Unit 6: Fraction, Positive/Negative #'s, Algebra, and Inequalities

Dear 5V Math Families,

How are you? I hope you are well. I wanted to give you an update on our next unit of study. We will be moving into unit 6 upon our return to school on  April 1st. The unit will span from the 1st to the 16th. Standards of study include relevance to algebra, negative and positive numbers, fractions, and inequalities.

For more specifics please refer to the standards and skill acquisition below:

Standards

 

CC.6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

CC.6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

CC.6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

CC.6.NS.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.

CC.6.NS.6a 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.

CC.6.NS.7 Understand ordering and absolute value of rational numbers.

CC.6.NS.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

CC.6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.

CC.6.NS.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

CC.6.NS.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

CC.6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

CC.6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

CC.6.EE.2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

CC.6.EE.2c Evaluate expressions at specific values for their variables. Include expressions that arise from formulas in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^3 and A = 6 s^2 to find the volume and surface area of a cube with sides of length s = 1/2.

CC.6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

CC.6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

CC.6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. ACQUISITION (Students will know):   • Vocabulary: absolute value, algebraic expression, coefficient, constant term, equation, evaluate, exponents, inequality, integers, inverse operations, open sentence, opposite of a number, order of operations, rational numbers, reciprocal, relation symbol, solution, solution set, solve, variable, variable term • Reciprocal fractions will have a product of one. • That (a/b)(c/d) is equal to ac/bd (algorithm). • To divide by any non-zero number (whole numbers or fractions) you can multiply by its reciprocal, including fraction. • The standard algorithm for addition, subtraction, multiplication (traditional) and division (long division) with multi-digit decimals. • Positive and negative numbers can be applied to real world examples. • Positive and negative numbers have a relationship with zero. • How integers fit on a number line or coordinate plane. • The opposite of the opposite of an integer is itself . • Inequalities are statements about positions on the number line. • That absolute value is the distance between any number and zero. • Exponents represent repeated multiplication of a given base. • Each part of an expression has a name and a purpose. • A given value for a variable can be used to solve a variable expression or formula. • An equation or an inequality can have an answer or set of answers that make the equation or inequality true. • A variable expression or equation can represent a real-world situation (word problem). • A variable can represent a real-world situation (word problem). • A variable is used to represent numbers and write expressions. • An inequality can represent a real-life situation. • Inequalities have an infinite number of solutions. • Inequalities can be graphed on a number line.

Unit 5 Begins!

Hello 5V Math Accel Families,

How are you? We are starting unit 5 on Monday, March 11th. I wanted to share with you the common core standards we are covering, the acquisition of knowledge, the learning the kids are doing, and the transfer of that learning: Please see the information below:

Math Accel. Grade 5

Vocabulary: absolute value, axis, coordinate, coordinate plane, midpoint, ordered (number) pair, origin, parallel, perpendicular, polygon, parallelogram, quadrant, quadrilateral, reflection, right angle, right triangle, sector, skewed, trapezoid, triangle, vertex/vertices.

·  CC.6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

·  CC.6.RP.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent.

·  CC.6.NS.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.

·  CC.6.NS.6b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

·  CC.6.NS.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

·  CC.6.NS.7 Understand ordering and absolute value of rational numbers.

·  CC.6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.

·  CC.6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

·  CC.6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

ACQUISITION (Students will KNOW):

• Ratios can be represented as a percentage.
• How integers fit on a number line or coordinate plane.
• The opposite of the opposite of an integer is itself.
• That an ordered pair is the location of a point on a coordinate grid and how it is related to the x and y axis .
• That reflections across one or both axes are created by changing the signs of the x and/or y coordinate in an ordered pair.
• Positive and negative numbers have a relationship with zero .
• Positive and negative numbers can be applied to real world examples.
• How integers fit on a number line or coordinate plane.
• Students will understand there is a relationship among integers.
• That absolute value is the distance between any number and zero.
• Points on the coordinate plane can represent the vertices of a polygon.

