Study of the Combination of Crizotinib and Dasatinib in Pediatric Research Participants With Diffuse Pontine Glioma (DIPG) and High-Grade Glioma (HGG)

Inclusion Criteria: ALL RESEARCH PARTICIPANTS * Diagnosis of high-grade glioma (HGG) or diffuse intrinsic pontine glioma (DIPG). If histologic confirmation was obtained, diagnosis must be one of the following: anaplastic astrocytoma (WHO grade 3), anaplastic oligodendroglioma (WHO grade 3), anaplastic oligoastrocytoma (WHO grade 3), anaplastic ganglioglioma (WHO grade 3), pleomorphic xanthoastrocytoma with anaplastic features (WHO grade 3), malignant glioneuronal tumor, glioblastoma, or gliosarcoma (WHO grade 4) * Age \> or = 2 years and \< or = 21 years * Performance score \> or = 50 (Lansky for research participants \< or = 16 years and Karnofsky for those \> 16 years). * Adequate organ function at the time of enrollment as follows: * Bone marrow: Hemoglobin \> or = 8g/dL \[may have received packed red blood cell transfusion\], absolute neutrophil count (ANC) \> or = 1000/mm\^3, platelets \> or = 100,000/mm\^3 \[transfusion independent\]) * Renal: Normal serum creatinine based on age as shown below or GFR \> 70ml/min/1.73m\^2: * Age \< or = 5 years: 0.8 mg/dL maximum * Age 5 to 10 years: 1.0 mg/dL maximum * Age 10 to 15 years: 1.2 mg/dL maximum * Age \> 15 years: 1.5 mg/dL maximum * Hepatic: SGPT and SGOT \< 3x the institutional upper limit of normal (ULN), total bilirubin concentration \< 1.5x the institutional ULN, albumin \> or = 2g/dL * Female research participants \> or = 10 years of age or post-menarchal must not be pregnant (confirmed by serum or urine pregnancy test within 1 week of study enrollment) or breastfeeding * Female research participants of childbearing age or males research participants of child fathering potential must agree to use safe contraceptive methods for the duration of the study and for 3 months thereafter Inclusion Criteria: STRATUM A * Diagnosis of recurrent or progressive HGG or DIPG. * Neurological deficits must be stable on a fixed or decreasing dose of dexamethasone for ≥7 days before study enrollment. * Recovery to ≤ grade 1 from all significant toxicities of previous therapies. * Irradiation: Interval from the last dose of local radiation therapy (RT), craniospinal RT, and palliative RT for symptomatic disease \> or = 3 months, \> or = 6 months, and \> or = 2 weeks before study enrollment, respectively * Myelosuppressive chemotherapy: Interval \> or = 6 weeks and \> or = 4 weeks from last dose of nitrosourea and other chemotherapy drugs before study enrollment, respectively. However, interval must be \> or = 1 week from last dose of oral etoposide and other drugs administered at low doses (metronomic regimen) before study enrollment * Small-Molecule Inhibitors: Interval \> or = 1 week from last dose before study enrollment. If a previously used agent has a prolonged half-life, the appropriate interval will be determined after consultation with the principal investigator * Monoclonal Antibodies: Interval \> or = 3 half-lives before study enrollment. Such cases will need to be discussed with the principal investigator * High-Dose Chemotherapy with Stem-Cell Rescue: Interval \> or = 3 months before study enrollment * Cancer Vaccines and Convection-Enhanced Therapies: Interval \> or = 1 month before study enrollment * Growth Factors: Interval \> or = 1 week and \> or = 2 weeks before study enrollment for standard and long-acting growth factors (e.g., pegfilgrastim), respectively Inclusion Criteria: STRATUM B * Completion of local RT with or without concomitant chemotherapy including temozolomide and radiosensitizing agents (e.g., carboplatin, vorinostat) outside the context of a clinical trial. Any agent administered during RT should have a short half-life so that it should already be completely eliminated by the start of this therapy. If other agents were used concurrently with RT, they will need to be discussed with the principal investigator to assess eligibility * Interval \> or = 4 weeks and \< or = 8 weeks from the completion of radiochemotherapy Exclusion Criteria: ALL RESEARCH PARTICIPANTS * Metastatic disease for stratum B only * Concomitant use of other anticancer (except for corticosteroids) or experimental agents * Use of enzyme-inducing anticonvulsants (EIACs). A minimum interval of 10 days between the last dose of EIAC and start of this therapy will be required for research participants who were previously receiving such medications. * Pregnant or lactating patients * Research participants with other clinically significant medical disorders that could compromise their ability to tolerate protocol therapy or would interfere with the study procedures or results * Prior therapy with a PDGFR or c-Met inhibitor * Original treatment design: Body surface area ≥ 1.8m2on dosage levels 3b, 4, and 5 * Modified treatment design: Body surface area \< 0.55 m\^2 for all dosage levels