Activity Name: Hit the Target
Activity Name: Square Roots Go Rational Grade level: 8
Skills/Goals: Students will model perfect squares and their roots, discover patterns in a table of roots and perfect squares, model and build a table of not-so-perfect squares and approximate their square root as a mixed number.
Students may also develop an algorithm to approximate the square root of not-so-perfect squares.
Assessment Anchor(s) & Eligible Content addressed:
M08.A-N.1.1.1 Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths).
M08.A-N.1.1.2 Convert a terminating or repeating decimal into a rational number (limit repeating decimals to thousandths).
M08.A-N.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144).
M08.A-N.1.1.4 Use rational approximations of irrational numbers to compare and order irrational numbers.
Students: whole class activity
Materials needed:
Square Roots Go Rational Activity Sheet
Square Roots Go Rational Answer Key
Directions:
To prepare students for the day's lesson, model the perfect square to the right for your students. 9 is a perfect square because 3 × 3 = 9. The length and the width of the square are the same.
Ask several students to draw several more perfect squares on the board. The length of the side of each of these squares is the square root of the square. Have the students organize the data of perfect squares and their square roots into a table. Allow your students time to explore patterns in this table. They might discover patterns in the unit digits, relationships between odd and even numbers, or the difference between consecutive squares. Discuss any of their discoveries.
A key pattern for this lesson is the sum of the outer new column and row as the squares grow in size is the difference of consecutive perfect squares.
The difference in the models of the consecutive perfect squares 16 and 9 is the addition of 7 X's in the outer column and row. Notice that 7 is the difference between 16 and 9. This can be expressed:
7 = 16 – 9 = 42 – 32
It is also the sum of the square roots of 16 and 9:
7 = 4 + 3
Main Activity
The key to the rational roots students will be exploring during the lesson lies in the outer column and row of each perfect square. Ask students to predict the square root of 7.
7 X's cannot fill a square, so 7 is not a perfect square. However, the square root of 7 can be estimated. The square root must be between 2 and 3 because 7 lies between the perfect squares 4 and 9.
From this model, the largest perfect square is a square with side lengths of 2 with three remaining units in the outer column and row. 5 X's are needed in the outer row and column to make the next larger perfect square. The missing units from the outer column and row are filled in with O's to reinforce the counting of these elements. Therefore, an approximation to the square root of 7 is very close to 2 3/5. The 3 represents the extra number of units (X's) in the outer row and column; and the 5 represents the total number of units (X's and O's) in the outer row and column. The meaning of a fraction is enforced with this model. The fractional part of the square root is the part of units we have divided by the whole number of units needed in the outside column and row.
Check for understanding by asking: "What two perfect squares would you expect 32 to be between?" [25 ................
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