# Making Sense of Math

### AISD Parent Newsletter for the 4th Six Weeks

## Fourth Six Weeks

I hope this newsletter is helpful for parents. If you would like more explanation on any of the topics mentioned below, please feel free to contact me. I would be happy to visit with you or give you examples.

Anna Holmgreen, Director of Instruction for Math

## Direct Mailing?

## Kindergarten

**16-20.**Students are expected to:

*count forward and backward*(with and without objects), as well as**read**,**write**, and**represent**the numbers.**compose and decompose numbers up to 10**using objects. They*instantly recognize the number being represented by a small quantity of objects*.**compare**sets of objects up to 20 and generate a set of objects and pictures that is*more than*,*less than*, or*equal to*a given number.**use comparative language**to describe the comparison of numbers represented using objects, pictures, or numerals.**recite numbers up to at least 100**by ones beginning with any number.**recite numbers by tens**up to at least 100 beginning with any multiple of 10 (e.g., 10, 20, 30, 40, etc.).

They also work on the** problem-solving process** by working repeatedly with real-world problem situations. Students continue to distinguish between joining, separating, and part-part-whole situations and the operation needed to solve the problem. Students:

- orally
**explain models and representations and justify solution strategies**. - use graphing situations to represent data with numbers 0-20. Students
**sort****and organize the data and the data is used to create real-object and picture graphs**. These graphs are examined to understand the components of graphing (e.g., title, labels of categories, what each cell or picture represents, etc.)

## 1st Grade

During this unit, students extend their understanding of the base-10 place value system to include the **hundreds** **place** as they continue exploring the foundations of whole numbers **up to 120**. Students:

- compose and decompose numbers through 120 as so many
**hundreds**, so many**tens**, and so many**ones**using*objects*,*pictures*, and*numbers* **generate numbers that are more or less than a given number**using tools (e.g., a hundreds chart, calendar, base-10 blocks, etc.).**compare whole numbers up to 120 and represent the comparison using comparison language and comparison symbols.**- extend using
*place value*and*open number lines*to order whole numbers up to 120. **recite numbers up to 120 forward and backward by ones and tens**;**skip****count**by**2s, 5s, and 10s**; and use place value patterns to determine a number that is*10 more*or*10 less*than a given number.**identify pennies, nickels, dimes, and quarters**by value and record the value using the cent symbol.**exchange coins or sets of coins**for other equivalent denominations.- apply skip counting by 2s, 5s, and 10s and compound counting to determine the value of a collection of pennies, nickels, and dimes up to 120 cents.
- use data represented in
**bar-type**and**picture graphs**to represent, generate, and solve problem situations involving sums and minuends up to 20 using spoken words, objects, pictorial models, and number sentences. - explore and explain strategies to solve one-step problems involving addition, subtraction, and comparison of the data. Students are expected to use a
*number sentence*with the unknown in*any position*to represent the situation.

## 2nd Grade

- During this unit, students
**add up to****four two-digi****t**numbers or**subtract two-digit numbers**that include calculating how money saved accumulates over time with deposits and withdrawals.These problems should include numbers that require*within 1,000*in real-world problem situations**regrouping**to solve the problem. In addition, students: - make
**connections between representing and solving addition and subtraction problems using***flexible methods*, concrete and pictorial models, and number sentences to mental strategies and algorithms based on knowledge of place value and properties of operations. **generate and solve problem situations for a given number sentence**involving addition and subtraction of whole numbers*within 1,000*.

- Students transition from
**bar-type graphs**to**bar graphs**and from**picture graphs**to**pictographs**. Both vertical and horizontal orientations of bar graphs and pictographs with up to*four categories*and*intervals of one or more*are experienced during this unit. Students**summarize**the existing data, or the inferential data, in bar graphs and pictographs to**draw conclusions**and**make predictions**. - Students also generate and solve one-step word problems based on the information in bar graphs and pictographs with intervals of one.

