What is Integrated Math?

Integrated Math is an alternative to the traditional high school math pathway, which typically separates Algebra, Geometry, Algebra 2/Trigonometry, and Precalculus into distinct courses. Instead, Integrated Math weaves together concepts from these areas so that students can see the connections between different math topics. This approach reflects the way math is used in real-world applications and makes it more relevant and engaging.

The Integrated Math Sequence

IM1: Building a Strong Foundation

IM1, or Integrated Math 1, is the starting point of this sequence. Generally, students start the sequence in 9th grade, but advanced learners can begin in 8th grade. IM1 introduces students to fundamental concepts across algebra, geometry, and statistics.

• Algebra: IM1 covers the basics of algebra, including linear equations, inequalities, and functions. Students learn to solve and graph equations, laying the groundwork for more complex problem-solving.
• Geometry: The geometry component of the course introduces students to basic shapes, congruence, and transformations. Students begin to understand how algebra and geometry are interconnected.
• Statistics: Basic data analysis and probability are also introduced in IM1. Students learn to interpret and present data and are taught skills that are increasingly important in a data-driven world.

IM1 is crucial for building a solid foundation. Success in this course sets a student up for more advanced mathematical thinking in the following years.

IM2: Expanding and Deepening Understanding

IM2, or Integrated Math 2, builds on the concepts learned in IM1 and introduces more complex topics.

• Algebra: Students delve deeper into quadratic functions, polynomials, and systems of equations. Other types of functions such as exponential functions with growth and decay models are introduced. The algebraic skills gained in IM1 are expanded to include more sophisticated methods of solving equations.
• Geometry: The geometry component covers more topics like similarity, basic triangle trigonometry, circles, and analytic/3D geometry (finding area and volume). Students begin to apply their algebraic skills to geometric problems, fostering a more integrated understanding.
• Statistics: IM2 expands on the statistics from IM1, focusing on probability and the interpretation of more complex data sets. This helps students develop a better understanding of randomness and variation.

IM2 is designed to solidify a student’s understanding of how mathematical concepts connect.

IM3: Preparing for Advanced Math

IM3, or Integrated Math 3, is the culmination of the Integrated Math sequence. It prepares students for higher-level math courses such as Pre-Calculus, AP Calculus, or AP Statistics.

• Algebra: IM3 covers more advanced algebraic concepts, including rational, exponential, and logarithmic functions, sequences and series, and advanced polynomial functions with transformations. These topics are essential for the math used in higher education.
• Geometry: In IM3, the focus shifts to the application of trigonometric identities and the exploration of more complex geometric properties. This course ties together all the geometric concepts learned in the previous courses.
• Statistics: IM3 often includes an introduction to inferential statistics. Students learn to make predictions or inferences based on data, a critical skill in many fields, from science to economics.

By the end of IM3, students should have a well-rounded mathematical education and be well-prepared for comes next.

How We Can Help

At MathTowne, we understand that the Integrated Math sequence can be challenging. Our experienced tutors are here to guide students through the integrated math sequence, ensuring they build the skills and confidence needed to succeed. Whether your student is just starting IM1 or needs support in IM3, we provide personalized tutoring that meets their unique needs.

Get in Touch! If you or your student needs help with any of the Integrated Math courses, don’t hesitate to reach out. We’re here to help make math less intimidating and more manageable, one concept at a time.

AP Precalculus

Pros:

1. College Credit: One of the most significant advantages of taking AP Precalculus is the potential to earn college credit. Scoring well on the AP exam can allow you to skip introductory college courses (depending on the college), saving time and tuition costs.
2. Standardized Curriculum: AP courses follow a standardized curriculum set by the College Board, ensuring a consistent level of rigor and quality. This can be beneficial for students who plan to apply to competitive colleges, as it demonstrates the ability to succeed in a nationally recognized program.
3. Advanced Content: According to College Board, AP Precalculus is designed to prepare students for subsequent AP Calculus courses or college-level calculus by ensuring students have a depth of understanding to be ready for calculus.
4. College Application Boost: Having AP courses on your transcript can strengthen your college applications. Admissions officers often view AP courses as a sign that a student is willing to take on challenging coursework.

