# 11th Grade Mathematics: Numbers and Operations Study Guide for the SBAC

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# Other Math Abilities Tested

The goal of this test is to prove you are college- and career-ready in the area of math. While assessing the above skills, and those in our other four math study guides, the SBAC Mathematics test will also assume you know how to do other things in math. Here is a list of these expectations:

## Concepts and Procedures

### You are expected to be fluent and accurate in performing mathematical operations and procedures and you can explain the process.

*Fifty percent of the questions assess this.* Questions on this test will not directly ask you the answer to a calculation such as \(7.5 \times 4.25\). However, you will need to do these calculations in order to answer the questions that are asked. You may also be asked questions that test your understanding of operations, such as the concept of inverse operations or why a particular step is needed in a calculation.

## Problem Solving

Problem solving involves looking at the information presented, seeing what is being asked, devising, and then carrying out a strategy to give the required answer. The answer you propose should be evaluated to be sure it makes sense in the context of the problem.

### You can solve problems using math knowledge.

Some of these will be complex problems requiring multiple steps and a variety of problem-solving strategies. Many of them will revolve around real situations that are possible in everyday life. Additionally, you can do the following:

### You can explain *why* a procedure works to solve a problem.

The problems may not only require a solution, but also require you to explain that solution to another person or convince someone that your answer is correct. This means explaining why you chose the steps and why they give the best possible answer to the problem.

### You can choose the correct operation and order the procedures to correctly solve problems.

The problems will require several steps and operations to give a result. You may need to consider concepts such as the **order of operations** to ensure that the result is correct. You may need to identify other quantities you need to determine before you can calculate the information required to answer the actual question.

### You can tell what the results mean (interpret).

This is a big part of being able to convince another person about your result. For example, you may make a graph with two intersecting lines. It is not enough to say “the values are equal here.” You have to be able to say something like, “this is the point where the business begins to turn a profit.”

### You can use visual aids in math, such as graphs, two-way tables, flowcharts, diagrams, and formulas.

Related to interpretation, while you may be able to get a solution purely with numerical calculations, the appropriate visual aid will help make the point of why the answer is correct. Also, you may need to examine a visual aid to get numerical information needed to solve the problem. Be sure you understand the advantages of the different visual aids.

## Communicating Reasoning

Twenty-five percent of the questions address this skill. You can effectively tell about the procedures and reasoning you use to solve problems and find errors in inaccurate reasoning. This might be tested in the following ways:

### Question asks for an argument in favor of a piece of reasoning.

Some questions will ask you to provide an explanation to prove or disprove a claim. This requires you to identify the quantities and determine the properties and relationships of these quantities. Use this information to construct a *logical and convincing* argument to make your point.

### Question asks for a proof of mathematical reasoning.

Mathematical proofs require you to understand steps in solving equations and geometric theorems. Each step in the proof must have a justification, either from a previous step, the given information, or a geometric theorem. This provides a *rigorous, deductive* proof.

### Question asks for justification of a solution to a problem.

As you explain or justify the solution to a problem, in addition to the properties and mathematical relationships, you can provide examples. This is especially useful when the goal is to disprove a statement. In this case, you provide a **counterexample** that shows a situation when the statement is wrong. This is usually enough to disprove a statement.

### Question asks for domains in which an argument applies.

Sometimes a statement is true but only in a limited domain. Some examples are properties that are true for rational numbers but not for irrational numbers, such as closure under multiplication. Rational numbers are closed under multiplication, but irrational numbers are not.

### Question asks you to use the definition of a math term to examine a claim.

You may need to use the definition of a term to examine a claim. For example, if you are told a figure is a rectangle, this gives you information about the angles in the figure, the congruence of certain sides, and the existence of parallel lines. Or, perhaps, there is a claim that a figure is a rectangle, and you can show that one of these requirements are not met.

### Question asks you to combine concepts from more than one math discipline to solve a problem and explain why you did so.

