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Be sure also to check the Secondary page for general support, and the Math Video series for parents.
Be sure also to check the Secondary page for general support, and the Math Video series for parents.
Algebra & Number: The meanings of, and connections between, each operation extend to powers and polynomials.
Algebra allows us to generalize relationships through abstract thinking.
Algebra & Number: The meanings of, and connections between, each operation extend to powers and polynomials.
Algebra allows us to generalize relationships through abstract thinking.
PRIME FACTORIZATION / POWERS
Visualize to explore mathematical concepts
Share the image on the right of the numbers 1-35 with your learners. (Source: Jo Boaler's Youcubed)
Invite them to look at how each number is represented visually.
"What patterns do you see? Find as many as you can."
Encourage students to express some of these numbers using prime factors and powers.
Ask students where they see these factors and powers in the picture of each number. For example:
"I see a pentagon of 2s (or a pair of pentagons) in 10. I see a pentagon of pentagons (i.e., five 5s or 5-squared) in 25."
"What could 37 look like? What about 36?"
See an "answer key" here.
PRIME FACTORIZATION (Factors of a Number)
Analyze and apply mathematical ideas using reason
Demonstrate strategic thinking about number
Instructions for students:
The numbers 18 and 20 each have 6 factors. What are the factors of each?
What is it about 18 and 20 that makes them have the same number of factors?
Sample response: Both 18 and 20 have two prime factors, one of one prime factor, and the square of the other.
What are some other numbers which have 6 factors?
Find some numbers which have 12 factors.
What is the smallest number that has 12 factors?
Can a number have an odd number of factors? Explain.
POWERS WITH INTEGRAL EXPONENTS
Demonstrate strategic thinking about number
Think creatively and with curiosity when exploring problems
Provide the image on the right to your students. Tell them they represent different ways to express 2^10. Ask them questions such as:
Which of these are most alike? What makes them alike?
Which of these are most different? What makes them different?
What other expressions can you write that are equivalent to 2^10? Challenge yourself to be creative!
MEANING OF BINOMIAL MULTIPLICATION
Represent mathematical ideas in pictorial and symbolic forms
Connect mathematical ideas with each other
Share the image on the right.
"What is the same? What's different?"
"Which two are most alike? Why?"
Encourage students to make as many connections as they can -- pictorially and symbolically, conceptual and procedural.
BINOMIAL MULTIPLICATION
Apply mathematical understanding through problem solving
Share the image on the right.
Ask your students:
How did you solve the problem?
How many different solutions can you find?
Have you found them all? How do you know?
Fill in the boxes to make a true statement.
COMMON FACTORS
Develop and demonstrate mathematical understanding
Explain mathematical ideas
Have your students choose to factor one or both of the following polynomials: 20x² – 15x or 18x⁴y + 30x³y² + 24x²y³
Ask:
Another polynomial can be factored using the same greatest common factor (GCF) as you used above.
What could it be? List several possibilities.What strategy did you use? Explain your thinking.
POLYNOMIAL FACTORING
Develop and demonstrate mathematical understanding
Explain mathematical ideas
Have your students factor both x² + 10x + 24 and x² + 14x + 40. Ask:
What binomial factor do these trinomials have in common?
How would thinking about common factors of 24 & 40 help you find possible common factors of x² + 10x + 24 and x² + 14x + 40?
Present the "product puzzles" (right) which mimics the question above. (The difference is in the table organization.)
Have students choose to complete a few of these puzzles and explain their thinking.
Source: https://donsteward.blogspot.com/2014/12/algebraic-product-puzzles.html. Note: (10) - (15) go beyond the intended scope of FPC10.
Patterns & Relations:
Constant rate of change is an essential attribute of linear relations and has meaning in different representations and contexts.
Representing and analyzing situations allows us to notice and wonder about relationships.
Patterns & Relations:
Constant rate of change is an essential attribute of linear relations and has meaning in different representations and contexts.
Representing and analyzing situations allows us to notice and wonder about relationships.
LINEAR FUNCTIONS
Explain and justify mathematical ideas and decisions
Connect mathematical concepts with each other
Share Graphs 14, 21 and 41 from wodb.ca, and the one on the right.
Ask students "Which One Doesn’t Belong?"
Challenge students to come up with as many reasons as they can why each graph, equation, table is different from all the others.