Cons:

1. High Pressure: As with any AP course, the pressure to perform well on the AP exam can be intense.
2. Pacing: The pace of AP courses is often faster than that of honors courses because the last month before the AP exam is usually spent on exam preparation. This can be challenging for students who need more time to grasp complex concepts.
3. Exam Focus: The curriculum is tailored to prepare students for the AP exam, which means there might be less flexibility in exploring topics outside of the exam’s scope.
4. Potential for Burnout: The workload can be significant, potentially leading to burnout, especially if a student is juggling multiple AP courses.

Precalculus Honors

Pros:

1. Rigorous Curriculum: Precalculus Honors courses still offer a challenging curriculum (compared to regular Precalculus) that prepares students well for calculus, but again, this depends on the school. The depth and complexity are sufficient for students who aim to excel in math without the added pressures of an AP exam.
2. Flexibility: Honors courses often allow teachers more flexibility in their teaching methods and the topics they cover. This can provide a more enriched and diverse learning experience.
3. Balanced Workload: Without the need to prepare for a standardized exam, the workload in an honors course can be more balanced, reducing the risk of burnout and allowing students to focus on understanding the material deeply.
4. Strong Foundation: Honors courses can provide a strong foundation in precalculus concepts, helping students succeed in future math courses, whether in high school or college.

Cons:

1. No College Credit: Unlike AP courses, honors courses do not offer the opportunity to earn college credit. This can be a drawback for students looking to get ahead in their college coursework.
2. Varied Curriculum: Since honors courses do not follow a standardized curriculum, the quality can vary significantly between schools and teachers.
3. Perception by Colleges: While honors courses are respected, they may not carry the same weight as AP courses to college admissions officers, particularly at highly competitive institutions.
4. Less Focus on Exam Skills: Honors courses may not place as much emphasis on developing the test-taking skills that are often cultivated in AP courses. For students who plan to take standardized tests such as the SAT or the ACT, developing these test-taking skills would be useful.

So while AP Precalculus aims to standardize the curriculum and offers the potential for college credit, its novelty means some teachers may still be getting accustomed to teaching it. Precalculus Honors, on the other hand, still provides a rigorous but potentially more varied and flexible curriculum, but without an AP exam to demonstrate sufficient knowledge. Ultimately, the choice between the two should be based on individual academic goals and learning preferences.

For more info, check out our blog post for a detailed breakdown of the AP Precalculus curriculum and exam.

1. Factoring

If the coefficient a in front of x is 1, usually factoring is the most efficient method to go with. The purpose is to turn the quadratic in standard form into x-intercept form so that the roots can be easily identified. For example, take the problem

$$x^2-6x+8=0$$

What we are looking for is two factors that multiply to 8 and add to -6. These will be -2 and -4. So the quadratic in factored form is

$$(x-2)(x-4)=0$$

$$x=2, 4$$

If the coefficient a is not 1, you can use the Diamond Method to factor it. For example, take $2x^2+7x-15=0$. In this problem, $a=2$, $b= 7$, and $c = -15$. Take a and c and multiply them together to get -30. The coefficient b is 7. Now we’re looking for two numbers that multiply to ac (-30) and add or subtract to b (7).

The simplest way to search for those two numbers is to write out the factors of -30 in pairs, starting with 1, as follows:

$$\pm 1, \pm 30$$

$$\pm 2, \pm 15$$

$$\pm 3, \pm 10$$

Here we can see that 10 and -3 will add up to 7, so these are the factors.

Next, split up the x term in $2x^2+7x-15=0$ into those factors:

$$2x^2+10x-3x-15=0$$

Then factor by grouping.

$$2x(x+5) – 3(x+5)=0$$

$$(2x-3)(x+5)=0$$

$x=3/2$ and $-5$.

However, sometimes you may not be able to find integer factors for ac that add up to b, in which case use the quadratic formula or complete the square.

