Making Sense of Math
Alice I.S.D. math newletter for parents
Second six weeks Newsletter!
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.
What's the Best Thing Parents Can do to Help Students with Math?
- Ask what they are learning in math.
- Ask students to teach you what they learned.
- Work on recall of basic facts addition (and subtraction) for K, 1, 2 and multiplication for 3rd grade and up. Often students difficulties in math start with not being able to recall those facts easily. That makes figuring out problems even more difficult.
- Ask student to explain their thinking. This makes a HUGE difference.
- If you have questions...ask your child's teacher.
Don't forget to contact me if I can help in anyway.
AISD Parent Conference - Be The Hero Every Child Needs
Saturday, Oct. 18th, 8am-1:30pm
Alice High School
I will be doing two math session: Decomposing Numbers Doesn't Mean They are Rotten, and Are You Smarter Than a Fifth Grader?.
Kindergarten
- They should understand that when counting numbers, the last number said names the number of objects and when objects are rearranged, the number stays the same.
- Students count forward and backward to 10 with and without objects, as well as read, write, and represent the numbers.
- They also compose and decompose numbers up to 10 using objects and pictures (see last six weeks newsletter for definitions of decompose). The goal is to have students instantly recognize a small quantity of numbers.
- Kinder students compare numbers and identify which is less than, more than or equal to a number. Kindergarteners are expected to recite numbers up to 60 by ones beginning with any number.
- Students also solve problems and determine if they are joining or separating and what operation to use and orally explain their strategies.
- Finally in the second six weeks, Kindergarteners begin to analyze data and graphs. They use the data to create both real-object graphs and picture graphs. Then they examine these graphs. Students draw conclusions to answer questions and summarize the data represented in real-object and picture graphs.
Decomposing a Number Breaking a number into parts or smaller values | Addend and Sum | Minuend, Subtrahend and Difference The minuend is the number from which another number is subtracted. The subtrahend is the number being subtracted. A difference is the remaining amount. |
Addend and Sum
1st Grade
- Students compose and decompose numbers through 20 as a sum of so many tens and so many ones.
- They use concrete objects which include base-10 blocks and place value disks, pictorial models, and numerical representations, which include expanded and standard forms.
- Students generate numbers that are more or less than a given number using tools (e.g., a hundreds chart, calendar, base-10 blocks, etc.). They use place value to compare whole numbers up to 20 and using comparison language and comparison symbols (< , >.)
- Students are also introduced to using place value and open number lines to order whole numbers.
During the second unit, students represent and solve real-world problem situations involving sums and minuends up to 20 using spoken words, objects, pictorial models, and number sentences.
- They explore and explain a variety of strategies to solve problems involving joining and separating problems, and problems involving comparing problems.
- Students are expected to use a number sentence to represent the problem and explain that the equal sign means that the values on each side are equivalent.
- Students begin to recognize basic fact relationships.They also use these relationships to determine the unknown whole number.
- Finally students develop addition and subtraction problems when given a number sentence of numbers within 20.
Digit Any numeral from 0 to 9. The number 26 is a two-digit number and 459 is a three-digit number. Students often confuse number and digit. | Expanded form The representation of a number as a sum of place values. | Standard form The representation of a number using digits. |
Open Number line An open number line is an empty number line where tick marks are added to represent landmarks of numbers, often indicated with arcs above the number line (referred to as jumps) demonstrating approximate proportional distances. | Associative Property of Addition The associative property of addition means that when more than two addends are used, they can be grouped in any order to get the same sum. (2 + 4) + 3 = 9 and 2 + (4 + 3) =9 | Equation and Expression Equations are mathematical statements with an equal sign. Student must realize that the value on one side is the same as the value on the other side. Expressions are mathematical statements without equal signs. |
Open Number line
Associative Property of Addition
(2 + 4) + 3 = 9 and
2 + (4 + 3) =9
2nd Grade
Students explore flexible methods and models to solve and represent addition and subtraction situations within 1,000.
- These including mental math, concrete models, pictorial representations, number sentences, and open number lines.
- Addition and subtraction situations, where the unknown may be any one of the terms in the problem, should include numbers that require regrouping to solve the problem.
- Students also experience developing addition and subtraction situations when given a number sentence involving addition or subtraction of numbers within 1,000.
- Students continue to work on automatic recall and fact fluency.
- They also determine the value of a collection of coins up to one dollar and exchange coins to create sets of equivalent values.
