(+2) + (+7) = (+9)? ( 3) + ( 4) = ( 7)? (+2) + ( 7) = ( 5)? (+9) + ( 4) = (+5)? (+2) (+7) = ( 5)? ( 3) ( 4) = (+1)?

Transcription

1 L1 Integer Addition and Subtraction Review.notebook L1 Adding and Subtracting Integer Review In grade 7 we developed rules for adding and subtracting integers... Do we remember the process for adding and subtracting Add the following integers: (2) (7) = (9) ( 3) ( 4) = ( 7) (2) ( 7) = ( 5) (9) ( 4) = (5) When adding two integers with the same signs we add the "numbers" and keep the sign the same. When adding two integers with different signs we keep the sign Click Here of the number we have the most, and find the difference of the two "numbers" Add the following integers: (2) (7) = ( 5) ( 3) ( 4) = (1) (2) ( 7) = (9) (9) ( 4) = (13) When subtracting integers the first number remains the same and we add the opposite of the second number. Then follow the addition rules above. Click Here Link to Worksheet

2 L2 Using Models to Multiply Integers (2.1).notebook 2.1 Using Models to Multiply Integers Model 1: Bank Model One of the models used to multiply integers is the bank model. Our bank is represented by a circle and we will always start with a balance of 0 in the bank. Step 1: When multiplying 2 integers the sign of the first number tells us the number of groups and if we are doing a deposit () or withdraw ( ). Step 2: When multiplying 2 integers the second number tells us the size of the groups if the groups are positive or negative. Example 1: (3) x ( 4) The (3) tells us we are putting in 3 groups. The ( 4) tells us we have 4 negatives in each group. Example 2: (2) x (3) Example 3: ( 2) x (3) The (2) tells us we are putting in 2 groups. The (3) tells us we have 3 positives in each group. After putting in 2 sets of (3) we have (6) in the bank. So (2) x (3) = (6) In this case we have to take out 2 sets of (3). We have to start with a balance of 0 in the bank so we add zero pairs until we have enough () in the bank to remove 2 sets of 3. After taking out 2 sets of (3) we have ( 6) in the bank. So ( 2) x (3) = ( 6) After putting in 3 sets of ( 4) we have ( 12) in the bank. So (3) x ( 4) = ( 12) Example 4: ( 2) x ( 2) In this case we have to take out 2 sets of ( 2). We have to start with a balance of 0 in the bank so we add zero pairs until we have enough () in the bank to remove 2 sets of ( 2). After taking out 2 sets of ( 2) we have (4) in the bank. So ( 2) x ( 2) = (4)

3 L2 Using Models to Multiply Integers (2.1).notebook Model 2: Number Line Model Another model used to multiply integers is the number line model. Step 1: When multiplying 2 integers the sign of the first number tells us the direction we are facing and the number of steps. Step 2: When multiplying 2 integers the second number tells us the size of the steps and the type of step, () is forward and ( ) is backward. Example 1: ( 2) x ( 5) The ( 2) tells us we are facing negative and taking 2 steps. The ( 5) tells us we are waking backwards in steps of size 5. Our walker stops at (10) on the number line so ( 2) x ( 5) = (10) Example 2: ( 1) x (5) The ( 1) tells us we are facing negative and taking 1 step. The (5) tells us we are waking forwards in a step of size 5. Example 3: (3) x ( 4) The (3) tells us we are facing positive and the ( 4) tells us we are waking backwards in steps of size 4. Our walker stops at ( 12) on the number line so (3) x ( 4) = ( 12) Example 4: ( 2) x ( 4) The ( 2) tells us we are facing negative and the ( 4) tells us we are waking backwards in steps of size 4. Our walker stops at (8) on the number line so ( 2) x ( 4) = (8) Example 5: ( 3) x (3) The ( 3) tells us we are facing negative and the (3) tells us we are waking forwards in steps of size 3. Our walker stops at ( 9) on the number line so ( 3) x (3) = ( 9) Our walker stops at ( 5) on the number line so ( 1) x (5) = ( 5)

4 L2 Using Models to Multiply Integers (2.1).notebook Example 4: The temperature fell 3 o C each hour for 6 hours. Use an integer number line to find the total change in the temperature. A fall of 3 o C can be represented by 3. A time of 6 h can be represented by 6. Using integers, we need to find (6) x ( 3) (6) x ( 3) = ( 3) ( 3) ( 3) ( 3) ( 3) ( 3) The (6) tells us we are facing positive and the ( 3) tells us we are waking forwards in steps of size Our walker stops at ( 18) on the number line so ( 2) x ( 4) = (8) The total change in temperature was 8 o C. Questions to try: Pp #s 5, 6, 7, 8, 9, 10, 11, 13, 16, 17

