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## Examples of Differentiated Planning for Achievable Challenge

Knowing a great deal is not the same as being smart; intelligence is not information alone but also judgment, the manner in which information is collected and used.
—Carl Sagan

To further illustrate how to plan activities and lessons that address students’ varying levels of achievable challenge, this chapter examines some specific approaches. In the following examples, notice how students are all learning the same concepts at different levels of challenge in order to maximize success and minimize feelings of frustration and inadequacy.

### Working with Shapes

An activity called “Draw My Picture” is especially enjoyable for Map Readers, but it also shows Explorers and students with high mastery of shape information the importance of communication. This activity involves the following steps:

1. Pair students with similar abilities regarding shape recognition and naming (or pair a student with high shape mastery and low communication skills with a student who has high communication skills and low shape mastery).
2. Give each pairing a set of variously shaped manipulatives (realia, drawings, or paper cutouts).
3. Have the partners sit opposite each other with a divider preventing them from seeing the other’s work surface.
4. One partner gives verbal instructions for drawing a selected shape. For example, one student might instruct his or her partner to draw a “long triangle with the pointy side down” and then draw a “semicircle, flat side down over the flat top of the triangle” (an ice cream cone).
5. For older children, the verbal instructions could include descriptive vocabulary words they are using in class, such as right angle, isosceles triangle, diagonal, or proportional sides.

The success of the speaker’s verbal communication and the artist’s careful following of instructions become immediate feedback when both students see the final drawing. The describer takes satisfaction in accurate communication, and the drawer is proud of his or her attention to the details of the description. They then discuss what was most helpful and what was confusing in the verbal instructions before switching tasks.

### Estimating Volume

In this activity, students estimate volumes at different degrees of achievable challenge in homogeneous groupings. The goal is to build their estimation/ prediction skills, ability to adapt to new evidence, math communication skills, number-sense skills, and conceptual awareness.

Group 1—Low Complexity. Have students form groups, and give each group a large pitcher of colored water. Explain that each member of the group will fill an 8-ounce cup from the pitcher and predict how high the level will reach when the water is poured into clear bottles of different diameters (e.g., empty soda or juice bottles or measuring cups of different sizes). When they come to a group consensus, students use a marker to indicate their predictions and then pour the colored water into the container and discuss the results. (For assessment purposes, individuals in each group can use differently colored markers to indicate their separate predictions.)

Possible discussion questions and extensions for this activity include the following:

• Why was the level lower/higher than your prediction?
• What do you predict the level will be when you add a second cup of water? (Students should then make a prediction, add the water, and discuss the results.)
• How many cups do you predict will be needed to fill up each of the containers? (Students should then make a prediction, add the water, and discuss the results.)
• Have students present their findings in the form of a diagram, chart, overhead transparency sketch, graph, other graphic organizer, or group discussion.

Group 2—Medium Complexity (Early Conceptual Thinking). The students in this group do a similar activity but should design the experiment themselves. The goal is still for students to make predictions, but students in this group are only provided with the materials (a pitcher of colored water, clear bottles of various diameters, and an 8-ounce cup), not the step-by-step procedure. Students make predictions and then discuss how they can gather evidence to estimate how many cups of water are needed to fill each container. Have them keep group (or personal) records of their predictions, results, and explanations. Students should be challenged to respond to questions, explain their reasoning for changing predictions, and even come up with any “rules” (i.e., big ideas, concepts) they think apply to the activity. Examples of such rules might include “the wider the diameter, the lower the water level” or, more specifically, “if the diameter of the container is twice that of the previous container, the water level will be exactly one-quarter as high.” Encourage more advanced group members to identify two or more ways to complete the experiment and solve the problem.

As you observe your students pouring water from the pitcher into the cup and from the cup into the bottles, invite them to stop after each step and write down any new estimates, based on the evidence they acquire along the way. If they revise their predictions, students should include an explanation about why they did so. Have them keep notes or diagrams about the results of the experiment (encourage them to use terms such as half, quarter, or percent).

Group 3—High Complexity (More Abstract Conceptual Thinking). Students in this group incorporate metric conversions and look at the ounce markings on a measuring cup. After performing the experiment they design, they pour water from the 8-ounce cup into a measuring cup with metric markings. In addition to the procedure used by Group 2, they also analyze, predict, test, adjust, and develop correlations about the relationship of cups, liters, ounces, and milliliters. They can then discuss ways to find a conversion factor for each of these comparisons (e.g., how many cups are in a liter) and see how that conversion factor also applies to milliliters and ounces.

