied by 245. I see that \(245 \times 30\) is the top part of the diagram and \(245 \times 5\) is the bottom part. Together that’s the whole diagram.)
Activity 2: Record Partial Products (15 minutes)
The purpose of this activity is for students to consider 2 different ways of recording partial products in an algorithm that they worked with in a previous course. The numbers are the same as in the previous activity to allow students to make connections between the diagram and the written strategies. Students examine two different ways to list the partial products in vertical calculations, corresponding to working from left to right and from right to left. Regardless of the order, the key idea behind the algorithm is to multiply the values of each digit in one factor by the values of each digit in the other factor.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
- Groups of 2
- “We’re going to look at two ways students recorded partial products for multiplying 245 by 35.”
- Display the image of Andre’s and Clare’s calculations.
- “How does this relate to what you just did?” (You can see they split it up into different partial products and listed the results to add them up.)
- 3 minutes: independent work time
- 5 minutes: partner work time
- Monitor for students who identify a pattern for how Andre and Clare list the partial products
- How are Andre’s and Clare’s strategies the same? How are they different?
- Create a list of equations to match the partial products Andre and Clare found.
Advancing Student Thinking
If students do not write the correct equations, refer to the individual partial products and ask, “Where is this partial product represented in the multiplication expression \(245 \times 35\)?”
- “Both of these strategies use an algorithm that lists the partial products. An algorithm is a set of steps that works every time as long as the steps are carried out correctly.”
- “How are both the approaches the same?” (They both multiply ones and tens by hundreds, tens, and ones.)
- “How are the approaches different?” (One starts with the hundreds and the other starts with the ones. One goes from left to right and the other goes from right to left.)
- “Why is it important to list the products in an organized way?” (That way I know I found all the partial products. I did not leave some out or take some twice.)
\(245 \times 35\)
- Display student work to show the list of equations from the second problem or use the list in the student responses.
- “How does each expression relate to the product \(245 \times 35\)?” (\(30 \times 200\) is the product of the 3 in the tens place of 35 and the 2 in the hundreds place of 245.)
“Today we found products of two-digit and three-digit numbers using partial products. We saw how diagrams can help us make sure we found all the partial products. We also saw we could list partial products using an algorithm.”
“How do you know that all the different ways to find the product give the same answer?” (You’re always adding up the same partial products, just calculating them and putting them together in different ways.)
“What is helpful to remember when you are using partial products to determine a full product?” (You have to make sure to find all of the partial products. You have to make sure you add them. Sometimes I can add them mentally and then don't need to list all of them.)