Fact Families, Number Lines, and Games

Positive and Negative Numbers:  

Fact Families, Number Lines, and Games

While researching the operation rules for positive and negative numbers, I came across the term fact families.  I looked it up and learned that it is a group of numbers used to demonstrate a mathematical principle.  Fact Families are used as a strategy in early elementary mathematics to make math easier to understand for students in first through third grade.  They build on a foundation of mathematical facts that the student already knows to be true.  

Fact Families

Fact families are introduced as a strategy for students to help them with addition and subtraction, then multiplication and division.  They begin with 3 numbers that can be represented in 4 different mathematical equations that are all true.  The numbers in the family either add or subtract to equal each other.  Students need to check their work to make sure that the fact belongs to the fact family.  Using fact families helps students fill in the blanks and make connections between the numbers.  They get comfortable manipulating numbers forwards and backwards, doing and undoing operations.  They are working with inverse operations with ease.
For example,  let's look at the fact family of 4, 5, and 9.
Students should be able to write 4 facts that are true statements about how these numbers relate to each other in terms of addition and subtraction.
                                                     4, 5, 9

                                            1.  4 + 5 = 9
                                2.  5 + 4 = 9
  *  After writing the two addition statements, students should take the sum and write two subtraction statements with numbers from the fact family.                          
                                    3.  9 - 5 = 4
                                            4.  9 - 4 = 5 

     *  They can continue by recognizing the statements that do not belong in the fact family.  While these statements are true, they do not help students connect the addition to the subtraction, therefore, they are not part of the fact family.

                                           5.  9 + 4 = 13
                                           6.  9 + 5 = 14  

Fact families work with positive and negative numbers.  If students are familiar with fact families from their early elementary math classes, introduce them as a strategy to understand the manipulation of positive and negative numbers.

    3, -4, -7                         -2, -3, -5

     Addition:          3 + -7 = -4                       -2 + -3 = -5

     Addition:          -7 + 3 = -4                       -3 + -2 = -5

     Subtraction:      -4 - 3 = -7                      -5 - (-2) = -3 

     Subtraction:    -4 - (-7) = 3                      -5 - (-3) = -2       

Number Lines

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Number Lines help students understand negative numbers by giving them a point of reference, a visual representation of a negative number.  Students aren't really taught about negative numbers until 6th grade.  They may have seen a science problem with negative degrees on a thermometer or heard that as you go further below sea level, you use negative numbers, but they do not use them in mathematical operations until 6th grade. Number lines are a great way to introduce negative numbers and their relationship to positive numbers.

Games   

To recognize the importance of games and learning math, I found a website hosted by the University of Cambridge and their NRICH Maths project.  The project is aimed at enriching mathematics for all math learners and to provide high-level engaging materials for teachers to use in their classrooms.   

Their game, Tug Harder!, involves playing a game along a positive and negative number line.

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 First Connect Three is another one of their games using positive and negative numbers:

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Visit their website for more game options.  They have resources for teachers and students.

 https://nrich.maths.org/

Common Core Standards³

CCSS.MATH.CONTENT.6.NS.C.5

CCSS.MATH.CONTENT.6.NS.C.6

CCSS.MATH.CONTENT.6.NS.C.6.A

Conclusion

Students need to learn how to work with positive and negative numbers.  These numbers have specific rules for each mathematical operation, and learning these rules will help make the life of a young mathematician so much easier.  Students might have to memorize the rules, but there are many ways to support students as they learn the rules.  Teachers should invest a little more time on this subject, hang posters in their rooms, use manipulatives, show videos and play games with their students.  I think this practice is the foundation for a major portion of  the mathematics curriculum going forward after 6th grade.     

Sources:

1- http://www.11plusforparents.co.uk/Maths/negativenum.html

2 - https://nrich.maths.org/5898

3 - http://www.corestandards.org/Math/Content/6/NS/

Multiplying and Dividing Positive and Negative Numbers

Continuing the Rules for Positive and Negative Numbers:

Multiplying and Dividing 

I wanted to continue with the rules for positive and negative numbers because the rules for adding and subtracting were tricky.   
Fortunately, the rules for multiplying and dividing are a little easier and after working with addition and subtraction hopefully students think so too.

Word Wall

In my research this week I can across the term fact family.  I liked what I learned about it because it reminded me of previous examples we've seen where the teacher selects specific numbers to teach a lesson.  Fact Family falls into this concept.  The fact family helps us understand the relationship between operations.  I'll delve deeper into this concept in my next blog.

Fact Family:  A group of math facts or equations using the same set of numbers.    

Multiplication Rules

1)  A positive times a positive equals a positive
                            5 x 5 = 25

2)  A negative times a negative equals a negative
                          -4 x -4 = 16

3)  A positive times a negative equals a negative
                          5 x -4 = -20

4)  A negative times a positive equals a negative
                          -5 x 4 = -20 

Division Rules

The rules for division are the same as the rules for multiplication.

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Flocabulary

Our social study teachers show the Flocabulary Week in Rap every Friday.  The students love it!  They use hip-hop rhythms and rap lyrics to get students excited about learning.  I had no idea they covered every subject and had lessons that aligned with state standards.  You have to pay for the service.  The attached video is 3:18 in length, but I don't have an account it only shows the first :40 seconds, you'll get a feel for what they do though.       

https://www.flocabulary.com/unit/multiplying-dividing-integers/  (3:18)

Poster

This poster demonstrates the rules for all of the operations.

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Conclusion

It is so important that students learn these rules.  Teachers need to demonstrate then in several ways to their students so they really grasp them.  They need to hear them, see them and use them in order to really understand them.  I'd recommend using manipulatives like chips or the - and + spacers we learned about in class.  Find interesting videos on the topic and allow students to work together to reason out why these rules work the way they do.  Also, have students apply them to real world problems to so they can connect meaning to them. 