3rd Grade
They use real-life experiences to construct multiplication models (concrete, pictorial, and area models), equal groups, arrays, strip diagrams, and equations in a relevant way. Students also explore the properties of multiplication to help learn and apply basic facts up to 10 x 10. These include:
- commutative (if the order of the factors are changed, the product remains the same),
- associative (if three or more factors are multiplied, they can be grouped in any order, and the product will remain the same), and
- distributive properties (if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together)
Although not expected to learn these properties of multiplication by name, students are expected to be able to analyze, describe, and represent these strategies.
- Students understanding of multiplication is strengthened by solving one-step problems, including problems involving area of rectangles, and making sense of meaningful, efficient representations and strategies.
Array An array is a set of objects arranged into rows and columns. | Area Model An area model is an arrangement of squares/rectangles into a grid format. | Factors and Product A factor is a number multiplied by another number to get a product. |
Associative Property of Multiplication Factors can be grouped and still get the same product. | Commutative Property of Multiplication Factors can be rearranged and still get the same product. | Distributive Property of Multiplication The Distributive Property says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. Ex: 3 x (2 + 4) = 3 x 2 + 3 x 4 -- the three is "distributed" |
Fourth Grade
- Students are expected fluently solve one and two step division problems including remainders.
- This unit also includes representing division problem situations using strip diagrams and equations with a letter standing for the unknown quantity.
- Students use estimation, rounding and compatible numbers, to find estimates and justify reasonableness for solutions to problems involving division.
During the second part of the six weeks, students apply previously learned concepts to solve one-, two-, or multi-step problems involving:
- addition and subtraction of whole numbers and decimals to the hundredths place,
- multiplication of whole numbers up to two-digit factors and up to four-digit factors by one-digit factors, and
- division of whole numbers up to four-digit dividends by one-digit divisors with remainders in appropriate contexts.
Students calculate a profit and learn to distinguish between fixed and variable expenses.
Fourth graders also represent problems using a input/output table and numerical expressions to generate a number pattern that follows a given rule.
Quotient Quotient - the size of each group or the number of items being divided. Dividend- the number being divided Divisor - the number the group of items is being divided by | Input- Output table Input - Output tables represent how the application of a rule results in a different value. |
5th Grade
- They define types of taxes including income tax, sales tax, property tax, and payroll tax.
- They investigate the differences between gross income and net income and compare and contrast various methods of payment including, checks, credit cards, debit cards, and electronic payments.
- Students develop a system for keeping financial records with regards to income and expenses and balance a simple budget.
This six weeks students are formally introduced to the thousandths place.
- Students understand that each place-value position is one-tenth the value of the place to its left and 10 times the position to the right.
- Students are expected to use expanded notation and numerals to represent the value of a decimal through the thousandths and use symbols (<,>, =) to compare and order decimals to the thousandths place and round decimals to the tenths or hundredths place.
- Students represent multiplication and division in problem situations with products and decimals with concrete objects, pictorial models and area models.
- Students are expected to estimate to determine products and quotients,
- solve for products and quotients, and
- simplify numerical expressions that include multiplication and division of whole numbers and decimals.
Division is limited to four-digit dividends and two-digit whole number divisors. For example 4567 / 26.
Sixth Grade
- They understand that absolute value can be represented on a number line as the distance a number is from zero, and that absolute value is always a positive value or zero.
- Students will work with concrete and pictorial models (not just memorizing the rule as their teachers and parents may have done). This is foundational to student understanding of operations with integers. The use of concrete and pictorial models for integer operations is intended to be a bridge between the abstract concept of operations with integers and their standardized algorithms.
- Sixth grade students are formally introduced to proportional reasoning with the building blocks of ratios, rates, and proportions.
- Students generate equivalent forms of fractions, decimals, and percents using ratios, including problems that involve money.
- They solve and represent problem situations involving ratios and rates with scale factors, tables, graphs, and proportions.
- Students also represent real-world problems involving ratios and rates, including unit rates, while converting units within a measurement system.
Students solve real-world problems using part, percent and whole.
- They find the whole when given part and percent.
- Find the part given the whole and the percent.
- Find the percent given the part and the whole.
The concept of proportionality is vital to future understanding of upcoming math concepts in this grade and beyond.
Absolute Value Absolute value is the distance of a value from zero on the number line | Integers Integers are the set of positive and negative whole numbers and zero | Ratio A ratio is a multiplicative comparison of two quantities. For instance, suppose you were going on a field trip and students and teachers were split into groups. There are 20 students and 4 teachers, so each teacher has 5 students. The teacher to student ratio is 1:5 or 1 to 5. |
Rate A rate is a multiplicative comparison of two different quantities where the measuring unit is different for each quantity. For instance, miles per gallon is considered a rate. The number of people per square mile is also a rate. | Scale Factor Scale factor is the multiplicative ratio between two sets of data. For example, suppose School A has 5 sixth grade classes and School B has 10 sixth grade classes. The scale factor comparing A to B is 2. |
Rate
The number of people per square mile is also a rate.
