CISD Math Department

CISD Math Department

New vocabulary! New verbs! We cannot just go to OLD questions and use them to represent NEW TEKS. New nouns! New grade level changes! New resources!

NEW 7.4A represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt This TEK is a combination of the old TEKS 7.4B, 7.4C, and 7.5B Changed strands from Patterns to Proportionality Changed relationships to real-world problems Changed words and problem situations to verbal relationships Changed symbols and equations to algebraic representations

Breaking Down the TEK 7.4.A represent constant rates of change in mathematical and real world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt

Water represents Bucket represents knowledge student school learning provides Escaping water represents knowledge not mastered

Just a few holes in Just a few holes in Just Intermediate few 3-4 holes in K-2 Tiny holes become gaping WOUNDS at secondary level

Proportionality has been described as the capstone of the elementary curriculum and the cornerstone of algebra and beyond. - Lesh, Post, and Behr

This situation is additive. Janet will still be 4 blocks behind, so 8 blocks. If incorrectly solved through multiplicative reasoning, you would have gotten 4 blocks.

This situation is constant. It will still take 8 weeks for the corn to grow, regardless of how many rows were planted. If solved through multiplicative reasoning, the incorrect answer would be 12 weeks.

This situation is multiplicative, and the answer is 12 ounces.

Ideas from Van de Walle: Help them understand that ratios are distinct entities representing a relationship different than the quantities they compare. Have them understand relationships in which two quantities vary together and help them see how the variation of one coincides with the variation of another. Help them recognize proportional relationships as distinct from non-proportional relationships in real-world contexts. Help them develop a wide variety of strategies for solving proportions or comparing ratios.

As teachers and researchers know, students' understanding of proportionality develops slowly over a number of years. - Cramer and Post

A Vertical Look at the Proportionality TEKS Where do the new proportionality TEKS fall?

Laying the Ground Work Early 3.5C describe a multiplication expression as a comparison such as 3 x 24 represents 3 times as much as 24 3.5D determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or a product

5 th Grade New TEKS 5.4D recognize the difference between additive and multiplicative numerical patterns given in a table or graph 5.4 C generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph

6 th Grade New TEKS 6.4A compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships 6.4C give examples of ratios as multiplicative comparisons of two quantities describing the same attribute 6.4D give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients 6.4D give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients

6 th Grade New TEKS 6.5A represent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions 6.5B solve real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole including the use of concrete and pictorial models 6.5C use equivalent fractions, decimals, and percents to show equal parts of the same whole

7 th Grade New TEKS 7.4A represent constant rates of change in mathematical and real world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt 7.4C determine the constant of proportionality (k = y/x) within mathematical and real world problems 7.4D solve problems involving ratios, rates, and percents, including multi step problems involving percent increase and percent decrease, and financial literacy problems 7.4E convert between measurement systems, including the use of proportions and the use of unit rates

7 th Grade New TEKS 7.5B describe pi as the ratio of the circumference of a circle to its diameter 7.6B select and use different simulations to represent simple and compound events with and without technology 7.6C make predictions and determine solutions using experimental data for simple and compound events 7.6D make predictions and determine solutions using theoretical probability for simple and compound events

7 th Grade New TEKS 7.6E select and use different simulations to represent simple and compound events with and without technology 7.6G solve problems using data represented in bar graphs, dot plots, and circle graphs, including part-to-whole and part-to-part comparisons and equivalents 7.6H solve problems using qualitative and quantitative predictions and comparisons from simple experiments 7.6I determine experimental and theoretical probabilities related to simple and compound events using data and sample spaces

8 th Grade New TEKS 8.3A generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dlation 8.3B compare and contrast the attributes of a shape and its dilation on a coordinate plane 8.3C use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation 8.4A use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values is the same for any two points on the same line

8 th Grade New TEKS 8.4B graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship 8.4C use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems. 8.5B graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship

8 th Grade New TEKS 8.5.A represent linear proportional situations with tables, graphs, and equations in the form of y = kx 8.5C contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation 8.5D use a trend line that approximates the linear relationship between bivariate sets of data to make predictions 8.5E solve problems involving direct variation

8 th Grade New TEKS 8.5F distinguish between proportional and non proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b 0 8.5G identify functions using sets of ordered pairs, tables, mappings, and graphs 8.5H identify examples of proportional and non proportional functions that arise from mathematical and real world problems 8.5I write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations

Ratios Comparing Same Types of Measures Comparing Different Types of Measures Part/Whole Part/Part Rate Different Types of Ratios

Examples of Ratios Part-to-Whole Ratios- the ratio of girls to all students in the class. Part-to-Part Ratios- the number of girls in the class can be compared to the number of boys. Rates as Ratios- compares two DIFFERENT types of measures such as miles per hour.

Will always go through the origin on a graph. (0,0) Graph will always be a straight line. Always write the constant as a quotient of y over x. Reduce of divide to find the constant ratio.

If two quantities are proportional, then they have a constant ratio. If the ratio is not constant, the two quantities are said to be non-proportional. We will make tables and look at the relationship between the variables to determine proportionality.

In order to tell from a table if there is a proportional relationship or not, you can check to see if the ratio of y over x is the same. The ratio of y divided by x is also known as the scale factor.

