Making Sense of Math
AISD Parent Newsletter • 4th Six weeks 2016-2017
Fourth Six Weeks
I hope this newsletter is helpful for parents. If you would like more explanation on any of the topics mentioned below, please feel free to contact me. I would be happy to visit with you or give you examples.
Anna Holmgreen, Director of Instruction for Math
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Kindergarten
- count forward and backward (with and without objects), as well as read, write, and represent the numbers.
- compose and decompose numbers up to 10 using objects. They instantly recognize the number being represented by a small quantity of objects.
- compare sets of objects up to 20 and generate a set of objects and pictures that is more than, less than, or equal to a given number.
- use comparative language to describe the comparison of numbers represented using objects, pictures, or numerals.
- recite numbers up to at least 100 by ones beginning with any number.
- recite numbers by tens up to at least 100 beginning with any multiple of 10 (e.g., 10, 20, 30, 40, etc.).
They also work on the problem-solving process by working repeatedly with real-world problem situations. Students continue to distinguish between joining, separating, and part-part-whole situations and the operation needed to solve the problem. Students:
- orally explain models and representations and justify solution strategies.
- use graphing situations to represent data with numbers 0-20. Students sort and organize the data and the data is used to create real-object and picture graphs. These graphs are examined to understand the components of graphing (e.g., title, labels of categories, what each cell or picture represents, etc.)
1st Grade
During this unit, students extend their understanding of the base-10 place value system to include the hundreds place as they continue exploring the foundations of whole numbers up to 120. Students:
- compose and decompose numbers through 120 as so many hundreds, so many tens, and so many ones using objects, pictures, and numbers
- generate numbers that are more or less than a given number using tools (e.g., a hundreds chart, calendar, base-10 blocks, etc.).
- compare whole numbers up to 120 and represent the comparison using comparison language and comparison symbols.
- extend using place value and open number lines to order whole numbers up to 120.
- recite numbers up to 120 forward and backward by ones and tens; skip count by 2s, 5s, and 10s; and use place value patterns to determine a number that is 10 more or 10 less than a given number.
- identify pennies, nickels, dimes, and quarters by value and record the value using the cent symbol.
- exchange coins or sets of coins for other equivalent denominations.
- apply skip counting by 2s, 5s, and 10s and compound counting to determine the value of a collection of pennies, nickels, and dimes up to 120 cents.
- use data represented in bar-type and picture graphs to represent, generate, and solve problem situations involving sums and minuends up to 20 using spoken words, objects, pictorial models, and number sentences.
- explore and explain strategies to solve one-step problems involving addition, subtraction, and comparison of the data. Students are expected to use a number sentence with the unknown in any position to represent the situation.
Expanded Form
the representation of a number as a sum of place values (e.g., 119 as 100 + 10 + 9)
This example is a bit larger than first graders will use, but it is an example.
Standard Form
the representation of a number using digits (e.g., 118)
This example is a bit larger than first graders will use, but it is an example.
Period
a three-digit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
Comparison symbols
> greater than
= equal to
Students read numbers left to right and should recognize the symbol by the correct name
2nd Grade
- During this unit, students add up to four two-digit numbers or subtract two-digit numbers within 1,000 in real-world problem situations that include calculating how money saved accumulates over time with deposits and withdrawals.These problems should include numbers that require regrouping to solve the problem. In addition, students:
- make connections between representing and solving addition and subtraction problems using flexible methods, concrete and pictorial models, and number sentences to mental strategies and algorithms based on knowledge of place value and properties of operations.
- generate and solve problem situations for a given number sentence involving addition and subtraction of whole numbers within 1,000.
- Students transition from bar-type graphs to bar graphs and from picture graphs to pictographs. Both vertical and horizontal orientations of bar graphs and pictographs with up to four categories and intervals of one or more are experienced during this unit. Students summarize the existing data, or the inferential data, in bar graphs and pictographs to draw conclusions and make predictions.
- Students also generate and solve one-step word problems based on the information in bar graphs and pictographs with intervals of one.
Bar Graph
a graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs
Pictograph
a graphical representation to organize data that uses a picture or symbol, where each picture or symbol may represent one or more than one unit of data, to show the frequency (number of times) that each category occurs
3rd Grade
During this six weeks students extend their understandings of multiplication and division using arrays, area models, strip diagrams, and equations.
- During this unit, students solve one- and two-step problems involving addition and subtraction within 1,000 and multiplication and division within 100.
- Students reason through and solve real-world problem situations. Students explain their reasoning and solution strategies using expressions, equations, and precise mathematical language.
- Students solve problems involving multiplication of a two-digit number by a one-digit number and develop fluency using standard algorithms to solve addition and subtraction problems within 1,000.
- Experience real-world situations that involve various operations, including decomposing composite figures to determine area of the original figure.
- Use input-output tables and explore number pairs in tables to determine additive and multiplicative patterns that exist and represent the pattern (or process) using equations and expressions.
- Summarize a set of data using a frequency table, dot plot, pictograph, or bar graph. Students use these data representations to solve one- or two-step problems involving the categorical data represented.
- Sort and classifying two- and three-dimensional figures that may vary in size, shape, and orientation based on attributes using formal geometric language.
