Tag Archives: multiplication

Just the Facts Again

A month ago I wrote a post called Just the Facts where I talked about the way my daughter has been practicing her multiplication facts with me at home. In case you don’t want to check out that post, here’s a quick recap of the way she’s practicing her facts:

  1. She answers as many multiplication flash cards as she can in one minute. She does two trials of one minute each.
  2. She counts the cards after each trial.
  3. I monitor and provide feedback as needed.
  4. After two trials, she graphs her higher score.

Here’s how she was doing after five sessions back in late August:

And here’s how she’s been doing after 16 total sessions between August and now:

I feel like I’m supposed to do some statistical analysis of this and talk about the median or mode or something, but I don’t really care to be so formal. These are our informal takeaways:

  • She has good days and better days. We don’t focus too much on her results for any given day.
  • She appreciates getting two trials each time because if she blows it on one trial she knows she can make up for it on the other trial.
  • She and I were both excited when she set a new record of 23 correct in one minute.

I’m so proud to see her flexibly use appropriate strategies for finding various products. She can double like no one’s business when she sees a factor of 2, 4, or 8, but she can just as quickly build up or down from a 5 or 10s fact when she sees a factor of 6 or 9.

I wrote earlier about how she struggles with mentally doubling numbers like 18 and 36. I offered to let her use a whiteboard to help her with that doubling when she needs it, and that has been a huge help. I’m especially proud that it hasn’t turned into a crutch for every problem. She really only uses it when she knows the number crunching in her head is too much for her to handle.

For a few sessions, I was doing some flash card practice with her around those challenging doubles, but I decided to move away from it. After I wrote my previous post, Michael Pershan shared this excellent post called What People Get Wrong About Memorizing Math Facts. He said something in the post that I needed reminding of:

“…the best practice for remembering something is practicing remembering it.”

This led me to change tactics. I created a set of flash cards of facts that my daughter is able to solve using a strategy, but I want to give her an opportunity to practice remembering the products.

When we practice these facts, it is untimed, though I only give her about 5 seconds per card, otherwise I can see her start using a strategy to derive the answer. I usually run through these cards once or twice before her two trials for that session. I told her the goal here is to practice remembering the products. I don’t want her to try and use a strategy. I just want her to see if her brain can pull up the answer from memory. I told her it’s fine if it can’t, but the act of trying to remember is a good thing to practice. I also made clear that during her two one-minute trials I still want her to try to remember these products, but if she doesn’t it’s totally okay to use a strategy at that point.

We’ve been using these cards for a while now, and I’m noticing that slowly she is starting to remember some of the products, though not all of them. What I’m really excited to see is that if she doesn’t quite remember the product I’ll ask, “What tens do you think the product is in?” and she is getting pretty good about knowing which ones are in the 50s or 60s even if she doesn’t remember if the exact product is 54 or 56.

So that’s where we are now. We only do this facts practice 1-2 times per week now that school has started. Here’s what I like about it:

  • It’s quick to do.
  • When we first started, she felt pretty down on the days when she only got 10-14 correct. But because we’ve continued doing it and graphing her results, she sees that those days are blips in an overall pattern of success.
  • She loves that there are two trials each time we do it. That feeling of getting a second chance is powerful.
  • I’ve been able to watch her work and provide support and modifications that specifically help her be successful and feel confident.
  • We’re able to work on dual goals of memorization and strategy use.

My next step is going to be introducing division flash cards into the mix. We’ve done some work relating multiplication and division already, and we’ve specifically talked about how thinking of a related multiplication fact can help her solve a division fact. I expect some bumps as that gets started, but I feel confident she’s on a good path.

Just the Facts

In my previous post (Link), I shared how I’ve recently starting doing math with my daughter to help her get warmed up for the start of 4th grade. In that post I talked about how I’m using the centers from the Illustrative Mathematics K-5 curriculum (Link) to revisit and practice working with multiplication and arrays.