I am going to show you a graphic for five seconds. Please do not talk out loud. Raise your hand when you have made sense of it. RCMPYMCAIBMGGBFBI

The objective is to find groups of letters that create abbreviations. Please do not talk out loud. Raise your hand when you have it. RCMPYMCAIBMGGBFBI Does the objective matter?

RCMPYMCAIBMGGBFBI Now can you apply this to a new situation? TEASTAARSSIRTIAIP

TEKS 7(6)(A) represent sample spaces for simple and compound events using lists and tree diagrams; 7(6)(C) make predictions and determine solutions using experimental data for simple and compound events; 7(6)(D) make predictions and determine solutions using theoretical probability for simple and compound events; 7(6)(I) determine experimental and theoretical probabilities related to simple and compound events using data and sample spaces.

Sample Space What is the total number of different combinations in which you could roll the sums listed above (# in the Sample Space)? List the possible sums you could roll with two number cubes.

THEORETICAL PROBABILITY Part 4 - continued

1. If two number cubes are rolled twice, calculate the theoretical probability of rolling two cubes with a sum of 6 and then rolling two cubes with a sum of 7. 2. If two number cubes are rolled twice, calculate the theoretical probability of rolling two cubes with a sum of 2 and then rolling two cubes with a sum of 3. 3. If two number cubes are rolled once, calculate the theoretical probability of rolling two cubes with a sum of 4 or a sum of 5. 4. If two number cubes are rolled once, calculate the theoretical probability of rolling two cubes with a sum of 9 or a sum of 10.

Which race cars have the same experimental and theoretical probabilities? What is the difference between the theoretical and experimental probabilities for Race Car 6? Using the class experimental probability for Car 6, predict the number of races out of 160 you would expect Race Car 6 to win. Using the theoretical probability for Car 6, predict the number of races out of 160 you would expect Race Car 6 to win. Using the class experimental probability for Car 3, predict the number of races out of 160 you would expect Race Car 3 to win. Using the theoretical probability for Car 7, predict the number of races out of 160 you would expect Race Car 7 to win.

DID WE ADDRESS THE TEKS? 7(6)(A) represent sample spaces for simple and compound events using lists and tree diagrams; 7(6)(C) make predictions and determine solutions using experimental data for simple and compound events; 7(6)(D) make predictions and determine solutions using theoretical probability for simple and compound events; 7(6)(I) determine experimental and theoretical probabilities related to simple and compound events using data and sample spaces.

New 8 th Grade TEKS Covered

FUNCTIONS Match the terms in the x-column to the corresponding terms in the y-column.

INPUT OUTPUT

INPUT OUTPUT

INPUT OUTPUT

INPUT OUTPUT

Use any x values desired. Recommend using at least one negative, one positive, and 0.

Choose your own equation. Use any x values desired. Recommend using at least one negative, one positive, and 0.

Work towards the idea of Growth rate Stage #

Stage 0 = 0 tiles Starting Value = 0 Graph (0, 0)

3 0

How many tiles in Stage 0?

Work towards the idea of Growth rate Stage # + Starting Value

6/3= 16/8= Stage 0 = 1 tile Starting Value = 1 Graph (0, 1)

6/3= 16/8=

At Stage 2, 8 tiles were used. 3 2 14 No, starting value = 2 /3=3 /8=3

6/3= 16/8=

Yes 5 0 No 4 1 Yes 3 0 Plot points on the graph to see the growth Plot points on the graph to see the growth 3 2 y = 3x + 2 2 4 y = 2x + 4

2 1 y = 2x + 1 3-1 y = 3x - 1 1 3 y = 1x + 3-1 4 y = -1x + 4 The x values jumped 3. The y values jumped 9. 9/3 = 3

SLOPE To Climb = A Monter (French) m Definition Same as Growth Rate Four Types Tiles per Stage

Cut out descriptor cards & glue

3/2 21/3=7 7/1=7-5/1=-5-10/2=-5 UND -3/5 2/1 = 2 1/2-1/1=-1 0 Use Similar Right Triangles for Slope 6/1=6 5/2-4/1=-4

1 When x increases by 1, y increases by 1. -1/2 When x increases by 2, y decreases by 1. 5/4 When x increases by 4, y increases by 5. When x, y -13/0 = undefined none 75 ft/min $2/min $5/pkg $9/shirt

SLOPE PRACTICE Graphs in Context Formula Rise Run

SLOPE PRACTICE Tables in Context Formula Δy Δx Starting value = Stage 0

y = mx + b y = slope x + y-intercept Starting value is the y-intercept y = growth rate x + starting value

y = mx + b y = slope x + y-intercept y = growth rate x + starting value

y intercept = starting value Slope = growth rate (y per x) Equation is y = mx + b $5; starting cost (for shoes) 3; $3 per game (cost per game) y = 3x + 5 $50 3(15) + 5

RULE OF 4: Multiple Representations Define x & y Fill out table; use mx+b for process MAKE CONNECTIONS!!! Look for Slope in Verbal, Equation, Table, & Graph Look for y-intercept in Verbal, Equation, Table, & Graph Answer questions Find m (per) & b (start); write y=mx+b Plot points for graph

M & B Match Students cycle through three stations At station 1, students find m & b from 4 tables on cards. At station 2, students find m & b from 4 graphs on cards. At station 3, students find m & b from 4 verbal descriptions on cards.

Did we cover these TEKS?