What Numbers Multiply To And Add To – Presentation on theme: “As 11 and 30 are two numbers: 5 and 6 because 6 & =11 and 5×6=30” — Presentation transcript:
1 which adds 2 numbers to give the first number and multiplies to give the second number…
What Numbers Multiply To And Add To
As 11 and 30 are two numbers: 5 and 6 because 6 & =11 & 5×6=30 9 and 14. 9 and 18. 7 and 10. 10 and 21. 11 and 10 11 and 28 13 and 40 13 and 36 11 and 24 1 and -6 1 and -12 4 and -21 7 and -18 2 and -24 1 and -30 2 and -35 6 and -27 -1 and -20 -4 and -12 -4 and -21 – 5 and -50 -8 and -9 -4 and -45 -5 and -24 -4 and -5
Part 1) Multiplying Two Binomials
3 Expansion (x+3)(x-6) FOIL is a mnemonic used to expand quadratic expressions (2 parentheses) F=First O=Outermost I=Inner L=Last (first ______)(first_____) ( x+3 )( x-6) (outer______)(_____outside) (x+3)(x-6) ( _____inner)(_____in) (x+3)(x-6) ( ______end) (______end) (x+ 3) (x-6)
Factorization is the opposite of expansion. You find the reason. (x+4)(x-2) = x2+2x-8 Which two numbers should be multiplied to give -8 and added to give +2? x2+8x adds to multiply two numbers and give +3 and +5 so the factor of x2+8x+15 is (x+3)(x+5) or (x+5)(x+3) You do: Factorize Do a) x2+9x+14 (x+4)(x-2)= +4 and -2 factors of 15 are 1×15 and 3×5
X2 + 2x – 24 are two numbers that multiply to give -24 and add to give +2. Write all the factors of 24: -1×24, -2×12, −3×8, -4×6, 1×-24, 2×-12, 3×-8, 4×-6. Which pair adds up to give +2? These are our factors. x2 + 2x – 24 = (x-4)(x+6) Your turn: x2 + 3x – 40
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In order to operate this website, we record user data and share it with processors. To use this website, you must agree to our Privacy Policy, including the Cookie Policy. 9x + 20 x2 + 11x + 24 = (x + 5)(x + 4) = (x + 8)(x + 3) What is the relationship between product form and factor form? x2 + 5x + 4 = (x + 4)(x + 1)
Many trinomials can be written as the product of 2 binomials. Remember: (x + 4)(x + 3) = x2 + 3x + 4x + 12 = x2 + 7x + 12 The middle term of a simple trinomial is the SUM of the last two terms of the binomial. The last term of a simple trinomial is the PRODUCT of the last two terms of the binomial. This is why this type of factorization is referred to as SUM PRODUCT!
For factoring triples, you ask yourself… “What numbers multiply the last terms and add to the middle terms?” x12 +7 x2 + 7x + 12 1, 2, 6 8 (x + 3)(x + 4) 3, 4 7
17 – 5t – 3t2 + 15 + 4t2 – 3 – 3t t2 – 8t +12 Factor: ( x – 2)( x – 6)
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Step 1: Combine terms like t2 – 8t +12 ( x – 2) ( x – 6) x 12 – 8 1, 12 13 -1, -12 -13 2, 6 8 -2, -6 -8
18 7q2 – 14q – 21 7 ( q2 –2q –3) 7 ( q – 3)( q + 1) Factor: -3 -2 -1, 3 2
Step 1: Find the GCF 7 (q2 –2q –3) 7 (q – 3)( q + 1) -3 -2 -1, 3 2 -3, 1 -2
Then see if you can find a common factor. Enter 2 sets of parentheses with x in the first position. Find 2 numbers whose sum is the middle coefficient, and whose product is the last term. Check by thwarting the factor. ie + = 7 x = 10 common factors? 5, 2 ie + = 1 x = -20 common factors? -4, 5
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20 Let’s see x2 + 5x + 6 How can we factor it using algebra tiles? Create a rectangle using the correct number of tiles in the given expression. Remember that a trinomial represents the area – two binomials multiplied by each other. What is the width and length of the rectangle? These are the factors of the original rectangle. To know that? (x+3)(x+2) x + 3 x + 2
Create a rectangle using the correct number of tiles in the given expression. Remember that a trinomial represents the area – two binomials multiplied by each other. What is the width and length of the rectangle? These are the factors of the original rectangle.
In order to operate this website, we record user data and share it with processors. To use this website, you must agree to our Privacy Policy, including the Cookie Policy. When students in grades 3 and up initially learn to work with addition, subtraction, multiplication, division, and basic numerical expressions, they begin by performing operations on two numbers. But what happens when an expression requires multiple operations? For example, do you count or multiply first? What about multiplication or division? This article explains the sequence of operations and gives you examples that you can also use with students. It also offers two lessons to help you introduce and develop the concept.
The order of operations is an example of mathematics that is very systematic. It’s easy to mess up because it’s less a concept you’ve mastered and more a list of rules you have to memorize. But don’t be fooled into thinking procedural skills can’t be deep! It is suitable for older students and can provide difficult problems suitable for class discussion:
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Over time, mathematicians agreed on a rule called the order of operations to determine which operations should be performed first. When an expression includes only four basic operations, here are the rules:
When simplifying an expression such as (12 div 4 + 5 times 3 – 6) first calculate (12 div 4) since the order of operations requires that any multiplication and division (whichever comes first) be evaluated first. left to right before evaluating either addition or subtraction. In this case, this means calculating (12 div 4) first, then (5 times 3). Once all multiplication and division are complete, proceed from left to right with addition or subtraction (whichever comes first). The steps are shown below.
Sometimes we want to make sure that addition or subtraction is performed first. Grouping symbols such as parentheses (( )), parentheses ([ ]), or parentheses (\), allow us to specify which operations are performed.
Order of operations requires that operations within the grouping symbols be performed before operations outside them. For example, suppose there were parentheses around the expression 6 + 4:
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Note that the expression has completely different values! What if we put brackets around (7 – 3) instead?
Because (4 times 4 = 16), and with no parentheses left, we proceed with multiplication before addition.
This set of brackets gives another answer. So, when parentheses are involved, the rules for ordering operations are:
Before your students use parentheses in math, they need to be clear about the order of operations without parentheses. Begin by reviewing the addition and multiplication rules for order of operations, then show students how parentheses can affect that order.
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Prerequisite Skills and Concepts: Students must be able to evaluate and discuss addition, subtraction, multiplication, and division.
This would be a good moment to discuss the mathematical practice of attention to precision. In mathematics, it is important that we act deliberately when writing mathematical expressions and making mathematical statements. Small mix-ups with mathematical rules of operations or parentheses can make drastic changes! Imagine that you evaluate an expression incorrectly when, for example, calculating the dosage or cost of a drug.
Give students a few more examples, showing an expression with and without brackets. Evaluate the expressions of student volunteers and compare their values. When students arrive at different values, avoid telling them they are right or wrong. Instead, ask them to find the similarities and differences between the strategies, and guide the discussion so students can see which strategies match the action rules.
Prerequisite Skills and Concepts: Students must be familiar with the sequence of operations and feel prepared to practice it.
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It is important that students remember the rules for ordering operations with and without parentheses Avoid giving extra practice worksheets. Instead, look for math problems that naturally lead to evaluating expressions, such as substituting values into a formula, and let students practice the sequence of operations in the context of other problems.
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