By viewing equivalent ratios and rates as derived from pairs of rows (or columns) in the multiplication table and by analyzing simple drawings showing the relative sizes of quantities, students connect their understanding of multiplication and division with ratios and speeds. For example, students expand the scope of problems where they can use multiplication and division to solve problems, and connect ratios and fractions. The mean measurement point is centered in the sense that it is the value that each data point would take if the total of the data values were equally redistributed, and also in the sense that it is an equilibrium point.
They prepare for work on drawing and scale construction in grade 7 by drawing polygons on the coordinate plane. For example, "The ratio of wings to beaks in the zoo birdhouse was 2:1 because for every 2 wings there was 1 beak." "For every vote that candidate A received, candidate C received almost three votes." Students are expected to: Mathematical practices Explanations and examples 6.RP.A.2. the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio.
Students will begin to notice that related unit prices are reciprocals, as in the first example. Make tables of equivalent ratios that relate quantities to whole number measurements, find the missing values in the tables, and plot the pairs of values on the coordinate plane.
The Number System (NS)
Extend the diagrams of number lines and coordinate axes known from previous classes to represent points on a line and in a plane with negative numerical coordinates. Recognize the opposite signs of numbers as marking the locations on opposite sides of 0 on the number line;. Plotting points and reflection across zero on a number line extends to plotting and reflecting points across axes on a coordinate grid.
The use of both horizontal and vertical number line models eases the move from number lines to coordinate grids. Find and place integers and other rational numbers on a horizontal or vertical number line chart; find and place pairs of integers and other rational numbers on a coordinate plane. Interpret statements about inequality as statements about the relative position of two numbers on a number line diagram.
For example, interpret –3 > –7 as saying that –3 is to the right of –7 on a left-to-right number line. Common models for representing and comparing integers include number line models, temperature models, and the gain-loss model. On the number line model, a number is represented by an arrow drawn from zero to the number's location on the number line; the absolute value is the length of this arrow.
The number line can also be viewed as a thermometer, where each point on the number line is a certain temperature. When working with number line models, students internalize the order of numbers; larger numbers on the right or upper side of the number line and smaller numbers on the left or lower side of the number line. They use order to correctly locate whole numbers and other rational numbers on a number line.
Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
Expressions and Equations (EE)
Arizona Department of Education - High Academic Standards for Students Arizona's College and Career Ready Standards - Approved by the State Mathematics Board June 2010 October 2013 Publication Page 19 of 45. If an expression is the product of a number and a variable, the number is called the coefficient of the variable. You can use the expression c + 0.07c to find the total cost of an item with a 7% sales tax, where c is the cost of the item before tax.
Arizona Department of Education – High Academic Standards for Students Arizona's College and Career Ready Standards – Mathematics State Board Approved June 2010 October 2013 Publication Page 21 of 45. Arizona Department of Education – High Academic Standards for Students Arizona's College and Career Ready Standards – Mathematics State Board Approved June 2010 October 2013 Publication Page 23 of 45. Students are expected to: Mathematical Practices Explanations and Examples 6.EE.B.5. comparison or inequality as a process of answering a question:. which values of a specified set, if any, make the comparison or inequality true.
Initial experiences in solving equations should require students to understand the meaning of the equation as well as the question being asked. Students ask themselves, "What number was added to 26 to get 100?" to help them. determine the value of the variable that makes the equation true. Use knowledge of inverse operations: Since subtraction "undoes" addition, then subtract 26 from 100 to get the numerical value of n. Scale pattern: There are 26 blocks on the left side of the scale and 100 blocks on the right side of the scale.
74 blocks must be added to the left side of the scale to balance the scale. Describe a problem situation that can be solved using the equation 2c + 3 = 15; where c represents the cost of an item. Arizona Department of Education – High Academic Standards for Arizona College Students and Career Ready Standards – State Mathematics Board Adopted June 2010 October 2013 Publication Page 25 of 45 .
Sample solution: Students might say, “I created the bar model to show the cost of the three pairs of jeans. Arizona Department of Education – High Academic Standards for Students Arizona's College and Career Ready Standards – Mathematics State Board Approved June 2010 October 2013 Publication Page 27 of 45. Arizona Department of Education – High Academic Standards for Students Arizona's College and Career Ready Standards – Mathematics State Council Approved June 2010 October 2013 Publication Page 29 of 45.
Geometry (G)
Apply the formulas V = l w h and V = b h to find volumes of rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Students need multiple opportunities to measure volume by filling rectangular prisms with blocks and looking at the relationship between the total volume and the area of the base. Through these experiences, students derive the volume formula (volume equals the area of the base times the height).
Students can explore the relationship between filling a unit cube box and the volume formula using interactive apps like NCTM's Cube Tool in Lights. http://illuminations.nctm.org/ActivityDetail.aspx?ID=6). In addition to filling in the boxes, students can draw diagrams to show the lengths of fractional sides, relating to the multiplication of fractions. Arizona Department of Education - High Academic Standards for Arizona College Students and Career Readiness Standards - State Mathematics Board Adopted June 2010 October 2013 Publication Page 31 of 45 .
Students construct models and meshes of three-dimensional figures and describe them based on the number of edges, vertices and faces. Students can create meshes of 3D figures with specified dimensions using the Dynamic Paper Tool on NCTM's Illuminations (http://illuminations.nctm.org/ActivityDetail.aspx?ID=205). Students make and test conjectures by determining what it takes to create a specific three-dimensional figure.
Create the mesh for a given prism or pyramid and then use the mesh to calculate its area. Arizona Department of Education – High Academic Standards for Students Arizona's College and Career Ready Standards – Mathematics State Board Approved June 2010 Published October 2013 Page 33 of 45.
Statistics and Probability (SP)
Two dot plots show the 6-trait writing scores for a group of students on two different traits, organization and ideas. Students consider the context in which the data were collected and identify clusters, peaks, gaps, and symmetry. For example, students will be able to see from the display of the two graphs that idea scores are generally higher than organization scores.
One observation that students can make is that the scores for organization are clustered around a score of 3, while the scores for ideas are clustered around a score of 5. Arizona Department of Education – High Academic Standards for Students Arizona's College and Career Ready Standards – Mathematics State Board Approved June 2010 October 2013 Publication Page 35 of 45. When measures of center (mean, median and mode) and range are used, students describe ' a data set in a single number.
The range provides a single number that describes how the values vary across the dataset. Consider the data shown in the dot plot of the six traits of organization for a group of students.
Statistics and Probability (SP) Summarize and describe distributions
The measure of center a student chooses to describe a data set will depend on the shape of the data distribution and context of data collection. In this case, the median or mean value of the data set may be more descriptive. In data sets that are skewed, the mean and median will differ, with the median often providing a better overall description of the data set.
The smoothing process can be associated with and used to develop understanding of the calculation of the mean. The number of letters in each of the names is used to create the data set. Students would understand the average distance between the pieces of data, and the mean of the data set expresses the spread of the data set.
To find the mean absolute deviation, students examine each data point and its difference from the mean. The mean of the absolute deviations is determined by summing the absolute deviations and dividing by the number of data points. The median is the middle number of a data set with half the numbers below the median and half the numbers above the median.
The quartiles divide the data set into four parts by dividing each of the halves of the data set in half again. The interquartile range of the data is the difference between the lower and upper quartiles (Q3 – Q1). If there are even numbers of values, the median is the average of the middle two values.
Q1 would be the average of the 2nd and 3rd values in the dataset or 4th.
Standards for Mathematical Practice (MP) Standards