Note: It’s possible to give a justification for each element in each set being the “odd one out.”
LINEAR FUNCTIONS
Use reasoning to analyze mathematical ideas
Share the following four representations of a linear relation.
Ask your learners:
What stays the same and what changes in this relationship? How does this appear in each representation? (or, more formally, Where do you see the constant term (y-intercept) and the rate of change (slope) in each representation?)
What effect would increasing the starting cost have on each representation? What about decreasing the price per topping?
Sometimes ordering a pizza from Christopher's Pizzeria is less expensive than ordering from Marco's Pizzeria. Sometimes it's more expensive. What could the equation for Christopher's Pizzeria be?
LINEAR FUNCTIONS
Explore and analyze mathematical ideas using reason
Explain and justify mathematical ideas and decisions
Share the list of properties (or "constraints') of linear functions below.
Challenge students to design a set of linear functions that, as a whole, satisfy all of these properties at least once.
Which properties can be satisfied at the same time by a single linear function? Which cannot?
Can you satisfy all of these properties using fewer functions? How do you know?
A handout, as well as a more detailed description of the task, can be found on Nat Banting's blog.
LINEAR FUNCTIONS WITH INDIGENOUS CONNECTIONS
Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities
Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts
Please read this blog post to learn about respectfully connecting cultural practices and perspectives of First Peoples to linear relations.
Cultural context as well as a detailed description of the math task is provided there.
LINEAR RELATIONS WITH INDIGENOUS CONNECTIONS
Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities
Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts
Please read this blog post to learn about respectfully connecting cultural practices and perspectives of First Peoples to linear relations.
Cultural context as well as a detailed description of the math task is provided there.
Phyllis Atkins at blessing ceremony for We Are All Connected to This Land
LINEAR SYSTEMS (Introduction)
Explore mathematical ideas using reasoning
Develop mathematical understanding through problem solving
Pose one or more logic puzzles (below & right).
Challenge to solve these puzzles using informal methods.
Ask: What strategy did you use? Can you solve the puzzle a different way?
Students may reason to a solution in ways that are similar to the algebraic methods of elimination and substitution. For example,
"In Row 1, 2 smiles and one frown are worth $40. In Column 2, 2 smiles and 2 frowns are worth $50. That means one extra frown is worth $10 and so..."
"In Round 3, I "tagged out" Ivan and replaced him with 2 grandmas and one acrobat (from Round 2) and so..."
Other strategies involving playing with numbers. This shows an understanding of what it means for a set of values to be considered a solution to a system.
"I knew that 2 smiles and 1 frown added to $40 so I first tried $10 for each smile and $20 for a frown and then..."
"I gave a value of 100 (or 1 or 20) to each side in Round 1. That makes acrobats worth 20 (or 1/5 or 4) each and grandmas worth 25 (or 1/4 or 5). So..."
Note: It is not intended that students set up and manipulate algebraic equations when solving these puzzles. Rather, in a follow-up lesson, you will build upon their mathematical thinking, connecting to more symbolic expressions of these same ideas.
Source: Van de Walle et al.
Source: NCTM Figure This!
Source: Marilyn Burns
CONNECTING GRAPHS AND CONTEXTS
Think with curiosity and wonder when exploring problems
Connect mathematical concepts with other areas and interests
Establish a "Graph of the Week" routine.
Share a graph that is both contextually and mathematically interesting with your students.
Turner's Graph of the Week is an excellent resource for this and the source of our choice: 3/23 The Netflix Generation.
Ask "What do you notice? What do you wonder? What's going on in this graph? What story does this graph tell?"
Additional questions prompts appear on the handout itself.
CONNECTING GRAPHS AND CONTEXTS
Develop, demonstrate, and apply mathematical understanding through story
Connect mathematical concepts with other areas
Have students choose one of the videos on www.graphingstories.com/
Have them graph the story of their chosen video.
DOMAIN AND RANGE
Demonstrate mathematical understanding
Provide your students with the graphs on the right.
Ask them to determine the domain and range of each.
Challenge students to demonstrate a deeper understanding by "going backwards":
Can you draw a graph that has:
domain -3 ≤ x < 2 and range -4 ≤ y < 3?
