You may have learned some rules for doing arithmetic with positive and negative numbers. Many people find these rules hard to remember and don’t understand where they come from. This story will help you understand how positive and negative numbers work. Many people remember the story many years after they first heard it. The memory helps them reconstruct the rules for positive and negative numbers. The story also helps make sense of the rules. The StoryIn a far-off place, there was once a team of amazing chefs who cooked up the most marvelous food ever imagined. They prepared their meals over a huge cauldron, and their work was delicate and complex. They frequently had to change the temperature of the cauldron to bring out the flavors and cook the food to perfection. They adjusted the temperature by adding either special hot cubes or special cold cubes to the cauldron or by removing some of the hot or cold cubes that were already in the cauldron. The cold cubes were similar to ice cubes, except they didn’t melt. The hot cubes were similar to charcoal briquettes, except they didn’t lose their heat. If the number of cold cubes in the cauldron was the same as the number of hot cubes, the temperature of the cauldron was 0 degrees on their temperature scale. Group ActivityThe Chefs’ Hot and Cold Cubescontinued Patterns: Communicating About Mathematics21 22 In t e r a c tI v e Ma t h eM a tI c s Pr o g r aMcontinued For each hot cube put into the cauldron, the temperature went up 1 degree. For each hot cube removed from the cauldron, the temperature went down 1 degree. Cold cubes worked the opposite way. Each cold cube put in lowered the temperature 1 degree. Each cold cube removed raised it 1 degree. The chefs used positive and negative numbers to keep track of the temperature changes. For example, suppose 4 hot cubes and 10 cold cubes were dumped into the cauldron. The temperature would be lowered by 6 degrees altogether. This is because 4 of the 10 cold cubes would balance out the 4 hot cubes, leaving 6 cold cubes to lower the temperature 6 degrees. To represent these actions and their overall result, the chefs would write14 1210 526Similarly, if they added 3 hot cubes and then removed 2 cold cubes, the combined result would raise the temperature 5 degrees. In that case, the chefs would write13 222 515The subtraction indicates that the 2 cold cubes were being taken out. If they wrote 25 216 5211, it would mean that 5 cold cubes were added and then 6 hot cubes were removed. The combined result was to lower the temperature 11 degrees. Sometimes the chefs wanted to raise or lower the temperature by a large amount, but they did not want to put the cubes into the cauldron one at a time. So, they would put in or take out bunches of cubes. For instance, to raise the temperature 100 degrees, the chefs might toss 5 bunches of 20 hot cubes each into the cauldron, instead of 100 cubes one at a time. This saved a lot of time because they could have assistant chefs do the bunching. When the chefs used bunches of cubes to change the temperature, they used a multiplication sign to record their activity. For example, to describe tossing 5 bunches of 20 hot cubes each into the cauldron, they would write. The chefs could also change the temperature by removing bunches. For example, suppose they removed 3 bunches of 5 hot cubes each. This would lower the temperature 15 degrees, because each time a bunch of 5 hot cubes was removed, the temperature went down 5 degrees. To record this change, they would write23 ?15 5215The 23 means that three bunches were being removed. The 15 shows that there were 5 hot cubes in each bunch. 1. Each problem describes an action by the chefs. Figure out how the temperature would change overall in each situation. Write an equation to describe the action and the overall result. a. Three cold cubes were added, and then 5 hot cubes were added. b. Five hot cubes were added, and then 4 cold cubes were removed. c. Two bunches of 6 cold cubes each were added. d. Four bunches of 7 hot cubes each were removed. e. Three bunches of 6 cold cubes each were removed. 2. Describe the action involving hot or cold cubes that is represented by each arithmetic expression. State how the temperature would change overall. a.14 223b.26 124c.210 ?25d.14 ?28Patterns: Communicating About Mathematics23

24 In t e r a c tI v e Ma t h eM a tI c s Pr o g r aMExplain each expression in terms of the “hot and cold cubes” model. Your explanations should describe the action and state how the temperature changes overall. 1.26 129 2.27 2210 3.15 ?22 4.24 216 5.13 127 6.26 ?19 7.23 ?24 8.18 2212 9.212 115ActivityDo It the Chefs’ Way24 In t e r a c tI v e Ma t h eM a tI c s Pr o g r aMExplain each expression in terms of the “hot and cold cubes” model. Your explanations should describe the action and state how the temperature changes overall. 1.26 129 2.27 2210 3.15 ?22 4.24 216 5.13 127 6.26 ?19 7.23 ?24 8.18 2212 9.212 115

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