Asking Better Questions: Algebra - Marian Small · Asking Better Questions: Algebra ©Marian Small 2019. Today •What is the difference between knowledge and understandingin math

Asking Better Questions:Algebra

©Marian Small 2019

Today

• What is the difference between knowledge and understanding in math Gr 7 - 9 • We will see lots of examples.• We will work together to create understanding questions.

Today

• What is the difference between applicaton and thinking in Gr 7 – 9 math?• We will see lots of examples.• We will work together to create understanding questions.

What is the difference?

• Knowledge: What is the general term of the pattern 7, 10, 13, 6,….?• Understanding: Draw a picture to show why the general term of this

pattern is of the form 3n + 4.

Maybe

To build understanding

• I ask for a picture that focuses on meaning and not on just how to get the answer.

How are they different?

• Knowledge: What is the general term of 5, 8, 11, 14,…?• Understanding: A linear pattern of integers includes the terms -6 and

18. What could the pattern rule be?

Maybe

• -6, -2, 2, 6, 10, 14, 18,…• -6, 0, 6, 12, 18,…• -10, -9, -8, -7, ….• -42, -30, -18, -6, 6, 18,….


• I focus students on seeing that if an integer linear pattern includes x and y, the growth must be a factor of y – x.

What’s the difference?

• Knowledge: Create a table of values to show the number of times your heart beats for different numbers of minutes and graph it.• Understanding Describe a situation where the second column of the

table of values would show only multiples of the first number in that column.


• We focus on what these sorts of constant rate situations are all about rather than the values in a particular one.

What’s the difference?

• Knowledge: What is the 10th term of each: 5, 10, 15, 20,…? 5, 10, 20, 40,….?• Understanding: WITHOUT creating the graphs, predict how the

graphs that show each of the patterns 5, 10, 15, 20.. and 5, 10, 20, 40,… will be different?


• Does it make sense to you why I would ask them not to graph it first?


• Knowledge: Evaluate 3x – 2 for x = 10, x = ½ and x = 2.3.• Understanding: An algebraic expression increases by 4 when x

increases by 1. What could it be?• OR

Maybe

• 4x+12• 4x+8• 10 + 4x


• Knowledge: Evaluate 3x – 2 for x = 10, x = ½ and x = 2.3.• Understanding: An algebraic expression decreases by 10 when x

increases by 2. What could it be?


• I don’t hope the kids see how the change relates to the coefficient of x or just tell them it does. They show me they see it.


• Knowledge: Evaluate 4x – 7 for …..• Understanding: An algebraic expression results in integer values

whenever you evaluate it for x = []/2 for any value of []. What could the expression be?

Maybe

• 6x + 3• 10x - 5• 32x + 8


• By being more “general”, it gets to the bigger picture.


• Knowledge: Translate what this says into “English”: 10 – 2n.• Understanding: When you describe an algebraic expression using

everyday words, you include the phrase “how much less”. What might the expression be?

So when I teach

• My focus is on what algebraic expressions really mean.


• Knowledge: Simplify 3x -4 + (2x + 5).• Understanding: You evaluate two different algebraic expressions and

get the same result no matter what value of x you use. What could the expressions be?• OR

Maybe

• 3x and 2x + x• 7(2 - x) and 14 - 7x• X2 -3x + 2 and x(x-3) + 2


• I am defining what equivalent expressions are and leading to the notion that simplification of expressions is about getting a simpler, equivalent form.• I started with a simpler form, though, to make the point.


• Knowledge: Simplify 3x -4 + (2x + 5).• Understanding: You add two binomials and end up with a monomial.

What could they have been?

Maybe

• (4x + 7) + (7x-7)• (2x + 8) + (4 - 2x)


• I am focusing on a generalization.


• Knowledge: What algebraic expression do these tiles model?

• Understanding: What algebraic expressions could you show with exactly 3 algebra tiles?


• I want students to see that the expression ax + b uses a+ b tiles (irrespective of colour)


• Knowledge: Where, in the pattern with the rule -2n + 5 is the value 47?• Understanding: In what situations might you use the equation -2n+5 =

47?

So when I teach

• My focus is on what situations this math is all about.


• Knowledge: Solve 3x – 2 = 5x + 4.• Understanding: You try to solve an equation, but nothing works. What

might the equation have been?


