Build Your Model: What’s Your Sign? CAMT 2016

Ally Zamora Klein ISD Math Instructional Specialist University of Texas at Arlington Interdisciplinary Studies in math and science(BS)

Kimi Williams Klein ISD Math Instructional Specialist Sam Houston State University Academic Studies with a minor in mathematics (BS) Curriculum and Instruction with a concentration in math (MEd)

Integer Operations with NO RULES  Integer Chips Integer Operations with NO RULES 

1st Step Have your students understand the following concepts: Opposites Zero pairs

You have red and yellow chips sitting on your desk. Red Chips = Negative Value Yellow Chips= Positive Value

The first concept we need to agree on is OPPOSITES. Place 1 red chip on your desk, what is the opposite of 1 red chip? Place 3 red chips on your desk, what is the opposite of 3 red chips? Place 4 yellow chips on your desk, what is the opposite of 4 yellow chips?

So now that we have agreed on what OPPOSITES look like… When you put an integer together with its opposite, it has a value of zero. Let’s give it a name…..a ZERO PAIR. We call it this because it has a value of ZERO.

Place 2 Red chips on your desk. How many yellow chips do you need to place on your desk to make a value of ZERO?

Place 4 Yellow chips on your desk. How many red chips do you need to place on your desk to make a value of ZERO?

Use 6 chips and create a value of ZERO.

Use 12 chips and create a value of ZERO.

Use 7 chips and create a value of ZERO. What is the value with 7 chips?

Use 9 chips and create a value of ZERO. What is the value with 9 chips?

Adding Integers

Model and Solve this problem using your integer chips Ok, now that you FULLY UNDERSTAND a ZERO PAIR, we can start solving some integer problems. Model and Solve this problem using your integer chips -3 + 4 = Teacher Talk: “Negative 3 combined with positive four” “Can we create any zero pairs?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips 6 + (-4) = Teacher Talk: “Positive 6 combined with negative four” “Can we create any zero pairs?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips -5 + 7 = Teacher Talk: “Negative 5 combined with positive 7” “Can we create any zero pairs?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips 3 + 5 = Teacher Talk: “Positive 3 combined with positive 5” “Do we need to create any zero pairs?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips -3 + -2 = Teacher Talk: “Negative 3 combined with negative 2” “Do we need to create any zero pairs?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips -4 + -5 = Teacher Talk: “Negative 4 combined with negative 5” “Do we need to create any zero pairs?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips 8 + (-3) = Teacher Talk: “Positive 8 combined with negative 3” “Can we create any zero pairs?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips -6 + (4) = Teacher Talk: “Negative 6 combined with positive 4” “Can we create any zero pairs?” “What is the value of the remaining chips?”

Let’s write a summary statement about Adding Integers in your own words. Sentence Starters: When I add integers with the same sign……. _________________________________________________________. When I add integers with different signs…… ___________________________________________________________.

Subtracting Integers

Sometimes when we subtract numbers we have to REGROUP. 100 23 For example: If I have 3 positive chips and I want to TAKE AWAY 5 negative chips, how do you CREATE negative chips to TAKE AWAY? ZERO PAIRS!!!!! +3 – (-5) = The same thing has to happen when we subtract with integers. The way we REGROUP with integers is with ZERO PAIRS. Putting these Zero Pairs into the problem does not change the value.

Model and Solve this problem using your integer chips 5 - 6 = How many Zero Pairs should we create, so we can remove 6 positive chips? When you create a zero pair and add it to the problem it DOES NOT change the VALUE of the problem!!!!! REMEMBER Teacher Talk: “Positive 5 TAKE AWAY positive 6” “Do we have enough chips to TAKE AWAY positive six?” “Could we create zero pairs first and then TAKE AWAY positive 6?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips 3 - 7 = How many Zero Pairs should we create, so we can remove 7 positive chips? When you create a zero pair and add it to the problem it DOES NOT change the VALUE of the problem!!!!! REMEMBER Teacher Talk: “Positive 3 TAKE AWAY positive 7” “Do we have enough chips to TAKE AWAY Positive 7?” “Could we create zero pairs first and then TAKE AWAY positive 7?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips 6 – (-4) = How many Zero Pairs should we create, so we can remove 4 negative chips? Teacher Talk: “Positive 6 TAKE AWAY negative 4” “Do we have enough chips to TAKE AWAY negative 4?” “Could we create zero pairs first and then TAKE AWAY negative 4?” “What is the value of the remaining chips?” When you create a zero pair and add it to the problem it DOES NOT change the VALUE of the problem!!!!! REMEMBER

