Teach students about methodical thinking with a simple 8-by-8 chessboard. Ask them to count the number of squares on the board. At first glance a student might answer that there are 64 squares on the board. Encourage him to count all sizes of squares, however, made up of smaller squares. For instance, there is one large square made up of all of the squares together. There are also many 2-by-2 squares within the chessboard, as well as 3-by-3 squares, and so on. Once students discover this pattern of finding squares, they should conclude that there are 204 total squares.
Give students a chance to practice using math in a real-world context by asking them to create and design their own amusement park. Tell students they will have a given amount of money to spend on creating the park. Provide them with a list of attractions they can include and the cost of each. Students will have to calculate the costs of the attractions they include in their park. Also provide students with a list of the daily costs of running each attraction. They will then have to use that information to decide on admission charges and calculate how much profit they can make at the amusement park each day.
Give students a scenario in which they are sharing a square pan of brownies but, in order to cut the brownies, each piece must be exactly square, and there cannot be any left over. Ask them to figure out different numbers of people they can share the brownies with according to this rule. For example, they could not just share the pan with one other person because a square cut in half would not make square pieces. However, if they cut the pan of brownies into four equal pieces, they would each be square. Make the investigation more challenging by introducing other rules, such as the square pieces do not all have to be the same size. Allow students to experiment with cutting squares to figure out each answer.
Give students an example of this type of problem and then let them explore to discover other problems that follow the same pattern. Start by showing them the product of 37 x 73, which is 2,701. Explain that the factors have reversed the position of their digits. Then give students another product without giving them the factors. For example, ask them to find the two-digit numbers that have reversed digits that equal 6,786 when multiplied. After trying out several pairs of numbers, they should find that 78 x 87 will equal 6,786. Continue giving them products and ask them to problem-solve to find which factors fit the pattern.