2. Square Roots Method

Whenever a quadratic equation has only $x^2$ terms (and no $x$ terms), the square roots method is most efficient. For example, to solve

$$2x^2 + 4 = x^2 + 20$$

Start by combining the like terms of $x^2$ together on one side, and the constants on the other (by subtracting $x^2$ and then 4 from both sides).

$$2x^2-x^2 + 4 = 20$$

$$x^2 + 4 = 20$$

$$x^2 = 16$$

$$x = \pm4$$

Don’t forget the $\pm$ since both are solutions that will give the positive square number!

Caution – if the last step with $x^2$ equals a negative number (for example $x^2 = -16$, then the equation has no real solutions, although your math class may ask you to include imaginary solutions (which will be $\pm4i$).

3. Completing the Square

Generally, the purpose of completing the square is to turn a quadratic in standard form into vertex form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola, since it is the least efficient way of solving an equation. But it can still be used for when the solutions may not be rational numbers. Let’s say we want to solve, by completing the square,

$$3x^2-12x+7=0$$

First, factor out the coefficient in front of $x^2$ and include the $x$ term in parentheses as well:

$$3(x^2-4x) + 7 = 0$$

Next, we want to find a number to add to $(x^2-4x)$ so that it will turn into the binomial squared form $(x-h)^2$. To find it, take the b coefficient in front of the $x$ term, divide by 2, and square it – in this case, the coefficient is -4.

$$(\frac{-4}{2})^2 = 4$$.

Add the 4 inside the parentheses

$$3(x^2-4x+4) + 7 = 0$$

Because we’ve effectively added 3$\times$4 = 12 to the left side of the equation, we have to subtract 12 in order to keep the original equation!

$$3(x^2-4x+4)+7-12 = 0$$

$$3(x^2-4x+4)-5 = 0$$

Now the $(x^2-4x+4)$ can be factored into $(x-2)^2$.

$$3(x-2)^2-5 = 0$$

From here, use the square roots method to solve.

$$3(x-2)^2=5$$

$$(x-2)^2=\frac{5}{3}$$

$$x-2=\pm\sqrt{\frac{5}{3}}$$

$$x=2\pm\sqrt{\frac{5}{3}}$$

The two solutions are irrational, but roughly 3.29 and 0.71.

If you find yourself struggling with quadratic equations or any other aspect of algebra, don’t hesitate to reach out for help. Our Algebra 1 and Algebra 2 tutors are here to guide you through your math class, providing personalized support tailored to your learning style and pace. Ready to take your algebra skills to the next level? Schedule a free consultation with us today!

The quadratic formula is a powerful tool used to solve quadratic equations – these are equations involving a variable squared, such as x² (“quad” meaning square), and no higher power. Generally, the problems you encounter will be in standard form $ax^2 + bx + c = 0.$

The solution(s) you get will be the roots, or x-intercepts of the graphed function which will be a parabola. The quadratic formula is as follows:

$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

When To Use the Quadratic Formula?

Whenever you see the words “solve for x” in a problem or “find the roots,” the quadratic formula is an option that you can use. For example, take the function $$y=2x^2-5x-3$$ with the problem asking you to find the roots. From the equation, $a=2, b=-5,$ and $c=-3.$

Plugging these into the quadratic formula,

$$x=\frac{-(-5)\pm\sqrt{(-5)^2-4(2)(-3)}}{2(2)}$$

$$x=\frac{5\pm\sqrt{25+24}}{4}$$

$$x=\frac{5\pm\sqrt{49}}{4}$$

$$x=\frac{5\pm 7}{4}$$

$$x=3,-\frac{1}{2}$$

These are the two solutions. As an aside, these roots will the cross the x-axis at the points $(3,0)$ and $(-\frac{1}{2},0).$

And there you have it! The quadratic formula is not the only way to solve a quadratic equation – the other methods are factoring and completing the square leading into the square roots method, alternative ways to be discussed in a later blog post.

Extra Practice

For extra practice on using the quadratic formula, try out these worksheets with the answer key included.

Worksheet 1

Worksheet 2

Worksheet 3