# Making Sense of Math

### 1st Six Weeks 2015-16 Newsletter for parents

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

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.

## Would you like this newsletter emailed to you directly?

I will add you to the email list!

## 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...)

## Kindergarten

**0-5**. Students:

- count forward and backward and reciting numbers up to 30 (beginning with any number).
- compose and decompose numbers up to 5, which means they break apart the number. For example 5 can be represented as 1 and 4, 2 and 3, 1 and 1 and 3, etc. Composing and decomposing numbers is a critical part of number sense.
- name one more or one less than a given number and compare and order numbers.
- model and explain strategies to add and subtract to 5.
- use concrete objects and pictures to act out joining and separating and 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. - use words, objects, pictorial models and number sentences to represent and solve real-world problems.
- 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

**numbers up to 1,200**. Students:

- compose and decompose numbers through 1,200 as a sum of so many one thousands, so many hundreds, so many tens and so many ones using concrete objects, picture models and numeral representations
- use hundreds charts and base-10 blocks and compare whole numbers
- use number lines to locate, name and order numbers

## 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

**numbers up to 100,000**. Students:

- identify their base-10 relationships through the hundred thousands place and compare and order these numbers.
- use concrete objects, pictorial models, and numerical representations to compose and decompose numbers as so many ten thousands, so many thousands, so many hundreds, so many tens and so many ones.
- begin to describe the mathematical relationship between digits in a number (one hundred is 10 tens, one thousand is 10 hundreds, etc.)
- estimate, solve and represent
**one- and two-step addition and subtraction problems**. - use number lines and place value relationships to round numbers to the nearest 10 or 100.
- 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

**place value patterns**. They:

- represent the value of digits through
**one billion**(whole numbers) and**decimals****to the hundredths using expanded notation and numerals**. - compare and order whole numbers and represent their comparisons using symbols (< or >). When ordering three or more numbers they will include descriptors (greatest/least, ascending/descending, fastest/slowest, etc.)
- locate numbers on a number line
- represent decimals using concrete and visual models (number lines, decimal grids, base-10 blocks) and money
- round numbers in the
using their choice of strategies.**hundred thousands**place - analyze real-life problem situations to identify vocabulary that indicates estimation (about, close, approximately, etc.).
- make connections between place value and the standard algorithms for adding and subtracting. This includes adding and subtracting decimals including
.**tenths and hundredths** - examine at the purpose of financial institutions in their financial literacy lessons.
- determine products of a number by 10 or 100 using properties of operations and place value understandings.
- 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**. - represent problem situations using strip diagrams and with equations with a letter standing for an unknown quantity.

## Decimals The digit to the right of the decimal place is the tenths place.The digit two places to the right of the decimal is the | ## Multiply a four-digit number by a one-digit number | ## Multiply two-digit by two-digit number |

## Intermediate -5th and 6th Grade

## Fifth Grade

- estimate to determine solutions using all operations with whole numbers in real-world problems.
**multiply a 3-digit number by a 2-digit number**and**divide a 4-digit number by a 2-digit divisor**.

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

**equivalent forms of fractions, decimals and percents**as well as solving real-world problems involving fractions, decimals and percents (including money). Sixth graders:

- use concrete and pictorial models to represent multiples of benchmark fractions and percents. Percents less than or greater than 100% are studied as well as percents with fractional or decimal values.
- work with various forms of numbers including
**negative numbers** - locate a number on a number line and use the location to compare and order a set of values
- classify numbers as
**counting numbers**,**whole numbers**,**integers**, or**rational**numbers

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

## Middle School - Gr. 7, 8

Students in grade 7 begin working the year working with sets of rational numbers and solving equations. 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. - 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.
- 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. - represent solutions to
**equations and inequalities on number lines**and given values are used to determine if they make an equation or inequality true. - write an equation or inequality to represent conditions or constraints within a problem
- 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.
- write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.
- 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.
- order a set of rational numbers that arise from mathematical and real-world situations.
- convert between standard decimal notation and scientific notation for both positive and negative numbers and with values greater than and less than one
- order numbers and find the mean to calculate the mean absolute deviation of up to 10 data points
- describe the data by comparing each data point to the mean absolute deviation
- students describe the spread and shape of data by looking at its difference from the mean
- develop the notion that random samples of a population with known characteristics is representative of a population from which it was selected
- 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 - write one-variable equations or inequalities with variables on both sides to represent problems using rational numbers coefficients and constants
- write a corresponding real-world problem for a one-variable equation or inequality with variables on both sides
- calculate and compare 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 |

## 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.