In the six and half years I worked as a district math curriculum coordinator, a common concern I heard from 4th and 5th grade teachers is that their students don’t come in knowing their multiplication facts. I can attest that my daughter learned a lot about multiplication and division in 3rd grade, but I’ll be honest, she hasn’t done a whole lot of multiplying or dividing this summer (not to mention fluency is something that tends to develop over a period of years, not months). It comes as absolutely no surprise to me that she’s rusty, particularly with knowing her multiplication facts. I’m going to go out on a limb and claim that a lot of kids are rusty at the start of a new school year. We need to give them grace, which means not saying things like, “Didn’t your teacher teach this last year?” We also need to intentionally build in opportunities to practice and dust off the mental cobwebs.

Today I’d like to share how my daughter and I have been practicing multiplication facts. What I like about what we’re doing is that (1) it only takes a few minutes a day, (2) it reinforces flexible use of strategies, and (3) it gives her a second chance everyday. I got this idea from a free math intervention called Pirate Math Equation Quest (Link), developed by Dr. Katherine Berry and Dr. Sarah Powell from the Meadows Center (Link) at The University of Texas at Austin. Their intervention includes a component called Math Fact Flaschards that goes like this:

  • Student completes two trials of Math Fact Flashcards, each for 1 minute
  • Teacher and student count cards after each timing
  • Teacher monitors and provides feedback as needed
  • After 2 trials, student graphs the higher score

Rather than use traditional flashcards, I created flashcards that show two facts per card, the initial fact and its turnaround. For example, the card with 2 × 5 also shows 5 × 2. I got this idea from the 4th grade Investigations 2nd edition curriculum. It reinforces the idea that every time you know the answer for one fact, you really know the answer for two (with the exception of square numbers).

Four multiplication flash cards.

Before we start a trial, I always remind her that she is going to “just know” some of the facts because she’s so familiar with them, but for the ones she doesn’t “just know” she can use one of the multiplication strategies she’s learned. The following poster is hanging on the wall next to where she’s sitting so she can turn and reference it as needed.

Table listing the multiplication thinking strategies

These are the thinking strategies developed by Origo Education (Link). If you’re not familiar with them, check out this YouTube playlist that includes one-minute videos explaining each strategy. (Link) If you want to see how a child uses one of the strategies, here’s a link to a short video of my daughter talking through the Build Down strategy she used to solve 9 × 7. (Link)

Please note, you can’t just throw strategies at your students. They have to be intentionally introduced and practiced, but it is well worth the time! Students who lack a robust toolbox of strategies have to rely solely on memorization (which is a big ask!) or inefficient strategies like skip counting. If you’re interested in learning more about how to teach these strategies, Origo has a great series called The Book of Facts that shares activities and games for teaching a set of fact strategies for each of the four operations. (Link)

During each trial, I present the flashcards one at a time. I put all of the ones she answers correctly in a pile and any she answers incorrectly in another pile. After the minute is over, she counts the number correct, and then we discuss the ones she answered incorrectly. Sometimes her incorrect answers are because of a simple mistake, and I reinforce that it’s fine because she has been able to recognize the error herself. However, sometimes it’s more than a simple error. I was able to pick up very quickly that she’s also rusty with doubling 2-digit numbers that involve bridging a ten. For example, to solve 4 × 7, she can easily double 7 to get 14 and double 14 to get 28. However, to solve 4 × 8, she can easily double 8 to get 16 but she gets stuck doubling 16. Her answer might be 26 or 36.

Based on this observation, I’ve added in practice with doubling 2-digit numbers. This practice is untimed for now, though I might eventually add these cards into the deck of multiplication flashcards.

Set of flashcards that say things like "Double 25" and "Double 18"

At the end of the two trials, we graph her higher score for the day. I really love this because if she blows the first trial for whatever reason, she knows she’s going to get a second chance to get a higher score. It really takes the pressure off.

We’ve only been doing it for a week, so there’s not a lot of data to look at, but I’ve already used her graph to talk about how we all have good days and better days. I also reinforce that while some days are lower, her rate of incorrect responses is consistently low. She only ever misses 0, 1, or rarely 2 cards during a trial. She’s also been really good about stopping and thinking of an appropriate strategy whenever she gets stuck, and she is doing a great job of executing her chosen strategy accurately.

