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C. Algebra · Algebra: Expressions, Equations and Coding

Algebra: Expressions, Equations and Coding: Unit Review

A1 A2 C1.1 C1.2 C1.3 C1.4 C1.5 C2.1 C2.2 C2.3About 40 minutes
Learning goals
  • I can write, simplify, expand and compare algebraic expressions.
  • I can solve and verify equations, including ones with brackets, fractions and decimals, and rearrange formulas.
  • I can create equations to model and solve real problems.
  • I can read, write and alter Python code that uses variables, functions, loops and inequalities.

Words to know

al-jabr
“Restoring”: adding a subtracted amount to both sides. It gave algebra its name.
like terms
Terms with the same variables and exponents, such as 3x² and −x².
distributive property
a(b + c) = ab + ac.
equivalent expressions
Expressions equal for every value of the variable.
verify
Substitute a solution into both sides of the original equation to check LS = RS.
parameter
A function's input variable, or a number in a model that you can change to describe a similar situation.
while loop
Code that repeats as long as a condition is True.

Let's review!

In this unit you explored where algebra came from, wrote expressions to generalize patterns and situations, tested expressions for equivalence four ways, simplified by collecting like terms, expanding and multiplying binomials, solved and verified many kinds of equations, rearranged formulas, created equations to make decisions, and wrote and altered Python code that does algebra.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

Expressionswrite from words,pictures, patternsSimplify & comparelike terms, distribute,binomials, equivalent?Equationssolve, verify, formulas,model real problemsCodevariables, functions, loops, if: algebra that runsHistory: these ideas grew over thousands of years in many cultures.
How the big ideas of this unit connect. Expressions are simplified and compared, then used in equations; code turns all of it into steps a computer can run.

How to use this review

  1. Do one set of questions at a time, on paper, showing your steps.
  2. Check each answer right away.
  3. If you miss two or more in one set, go back to that lesson and redo its “Your turn” questions.
  4. For coding questions, trace by hand first. Running the code afterwards (if you can) is a great way to check.

The Story of Algebra: Ideas from Many Cultures

1

What does al-jabr mean, and which mathematician's book gave us the word “algebra”?

2

Nine Chapters style: if each person pays 6 coins there are 5 too many; if each pays 5 coins they are 3 short. Find the number of people and the price.

3

Who introduced the equals sign, and in what year?

Variables, Terms and Writing Algebraic Expressions

1

Write an expression for 8 less than 3 times a number.

2

Classify 5x²y − 2x + 1 and state its degree.

3

A hockey tournament charges $150 per team plus $12 per player. Write an expression and find the cost for 15 players.

Generalizing Patterns with Algebraic Expressions

1

Write an expression for 9, 14, 19, 24, ... and find term 30.

2

Hexagons in a row made of toothpicks use 6, 11, 16, ... toothpicks. Write the expression.

3

Write an expression for 0, 3, 8, 15, ...

Are They Equivalent? Comparing Expressions Four Ways

1

Are 3(x − 2) + 6 and 3x equivalent?

2

Give a counterexample to show (x − 2)² is not equivalent to x² − 4.

3

What would the graphs of y = 4x + 1 and y = 4(x + 1) look like? Are the expressions equivalent?

Collecting Like Terms: Adding and Subtracting Polynomials

1

Add (4x² − x + 3) + (−x² + 5x − 8).

2

Subtract (6x − 2) − (4x − 7).

3

Simplify 3ab − 2a + ab + 5a.

The Distributive Property: Multiplying Monomials, Expanding and Common Factors

1

Expand −3x(2x − 5).

2

Expand and simplify 4(x − 1) − 2(x + 3).

3

Factor 9x² + 12x completely.

Multiplying Binomials with Tiles and Area Models

1

Expand (x + 6)(x − 2).

2

Expand (3x − 1)².

3

Expand (x + 9)(x − 9).

Solving Multi-Step Equations and Verifying Solutions

1

Solve 7x − 3 = 4x + 12.

2

Solve 3(x + 2) − 5 = 2(x − 1) + 9 and verify.

3

How many solutions does 5(x − 1) = 5x + 2 have?

