# Making Sense of Math

### AISD Parent Newsletter for the 3rd six weeks

This newsletter is intended to give parents an idea of what is being covered each six weeks in math and what their students should be learning.

Here are links to the first two six weeks newsletters. Please visit them to access vocabulary you may need.

First Six Weeks: https://www.smore.com/h71kx

Second Six Weeks: https://www.smore.com/v8edc

Please contact Anna Holmgreen, Director of Instruction for Math, if you have questions.

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## Kindergarten

- Students count forward and backward to 15 with and without objects, as well as read, write, and represent the numbers.
- Students also compose and decompose numbers up to 10 using objects and pictures
- They compare sets of objects up to 15 using comparative language and generate a set of objects and pictures that is more than, less than, or equal to a given number.
- When given a number up to 15, students are expected to generate a number that is one more than or one less than the number.
- Students are expected to recite numbers up to 90 by ones beginning with any number.

In the second unit students develop the foundation of operations.

- Students use concrete objects, pictorial models, and acting out a situation to model and represent joining and separating problems involving sums and minuends up to 10.
- Students record their solution using a number sentence and orally explain their solution strategy.
- As students model, represent, and solve addition and subtraction problems, they begin to develop an understanding of the problem solving process that includes
*understanding the context of the problem situation and the question being asked*,*forming a plan or strategy*, and*using the plan or strategy to determine a solution*.

## Addend The number being added or joined with another number. | ## Minuend The number from which another number will be subtracted. | ## Number Sentence A mathematical statement consisting of numbers, an operation and an equality or inequality symbol. |

## First Grade

- Students
**compose and decompose numbers through 99**(tens and ones) using concrete objects (base-10 blocks, place value disks, etc.), pictorial models, and numerical representations (e.g., expanded form and standard form). - Students use place value relationships to generate numbers that are more or less than a given number using tools such as a hundreds chart and/or base-10 blocks.
- Students
**compare whole numbers up to 99**and represent the comparison using comparative language and symbols. - Students use
**open number lines**to represent the order of numbers.

During the next unit, students continue delving deeply into the place value system.

- Discover numerical patterns in the number system using Various representations (e.g., linking cubes, straw bundles, base-10 blocks, place value disks, hundreds charts, and open number lines)
- Students use place value patterns to determine the sum up to 99 of a multiple of 10 and a one digit number, as well as determining a number that is 10 more or 10 less than a given number.
- Students develop the understanding of cardinal numbers (numbers that tell "how many"

## Open Number Line An empty number line where tick marks are used to indicate landmarks of numbers. | ## Decomposing Numbers Click the link below for a short video. | ## Open Number Line Click the link below for a short video. |

## Second Grade

- classify and sort polygons with 12 or fewer sides by identifying the number of sides and number of vertices.
- examine if the sides are equal in length, and if the corners are square.
- use attributes based on formal geometric language to classify and sort three-dimensional solids, including spheres, cones, cylinders, rectangular prism (including cubes as special rectangular prisms), and triangular prisms.
- compose two-dimensional shapes and three-dimensional solids with given properties or attributes.
- decompose two-dimensional shapes into equal or unequal parts and use geometric attributes to identify and name the resulting parts.

In the next unit, students continue to develop spatial visualization skills. They:

- analyze the resulting parts to determine if equal parts exist and name the fractional parts using words rather than symbols (e.g., one-half or 1 out of 2 equal parts rather than).
- discover and explain the relationship between the number of fractional parts used to make a whole and the size of the parts.
- Using concrete models, students recognize how many parts it takes to equal one whole, and use this understanding to count fractional parts beyond one whole.
- Students make connections between counting whole numbers and counting fractional parts as well as extend their understanding of hierarchical inclusion, meaning each prior number in the counting sequence is included in the set as the set increases, to include the sequence of fractional parts.

## 3rd Grade

- Fractions are represented using concrete objects, pictorial models (including strip diagrams), and number lines.
- Students explain the unit fraction as one part of a whole that has been partitioned into 2, 3, 4, 6, or 8 parts and use this understanding to compose and decompose a fraction as a sum of unit fractions.
- Additionally, students represent fractions as halves, fourths, and eighths as distances from zero on the number line. They must also determine a corresponding fraction when given a specified point on a number line.
- Students also solve problems that involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions.