ACQUISITION (Students will be able to):

• Determining a missing value in a percent proportion.
• Finding the opposite of a given number.
• Plotting rational numbers on a number line.
• Plotting ordered pairs on a coordinate plane.
• Graphing the reflection of a given ordered pair on a number line.
• Interpreting absolute value.
• Graphing points in all four quadrants of the coordinate plane.
• Using coordinates and absolute value to find distances between points with the same first coordinates or the same second coordinates.
• Determining the distance between two points on the coordinate plane given that the points have the same x or y value.
• Solving real-world problems using distances on the coordinate plane.

 

  

Math Update: Week of 2/11-2/15

Hi 5V Math Families,

How are you? We are journeying through the middle of Unit 4 this week. Last week we studied what factors are, prime factorization, how to find the greatest common factor, how to convert between fractions, decimals, & percents, how to find the part of a whole number by understanding that a percent is just that, and finally we ended the week by trying to learn what a down payment on a home represents and how the percent of a whole is a significant factor in buying a home.

For this week here is what we have on tap:

Monday: Study strategies for how to add fractions with unlike denominators.

Tuesday: There is no math because of an assembly.

Wednesday: Understanding percent and parts in authentic problems: Part 1

Thursday: Understanding percent and parts in authentic problems: Part 2

Friday: Analyzing the relationship between down payment, loan, and mortgage

 Please refer to the Common Core Math Standards Guide I emailed at the beginning of the unit for more specific information on the standards we are covering and studying this unit and for this week!

Finishing up Unit 3: Update 1/21-1/25

Hi 5V Families,

How are you? It's been a great unit 3. And now we are just about finished. We have studied and learned quite a bit at a high level. Some of that learning includes:

* Learning what expressions are.
* Learning how to write expressions.
* Learning what rate is.
* Learning how to calculate rate.
* Learning how to calculate distance, rate, and speed using formulas.
* Learning what unit rate is and how to find it.
* Learning what ratios are.
* Learning how to calculate ratios
* Learning how to solve rate and ratio story problems.
* Learning how to plot coordinate pairs on a coordinate plane.
* Learning what linear relationships are.
* Learning how to graph linear relationships.
* Learning how to interpret linear relationships.

The last thing that we will study formally as part of this unit includes:

* Learning how to solve for equations. (We've studied this in units 1 and 2 as well)
* Standard: 6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Learning how to solve equations is a standard that we see a few times this year in math and we will wrap up unit 3 with this standard!

Discovery Learning: Update 1/7-1/11

I titled this blog post "discovery learning". I talked to the chief (Mr. LeCrone) and we had an enlightening conversation about giving students more opportunities to at least begin their learning through a process of discovery. This doesn't mean we leave them to their own devices. It means that when we begin studying a skill (writing ratios), strategy (making proportions), content (taxation), or concept (sales tax) we give the students more time to discuss what they already know, more time to experiment with methods, more time to talk to each other about how they would solve a problem, more time for the kids to experiment with multiple options, and demonstrate their understanding in varied and multiple ways. It also means that I step in and model possible or correct methods afterwards, and assess for learning. 

Ratio of orange to grey is 3:1

Discover

Model

Assess

We tried this out on the Friday before winter break and it was quite meaningful and it allowed the children to demonstrate their understanding in their varied ways and multiple ways. The way I envision our math class utilizing this model of instruction is as follows: (Below is just a scenario) 

Monday in class: Discover before actual instruction. Kids will discuss what they already know, answer entry level questions about what they're trying to learn, use manipulatives to try demonstrate their understanding in varied or multiple methods, try to rationalize their work, and inquire about what they want to further learn. 

Monday night: Kids watch instructional video that will model how a strategy is used to solve a problem or how to use a strategy to become more adept at a skill. The modeling can also include the transmitting of simple information such as vocabulary or content knowledge. *If there is no video lesson kids will receive modeling the next day in class. 

Tuesday in class: Students will have more opportunity to demonstrate their understanding using their own methods or the testing of methods I have introduced. They will work together, or sometimes independently to demonstrate their understanding and I will assess their work or have a conference to assess them. Kids will also assess the success of the group and themselves.  


This week's standards focus:

6.RP.1
6.RP.2
6.RP.3 

• Defining and writing ratios
• Understanding equivalent ratios
• Using ratios to solve real world mathematical problems via reasoning. 
• Examining golden ratios