Every discipline in mathematics provides tools to solve problems. Often, real-world problems require tools from more than one discipline. This is frequently seen in geometry, where geometric knowledge is used to set up an algebraic equation to solve. In communicating, the key is to understand and explain why you chose and use these tools.

## Modeling and Data Analysis

Modeling and data analysis uses mathematics to solve problems. Often, this means finding a **relevant equation** that describes the situation and makes predictions of outcomes.

### Relevance of Information

The first step in data analysis is to examine all of the information in the problem. For each item, determine whether it is relevant, and then decide how to represent it. Commonly, information that describes “how much” of some quantity will be represented as a variable in a model.

### Modifying the Model

You may be given a model for a situation that does not work, or be asked to modify a model to fit a similar situation with some different details. In either case, you will need to determine exactly why the model isn’t working and then make adjustments. Perhaps the wrong variables are modeled, some coefficients need adjusting, or a different type of model is required.

## Assorted Other Expectations

Although you are in a specific grade when taking this test, the content on the test will require you to apply content that you may have covered in depth in previous grades. When you find a topic you need to review, use that as a cue to review other content from that same year of study. Here are some math concepts for which your competency is assumed in this test:

### Use Ratio and Proportion

A **ratio** describes a numerical relationship between two quantities.

A ratio of 1 part milk to 2 parts water can be written as 1:2 or \(\frac{1}{2}\).

A **proportion** is a statement that two different ratios are equal to each other.

Often, one of the quantities is unknown. For example, how much milk is needed if 5 cups of water are required? It is best to write the ratios as fractions and set them equal, like:

\[\frac{1}{2} = \frac{x}{5}\]Solve by **cross-multiplication**.

### Apply Percentages

A **percent** is a ratio or fraction expressed with a denominator of 100. For example:

\(63\%\) means “63 out of 100” or \(\frac{63}{100}\)

It can also be written as its **decimal** equivalent of \(0.63\).

You always use the decimal equivalent or fraction when performing calculations with percents.

### Apply Unit Conversions

When converting units, the technique of **dimensional analysis** is very helpful.

For example, you may need to convert meters per second (m/sec) into kilometers per hour (km/hr).

You know that there are 1000 meters per kilometer and 3600 seconds per hour, but you don’t know what to multiply or divide.

Do the calculation just with the units:

\(\require{cancel}\)

This tells you how to multiply the numbers.

### Use Basic Function Concepts

A **function** is a way of specifying an output value for a given input value. The most common function you will encounter is the linear function \(y = mx +b\), which graphs as a straight line. Be prepared to interpret the **slope** and **intercept** in the context of a problem. For example, the intercept may represent fixed costs and the slope represent dollars per month in a business problem.

### Utilize Geometric Measurement

Be familiar with geometric measurements such as **perimeter**, **area**, **surface area**, and **volume** for common geometric figures. Use the **properties of the geometric figure** when analyzing a diagram. For example, you may be told that a triangle is isosceles, but only one side is labeled. From the properties of isosceles triangles, you know that the other side is the same measurement.

### Apply Basic Statistics and Probability Skills

You should be familiar with statistics concepts such as **mean**, **median**, **mode**, and **standard deviation**, and be able to apply them to a problem situation. Also review the basic calculation of **probabilities** and the **laws governing probabilities of more than one event**, such as **independent events**, where the probability of each individual event is multiplied by that of other events to determine the probability of all of the events happening.

### Fluently Perform Rational Number Arithmetic

After you have analyzed a problem and applied concepts, you will still need to calculate an answer. You still need to be fluent in **arithmetic facts** and able to apply them in calculations with rational numbers. Although calculators may be available or permitted, do not use a calculator as a substitute for the ability to quickly perform simple calculations and be sure to develop the **number sense** required to know if a calculated answer is reasonable.

## Tackling Differently Formatted Test Items

There is important information about differently formatted test items on the SBAC exam. Go to our home page for the SBAC to read it as you prepare. Scroll down to “Tips and Tricks.”

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