Bar graph showing the number of multiplication flash cards answered correctly each day. Bars range in height from 14 to 21.

For full transparency, her deck of flashcards includes all of the facts including the “easy” ones like 0s and 1s facts, and I’m okay with that. They’re still facts and she needs to know them. The important thing is that I continue to monitor to uncover any issues where I can support her, like with doubling 2-digit numbers. Eventually I might ween the deck down to the ones that need more intensive practice.

I like that this practice doesn’t take a lot of time, only about 3-5 minutes. If you’d like to try this out in your classroom, you might consider doing it in small groups, which is an idea shared in the Pirate Math Equation Quest intervention I mentioned earlier. During the one-minute trial, the teacher goes around the group round robin style, showing one flashcard to each student. All of the flashcards are placed in one pile and the total correct is the group’s score. The goal as a group is to try to get more and more correct each time. I like that this allows for a bit of a tradeoff. The teacher doesn’t have to feel pressured to run this activity individually with every student, but at the same time, she can learn something about each student as she conducts these trials in small groups. I’m doing this with my daughter everyday, but a teacher might be able to make small groups such that she ends up seeing every student every 3-4 days.

As I was reading over the small group directions, I realized they recommend letting the student continue trying until they get the answer correct. If the student answers incorrectly, the teacher intervenes with a suggestion such as a strategy a student might use. I think I might try that with my daughter rather than setting aside incorrect answers. Helping in the moment seems much more powerful than helping at the end. It also does a better job of validating the power of identifying and correcting mistakes. I like forward to seeing how it goes next week!

Can You Build It?

This week I’m starting to do a little math with my daughter everyday to dust off the cobwebs before 4th grade starts in September. One of the resources I’m using is the centers from the Illustrative Mathematics K-5 curriculum (Link to Kendall Hunt’s version of IM K-5 Math).

We kicked things off on Monday with a center called Can You Build It? (Link) One thing I like about the IM centers is that they often contain multiple stages within the same center, so you can choose just the right starting point within a given concept. Since my goal was to revisit arrays and the meaning of multiplication, we started with Stage 1. In the original IM version, one person builds an array secretly and then describes it to their partner and the partner tries to recreate it.

I changed this stage into a cooperative game that turned out to be really fun for my daughter. Here’s how it works:

  1. Draw a target area card. (I created a deck of cards that have the numbers 10 – 27 on them. This means there are 18 possible target areas, which feels like a good range. The numbers are also small enough that you won’t spend all your time counting out the tiles you need before making your array.)
  2. Each player secretly makes an array with that target area.
  3. Share your arrays. If you made the same array, you collectively earn 1 point. If you each made a different array, you collectively earn 2 points. (To clarify, a 2 by 6 array is the same as a 6 by 2 array.)
  4. Earn 5 points in as few rounds as possible.

If you don’t have square tiles handy, you could use a free app like Number Frames from the Math Learning Center (Link) which can be used in a browser or downloaded onto a tablet.

Screenshot of Number Frames app. The workspace of the app shows 3 arrays: a 1 by 12, a 2 by 6, and a 3 by 4.

Or if you still want something hands-on, you could always use some crackers!

Source

After a couple of days playing Stage 1 and revisiting how to build and describe arrays, we moved on to Stage 2. There are a couple of key differences here:

  1. Instead of secretly making only one array, the goal now is to make as many different arrays as possible with the target area.
  2. The game is competitive now. The player who makes more arrays earns 2 points and the other player earns 0. If both players make the same number of arrays, they both earn 1 point. The winner is the first to 5 points. (The original IM center used a slightly different scoring scheme. I opted for something similar to the game we played for Stage 1.)

My daughter immediately started bumping into ideas related to prime numbers. Here are some highlights from our conversation as we played for the first time:

1/ Daddy: Today our game is slightly different. This time when we draw a target area, our goal is to make as many different arrays as possible. If we get the same number of arrays, we each earn 1 point. If one of us makes more than the other, that person earns 2 points.