5 L3 Developing Rules to Multiply Integers (2.2).notebook 2.2 Developing Rules to Multiply Integers Lets review the products in the examples given in the last lesson: (3) x ( 4) = ( 12) (2) x (3) = (6) ( 2) x (3) = ( 6) ( 2) x ( 2) = (4) ( 2) x ( 5) = (10) ( 1) x (5) = ( 5) ( 2) x ( 4) = (8) ( 3) x (3) = ( 9) (6) x ( 3) = (18) Find each product: ( 3) x (6) (2) x (4) 3 x 6 = 18 and the integers have opposite signs, so the product is negative ( 3) x (6) = ( 18) 2 x 4 = 8 and the integers have the same sign so, the product is positive (2) x (4) = (8) What do you notice about the products when we multiply two numbers with the same sign ( 5) x ( 3) (2) x ( 7) When we multiply two numbers with the same sign () x () = () or ( ) x ( ) = () the answer will be positive. What do you notice about the products when we multiply two numbers with different signs 5 x 3 = 15 and the integers have the same sign, so the product is Positive ( 5) x ( 3) = (15) 2 x 7 = 14 and the integers have opposite signs, so the product is negative (2) x ( 7) = ( 14) When we multiply two numbers with different signs () x ( ) = ( ) or ( ) x () = ( ) the answer will be negative.

6 L3 Developing Rules to Multiply Integers (2.2).notebook We may also use some properties of whole numbers to assist in multiplying integers. Example 2 Method 2 We can also multiply using our more traditional method. First multiply the number as if they were positive. Example 1 25 x The integers have the same sign, so the product is positive. ( 25) x ( 48) = (1200) Questions to try Pp #s 3, 4, 6 (a,c,e,g), 7 (a, c, e, g), 8, 9, 10, 15, 17

7 L4 Using Models to Divide Integers (2.3).notebook 2.3 Using Models to Divide Integers We can use the same models to divide integers that we used to multiply integers. Model 1: Bank Model One of the models used to divide integers is the bank model. Our bank is represented by a circle and as with multiplication we will always start with a balance of 0 in the bank. Step 1: When dividing 2 integers tells us what we want to end up having in the bank. Step 2: When dividing 2 integers the second number tells us the size of the groups we are moving. Example 2: The ( 6) tells us we want to end up with 6 negatives in the bank at the end. The (2) tells us we are moving sets of 2 positives. We cannot add sets of (2) to the bank and end up with ( 6) at the end! What can we do We can add zero pairs to the bank until there is a total of 6 negatives in the bank. We are removing 3 sets of 2 positives from the bank. Withdrawing from the bank (subtracting) is negative, so Example 1: The ( 12) tells us we want to end up with 12 negatives in the bank at the end. The ( 4) tells us we are moving sets of 4 negatives. What do we do with our sets of 4 negatives to end up with 12 negatives in the bank We have to add 3 sets of negative 4 to the bank! Depositing in the bank (adding) is positive, so Example 3: We cannot add sets of ( 4) too make (8) in the bank. Instead we add zero pairs until there is a total of 8 negatives in the bank. We remove 2 sets of 4 negatives from the bank and there will be 8 positives remaining. Withdrawing from the ban (subtracting) is negative, so

8 L4 Using Models to Divide Integers (2.3).notebook Model 2: Number Line Model Another model used to divide integers is the number line model. Step 1: When dividing 2 integers the first number is the final location on the number line. Step 2: When dividing 2 integers the second number is the step size and direction we are moving, forwards or backwards Example 2: The ( 2) means walking backwards with a step size of 2. In order to get to (6) while walking backwards the person must be facing the negative direction. Making 3 steps and facing the negative give an answer of 3. Example 1: The ( 8) is where we are going on the number line. The ( 4) means walking backwards with a step size of 4 Example 3: In order to get to ( 8) while walking backwards then the person must be facing towards the positive end of the number line. It would take two steps to get to ( 8). Two steps while facing positive gives and answer of (2). The (5) means we are walking forwards with a step size of 2. In order to get to ( 10) while walking forwards we must be facing the negative direction. Making 2 steps and facing negative gives an answer of 2.

