# Making Sense of Math

### Kinder - Algebra math summary for Alice I.S.D. parents --

## Why this newsletter? - A message from Anna Holmgreen, Director of Instruction for Math

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.

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

## Some helpful vocabulary....

## Decomposing a number Decomposing a number is simply breaking it up into parts or smaller values. For instance 35 could be 30 + 5 or 10 + 10 + 10 + 5. Sometimes decomposing numbers can help students when adding. | ## Place value chart Our place value is based on the Base-10 system. Each place has a value (ones, tens, hundreds, etc.) and each place value to the right is 10 times the value of the place to the left. (1 thousand is 10 hundreds, 1 hundred is 10 tens...) | ## Fact Families Related number sentences using the same set of numbers. Fact families are used to represent addition and subtraction or multiplication and division facts. |

## Decomposing a number

10 + 10 + 10 + 5.

Sometimes decomposing numbers can help students when adding.

## Place value chart

Each place has a value (ones, tens, hundreds, etc.) and each place value to the right is 10 times the value of the place to the left. (1 thousand is 10 hundreds, 1 hundred is 10 tens...)

## Kinder - Grade 4

## Kindergarten

Students also begin modeling and explaining strategies to add and subtract to 5. Kindergartenders are using concrete objects and pictures to act out joining and separating. They also begin using a number sentence (2 + 3 = 5).

## First Grade

Next students move on to subitizing, which means instantly recognizing a quantity of a small group of objects. Students explore composing and decomposing quantities to 10 to investigate addition and subtraction operations. Students are using words, objects, pictorial models and number sentences to represent and solve real-world problems. Students should understand that the expressions on each side of an equals sign represent the same value.

Finally in the first six weeks, students begin telling time to the hour using both digital and analog clocks. This is the first time students have been introduced to the concept.

## Second Grade

The second unit has students addressing patterns and relationships in numbers. This includes basic addition and subtraction facts, fact families, odd and even numbers, place value and value of coins.

## Expanded Form Representing a number as a sum of place values 56,789 50,000 +6,000 + 700 + 80 + 9 | ## Expanded Notation Representing a number as a sum of place values where each term is shown as a digit times its place value 56,789 (5 x 10,000) = (6 x 1,000) + (7 x 100) + (8 x 10) + (9 x 1) | ## Compatible numbers Compatible numbers can be thought of as "friendly" numbers or well matched numbers that can help in performing computations mentally. For example in addition compatible numbers can be those that equal 10 such as 6 and 4, 3 and 7, etc. You can use compatible numbers in other ways, too. |

## Expanded Notation

56,789

(5 x 10,000) = (6 x 1,000) + (7 x 100) + (8 x 10) + (9 x 1)

## Compatible numbers

For example in addition compatible numbers can be those that equal 10 such as 6 and 4, 3 and 7, etc.

You can use compatible numbers in other ways, too.

## Third Grade

Students begin to describe the mathematical relationship between digits in a number. (one hundred is 10 tens, one thousand is 10 hundreds, etc.)

In the next unit students estimate, solve and represent one- and two-step addition and subtraction problems. Number lines and place value relationships are used to round numbers to the nearest 10 or 100. Students also use their estimation and mental math strategies to justify the reasonableness of their solutions. These concepts are extended by determining the value of a collection of coins and bills and determining the perimeter of a polygon.

## Fourth Grade

Next students move on to rounding numbers in the hundred thousands place using their choice of strategies. Real-world problem situations are analyzed to identify vocabulary that indicates estimation (about, close, approximately, etc.). Students also make connections between place value and the standard algorithms for adding and subtracting. This includes adding and subtracting decimals including tenths and hundredths. Students also examine at the purpose of financial institutions in their financial literacy lessons.

Students end the six weeks studying multiplication. They will be determining products of a number by 10 or 100 using properties of operations and place value understandings. Fourth graders use strategies (mental math, partial products, and the commutative, associative and distributive properties) as well as algorithms to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number.

They represent problem situations using strip diagrams and with equations with a letter standing for an unknown quantity.

## Intermediate -5th and 6th Grade

## Fifth Grade

Students who are not already fluent in their multiplication facts need to practice those to help in problem solving.

In the second unit, students are developing algebraic understanding. They will be introduced to properties of prime and composite numbers. They also examine grouping symbols and solve multi-step problems using an equation with a letter representing an unknown.

These first two units develop important concepts which will be used all year.

## Division terms Dividend - the number being divided Divisor - the number the dividend is divided by Quotient - the number in the group (the answer to the problem) | ## Factors Factors are what you multiply to get a number. For example the factors of 12 are 1 and 12, 2 and 6, 3 and 4 | ## Prime and Composite Numbers Prime numbers are those which have only 1 and itself as factors. Composite numbers have more than two factors. |

## Division terms

Divisor - the number the dividend is divided by

Quotient - the number in the group (the answer to the problem)

## Factors

For example the factors of 12 are 1 and 12, 2 and 6, 3 and 4

## Sixth Grade

In the second unit students continue working with various forms of numbers and expand this to include negative numbers. They must locate a number on a number line and use the location to compare and order a set of values. Students classify numbers as a counting number, whole number, integer or rational number.