## Bar Grapha graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs | ## Pictographa graphical representation to organize data that uses a picture or symbol, where each picture or symbol may represent one or more than one unit of data, to show the frequency (number of times) that each category occurs |

## Bar Graph

a graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs

## 3rd Grade

- During this unit, students solve one- and two-step problems involving addition and subtraction
**within 1,000**and**multiplication and division within 100.** - Students reason through and solve real-world problem situations. Students explain their reasoning and solution strategies using
**expressions**,**equations**, and**precise****mathematical****language**. - Students solve problems involving
**multiplication**of a**two-digit number by a one-digit number**and develop fluency using standard algorithms to solve addition and subtraction problems*within 1,000*. - Experience real-world situations that involve various operations, including decomposing composite figures to
**determine area**of the original figure. - Use
**input-output tables**and explore**number pairs**in tables to determine additive and multiplicative patterns that exist and represent the pattern (or process) using equations and expressions. - Summarize a set of data using a
**frequency table**,**dot plot**,**pictograph**, or**bar graph**. Students use these data representations to solve one- or two-step problems involving the categorical data represented.

- Sort and classifying
**two**- and**three-dimensional**figures that may vary in size, shape, and orientation based on attributes using formal geometric language. - Explore two-dimensional figures and subcategories of
**quadrilaterals**, including**rhombuses**,**parallelograms**,**trapezoids**,**rectangles**, and**squares**. - Decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole.

## Composite Figurea figure that is composed of two or more two-dimensional figures | ## Decompose Figuresto break a geometric figure into two or more smaller geometric figures | ## Equationa mathematical statement composed of equivalent expressions separated by an equal sign |

## Expressiona mathematical phrase, with no equal sign, that may contain a number(s), an unknown(s), and/or an operator(s) | ## Input-Output Tablea table which represents how the application of a rule on a value, input, results in a different value, output | ## Polygona closed figure with at least 3 sides, where all sides are straight (no curves |

## Expression

a mathematical phrase, with no equal sign, that may contain a number(s), an unknown(s), and/or an operator(s)

## Input-Output Table

a table which represents how the application of a rule on a value, input, results in a different value, output

## 4th Grade

- During this unit, students begin the formal introduction to
**formulas**to determine the**perimeter and area of rectangles and squares**. Students use models to determine the formulas for the perimeter of a rectangle*(l + w + l + w*or2*l +*2*w)*, including the special form for perimeter of a square (4*s*) and the area of a rectangle*(l x w)*. - Identify
**parallel**and**perpendicular**lines. - Students are expected to
*solve problems related to perimeter and area*of rectangles where dimensions are whole numbers. - In addition to solving problems involving length, students also
*solve problems*that deal with measurements of intervals of*time*,*liquid volumes, mass, and money*using addition, subtraction, multiplication, or division as appropriate. - Identify relative sizes of measurement units within the customary and metric systems, and apply this knowledge to
*conversion of measurements within the same measurement system*, customary or metric.

- Identify
**points**,**lines**,**line segments**,**rays**,**angles**, and perpendicular and parallel lines. These concepts are essential for the ability to classify two-dimensional figures. - Use formal geometric language such as
**parallel**,**perpendicular**,**acute**,**obtuse**, and**right angle**to classify two-dimensional figures. - Additionally, students apply knowledge of right angles to identify
**acute**,**right**, and**obtuse****triangles**. - Identify and draw one or more
**lines of symmetry**, if they exist, for two-dimensional figures.

## Parallel Lines Lines are parallel if they are always the same distance apart and never meet. | ## Perpendicular Lines Perpendicular lines are those at right angles (90º) to each other. | ## Line Segment A line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points. |

## Lines of symmetryIf you can reflect (or flip) a figure over a **line**and the figure appears unchanged, then the figure has reflection**symmetry**or**line symmetry**. The**line**that you reflect over is called the**line of symmetry**. A**line of symmetry**divides a figure into two mirror-image halves.
| ## Right angle A right angle is one that measures 90º. See the definitions below 5th grade for examples of acute, right and obtuse angles. |

## Lines of symmetry

If you can reflect (or flip) a figure over a

**line**and the figure appears unchanged, then the figure has reflection**symmetry**or**line symmetry**. The**line**that you reflect over is called the**line of symmetry**. A**line of symmetry**divides a figure into two mirror-image halves.