9 L4 Using Models to Divide Integers (2.3).notebook Example 4: At 5 pm the temperature outside was 20 o C, 4 hours later at 9 pm the temperature had decreased to 32 o C for a total decrease in temperature of 12 o C. How much did the temperature fall each hour A decrease of 12 o C is represented by ( 12) The four hours later is represented by (4) To find the change per hour we divide the temperature change by the number of hours The (4) means we are walking forwards with a step size of 4. In order to get to ( 12) while walking forwards we must be facing the negative direction. Making 3 steps and facing negative gives an answer of ( 3) Questions to try: Pp #s: 3, 4, 6, 9, 11, 12

10 L5 Developing Rules to Divide Integers (2.4).notebook 2.4 Developing Rules to Divide Integers Lets review the quotients in the examples given in the last lesson: ( 12) ( 4) = (3) ( 6) (2) = ( 3) (8) ( 4) = ( 2) ( 8) ( 4) = (2) (6) ( 2) = ( 3) ( 10) (5) = ( 2) ( 12) (4) = ( 3) Find each product: ( 20) (5) (15) (3) 20 5 = 4 and the integers have opposite signs, so the product is negative ( 20) (5) = ( 4) 15 3 = 5 and the integers have the same sign so, the product is positive (15) (5) = (5) What do you notice about the quotients when we divide two numbers with the same sign ( 9) ( 3) (14) ( 7) When we divide two numbers with the same sign () () = () or ( ) ( ) = () the answer will be positive. What do you notice about the products when we multiply two numbers with different signs 9 3 = 3 and the integers have the same sign, so the product is Positive ( 9) ( 3) = (3) 14 7 = 2 and the integers have opposite signs, so the product is negative (14) ( 7) = ( 2) When we divide two numbers with different signs () ( ) = ( ) or ( ) () = ( ) the answer will be negative. Note: The rules for multiplying and dividing integers are the same Example A sprinkler was 80 cm below ground level. Mr. King has a machine that digs down 16 cm at a time.how many digs does he need to make in order to reach the sprinkler 80 cm below ground can be represented by 80. Digging down 16 cm can be represented by 16 Using integers, we need to find ( 80) ( 16). ( 80) ( 16) = (5) Mr. King must make 5 digs.

11 L5 Developing Rules to Divide Integers (2.4).notebook Divide the following integers. Note: When there is no bracket around a number and no signed shown in front of the number, the number is considered to be positive. 4 1 Question to try: Pp , 5, 7, 8, 9, 10, 11, 12, 13, 15,

12 L6 Order of Operations with Integers (2.5).notebook 2.5 Order of Operations with Integers Evaluate the expression below on your notebook and record your answer. Let's look at our questions again making sure we follow the order of operations 9 x Multiply / divide left to right... Multiply first here 9 x Let's record some of the answers in the space provided below Division before adding and subtracting Add / subtract left to right... Add first here 63 1 Subtract your numbers 62 When you performed your calculation, did you remember to follow the order of operations. The order of operations does not change, it is the same for whole numbers, decimals, integers and fractions. Note: B Brackets When simplifying the expression I stacked my steps vertically. It is much easier than doing it across the page in 1 line as some students try to do, it is too confusing. E Exponents (We will do these in grade 9) 9 x = = = 63 1 = 62 DM AS Division and Multiplication in the order they appear (left to right) Addition and Subtraction in the order they appear (left to right) The division could be written in another forms. The 9 and the 6 written adjacent to each other means multiplication. The the 36 over 4 is a fraction, but it also means division!

13 L6 Order of Operations with Integers (2.5).notebook For you to try: Example 1: (10 8) x ( 4) 2 x 4 2 x ( 4) 2 x 4 ( 8) 2 x 4 ( 4) x 4 16 Example 2: The subtraction is done first in this problem because it is inside the brackets. After the brackets the only operations left are multiplication and division, so we complete those in order left to right. Example 4: Evaluate the numerator and denominator separately. After you have evaluated both then you divide them to get your final answer. Note that in the numerator, the square brackets take priority over anything inside "regular" brackets. ( 10) x ( 1) ( 8) 4 10 ( 8) 4 10 ( 2) Example 3: ( 1) 8 4 ( 1) ( 1) 2 ( 1) ( 1) ( 2) ( 1) 2 1 The multiplication and division were done first, left to right, before we did the subtraction. Note that we changed subtracting a negative to adding a positive. The division is done first, left to right, in the order they appear. The subtraction was done last after we changed it to adding the opposite. Questions to try: Pp #s: 3, 7, 8, 9, 10, 13, 15