Integers

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Problems about integers.

Describe the set of integers. Give some relevant and revealing examples/nonexamples.

The integers are the counting numbers, 0, and the opposites of the counting numbers. $\{ \dots , -3, -2, -1, 0, 1, 2, 3, \dots \}$

Give some examples of contexts we might use to describe integers. What relevant properties should your context have?

Common contexts include temperature, bank balance, and elevation. A good context should have a sense of positive and negative, or a quantity and its opposite.

For the next few problems, identify which operation should be used to solve the problem.

Alison has $34$ head bands. April has $38$ headbands. How many more headbands does April have? The problem above is an example of:

Addition. Take-away subtraction. Comparison subtraction. Missing-addend subtraction.

Gabe has a bank balance of $\$17$ . After he spends $\$22$ , what will Gabe’s balance be? The problem above is an example of:

Addition. Take-away subtraction. Comparison subtraction. Missing-addend subtraction.

Yesterday, Ollie was $9$ miles west of home. Today, Ollie is $6$ miles east of home. How far did Ollie travel overnight? The problem above is an example of:

Addition. Take-away subtraction. Comparison subtraction. Missing-addend subtraction.

The problem is asking what was added to Ollie’s distance from home yesterday in order to get to today’s location.

Last week, the average temperature was $-2$ degrees Celsius. This week, the average temperature is $4$ degrees warmer. What is the average temperature this week? The problem above is an example of:

Addition. Take-away subtraction. Comparison subtraction. Missing-addend subtraction.

The next few problems use the red and black chips we discussed for modeling integers.

What is the total value of the chips in the following picture?

The total value is $\answer [given]{1}$ .

What is the total value of all of the chips in the picture below?

The total value of the chips is $\answer [given]{2}$ .

Suppose you would like to use chips to represent a total value of $8$ . Which of the following combinations of chips would give you this value?

$8$ black chips $8$ red chips $10$ black chips and $2$ red chips $2$ black chips and $10$ red chips $24$ black chips and $16$ red chips $3$ black chips and $11$ red chips $5$ black chips and $3$ red chips

Suppose you would like to use chips to represent a total value of $-6$ . Which of the following combinations of chips would give you this value?

$-6$ black chips $6$ red chips $6$ black chips and $6$ red chips $2$ black chips and $4$ red chips $10$ black chips and $6$ red chips $24$ black chips and $30$ red chips $18$ black chips and $12$ red chips

You are working on the subtraction problem $10 - (-2)$ , and you would like to solve this problem using red and black chips from a take-away perspective. First, you put your chips down on the table. What chips will you need?

You need to model a total value of $\answer [given]{10}$ using the chips. You will need at least $\answer [given]{2}$ red chips in your model so that you have enough to take away.

You decide to use a model with $12$ black chips and $2$ red chips. How many chips do you take away, and what is left?

You take away $\answer [given]{2}$ chips, leaving $\answer [given]{12}$ black chips and $\answer [given]{0}$ red chips.

What is $10 - (-2)$ , from your result with the chips?

$10 - (-2) = \answer [given]{12}$

You are working on the subtraction problem $(-4) - (-7)$ , and you would like to solve this problem using red and black chips from a take-away perspective. First, you put your chips down on the table. What chips will you need?

You need to model a total value of $\answer [given]{-4}$ using the chips. You will need at least $\answer [given]{7}$ red chips in your model so that you have enough to take away.

You decide to use a model with $6$ black chips and $10$ red chips. How many chips do you take away, and what is left?

You take away $\answer [given]{7}$ chips, leaving $\answer [given]{6}$ black chips and $\answer [given]{3}$ red chips.

What is $(-4) - (-7)$ , from your result with the chips?

$(-4) - (-7) = \answer [given]{3}$

You are working on the subtraction problem $1-5$ , and you would like to solve this problem using red and black chips from a missing-addend perspective. First, you change your problem into the related addition problem. What is the result?

The related addition problem is $\answer [given]{1} = \answer [given]{5} + ?$ .

We now need to begin with a model for $5$ and decide how to get to $1$ . What chips would we lay down to do this?

We lay down $\answer [given]{4}$ black red chips.

Using our work with the chips, what is $1-5$ ?

$1-5 = \answer [given]{-4}$

You are working on the subtraction problem $(-8)-7$ , and you would like to solve this problem using red and black chips from a missing-addend perspective. First, you change your problem into the related addition problem. What is the result?

The related addition problem is $\answer [given]{-8} = \answer [given]{7} + ?$ .

We now need to begin with a model for $7$ and decide how to get to $-8$ . What chips would we lay down to do this?

We lay down $\answer [given]{15}$ black red chips.

Using our work with the chips, what is $(-8)-7$ ?

$(-8)-7 = \answer [given]{-15}$

In the picture below, use our definition of multiplication to find the total value of the chips.

The total value of the chips is $\answer [given]{3} \times \answer [given]{-4}$ .

Imagine using a number line like the one below to solve the subtraction problem $-8 - 4$ .

We begin by standing on the number line at the tick marked with $\answer [given]{-8}$ . Since we are subtracting, we face towards the . We will move $\answer [given]{4}$ spaces , since $4$ is positive. Where on the number line are we now?

We are located at the tick labeled $\answer [given]{-12}$ .

Imagine using a number line like the one below to solve the subtraction problem $(-22) - (-54)$ .

We begin by standing on the number line at the tick marked with $\answer [given]{-22}$ . Since we are subtracting, we face towards the . We will move $\answer [given]{54}$ spaces , since $54$ is negative. Where on the number line are we now?

We are located at the tick labeled $\answer [given]{32}$ .

In your own words, what is the difference between subtraction and negation?

Subtraction is an operation, requiring two numbers. With subtraction, we have two quantities that we are trying to relate. With our models, subtraction requires some movement: removing chips, adding chips, or walking on the number line. Negation, on the other hand, only requires a single value. With negation, we are essentially just asking, “What is the opposite?” With our models, this is essentially a switching rather than a moving: we switch from black chips to red, or we face backward instead of forward.