After all three groups have completed their experiments, they share their experiences and results with the class, emphasizing how they altered predictions based on new evidence. Group 1 students should report first, as this might be their only reportable information, and each subsequent group adds new information to the discoveries of the previous group. Record the class information on a chart or other graphic organizer. The class can then make further predictions from the gathered data, test their conversion factors, and confirm these factors with formal conversion charts.

Ultimately, all students can participate by considering how the information can be transferred—used for purposes other than just the class experiment— such as how different bottle designs create the illusion of a larger volume or the factors to consider when determining cost and the ecologic impact of selling items in large-volume containers. Taking into consideration their different levels of achievable challenge, students can be given differentiated homework assignments. For example, lower-level students can look around the supermarket or in newspaper ads to evaluate how different companies persuade people to buy their products based on the size or design of a container, while higher-level students can determine which is a better value in terms of cost and quantity: a six-pack of 12-ounce soda cans or a liter bottle at the same price. An even better approach is to have students design their own questions that they want to evaluate—an approach that assigns ownership and meaning to their work.

### Exploring Number Lines

Number lines are helpful constructions for both Explorers and Map Readers. Explorers can move about the line, and Map Readers can examine and evaluate the designations and patterns in the line. This strategy also allows your students to gain some experience using KWL charts. The KWL strategy activates students’ prior knowledge by asking them to identify what they already know about a particular topic and write their responses on a chart. Students then set goals specifying what they want to learn. At the culmination of the unit, students discuss what they have learned and complete the chart, correcting any errors in the K column. Additionally, if students create individual KWL charts, they can build their own goals—things they particularly want to know that are relevant to the coming unit of study.

As a preliminary activity for the number line strategy, the whole class (or the class divided into small groups) is free to explore number lines without a specific assignment. Begin by creating several number lines on large sheets of paper to roll out on the floor, or use masking tape for more permanent lines. After exploring the lines, students meet in mixed groups to create their K and W ideas for a KWL chart. Students then share their ideas for a master chart that you create. As you walk around and listen to students contribute ideas to their first group meeting, you will gain insights about background knowledge that may be different from your own predictions for each student’s mastery of the topic. This additional source of differentiation information can be more formalized if each student is assigned a colored pencil with which he or she can add to the group KWL chart.

During subsequent lessons, students can be moved to groups of higher or lower achievable-challenge levels as you observe their participation and understanding within the original heterogeneous groups. Groups well matched to their achievable-challenge levels are guided to progress further along the continuum of activities; students with higher mastery/background levels will eventually reach a conceptual analysis of what the number line means regarding integers. Groups still working at the more basic levels of exploration should be engaged at their appropriate levels of achievable challenge, but they will also benefit from what is going on around them. As they observe other groups engaged in apparently enjoyable activities, they are motivated to continue their explorations and discussions to achieve higher levels and engage in these same activities.

The following activity outlines the differentiation that is possible within homogeneous groupings that capitalize on various levels of achievable challenge.

Group 1—Low Complexity. Do a “modeling demonstration” in which you count aloud as you step forward along the number line from 0 to 5, looking down and counting the numbers as you walk. Students then take turns counting the steps aloud as they walk, stopping whenever they choose and announcing the number of units they walked. The rest of the group confirms this number by looking at the numeral that the “walker” is standing on. When each student has had an opportunity to participate, encourage the group members to discuss what they noticed. Then have them write or sketch their conclusions individually and on a group chart.

Group 2—Medium Complexity (Early Conceptual Thinking). Those who show mastery early on can be in a flexible group for the next section of the unit and advance to addition and subtraction. The student who walks the number line can stop at any point, and the group can take turns asking him or her to take one more step and announce the new number. He or she might choose to look down at the number or to predict the answer before checking.

Extend the activity by asking group members to predict where the volunteer “walker” would be if he or she took one more step, two more steps, three more steps, and so on. The student should then take the appropriate number of steps so the group can check the prediction. The volunteer can also independently make a prediction before taking the steps and checking his or her own prediction.

When every group member has had an opportunity to be the “walker,” the group members can turn their observations into a diagram or chart. Students should then use their own words to describe the action. Some may use words that indicate addition, such as, “She moved up 3 squares from 3 and ended on square 6” or “She went forward (or to the right) 3 more squares from 3 and was on number 6.” On the board, keep a list of the relevant mathematical terms that students use in their descriptions.