During the next unit, students extend their understanding of multiplication and division and the mathematical relationships between these operations.

- They represent multiplication and division of a two-digit number by a one-digit number using arrays, area models, strip diagrams, and equations.
- Students use the
**commutative**(3 x 2 = 2 x 3),**associative**[ (2 x 3) x 4 = 2 x (3 x 4) ], and**distributive**[ 2 x (3 +2) = (2 x 3) + (2 x 2) ]**properties**of multiplication, - mental math,
- partial products, and the
- standard multiplication algorithm to represent and solve one- and two-step multiplication and division problem situations within 100.

Multiplication problems include determining area of rectangles.

The last unit deals with future financial security as students explore how skills and education needed for jobs may impact potential income. They also deal with how scarcity of resources may impact cost. Students discuss decisions related to planned and unplanned spending including the responsibilities of using credit.

Students list reasons to save and explain the benefits of saving (e.g., saving for college, surviving hard financial times, being prepared for unforeseen expenses, setting goals, etc.). Students also identify decisions regarding charitable giving.

## Denominator The part of the fraction written below the fraction bar that tells the total number of equal parts in a whole or set. | ## Numerator The part of the fraction written above the fraction bar that tells the number of fractional parts being specified. |

## Dividend, Divisor, Quotient The dividend is the number being divided. The divisor is the number you divide by and the quotient is the answer after you divide. | ## Charitable giving Charitable giving is donating to an organization that collects money, goods, or services to groups in need/ |

## Fourth Grade

**tenths**and

**hundredths**, and represent both types of numbers as

**distances from zero on a number line.**

- Along with representing fractions (including those that represent values greater than one) as sums of unit fractions, students decompose fractions into sums of fractions with the
**same denominator**using concrete and pictorial models and record the results with symbolic representations. - Students represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations.
- Students evaluate the reasonableness of those sums and differences using benchmark fractions 0, , and 1, referring to the same whole.
- Using a variety of methods to determine equivalence of two fractions underlies students’ abilities to compare two fractions with different numerators and different denominators and represent those comparisons using symbols.

During the second unit, students represent data on a **frequency table, dot plot, or stem-and-leaf plot** marked with whole numbers and fractions.

Students examine the characteristics of each data representation, as well as compare the similarities and differences between them. This allows students to solve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot.

## Improper Fraction A fraction where the numerator is larger than the denominator | ## Least Common Denominator (LCD)Least common denominator (LCD) – the least common multiple of the denominators of two or more fractions | ## Least Common Multiple (LCM)Least common multiple (LCM) – the smallest multiple that two or more numbers have in common |

## Least Common Denominator (LCD)

**Least common denominator (LCD)**– the least common multiple of the denominators of two or more fractions

## Dot PlotDot plot – a graphical representation to organize data that uses dots (or Xs) to show the frequency (number of times) that each category or number occurs | ## Frequency TableFrequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs | ## Stem and Leaf PlotStem-and-leaf plot – a graph used to analyze and compare groups or clusters of data by separating one place value from another place value of a data set. The larger of the two place values is called the stem and the smaller of the two place values is called the leaf. Click the link to see a video. |

## Dot Plot

**Dot plot**– a graphical representation to organize data that uses dots (or Xs) to show the frequency (number of times) that each category or number occurs

## Frequency Table

**Frequency table**– a table to organize data that lists categories and the frequency (number of times) that each category occurs

## Stem and Leaf Plot

**Stem-and-leaf plot**– a graph used to analyze and compare groups or clusters of data by separating one place value from another place value of a data set.

The larger of the two place values is called the stem and the smaller of the two place values is called the leaf.

Click the link to see a video.

## Fifth Grade

- This unit has them add and subtract positive rational numbers fluently to include fractions.
- Students continue to estimate solutions in mathematical and real-world problems. Concrete objects, pictorial models, and properties of operations are used to represent and solve problems involving adding and/or subtracting fractions with
**unequal denominators**referring to the same whole. - Students also continue to simplify numerical expressions that involve adding and subtracting fractions with unequal denominators.

