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C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling: Unit Review

A1 C1.1 C1.2 C1.3 C1.4 C2.1 C2.2 C2.3 C2.4 C3.1 C3.2 C4About 40 minutes
Learning goals
  • I can identify, compare, create and extend patterns with rational numbers, and use pattern rules to solve for unknowns.
  • I can add and subtract monomials, add binomials, and evaluate expressions with rational numbers.
  • I can solve equations with integers and decimals, and solve and graph inequalities with integers.
  • I can write and alter code that analyses data, and use the modelling process to make a recommendation.

Words to know

linear pattern
A pattern with a constant difference between terms.
initial value
The value at term 0.
rational number
A number that can be written as one integer divided by another.
zero pair
A positive and a matching negative tile: together they make 0.
binomial
An expression with two unlike terms, like 3x − 2.
reverse the inequality
Turn the sign around when multiplying or dividing both sides by a negative.
FOR EACH loop
A loop that runs once for every item in a list.
assumption
Something treated as true so a model can be built.

Let's review!

In this unit you checked whether patterns are linear, compared linear patterns by their rates and initial values, created and translated patterns with rational numbers, found pattern rules and unknown terms, and used patterns to explain how rational numbers are related. You added and subtracted monomials and added binomials with algebra tiles, evaluated expressions with negatives and fractions, solved equations (including variables on both sides) and inequalities with integers, wrote and altered code that analyses data, and modelled a maple syrup project.

This review has a few questions from every lesson. Try each one, then press Show answer to check. If a part feels tricky, go back to that lesson for more practice.

Zero pairs make 0+−= 0+−x + (−x) = 0 and 1 + (−1) = 0Divide by a negative? Flip!−2x > 6÷ (−2)x < −3Initial value = term 07, 10, 13, ... (rate 3)go back one jump: 7 − 3 = 4t = 3n + 4Need "enough"? Round UPtaps: t ≥ 7.067 taps is not quite enoughso use 8 taps
Four tools from this unit: zero pairs, flipping the sign when you divide an inequality by a negative, finding the initial value at term 0, and rounding up when you need enough.
find 3 or moredifferencesall thesame?yesnoLINEARrate = that differenceNON-LINEARdifferences changeThen check the sign: + means growing, − means shrinking.
The difference test in one picture: constant differences mean linear; changing differences mean non-linear.

Patterns in Real Life: Linear or Not?

1

Is −9, −6.5, −4, −1.5 linear? Growing or shrinking?

2

Is 3, 6, 12, 24 linear?

3

A repeating pattern has the core (snow, snow, rain). What is the 40th term?

Comparing Linear Patterns: Rates, Initial Values and Graphs

1

Write an equation for −1, 1.5, 4, 6.5, ... (term 1 is −1).

2

Plan A: $50 plus $5 per month. Plan B: $15 per month. When do they cost the same?

3

Which line is steeper: t = −6n + 2 or t = 4n − 10?

Creating and Translating Patterns with Rational Numbers

1

Write the first 4 terms of t = (3/4)n − 1.

2

Write an equation for the table n: 0, 1, 2, 3 and t: 1.2, 0.7, 0.2, −0.3.

3

Which point is on y = 3x − 2: (−1, −5) or (−1, −1)?

Pattern Rules: Predict, Find Missing Terms and Solve for Unknowns

1

Find the missing terms: −2, __, __, 5.5 (linear).

2

Which term of t = 5n − 12 has the value 68?

3

Is 40 a term of t = 3n + 2? Justify.

Patterns That Show How Rational Numbers Are Related

1

Continue: (−6) × 1 = −6, (−6) × 0 = 0, (−6) × (−1) = ?

2

Use the elevenths pattern to write 4/11 as a decimal.

3

What is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64?

Adding and Subtracting Monomials, and Adding Binomials, with Algebra Tiles

1

Simplify −6x + 2x − (−3x).

2

Simplify (3x − 8) + (−5x + 2).

3

Simplify (−x + 4) + (x − 4).

Evaluating Algebraic Expressions with Rational Numbers

1

Evaluate −2a + 3b when a = −4 and b = −1.5.

2

Evaluate x² − 2x when x = −3.

3

Change −20°C to °F with F = 1.8C + 32.

Solving Equations with Integers and Decimals

1

Solve 5x + 12 = −8.

2

Solve −3y + 7y − 2 = 14.

3

Solve −0.6k + 1.5 = 4.5.

Equations with Variables on Both Sides: When Are Two Plans Equal?