When students look over the words they used to describe moving ahead, forward, or to the right, they may use the word added, but this is not as important as seeing the process and describing the predictable result. Along the way, they will discover the process of addition, and they will eventually use this term after they understand its meaning. In subsequent number line walks, encourage students to use the word add and write their results in sentences such as, “I was on number 7, then I added 3 more and was on number 10.”

It won’t be long before students instinctively start experimenting with movement in the opposite direction, and in doing so they will discover the concept of subtraction along a number line. Many options can follow that will help students transfer this new information by mental manipulation to other representations of addition and subtraction, including organizing collections of objects and writing sentences with appropriate conceptual vocabulary.

They can also continue playing the walk-the-line game; for example, either you or group members can request that the volunteer “walker” start on number 9 and move five spaces to the left. Have students make predictions before the volunteer moves, but if the prediction is off, ask students to make a prediction of where the “walker” will be if he or she only moves one or two steps to the left. Then have students make a new prediction about the original assignment (move five spaces to the left), and have the “walker” take the appropriate number of steps. If the prediction is still incorrect, tell students to confer about how they can rethink the process they observed and make more accurate predictions next time. If necessary, have them go back to the earlier practice of counting while watching the squares. The “walker” starts at number 9 and, without first predicting, counts up to 5 aloud as he or she takes five steps along the number line to the left. Encourage students to report on the process aloud, for example, “I started on number 9, walked back 5, and I’m now on number 4.”

Encourage group members to follow your coaching model when their classmates make prediction errors. Such encouragement will help students become more skilled at supportive instruction and avoid being critical or taking over the solution from the “walker.” Both the “walker” and the guiding group member will enjoy the satisfaction of supportive collaboration and even the dopamine pleasure that accompanies kindness, optimism, intrinsic satisfaction, generosity, and positive peer interactions. Stress will remain low, and students will return to the number line activities with positive feelings, thanks to the activation of their dopamine-pleasure circuit in anticipation of a positive experience.

I’ve observed students engaged in this activity on many occasions. They discussed why some of their predictions were correct and some were not. Their communication was enthusiastic, but rarely pushy. They felt the comfort of sharing ideas and a communal responsibility to communicate patiently so the group would continue to explore and build knowledge together.

The value of not forcing the words addition or subtraction or the symbols + or – at this time is that the students are able to construct their own concepts, rather than memorize an abstract formula. When the correct words and symbols are added later, they will create an accurate and durable neural pathway, and they will have a tangible meaning because students already created the concept on their own and experienced the intrinsic reward of achieving a challenge they valued.

Group 3—High Complexity (More Abstract Conceptual Thinking).
For those students whom you observe developing mastery in the middle level of complexity, such that they are now below their appropriate achievable-challenge level, or for students who have mastery of the foundational material, the activities described here are more appropriately challenging. These students work with peers to brainstorm what other things they might do with a number line, or they follow your guidance to lead them to the concept of multiples and multiplication. For example, you can ask, “Can you find a way to see what happens and how to predict where the ‘walker’ will be if he walks two squares at a time?”

Eventually, when students have mastered subtraction along the number line, they will want to know how to write or label the negative integers below or to the left of the 0. They will ask you to add more tape or paper to the left of the 0 (or you could preemptively add it before the next math session) to explore negative integers. Discussing, predicting, and asking you for direct instruction (now they want to know what you have to teach them), they are now in an ideal situation for motivated learning and perseverance through mistakes or obstacles. First, encourage them to confirm their understanding using prediction. Ask, “Where will the ‘walker’ be if she starts at 7 and takes 7 steps, 9 steps, or 11 steps to the left?” Students should take turns standing on a number and predicting, along with the group, where they will be if they take various numbers of steps to the left before they actually move.

At this stage, continued use of prediction and checking for understanding builds mathematical reasoning and is a more comfortable (i.e., lower-risk) and enjoyable (because it involves movement, dopamine, and peers) way to practice than with paper-and-pencil drills. This approach is also more appropriate, because the task is not to use abstract symbols for negative or
subtraction, but to build the concepts that students will later connect with those symbols. This form of concept construction avoids the inevitable confusion concerning the difference between the symbol – as a negative sign and the same symbol as a sign for subtraction.

Encourage students to find words to describe what these negative numbers represent to them. They may say, “Three down from 1” or “Three on the other side of the 0.” As with subtraction, these created words become owned concepts because students develop their own understanding. They now have valuable tools for mentally manipulating and strengthening the developing concept of negative numbers.