2/ Daddy: (draws card) Our first target area is 20.
(both make arrays in secret)
Daddy: I made a 2 by 10 and a 4 by 5. How about you?
Me: I made those, and I made a 1 by 20.
Daddy: Oh! I forgot that one!
Me: You have to remember you can *always* make a 1 by array!

3/ Daddy: (draws card) Okay, this time our target area is 13.
(both make arrays in secret)
Me: Ugh! I can only make one.
Daddy: Me, too. What did you make?
Me: 1 by 13.
Daddy: Hmm, I wonder why we could only make one array.
Me: Maybe because it’s an odd number.

4/ Daddy: (draws card) Now our target area is 11.
(both make arrays in secret)
Me: No! You can only make one again.
Daddy: Huh, is this an odd number, too?
Me: Yeah.
Daddy: That’s interesting.

5/ Daddy: (draws card) Ok, our target area is 10.
Me: I’m just going to write down the 1 by array. I don’t even need to make it.
(both make arrays in secret)
Daddy: What did you make?
Me: A 1 by 10 and a 2 by 5.
Daddy: Same here. Is 10 odd?
Me: No, it’s even.
Daddy: Hmm…

6/ Daddy: You made two really interesting observations today. Do you remember what they were?
Me: …if a number is odd you can probably only make one array?
Daddy: What else?
Me: …and you can always make a 1 by array for every number!

Originally tweeted by Splash (@SplashSpeaks) on August 18, 2021.

I love how this game has a simple premise – make arrays – but it creates opportunities for students to notice deeper ideas about numbers and multiplication. If you woudl like to try this game out with your own child or students, here’s a link to the center. (Link)

If you work in a grade level that introduces prime and composite numbers, I also recommend checking out 4th Grade Unit 1 of the IM curriculum for well-designed, ready-to-go lessons. (Link)

[UPDATE] Alyson Eaglen shared a great idea on Twitter. She said that instead of using cards with pre-printed target areas, she suggests rolling three 9-sided die and the sum is the target area. What a great way to bring in some bonus addition practice! If you don’t have 9-sided dice, you could always use five 6-sided dice or whatever combination of dice yields the range of target areas you’re interested in for the game. If you don’t have physical dice handy, Polypad’s free virtual manipulatives (Link) include a variety of dice under the Probability and Statistics menu.

Inspiration – Summer Edition

As you may or may not know, I have a tendency to roam the seasonal aisle at Target, looking for mathematical inspiration. So far I’ve shared photos I’ve taken at Halloween, Valentine’s Day, and Easter. You can find them all here.

Today I was stopping by Target for some bug spray which just so happens to be next to the summer seasonal aisle. I couldn’t resist the urge to take a stroll and take some pictures. Here’s what I’ve got for you today.

How many large wooden dice are in the package?

20170601_19145920170601_191506

It’s totally obvious, right? For younger students, maybe not so much. But even after everyone is in agreement that it’s 6, what do you think they’re going to say once you reveal the answer?

20170601_191509

Not what you were expecting, is it? You probably thought I was wasting your time starting with such a simple image. So now you get to wonder, “Why/How are there only 5 dice in this package?” Perhaps this will help:

20170601_191516

That burlap bag has to fit somewhere!

Let’s move on to another large wooden product. How many dominoes are in this pack?

20170601_191256

It might be a little hard to tell from this perspective. Let’s look at it another way.

20170601_191337

Barring any more burlap sacks, you might just have the answer. Before we find out, stop and think, what answers are reasonable? What answers are not reasonable?

Ok, time to check if you’re right.

20170601_191348

No surprises here. Although after the first image, I probably had you second guessing yourself. There’s something to be said about the importance of how we sequence tasks.

Speaking of sequencing tasks, let’s move on to another one. How many light bulbs on this string of lights?

20170601_191802

I really like this box because you get this tiny 2 by 3 window, and yet it’s such a perfect amount to be able to figure out the rest. This would be one I’d love to give students a copy of the picture and let them try to show their thinking by pointing or drawing circles on it.

Again, this is a great time to ask, what answers are reasonable? What answers are not reasonable? Assuming the light bulbs do create a rectangular array, there are definitely some answers that are more reasonable than others.

