# Pumpkin Math: A Week of Investigations

Harvest (and Halloween!) season is here and, likely, your students have pumpkins on their minds. These lessons build upon their interests and further develop their mathematical thinking and reasoning skills. Focus includes: describing attributes, comparing, and ordering, as well as counting & reasoning.

While these lessons document a classroom investigation, they can easily be incorporated into a homeschool program or integrated into conversations with your own kids this Halloween season.

Day 1: How can you describe this pumpkin? (Attributes)

Day 2: How can we compare these two pumpkins? (Comparing)

Day 3: How could we put these pumpkins in order? (Ordering)

Day 4: Which Pumpkin has more seeds? (Reasoning, Counting)

__Prep:__ Pick out three pumpkins of various sizes. I try to pick out a few that will lead to interesting conversations around comparing pumpkins (i.e. a tall thin pumpkin and a wider/rounder short pumpkin leads to interesting discussion around the word “big”.) I look at the stems (some that are longer, some that are curvier, etc.). I try to consider attributes that will lead to ordering the pumpkins in a variety of ways. I also try to pick out pumpkins in varying shades of orange (notably lighter to darker).

Click __HERE__ to see these investigations play out in the classroom.

__Day 1:__ Describing A Pumpkin

Select one of the pumpkins to present to the group. “How would you describe this pumpkin?” Make a list of ALL of the ways that the kids come up with. After recording initial ideas based on what the kids see, pass the pumpkin around the circle. What else should they add to the list? How does it feel?

After the group discussion, kids will need a clipboard and drawing materials to complete an observational drawing - making sure that their drawing clearly shows all of the appropriate attributes of the pumpkin.

__Day 2:__ Comparing Pumpkins

Today, bring out a second pumpkin and put the two pumpkins next to one another for the kids to compare. How might you compare these two pumpkins? What comparisons can you make? As the ideas begin to be shared (one pumpkin is taller than the other; one is darker in color than the other, etc.), you will likely come to a problem: how do we know which pumpkin is which? This naturally leads to a discussion about labeling or naming each pumpkin. Take ideas from the group - how could we label or name the pumpkins so it is clear which one each person is talking about? Why is this important? You may decide to give simple labels for now (i.e. pumpkin 1 and pumpkin 2 and come back later in the day to brainstorm and vote on actual names if this is an idea brought up in the group… it often is). Make a LONG list of all of the ways to compare the pumpkins. We want kids to see the variety of ways in which we can describe and compare two things (not just by quantity or size).

As a follow up activity on this day, you may send the kids off with a partner to explore a collection of materials (leaves, rocks, etc.). After exploring the collection - describing the different items as they look through it - each group picks two items to compare. Play the comparison game: go back and forth as many times as you can stating a comparison of the two items. (I.e. The one is wider than the other, the one has more holes in it, etc.) The kids should have the opportunity to record their comparison both on paper (drawing the items they chose to compare and at least one way in which they compare) and then with the larger group. (Sharing on Seesaw or a similar platform is another option as well.) How did you describe your button? How do the two buttons compare? Coming back together as a group to reflect on the learning from the day is a powerful habit to build reflection into the learning process, deepen understanding, and lead towards next questions to continue pursuing.

As an alternative, you can have the kids create a second day observational drawing - completing detailed observational drawings of the two pumpkins side by side. These drawings also could be completed during centers/exploration time later that day or the next morning. __HERE__ is the recording page.

__Day 3:__ Ordering Pumpkins

Today, bring out the third pumpkin. The first question to put to the group is what to call it (as a follow-up from the discussion the previous day). Will it be pumpkin number 3? Have you named the previous two pumpkins Ned and Fred? What will this third pumpkin be called so that we can be CLEAR about which pumpkin we are referring to? Once that matter is settled, ask the class, “How might we put these pumpkins in order?” Give the class plenty of time to order the pumpkins in a variety of ways. Let volunteers come up to put the pumpkins in order and explain how they are ordering their pumpkins. Try to avoid restating their words. Let their voices be heard (let the kids learn to listen to and learn from one another, not just from you, the teacher.) Give lots of thinking time. Remember, we often need to back up to let our brains see it or think about it in a new direction.

As a next step, prepare collections of items for the children to work in pairs to put in order. Can they come up with at least two ways to order their collections? Children should be responsible for recording their thinking on paper - showing the two (or more!) ways that they ordered their collection (they could also record on a platform like Seesaw prior to returning to the rug to share with the group). The kids should have the opportunity to share back with the group at the end of the lesson. Collections might include: four or five months of the year or days of the week, four or five names from the class, four or five pencils, four or five crayons in different shades of one color, bottles of colored water (diluted to different shades of the same color and filled to different levels), four or five cylinders from the block center, etc. When making the collections, consider a collection that has multiple ways of being ordered. The kids may come up with ways that you haven’t considered!

__Day 4:__ Which pumpkin do you think has the most seeds in it? (Counting Pumpkin Seeds) *pick out two of the pumpkins for this investigation.

Pose this question to the group and let volunteers share their reasoning. Do kids think that the bigger pumpkin will have more seeds? Do some wonder whether one will have thicker walls and less room for seeds? What other reasons shape their responses? Remember, going back a quick answer to a focus on the why (the reasoning) is our goal. Once ideas have been shared, ask, “How could we find out? What would we need to do to discover which pumpkin has the most seeds?” Kids will likely come up with the idea of cutting open and taking out the seeds.

