Multiple Strategies
Quadratic word problems are my favourite kind of problem because there are so many strategies to solve them. Here’s an example of one we solved today.
It’s a question about when a farmer should harvest her potato crops to maximize revenue. There are initially 100 bushels that can be sold for $2/bushel. For each week she waits the price per bushel drops by $0.10, and the number of bushels grows by 100.
My students were unsure of how to start the analysis, so we looked at a strategy of making a table to help us organize our thoughts and generalize patterns. Some students really like making a table as a solution method. It’s time consuming, but would work eventually. We talked about how to make the solution more efficient by making an equation for revenue using x as the number of weeks that the farmer waits.
Once we modelled the situation with an equation we split into groups to try to solve for the number of weeks x to wait to have a maximum revenue.
some groups decided to multiply the binomials to start, and then did the quadratic formula to solve for the x intercepts. Most got bogged down after doing all this math and forgot that they were looking for the vertex (which falls always between the 2 x intercepts). Sometimes our solutions are looking for x intercepts, some for the vertex, we’re looking at interpreting word problems correctly and looking for words like maximum, maximize, minimum, minimize to hint at when we need to find the vertex.
Once we noticed (by looking around the room) that some groups concluded that the farmer should wait 20 weeks and others got a different, smaller, answer I prompted them to substitute in their weeks number for x and to calculate the revenue. When groups noticed that when they subbed in 20 the revenue was zero, they realized they needed to rethink some things.
Some groups noticed we had started with a quadratic in factored form, and that we could solve for the x intercepts by looking at the cases that give revenue=0
Prompting the groups to do a little sketch was helpful for them to start to find the vertex.
Groups around the room started to realize that they could get to the x intercepts in 2 ways, by solving for what makes the brackets=0 from factored form, or from the quadratic formula. Both methods require a quick calculation to find the axis of symmetry.
While these methods were percolating and spreading around the room, another group headed down the road of completing the square to locate the vertex (before getting side tracked).
While consolidating we looked at how this strategy also would work nicely. As long as we correctly managed the negative sign at the start.
In this example we can factor out -10 and the numbers are still nice to work with, so we factored -10 then completed the remaining single square, and then multiplied the -10 back in the end. This method directly finds the vertex, which allows us to find the number of weeks the farmer should wait AND the maximum revenue she’d earn.
We had a nice chance to debrief and talk about choosing efficient strategies to solve. Based on what we are given and what we are asked for it might make more sense to use one method over another, so it’s great to have many strategies in your back pocket.
Parallelepiped
Today we explored applications of the cross product, including in areas of triangles, parallelograms and volumes of parallelepipeds.
Here’s a neat desmos 3D graph that I made to demonstrate a parallelepiped. I like that you can view the graph from various perspectives, and rotate it to get a sense of the 3D nature of the form.
It’s amazing how far technology has come!
Vector Fun
Grade 12s are getting into some fun vectors problems these days. We have learned how to do the dot product and the cross product now
We’ve now seen how to calculate angles between vectors, and we were able to show that the magnitude of the cross product of a and b is equivalent to the magnitudes of a and b times sine of the angle between them.
it’s fun to see people proud of their work!
We had a fun time figuring out how to use our hands to do the right hand rule today as well
We’re keeping our motivation into the last month of school!
keep it up!
Productive Struggle
In grade 10 today we were working on solving quadratic word problems. We also were working on thriving through productive struggle.
Here’s an example of one of the questions. We know the sidewalk area equals the garden area, and we’re solving for the value of x, the uniform width of the sidewalk.
Groups worked together at the walls. We reflected a bit about how we all know each other’s names and we can all work together reasonably well. It’s been a semester of growth in teamwork!
Many groups used area model approaches to divide the sidewalk into sections. There was some struggle about making equations for area vs perimeter. We worked through that struggle for quite a while! Groups helped groups, we tried several approaches.
This group landed on an approach where they found outside dimensions of (42+2x) and (20+2x). They knew they needed to subtract the garden area, and then that equals the same as the garden area. Once we get to a trinomial=0 we can sig the quadratic formula and solve.
This group kept with the area model, and had the 4 corner x squares, and added 2 rectangles that were 20x, and 2 rectangles that were 42x. The total would be 840. This led to the same trinomial in the end.
We did a few more problems after this one. It’s important to practice a lot of different types of problems so that we get confidence in thinking through the process.
Math Buddies
Today in grade 12 we invited the grade 11s from across the hall to join us to solve a problem. We worked together in random groups of 3, decided by drawing playing cards. The question we were working on was determining how many ways there were to “win” a game of 3D tic tac toe.
The representations used were pretty neat to see. We have been working on 3D vectors lately, and talking about the x,y,z axis and various planes, so grade 12s were jumping in to represent in pictures, and have good conversations with grade 11s to understand the problem and come up with a plan to represent their thinking.
Having blocks to hold and manipulate helped ground the conversations, and were useful when justifying conclusions. Groups saw and organized the “wins” in several different ways, which was neat to see. Some groups after doing a 3x3x3 case, were asked about a 4x4x4 or 5x5x5, and one group dove in and tried to model a general case of a cube of length w.
After coming up with models, they were asked to graph them, look for features of the graph, explore the domain, and talk about what set of numbers made sense for this task. One group even wrote a piecewise function as a model.
Grade 12s were great collaborators, bringing some additional knowledge from advanced functions, and vectors, while grade 11s had some knowledge about functions, domain and range at the top of their minds. It was a successful collaboration and the room was buzzing for the last period on a Friday before a long weekend. Way to end the week on a high note. Thanks to all for participating. Thanks to Al Overwijk for the inspiration of this problem.
Completing the Square
For the last 2 days I’ve been working with a grade 10 class on completing the square. We’ve used algebra tiles and visual representations to make sense of the algebra that is required, and we’ve made connections between vertex form and graphs of parabolas.
