Chef’s Hot and Cold cubes story (Taken from the Interactive Mathematics Program)
Do Now: a) -2 + -9 =
b) -5 + 7 =
c) 12 + (-2) =
Motivation: Suppose one of the chefs accidently tossed is 5 cold cubes and then removed 2 hot cubes to adjust the temperature. How would these actions have changed the temperature of the soup?
Development/ Instructional Activities:
Recap the Chef’s Hot and Cold cubes story with the class. (The story is first introduced when learning integer addition and can be used for subtraction and division.) Discuss the different ways that they could “warm up” or “cool down” the soup.
Sometimes the chefs wanted to change the temperature of the soup without adding any cubes in. To do this they simply used their large ladle and scooped out hot or cold cubes. If they wanted to warm the soup up, they scooped out a ___________ cube. If they wanted to cool the soup down they scooped out a _____________ cube.
Present the motivation to the class. Discuss how this could be modeled using the red and yellow chips. ** Use this example to discuss the need to add in zero pairs in order to model the subtraction – students will need to do this to complete the investigation worksheet.
Have students work with a partner to complete the investigation.
Investigation: In pairs model each cooking action that is on your worksheet. Then write a number sentence to represent what happened. Record the overall "temperature change" of the soup. Each problem is independent of the one that comes before it!
When students have completed all 9 have them make a conjecture about what they think a rule for subtracting integers is. They should explain their rule and be able to give a justification for it. *Alternatively, you could determine the rule as a class.
Discuss the results of the investigation with the class. Agree on a general rule that can be used to subtract integers and have students copy it into their notes. (Ex: To subtract integers we change the operation to addition and add the opposite of the second number to the first number.)
In general, how do we subtract integers? Why does this rule work?
Cooking With Integers Directions: Read the Chef’s Hot and Cold Cubes story. Each of the problems below describes an action by the chefs. Write an equation to describe the action and find the overall result. Each problem is unrelated to the previous.
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 the rules come from. The following mythical story provides a context for understanding how positive and negative numbers work. Many people find it easy to remember the story many years after they first heard it, and the memory of the story enables them to reconstruct the rules. The story also helps some people make sense of the rules.
The Story In 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 very delicate and complex. During the cooking process, they frequently had to change the temperature of the cauldron in order to bring out the flavors and cook the food to perfection.
They adjusted the temperature of the cooking either by adding special hot cubes or 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 that they didn’t melt, and 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.
For each hot cube that was put in the cauldron, the temperature went up one degree; for each hot cube removed, the temperature went down one degree. Similarly, each cold cube put in lowered the temperature one degree and each cold cube removed raised it one degree.
The chefs used positive and negative numbers to keep track of their changes they were making to the temperature.
For example, suppose 4 hot cubes and 10 cold cubes were dumped into the cauldron. Then the temperature would be lowered by 6 degrees altogether, since 4 of the 10 cold cubes would balance out the 4 hot cubes, leaving 6 cold cubes to lower the temperature 6 degrees. They would write
+4 + -10 = -6
to represent these actions and their overall result.
Similarly, if they added 3 hot cubes and then removed 2 cold cubes, the combined result would be to raise the temperature 5 degrees. In that case, they would write
+3 - -2 = +5
And if they wrote -5 - +6 = -11, it would mean that first 5 cold cubes were added and then 6 hot cubes were removed, and that the combined result was to lower the temperature 11 degrees.
Sometimes they wanted to raise or lower the temperature by a large amount, but did not want to put the cubes into the cauldron one at a time. So for large jumps in temperature, they would put in or take out bunches of cubes.
For instance, if the chefs wanted to raise the temperature 100 degrees, then they might toss five 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 five bunches of 20 hot cubes each into the cauldron, they would write
+5 +20 = + 100
where the +5 meant that the five bunches were being added, and the +20 showed that there were twenty hot cubes in each bunch.
The chefs could also change the temperature by removing bunches. For example, if they removed three bunches of 5 hot cubes each, the result was to 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 write
-3 +5 = -15
where the -3 meant that three bunches were being removed, and the +5 showed that there were five hot cubes in each bunch.