Guided Lessons

# Area Models and Mixed Numbers

Help students visually represent multiplication with mixed numbers and whole numbers. Use this lesson as a standalone lesson, or as support for the lesson Multiplication of Mixed Numbers with Area Models.
This lesson can be used as a pre-lesson for theMultiplication of Mixed Numbers with Area ModelsLesson plan.
This lesson can be used as a pre-lesson for theMultiplication of Mixed Numbers with Area ModelsLesson plan.

Students will be able to use an area model to solve multiplication problems with mixed numbers.

##### Language

Students will be able to review their thinking on how to decompose mixed numbers in multiplication problems using visuals and area models.

(5 minutes)
• Review area as additive with students by having them find the combined area of your classroom and your neighbour's classroom (e.g., (23 x 10) + (16 x 10) = (39 x 20) = 780 sq ft.).
• Use key vocabulary, such as "area" and "by" to represent multiplication. Use sequencing words as you solve this problem together, such as:
• "First, I ____."
• "After I ____, I ____."
• "Then, I ____."
• "My next step was ____."
• Model how to transfer the information to an area model with two rectangles and label each of the rectangles as the prospective room (e.g., the first rectangle would be your classroom dimensions and the second rectangle would be the neighbour's classroom dimensions). Ask for volunteers to provide the next steps to find the area of both of the classrooms.
• Tell students that today they will work on finding the area of a mixed number and whole number in a multiplication problem.
(7 minutes)
• Tell students that area models can help them visually understand the maths behind multiplying mixed numbers by a whole number.
• Adjust the areas of each of the rooms from the introduction by saying the original measurements were rounded, but now you have more precise dimensions (e.g., (23 x 9 ½) + (16 x 8 ⅔)).
• Model finding the area for your room (e.g., 23 x 9 ½ = 103 ⅓ sqft) next to the area model from the introduction so students can see the slight adjustments. Draw one rectangle with the sides labeled 23 x 9. Then, add an additional rectangle attached to the first rectangle to accommodate the fraction (e.g., ½ x 23).
• Choose a student to restate the process for solving the multiplication problem. Allow other students to give input and adjust the presenter's sequenced explanation.
• Tell students to solve the problem again in partners using their whiteboards.
(7 minutes)
• Distribute the Review Your Thinking! worksheet. Read the directions and explain they'll use the worksheet throughout the lesson to review their understanding of how to use area models to solve multiplication problems with mixed numbers.
• Ask students to think about how they will solve the next multiplication problem to find the neighbour's area (e.g., 16 x 8 ⅔). Have students write their thoughts on the first cell of the Review Your Thinking! worksheet. Model sharing your answer for the Beginning section using the following sentence frames:
• "I hadn't considered that ____."
• "This makes me realise that ____."
• Assign partnerships to talk about their Beginning section and the strategy they'd use to solve the next area problem.
• Have volunteers share their ideas on how to solve the problem. Allow other students to correct misconceptions. Make sure everyone has an idea of how to solve the problem before allowing them to separate into partnerships again to solve it.
• Have partners solve for the area of the neighbour's classroom (e.g., 16 x 8 ⅔) on their copy paper.
(7 minutes)
• Present the next multiplication problem and have partners solve it on their whiteboards (e.g., 10 x 6 27). Choose volunteers to share their answers.
• Have students talk in partners and reassess their understanding from the beginning of the lesson on the Review Your Thinking! worksheet. Allow them to discuss their new ideas about solving area models for fraction multiplication.
• Tell students to write their answers down in the Middle section of the worksheet Review Your Thinking!

Beginning

• Allow students to use their home language (L1) or their new language (L2) in all discussions. Provide bilingual reference materials to assist in their vocabulary word acquisition.
• Encourage them to use the vocabulary cards and terms in their conversations and writing. Allow them to draw pictures to support their understanding of the terms.
• Pre-teach how to decompose mixed numbers and use area models and key vocabulary if necessary.
• Shorten their assignments. For instance, allow them to choose to answer one of the three questions in the closing section.

• Ask students to share language frames for the questions in the Review Your Thinking! worksheet and when they are sequencing their processes.
• Pair students with mixed ability groups so they can offer explanations and provide feedback to beginning ELs when appropriate.
• Have them write some of their answers on the board before students complete their own writing.
• Choose them to lead a small group to review some of the key terms with students with their L1.
(7 minutes)
• Refer to the End section of the Review Your Thinking! worksheet and ask students to write down one thing they learned about multiplying mixed numbers by fractions and something they still wonder about the topic.
• Model answering the questions with a student volunteer serving as your partner (e.g., "I think using area models is helpful with smaller numbers, such as fractions, and larger, three-digit numbers. Before I thought area models were just for whole numbers, but now I know I can use them with fractions too.").
• Provide the following sentence frames for student discussions and read them aloud:
• "I learned ____."
• "Before I thought ____, now I think ____."
• "I wonder ____."
• Have students share their ideas with a partner and then write them down on their paper using the language frames from the board.
• Assign students one more multiplication problem for them to solve to model their understanding of using area models with mixed numbers (e.g., 4 x 1 ½).
(7 minutes)
• Tell students to review the answers for the assessment section in partners.
• Conduct a "Connect, Extend, and Challenge" exercise in pairs where students answer the following three questions:
• How do the new ideas about area models ConnectTo what you already knew? ("These new ideas are similar to the way I solved for area in the past because I still draw the rectangle, but I add more rectangles for each number I decompose.")
• What new ideas did you get that ExtendOr push your thinking in new ideas? ("I wonder if I can use the same strategy for larger numbers or mixed number multiplication.")
• What is still ChallengingFor you to understand? ("I still don't understand...")
• Choose volunteers to share one answer to the questions.