Build Square Roots

Overview

Students build and identify perfect squares from 1 through 15 using tiles, counters, drawings, and repeated oral practice. They then connect squaring with square roots as opposite operations, using simple “What number times itself?” reasoning. This first-week routine emphasizes predictable steps, hands-on learning, and confidence-building.

Learning intentions

Students will be able to:

• Explain that squaring means multiplying a number by itself.
• Build and record perfect squares from 1 through 15.
• Explain that a square root asks for the number multiplied by itself.
• Use a perfect-square fact to find a square root.

Success criteria

• I can write a number squared as a multiplication sentence.
• I can show why a number is called a perfect square.
• I can find square roots from 1 through 225.
• I can check my answer by multiplying the number by itself.

Curriculum links

• Expressions and Equations — apply properties of integer exponents to generate equivalent numerical expressions, including squares.
• Expressions and Equations — connect numerical expressions and proportional reasoning through accurate interpretation of mathematical representations.
• This lesson directly supports the stated Texas focus on perfect squares and square roots through concrete models, equivalent expressions, and mathematical explanation.

Lesson structure (45 minutes)

1. 0–4 min · Welcome and math routine. Teacher greets students by name, previews the predictable sequence on the visual lesson sequence, and says, “Today we will build squares, write square facts, and discover what a square root asks.” Students repeat the goal and practice the response, “I can show it, say it, and write it.” Briefly model asking for help: “Please show me the next step.”

2. 4–12 min · Teaching: build a square. Teacher displays square tiles or counters and says, “A square has the same number of rows and columns. Squaring a number means multiplying the number by itself. Four squared is four times four: 4² = 4 × 4 = 16.” Build a 4-by-4 array and draw it on the square-array teaching slides. Emphasize, “The little 2 tells us to use four two times. It does not mean 4 × 2.” Students build 1-by-1, 2-by-2, and 3-by-3 arrays, count each total, and say the complete sentences: “Three squared equals three times three equals nine.”

3. 12–19 min · Model: small fact groups. Teacher models the first group on the board: 1² = 1 × 1 = 1 2² = 2 × 2 = 4 3² = 3 × 3 = 9 4² = 4 × 4 = 16 5² = 5 × 5 = 25 Say, “We will learn five facts at a time, not all fifteen at once.” Then model groups 6–10 and 11–15 using an array sketch for smaller numbers and a fact chart for larger numbers: 6²=36, 7²=49, 8²=64, 9²=81, 10²=100; 11²=121, 12²=144, 13²=169, 14²=196, 15²=225. Students echo each fact and trace the matching row on the perfect-squares practice sheet.

4. 19–28 min · Ask Students: connect and check. Teacher asks one question at a time: “What does 5² mean?” “Is 6² equal to 6 × 2? Why not?” “How many rows and columns would a 9-by-9 square have?” “Which fact helps you find 144?” Students answer with a partner, use counters or graph paper to prove one fact, and hold up or point to their answer. Correct immediately: if a student says 4² = 8, respond, “You multiplied by two. Look at the exponent: use four as a factor two times. Say 4 × 4.” If a student reverses a fact, have the student rebuild or draw the array before repeating it.

5. 28–36 min · Guided Practice: square roots. Teacher says, “A square root is the opposite of squaring. The question is: What number multiplied by itself equals the number under the radical?” Write and model: √16 = 4 because 4 × 4 = 16. Students use the chart and tiles to solve with the teacher: √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, and √100=10. Continue with √121=11, √144=12, √169=13, √196=14, and √225=15. Use the repeated frame: “The square root of ___ is ___ because ___ × ___ = ___.” For a mistake such as √25 = 10, say, “Check the matching square fact. What number times itself makes 25?”

6. 36–41 min · Independent Practice. Distribute the perfect-squares practice sheet. Students complete a short mixed set: write five squared expressions, match five arrays or numbers to facts, and find five square roots. Teacher works beside students who need support, covering unused questions and prompting, “Find the number that repeats.” Students may use counters, graph paper, and the displayed fact chart; require each answer to include a multiplication check.

7. 41–45 min · Exit Ticket and close. Students complete the final box on the square-and-square-root exit questions: (1) write 7² as multiplication and find the answer, (2) find √81 and explain with multiplication, and (3) circle the correct statement: “3² means 3 × 3” or “3 × 2.” Teacher says, “Show your thinking, not just a number,” collects responses, and previews that the next lesson will use these facts for estimation and problem solving.

Resources

• the perfect-squares visual slide deck
• the perfect-squares practice sheet
• Square tiles, linking cubes, or counters
• Graph paper and pencils
• Dry-erase boards and markers
• Large perfect-squares reference chart from 1 through 15
• Radical and exponent symbol cards
• Six individual workspaces or trays for manipulatives

Assessment

• Listen for accurate explanations during partner responses and check whether students build equal rows and columns.
• Use whiteboards and guided questions to identify the misconception that an exponent means multiplication by the exponent.
• Score the exit ticket for correct expressions, square roots, and multiplication checks; use results to group students for the next lesson.

Differentiation

• Support students with a color-coded chart, pre-drawn arrays, number lines, reduced choices, and the sentence frame “___ × ___ = , so √ = ___.”
• Read every direction aloud, present one item at a time, use large uncluttered print, and allow students to point, build, say, or write an answer.
• Pair students strategically and provide repeated oral rehearsal without requiring public performance before they are ready.
• For students ready for a challenge, ask them to explain why √144 is 12 and not 6, or create a square-root question for a partner using a fact from 11 through 15.