Seventh Grade
Seventh graders are studying proportional reasoning with ratios and rates through the lens of constant rates of change.
- Students are expected to represent constant rates of change given pictorial, tabular, verbal, numeric, graphical and algebraic representations, including d = rt.
- Students solve problems involving ratios, rates, and percents.
- They calculate unit rates from rates and determine the constant of proportionality within mathematical and real-world problems.
- Students use proportions and unit rates to convert units between customary and metric measurements.
- Students are also solving problems involving percent increase, percent decrease, and financial literacy and identify the components of a personal budget.
- Students use data with two variables, to determine constant rates of change given pictorial, tabular, verbal, numeric, graphical and algebraic representations.
- Students are formally introduced to the slope intercept form of equations, y = mx + b, to represent linear relationships.
Constant rate of change Constant rate of change is a ratio when the dependent, y-value, changes at a constant rate for each independent, x-value | Constant of Proportionality Constant of proportionality is a constant positive ratio between two proportional quantities denoted by the symbol k | Variable Expenses Variable expenses change from time period to time period, such as groceries and clothing, or even cell phone expenses. Fixed expenses are those which stay the same month to month such as rent, or car payment. |
Constant rate of change
Constant of Proportionality
Coefficient A coefficient is a number that is multiplied by a variable. Ex: 3x....3 is the coefficient | Constant The constant is a fixed value that does not appear with a variable | Linear Relationship A linear relationship is a relationship with a constant rate of change represented by a graph with a straight line. |
Coefficient
Ex: 3x....3 is the coefficient
Similar Shapes Similar shapes are those whose angles are congruent and whose side lengths are proportional. | Slope The slope is the rate of change in y (vertical) compared to the rate of change in x (horizontal), or or , denoted as m in y = mx + b | Y-intercept The y-intercept is y-coordinate of a point at which the relationship crosses the y-axis meaning the x-coordinate is equal to zero, denoted as b in y = mx + b. |
Similar Shapes
Slope
Eighth Grade
- Students use similar right triangles to develop an understanding of slope using the change in y-values and x-values.
- Students use data from a table or graph to determine the rate of change or slope and the y-intercept.
In the next unit students extend their previous understandings of slope and y-intercept to represent proportional and non-proportional linear situations. Students
- distinguish between proportional and non-proportional linear situations.
- examine the relationship between the unit rate and slope of a line that represents a proportional linear situation
- Students examine graphs of linear equations and the intersection of graphed equations and identify the values of x and y. Students must also verify these values algebraically with the equations.
They use this understanding to explain how small amounts of money, without interest, invested regularly grow over time.
Students also examine how periodic savings plans can be used to contribute to the cost of attending a two-year or four-year college after estimating the financial costs associated with obtaining a college education.
Students are formally introduced to functions and must identify functions using sets of ordered pairs, tables, mappings, and graphs.
Direct Variation When two variables are related so that their ratios are always the same, they are said to be in direct variation. For example, if a gallon of milk costs $3 and I buy 10 gallons, the cost is $30. The total cost of milk and the number of gallons are in direct variation--the ratio of cost to gallons is always 3. If y varies directly as x, then the graph of all points that describe this relationship is a line going through the origin (0,0) whose slope is called the constant of the variation. | Functions A function is a relation in which each element of the input (x) is paired with exactly one element of the output (y) |
Direct Variation
For example, if a gallon of milk costs $3 and I buy 10 gallons, the cost is $30. The total cost of milk and the number of gallons are in direct variation--the ratio of cost to gallons is always 3.
If y varies directly as x, then the graph of all points that describe this relationship is a line going through the origin (0,0) whose slope is called the constant of the variation.
Algebra 1
Representative equations and inequalities are formulated for problem situations, solved by a method of choice, and solutions justified in terms of the problem situation.
The next unit, Investigating Linear Functions, has students investigate the linear parent function and the effects of parameter changes on the linear parent function.
- Students represent problem situations involving linear functions and linear inequalities with tables, graphs, and algebraic representations.
- Students also identify characteristics such as domain, range, slope, and intercepts. Connections are made between representations and characteristics in terms of the problem situation.
- Comparisons are made between direct variation and linear functions.
Parent Function This is the simplest graph of a function family with similar characteristics (e.g., linear parent function y = x) We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin (0, 0). | Slope The rate of change in y (vertical) compared to change in x (horizontal), |
Parent Function
We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin (0, 0).