• I am focusing on the difference between variables as unknowns and what I would call algebraic identities or algebraic inconsistencies.• Compare, for example, the equation • 2x = x + 8 vs• 2x = x + x vs• 2x +1 = 2x + 2


• Knowledge: Solve 9x – 2 = -58.• Understanding: WITHOUT SOLVING or writing anything down,

estimate the solution.


• I want students to use estimation and to see that you use a related simpler equation to do that.


• Knowledge: Solve 5x – 3 = 12x + 8?• Understanding: I solved an equation by subtracting from both sides,

then adding to both sides and then dividing by 4. What might the equation have been?

Maybe

• 6x - 7 = 5 + 2x• 5x - 7 = 10 + x• 3x + 7 = 10 – x


• I am generalizing the process we use to solve equations and helping them notice that we added if there was a subtract somewhere, we subtracted if there was an odd somewhere and we divided if there was a multiplication somewhere--- lots of opposite operation work.


• Knowledge: Solve 5x – 3 = 12x + 8?• Understanding: What could you graph to help you solve this

equation?


• I am focusing on the relationship between graphing and solving equations.


• Knowledge: Solve 4x – 3 = -3x + 32• Understanding: How could you model 4x – 3 = -3x + 32 with algebra

tiles to help solve it?

47


• I am focused on what the equation says and how opposite operations look concretely.


• Knowledge: Solve -3x – 4 = x• Understanding: Model an equation using exactly 8 algebra tiles and

then solve it using the tiles.

Maybe

Maybe

Maybe


• I am focusing on the fact that if 8 tiles are used to solve ax + b = cx + d, then the absolute values of a, b, c and d add to 8


• Knowledge: Solve -3x – 4 = x• Understanding: The solution to an equation is x = -1. What could the

equation be if there are two terms on each side?

Maybe

• 4x - 7 = 9x – 2• 17x – 4 = -10x - 31


• I am focused on the fact that linear equations with the same solution are equivalent.


• Knowledge: Solve 3x + 4 = 2x + 9• Understanding: How does this picture show that 3x + 4 = 2x + 9?

How could it help you solve the equation?

x x x

x x 9

4


• Knowledge: Solve 5x – 3 = 3x + 17?• Understanding: How does this picture show that 5x – 3 = 3x + 9? How

could it help you solve the equation?

x x x x x

x x x 17

3


• Knowledge: Solve 100 x + 6 = 87x + 2• Understanding: WITHOUT SOLVING, tell why the solution to 100 x + 6

= 87 x + 2 HAS TO be negative.

So when I am teaching

• I am focusing on the notion of an equation as a balance, the value on the left has to match the value on the right.• If the left is 5x + 4, and the solution is positive, would the right be 4x +

more than 4 or 4x + less than 4?


• Knowledge: Evaluate 3x – 2y – 4 when x = ½ and y = –3. • Understanding: []x – []y – 4 is negative when x = 1 and y = –2. What

could the missing values be?

Maybe

• []x – ∆y – 4 is negative when x = 1 and y = –2, so• [] +2∆ - 4 is negative• Maybe [] = -100 and ∆ = 1• Maybe [] = -10 and ∆ = 6 etc.


• Knowledge: What is polynomial 1 + polynomial 2?• Understanding: You add two polynomials and the result is 3x2 – 2x +

[]. What could the polynomials have been?


• Knowledge: What is 4x2 – 3 - (x2 -3x + 5)?• Understanding: How could you figure out 4x2 – 3 - (x2 -3x + 5) by

starting with the model of x2 -3x + 5?


• I think of subtraction as how much more do I need?• Start: x2 -3x + 5• End: 4x2 – 3

To build on understanding

• I am focused on how subtractions can be solved by adding up.


• Knowledge: What is (x2 – 2x – 1) – (–3x +2)?• Understanding: When you subtract a binomial from a trinomial, can

your answer be a binomial? A trinomial? Explain


• My focus is not on what the answer is but the structure of the answer and when we “get rid” of term types.


• Knowledge: Factor 3x2 -9x.• Understanding: An algebraic expression is factored by “pulling out

3x”. What might the expression have been and how would it be factored?


• Working backwards is often a test on understanding.

How are these different?