## 5th Grade

- Explore volume as a three-dimensional measure. Students use objects and pictorial models to develop the formulas for volume of a rectangular prism (
*V*=*l*x*w*x*h*and*V*=*Bh*), including the special form for the volume of a cube (*V*=*s*x*s*x*s*). - Students use Reference Chart formulas to represent and solve problems related to perimeter and/or area and volume.
- Classify two-dimensional figures based on the presence or absence of
**parallel**or**perpendicular**lines or angles of a specified size to formally classify two-dimensional figures into sets and subsets using graphic organizers. - Solve problems by calculating conversions within a measurement system.
- Students are introduced to the coordinate plane and its key attributes including the axes and origin. Students graph ordered pairs in the first quadrant of the coordinate plane. Although graphing is limited to the Quadrant I of the coordinate plane, ordered pairs may include any positive rational number, including fractions and decimals.
- Students are expected to graph ordered pairs in the first quadrant of the coordinate plane that are generated from number patterns or an input-output table.
- Number patterns are examined closely as students recognize the difference between
*additive*and*multiplicative*numerical patterns when given in a table or graph. Students use input-output tables and graphs to generate numerical patterns when given a rule in the form*y = ax*(multiplicative numerical pattern) or*y = a + x*(additive numerical pattern).

## Acute Angle an angle that measures less than 90° | ## Obtuse Angle an angle that measures greater than 90° but less than 180° | ## Quadrants any of the four areas created by dividing a plane with an x-axis and y-axis |

## Ordered Pair ordered pairs are pairs of numbers. When plotting ordered pairs, the first number is the x-coordinate and the second number is the y-coordinate. | ## Multiplicative and Additive Relationships Multiplicative and Additive Relationships are identified often in input-output tables. To determine the relationship, you must look at the relationship between the x and y variables. |

## Ordered Pair

## 6th Grade

- Students graph both positive and negative rational numbers in all four quadrants of the coordinate plane.
- Students are now expected to compare two rules (ex.
*y*=*ax*or*y*=*x*+*a)*verbally, graphically, or symbolically in order to differentiate between additive and multiplicative relationships. - Students identify
**independent**and**dependent**relationships and quantities. Within this unit, students represent a given situation using**verbal descriptions**,**tables**,**graphs**, and**equations**. Also, given one representation, students should be able to create one or all of the different representations for the problem situation. For this grade level, problem situations for additive relationships may include both*positive and negative*rational numbers, whereas multiplicative relationships may only include integers or positive fractions or decimals. - Students extend their knowledge of triangles and their properties to include the sum of the angles of the triangle, and how those angle measurements are related to the three side lengths of the triangle.
- Students examine and analyze the relationship between the three side lengths of a triangle and determine whether three side lengths will form a triangle using the
*Triangle Inequality Theorem*. - Students also decompose and rearrange parts of
**parallelograms**(including rectangles),**trapezoids**, and**triangles**in order to model area formulas for each of the figures. - Students write equations and determine solutions for problems dealing with area of rectangles, parallelograms, trapezoids, and triangles.
- Students expand previous knowledge of converting units within the same measurement system when determining solutions to problems involving length. Conversion processes for measurement extend beyond the use of proportions to now include dimensional analysis and conversions graphs.

## Dependent Variable the variable in an equation or rule which represents the output value | ## Independent Variable the variable in an equation or rule which represents the input value | ## Triangle Inequality Theorem The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides. |

## 7th Grade

- Students
**convert between measurement systems**, including the use of*proportions*and the use of*unit rates*. - Students are introduced to the
**measurement of circles**as they describe pi as the ratio of the circumference of a circle to its diameter. - The
**formulas for circumference and area of a circle**are used to solve problems. - Determine the area of composite figures consisting of rectangles, triangles, parallelograms, squares, quarter circles, semicircles, and trapezoids.
- Students
**model the relationship between the volume of rectangular prism and a rectangular pyramid**. - Students solve problems involving
**volume**, including the volume of**rectangular****prisms**,**triangular****prisms**,**rectangular****pyramids**, and**triangular****pyramids**. - Students also solve problems involving
**surface area**by determining the area of the shape's net, including**lateral and total surface****area**of a**rectangular prism**,**rectangular pyramid**,**triangular prism**, and**triangular pyramid**.

## Pi The ratio of a circle's circumference to its diameter. 3.14 is used at the value for pi at 7th grade. | ## Prism vs. Pyramid A prism has two ends that are exactly the same size and shape and are parallel to each other. A pyramid has triangular sides which meet at a point called a vertex. The base is flat. | ## Lateral and Total Surface Area The lateral surface area does not include any bases. Total surface area does include bases and sides.In the hexagonal prism to the left, the lateral surface area would be the sum of the area of the blue surfaces. The total surface area would be those |

## Pi

3.14 is used at the value for pi at 7th grade.

