# Making Sense of Math

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

There are many new math standards being taught this year. The state has rearranged the skills to various grade levels.

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.

aholmgreen@aliceisd.esc2.net

## Kindergarten

In Kindergarten during the first six weeks, students are developing an understanding of the numbers 0-5. They are counting forward and backward and reciting numbers up to 30 (beginning with any number). Students are also composing and decomposing 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. Students also name one more or one less than a given number and compare and order numbers.

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 graders begin the year collecting, sorting and organizing data and representing that information using picture and bar-type graphs to draw conclusions and answer questions from the data.

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.

Students will be working on extending their understanding of the base-10 system to include the thousands period and 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. They also use hundreds charts and base-10 blocks and compare whole numbers. Students use number lines to locate, name and order numbers.

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.

Third grade also addresses composing and decomposing numbers, but they are responsible for numbers up to 100,000. They must identify their base-10 relationships through the hundred thousands place and compare and order these numbers. Third graders 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.

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.

This six weeks fourth graders begin the year by extending their understanding of place value patterns. They will represent the value of digits through one billion (whole numbers) and decimals to the hundredths using expanded notation and numerals. They 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.) They will also locate numbers on a number line. Decimals are represented using concrete and visual models (number lines, decimal grids, base-10 blocks) and money.

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

During the first six week students in fifth grade work on becoming fluent in solving whole number addition, subtraction, and multiplication problems and proficient in solving whole number division problems. Students also estimate to determine solutions using all operations with whole numbers in real-world problems. Fifth graders will be multiplying a 3-digit number by a 2-digit number and dividing 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.

Students in sixth grade begin their year generating equivalent forms of fractions, decimals and percents as well as solving real-world problems involving fractions, decimals and percents (including money). Sixth graders are using 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.

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.

## 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.

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.

## Algebra 1

Students in Algebra 1 begin their year with the study of functions, which is a special relationship between values. A graphing calculator is used to collect and analyze data

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.