Authentic Lessons for 21st Century Learning

Multiplying Polynomials

Beyond FOILing

Samaya Williams, Michell Eike | Published: November 1st, 2023 by K20 Center

  • Grade Level Grade Level 10th, 11th
  • Subject Subject Mathematics
  • Course Course Algebra 2
  • Time Frame Time Frame 90-100 minutes
  • Duration More 2-3 class periods


Students will recall properties of exponents and explore different methods of multiplying binomials. Students will apply this knowledge to multiplying more advanced polynomial expressions or multiplying polynomial expressions in the context of a geometry setting.

Essential Question(s)

How can you use multiplication to simplify polynomial expressions?



Students activate prior knowledge using the properties of exponents to complete a puzzle.


Students explore multiplying polynomials and create a poster illustrating the steps.


Students learn and reflect on their peers’ posters and find a favorite method for multiplying polynomials.


Students apply their understanding to new scenarios through a Choice Board.


Students reflect on their learning using the What Did I Learn Today? strategy.


  • Lesson Slides (attached)

  • Algebra Two Ways handout (attached; one per student; printed front only)

  • Multiplying Methods handout (attached; one page per student; printed front only)

  • Pencil

  • Paper

  • Markers

  • Chart Paper

  • Sticky notes

  • Desmos account

  • Device with internet access

  • Clipboards (optional)


10 Minute(s)

Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s essential question. Slide 4 identifies the lesson’s learning objective. Review each of these with your class to the extent you feel necessary.

Transition to slide 5 and give each student a copy of the attached Algebra Two Ways handout. Preview the activity with the students by explaining that they will be given a puzzle on the next slide and they must use prior knowledge of the exponent rules of multiplication to complete the puzzle.

Move to slide 6. Have students add the given information to their Puzzle 1 on their handout then independently solve their puzzle.

Transition to slide 7 to reveal the answer to Puzzle 1.

Next, have students find a partner and show slide 8. Have pairs copy the given information to Puzzle 2 from the slide then work together to solve the puzzle. Have pairs repeat this collaboration for Puzzle 3.

After a few minutes, transition to slide 9 and have groups share their process and support different ways the students thought about solving both puzzles.


35 Minute(s)

Display slide 10 and divide the class into 4 or 8 equal groups. If possible, try to have no more than 4 students in a group. Numbering the students off 1-4 could help with this. This may result in multiple groups working on the same method at one time, but keeping groups smaller will ensure greater individual ownership. The attached Multiplying Methods handout has one multiplication method per page. Make enough copies such that everyone in group 1 (and 5) gets a copy of page 1, everyone in group 2 (and 6) gets a copy of page 2, everyone in group 3 (and 7) gets a copy of page 3, and everyone in group 4 (and 8) gets a copy of page 4. Preview the activity with students by explaining to them that each group will be assigned a method to explore multiplying polynomial expressions: geometric area (algebra tiles), the distributive property, vertical (standard) multiplication, or the box method. In each group, for their assigned method, students will have four different expanded expressions that they will need to analyze to see a pattern. For the pattern, students need to think about how the expression was expanded and what characteristics each problem has in common. Students are to then use the recognized pattern to come up with generalizations or steps to simplifying polynomial expressions using multiplication. Students will use these generalizations to multiply a binomial by a trinomial.

After about 10 minutes, transition to slide 11 and introduce the Anchor Charts strategy. Provide students with sticky chart paper and markers or direct them to a location in the room where they can grab these supplies when finished with the handout. Explain to students that after they complete their handout they will create a poster to use for teaching their classmates about how to simplify polynomials using their method of multiplication and what they noticed from the exploration. The posters should have the following information: 

  • The name of their multiplying method

  • The completed problem from the Applying section of their handout showing how they used their method to simplify the polynomials

  • Notes or numbered steps someone can follow if they choose to use that method


25 Minute(s)

Give groups 5 minutes to review the posters they created on day 1 and to hang their posters around the room. If you have 8 groups, have groups 1-4 hang their posters in one location and groups 5-8 in a different location to prevent students from seeing the same method twice when rotating during the four rounds of the following activity.

Display slide 12 and introduce the Expert Stay and Stray strategy. Consider giving students clipboards to more easily take notes. Give each group some sticky notes and have them take out paper to take notes on.  Explain to students that one student from the group will stay at the poster as the expert and explain their group’s multiplying method to their audience. Everyone else from that group rotates (strays) to the next poster to listen and learn from the expert of that poster. Those who are listening need to pay careful attention because each round a new person stays and becomes the new expert/presenter. Continue previewing the activity by explaining that there will be four rounds, and that everyone will have a turn as the expert. There will be 3 minutes per round. The expert will also use a different color marker to add important information to the poster as they answer clarifying questions from their audience. Alternatively, sticky notes can be an option to add information instead of writing directly on the poster.

Within each group, assign each student a number. Show slide 13 and begin round 1. Direct all number 1s to stay at their posters, ready to present. Direct everyone else to move (stray) to the next poster and take notes and ask questions. Remind them that they are preparing to be the next expert. Stress to students that there are two goals by them learning different methods: They need to decide for themselves (1) when would this method be most efficient to use for multiplying polynomial expressions and (2) which method is their favorite and why.

Display slide 14 and start the 3-minute timer.

When the timer expires, direct students to “freeze,” and show slide 15 for round 2. Direct all number 4s to stay at the posters they are at. They are the new experts of that poster. Direct everyone else to move (stray) to the next poster and take notes, ask questions and listen carefully enough to be ready to be the next expert. Remind students of what they should be considering: when would this method be efficient and is this their favorite method and why.

Display slide 16 and start the 3-minute timer to begin round 2.

When the timer expires, repeat these steps using slides 17-20, where all number 2s are the experts for round 3, and all number 3s are the experts for round 4. 

When the timer expires the final time, direct students to find their original group.

Display slide 21 and have the students answer the questions on the slide and write their responses on the back of their Multiplying Methods handout.

  • What did you notice that all the posters had in common?

  • Was one method more efficient than the others? When and why?

  • Which method is your favorite and why?

Once you discuss the questions above as a whole class, consider the misunderstandings that you heard and take time to clear up these misunderstandings for the students. Ask the class if there are any questions. Use this time to make sure students understand how to simplify polynomial expressions using multiplication. 


15 Minute(s)

Show slide 22. Provide students with your session code. Then, have students go to and enter the session code.

Direct students’ attention to screen 1 and introduce students to the Choice Boards strategy. Let students know that they are going to complete one of three card matches. They can pick the card set on screen 2, screen 3, or screen 4 - their choice. Let students know that when they correctly match the cards they will notice the directions at the top of the screen change to read “Well done! Go to screen 5.”

Once students complete their card match, they are expected to go to screen 5 and answer the area question on that screen. Once they correctly answer that question, students are prompted to go to the next screen.

On screen 6, students are again given a choice. They can either answer the volume question on screen 7 or the one on screen 8. These volume questions are written mostly as word problems. After answering either question correctly, students are prompted to go to screen 9.

On screen 9, students are again given a choice: to answer a volume question on screen 10, screen 11, or screen 12. These volume questions are presented with diagrams. After answering one of these questions correctly, students are prompted to go to the last screen.


5 Minute(s)

On screen 13, students are directed to use the slider on the screen to answer the following:

  • How well did you understand the math today?

  • How did you feel about learning math today?

Then students are asked to elaborate those responses.

Lastly, students are prompted to reflect on their learning using the What Did I Learn Today? strategy and write 2-3 sentences about what they learned during this lesson.