Students will learn how to simplify square roots of decimal and rational numbers. Students can practice here.
Next, give your students these challenges:
- Hand Span by NRICH
Neighbours by NRICH: Here's the solution: Suppose house 3 is somewhere along the top row, and house 10 is somewhere along the bottom. Setting houses 3 and 10 to be in the rightmost-column, and reading the house numbers clockwise, gives us [4, 5, 6, 7, 8, 9]. Notice the right column of houses has 6 houses in it, so the total number of houses in the square is \(6 \cdot 4 = 24.\) If we move houses 3 and 10 to the left one column, we get [3, 4][5, 6, 7, 8][9, 10]. In this case, the number of houses in the right column is 4, so the total number of houses in the square is \(4 \cdot 4 = 16.\) Suppose we try to move houses 3 and 10 another column to the left. This would give us [3, 4, 5][6, 7][8, 9, 10]. But this is no longer a square, it's a rectangle. So the smallest number of houses in the square is \(16,\) and the largest number of houses is \(24.\) Ask students why there's no solution with an odd number of houses along each edge? Here's the answer: First, notice that there are exactly 6 numbers between 3 and 10. If we look at our second solution, from before, we see 3 is opposite 10, and 4 is opposite 9. Because each number along the top strip is opposite some number along the bottom strip, the total number of houses along the top and bottom strip, to the right of 3 and 10, will be some even number. We want to add the number of houses per strip to this even number to get 6. But 6 is an even number, and even plus odd is odd. Thus, the number of houses per strip cannot be odd.
Conclude by leading this investigation:
12 Days (addition, logic, factoring)