In the next unit, students represent a **product of whole number and a fraction** referring to the same whole using concrete and pictorial models, including **area models and strip diagrams. **

- Students continue to estimate solutions in mathematical and real-world problems.
- Concrete and pictorial models, including area models and strip diagrams, are also used to represent the
**division of a whole number by a unit fraction**and the**division of a unit fraction by a whole number**. - Students also continue to simplify numerical expressions that involve all operations with whole numbers, decimals, and fractions, respectively.

## Sixth Grade

**order of operations**

**without exponents**, to simplifying numerical expressions using order of operations

**with**

**exponents**, and to generating equivalent numerical expressions.

- Prime factorization is introduced as a means to generate equivalent numerical expressions. Students should recognize that when a number is decomposed into prime and composite factors, the product of the factors is equivalent to the original number.

In this unit they are formally introduced to **algebraic expressions**.

- Students investigate generating equivalent numerical and algebraic expressions using the properties of operations which include the
**inverse**,**identity**,**commutative**,**associative**, and**distributive**properties. - Concrete models, pictorial models, and algebraic representations are used to determine if two expressions are equivalent.
- Students should distinguish between and expression and an equation.
- Equations within this unit are limited to
**one-variable, one-step equations**. - Constants or coefficients of one-variable, one-step equations may include positive rational numbers or integers.
- Students are expected to write a one-variable, one-step equation as well as write a corresponding real-word problem when given a one-variable, one-step equation.

Concrete models, pictorial models, and algebraic representations are used again as students model and solve one-variable, one-step equations that represent problems, including geometric concepts.

- Students are expected to represent their solution on a number line as well as determine if a given value(s) make(s) the one-variable, one-step equation true.

- During this unit students are introduced to
**algebraic inequalities**. - Constants or coefficients of one-variable, one-step inequalities may include
**positive**rational numbers or**integers**. - Students are expected write a one-variable, one-step inequality as well as write a corresponding real-word problem when given a one-variable, one-step inequality.
- Concrete models, pictorial models, and algebraic representations are used again as students model and solve one-variable, one-step inequalities that represent problems, including geometric concepts.
- Students are expected to represent their solution on a number line as well as determine if a given value(s) make(s) the one-variable, one-step inequality true.

## Coefficient, Constant, Variable Coefficient - a number multiplied by a variable (4x - 4 is coefficient) Constant - a value that doesn't appear with a variable Variable - a letter or symbol that represents a number | ## Exponent The exponent of a number says how many times to use that number in a multiplication. It is written as a small number to the right and above the base number. In this example: 82 = 8 × 8 = 64 (The exponent "2" says to use the 8 two times in a multiplication.) | ## Prime Factorization Decomposing a number to show it as the product of only prime numbers. Remember, a prime number is one whose only factors are one and itself. (7 is prime because its only factors are 1 and 7.) |

## Coefficient, Constant, Variable

Constant - a value that doesn't appear with a variable

Variable - a letter or symbol that represents a number

## Exponent

It is written as a small number to the right and above the base number.

In this example: 82 = 8 × 8 = 64

(The exponent "2" says to use the 8 two times in a multiplication.)

## Seventh Grade

**proportionality**to

**two-dimensional figures**as they solve mathematical and real-world problems involving

**similar shapes**and

**scale**

**drawings**. Students generalize the critical attributes of similarity which include examining the

**multiplicative relationship**within and between similar shapes.

During this unit, students extend the use of proportional reasoning to solve problems as they are formally introduced to **probability** concepts.

- Students use various representations including
**lists**,**tree****diagrams**,**tables**, and the**Fundamental Counting Principle**to represent the sample spaces for simple and compound events. - Students select, design, develop, and use various methods to simulate simple and compound events. Students are expected to distinguish between
**theoretical**and**experimental**data and find the probabilities of a simple event. - Students analyze and describe the relationship between the probability of a simple event and its
**complement**. Probabilities may be represented as a**decimal**,**fraction**, or**percent**. - Data and sample spaces are used to determine experimental and theoretical probabilities to simple and compound events.
- Data from experiments, experimental data, theoretical probability, and random samples are used to make qualitative and quantitative inferences about a population. Qualitative and quantitative predictions and comparisons from simple experiments are used to solve problems.