## Prism vs. Pyramid

A pyramid has triangular sides which meet at a point called a vertex. The base is flat.

## Lateral and Total Surface Area

**does**include bases and sides.

In the hexagonal prism to the left, the lateral surface area would be the sum of the area of the blue surfaces. The total surface area would be those **plus** the area of the two hexagonal bases.

## Gr. 8

- Students extend previous knowledge of sets and subsets to
**order and describe relationships between sets of real numbers**, which includes rational numbers and their subsets as well as irrational numbers. - Students
**approximate the value of irrational**numbers*less than 225*and**locate those approximations on a number line.** - Establish facts about the
**angle sum and exterior angle of triangles**, the angles created when parallel lines are cut by a transversal, and the**angle-angle criterion for similarity of triangles.** - Right triangles are examined more closely within this unit as students use models to explain the
**Pythagorean theorem**. Students use the Pythagorean theorem and its converse to solve problems and*apply these understandings to the coordinate plane as they determine the distance between two points on the coordinate plane.* - Determine the
**volume of a cylinder**in terms of its base area and height. Students are expected to model the relationship between the**volume of a cylinder and a cone**having both congruent bases and heights. Students**connect these models to the actual formulas**for determining the volume of a cylinder and cone, which directly coincides with formulas used for determining the volume of prisms and pyramids on the STAAR Grade 8 Mathematics Reference Materials. - Students
**solve problems involving the volume of cylinder, cones, and spheres.** - The concept of surface area is extended from finding the sum of the areas of the faces from the net to
**abstract formulas for lateral and total surface area**. - Students are expected to use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders.

## Angle-Angle Criterion for Similarity We know that the angles of a triangle must add up to 180°. This means that if a triangle has two angle measurements of 40° and 80°, then the third angle must be 60°. Now if a second triangle has two angle measurements of 40° and 60°, we know the third angle must be 80°. This means the two triangles are the same shape, but not necessarily the same size. Alternately we may think of one as a dilation of the other. Either way we know that the triangles are similar. We call this the angle-angle criterion for similarity | ## Pythagorean Theorem
a2 + b2 = c2 |

## Angle-Angle Criterion for Similarity

Now if a second triangle has two angle measurements of 40° and 60°, we know the third angle must be 80°. This means the two triangles are the same shape, but not necessarily the same size. Alternately we may think of one as a dilation of the other. Either way we know that the triangles are similar. We call this the angle-angle criterion for similarity

## Algebra I

- Solve problems using systems of equations.
- Linear systems are formulated for the problem situations, solved by a method of choice (tables, graphs, or algebraic methods including substitution and elimination), and the reasonableness of the solutions justified in terms of the problem situations.
- Graphing calculator technologies continue to be utilized to explore the relationships within systems of equations.
- During this unit, the concepts of
**exponents**are extended to generate and**apply rules of exponents.**Simplification and combination of expressions are extended to include multiplication and**factoring with more complex polynomial expressions**. These concepts are applied in problem situations.

## Polynomial Expressions Polynomials are algebraic expressions that include real numbers and variables. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication. Polynomials contain more than one term. Polynomials are the sums of monomials. A monomial has one term: 5y or -8 x2 or 3.A binomial has two terms: -3 x2 2, or 9y - 2y2A trinomial has 3 terms: -3 x2 2 3x, or 9y - 2y2 yThe degree of the term is the exponent of the variable: 3 x2 has a degree of 2.When the variable does not have an exponent - always understand that there's a '1' e.g., 1x | ## Exponent Rules Watch these videos from Khan Academy for Exponent Rules. https://www.youtube.com/watch?v=kITJ6qH7jS0 |

## Polynomial Expressions

Polynomials contain more than one term. Polynomials are the sums of monomials.

A monomial has one term: 5y or -8

*x*2 or 3.

A binomial has two terms: -3

*x*2 2, or 9y - 2y2

A trinomial has 3 terms: -3

*x*2 2 3x, or 9y - 2y2 y

The degree of the term is the exponent of the variable: 3

*x*2 has a degree of 2.

When the variable does not have an exponent - always understand that there's a '1' e.g.,

*1x*

## Exponent Rules

https://www.youtube.com/watch?v=kITJ6qH7jS0