- Explore two-dimensional figures and subcategories of quadrilaterals, including rhombuses, parallelograms, trapezoids, rectangles, and squares.
- Decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole.
Composite Figure
a figure that is composed of two or more two-dimensional figures
Decompose Figures
to break a geometric figure into two or more smaller geometric figures
Equation
a mathematical statement composed of equivalent expressions separated by an equal sign
Expression
a mathematical phrase, with no equal sign, that may contain a number(s), an unknown(s), and/or an operator(s)
Input-Output Table
a table which represents how the application of a rule on a value, input, results in a different value, output
Polygon
a closed figure with at least 3 sides, where all sides are straight (no curves
4th Grade
Fourth graders begin work with data this six weeks. Students represent information 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. Students also use this data to solve one- and two-step problems.
Next students work to relate their understanding of decimals to fractions that name tenths and hundredths. These are also represented on a number line as distances from zero.
Students also represent and decompose fractions (including values greater than one) as sums of unit fractions with the same denominator using concrete models and representations.
Frequency Table
Dot Plot
Stem and Leaf Plot
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 explaining these:
5th Grade
Students compare various forms of payment including checks, credit cards, debit cards, and electronic payments.
They work on a system for keeping financial records and balance a simple budget. The concept of negative values is discussed when balancing budgets.
Next students move on to adding and subtracting whole numbers and decimals. They:
- add and subtract positive rational numbers including fractions
- estimate solutions to determine reasonableness
- use concrete objects and pictorial models and properties of operations to represent and solve problems including adding/subtracting fractions with unequal denominators
Fifth graders then move on to representing the product of a whole number and a fraction using concrete and pictorial models (including area models and strip diagrams)
Students also use these to represent the division of a whole number by a unit fraction and the division of a unit fraction by a whole number.
Students continue to estimate solutions to determine reasonableness and simplify numerical operations involving all operations with whole numbers, decimals and fractions.
6th Grade
- Students graph both positive and negative rational numbers in all four quadrants of the coordinate plane.
- Students are now expected to compare two rules (ex. y = ax or y = x + a) verbally, graphically, or symbolically in order to differentiate between additive and multiplicative relationships.
- Students identify independent and dependent relationships and quantities. Within this unit, students represent a given situation using verbal descriptions, tables, graphs, and equations. Also, given one representation, students should be able to create one or all of the different representations for the problem situation. For this grade level, problem situations for additive relationships may include both positive and negative rational numbers, whereas multiplicative relationships may only include integers or positive fractions or decimals.
- Students extend their knowledge of triangles and their properties to include the sum of the angles of the triangle, and how those angle measurements are related to the three side lengths of the triangle.
- Students examine and analyze the relationship between the three side lengths of a triangle and determine whether three side lengths will form a triangle using the Triangle Inequality Theorem.
- Students also decompose and rearrange parts of parallelograms (including rectangles), trapezoids, and triangles in order to model area formulas for each of the figures.
- Students write equations and determine solutions for problems dealing with area of rectangles, parallelograms, trapezoids, and triangles.
- Students expand previous knowledge of converting units within the same measurement system when determining solutions to problems involving length. Conversion processes for measurement extend beyond the use of proportions to now include dimensional analysis and conversions graphs.
Dependent Variable
Independent Variable
Triangle Inequality Theorem
7th Grade
- 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.
Next, students work with the concept of similarity.
- During this unit, students extend concepts of 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.
The next unit deals with probability.
- 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.
Similar Figures
Two figures that have the same shape are said to be similar. When two figures are similar, the ratios of the lengths of their corresponding sides are equal. To determine if the triangles below are similar, compare their corresponding sides.
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.
Complement of an Event
The complement would be 3 out of 5.
In the picture above, if the probability of event A is 5 or 6, the complement would be 1, 2, 3 or 4.
Gr. 8
Eighth graders develeop 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.
Next, students extend previous knowledge of sets and subsets to order and describe relationships between sets of real numbers, which includes rational numbers and their subsets as well as irrational numbers.
- Students approximate the value of irrational numbers less than 225 and locate those approximations on a number line.
- Establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
- Right triangles are examined more closely within this unit as students use models to explain the Pythagorean theorem. Students use the Pythagorean theorem and its converse to solve problems and apply these understandings to the coordinate plane as they determine the distance between two points on the coordinate plane.
Angle-Angle Criterion for Similarity
Now if a second triangle has two angle measurements of 40° and 60°, we know the third angle must be 80°. This means the two triangles are the same shape, but not necessarily the same size. Alternately we may think of one as a dilation of the other. Either way we know that the triangles are similar. We call this the angle-angle criterion for similarity
Pythagorean Theorem
When the triangle has a right angle (90°) ...
... and squares are made on each of the
three sides, then ...
... the biggest square has the exact same area as the other two squares put together!
a2 + b2 = c2
Hypotenuse
Dilation
Exterior Angle of a Triangle
Alice I.S.D. Math Curriculum
Email: aholmgreen@aliceisd.esc2.net
Website: aliceisd.net
Location: 2 Coyote Trail, Alice, TX, United States
Phone: 361-664-0981
Facebook: https://www.facebook.com/AISDMath