Model and Solve this problem using your integer chips -5 – (3) = How many Zero Pairs should we create, so we can remove 3 positive chips? Teacher Talk: “Negative 5 TAKE AWAY positive 3” “Do we have enough chips to TAKE AWAY positive 3?” “Could we create zero pairs first and then TAKE AWAY positive 3?” “What is the value of the remaining chips?” When you create a zero pair and add it to the problem it DOES NOT change the VALUE of the problem!!!!! REMEMBER

Model and Solve this problem using your integer chips -7 - (-4) = Do we need to create ZERO PAIRS to be able to remove 4 negative chips? Teacher Talk: “Negative 7 TAKE AWAY negative 4” “Do we have enough chips to TAKE AWAY negative 4?” “Do we need to create zero pairs to TAKE AWAY negative 4?” “What is the value of the remaining chips?”

Model and Solve this problem using your integer chips -4 - (-5) = Do we need to create ZERO PAIRS to be able to remove 5 negative chips? Teacher Talk: “Negative 4 TAKE AWAY negative 5” “Do we have enough chips to TAKE AWAY negative 5?” “Do we need to create zero pairs to TAKE AWAY negative 5?” “What is the value of the remaining chips?”

Let’s write a summary statement about Subtracting Integers in your own words. Sentence Starters: When I subtract integers with the same sign……. _________________________________________________________. When I subtract integers with different signs…… _________________________________________________________

Multiplying and Dividing Integers

We can use this same array to learn how to multiply integers. In elementary school you might have used an array (also known as area model) for learning multiplication. If not let’s do a quick review: The array below can be used to show 2 groups of 3 or 3 groups of 2. 3 x Positive 6 2 We can use this same array to learn how to multiply integers.

The only term we need to learn for multiplying integers is “the opposite of” Read these below statements: the opposite of 5 (-5) The opposite of -4 (4) The opposite of 3 groups of 5 ( 3 x -5) The opposite of 6 groups of 8 ( -6 x 8) The opposite of the opposite of 2 groups of 4 (-2 X -4) Do you understand the meaning of the phrase “the opposite of”?

Positive 6 Negative 6 Positive 6 Let’s make some arrays (models) and see how this applies to multiplying integers: 2 groups of 3 the opposite of 2 groups of 3 The opposite of the opposite of 2 groups of 3 -3 3 -3 2 2 -2 Positive 6 Negative 6 Positive 6

1.) -5 X 4 = The opposite of 5 groups of 4 Let’s say each of these phrases and model it with our integer chips: 1.) -5 X 4 = The opposite of 5 groups of 4 2.) 6 X -7 = The opposite of 6 groups of 7 3.) -3 X -2 = the opposite of the opposite of 3 groups of 2 4.) 9 X 7 = 9 groups of 7 5.) -7 X -5 = the opposite of the opposite of 7 groups of 5

Does this work for DIVISON??? We can use the OPPOSITE strategy if we know about FACT FAMILIES. Fact Family: 2 X 3 = 6 3 X 2 = 6 6 ÷ 3 = 2 6 ÷ 2 = 3 Fact Family: -2 X 3 = -6 3 X -2 = -6 -6 ÷ 3 = -2 -6 ÷ -2 = 3 The opposite of 2 groups of 3 The opposite of 3 groups of 2 The opposite of 6 divided by 3 The opposite of 6 divided by 2

-5 X -4 = 20 -4 X -5 = ______ 20 ÷ -4 = ______ 20 ÷ -5 = ______ Let’s See if you can complete these FACT FAMILIES: Remember to say “the opposite of” -5 X -4 = 20 -4 X -5 = ______ 20 ÷ -4 = ______ 20 ÷ -5 = ______ -3 X -4 = 12 -4 X -3 =_______ 12 ÷ -4 = ______ 12 ÷ -3 = ______ -2 X 8= -16 8 X -2 = _______ -16 ÷ -2 = ______ -16 ÷ 8 = ______ -6 X 7 = -42 7 X -6 = ______ -42 ÷ -6 = ______ -42 ÷ 7 = ______

Additional Strategies to Build Integer Fluency

Integer Index Cards Self-check activities (Color Match) Integer Bingo

Ally Zamora azamora1@kleinisd.net Kimi Williams kwilliams6@kleinisd.net