Day 1 we worked on writing expressions in standard form and vertex form
We worked on building squares, and writing them as perfect squares, and including any extra tiles as “extras” outside the perfect square. We know that we can put red unit tiles in the bottom right corner because they “complete” the square, but we can’t ever put blue ones there because (+)(+)=(+) and (-)(-)=(+), ans squares have identical sides.
Sometimes when we don’t have enough to complete the square, we need to add more unit tiles. We have to do this carefully though, and add zero pairs. The positive of each pair can fill the square, and the negative of each pair will be outside.
If there are only negative unit tiles they can’t go in the square, so we fill the square completely with positive tiles, and have the negative part of the zero pair join with the negative unit tiles.
Here we complete the squareS. This was my big algebra tile breakthrough, when I understood how to complete the square with a leading coefficient. We need to create identical squares. We do the same steps. It’s beautiful!
Students got really good at completing the square(s) and stating the vertex and direction of opening, and using their logic to see if there would be x intercepts or not.
Day 2
we did some review of day 1 material with students in random groups at the whiteboards around the room, and then I slid in a challenge to cause some productive struggle. These numbers don’t work out nicely.
students had excellent conversations about how to use all the x tiles and create squares. Once they realized that x tiles could be “cut” we had some progress! This was not an easy question, but wrestling with it helped us sew if the students were understanding what they were doing, or if they were following a rote procedure.
Next we looked at completing the square if the leading coefficient is negative. To do so, we common factored a -1 from each term, held that aside, completed the square of what remained, then multiplied the -1 back into the question.
For fun, we kept going, and explores how to calculate the x intercepts when we have the equation in vertex form. We set y=0 and used opposite operations to isolate x.
we even saw what happens when we have no x intercepts when we use this method. We’d end up trying to take the square root of a negative, which we know doesn’t work, so there are no x intercepts. It is fun to learn how to interpret situations where the math says “no”.
it was a busy 2 days with lots of good learning and participation. Many thanks for letting me join and work with the class!
Locker problem
Today our friends from across the hall came to join us and we worked in groups on solving a problem about lockers.
“There are 100 lockers, and 100 people. The first person walks by and opens all the lockers. Person 2 walks by and closes every 2nd locker (2,4,6,8…). Person 3 walks by and changes the state of each 3rd locker (if it’s open they close it, if it’s closed they open it). This continues until all 100 people have walked by. Which lockers remain open?”
Groups modelled the problem with many strategies: a drawing
a list
Some groups used smaller sets of lockers, and watched for patterns that could extend to the larger set.
It was lots of fun to see the different solution methods help groups arrive at an answer.
It was good to get our classes talking and interacting and solving together. Groups were explaining their thinking, and working hard justifying their conclusions to convince their peers.
Quadratic Formula
In grade 10 we’ve been working on parabolas all year. In our first lap we did factoring and expanding with algebra tiles. In Lap 2 we did all things factored. Now that we’re in lap 3 we did vertex form, and also derived the quadratic formula.
we sing the quadratic formula to the tune of pop goes the weasel. Here’s a youtube video
We’re connecting back to all we know about quadratics, and finding the axis of symmetry, the vertex, and the y intercept as well.
Students are enjoying working together at the walls “thinking classroom” style. It gets us up and moving, and collaborating. At this time of year it’s nice to see that there are so many people in the room who are getting confident and able to help others out. We’ve come a long way since February!
Pretty soon we’ll be solving word problems!
OAME 2025
What a fun time we’ve had at OAME! Many thanks to the wonderful presenters for sharing their skills, and for inspiring us to think about things in new ways. I’ve got more good problems to try with my classes now, and some new ideas for running lessons.
I was part of a presenter team this year. Our session was titled Fostering Confident Math Learners: Cultivating a Growth Mindset through Engaging Activities. Here are our slides
Our time-slot was the last session on Friday afternoon, so we were excited to get hands on with some manipulatives and try to represent some math.
We had 2 different concept circle stations happening, one with relational rods and one with toothpicks and plasticine building models of exponents.
There were lots of great conversations among the groups about what the exponents of 2 and 3 mean physically, and what the brackets mean, in relation to changing the side length of the object, and how that can change the surface area or volume.
We then had the great cheerio stacking competition, where groups aimed to see who was the fastest cheerio stacker, and then we looked at connections to rates and slopes, and units of measuring speed. We determined who was the fastest stacker in the room and then, if time allowed, we would have created a photo finish by having everyone determine the headstart (in number of cheerios) that they’d need to have a full skewer at the same time as the fastest person.
this activity is good for grade 9 and 10, looking at rates, graphing, points of intersection and determining the b value (y intercept).
It was fun to share some of our ideas with the participants, and hear their ideas of what we could try next.
Many thanks to all who came to our presentation!
Painted Cube (again!)
Today another group of grade 10s worked on the painted cube rich task (blogged earlier).
As always when doing this problem, something new shows up as students interpret it differently, or work to make connections to math that they know.
This group decided to represent the blocks as a fraction of the whole. This makes me see all kinds of connections to fractions and percents, or patterning about what happens to these fractions or percents as the cube’s side length increases.
Something I’d be curious to explore with a class is modelling the patterns you’d see in the fractions. This would be an interesting task for grade 11s and 12s I think, as an inquiry into the rational functions. There could be discussion about vertical asymptotes and end behaviours.
These functions could be graphed, and then the intersections between them could be calculated too.
We didn’t go down that road with these grade 10s because they were struggling a bit with the initial task and building general equations for a cube with side length x, and were not ready to push into new territory.
It’s so refreshing to see new ideas and approaches pop up with each new iteration of the task.
Good teamwork today!
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