(Aside: Graphs 4, 18, and 32 on wodb.ca evoke domain and range as does this one from Michelle Rinehart.)
FUNCTIONS (Defining)
Use reasoning to explore and apply mathematical ideas
Connect different representations of a math concept
Provide your students with the image on the right. Ask them to explore the image and answer the question, "What is a function?". Explain their reasoning.
Ask them to apply their definition of function to the tables and graphs below.
Which are functions?
How do you know?
FUNCTION NOTATION
Use reasoning to explore and apply mathematical ideas
Connect different representations of a math concept
The purpose of this task for students to figure out the meaning of function notation rather then being provided with the meaning.
Provide your students with the image on the right.
Tell them:
There is a special kind of notation used for functions. It is called function notation. In the graph of a function provided, the following statements are true:
f(1) = 2
f(5) = -1
Ask, based upon the given information:
What do you think function notation means?
What is f(0)? f(3)?
If f(n) = 1, what could n be?
Trigonometry: Proportional reasoning helps us to solve indirect measurement problems.
Trigonometry: Proportional reasoning helps us to solve indirect measurement problems.
RATIOS
Analyze and apply mathematical ideas through reason
Think with curiosity and wonder when exploring problems
Provide the image on the right and ask students to observe the two triangle. Ask:
How are the two triangles related?
How does the ratio BC:AB compare to EF:DE?
Explore other ratios within each triangle.
What do you notice?
TRIGONOMETRIC RATIOS
Analyze and apply mathematical ideas through reason
Give students a right triangle with all 3 side lengths shown. An example is on the right.
Ask questions such as:
Determine sinC and tanA.
Which trigonometric ratio is 20/29? Is there more than one?
Which trigonometric ratio is the largest value?
TRIGONOMETRY: Determining Unknown Lengths
Model with mathematics in situational contexts
Reflect on mathematical thinking
Give students one or both of the images shown.
Ask them, "How tall is the tree?""How did the first problem help you with the second problem?"
TRIGONOMETRY: Determining Unknown Angles
Model with mathematics in situational contexts
Connect mathematics to personal interests
Provide the image on the right to your students.
Explain that the angle of an inclination for a ladder that is considered to be safe is close to 75° (say between 71° and 79°).
Ask:
Which of these ladders are at a safe angle?
Why do you think they have what's called a 4-to-1 Rule for using a ladder?
Financial Literacy: Gross & Net Pay
Financial Literacy: Gross & Net Pay
GROSS PAY
Explain and justify mathematical decisions
Connect mathematical concepts to personal interests
Task 1:
Would you rather…
(A) Earn an annual salary of $30 000, or
(B) Work full-time earning $15/hour?
Why? Justify your decision.
What assumptions are you making?
(Adapted from www.wouldyourathermath.com.)
Task 2:
Would you rather…
(A) Be paid $1000 per month plus 3% commission on sales, or
(B) Earn 5% commission on sales but not be paid a base salary?
Why? Defend your decision.
What assumptions are you making? When would the other option be a better choice?
Please see the related WYR…? Numeracy Task on the Secondary Homepage.
GROSS PAY WITH OVERTIME
Apply mathematical understanding through problem solving
Connect mathematical concepts to personal interests
Provide a selection of problems involving gross pay with overtime. Some ideas are below. For all problems, assume the BC standard of overtime which is time-and-a-half for hours worked over 40 hours in a week.
What is the gross pay if someone works 48 hours one week and makes $16/hr?
What is the gross pay each week if someone works 36 hours one week and 45 hours the next week. They make $17/hr.
Who had the higher gross pay this week, someone who worked 40 hours at $25/hr or 46 hours at $21/hr? Which job would you rather have? Why?
Someone makes $22/hr. How many hours did they work one week if their gross pay was $1078?
What is someone's hourly wage if they worked 50 hours and their gross pay was $1017.50?
Write a scenario involving gross pay with overtime (ie. hourly wage and number of hours worked) that results in $1000 gross pay for one week.
NET PAY
Apply mathematical understanding through problem solving
Connect mathematical concepts to personal interests
Tell students to use the chart on the right to calculate the gross pay and net pay for one week if one earns:
$26/hr and works 36 hours that week
$500/week and 6% commission on $12 000 sales that week
If the second person worked 40 hours that week, how much did they earn per hour on average?
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