Intent

This activity gives students more experience with the model of hot and cold cubes for integer arithmetic.

Mathematics

Students use the hot and cold cubes model to understand arithmetic with integers. This is also a good time to introduce the concept of absolute value and to explore patterns in operations with integers.

Progression

Students have worked in their groups to make sense of the hot and cold cubes model. Now they will spend some individual time practicing with and confirming their understanding of the model. After comparing their work with one another and discussing questions that arise, students review a few more basic ideas related to operations with integers.

Approximate Time

20 minutes for activity

30 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Materials

Manipulatives, such as cubes or tiles, in two colors

Doing the Activity

To introduce the activity, emphasize that students are to explain each expression in terms of the hot and cold cubes model.

Discussing and Debriefing the Activity

Have a volunteer from each group explain one of the answers. Insist on the use of the hot and cold cubes model, even if the student prefers to quote arithmetic rules.

For example, for Question 4, a student should say something like, “The term −4−4 size 12{ {} rSup { size 8{-{}} } 4} {} [read “negative four”] means the chefs put in four cold cubes, which lowered the temperature 4°. The −+6−+6 size 12{- rSup { size 8{+{}} } 6} {} [read “subtract positive six”] means they took out six hot cubes, which lowered the temperature 6°. Altogether the temperature went down 10°. In terms of the model, you would write −4−+6=−10−4−+6=−10 size 12{ {} rSup { size 8{-{}} } 4- rSup { size 8{+{}} } 6= rSup { size 8{-{}} } "10"} {}.”

Some students may comment that taking out hot cubes has the same effect as putting in cold cubes. Such alternative explanations of the expressions, in terms of the model, should be encouraged.

In a problem like +5⋅−2+5⋅−2 size 12{"" lSup { size 8{+{}} } 5 cdot "" lSup { size 8{-{}} } 2} {} (Question 3), students may say that the answer is negative “because a positive times a negative makes a negative.” Encourage a reinterpretation of such an expression in terms of the model. For example, they might say, “This is as if the chefs were putting in bunches of cold cubes, which lowers the temperature.”

If some students continue to resist learning the model, insisting that they can get the answers more easily from rules, mention that part of learning mathematics is being able to explain it and that being able to explain simple situations like this is good practice for explaining more complex problems later. Point out that students do not have to use the model for every computation they do now or in the future, but that they should be prepared to justify their work in terms of the model when asked to do so.

Absolute Value

Working with the hot and cold cubes model is an ideal opportunity to introduce the term absolute value. Tell students that the absolute value of an integer is the number of cubes it represents. Help them to see that any integer except zero is a combination of a sign and an absolute value.

Also introduce the notation for absolute value through examples, such as |5| = 5, ∣−7∣=7∣−7∣=7 size 12{ \lline rSup { size 8{-{}} } 7 \rline =7} {}, and |0|=0.

Ask students, What’s the difference between the operation of subtraction and the negative sign?

If students develop their own general rules about the relationship between sign and operation, such as “adding a negative gives the same result as subtracting a positive,” that’s fine. However, tell them that familiarity with the hot and cold cubes model will give them something to fall back on if they happen to forget the rule.

At this point you can announce that you (and their books) will generally omit the raised plus sign prefix for positive numbers and will write the negative sign the same way as a subtraction sign.

Explain that in order to avoid seeming to write two arithmetic operation symbols next to each other, it is common to insert parentheses. For example, instead of 10 + –7 or 8 – –4, we might write 10 + (–7) or 8 – (–4). Also, for 5 . –3, we might write 5 . (–3), but we often omit the multiplication sign when there are parentheses to indicate multiplication: 5(–3). Students are probably familiar with the use of parentheses for multiplication in such expressions as 5(2 + 7), but may not have seen it used in conjunction with a symbol immediately inside the parentheses that could be interpreted as an arithmetic operation, such as 5(–2 + 7).

Patterns in Integer Arithmetic

At this point, you may want to present the following pattern approach to operations with integers. Begin by writing this sequence of addition equations.

Ask students to look for a pattern and use it to explain what number belongs in place of the question mark. What should come next in this sequence? Presumably they will see that the sequence of answers suggests that 7+(-1) should equal 6.

Continue with 7+(-2)=? and similar problems. Ask students what is happening. They should see that as negative numbers of greater magnitude are added, the resulting sum gets smaller. You may want to continue through 7+(-7) and on into examples that give a negative sum, such as 7+(-8).

After the pattern has been described, ask the class whether this pattern gives the same answers as the hot and cold cubes model. Students should be able to explain how to get the same results from the model.

Present the next series of equations, which relate to subtraction, and ask, What should come next in this sequence?

From the continuation of this pattern, students should see that if a greater number is subtracted from a lesser number, the result is a negative number. Some may also notice that the result is the opposite of the result when the two numbers are reversed.

Continue the pattern above in the opposite direction, subtracting smaller and smaller numbers from 7. What should come next in this sequence?

Students should notice that as the number being subtracted grows smaller, the result increases. Some may recognize that subtracting a negative number gives the same result as adding the corresponding positive number, so that there is a related addition equation for each subtraction equation. For example, the subtraction equation 7-(-5)=12 relates to the addition equation 7+5=12.

Finally, have students look for patterns in the products of integers, as in this sequence.

Students should observe that as the second factor decreases by 1, the products decrease by 6.

Key Questions

What’s the difference between the operation of subtraction and the negative sign?

What should come next in this sequence?

How does this pattern relate to the hot and cold cubes model?