1

Solve 6x − 5 = 2x + 7.

2

Solve x + 9 = 4x − 3.

3

Solve 2.5n − 1 = n − 7.

Solving and Graphing Inequalities with Integers

1

Solve x − 6 ≥ −8 and describe the graph.

2

Solve −4x > 20.

3

Solve −2n + 3 ≤ 11.

Coding to Analyse Data and Make Decisions

1

SET total = 0 / FOR EACH v IN [−4, 7, −1, 2] / ADD v to total / SAY total. What is said?

2

What is the mean of [−4, 7, −1, 2]?

3

Code to find the highest value starts with SET highest = 0 and runs on [−5, −2, −9]. What goes wrong?

Reading and Altering Data Code: Outcomes and Efficiency

1

In the eco-club code, the goal changes to IF mean ≥ 12. With the original data (mean 13), what is the decision?

2

The list gets a 7th week of 13 kg. What is the new mean?

3

Why is one loop that finds both the total and the highest value more efficient than two loops?

Math Modelling: Planning a Maple Syrup Project

1

How much sap makes syrup for 80 people at 30 mL each (40 to 1)?

2

At 8.5 L of sap per day, how many days does it take to collect 96 L?

3

Why is checking the model against real data important?

Just for fun

Riddle: I'm the value of every zero pair, the initial value of t = 3n, and the one number the mean code must never divide by. What am I?

Zero!

How to use this review

  • Work through one section at a time and write your steps, not just the answers.
  • Put a check mark beside each question you got right on your own. Put a star beside any you needed help with.
  • If you starred two or more questions in a section, go back to that lesson and redo its "More practice" questions.
  • For every equation and inequality, verify your answer by substituting it back in.
  • Finish with the quiz. For every quiz mistake, read the explanation and find the lesson it came from.
Check your understanding

Quiz

1.

Which pattern is linear?

2.

What is the initial value of 4.5, 6, 7.5, 9, ... (term 1 is 4.5)?

3.

Which term of t = 3n − 8 has the value 40?

4.

Using a pattern, (−7) × (−2) = ?

5.

Simplify (4x − 3) + (−6x + 5).

6.

Evaluate 3x − y when x = −2 and y = −5.

7.

Solve −4x + 9 = 25.

8.

Solve 5n + 2 = 2n − 10.

9.

Solve −3x ≥ 12.

10.

A code runs SET count = 0 / FOR EACH t IN [−2, 5, −8, 0, 3] / IF t < 0 THEN ADD 1 to count. What is count?

For parents and teachers

Use mistakes to choose what to revisit. Each review section matches one lesson.

  • Calls a pattern linear after checking only one difference, or thinks a pattern with negatives can't be growing: revisit Patterns in Real Life: Linear or Not?.
  • Uses term 1 as the initial value, or can't find where two plans are equal: revisit Comparing Linear Patterns: Rates, Initial Values and Graphs.
  • Struggles to move between a table, graph and equation with fractions or decimals: revisit Creating and Translating Patterns with Rational Numbers.
  • Counts blanks instead of jumps, or accepts a non-whole term number: revisit Pattern Rules: Predict, Find Missing Terms and Solve for Unknowns.
  • Quotes "two negatives make a positive" without being able to explain it, or adds denominators: revisit Patterns That Show How Rational Numbers Are Related.
  • Combines unlike terms, or makes sign errors with constants: revisit Adding and Subtracting Monomials, and Adding Binomials, with Algebra Tiles. Use the paper tiles.
  • Drops brackets when substituting negatives, especially with exponents: revisit Evaluating Algebraic Expressions with Rational Numbers.
  • Divides by the wrong sign, or changes only one side: revisit Solving Equations with Integers and Decimals.
  • "Moves" a variable term without changing its sign: revisit Equations with Variables on Both Sides: When Are Two Plans Equal?. Try the envelope game.
  • Forgets to reverse the sign when dividing by a negative, or mixes up open and closed circles: revisit Solving and Graphing Inequalities with Integers.
  • Starts the highest value at 0, or loses track of a running total: revisit Coding to Analyse Data and Make Decisions.
  • Can't predict how changing < to ≤ affects a count, or thinks shorter code is always better: revisit Reading and Altering Data Code: Outcomes and Efficiency.
  • Rounds down the number of taps or days, or treats assumptions as facts: revisit Math Modelling: Planning a Maple Syrup Project.

A few mistakes are normal. Praise the strategy your child explains and the habit of verifying, not only the right answer.