After some fun discussion about arrays, it’s time to check the actual amount.

20170601_191808

So fun! Like I said, I love this image. Let’s look at another package that caught my eye.

How many pieces of sidewalk chalk in this box?

20170601_190947

I was pleasantly surprised to find that Crayola put arrays on top of all their summer art supplies. It’s like they were designed to inspire mathematical conversation! Granted, the box doesn’t give it away that the dots represent the pieces of chalk, I wouldn’t point it out to students. I’d let them wonder and make assumptions about it. It’ll turn out that their assumptions are completely right, and how satisfying that will be for them!

20170601_190950

Since we’re talking about arrays, which means we’re talking about multiplication, let’s shift gears a bit to look at some equal groups.

How many plastic chairs in this stack?

20170519_150218

And to throw a wrench into what looks to be a simple counting exercise, how much would it cost to buy the whole stack?

20170519_150227

Now students have got some interesting choices about how they calculate the cost. The fact that half the stack is blue and half the stack is red is just icing on the mathematical discussion cake.

My final image from the summer seasonal aisle has been a real head scratcher for me.

How many water balloons do you estimate are in this package?

WaterBalloon

 

What is an estimate that is too low?

What is an estimate that is too high?

What is your estimate? How did you come up with that?

Take a look at the box from another angle, and see if you want to revise your estimate at all.

WaterBalloon2

 

We clearly have groups – eight of them to be precise – but the question I’m not entirely sure about is whether there are eight equal groups. Maybe? And if there are equal groups, then there are certain answers that are more reasonable than others.

So how do you wrap your head around this?

20170519_150720

I’ll give you a moment to think about why this is confusing me a bit.

Assuming there is an equal amount of each color, this doesn’t make any sense! But then I noticed the small white tag on the set of purple balloons.

20170519_150746

Oh! That explains it. There’re only 260 balloons in here so…no, that still doesn’t work if these are eight equal groups.

Oh, then maybe it’s 5 more than 265 so it’s actually 270 so…no, that doesn’t work either. So I’m left to conclude that either this is not a pack with eight equal groups or there is some funny math going on! Sadly, $25 is a bit steep to satisfy my curiosity. If any of you purchase this pack and want to count balloons, I’d love to get the full story.

And with that, my tour of the summer seasonal aisle comes to an end. If you’re just finishing the school year, bookmark this post to revisit when school gets back in session. What a fun way to start the year! If you’re still going strong, then I hope you’re able to use these to spark some fun, mathematical discussions in your classrooms.

 

 

Inspiration

Tonight I hosted #ElemMathChat and our topic was inspiration. Specifically, what inspires you as you’re planning for and teaching math?

One place I’ve found a great deal of inspiration is the seasonal aisle at Target. Honestly, inspiration can be found at just about any store, but the seasonal aisle is a particularly rich source of inspiration because it taps into the novelty and appeal of holidays.

My closing thought for the chat was this:

Q0-FinalThought

“What can I do with this?” That’s the question I carried with me as I wandered the Easter aisle this week, wondering what mathematics I could draw out of the colorful assortment of products around me. I shared a few examples during #ElemMathChat tonight. I’ll share those here along with several more examples I couldn’t squeeze into the hour-long chat.

If you’d like even more examples, check out these posts I wrote around Halloween and Valentine’s Day:

As you’re reading this post, I challenge you to continually ask yourself “What can I do with this?” because you might notice something I didn’t and be inspired to ask a different question or draw out different mathematical ideas. If that’s the case, I’d love to hear about it in the comments!

Let’s get started!

Jelly Beans

How many jelly beans are in this bag? What is an estimate that is too HIGH? Too LOW? Just right?

Q1

When estimating, our goal is to come up with a reasonable guess. The reasonableness comes from our guess lying within a particular range of numbers that makes sense. You could easily say that your “too low” guess is 1 because you know there is more than 1 jelly bean in the bag. You could also say your “too high” guess is 10,000 because it is unlikely there are 10,000 jelly beans in this one bag. But those are just cop out answers, not reasonable estimates. They don’t demonstrate any understanding of what makes sense given the picture of the bag and the window showing some of the jelly beans.