Equations with Fractions and Decimals, and Rearranging Formulas

1

Solve x/2 + x/5 = 14.

2

Solve (x − 3)/4 = (x + 1)/6.

3

Rearrange V = lwh for h.

1234567820406080(4, 40)cost ($)classesPlan A: 20 + 5nPlan B: 10n
The two dance-studio plans in the next question set. The lines cross at (4, 40): after 4 classes, Plan A's line is lower, so it is cheaper.

Creating Equations to Model and Solve Problems

1

A dance studio offers Plan A: $20 plus $5 per class, or Plan B: $10 per class. When do they cost the same? (See the graph just above.)

2

A rectangle's length is 4 m less than 3 times its width. Its perimeter is 72 m. Find its dimensions.

3

Three consecutive integers add to 135. Find them.

Coding with Variables, Expressions and Functions

1

Predict: x = 4, then x = 3 * x - 2, then print(x).

2

Predict: print(2 ** 3 + 1).

3

After def g(t): return 5 * t - 1, what does print(g(3) == 14) show?

Decomposing Problems: Code with Loops, Equations and Inequalities

1

Predict: for n in range(1, 5): print(n * n)

2

Savings start at $50 and grow by $20 a week. A while loop runs while savings < 200: adding 20 and counting weeks, then prints weeks and savings. Predict the output.

3

What does solve_both(4, 1, 2, 9) return? (It solves ax + b = cx + d.)

Reading, Predicting and Altering Code

1

Alter the fence code (40 m: area = w * (20 - w), range(1, 20)) for 48 m of fencing. What is the new output?

2

In the inequality loop over x = −5 to 5, the condition 2 * x + 1 > 5 is changed to 2 * x + 1 >= 5. What is printed now?

3

Predict: x = 1, then for i in range(3): x = x * 4, then print(x).

Just for fun

Riddle: I'm named after a mathematician from Baghdad. I'm a clear, step-by-step plan. Every program and every equation-solving method in this unit is one of me. What am I?

An algorithm! (From al-Khwarizmi.)

Check your understanding

Quiz

1.

The word algebra comes from:

2.

Which expression means 5 more than twice a number?

3.

Simplify 5x − 3 + 2x + 8.

4.

Expand −2(x − 6).

5.

Expand (x + 2)(x + 7).

6.

Solve 4x + 3 = x + 18.

7.

Solve x/3 − 1 = 4.

8.

Company A charges $100 plus $20 per visit; Company B charges $40 per visit. When do they cost the same?

9.

What does print(3 * 2 ** 2) show in Python?

10.

Which values does range(0, 3) give?

For parents and teachers

Use the results to decide what to revisit:

  • History questions (al-jabr, Recorde, Nine Chapters problems): revisit The Story of Algebra.
  • Reversed “less than” or missing brackets when writing expressions: revisit Variables, Terms and Writing Algebraic Expressions.
  • Using figure 1 as the constant, or missing non-linear patterns: revisit Generalizing Patterns.
  • Saying expressions are equivalent after one matching value, or not finding a counterexample: revisit Are They Equivalent?.
  • Sign errors when subtracting a polynomial, or x² + x² = x⁴: revisit Collecting Like Terms.
  • Multiplying only the first term, or sign errors with a negative outside the bracket: revisit The Distributive Property.
  • (x + a)² written as x² + a², or forgetting to collect the middle terms: revisit Multiplying Binomials.
  • Mistakes solving, or skipping the LS/RS check: revisit Solving Multi-Step Equations.
  • Not multiplying every term by the LCD, or undoing a formula in the wrong order: revisit Equations with Fractions and Decimals, and Rearranging Formulas.
  • Trouble setting up word problems: revisit Creating Equations, especially the let statement and verifying in context.
  • Confusing = and ==, or writing 3x in code: revisit Coding with Variables, Expressions and Functions.
  • Off-by-one errors with range, or while loops that stop one step early or late: revisit Decomposing Problems and Reading, Predicting and Altering Code.

Praise clear steps and checking, not only correct answers. The next algebra unit, on linear and non-linear relations, uses all of these skills.