Equal Groups to Arrays

Overview

In Lesson 3 of 10 in “Build, Model, Solve,” students connect equal groups to arrays and area models. They review familiar multiplication facts and properties, then use place-value patterns to represent and explain larger products efficiently.

Learning intentions

Students will be able to:

• Represent multiplication with equal groups, counters, arrays, and area models.
• Connect repeated addition, multiplication equations, and rectangular arrangements.
• Use the distributive property and place-value patterns to solve larger products.
• Explain how different models show the same multiplication expression.

Success criteria

• I can build and draw equal groups and arrays for a multiplication equation.
• I can label the rows, columns, factors, and product in an array or area model.
• I can use a known fact or split a factor to solve a harder multiplication problem.
• I can explain why two different models represent the same product.

Curriculum links

• Number and Operations in Base Ten — multiply a whole number of up to four digits by a one-digit number using place value and properties of operations.
• Number and Operations in Base Ten — illustrate and explain multiplication with equations, rectangular arrays, and area models.
• Number and Operations in Base Ten — use place-value understanding to recognize how a digit’s value changes by a factor of ten.
• Number and Operations in Base Ten — read, write, compare, and express multi-digit numbers in expanded form.

Lesson structure (60 minutes)

1. 0–7 min · Hook and retrieval. Open with the equal-groups hook slide showing 24 counters arranged randomly and ask, “How could we organize these so the total is easy to find?” Students sketch or describe one arrangement, solve a quick fact such as 6 × 4, and share how they know the answer.

2. 7–17 min · Model the connections. Use the model-connection slides to demonstrate 4 groups of 6 with counters, a repeated-addition equation, a 4-by-6 array, and an area model. Think aloud: “The factors tell how many groups and how many in each group; the product is the total.” Students identify rows, columns, factors, and product, then turn and explain why 4 × 6 and 6 × 4 have the same product.

3. 17–30 min · Build and compare. Place students in groups of three or four and provide counters and grid paper. Display directions from the group-task slides. Call out or display multiplication expressions such as 3 × 8, 5 × 7, and 6 × 9. Students build each expression, draw an array, write a matching equation, and compare their model with another group’s model. Encourage use of the sentence frame, “Our array shows __ rows of __, so __ × __ = __.”

4. 30–42 min · Develop efficient strategies. Show 6 × 14 using an area model split into 6 × 10 and 6 × 4. Connect the model to 60 + 24 = 84 and the distributive property. Then show a place-value pattern such as 6 × 4 = 24 and 6 × 40 = 240. Students solve 7 × 13 and 4 × 21 with an area model or expanded equation, explaining how a known fact helps them. Address the misconception that 6 × 40 is 24 rather than 240.

5. 42–54 min · Independent application. Distribute the equal-groups, arrays, and area-model worksheet. Students complete tasks that ask them to match models and equations, draw an array for a product, solve two-digit by one-digit problems using a split area model, and explain one strategy in words. Circulate and check whether students preserve the meaning of both factors when drawing models. Invite early finishers to solve the same problem in two ways and compare the methods.

6. 54–60 min · Share and exit check. Display the discussion and closure slides and ask two students to share different solutions for 6 × 14. Students finish with the Success Criteria exit ticket slips by representing 5 × 12 with an area model or labeled array and writing one sentence explaining their strategy. Collect responses to plan the next lesson’s small groups.

Resources

• the complete equal-groups-to-arrays slide deck
• the equal-groups, arrays, and area-model worksheet
• Counters or linking cubes, one set per group
• Grid paper
• Whiteboards and markers
• Pencils, crayons, or colored pencils
• Chart paper or classroom display
• Exit ticket slips

Assessment

• During modeling and group work, ask: “What does each factor represent?” and “How does your drawing prove the equation?” Note students who confuse rows and columns or omit labels.
• Review worksheet models for accurate arrays, decomposed products, and explanations connecting addition or multiplication to the model.
• Use the exit ticket to identify whether students can represent a two-digit by one-digit product and apply a place-value or distributive strategy.

Differentiation

• For support, provide pre-drawn grids, counters, a multiplication fact chart, and the sentence frames “I see __ groups of __” and “I split __ into __ and __.”
• For students developing English, pair spoken directions with the visual models and explicitly rehearse words such as factor, product, row, column, equal groups, and area.
• For students with learning needs, reduce the number of worksheet problems while keeping the representation and explanation requirements; allow students to build before drawing.
• For extension, ask students to solve 8 × 17 in two different ways, draw both area models, and explain which strategy they prefer and why.