Along the way, observe how different students walk the line, and ask them to describe their thinking. You will find evidence identifying Explorers who first take the steps, then develop unifying rules or concepts that build their understanding of negative numbers. You’ll also observe Map Readers drawing diagrams or notes, making their own small number lines, or verbalizing the steps sequentially before considering a unifying concept.

After students in Group 3 understand movement along a number line, the next step is to have them manipulate objects to successfully demonstrate that 3 plus 4 adds up to 7, and later that 7 “take away” 3 is 4. As they make these constructions, prompt them to think aloud and describe what they are doing in their own words. Groups should operate with the ground rule that members can disagree, agree, or add to the speaker’s explanation only after he or she is finished. Listening to these discussions provides you with powerful formative assessment information as you make checkmarks next to the names of students who are ready to move to a higher level of achievable challenge.

Students will vary in the amount of time they need to predict, correct, practice, and observe before they are very clear on a new process or procedure and are ready to simplify and unify the concept with the new words used in the formal description of negative numbers. When the physical activity makes sense and they can communicate their understanding, they are ready to grasp the higher conceptual and abstract representations of mathematics with numbers, symbols, operational signs, formulas, and equations. Without these intermediate connecting steps, they might get correct answers, but they will be memorized rote answers without the conceptual comprehension that builds the neural networks necessary for math knowledge.

As you would with most group collaborations, call a warning time a few minutes before students are to stop gathering new data, and have them discuss what they noticed. This time allows students who did not get a chance to participate to do so. When time is called again, they should write or sketch their conclusions individually, then on a group chart. Before they present their findings, however, review the group’s presentation material to see if it is the right time for the rest of the class to hear the information. It might be beyond the achievable-challenge comprehension level of other students until they progress further in their own investigations.

If this is the case, explain your reason to the group, go over any errors with them, support further investigations they propose if other groups need more time, and challenge them to prepare their presentation to the whole class with clarity for students who are just beginning to understand the concept. One option to build their depth of understanding and build communication skills is for them to prepare presentations of the material in several ways, including different modes of sensory input (e.g., visual, auditory, demonstration of the steps they took) so they connect with the learning strengths of Explorers and Map Readers.

### Understanding Division

After your unit opening to assess prior knowledge of the topic and to ignite motivation, model the following activity with manipulatives for the class. The goal is for students to understand the concept of division as a means of breaking larger quantities into specified numbers of portions and to recognize that the process (which, early on, does not need to be called division) is a tool for predicting how many objects will be in each new grouping based on the goal (i.e., how many smaller groupings are needed).

Working back from that goal, this activity uses manipulatives as a concrete representation of the division process. One way to assess prior knowledge is to demonstrate a few examples with manipulatives and count the number of objects in each of the smaller groups; after a few demonstrations, ask students to write their predictions for the outcome of the next demonstration on a piece of paper or their personal whiteboards. Feedback from this exercise will help you plan the flexible groups for the following activity.

Group 1—Low Complexity. This group consists of the students whose predictions, after your modeling with manipulatives, were almost all incorrect, even after you gave corrective feedback on the early questions. This group will develop predivision skills by playing games or “sharing” activities. Start by giving the group 10 manipulatives, such as small cubes, and asking them how they would share the cubes among their group of five students (numbers can be adjusted for the size of the group). They can follow their learning strengths and work individually or in pairs at first. Map Readers may want to think first, then discuss their idea, and finally distribute the cubes accordingly. Explorers who like to move or manipulate before sharing their thoughts can do so. When the group meets to discuss, have individual students or pairs of students explain why they did what they did.

Next, using 15 cubes, have each student predict how many cubes each group member will get if the cubes are evenly shared. Students then check their predictions by manipulating the cubes to see if they were correct. Continue this procedure with different numbers of cubes, then with different numbers of group members (pairs or triplets, for example). Students record their data, discuss their findings, and prepare a summary or diagram of their interpretations.

Group 2—Medium Complexity (Early Conceptual Thinking). This group consists of those students who, after some corrective feedback, made correct predictions and could explain their reasoning while you modeled the manipulative division for the whole class. Members of this group can start with their own manipulatives, but they will soon be ready to move on to a greater challenge.

Using real or plastic pennies and groups of five students, ask students how many 10-cent pencils, for example, each of the five group members could “buy” if 100 pennies are distributed equally. Continue with other questions about purchasing things that cost 10 cents, then 20, to determine how many items each group member could “buy.” After using their preferred method (e.g., orally, in a graphic organizer) to show their success with the 10- or 20-cent items, students can experiment with the challenge of buying 15-cent items, following the rule that every group member must get exactly the same number of items.