## Congruent Having the same size and shape. Two shapes are congruent when you can Turn, Flip and/or Slide one so it fits exactly on the other. | ## Complement of an event The probability of the opposite of what is asked. For example if there are 5 marbles and 2 are red, the probability of picking a red marble is 2 out of 5. The complement would be 3 out of 5. In the picture above if the probability is 5 or 6, the complement would be 1, 2, 3, or 4. | ## Fundamental Counting Principal The fundamental counting principal is about choices and options. For example, if you have a red shirt, a blue shirt and a green shirt and jeans and shorts, you could have the following combinations: Red shirt - jeans Red shirt - shorts Blue shirt - jeans Blue shirt - shorts Green shirt - jeans Green shirt - shorts That is 6 possibilities or 3 shirts x 2 pants. |

## Congruent

Two shapes are congruent when you can Turn, Flip and/or Slide one so it fits exactly on the other.

## Complement of an event

The complement would be 3 out of 5.

In the picture above if the probability is 5 or 6, the complement would be 1, 2, 3, or 4.

## Fundamental Counting Principal

For example, if you have a red shirt, a blue shirt and a green shirt and jeans and shorts, you could have the following combinations:

Red shirt - jeans

Red shirt - shorts

Blue shirt - jeans

Blue shirt - shorts

Green shirt - jeans

Green shirt - shorts

That is 6 possibilities or 3 shirts x 2 pants.

## Eight Grade

**trend**

**lines**that approximate the relationship between

**bivariate**sets of data.

- Students contrast graphical representations of linear and non-linear relationships.
- Scatterplots are constructed from bivariate sets of data and used to describe the observed data.

Observations include questions of association such as **linear** (**positive or negative trend**), **non-linear**, or **no association.** Students continue to represent situations with tables, graphs, and equations in the form *y* = *kx * or *y* = * mx *+ *b*, where *b *≠ 0. Within a scatterplot that represents a linear relationship, students use the trend line to make predictions and interpret the slope of the line that models the relationship as the unit rate of the scenario.

During this unit, students develop transformational geometry concepts as they examine **orientation** and **congruence** of **transformations**.

- Students extend concepts of similarity to
**dilations**on a coordinate plane as they compare and contrast a shape and its dilation(s). - Students generalize the ratio of corresponding sides of a shape and its dilation as well as use an algebraic representation to explain the effect of dilation(s) on a coordinate plane.
- Students generalize the properties as they apply to rotations, reflections, translations, and dilations of two-dimensional figures on a coordinate plane. Students must distinguish between transformations that preserve congruence and those that do not.
- Students are expected to use an algebraic representation to explain the effect of translations, reflections over the
*x*- or*y*- axis, dilations when a positive rational number scale factor is applied to a shape, and rotations limited to 90°, 180°, 270°, and 360°. - The relationship between linear and area measurements of a shape and its dilation are also examined as students model the relationship and determine that the measurements are affected by both the scale factor and the dimension (one- or two-dimensional) of the measurement.
- Students are expected to generalize when a scale factor is applied to all of the dimensions of a two-dimensional shape, the perimeter is multiplied by the same scale factor while the area is multiplied by the scale factor squared.

## Bivariate data Data for two variables---often represented by a scatterplot. Example: Ice cream sales versus the temperature on that day. The two variables are Ice Cream Sales and Temperature. | ## Discrete data Data that can only take certain values. There is no in-between (for example, you can't have half a student). | ## Linear Relationship A relationship with a constant rate of change represented by a graph that forms a straight line. |

## Bivariate data

Example: Ice cream sales versus the temperature on that day. The two variables are Ice Cream Sales and Temperature.

## Discrete data

## Scatterplot The data is displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis | ## Scale Factor An image that is transformed by either enlarging or reducing, depending on the scale factor. | ## Scale Factor The common multiplicative ratio between two sets of data. In the picture, the scale factor from the smaller to the larger would be 2, because the corresponding side on the larger is two times the smaller. |

## Scatterplot

## Scale Factor

## Algebra 1

**linear functions**focusing on data collection and the analysis of

**scatterplots**,

**trend lines**, and

**linear correlation**. Students make interpretations, predictions, and critical judgments from the functional relationships based on the data.

During the next unit, students are introduced to solving linear systems using concrete models, tables, graphs, and algebraic methods including **substitution** and **combination**. Reasonableness of the solution is justified using various methods.

## Correlation The description between two sets of data. In a linear equation there are three possibilities: positive linear correlation, negative linear correlation and no correlation. | ## Trend Line The line that best fits the data on a scatterplot. |