Counting all of the seeds in the pumpkin would likely be a very high number (beyond much meaning for most in the group) and, as some kids might point out, would take a very long time AND we might get lost in all the goop. Instead of counting all of the seeds as a group, talk about splitting up the job. Let the kids know that they will each be responsible for counting some of the pumpkin seeds. Show kids the pumpkin counting recording page and talk about how each of them will be responsible for filling their inventory page with seeds - each pumpkin on the page has the numeral for the number of seeds that belong in that pumpkin. The pumpkins are labeled 1-10. One seed gets glued in pumpkin 1, 2 seeds in pumpkin 2, three seeds in pumpkin 3, and so on.

*So, if we want to compare the number of seeds in the two pumpkins, how can we organize our work?* Discuss how it will be important to keep track of who counts seeds from which pumpkin. Split the class into two groups - half of the class will count seeds from pumpkin 1 and half of the class will count seeds from pumpkin 2. (This is an important piece in order to raise the ceiling of the thinking that can happen.) Once these pieces are established, use established routines so that kids can get started counting out the seeds. Kids will need: a squeeze glue. A pumpkin counting seed page. A paper towel. A handful of pumpkin seeds. I find it works well for kids to start with one quick handful/grab of seeds (we have been playing grab and count quite a bit which makes this a familiar routine). Once they grab a handful, they can go get themselves set up to begin (put their pumpkin seeds down at a spot, go get a squeeze glue and a pumpkin seed counting page). Kids can always return for more seeds (from the same pumpkin) if they need to once everyone is working.

While kids are working, observe kids ability to count out the accurate number of seeds in each pumpkin, as well as their organizational skills. Note students who are beginning to organize their work on the page (how are the seeds arranged on the page). If kids fill all of their pumpkins, you can ask, “How many seeds are on the page all together?” Note the strategies children use to arrive at a solution. This is a big number - be sure to double check and double check with a neighbor. It is likely that you will hear two different solutions within the group - name this and ask the two children to get together and see if they can figure out why they got a different solution and see if they can come to agreement. As always, scrap paper and pencils should be available for kids to work out their thinking.

*If one of the pumpkins runs out of seeds, stop the group and problem solve together. What could they do? Should those kids stop counting? Should they take seeds from the other pumpkin? Should they open up the third pumpkin? Remember, if the bigger picture goal is to compare the seeds in each, it would cause problems to take seeds from the other pumpkins. Come up with a solution as a group - there are lots of possibilities as long as the group keeps count of what is done.

Let the kids clean up and set their pumpkin counting pages aside to dry overnight. Plan to return to the discussion the following day..

__Day 5:__ So which pumpkin had more seeds?

Bring all of the pumpkin seed counting pages to the rug, organized in two groups - all of the seeds from pumpkin A in one row and all of the seeds from pumpkin B in another row. Begin by letting kids share what they found. Questions to consider to invite participation from diverse participants: *How did you arrange your seeds on the page? What was challenging about counting the seeds? What strategies were helpful? Was anyone able to figure out how many seeds were on the page all together? How did you figure that out? If Connor and Abby found that there were 55 seeds on each of their pages, what do you think that means about the number of seeds on everyone’s page? *This variety of questions provides a way for different children to engage in the conversation (just as they were able to engage in the investigation) - whether they are children who are working on accurately counting collections to ten, who are able to count collections up to 50 or beyond, or who are working on ideas of how to combine parts to find totals.

Then, connect the collections pages to the bigger question - how does this help us? Can we tell from what we’ve done which pumpkin has more? The next steps will depend on what has happened along the way. Did one of the pumpkins run out? What might that mean? These are high level connections, but they are the type of questions we WANT our kids considering and reasoning through. What does it mean if seven people counted seeds from each pumpkin and all of the pages are filled? Are there more pumpkin seeds left in the pumpkins? There isn’t a necessary end point - all of the conversations and connections and reasoning shared is a great piece of helping the children build their mathematical reasoning skills. This bigger question provided motivation to the investigation - beyond the activity of filling pumpkins with 1-10 seeds - and provides children with the model of mathematical thinking that drives us.

Here is sample a wrap up of the investigation, with ideas about how to provide open, high level thinking avenues for the students to go beyond the basic counting skills involved:

*Before cutting the pumpkins open, we considered which pumpkin (either Ted or Fred) we thought might have more seeds. Kids shared their mathematical reasoning - some thought Ted would have more because it was taller, while others thought they would have about the same amount because, although Ted was taller, Fred was wider and so they might have about the same amount of space inside. After listening to the different ideas, we dug in and got counting! (Some kids were more excited than others were to reach inside!) The kids decided that, in order to keep track of and compare the amount of seeds from each pumpkin, 7 kids should count seeds from Ted and 7 kids should count the seeds from Fred. The kids worked to fill their pumpkin counting pages (organizing the seeds in groups according to the numbered pumpkins (4 in the #4 pumpkin, 5 in the #5 pumpkin, and so on). Some kids figured out that they each had 55 seeds on their counting page so, as one kid reasoned, “Ted had lots of groups of 55 seeds.” As the kids counting Fred’s seeds discovered, there weren’t enough seeds to fill all of their counting pages. As a group, the kids agreed that it was clear that Ted had more seeds. The same number of kids were counting the two pumpkins and, although there were enough seeds from Ted to fill all of the counting pages, there weren’t enough seeds from Fred. The kids wondered whether Fred was a “younger pumpkin” or got cut from the vine before it had enough time “to make all of its seeds.”*

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