As with all group-learning activities, prepare students by sharing with them the rules of group work, such as the rule that every person in the group must be able to explain what the group is doing, and why the group is doing it, or else the entire group’s work is not considered successful.

Group 3—High Complexity (More Abstract Conceptual Thinking).
This group is likely to have students who already know how to do division with remainders. These students may form a group that breaks off from the medium-complexity group when you see that they need a higher level of achievable challenge, or you may have done a preassessment that revealed background knowledge that logically puts them in this high-complexity group from the beginning.

These students can use real or plastic pennies to answer questions about purchasing items (e.g., small blocks worth one price and large blocks worth a higher price) that cost 10 and 20 cents. They then evaluate the worth of varying numbers of small blocks, then large blocks, then mixed blocks. They eventually move on to distributing the pennies equally, then the blocks representing the value of a specific number of pennies.

Soon, you may find that some group members begin writing calculations with the appropriate mathematical symbols. They are also likely to use the term remainder when they use a small block and a large block to represent 30 cents and follow the requirement that every group member gets exactly the same value in blocks.

Numerous options are available for these students. You can ask them to do a fair division of products that cannot be done with the limitations of 10- and 20-cent blocks and incorporate pennies, along with blocks, to create portions of equivalent value. Students can discuss the concept of a
remainder using their own words and then write a narrative or play script depicting the remainder as an animate object. This can become part of the subsequent class instruction about remainders or fractional answers for division problems. Skits can be videotaped for next year’s class.

Another option is to give this group newspaper advertising sections or supermarket flyers with ads offering items at two different prices depending on whether, for example, you purchase one for \$.50 or three for \$1.00. Have students cut out ads for things they like and explore methods (practical or fanciful) to find out how much the items will individually cost at the new price and how much money they would need to purchase one for each member of the class. Students can then make posters using the original ad and the ideas they had for solving the problems you suggested. These can also become part of the class instructional material when the other students are ready to make these same kinds of calculations. The experience will be motivating for students in this group because it is at their achievable-challenge level, requires them to be creative, has options for different learning strengths, and is valued for its usefulness for future classes.

A Whole-Class Activity for All Groups. Through this activity, students develop their own ideas about remainders and experience the valuable social lesson of fairness. This activity is also an opportunity to demonstrate situations in which multiple opinions can all be correct.

Divide the class into three-person groups, and ask students how they would divide seven large blocks so each group member receives an equal share. Impractical but fanciful solutions, such as pretending to cut remainder blocks into pieces, are great opportunities for fun and creativity. Although the groups are working on background understanding for division with remainders, the concepts they build and then write, draw, or perform become motivating “advertisements” for subsequent lessons about fractions without using the word fraction.

The words and representations that students use become memory links—as do the positive feelings and fun (reinforced by dopamine) they have while exploring the block problems with their groups. As a result, the subsequent fraction unit will have an established neural network with which to link new learning, and students will approach fractions with more optimism and resilience.

I have groups keep word lists or make sketches during the inquiry about the extra block so that I can reestablish a link to this block activity in later lessons. Because they can be creative and not necessarily practical during the block exploration, my students’ ideas reflect many ideas, including sawing the blocks into pieces, throwing them away, buying extras from other groups, and sharing with another group on a rotating cycle (each would take the extra ones home on alternate days). They draw delightfully creative sketches and write words that I later incorporate onto a class chart, such as
borrow, lend, divide, break up, cut up, equal parts, pieces, whole, half, and quarter. I post this chart, and their sketches, when we begin our fraction unit. Further along in the fraction unit, I return the original charts and sketches to the individual groups. For groups 1 and 2, I write the more “formal” math terms below their original words, and I invite group 3 students to write the terms or symbols they believe are appropriate.

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At the end of the day, take time to give your own brain a chance to acknowledge your successes. Did you reduce a student’s math negativity or engage even one student through his or her individualized level of achievable challenge? Did you demonstrate something students valued and enjoyed about mathematics? Did a reluctant student feel comfortable participating and risk making a mistake? If so, you moved a student closer to a positive math attitude and much more. If your students experienced the optimism of progress in their range of achievable challenge, that positive experience strengthened one of their neural math networks. The day was a success because this neural pathway became stronger and closer to becoming their default pathway— the one to which their brains will revert when they approach new challenges with resilience. Good for you!