If you share this picture with your students, see if you can get them to take risks as they estimate. For example, I can count about 12 jelly beans in the bag’s window. I’m going to guess there are at least 10 groups of 12 jelly beans in the entire bag for a low-ball estimate of 120 jelly beans. However, I don’t think there’s enough room for 25 groups of 12 jelly beans in the bag, so my high-ball estimate is 300 jelly beans. I think the actual number is somewhere in the middle around 200 jelly beans.

See how much more narrow my range is? I think the number of jelly beans is somewhere between 120 and 300 jelly beans. In some ways that’s still a fairly broad range, but it’s so much more reasonable (and riskier!) than saying there are between 1 and 10,000 jelly beans in the bag.

And now for the reveal:

Q1-Answer

Notice I didn’t give the actual answer. I’d want my students to use the information provided to find out about how many jelly beans are actually in the bag. Depending on the grade, this could be a great impromptu number talk to find the product of 23 × 9.

We’ve talked about one bag of jelly beans, but let’s compare that to some others. Which of these bags do you think has the least jelly beans? The most? How do you know? (Click the pictures to enlarge them.)

After some discussion and estimating, reveal this image for the SweetTarts bag. How does this bag compare to the Nerds jelly beans? Can you compare without calculating?

Some students will likely calculate the products regardless, but I would want to make sure it also came out that both packages have 9 servings. The serving size in the SweetTarts bag is larger so the total amount of jelly beans in that bag is greater than in the Nerds bag. In other words, 31 × 9 > 23 × 9 because you are multiplying 9 by a greater number in the first expression, so the resulting product will be greater.

After that discussion, it’s time to reveal the answer for the third bag. A challenge to students: Can you compare the quantity in this bag to the other two without calculating the actual product?

Q2-Answer2

Which One Doesn’t Belong?

If you’ve never checked out the site Which One Doesn’t Belong?, I highly recommend it. The basic gist is that students are presented four images and they have to choose one and justify why it doesn’t belong with the other three. The twist is that there isn’t one right answer. You can make a case for why any of the four pictures doesn’t belong with the other three.

Look at the four pictures below. Find a reason why each one doesn’t belong.

Q3

And here’s another example, this time involving candy:

Q6

You’ll notice I’m not providing answers, because there isn’t one right answer! To quote Christopher Danielson, “It’s not about being right. It’s about being true.”

Chocolate Bunnies

Take a look at this picture of two boxes of chocolate bunnies and ask yourself, “What questions could I ask about this?”

Q4

Here are some questions that came to my mind:

  • How many chocolate bunnies are left? Can you find the number in another way?
  • How many chocolate bunnies have been sold? Can you find the number in another way?
  • If each bunny costs 75¢, how much will it cost to buy the remaining bunnies?
  • What fraction of each package has (not) been sold?

Peeps

How many Peeps are in this package? What is an estimate that is too HIGH? Too LOW? Just right?

Q5a

The quantity is smaller and you can see so many of them that I would want students to be very narrow in their range of estimates and very clear in their justifications.

We know it’s a number divisible by 3 because there are three rows. We also know there are at least 3 Peeps in each row – we can see those! I would estimate 12 (four per row) is too low and 18 (six per row) is too high. My just right estimate therefore is 15 because I think there’s room for more than 4 in each row but not enough room for 6.

This might be a tad controversial because some folks associate estimating with numbers that end in 0 or 5, such as 25, 75, 100, 900. However, given the facts – three rows – I know the total number has to be divisible by 3. That means estimates like 12, 15, and 18 make much more sense to me than 10 or 20. That’s not to say that 10 and 20 are unreasonable estimates – they’re decent in this example – but I’m not going to limit myself to just those numbers given what I know about the configuration of Peeps.

And here’s the reveal:

Q5a-Answer

But it doesn’t end there! Now that you know the quantity in one package, what can you tell me about the number of Peeps in this case?

Q5b

And to take it another step further, here’s the price of one package. How much would it cost to buy half the case? How many Peeps would I be getting?