## Counting Numbers Numbers starting with 1 and increasing by one each time. | ## Integers The set of counting numbers, their opposites and zero. | ## Whole Numbers The set of counting numbers and zero. |

## Middle School - Gr. 7, 8 Alg.

Students in grade 7 begin working the year working with sets of rational numbers and solving equations.During Unit 1, students use a visual representation to organize and display the relationship of the sets and subsets of rational numbers, which include counting (natural) numbers, whole numbers, integers, and rational numbers. Grade 7 students are expected to fluently add, subtract, multiply, and divide various forms of positive and negative rational numbers which include integers, decimals, fractions, and percents converted to equivalent decimals or fractions for multiplying or dividing. Students also begin working with the new personal financial literacy skills by creating and organizing a financial assets and liabilities record, constructing a net worth statement, calculating sales tax for a given purchase, and calculating income tax for earned wages.

In the next unit, students work with equations and geometry concepts. Students model and solve one-variable, two-step equations and inequalities with concrete and pictorial models and algebraic representations. They represent solutions to equations and inequalities on number lines and given values are used to determine if they make an equation or inequality true. They are expected to write an equation or inequality to represent conditions or constraints within a problem and then, when given an equation or inequality out of context, they are expected to write a corresponding real-world problem to represent the equation or inequality. Students write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships. Students also use equations and inequalities to include problem situations involving monetary incentives such as sales, rebates, or coupons. Financial literacy aspects such as calculating and comparing simple and compound interest as well as utilizing a family budget estimator to determine the minimum household budget needed for a family to meet its basic needs is also explored.

## Financial Assets An object or item of value that one owns. | ## Financial Liability An unpaid or outstanding debt. | ## Net Worth An individual's assets after all debts are paid. |

## Coefficient A number multiplied by a variable. | ## Complementary Angles Two angles whose measurements added together equal 90 degrees. | ## Supplementary Angles Two angles whose measurements added together equal 180 degrees. |

## Order of Operations The order in which calculations are performed when solving a mathematical equation or simplifying an expression. | ## Principal and Simple Interest Principal - the original amount invested or owed. Interest - interest paid on original principal minus any previously earned interest | ## Variable A letter or symbol that represents a number |

## Order of Operations

## Principal and Simple Interest

Interest - interest paid on original principal minus any previously earned interest

## Eighth Grade

The 8th graders also begin the year studying sets and subsets of rational numbers.They use a visual representation, such as a Venn diagram, to describe the relationships between the sets and subsets. Students order a set of rational numbers that arise from mathematical and real-world situations. Students extend previous understandings of the relationships within the base-10 place value system as they convert between standard decimal notation and scientific notation for both positive and negative numbers and with values greater than and less than one

In unit 2, students extend their knowledge of ordering numbers and finding the mean to calculate the mean absolute deviation of up to 10 data points and describe the data by comparing each data point to the mean absolute deviation. Data with one variable (or univariate data), is examined as students describe the spread and shape of data by looking at its difference from the mean. Additionally, students develop the notion that random samples of a population with known characteristics is representative of a population from which it was selected. Students explore appropriate methods for simulating such samples.

Finally in unit 3, students extend their understanding of solving equations to model and solve one-variable equations with variables on **both** sides of the equal sign. Students write one-variable equations or inequalities with variables on both sides to represent problems using rational numbers coefficients and constants. When given a one-variable equation or inequality with variables on both sides, students are expected to write a corresponding real-world problem. Students also begin their personal financial literacy study by calculating and comparing simple and compound interest rates and how those rates affect earnings and total cost in repaying a loan or credit card.

## Base A number that is going to be razed to a power. | ## Power The power of a number shows how many times you multiply the base by itself. For example in the picture 8 to the second power means 8 x8. 8 to the third power would be 8 x 8 x 8. If the power is 0, the simplified form will equal 1. | ## Scientific notation A way of writing a very large or very small number using powers of ten that is written as a decimal with exactly one non-zero digit to the left of the decimal. |

## Power

8 to the third power would be 8 x 8 x 8.

If the power is 0, the simplified form will equal 1.

## Mean The average of numbers; to calculate, add up the numbers and divide by how many numbers there are | ## Mean Absolute Deviation The measure of the distance between each data point and the mean | ## Collateral Something pledged when a loan is taken out to secure the loan. If the loan is not paid the collateral is forfeited. |

## Mean

## Payday Loan A high-interest, short-term loan of cash for which collateral, such as a car title, is required. | ## Annual Percentage Rate (APR) Annual percentage rate applied to the balance on a loan compounded monthly |

## Algebra 1

It is often written as "f(x)" where x is the value you give it.

In a function each number that goes in (domain) is paired with exactly one number of the output (range).

Example:

f(x) = x/2 ("f of x is x divided by 2") is a function, because for every value of "x" you get another value "x/2", so:

* f(2) = 1

* f(16) = 8

* f(-10) = -5

In Unit 2 students review these methods for solving one- and two-step equations and also solve one- and two-step equations by analyzing tables and graphs of functions. The properties of algebra are used to simplify algebraic expressions. Students explain and difference between expressions and equations.

## Expressions Numbers, symbols and operators (such as + and ×) grouped together that show the value of something. Example: 2×3 is an expression | ## Domain and Range of a Function All the values that go into a function The output values are called the range. Domain → Function → Range Example: when the function f(x) = x2 is given the values x = {1,2,3,...} then {1,2,3,...} is the domain. | ## Equations An equation says two things are equal. It is like a statement: "this equals that" Here is an example of an equation: 4x -7 = 5 |

## Expressions

Example: 2×3 is an expression

## Domain and Range of a Function

The output values are called the range.

Domain → Function → Range

Example: when the function f(x) = x2 is given the values x = {1,2,3,...} then {1,2,3,...} is the domain.