Sources:

1 - https://www.youtube.com/watch?v=L1ZNw8Antv8

2 - http://www.loisterms.com/lois21.htm

Adding and Subtracting Positive and Negative Integers

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Rules for Adding and Subtracting Positive and Negative Numbers 

     Students come into sixth grade knowing what numbers are and many way to work with them.  Now they are asked to recognize a letter as a number and a number with a negative sign in front of it.  They will work extensively with positive and negative numbers during their sixth grade year and the better they feel about them going into 7th grade they better they will do.  There are specific rules for each operation with these numbers.  How can math teachers help students conceptualize the rules of positive and negative numbers so that they can be successful as they move on in math? 

Word Wall

Integers:  The set of whole numbers.  They can be positive, negative and zero.  No fractions.

Rational Numbers: A set of numbers that can be written as a  fraction or a whole number.  The quotient of the fraction either ends or repeats in a pattern. 

Negative Number:  A number less than zero

Positive Number:  A number greater then zero

Zero:  The number between the set of negative numbers and the set of positive numbers.  Zero is neither positive or negative. 

Absolute Value:  The distance from zero on the number line.  The absolute value is always positive. 

Positive and Negative Number Rules

There are 3 rules for addition:

1)  Adding two positive numbers, add and keep the sign.
                             6 + 3 = 9

2)  Adding two negative numbers, add and keep the sign.

                             -6 + -3 = -9

3)  When adding a positive number and a negative number, subtract the numbers and keep the sign of number with the largest absolute value.                  -6 + 3 = -3

Subtraction:

1)  Subtracting two positive numbers is simple subtraction.
                                    7 - 4 = 3

2)  Subtracting a positive number from a negative number, change the subtraction to addition and change the number from positive to negative and add.        -7 - 4
                                    -7 + (-4) = -11

3)  Subtracting a negative number:    7 - (-4) 
     Two negatives make a positive:    7 + 4 = 11

Poster and Chip Models 

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The following video explains how to subtract integers using positive and negative chips.  I love this idea and would use it in the classroom.  It really helps to see the idea of zeroing out 

Subtracting Integers Using a Chip Model (5:55)

https://www.youtube.com/watch?v=_77vO0uzBfA

Chip Models are great manipulatives to use in a classroom to help solidify this idea.  They provide a practical way to see what is happening with the positive and negative numbers.  

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Common Core Standards⁴

CCSS.MATH.CONTENT.6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

CCSS.MATH.CONTENT.6.NS.C.6.A
Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.

Conclusion

These rules are so specific to mathematical operations, that students must know them if they are going to be successful in the classroom.  I think it is important for teachers to slow down at this point in their curriculum and pull out all the manipulatives and support they can find to help students get these rules down.  From what I've seen in middle school, 7th grade math teachers expect the students to already know the rules and spend very little time, if any, reviewing them.  I'd recommend displaying posters, have students make note cards, watch different video's and use the chip model.  Looking at the rules from different perspectives and practicing them will help students understand how they work.     

Sources:

1 - https://i.pinimg.com/originals/1d/0e/95/1d0e953c1db6411ac3d1a18e1d82f789.jpg

2 - https://www.pinterest.com/pin/418201515371741167/?lp=true

3 - https://www.slideshare.net/aeherzog/adding-integers-ppt-3331211

4 - http://www.corestandards.org/Math/Content/6/NS/

Translating Algebraic Expressions

     As students progress into algebra, they need to be able to understand and interpret algebraic expressions.  Even though the expressions are usually short and relatively simple to read, they can be tricky for students to translate.  Students need to remember their mathematical terms for each operation.  They also need to be aware of "stitch order" and the use of parenthesis.  

Vocabulary

Algebraic Phrase:  A short written statement that contains numbers, variables, and mathematical operations

Need To Know

Algebraic expressions are like short word problems. 

Recall all key operation vocabulary:  

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Study and Practice²

Flash Cards
To help students remember the key words they could make flash cards to study with.  

Using 3x5 index cards they would write the operation and its symbol (+, -, x, /) on the front of the card and the associated key terms on the back, lined side of the card.  They can study independently or with a partner.  It is important that they know these words to help them translate a phrase properly.  

"Math Coach"

Once students have a set of 3x5 cards, they can start translating algebraic expressions.  Students can work in groups practicing how to translate an algebraic phrase.  Have one student read a phrase aloud to the group.  Students should underline any key terms that they find. The "math coaches" in the group would use their flash cards to identify the vocabulary in the phrase and follow the flash card operation.  The group should switch readers and coaches after they translate each phrase.
 
Example:
1.)  Break the phrase up into sections:  three times a number plus twelve    

                                                   three times a number plus twelve

2.)  Translate the identified key words:    3 * x + 12    

3.)  Rewrite in a variable phrase:              3x + 12

Switch Order Example:  six taken away from 5 and a number

1.)  Identify                 six taken away from 5 and a number

2.)  Translate                             6      -       5x   
     *This order will not work.  Students need to recognize they need the 5x first before they can take 6 away.  

3.)  Switch Order                       5x-6                                                  

Common Core Standard ³

CCSS.Math.Content.6.EE.A.2
Write, read, and evaluate expressions in which letters stand for numbers.

CCSS.Math.Content.6.EE.A.2.a
Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" as 5 - y.

Sources:

1 - https://img1.etsystatic.com/046/0/10046090/il_570xN.677510387_py3b.jpg

2 - https://www.vocabulary.com/articles/lessons/using-key-words-to-unlock-math-word-problems/

3 - http://www.corestandards.org/Math/Content/6/EE/