Q5c

I love the layering in this example because it starts out so simple – estimating how many Peeps in one pack – but it really takes off from there with a few added details.

Easter Eggs

How many eggs in my hand? What is an estimate that is too HIGH? Too LOW? Just right?

Q7

This one is trickier because the eggs are not arranged neatly like the Peeps. In this case I’m probably going to use numbers like 5, 10, or 20 to make my estimates.

However, this question is also a bit tricky because of how I worded the question. Did you notice?

Let’s take a look at the front of the package.

Q7-Answer

Students might be drawn quickly to 18 as the answer, but that’s not quite it. If you read carefully, it says “18 colored eggs and one golden egg” which brings the total to 19. But that’s not quite right either. I asked how many eggs in my hand, and if you’re noticing the shape of the container, there are actually 20 eggs in my hand. Sneaky!

So, if there are 20 eggs in my hand, how many colored eggs inside these 5 containers? (I would say “on this shelf” but students might get caught up in the fact that you can see there are more containers in the back. I want to focus just on the five up front.)

Q7b

This is another chance for an impromptu number talk. I especially like how it can build off the discussion about the number of eggs from the previous image. You can start with 20 × 5 and back up to remove the 5 large egg containers (I asked about the colored eggs inside) and the 5 golden eggs (I asked about the colored eggs, and the packaging does not include gold as a colored egg. This is semantics though, so I might accept these in the total since gold is a color.)

Now that we’ve talked a bit about this package, let’s do some comparing. Which would you rather buy – one package of the eggs we just talked about or two packs that each have 12 eggs in them.

Q7c

In case you missed it, the price for the package on the left is $5.00. It’s printed on the label. The price for the packages on the right is 89¢ each. (I would probably ignore the Buy One, Get One 50% Off unless you wanted to take into account that wrinkle.)

Notice I didn’t ask, “Which is cheaper?” I asked, “Which would you rather buy?” On cost alone the two dozen eggs is significantly cheaper, but there are some definite perks to the $5.00 package. Again, it’s about being true, not correct. So as long as students are able to defend their choice, that’s what matters.

For this next one I would probably change up the question and ask, “Which is the better deal – 1 pack of 48 eggs or 4 packs of 12 eggs?”

Q9

The price you see in the left picture – $2.50 – is the cost of 1 pack of 48 eggs. Ignoring the buy one, get one 50% off, the left picture is a clear bargain. However, this might be a good time to tell students that for every one pack of 12 eggs, you get a second for half off. Then I would challenge them to determine the price of 4 packs given that discount. It’s definitely a closer answer when you take that into account!

Coconut Macaroons

I don’t know that I associate coconut with Easter, but I had to share these packages of coconut that caught my eye in the Easter aisle.

How many cups of shredded coconut in this package? What is an estimate that is too HIGH? Too LOW? Just right?

Q8a

Here’s the reveal, which is why these packages caught my eye:

Q8a-Answer

Such an oddly specific amount! So if I bought all of these bags of coconut, how many cups of coconut would I be getting? How much would the three bags cost?

Q8b-2

There’s a recipe for coconut macaroons on the back of the package. If I bought three bags of coconut, how many cookies could I make?

Q8b-1

I like this because students have to wade through a lot of information to find what they need. Oftentimes in math problems we make needed information stand out or we don’t provide any distractions at all. It’s good to make students work for it a bit like they would have to do in the real world if they wanted to bake these cookies.

Another question I thought of is, “How long does it take to make 3 dozen macaroons?” This provides another opportunity for reading the recipe to search for relevant information. Students might just add 15 minutes and 20 minutes, but that’s only if you can fit all 36 cookies in the oven at the same time. If you only have one baking sheet that can hold 12 cookies at a time – which is about all I can do at home – then how long will it take? What if you could squeeze 18 cookies on a cookie sheet? How much time would you save?

Miscellaneous

I’m going to close out this post with a final set of pictures that might inspire you to share them with your students and prompt some mathematical discussions. (Click the pictures to enlarge them.)