• Knowledge: You know that A = (b1+b2)h/2 for a trapezoid. If you know that A = 500, b1 = 6 +b2= 8, what is h?• Understanding: One way to write the formula for the area of a

trapezoid is like this: A = (b1+b2)h/2 . Write the same formula at least several other ways. OR


• Knowledge: You know that A = (b1+b2)h/2 for a trapezoid. If you know that A = 500, b1 = 6 +b2= 8, what is h?• Understanding: If you know the values of A, b1 and b2, when would

you just plug the values in to figure out the volume for h and when would you rewrite the formula in the form h = 2A÷ (b1+b2)


• I focus on when you might realistically rearrange formulas.


• Knowledge: Is this a partial or direct variation? Dina earns $15 per hour but an additional $25 each day.• Understanding: Describe a linear relation where if x triples, so does y.


• Knowledge: What is the slope of the line 2x – 3y = 12?• Understanding: The point (5,3) is on a line with slope –2. Name three

other points on that line.


• I focus on slope meaning rise/run and that a negative slope either means over and down or back and up.


• Knowledge: A line is perpendicular to y = -2x -1 at the point (4, -9). What is its equation?• Understanding: Two lines are both perpendicular to y = –2x – 1. Could

they both go through (5, -1)? Explain.


• Knowledge: Graph y = 2x – 7.• Understanding: A line goes through three quadrants.• What could its equation be?• What could it not be?

Some strategies you might use

• Show… without getting the answer (or solving)• Focus on meaning, sometimes by drawing a picture to show why• Build toward a generalization• Build a sense of why we do something

You try

• Turn one of these Knowledge questions and 2 of your own related to algebra into Understanding ones.

Maybe

• What is the general term of 5, 9, 13, 17, 21,…?• Evaluate 5x – y when x = -0.5 and y = 1.3• Solve 3x – 8 = 0.5x – 2• Simplify (2x – 3) – x(5 – 2x).

Now

• Now let’s look at the differences between application and thinking.


• Application: Dan decided to take his Mom’s challenge of doing 2 chores on Monday, 5 on Tuesday, 8 on Wednesday, etc. for 10 days, doing 3 extra chores each day. How many chores will he have completed?• Thinking: Create a problem that is solved by adding the first 10 terms

of a pattern.


• Application: The school organized a telephone tree. The first caller phones three parents. Those each call three more, etc. How many will receive a call on the third round?• Thinking: If there are 363 parents to be called, how many rounds of

calls will be needed?


• Application: You bought 4 kg of steak plus a jar of sauce. You spend $23.19 in total, with $3.79 for the sauce. How much did 1 kg of steak cost?• Thinking: Complete this magic square:


• Application: It costs $3 to sharpen figure skates and $2.50 to sharpen hockey skates. If he earned $240 and sharpened about the same number of each kind of skates, how many of each?• Thinking: The perimeter of a rectangle is 3 times its length. What is

the relationship between the width and the length?


• Application: After driving for 2 hours, someone was 134 km from home; after 4.5 hours, he was 301.5 km from home. If he were driving at a constant speed, what would it be?• Thinking: You drive 200 km at a certain speed. Going back, you drive

the same distance 10 minutes faster. How fast might you have been driving each way?


• Application: It takes Ann 4 hours to paint a room and Samantha 3 hours to paint the same room. How long would it take them to do it together?• Thinking: It takes 2 kids a total of 1 7/8 hours to paint a room. If each

painted at a constant rate, but different from the other, how long would it take each of them to do it alone?


• Application: Two lines through (4, -2) are perpendicular. What could their equations be if one goes through (-2,5). • Thinking: A right triangle has one side on the line y = -2. What could

be the lines the other two sides lie on?


• Application: You have $150 in $5 bills and toonies. If you have 20 toonies, how many $5 bills do you have?• Thinking: You buy 35 of one kind of item and twice as many of

another and you spend $350. How much could each item have cost if the prices are whole numbers?


• Application: You save an extra $15 if the discount at a store goes from 15% to 20%. What did the item originally cost?• Thinking: You save 10% on one item and 30% on another. The total

savings was the same as if you saved 25% on the first item. How are the item prices related?


• Application: A drink and a candy bar costs $8. If the candy bar costs $2 more, how much does each cost?• Thinking: 2 drinks and 3 candy bars cost $1 less than 4 drinks and 2

candy bars. How much might each cost?

You try

• Start with an application question and change it to thinking.• If you wish, you might get some ideas from Desmos or three- act

tasks.

Asking Better Questions: Algebra - Marian Small · Asking Better Questions: Algebra ©Marian Small 2019. Today •What is the difference between knowledge and understandingin math

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