 

Bunnies

I get a kick out of this last one because it’s pretty easy to tell how many candies are in the package.

Q9d

I can foresee some really interesting discussion when you reveal what the packaging says about the number of candies contained within.

Ferrero

Final Thoughts

Please feel free to use these pictures with your students. I’d love to hear about the conversations they spark. If you get inspired to use them in ways I didn’t think about, please share in the comments. That way we can all learn and get ideas from one another!

My Favorite: Holidays at Target

myfav

Here we are in Week 2 of the ExploreMTBoS 2016 Blogging Initiative! This week’s challenge is to blog about one of my favorite things. During this school year, one of my favorite things has been visiting Target during the holidays. The holiday-themed merchandise is rich with mathematical possibilities! I already wrote three posts about a treasure trove of images from Halloween:

Valentine’s Day is around the corner, and I snapped some photos this evening to share with you. I’m going to cover a range of mathematical skills – mostly centered around estimation –  from Kinder through about Grade 6 to show you just how versatile this stuff is!

These first two images are good for estimating quantity. You can estimate the quantities individually. Don’t forget to ask students to estimate an answer that is TOO HIGH and one that is TOO LOW in addition to their actual estimate. Coming up with a reasonable range takes a lot of practice! You could also show students both images at the same time and ask, “Which package has more?”

I forgot to snap a picture of the answers, but I can tell you there are 15 bouncy balls and 24 eraser rings.

Here’s another one. How many Kisses are in the box?

IMG_2053

I was kind of surprised that the answer wasn’t an even number like 10 or 12. This just seems oddly specific.

KissesHeart-Reveal

Students tend to estimate better when the quantities are smaller. Here’s a larger quantity package to up the challenge a bit. How many gumballs are in the bag?

Gumballs-Estimate

I was kind of surprised to find out the answer myself.

Gumballs-Reveal

This next one is tricky! How many truffles are in the box? Go ahead and make an estimate.

TruffleHeart-Estimate

Now that you’ve made your estimate, I’d like to show you how deceptive product packaging can be. Would you like to revise your estimate?

TruffleHeart-NewInfo

And now for the reveal. How does your estimate compare to the actual amount?

TruffleHeart-Reveal

The first few images dealt with disorganized quantities. Once we move into organization, the thinking can extend into multiplicative reasoning. The great thing is that it doesn’t have to! Students can find the total by counting by 1s, skip counting, and/or using multiplication.

There are several questions you can ask about these pictures. They’re of the same box. I just gave different perspective. I’d probably show the almost-front view first to see what kids think before showing the top-down view.

  • How many boxes of chocolate were in the case when it was full?
  • How many boxes of chocolate are left?
  • How many boxes of chocolate are gone?

Here’s another package that could prove a bit tricky for some students. How many heart stickers are in this package?

HeartStickers-Estimate

Students might notice that the package says 2 sheets. If they don’t, you might show them the package from a different perspective.HeartStickers-NewInfo

And finally, you can reveal the total.

HeartStickers-Reveal

This next package can be shown one of two ways depending on how much challenge you want to provide the students. Even with some of the hearts covered, students can still reason about the total quantity.

This next one could simply be used to ask how many squares of chocolate are in the box, but what I’d really like to know is how many ounces/grams of chocolate are in the box.

ChocoSquares-Estimate

After some estimating, you could show your students this and let them flex their decimal computation skills to find the total.

ChocoSquares-One

However, the reveal is likely to raise some eyebrows.

And finally, you can do some more decimal calculations with this final product. How much would it cost to buy all of the boxes shown?

BigHeart-Estimate

And if you bought all 6 boxes, how many ounces of chocolate would you be getting?

BigHeart-Estimate2

Ten minutes in the holiday aisle and my iPhone are all it took to gather this wealth of math questions can now be shared with students. Even better, I didn’t have to purchase any of these products! Even better than that, I can go back for every major holiday to capture new images that will feel timely and relevant!

By the way, feel free to use any and all of these images with your own students. They’re fairly low quality so I don’t recommend printing them, but they should look just fine projected or shown on a screen.

Happy Valentine’s Day!